Proceedings_of_a_Second_Symposium_on_Large Scale_Digital_Calculating_Machinery_Sep49 Proceedings Of A Second Symposium On Large Scale Digital Calculating Machinery Sep49
Proceedings_of_a_Second_Symposium_on_Large-Scale_Digital_Calculating_Machinery_Sep49 Proceedings_of_a_Second_Symposium_on_Large-Scale_Digital_Calculating_Machinery_Sep49
User Manual: Proceedings_of_a_Second_Symposium_on_Large-Scale_Digital_Calculating_Machinery_Sep49
Open the PDF directly: View PDF 
.
Page Count: 431
| Download | |
| Open PDF In Browser | View PDF |
THE ANNALS OF'THE COMPUTATION LABORATORY
OF HARVARD UNIVERSITY
VOLUME XXVI
THE ANNALS :OF'THE COMPUTA.TION LABORATORY
OF HARVARD UNIVERSITY
I
A Manual of Operation for the Automatic Sequence Controlled Calculator
1946
Tables of the Modified Hankel Fun~tion~ of Order One-Third and of Their
Derivatives
1945
III
Tables of the Bessel Functions of the First Kind of Orders Zero and One .
1947
IV
Tables of the Bessel Functions of the First Kind of Orders Two and Three
1947
II
V
Tables of the Bessel Functions of the First Kind of Orders Four, Five, and Six.
1947
VI
Tables of the Bessel Functions of the First Kind of Orders Seven, Eight, and
Nine
1947
VII
Tables, of the Bessel Functions of the First Kind of Orders Ten, Eleven, and
Twelve ,
1947
VIII
Tables of the Bessel Functions of the First Kind of Orders Thirteen, Fourteen,
and Fifteen
1947
IX Tables of the Bessel Functions of the First Kind of Orders Sixteen through
Twenty-Seven
1948
X
Tables of the Bessel Functions of the First Kind of Orders Twenty-Eight through
Thirty-Nine .
1948
XI
Tables of' the Bessel Functions of the First Kind. of Orders Forty through
Fifty-One
1948
XII
Tables of the Bessel Functions of the First Kind of Orders Fifty-Two through
Sixty-Three .
1949
Tables of the Bessel Functions of the First Kind of Orders Sixty-Four through
Seventy-Eight
1949
Tables of the Bessel Functions of the First Kind of Orders Seventy-Nine through
One Hundx:ed Thirty-Five .
'
1951
I
XIII
XIV
XV
XVI
XVII
XVIII
(In preparation)
Proceedings of a. Symposium on Large-Scale Digital Calculating Machinery
1948
Tables for the Design of Missiles
1948
Tables of Generalized Sine- and Cosine-Integral Functions: Part I
1949
XIX Tables of Generalized Sine- and Cosine:-Integral Functions: Part II
XX Tables of Inverse Hyperbolic Functions
XXI
XXII
.
1949
1949
Tables of Generalized Exponential-Integral Functions
1949
Tables of the Function si: cP and of its First Eleven Derivatives
1949 .
XXIII . (In preparation)
,XXIV Description of a Relay Calculator
XXV
XXVI
XXVII
1949
(In preparation)
Proceedings of a Second Symposium on Large-Scale Digital Calculating
Machinery
1951
Synthesis of Electronic Computing and Control Circuits
1951
PROCEEDINGS OF A SECOND
SYMPOSIUM ON LARGE-SCALE
DIGITAL CALCULATING
MACHINERY
Jointly Sponsored by The Navy Department
Bureau of Ordnance and Harvard University
at The Computation Laboratory
13-16 September 1949
CAMBRIDGE, MASSACHUSETTS
HARVARD UNIVERSITY PRESS
~95I
LONDON : GEOFFREY CUMBERLEGE
OXFORD UNIVERSITY PRESS
The opinions or assertions contained herein are the private ones
of the writers and are not to be, construed as official or reflecting
the views of the Navy Department or the naval service at large.
Composition by The Pitman Press, Bath, England
Printed by offset lithography by the Murray Printing Company, Wakefield, Massachusetts, U.S.A.
PREFACE
InJanuary 1947 the Bureau of Ordnance of the United States Navy and Harvard University together sponsored a Symposium on Large-Scale Digital Calculating Machinery as a
means of furthering interest in the design, construction, application, and operation ofcomputing
machinery. This meeting was attended by over three hundred people, nearly four times the
originally expected attendance, and by popular aemand the proceedings were published as
Volume XVI of the Annals of the Computation Laboratory.
At the Oak Ridge meeting on computing machinery in April 1949, Mina Rees and John
Mauchly, representing the Association for Computing Machinery, suggested that another
symposium should be held at Harvard summa~izing. recent and current developments. The
staff of the Computation Laboratory had already considered this possibility in connection
with the announcement of the completion of Mark III Calculator, and were delighted with
the suggestions of Dr. Rees and Dr. Mauchly. Accordingly, the Bureau of Ordnance was
again invited to join Harvard University in sponsoring a second symposium with emphasis
on the application of digital calculating machinery.
From experience with the first symposium, it was expected that. perhaps three hundred
people might attend. The response of more than seven hundred participants clearly indicated
the rapidity with which the field of automatic computation is growing.
T~is volume, the twenty-sixth of the Annals of the Computation Laboratory, contains all
the papers presented at the second symposium except one. Two of the speakers, Manuel S.
Vallarta and Frederick V. Waugh, found at the last minute that they were unable to attend.
However, their papers were received and were read by J. Curry Street and Leon Moses,
respectively, both of Harvard University. Because of the tremendous editorial difficulties
experienced with the proceedings of the first symposium, each speaker at the second was
requested to supply his manuscript in advance, in order to avoid dependence upon transcription
from sound recording. Thirty-nine papers are herein published essentially as submitted.
Thus the work required to prepare this volume for publication was greatly reduced. However,
it was necessary to redraw many of the illustrations for offset reproduction; this was done by
Carmela M. Ciampa, assisted by Paul Donaldson, photographer of Cruft Laboratory, Harvard
University.
Since the symposium was held iIi September, prior to the opening of the fall term, it was
possible to make use of the dormitories in the Harvard Yard and the dining facilities of the
Harvard Union. Arthur Trottenberg of Harvard University supervised arrangements for
the use of these facilities and other accommodations. Preparation of the program and registration lists and the registration of the members of the symposium after their arrival were
carried out by Betty Jennings, Jacquelin Sanborn, Jean Crawford, and Holly Wilkins. It is
v
PREFACE
a pleasure to acknowledge the cooperation of Edmund C. Berkeley, secretary of the Association
for Computing Machinery, in this connection.
The staff of the Computation Laboratory wishes to express its appreciation to the members
of the symposium for their attendance and for their participation in the discussions, to the
chairmen of the several sessions for their assistance, and to the speakers not only for their
addresses during the symposium but also for their cooperation in preparing the manuscripts
of their papers.
The staff also wishes to express its gratitude to the Bureau of Ordnance and to its representatives, Captain G. T~ Atkins and Mr. Albert Wertheimer, for many years of pleasant
association throughout the building of Mark II and Mark III Calculators, for their continued
i~terest and help, and for making possible both the Second Symposium on Large-Scale Digital
Calculating Machinery arid the publication of its proceedings.
HOWARD H. AIKEN
Cambridge, Massachusetts
May 1950
VI
CONTENTS
. ix
PROGRAM OF THE SYMPOSIUM
xv
MEMBERS OF THE SYMPOSIUM.
FIRST SESSION: Opening Addresses
I
SECOND SESSION: Recent Developments in Computing Machinery
9
BANQUET
71
THIRD SESSION: Recent Developments in Computing Machinery
81
FOURTH SESSION: Numerical Methods.
135
FIFTH SESSION: Computational Problems in Physics
.
21 3
SIXTH SESSION: Aeronautics and Applied Mechanics .
261
SEVENTH SESSION: The· Economic and Social Sciences
321
EIGHTH SESSION: Discussion and Conclusions
363
Vll
PROGRAIvI
FIRST SESSION
Tuesday, September 13, 1949
10: 30 A.M. to 12: 00 P.M.
OPENING ADDRESSES
Presiding
Howard H. Aiken
Director
of the Computation Laboratory
7
Edward Reynolds
.3
Administrative Vice President of Harvard University
Rear Admiral.F. I. Entwistle, USN
Director of Research, Bureau of Ordnance
S-
SECOND SESSION
Tuesday, September 13, 1949
2:00 P.M. to 5:00 P.M.
RECENT DEVELOPMENTS IN COMPUTING MACHINERY
Presiding
Mina Rees, Office of Naval Research
1. The Mark III Calculator
II
Benjamin L. Moore
Harvard University
2. The Bell Computer, Model VI
'),0
Ernest G. Andrews
Bell Telephone Laboratories
3. An Electrostatic Memory System
.3'1>-
J. Presper Eckert, Jr.
Eckert-Mauchly Computer Corporation
4. The Digital Computation Program at Massachusetts Institute of Technology
vv wI:.Jay W. Forrester
Massachusetts Institute of Technology
IX
Lfl-
SECOND SESSION--CONTINUED
_b (J
5. The Raytheon Electronic Digital Computer
. Richard M. Bloch
Raytheon Manufacturing Company
6. A General Electric Engineering Digital Computer
t 5""
Burton R. Lester
.General Electric Company
BANQUET
Tuesday, September 13, 1949
7:00 P.M.
Toastmaster
Edward·A. Weeks, Jr.
Editor of The Atlantic Monthly
Speaker
William S. Elliott
Research Laboratories of Elliott Brothers (London) Limited
"The Present Position of Computing-Machine Development in England'· 7
THIRD SESSION
Wednesday, September 14, 1949
9:00 A.M. to 12:00 P.M.
RECENT DEVELOPMENTS IN COMPUTING MACHINERY
Presiding
E. Leon Chaffee, Harvard University
1. Semiautomatic Instruction on the Zephyr
4' ~
H. D. Huskey
National Bureau of Standards, Institute for Numerical Analysis
2. Static Magnetic Delay Lines
q1
. Way Dong Woo
Harvard University
x
If
THIRD SESSION-cONTINUED
3. Coordinate Tubes for Use with Electrostatic Storage Tubes
'1 b
R. S. Julian and A. L. Samuel
University of Illinois
4. Basic Aspects of Special Computational Problems I JE)'
Howard T. Engstrom
Engineering Research Associates, Inc.
5. Electrochemical Computing Elements
111
John R. Bowman
Mellon Institute
'-ot)~~(tf
.s~~,rl(rx
,111111
6. EDVAC Transformation Rules
I ~,
GeoJ;ge W. Patterson
University of Pennsylvania
FOURTH SESSION
Wednesday, September 14, 1949
2:00 P.M. to 5:00 P.M.
NUMERICAL METHODS
Presiding
Raymond C. Archibald, Brown University
1. Notes on the Solution of Linear Systems Involving Inequalities I -S 7
George W. Brown
Rand Corporation
2. Mathematical Methods in Large-scale Computing Units
1£.1-)
D. H. Lehmer
University of California
3. Empirical Study of Effects of Rounding Errors
I if 7
C. Clinton Bramble
U.S. Naval Proving Ground, Dahlgren, Virginia
". Numerical Methods Associated with Laplace's Equation IS" 1. W. E. Milne
Institutefor Numerical Anafysis, UCLA and Oregon State College
Xl
FOURTH SESSION-cONTINUED
5. An Iteration Method for the Solution of the Eigenvalue Problem of Linear Differentia
and Integral Operators
Ih 4
Cornelius Lanczos
Institute for Numerical Anarysis, UCLA
1--v7
6. The Monte Carlo Method
S. M. Ulani
Los Alamos Scientific Laboratory
FIFTH SESSION
Thursday, September 15, 1949
9: 00 A.M. to 12: 00 P.M.
COMPUTATIONAL PROBLEMS IN PHYSICS
Presiding
Karl K. Darrow, Bell Telephone Laboratories
1. The Place of Automatic Computing Machinery in Theoretical Physics
vIS-
Wendell H. Furry .
Harvard University
2. Double Refraction of Flow and the Dimensions of Large Asymmetric Molecules 7--1
q
Harold A. Scheraga, John T. Edsall, and J. Orten Gadd, Jr.
Cornell University, Harvard Medical School, and Computation Laboratory of Harvard University
3. L-Shell Internal Conversion
').. Lf'O
Morris E. Ro~e
Oak Ridge National Laboratory
4. The Use of Calculating Machines in the Theory of Primary Cosmic Radiation
Manuel S. Vallarta
University of Mexico
(read by J. C. Street, Harvard University)
.-. 5. Computational Problems "in Nuclear Physics
,,50
Herman Feshbach
Massachusetts Institute qf Technology
XlI
7. . t.!',{-
SIXTH SESSION
Thursday, September 15, 1949
2: 00 P.M. to 5: 00 P.M.
AERONAUTICS AND ApPLIED MECHANICS
Presiding
Harald M. Westergaard, Harvard
1. Computing Machines in Aeronautical R;esearch
Universi~v
1--l3
R. D. O'Neal
University of Michigan
2. Problem of Aircraft Dynamics
"1. 71
Everett T. Welmers
Bell Aircraft Corporation
3. A Statistical Method for Certain Nonlinear Dynamical Systems J... ~ I
George R. Stibitz
Consultant in Applied Mathematics, Burlington, Vermont
4. Combustion Aerodynamics
"')..... , ~~
Howard W. Emmons
Harvard University
5. Application of Computing Machinery to Research of the Oil Industry '2 D_r'
Morris Muskat
Gulf Research & Development Company
6. The 603-405 Computer
"3 ) ~
William W. Woodbury
Northrop Aircraft, Inc.
SEVENTH SESSION
Friday, September 16, 1949 .
9:00 A.M. to 12 :00 P.M.
THE ECONOMIC AND SOCIAL SCIENCES
Presiding
Edwin B. Wilson, Office of Naval Research
1. Application of Computing Machinery to the Solution of Problems of the Social Sciences
Frederick Mosteller
Harvard University
XUl
J J.. 3
SEVENTH SESSION-cONTINUED
2. Dynamic Analysis of Economic Eq~ilibrium
']. "3 '3
Wassily W. Leontief
Harvard Univer-fity
3. Some Computational Problems in Psychology
'33 r:r
Ledyard R. Tucker
Educational Testing Service, Princeton, New Jersey
4. Computational Aspects of Certain Econometric Problems ~~/-r
Herman Ohernoff
University of Chicago
~ 51
5. PhysIology and Computing Devices
William J. Crozier
Harvard University
-~ fj 7
Frederick V. Waugh
Council of Economic Advisers
. (read by Leon Moses, Harvard University)
6. The Science of Prosperity
EIGHTH SESSION
Friday, September 16, 1949
2:00 P.M. to 4:00 P.M.
DISCUSSION AND CONCLUSIONS
Presiding
Willard E. Bleick, U.S. Naval Academy Post Graduate School
1. The Selectron
"] bb
Jan Rajchman
Radio Corporation of America
2. Traits Caracteristiques de la Calculatrice de la Machine
l'Institut Blaise Pascal
-; 7 tf
Louis Couffignal
lnstitut Blaise Pascal
(read by Leon Brillouin, Harvard University)
3. The Future of Computing Machinery
J cr 7
Louis N. Ridenour
University qf Illinois
*
*
*
OPEN DISCUSSION
XIV
a Calculer
Universelle. de
MEMBERS OF THE SYMPOSIUM
MATTHEW C. ABBOTT, Engineer, W. S. MacDonald Company, Inc., Cambridge
MILTON ABRAMOWITZ, Numerical Mathematics Service, New York
CHARLES W. ADAMS, Massachusetts Institute of Technology, Cambridge
HOWARD H. AIKEN, Professor of Applied Mathematics and Director of the Computation
Laboratory, Harvard University, Cambridge
MOE LAWRENCE AITEL, Engineer, Raytheon Manufacturing Company, Waltham, Massachusetts
J. CHARLES AJEMIAN, Massachusetts Institute of Technology, Cambridge
MILTON ALDEN, President, Alden Products Company, Brockton, Massachusetts
SAMUEL N. ALEXANDER, Chief, Electronic Computers Section, National Bureau of Standards,
Washington, D.C.
WILLIAM R~ ALLEN, Computation Laboratory, Harvard University, Cambridge
JAMES C. ALLER, LT., USN, Medford, Massachusetts
R. K. ALLERTON, JR., Public Relations, Underwood Corporation, New York
FRANZ L. ALT, National Bureau of Standards, Washington, D.C.
BIAGIO F. AMBROSIO, Engineer, National Bureau of Standards, Los Angeles, California
FREDERICK J. ANDERSON, Development Engineer, Sylvania Electric Products, Inc., Boston
LOWELL O. ANDERSON, Physicist, Naval Ordnance Test Station, China Lake, California
RUTH K. ANDERSON, Mathematician, Naval Ordnance Test Station, China Lake, California
ERNEST G. ANDREWS, Technical Staff, Bell Telephone Laboratories, New York
.
THOMAS B. ANDREWS, JR., Aeronautic Research Sci~ntist, National A~visory Committee for
Aeronautics, Langley Aeronautical Laboratory, Hampton, Virginia
FRANK H. ANDRIX, Engineer, Bell Aircraft Corporation, Buffalo, New York
LEONARD J. ANGUS, Consultant, Manhasset, Long Island, New York
RAYMOND C. AItCHIBALD, Professor, Brown University, Providence, Rhode Island
WALTER E. ARNOLDI, Project Engineer, Hamilton Standard Propellers, Wethersfield, Connecticut
ELEANOR ASMUTH, Badger, Wi"sconsin
JOHN ASMUTH, Instructor, University of Wisconsin, Madison, Wisconsin
ALBERT A. AUERBACH, Design Engineer, Eckert-Mauchly Computer Corporation, Philadelphia,
Pennsylvania
ISAAC L~ AUERBACH, Senior Engineer, Burroughs Adding Machine Company, Philadelphia,
PennsyIvania
DONALD E. BABCOCK, Republic Steel Corporation, Youngstown, Ohio
PAUL H. BACKUS, Lt.-Cdr., USN, Chief, Ballistics Research Section, Bureau of Ordnance,
Washington, D.C .. '
XV
MEMBERS OF THE SYMPOSIUM
GEORGE A. BALL, Research Assistant, Harvard University, Cambridge
STANLEY S. BALLARD, Professor of Physics, Tufts College, Medford, Massachusetts
MELVIN D. BALLER, Engineer, Air Force Cambridge Research Laboratories, Cambridge
ROBERT M. BARRETT, Electronic Engineer, Air Force Cambridge Research Laboratories,
Cambridge
JEAN J. BARTIK, Eckert-Mauchly Computer Corporation, Philadelphia, Pennsylvania
STEPHEN D. BATCHELOR, Engineer, Raytheon Manufacturing Company, Waltham, Massachusetts
DWIGHT W. BATTEAU, Research Assistant, Harvard University, Cambridge
D. T. BELL, Bell Telephone Laboratories, Inc., New York
ALBERT 1. BELLIN, Assistant Professor, Harvard University, Cambridge
LAWRENCE W. BE,LOUNGIE, Raytheon Manufacturing Company, Waltham, Massachusetts
J. L. BELYEA, Lt., Royal Canadian Navy, Ottawa, Ontario, Canada
ALBERT A. BENNETT, Professor of Mathematics, Brown University, Providence, Rhode
Island
ROBERT J. BERGEMANN, JR., Electronics Engineer, Office of Naval Research, Boston
STEFAN BERGMAN, Harvard University, Cambridge
EDMUND C. BERKELEY, President, E. C. Berkeley and Associates, New York
ERIC W. ,BETH, Physicist, Geophysical Research Directorate, Air Force Cambridge Research
Laboratories, Cambridge
VICTOR E. BIEBER, JR., Aeronautical Engineer, Bureau of Aeronautics, Washington, D.C.
WALTER J. BINGEL, Mechanical Engineer, Raytheon Manufacturing Company, Waltham,
Massach usetts
ROBERT W. BIRGE, Nuclear Laboratory, Harvard University, Cambridge
ERNEST W. BIVANS, Air Force Cambridge Research Laboratory, Cambridge
GERRIT A. BLAAUW, Computation Laboratory, Harvard University, Cambridge
PAUL B. BLACK, Head of Equipment Engineering, Sylvania Electric Products, Inc., Boston
BARTOLOME C.' BLANCO, Harvard University, Cambridge
WILLARD E. BLEICK, Professor, U.S. Naval Postgraduate School, Annapolis, Maryland
ALAN BLOCH, Arma Corporation, Brooklyn, N ew York
RICHARD M. BLOCH, Manager, Analytical Section, Raytheon Manufacturing Company,
Newton, Massachusetts
H. W. BODE, Bell Telephone Laboratories, New York
GEORGE A. W. BOEHM, Science Editor, Newsweek, New York
, JOHN M. BOERMEESTER, John Hancock Mutual Life Insurance Company, Boston
MORTON BOISEN, Statistician, Bureau of the Census, Washington, D.C.
ROBERT N. BONNER, Junior Research Chemist, Carter Oil Company, Tulsa, Oklahoma
RICHARD C. BOOTON; JR., Research Assistant, Differential Analyzer Computer Laboratory',
Massachusetts Institute of Technology, Cambridge
GUY F. BOUCHER, Computation Laboratory, Harvard University, Cambridge
XVI
MEMBERS OF THE ,SYMPOSIUM
JOHN R. BOWMAN, Head, Department of Research in Physical Chemistry, Mellon Institute,
Pittsburgh, Pennsylvania
.
HUGH R. BOYD, Research Engineer, Massachusetts Institute of Technology, Cambridge
HENRY B. BRAINERD, Massachusetts Institute of Technology, Cambridge
C. CLINTON BRAMBLE, Director of Computation Ballistics, U.S. Naval Proving Ground,
Dahlgren, Virginia
C. S. BRAND, Arnold Engineering Company, Chicago, Illinois
LEON BRILLOUIN, International Business Machines Corporation, New York
PAUL BROCK, Reeves Instrument Corporation, New York
DOUGLAS A. BROWN, Assistant Director, Harvard University News Office, Harvard University,
Cambridge
GEORGE W. BROWN, Rand Corporation, Santa Monica, California
J. H. BROWN, Research Engineer, Massachusetts Institute of Technology, Cambridge
ROBERT G. BROWN, Operations Evaluation Group, CNO, Navy Depar.tment, Washington,
D.C.
T. H. BROWN, Professor, Harvard Business School,Harvard University, Boston
T. WISTAR BROWN, Sales Manager, Eckert-Mauchly Computer Corporation, Philadelphia,
Pennsylvania
WILLIAM FULLER BROWN, JR., Research Physicist, Sun Oil Company, Newtown Square,
Pennsylvania
JOSEPH A. BRUSTMAN, Chief Engineer, Remington Rand, Inc., South Norwalk, Connecticut
EDWARD C. BRYANT, Student, Boston University, Boston
RALPH W. BUMSTEAD, Patent Attorney, Westfield, New Jersey
RICHARDS. BURINGTON, Director, Evaluation and Analysis Group, Bureau of Ordnance,
Washington, D.C.
WILLIAM BURKHART, Monroe Calculating Machine Company, Orange, New Jersey
ROBERT J ..BURNS, Computation Laboratory, Harvard University, Cambridge
ROBERT R. BUSH, Research Fellow, Harvard University, Cambridge
SAMUEL H. CALDWELL, Professor of Electrical Engineering, Massachusetts Institute of Technology, Cambridge
FRANK J. CAMPAGNA, Computation Laboratory, Harvard University, Cambridge
ELIZABETH JEAN CAMPBELL, Computer,. Research Laboratory of Electronics, Massachusetts
Institute of Technology, Cambridge
R. E. CAMPBELL, Liaison Engineer, Signal Corps _Engineering Laboratories, Massachusetts
Institute of Technology, Cambridge
ROBERT V. D. CAMPBELL, Department Supervisor, Burroughs Adding Machine Company,
Philadelphia, Pennsylvania.
'
E. W. CANNON, Mathematician, National Bureau of Standards, Washington, D.C.
JOEL CARROLL, Geodetic Engineer, U.S. Geological Survey, Washington, D.C.
JOHN B. CARROLL, Assistant Professor of Education, Harvard University, Cambridge
XVll
MEMBERS OF THE SYMPOSIUM
ELBERT P. CARTER,Senior Engineer, Transducer Corporation, Boston
CARL C. CHAMBERS, A~.ting Dean, University of Pennsylvania, Philadelphia, Pennsylvania
GEORGE C. CHASE, Research Engineer, Monroe Calculating Machine Company, Orange,
New Jersey
JOSEPH CHEDAKER, Senior Engineer, Burroughs Adding Machine Company, Philadelphia,
Pennsylvania
T. C. CHEN, Senior Engineer, Burroughs Adding Machine Company, Philadelphia, Pennsylvania
HERMAN CHERNOFF, Research Associate, Cowles Commission, University of Chicago, Chicago,
Illinois
ALFRED C. CHEVERIE, Computation Laboratory, Harvard University, Cambridge
BENJAMIN F. CHEYDLEUR, Mathematician, U.S. Bureau of Ordnance, Washington, D.C.
HENRY W. F. CHIN, Assistant Mechanical Engineer, Raytheon Manufacturing Company,
Newton, Massachusetts
MANUEL P. CHINITZ, Mathematician, U.S. Naval Proving Ground, Dahlgren, Virginia
ALLEN G. CHRISTENSEN, Computation Laboratory, Harvard University, Cambridge
THOMAS J. CHRISTMAN, Lt., USN, Post-graduate Student, Massachusetts Institute of Technology, Cambridge
J. C. CHU, Senior Scientist, Argonne National Labora.tory, Chicago, Illinois
CARMELA M. CIAMPA, Computation Laboratory, Harvard U.niversity, Cambridge
A. G. CLAVIER, Assistant Technical Director, Federal Telecommunication Laboratories,
Nutley, New Jersey
GERALD M. CLEMENCE, Director, Nautical Almanac, U.S. Naval Observatory, Washington,
D.C.
R.F. CLIPPINGER, Computing Laboratory, Ballistic Research Laboratory, Aberdeen Proving
Ground, Maryland
A. M. CLOGSTON, Bell Telephone Laboratories, Murray Hill, New Jersey
RICHARD P. COATES, Computation Laboratory, Harvard University, Cambridge
TODD D. COCHRAN, JR., Development Engineer, Eastman Kodak Company, Rochester, New
York
R. C. COILE, Operations Evaluation Group, Office of the Chief of Naval Operations, Navy
Department, Washington, D.C.
CHARLES F. COIT, Senior Engineer, Raytheon Manufacturing Company, Newton, Massachusetts
ARNOLD A. COHEN, Senior Engineer, Engineering Research Associates, Inc., St. Paul, Minnesota
CHARLES J., COHEN, Mathematician, Naval Proving Ground, Dahlgren, Virginia
JOHN J. CONNOLLY, Engineering Supervisor, Teleregister Corporation, New York
CHARLES A. COOLIDGE, JR., Computation Laboratory, Harvard University, Cambridge
JOHN M. COOMBS, Director of Development, Engineering Research Associates, St. Paul.
Minnesota
XVlll
MEMBERS OF THE SYMPOSIUM
GERALD COOPER, Research Assistant, Massachusetts Institute of Technology, Cambridge
CHARLES L. CORDERMAN, Research Assistant, Massachusetts Institute of Technology, Cambridge
DOUGLAS S. CRAIG, Second Vice-President, Metropolitan Life Insurance Company, New York
JEAN CRAWFORD, Computation Laboratory, H;;trvard University, Cambridge
JOHN H.· CREDE, Associate Director of Research, Allegheny Ludlum Steel Corporation,
Brackenridge, Pennsylvania
. L. P. CROSMAN, Director, Electronic Calculator Research, Remington Rand, Inc., South
Norwalk, Connecticut
EDWARD D. CROSS, Director of Engineering, Alden Products Company, Brockton, Massachusetts
W. J. CROZIER, Professor, Harvard University, Cambridge
WILLIAM H. CUMMINS, Chief, Classification and Coding Branch, Federal Security Administration, Washington, D.C.
HASKELL B. CURRY, Professor of Mathematics, The Pennsylvania State College, State College,
Pennsy lvania
W. W. CURTIS, Office Supervisor, Aluminium Company of America, Boston
R.J. CYPSER, Instructor, Servomechanisms Laboratory, Massachusetts Institute of Technology,
Cambridge
JOHN A. DAELHOUSEN, Staff Member, Massachusetts Institute of Technology, Cambridge
EVERETT J. DANIELS, Staff Member, Massachusetts Institute of Technology, Cambridge
M. DANILOFF, Senior Engineer, Raytheon Manufacturing Company, Waltham, Massachusetts
GEORGE B. DANTZIG, Mathematician, US. Air Force Comptroller, Washington, D.C.
KARL K.,DARROW, Physicist, Bell Telephone Laboratories, New York
GERALD W. DAVIS, Electronic Scientist, National Bureau of Standards, Washington, D.C.
MALVIN E. DAVIS, Actuary, Metropolitan Life Insurance Company, New York
PHILLIP DAVIS, Mathematician, Harvard University, Cambridge
CHRISTOPHER DEAN, Teaching Fellow, Harvard University, Cambridge
FRANKLIN R. DEAN, Engineer, Raytheon Manufacturing Company, Waltham, Massachusetts
J. T. DEBETTENCOURT, Section Manager, Raytheon Manufacturing Company, Waltham,
Massach usetts
L. S. DEDERICK, Associate Director, Ballistic Research Laboratory, Aberdeen Proving Ground,
Maryland
GEORGE H. DEPINTO, Harvard University, Cambridge
HENRY DESTEFANO, Harvard University, Cambridge
.
JOHN E. DETURK, Senior Engineer, Raytheon Manufacturing Company, Waltham, Massachusetts
M. L. DEUTSCH, Physicist, Socony-Vacuum Research Laboratory, Paulsboro, New Jersey
A. J. DEVAUD, Engineer, Raytheon Manufacturing Company, 'tValtham, Massachusetts
R. L. DEVEER, C. P. Clare and Company, Boston
XIX
MEMBERS OF THE SYMPOSIUM
GEORGE C. DEVOL, Manager, Magnetic Devices Department, Research Laboratory, Remington Rand, Inc., South Norwalk, Connecticut
''''ALTER L. DEVRIES, Actuarial Supervisor, Equitable Life Assurance Society, New York
C. B. DEWEY, Vice-President, Reeves Instrument Corporation, New York
ERNEST J. DIETERICH, Junior Engineer, Raytheon Ma.nufacturing Compariy, Waltham,
Massachusetts
JOHN D. DILLON,Assistant Chief, Research Services, Air Force, Cambridge Research Laboratories, Cambridge
BERNARD DIMSDALE, Mathematician, Ballistic Research Laboratory, Aberdeen Proving
Ground, Maryland
L. P. DISNEY, Chief, Section of Predictions, U.S. Coast and Geodetic Survey, Washington,
D.C.
NATHAN DIVINSKY, Research Associate, Cowles Commission, Chic~go, Illinois
STEPHEN H. DODD, Research Engineer, Massachusetts Institute of Technology, Cambridge
CHARLES H. DOERSAM, JR., Mechanical Engineer, Office of Naval Research, Section of
Development Contract, Port Washington, Long Island, New York
FRANCIS W. DRESCH, Assistant Director, Computation and Ballistics, U.S. Naval Proving
Ground, Dahlgren, Virginia
L. B. DUMoNT,Engineer, General Electric Company, Lynn, Massachusetts
ROBERT F. DUNCAN, Vice-President, John Price Jones Company, New York
S. W. DUNWELL, Future Demands Department, International Business Machines Corporation,
New York
J. R. DYER, Colonel, USAF, Staff Officer, Munitions Board, Department of Defense, Washington, D.C.
E. C. EASTON, Dean of Engineering, Rutgers University, New Brunswick, New Jersey
J. PRESPER ECKERT, JR., Vice-President, Eckert-:NlauchlyComputer Corporation, Philadelphia, Pennsylvania
W. J. ECKERT, Director, Pure Science, International Business Machines Corporation, New
York
ROBERT P. EDDY, Mathematician, Naval Ordnance Laboratory, Silver Spring, Maryland
NIELSE. EDLEFSEN, Associate Technical Director, North American Aviation, Santa Monica,
California
JOHN T. EDSALL, Associate Professor of Biological Chemistry, Harvard Medical School, Boston
MILTON EFFROS, Management Research, Metropolitan Life Insurance Company, New York
ROBERT D. ELBOURN, National Bureau of Standards, ""Vashington, D.C.
PETER ELIAS, Teaching Fellow, Harvard University, Cambridge
WILLIAM S. ELLIOTT, Research "Laboratories of Elliott Brothers (London), Ltd., Borehamwood,
Hertfordshire, England
MURRAY ELLIS, Junior Engineer, Raytheon Manufacturing Company, Waltham, Massachusetts
xx
MEMBERS OF THE SYMPOSIUM
GEORGE V. ELTGROTH, Vice President, ~ckert-Mauchly Computer Corporation, Philadelphia,
Pennsylvania
JOHN O. ELY, Research Associate, Massachusetts Institute of Technology, Cambridge
CLAUDE L. EMMERICH, Senior Physicist, Martin-Hubbard Corporation, Cambridge
HOWAR~ EMMONS, Associate Professor, Harvard University, Cambridge
HOWARD T. ENGSTROM, Vice President, Engineering Research Associates, Inc., St. Paul,
Minnesota
F. I. ENTWISTLE, Rear Admiral, USN, Director of Research, Bureau of Ordnance, Navy
Department, Washington, D.C.
HANS H. ESTIN, Brookline, Massachusetts
HELENTY ESTIN, Brookline, Massachusetts
ROBERT R. EVERETT, Research Engineer, Servomechanisms Laboratory, Massachusetts
Institute of Technology, Cambridge .
ROBERT G. EVERSEN, Computation Laboratory, Harvard University, Cambridge
HARRIS FAHNESTOCK, Servomechanisms Laboratory, Massachusetts Institute of Technology,
Cambridge
R. M. FAIRBROTHER, l\fassachusetts Institute of Technology, Cambridge
R. S. FALLOWS, Engineer, Sylvania Electric Products, Inc., Boston
JOHN T. FARREN, Computation Laboratory, Harvard University, Cambridge
LOUIS FEIN, Senior Engineer, Raytheon Manufacturing Company, Waltham, Massachusetts
]. H. FELKER, Member of Technical Staff, Bell Telephone Laboratories, Wilippany, New
Jersey
SAMUEL FELTMAN, Ordnance Engineer, Office of Chief of Ordnance, Washington, D.C.
FRED G. FENDER, Professor of Mathematics, Rutgers University, New Brunswick, New
Jersey
DAVID T. FERRIER, Lawrence, Long Island, New York
HERMAN FESHBACH, Associate Professor, Massachusetts Institute of Technology, Cambridge
F. A. FICKEN, Research Associate, New York University, New York
L. R. FINK, Manager, Electronics Laboratory, General Elec'tric Company, Syracuse, New
York
HAROLD A. FINLEY, Manager, Management Research, Metropolitan Life Insurance Company,
New York
LYMAN C. FISHER, Naval Ordnance Laboratory, White Oak, Silver Spring, Maryland
HARLAND W. FLAGG, Marchant Calculating Machine Company, Boston
DONALD A. FLANDERS, Senior Mathematician, Argonne National Laboratory, Chicago,
Illinois
MARGARET I. FLORENCOURT, Research Engineer, Massachusetts Institute of Technology,
Cambridge, Massachusetts
WILLIAM B. FLOYD, Sears Roebuck and Company, Chicago, Illinois
J. A. FLYNN, Mechanical Engineer, Cambridge Research Laboratory, Cambridge
XXI
MEMBERS OF T1{E SYMPOSIUM
. JAMES W. FOLLIN, JR., Physicist, Applied Physics Laboratory, Johns Hopkins University,
Silver Spring, Mary land
G. DONALD FORBES, Electronics Consultant, Sudbury, l\tlassachusetts
.
RICHARD E. FORBES, Computation Laboratory, Harvard University, Cambridge
JAY W. FORRESTER, Associate Director, Servomechanisms Laboratory, Massachusetts Institute
of Technology, Cambridge
GEORGE E. FORSYTHE, National Bureau of Standards, University of California at Los Angeles,
Los Angeles, California
FRANKLIN H. FOWLER, JR., Associate Editor, Product Engineering, New York
PHILIP FRANKLIN, Professor, Massachusetts Institute of Technology, Cambri~ge
WALTER J. FRANTZ, Research Engineer, Boeing Aircraft, Seattle, Washington
WILLIAM H. FRATER, Director, Wage Analysis, General Motors Corporation, Detroit, Michigan
DAVID FRAZIER, Research Chemist, Standard Oil Company, Cleveland, Ohio
R. 0. FREDETTE,Mathematician, Bureau of Ordnance, Navy Department, Washington, D.C.
F. N. FRENKIEL, Naval Ordnance Laboratory, White Oak, Silver Spring, Maryland
A. W. FRICK, Radio Corporation of America, Camden, New Jersey
W. BARKLEY FRITZ, Matherriatician, Ballistics Research Laboratory, Aberdeen, Maryland
A. E. FROST, Assistant Equipment Research Engineer, Western Union Telegraph Company,
New York
JOSEPH FUCARILE, Computation Laboratory, Harvard University, Cambridge
W. H. FURRY, Associate Professor of Physics, Harvard University, Cambridge
IRVING J. GABELMAN, Engineer, Watson Laboratories, USAF, Long Branch, New Jersey
EDWIN GABRIEL, Associate Engineer, Air Force Cambridge Research Laboratories, Cambridge
j. ORTEN GADD, JR., Computation Laboratory, Harvard University, Cambridge
JAMES A. GEAN, Chief Analytical Engineer, Parsons Corporation, Traverse City, Michigan
ARTHUR A. GENTILE, Computation Laboratory, Harvard University, Cambridge
EUGENE M. GETTEL, Research Engineer, United Aircraft Corporation, East Hartford, Connecticut
N. ELIOT GIBBS, Senior Engineer, Raytheon Manufacturing Company, Waltham, Massachusetts
RICHARD C. GIBSON, Assistant Professor of Electrical Engineering, USAF Institute of Technology, Wright-Patterson Air Force Base, Dayton, Ohio
DONALD· B. GILLIES, Computation Centre, Toronto, Canada
DOROTHY F. GILLETTE, Physicist, Air Force Cambridge Research Laboratories, Cambridge
H. F. GINGERICH, U.S. Navy Department, Office of Naval Research, Washington, D.C.
G. GLINSKI, Vice-President, Computing Devices of Canada, Ltd.,Ottawa, Canada
SIMON E. Gluck, Research Associate, Moore School of Electrical Engineering, University of
Pennsylvania, Philadelphia, Pennsylvania
THOMAS N. K. GODFREY, National Bureau of Standards, Massachusetts Institute of Technology!
Cambridge
XXll
MEMBERS OF THE SYMPOSIUM
STANFORD GOLDMAN, Prof~ssor of Electrical Engineering, Syracuse University, Syracuse,
New York
HARRY H. GOODE, Special Devices Center, Office of Naval Research, Sands Point, Long
Island, New York
C. C. GOTLIEB, Assistant Professor, University of Toronto, Toronto, Ontario, Canada
R. S. GRAHAM, Bell Telephone Laboratories, Murray Hill, New Jersey
E. F. GRANT, Chief, Applied Mathematics Branch, Air Force Cambridge Research Laboratories, Cambridge
RANULF W. GRAS, Massachusetts Institute of Technology, Cambridge
HARRY J. GRAY, JR., Instructor of Electrical Engineering, University of Pennsylvania, Moore
Sch<;>ol of Electrical Engin.eering, Philadelphia, Pennsylvania
WALTER H. GRAY, JR., Engineer, Raytheon :rvfanufacturing Company, Waltham, Massachusetts
BEN F. GREENE, Radio Engineer, Air Force Cambridge Research Laboratories, Cambridge
H. VOSE GREENOUGH, JR., Director, Tec9nichordRecords, Brookline, Massachusetts
DARRIN H. GRIDLEY, Naval Research Laboratories, Washington, D.c.
D. D. GRIEG, Division Head, Federal Telecommunication Laboratories, Nutley, New Jersey
B. A. GRIFFITH,Assistant Professor, University of Toronto, Toronto, Canada
IRVING 1. GRINGORTEN, Meteorologist, Air Force Cambridge Research Laboratories, Cambridge
H. R. J. GROSCH, Watson Scientific Computing Laboratory, New York
E. V. GULDEN, Research Engineer, National Cash Register Company, Dayton, Ohio
WILLIAM F. GUNNING, Engineer, Rand Corporation, Santa Monica., California
DANIEL HAAGENS, Electronics Engineer,Underwood Corporation, Hartford, Connecticut
JAMES V. HAGGERTY, Procedural Consultant, Social Security Administration, Baltimore,
Maryland
GILBERT O. HALL, Electronic Scientist, Air Force Cambridge Research Laboratories, Cambridge
W. K. HALSTEAD, Chief Engineer, W. S. MacDonald Company, Cambridge
F. E. HAMILTON, Engineer, International Business Machines Corporation, Endicott, New York
PRESTON C. HAMMER, Los Alamos Scientific Laboratory, Los Alamos, New Mexico
R. W. HAMMING, Bell Telephone Laboratory, Murray Hill, New Jersey
MRS. R. W. HAMMING, Morristown, New Jersey
GARNET HANES, Computation Centre,Universityof Toronto, Toronto, Canada
KENNETH C. HANNA, Computation Laboratory, Harvard University, Cambridge
GEORGE A. HARDENBERGH, Assistant Research Engineer, Engineering Research Associates;,
St. Paul, Minnesota
E. L. HARDER, Consulting Transmission Engineer, Westinghouse Electric Corporation, East
Pittsburgh, Pennsylvania
JOHN A. HARR, Computation Laboratory, Harvard University, Cambridge
XXlll
MEMBERS OF THE SYMPOSIUM
ROBERT W. HART, Electronics Engineer, Office of Naval Research, Boston
M. L. HASELTON, Vice President, TeleregisterCorporation, New York
SETH HASTINGS, Mutual Life Insurance Company, New York
WILLARD D. HATCH, Ohio State University, Cleveland, Ohio
ROBERT L. HAWKINS, Computation Laboratory, Harvard University, Cambridge
MILES V. HAYES, Computation Laboratory, Harvard University, Cambridge
INEZ HAZEL, Assistant Engineer, Raytheon Manufacturing Company, Newton, Massachusetts
VERNON H. HEAD, National Advisory Committee for Aeronautics Project, Gordon McKay
Laboratory, Harvard University, Cambridge
SAUL D. HEARN, Chief, Employee Statistics Section, Social Security Administration, Baltimore,
Maryland
.I. C. HEBARD, JR., Mechanical Engineer, Raytheon Manufacturing Company, Newton,
Massach usetts
MAURICE H. HELLMAN, Engineer, .Air Force Cambridge Research Laboratories, Cambridge,
Massachusetts
L. L. HENKEL, Industrial College of Armed Forces, Fort Lesley McNair, Washington, D.C.
H. A. HENNING, Bell Telephone Laboratories, Inc., New York
PAUL HERGET, Director, University of Cincinnati Observatory, Cincinnati, Ohio
FRANK C.HERNE, Computation Laboratory, Harvard University, Cambridge
ROGER W. HICKMAN, Lecturer, Harvard University, Cambridge
JOHN L. HILL, Engineer, Engineering Research Associates, St. Paul, Minnesota
GEORGE W. HOBBS, Electronics Engineer, General Electric Company, Schenectady, New
York
GEORGE G. HOBERG, Research Associate, Burroughs Adding Machine Company, Philadelphia,
Pennsylvania
RICHARD HOFHEIMER, Computation Laboratory, Harvard University, Cambridge
MURRAY HOFFMAN, Engineer, Raytheon l\tlanufacturing Company, Waltham, Massachusetts
DOUGLAS L. HOGAN, Electronics Engineer, Navy Department, Washington, D.C.
JOHN V. HOLBERTON, Mathematician, Ballistic Research Laboratories, Aberdeen Proving
Ground, Maryland
A. B. HOLLISTER, Engineer, Underwood Corporation, Hartford, Connecticut
MARVIN R. HOLTER, Research Assistant, University of Michigan, Ann Arbor, Michigan
LAWRENCE F. HOPE, Assistant Head, Department ME-6, Research Division, General Motors,
Detroit, Michigan
RALPH HOPKINS, Special Representative, International Business Machines Corporation,
Washington, D.C.
GRACE HOPPER, Chief of Application Department, Eckert-Mauchly Computer Corporation,
Philadelphia, Pennsylvania
VIRGIL M. HORN, Assistant Section Head, Actuary Planning, Metropoiitan Life Insurance
Company, New York_
XXIV
MEMBERS OF THE SYMPOSIUM
JACOB HOROWITZ, Engineer, Raytheon Manufacturing Company, Newton, Massachusetts
JOHN H. HOWARD, Senior Project Engineer, Sperry Gyroscope Company, Great Neck, Long
Island, N ew York
BRADFORD HOWLAND, Graduate Student, Department of Physics, Harvard University,
Cambridge
JOHN F. HUBBARD, Treasurer, Martin-Hubbard Corporation, Boston
WILLIAM HULME, Computation Laboratory, Harvard University, Cambridge
j. STUART HUNTER, Statistician, Institute of. Statistics, University of North Carolina, Raleigh,
North Carolina
ALLEN HUNTINGTON, Electronics Engineer, U.S. Navy Electronics Laboratory, San Diego,
California
C. C. HURD, International Business Machines Corporation, New York
HARRY D. HUSKEY, Mathematician, National Bureau of Standards, Los Angeles, California
W. R. HYDEMAN, Mathematician, U.S. Navy, Washington, D.C. .
FR~NK T. INNES, Research Engineer, The Franklin Institute, Philadelphia, Pennsylvania
EUGENE ISAACSON, Assistant Professor, Institute for Mathematics 'and Mechanics, New York
University, New York
DAVID R. ISRAEL, Research Assistant, Massachusetts Institute of Technology, Cambridge
ARVID W.jACOBSON, Assistant Professor of Mathematics, Wayne University, Detroit, Michigan
BETTY JENNINGS, Computation Laboratory, Harvard University, Cambridge
PAUL V. JESTER, Vice President, A. C. Nielsen Company, Chicago, Illinois
G. D. JOHNSON, Bell Telephone Laboratories, Ivlurray Hill, New jersey
PAUL A. JOHNSON, Engineer, Boeing Airplane Company, Seattle, Washington
STANLEY A. JOHNSON, Computation Laboratory, Harvard University, Cambridge,
R. F. JOHNSTON, Computation Centre, University of Toronto, Toronto, Canada
R. CLARK JONES, Senior Physicist, Polaroid Corporation, Cambridge
ROBERT HUDSON JONES, Head Engineer, Radar Design Branch, Navy Department, Bureau
of Ships, Washington, D.C.
THOMAS F. JONES, JR., Assistant Professor, Differential Analyzer Computer Laboratory,
Massachusetts Institute of Technology, Cambridge
WILLARD C. JONES, Assistant Chief Engineer, Underwood Corporation, Hartford, Connecticut
WILLIAM B. JORDAN, Engineer, General Electric Company, Schenectady, New York
THEODORE A. KALIN, Computation Laboratory, Harvard University, Cambridge
SIDNEY KAPLAN, Mathematician, Naval Ordnance Laboratory, Washington
MIDA KARAKASHIAN, Member, joint Computing Group, Massachusetts Institute of Technology, Cambridge
ARTHUR A. KATZ, Mathematician, Eckert-Mauchly Computer Corporation, Philadelphia,
Pennsylvania
j. KATZ, Engineer, University of Toronto, Canada
MARTIN KATZ IN, Consultant, Naval Research Laboratory, Washington, D.C.
xxv
MEMBERS OF THE SYMPOSIUM
ALLEN KELLER, Assistant Division Engineer, Turbine Engineering Division, General Electric
.
Company, Lynn, Massachusetts .
E. G. KELLER, General Electric Company, Schenectady, New York
DANIEL W. KELLIHER, Computation Laboratory, Harvard University, Cambridge
JACQUELIN KELLY, Assistant Engineer, Raytheon Manufacturing Company, Newton, Massachusetts
JOHN P. KELLY, Head, Central Statistics Laboratory, Atomic Energy Commission, Oak
Ridge, Tennessee
WENTWORTH KENNARD, Technical Editor, Raytheon Manufacturing Company, Waltham,
Massach usetts
HARRY KENOSIAN, Research Engineer, Massachusetts Institute of Technology, Cambridge
CHESTER H. J. KEPPLER, Captain, USN, Retired, Counsellor for Foreign Students, Harvard
University, Cambridge
J. H. KERFOOT, Civilian Technical Officer, Royal Canadian Navy, Ottawa, Ontario, Canada
MARSHALL KINCAID, Computation Laboratory, Harvard University, Cambridge
GILBERT W. KING, Arthur D. Little, Inc., Cambridge
R. W. P. KING, Professor, Harvard University, Cambridge
HENRY KINZLER, Supervisor, Procedure Planning, Metropolitan Life Insurance Company,
New York
HANS KLEMPERER, Research Engineer, Massachusetts Institute of Technology, Cambridge
HERBERT M. KNIGHT, Electronic Engineer, Air Force Cambridge Research Laboratories,
Cambridge
RALPH J. KOCHENBURGER, Instructor, Servomechan:isms Laboratory, Massachusetts Institute
of Technology, Cambridge
.
FLORENCE K. KOONS, Mathematician, National Bureau of Standards, Washington, D.C.
ZDENEK KOPAL, Massachusetts ,Institute of Technology, Cambridge
JOHN J. KORZDORFER, Project Engineer, Red Bank Division, Bendix Aviation Corporation,
Red Bank, New Jersey
HANS KRAFT, Aerodynamicist, General Electric Company, Schenectady, New York
H. P. KUEHNI, Division Engineer, General Electric Company, Schenectady, New York
JosEPH H. KUSNER, Munitions Board, Department of Defense, Washington, D.C.
N. L. KUSTERS, Research Engineer, National Research Council, Ottawa, Ontario, Canada
EDWARD LACEY, Electrical Engineer, National Bureau of Standards, Los Angeles, California
H. N. LADEN, LT., USN, Research and Development Engineer, Bureau of Ships, Washington,
·D.C.
LEON J. LADER, Radio Engineer, Watson Laboratories, Red Bank, New Jersey
CORNELIUS LANCZOS, Mathematician, Institute for Numerical Analysis, University of Cali·fornia at Los Angeles, California
CARNEY LANDIS, Professor of Psychology, Columbia University, New York
GERALD W. LAVIGNE, Computation Laboratory, Harvard University, Cambridge
XXVI
MEMBERS OF THE SYMPOSIUM
TIMOTHY LEARY, Division of Industrial Cooperation, Massachusetts Institute of Technology,
Cambridge
PHILIPPE E. LECORBEILLER, Professor of Applied Physics, Harvard University, Cambridge
HARRY S. LEE, Engineer, Massachusetts Institute of Technology, Cambridge
DERRICK H. LEHMER, Professor of Mathematics, University of California, Berkeley, California
R. A. LEIBLER, USN, Washington, D.C.
POLLY LEIGHTON, Supervisor, Computing Group, Massachusetts Institute of Technology,
Cambridge
ALAN L. LEINER, Physicist, National Bureau of .Standards, Electronic Computers Section,
Washington, D.C.
MRS. HENRIETTA C. LEINER, Physicist, National Bureau of Standards, Electron Tube Laboratory, Washington, D.C.
J. PLUMER LEIPHART, Electronic Engineer, Naval Research Laboratory, Washington, D.C.
WASSILY W. LEONTIEF, Professor of Economics, Harvard University, Cambridge
BURTON R. LESTER, Section Engineer, Electronics Laboratory, General Electric Company,
Syracuse, N ew York
JOSEPH H. LEVIN, Mathematician, Computation Laboratory, National Bureau of Standards,
Washington, D.C.
ARNOLD M. LEVINE, Department Head, Federal Telecommunications Laboratories, Nutley,
New Jersey.
FRED C. LEWIS, Engineer, Raytheon Manufacturing Company, Waltham, Massachusetts
C. C. LIN, Massachusetts Institute of Technology, Cambridge
PETER L. LINDLEY, Computation Laboratory, Harvard University, Cambridge
HERBERT G. LINDNER, Radio Engineer, Coles Signal Corps Laboratory, Red Bank, New
Jersey
S. B. LITTAUER, Associate Professor, Columbia University, New York
ELBERT P. LITTLE, Chief, Computation Section, Office of Air Research, USAF; Computation
Laboratory, Harvard University, Cambridge
HUBERT M. LIVINGSTON, Eckert-Mauchly Computer Corporation, Philadelphia, Pennsylvania
MRS. HUBERT M. LIVINGSTON, Eckert-Mauchly Computer Corporation, Philadelphia, Pennsylvania
CHARLES J. LODA, Physicist, U.S. Navy Underwater Sound Laboratory, New London, Connecticut
SAMUEL LUBKIN, Electronic Scientist, National Bureau of Standards, Washington, D.C:
E ..E. LUCCHINI, Computation Laboratory, Harvard University, Cambridge
DUNCAN LUCE, Research Center for Group Dynamics, University of Michigan, Ann Arbor,
Michigan
W.F. MACDoNAL!l, S. D. Leidesdorf and Company, New York
JOHN H. MACNEILL, Senior Engineer, Raytheon Manufacturing Company, Waltham, Massachusetts
XXVll
MEMBERS OF THE SYMPOSIUM
H. M. MACNEILLE, Atomic Energy Commission, Washington, D.C.
W. H. MACWILLIAMS, Jr., Bell Telephone Laboratories, Murray Hill, New Jersey
P; J. MAGINNISS, Engineer, General Electric Company, Schenectady, New York
ALFRED T. MAGNELL, CDR., USN, Office of Naval Research, Washington, D.C.
HENRY M. MALLON, Assistant Engineer, Raytheon Manufacturing Company, Waltham,
Massachusetts
HARLAND MANCHESTER, Roving Editor, Reader' s D~gest, New York
ROLAND A. MANGINI, John Hancock Life Insurance Company, Boston
CHARLES S. MANNING, Electrical Engineer, Navy Electronics Laboratory, San Diego, California
ETHEL Cox MARDEN, Mathematician, National Bureau ofSta:ndards, Washington, D.C.
EMIL MARECKI,Assistant Chief, Machine Tabulation Division, Bureau of the Census, Washington, D.C.
ARTHUR E. MARl, Mari & RentelCompany, Inc., Boston
JOSEPH MARKSTEINER, Computation Laboratory, Harvard University, Cambridge
LAWRENCE MARKUS, Harvard University, Cambridge
ELBERT W. MARLOWE, Engineer, Union Switch & Signal Company, Pittsburgh, Pennsylvania
H. W. MARSH, Consultant, USN Underwater Sound Laboratory, New London, Connecticut
ROBERT H. MARSH, Engineer, Massachusetts Institute of Technology, Cambridge
BYRON O. MARSHALL, Physicist, Air Force Cambridge Research Laboratories, Cambridge
DAVID W. MARSHALL, Procedures Analyst, Metropolitan Life Insurance Company, New York
W. T. MARTIN, Professor, Massachusetts Institute of Technology, Cambridge
EDWARD MASSELL, Engineer, Electronic Associates, Inc., Long Branch, New Jersey
MORTON P. MATTHEW, Research Engineer, Friden Calculating Machine Company, San
Leandro, California
J. W.MAUCHLY, President, Eckert-Mauchly Computer Corporation, Philadelphia, Pennsylvania
HAROLD F. MAY, Assistant Director, Teleregister Laboratories, New York
WILLIAM E. MAY, Department of the Army, Washington, D.C.
J. P. MAYBERRY, Computation Centre, University of Toronto, Toronto, Ontario, Canada
H. C. MAYER, Research and Development, Radio Corporation of America, Camden, New
Jersey
ROLLIN POWELL MAYER, Research Engineer,Massachusetts Institute of Technology, Cambridge
EVERETT" E. MCCOWN, Electronics Engineer, U.S. Navy Electronics Laboratory, San Diego,
California
JAMES O. McDONOUGH, Research Engineer, Massachusetts Institute of Technology, Cambridge
WILLIAM D. MCGUIGAN, Engineering Coordinator, Raytheon Manufacturing Company,
Waltham, Massachusetts
E. C. McKAY, Chief, Section of Tides,U.S. Coast and Geodetic Survey, Department of
Commerce, Washington, D.C.
XXV111
MEMBERS OF THE SYMPOSIUM
R. O. McMANUS, Mechanical Engineer, Air Force Cambridge Research Laboratory, Cambridge
JAMES L. MCPHERSON, Machine Development Officer, Bureau of the Census, Washington, D.C.
JOHN C. MCPHERSON, Vice President, International Business Machines, New York
KENNETH E. MCVICAR, Research Assistant, Electrical Engineering Department, Massachusetts
Institute of Technology, Cambridge
LEONARD C. MEAD, Research Coordinator, Tufts College, Medford, Massachusetts
RALPH 1. MEADER, Vice President, Engineering Research Associates, St. Paul, Minnesota
R. E. MEAGHER, Assistant Professor, University of Illinois, Urbana, Illinois
EUGENE A. MECHLER, Re'search Engineer, Franklin Institute, Philadelphia, Pennsylvania
C. S. MERCER, Sales Engineer,' Aluminium Company of America, Boston
DAVID MIDDLETON, Associate Professor of Applied Physics, Harvard University, Cambridge
JAMES G. MILES, Engineer, Engineering Research Associates, Inc., St~ Paul, Minnesota
E. J. MILLER, Technical Officer, National Defence, Ottawa, Canada
FREDERICK G. MILLER, Electrical Engineer, Naval Proving Ground, Dahlgren, Virginia
HAROLD C~ MILLER, Assistant Chairman, Physics Research, Armour Research Foundation,
Chicago, Illinois
BURTON E. MILLS, Air Force Cambridge Research Laboratories, Cambridge
W. E. MILNE, Professor, Oregon State College, Corvallis, Oregon
HARRY R. MIMNO, Professor of Applied Physics, Harvard University, Cambridge
MILTON J. MINNEMAN, Electronic Group Engineer, Glenn L. Martin Company, Baltimore,
Maryland
WILLIAM G. MINTY, Computation Laboratory, Harvard University, Cambridge
HERBERT F. MITCHELL, JR., Webb Institute of Naval Architecture, Glenn Cove, Long Island,
New York
SAMARENDRA K. MITRA, UNESCO, Calcutta, India
WILLIAM MITTELMAN, Griffiss Air Force Base, Rome, New York
CARLTON A. MIZEN, Project Engineer, Raytheon Manufacturing Company, Waltham,
Massachusetts
ELMER B. MODE, Professor of Mathematics, Boston University, Boston
BRUCE MONCREIFF, Systems Reviewer, Prudential Insurance Company of America, Newark,
New Jersey
ROBERT J. MONROE, Institute of Statistics, North .carolina State College, Raleigh, North
Carolina
N. F. MOODY, National Research Council, Ontario, Canada
CALVIN N. MOOERS, President, Zator Company, Boston
C. M. MOONEY, International Business Machines Corporation, New York
BENJAMIN L. MOORE, Assistant Director, Computation Laboratory, Harvard University,
Cambridge
.
EDWARD F. MOORE, Graduate Student, Brown University, Providence, Rhode Island
XXIX
MEMBERS OF THE SYMPOSIUM
JOHN MORRIS, Junior Engineer, Raytheon Manufacturing Company, Waltham, Massachusetts
REEVES MORRISSON, Head, Analysis Section, Research Department, United Aircraft Corporation, East Hartford, Connecticut
PAUL L. MORTON, Associate Professor of Electrical Engineering, University of California,
Berkeley, California
LEON MOSES, Graduate Student, Harvard University, Cambridge
Z. 1. MOSESSON, Senior Actuarial Assistant, Prudential. Insurance Comp~ny of America,
Newark, New Jersey
ROBERT G. Moss, Secretary, Electronic Calculator Committee, M~tropolitan Life Insurance
Company, New York
FREDERICK MOSTELLER, Assistant Professor of Mathematical Statistics, Harvard University,
Cambridge
HAROLD M. MOTT-SMITH, Assistant .Chief, Reactor Branch, Atomic Energy Commission,
Washington, I?C.
CLIFTON F. MOUNTAIN, Director, Office of Statistical Research, Boston University, Boston
J. D. MOUNTAIN, International Telephone and Telegraph Company, New York
CARL F. MUCKENHOUPT, Head, Scientific Section, Office of Naval Research, Boston
RALPH E. MULLENDORE, Bureau of the Census, Washington, D.C.
G. G. MULLER, Bell Telephone Laboratories, New York
ROBERT E. MUMMA, Manager, Electrical Development Laboratory, National Cash Register
Company, Dayton, Ohio
J. H. MUNCY, Electronic Engineer, Naval Air Development Station, Johnsville, Pennsylvania
MORRIS MUSKAT, Director of Physics Division, Gulf Oil Corporation, Pittsburgh, Pennsylvania
FRANKLIN G. MYERS, Design Specialist, Glenn L. Martin Company, Baltimore, Maryland
ROBERT A. NELSON, Re~earch Engineer, Servomechanisms Laboratory, Massachusetts Institute of Technology, Camb~idge
A. J. NEUMANN, Research Associate, University of Pennsylvania, Philadelphia, Pennsylvania
HERBERT B. NICHOLS, Science Editor, Christian Science Monitor, Boston
NATALIE N: NICHOLSON, Librarian, Harvard University, Cambridge
R. F. NICHQLSON, Data Utilization Laboratory, Griffiss Air Force Base, Rome, New York
EDWIN N. NILSON, Assistant Professor of Mathematics, Trinity College, Hartford, Connecticut
WILLIAM J. NOLAN, JR., Division of Industrial Cooperation, Massachusetts Institute of Technology, Cambridge
R. H. NOYES, Radio Engineer, Signal Corps Engineering Laboratories, Coles Laboratory,
Red Bank, New Jersey
ALEXANDER NYMAN, Technical Advisor, Alden Products Company, Brockton, Massachusetts
H. NYQuIsT, Bell Telephone Laboratories, New York
JAMES J. O'B~IRNE, Branch Chief, Control, Social Security Administration, Baltimore, Maryland
JOHN A. O'BRIEN, Research Engineer, Massachusetts Institute of Technology, Cambridge
!
xxx
MEMBERS OF THE SYMPOSIUM
L. A. OHLINGER, Project Engineer, Northrop Aircraft, Inc., Hawthorne, California
JOHN A~ O'KEEFE, Mathematician, Army Map Service, Washington, D.C.
BRUCE OLDFIELD, Mathematician, U.S. Naval Ordnance Test Station, China Lake, California
THOMAS K. OLIVER, Acting Chief, Office of Air Research, Wright-Patterson Air Force Base,
Ohio
R. D. O'NEAL, Aeronautical Research Center, University of Michigan,Ann Arbor, Michigan
ALLAN H. O'NEIL, Mathematician, U.S. Naval Proving Ground, Dahlgren, Virginia
ALEXANDER ORDEN, Research Associate, Massachusetts Institute of Technology, Cambridge
SIDNEY OVIATT, John Price Jones Company, Inc., New York
CHESTER H. PAGE, Electronics Consultant, National Bureau of Standards, Washington, D.C.
HENRY E. PAGE, Computation Laboratory, Harvard University, Cambridge
RALPH L. PALMER, Engineer, International Business Machines Corporation, Poughkeepsie,
New York
WILLIAM N. PAPIAN,Research Assistant, Servomechanisms Laboratory, Massachusetts Institute of Technology, Cambridge
JOHN E. PARKER, President, Engineering Research Associates, Inc., St. Paul, Minnesota
G. B. PARKINSON, Senior Engineer, Raytheon Manufacturing Company, Newton, Massachusetts
JAMES PASTORIZA, Electronic Engineer, Air Force Cambridge Research Laboratory, Cambridge
GEORGE W. PATTERSON, Assistant Professor, Moore School, University of Pennsylvania,
Philadelphia, Pennsylvania
ALFRED M. PEISER, Mathematician, Hydrocarbon Research, Inc., New York
C. L. PEKERIS, Institute for Advanced Study, Princeton, New Jersey
DAVID P. PERRY, Student, University of Pennsylvania, Philadelphia, Pennsylvania
C. G. PETERSON, District Agent, Marchant Calculating lYfachine Company, Boston
B. L. PFEFER, Project Manager, General Electric Company, Syracuse, New York
HARRY PFORZHEIMER, Head, Economic Section, Standard Oil Company (Ohio), Cleveland,
Ohio.
EDWARD W. PHARO, JR., Engineer, Philadelphia, Pennsylvania
J. R. PIERCE, Bell Telephone Laboratories, Murray Hill, New Jersey
ROBERT P. PINCKNEY, Student, Massachusetts Institute of Technology, Cambridge
SAMUEL PINES, Design Engineer, Republic Aviation, New York
CHARLES A. PIPER, Sup~rvising Engineer, Bendix Aviation Research Laboratories, Detroit,
Michigan
H ..POLACHEK, Mathematician, Naval Ordnance Laboratory, 'Vhite Oak, Silver Spring,
Maryland
ROBERT E. POPIEL,Computation Laboratory, Harvard University, Cambridge
H. PORITSKY, Consulting Engineer, General Electric Company, Schenectady,' N ew York
WILLIAM A. PORTER, Computation Laboratory, Harvard University, Cambridge
JOHN T. POTTER, President, Potter Instrument Company, Inc., Flushing, New York
XXXI
MEMBERS OF THE SYMPOSIUM
F. D. POWELL, Dynamics Engineer, Bell Aircraft Corporation, Buffalo, New York
BETTE PREER, Librarian, Boston Public Library, Boston
GEORGE G. PROULX, Computation Laboratory, Harvard University, 'Cambridge
C. C. PYNE, Harvard University, Cambridge
E. J. QUINBY, Director, Electronic Research and Development," Monroe Calculating Machine
Company, Orange, New Jersey
JAN RAjCHMAN, Research Physicist, Radio Corporation of America, Princeton, New Jersey
JOHN H. RAMSER, Senior Physicist, Atlantic Refining Company, Philadelphia, P~nnsylvania
EUGENE A. RASOR, 'Actuarial Mathematician, Social Security Administration, Washington,
D.C.
ROBERT RATHBONE, Division of Industrial Cooperation, Servomechanisms Laboratory,
Massachusetts Institute of Technology, Cambridge
A. G. RATZ, Research Engineer, University of Toronto, Toronto, Ontario, Canada
KONRAD RAUCH, Special' Application Engineer; National Cash Register Company, Dayton,
Ohio
RICHARD W. READ, Massachusetts Institute of Technology, Cambridge
DONALD REAM, Electronics Engineer, Navy Department, Washington, D.C.
MINA REES, Director, Mathematic Sciences Division, Office of Naval Research, Washington,
D.C.
K. M.REHLER, Engineer, Raytheon Manufacturing Company, Waltham, Massachusetts
GEORGE W. REITWIESNER, Mathematician, Ballistic Research. Laboratories, Aberdeen Proving
Ground, Aberdeen, Maryland
THEODORE V. RENTEL, Mari and RentelCompany, Inc., Boston
EDWARD REYNOLDS, Administrative Vice President, Harvard University, Cambridge
GEORGE E. REYNOLDS, Mathematician, Air Force Cambridge Research Laboratories, Cambridge
JAMESR.REYNOLDS, Office of Special Advisor to the President, Harvard University,
Cambridge
. ROBERT R. REYNOLDS, Assistant Professor, Oklahoma Agricultural and Mechanical College,
Stillwater, Oklahoma
CHARLES RHEAMS, Senior Engineer, Raytheon Manufacturing Company, Boston
IDA RHODES, Mathematician, National Bureau of Standards, Washington, D.C.
HOPE RICE, University of North Carolina, Chapel Hill, North Carolina
OSCAR K. RICE, Professor of Chemistry, University of North Carolina, Chapel Hill, North
Carolina
EDWIN S. RICH, Engineer, Massachusetts Institute of Technology, 'Cambridge
C. ,H. RICHARDS, Computation Laboratory, Harvard University, Cambridge
LOUIS N. RIDENOUR, Dean, Graduate College, University of Illinois, Urbana, Illinois
DONALD V. RIDER, Electronics Engineer, Raytheon Manufacturing Company, Waltham,
Massachusetts
XXXll
MEMBERS OF THE SYMPOSIUM
LEONARD D. RINALDI, Research Engineer, Cornell Aeronautical Laboratory, Buffalo, New
York
E. K. RITTER, Research Engineer, University of Michigan, Ann Arbor, Michigan
GORDON A. ROBERTS, Manager, Future Demands Department, International Business Machines
Corporation, New York
REX ROBERTS, Vice President, Transducer Corporation, Boston
JOHN W. ROCHE, Computation Laboratory, Harvard University, Cambridge
NATHANIEL ROCHESTER, Engineering Laboratory, International Business Machines Corporation, Poughkeepsie, New York
HAROLDO AZEVEDO RODRIGUES, Rio de Janeiro, Brazil
EDWARD ROGAL, President, Central Records, Inc., Boston
STANLEY ROGERS, Research Engineer, CC?nvair, San Diego, California
THOMAS A. ROGERS, Associate Professor of Engineering, University of California at Los Angeles,
Los Angeles, California
E. ROOT, Springfield, Vermont
_
MORRIS E. ROSE, Principal Physicist, Oak Ridge National Laboratory, Oak Ridge, Tennessee
JOSHUA H. ROSENBLOOM, Physicist, Naval Ordnance Laboratory, Washington, D.C.
CLARENCE Ross, U.S. Naval Proving Ground, Dahlgren, Virginia
JOHN ROTHERY, Air Force Cambridge Research Laboratories, Cambridge
WALTER ROTMAN, Electronic Engineer, Air Force Cambridge Research Laboratories, Cambridge
B. G. H. ROWLEY, LT.-CDR., British Naval Staff, Electronics Liaison Officer, Washington,
D.C.
D. M. RUBEL, LT.-CDR., USN, Office of the Secretary of Defense, Washington, D.C.
MILTON D. RUBIN, Research Director, G. A. Philbrick Researches, Inc., Boston
M. RUBINOFF, Institute for Advanced Study; Princeton, New Jersey
PHILIP RULON, Professor of Education and Acting Dean of the Faculty of Education, Harvard
University, Cambridge
HEINZ RUTISHAUSER, S~iss Federal Iri~titute of Technology, Zurich, Switzerland
DAVID RUTLAND, Engineer, National Bureau of Standards, Los Angeles, California
HERBERT E. SALZER:, M'athematician, Computation Laboratory, National Bureau of Standards,
Washington, D.C.
JOHNM. SALZER, Research Associate, Massachusetts Institute of Technology, Cambridge
ARTHUR L. SAMUEL, International Business Machines Corporation, Poughkeepsie, New York
PAUL A. SAMUELSON, Professor of Economics, Massachusetts Institute of Technology, Cambridge
JACKIE SANBORN, Computation Laboratory, Harvard University, Cambridge
BERNARD L. SARAHAN, Mathematician, Naval Research Laboratory, Washington, D.C.
EMIL D. SCHELL, Chief, Mathematic and Electronic Computer Branch, U.S. Air Force,
Washington, D.C. XXXlll
MEMBERS OF THE SYMPOSIUM
HAROLD A. SCHERAGA, Instructor, Cornell University, Ithaca, New York
MAX G. SCHERBERG, Chief of Applied Mathematics, Office of Air Research, Wright-Patterson
Air Force Base, Dayton, Ohio
ROBERT SCHILLING, Department Head, General' Motors Research Laboratory, Detroit,
Michigan
KURT SCHNEIDER,Chemist-Metallurgist, Salem, Massachusetts
HENRY W. SCHRIMPF, Methods Analyst, Prudential Insurance Company of America, Newark,
New jersey
R. E. SCHUETTE, Project Engineer, Barber-Colman Company, Rockford, Illin9is
WILLIAM T. SCOTT, Associate Scientist, Brookha:ven National Laboratory, Upton, Long Island,
New York
JOHN F. SCULLY, Radio Engineer, Air Force Cambridge Research Laboratories, Cambridge
RAYMOND j. SEEGER, Naval Ordnance Laboratories, White Oak, Silver Spring, Maryland
. ROBERT R. SEEKER, jR., International Business Machines Corporation, New York
WARREN L. SEMoN,Computation Laboratory, Harvard University, Cambridge
AMIR H. SEPAHBAN, Student, Harvard University, Cambridge
RO:sERT SERRELL, Radio Corporation of America Laboratories, Princeton, New jersey
EDGAR D. SEYMOUR, Development Engineer, Eastman Kodak Company, Rochester,New York
JACOB SHAPIRO, Physicist, Atomic Energy Commission, Elmhurst, N ew York
ROBERT F. SHAW, Engineer, Eckert-Mauchly Computer Corporation, Philadelphia, Pennsylvania
C. BRADFORD SHEPPARD, Engineer, Eckert-Mauchly Computer Corporation, Philadelphia,
Pennsylvania
HERBERT SHERMAN, Chief, Systems Branch, Plans Office, AMC, Watson Laboratories, Red
Bank, New jersey
JACK SHERMAN, Mathematician, The Texas Company, Beacon, New York
CHARLES E. SHINN, Electronics Department, Burroughs Adding Machine Company, Philadelphia, Pennsylvania
BERNARD SHOOR, Research Engineer,. Northrop Aircraft Company, Hawthorne,California
GEORGE SHORTLEY, Operations Research Office,Department of the Army, Ft. Leslie j.
McNair, Washington, D.C. '
ARNOLD SHOSTAK, Engineer, Office of Naval Research, Washington, D.C.
BYRON SHREINER, Assistant to Vice President, A. C. Nielsen Company, Chicago, Illinois
T. R. SILVERBERG, Engineer, Raytheon Manufacturing Company; Waltham, Massachusetts
THEODORE SINGER, Computation Laboratory, Harvard University, Cambridge
ROGER L. SISSON, Research Assistant,Massachusetts Institute of Technology, Cambridge
MRS. ROGER L. SISSON, Waltham, Massachusetts
C.j. SITTINGER, Consulting Engineer, Winchester, Massachusetts
MORTON L. SLATER,Senior Mathematician, Ordnance Research, Chicago, Illinois
DAVID S1.EPIAN, Harvard University, Cambridge
XXXIV
MEMBERS OF THE SYMPOSIUM
ALBERT E. SMITH, Physicist, Office of Naval Research, Washington, D'.C.
BRUCE K. SMITH, Junior Engineer, Raytheon Manufacturing Company, Waltham, Massachusetts
CHARLES V. L. SMITH, Head, Computer Branch, Office of Naval Research, Washington, D.C.
DEXTER SMITH, Computation Laboratory, Harvard University, Cambridge
LEONARD W. SMITH, Computation Laboratory, Harvard University, Cambridge
H. P. SMITH, Research, Underwood 'Corporation, Hartford, Connecticut
V. G. SMITH, Professor of Electrical Engineering, University of Toronto, Toronto, Ontario,
Canada
R. M. SNOW, Senior Physicist, Martin-Hubbard Corporation, Boston
FRANCES E. SNYDER, Program Analyst, Eckert-Mauchly Computer Corporation, Philadelphia,
Pennsylvania
R. L. SNYPER, Branch Chief, Ballistic Research Laboratory, Aberdeen Proving Ground,
Aberdeen, Maryland
SAMUEL S. SNYDER, Research Analyst, Department of the Army, Washington, D.C.
THOMAS G. SOFRIN, Engineer, Pratt & Whitney Aircraft Corporation, Hartford, Connecticut
AARON S. SOLTES, Electronics Engineer, Air Force Cambridge Research Laboratories, Cambridge
G. WALTER SPAHR, Manager of Engineering, Underwood Corporation, Hartford, Connecticut
AMBROSE P. SPEISER, Swiss Federal Institut~ of Technology, Zurich, Switzerland
JOHN W. H. SPENCER, Chief, Computing Branch, Geodetic Division, Army Map Service,
Washington, D.C.
DANIEL SPILLANE, Computation Laboratory, Harvard University, Cambridge
BERNARD I. SPINRAD, Associate Physicist, Argonne National Laboratory, Chicago, Illinois
H. P. STABLER, Professor of Physics, Williams College, Williamstown, Massachusetts
JACK J. STALLER, Mechanical Engineer, ,Raytheon Manufacturing Company, Newton,
Massachusetts
E. D. STANLEY, JR., CDR., USN, Munitions Board, Department of Defense, Washington, D.C.
S. W. STARK, Research Staff Member, Optical Research Laboratory, Boston University, Boston
CHARLES L. STEC, Civilian Assistant, Electrical Design Division, Bureau of Ships, USN,
Washington, D.C.
HAROLD STEIN, Physicist, Signal Corps Engineering Laboratories, Long Branch, New Jersey
HAROLD H. STEIN, Engineer, University of Toronto, Toronto, Ontario, Canada
RICHARD STEPHENSON, Bell Telephone Laboratories, Murray Hill, New Jersey
D. L. STEVENS, Head of Computer Group, Sylvania Electric Products, Inc., Boston
GEORGE R. STIBITZ, Consultant, Burlington, Vermont
W. W. STIFLER, JR., Physicist, Engineering Research Associates, Washington, D.C.
E. E. ST. JOHN, Electronic Engineer, Nepa Project, Oak Ridge, Tennessee
MORTON J. STOLLER, Aeronautical Research Scientist, National Advisory Committee for
Aeronautics, Langley Air Force Base, Virginia
xxxv
MEMBERS OF THE SYMPOSIUM
JOSEPH J. STONE, JR.~ Associate Electrical Engineer, Fairchild Engine and Aircraft Corporation, Oak Ridge, Tennessee
JOHN STRAND, Research Engineer, University of Michigan, Ann Arbor, Michigan
H~ L. STRAUS, Chairman of the Board, Eckert-Mauchly Computer Corporation, Philadelphia,
. Pennsylvania
C. STREET, Professor of Physics, Harvard University, Cambridge
PETER F. STRONG, Computation Laboratory, Harvard University, Cambridge
DANIEL R. STULL, Physical Research Laboratory, The Dow Chemical Company, Midland,
Michigan
GEORGE C. SUMNER, Massachusetts Institute of Technology, Cambridge
ALFRED K. SUSSKIND, Research Assistant, Project Whirlwind, Massachusetts Institute of
Technology, Cambridge
LOUIS SUTRO, Instructor, Department' of Electrical Engineering, Tufts College, Medford,
Massach usetts
RUTH W. SUTRO, West Medford, Massachusetts
. HENRY W. SYER, Assistant Professor of Education, Boston University, Boston
A. H. T AUB, .Professor, University of Illinois, Urbana, Illinois
NORMAN H. TAYLOR, Research Engineer, Massachusetts Institute of Technology, Cambridge
RICHARD TAYLOR, Assistant Professor, Department of Electrical Engineering, Massachusetts
Institute of Technology, Cambridge
J. B. TEPE, Research Engineer, DuPont Experimental Station, Wilmington, Delaware
BENJAMIN J. TEPPING, Statistician, Bureau of the Census, Washington, D.C.
P. J. THEODORIDES, Visiting Lecturer on Aeronautical Engirieering, Harvard University,
Cambridge
HELENE THOMAN, Computation Laboratory, Harvard University, Cambridge
L. H. THOMAS, International Business Machines Corporation, New York
PAUL D. THOMAS, Mathematician, U.S. Coast and Geodetic Survey, Washington, D.C.
GEORGE W. THOMSON, Senior Chemical Mathematician, Ethyl Corporation, Detroit,Michigan
R. THORARENSEN, Electronic Laboratory, General Electric Company, Syracuse, New York
W. K. TRAUGER, Professor, State Teachers College, Potsdam, New York
IRVEN TRAVIS, Director of Research, Burroughs Adding lVfachine Company, Philadelphia,
Pennsylvania
G. W. TRICHEL, Chrysler Corporation, Detroit, Michigan
ARTHUR D. TROTTENBERG, Assistant to Administrative Vice President, Harvard University,
Cambridge
LEDYARD R. TUCKER, Director of Statistical Analysis, Educational Testing Service, Princeton,
New Jersey
JOHN W~ TUKEY, Bell Telephone Laboratories, Murray Hill, New Jersey
L. R. TURNER, Aeronautical Research Scientist, National Advisory Committee for Aeronautics, Cleveland, Ohio
'1.
XXXVI
MEMBERS OF THE SYMPOSIUM
MRS. L. R. TURNER, Cleve!and, Ohio
ARTHUR W. TYLER, Physicist, Eastman Kodak Company, Rochester, New York
STANISLAUS M. ULAM, Los Alamos Scientific Laboratory, Los Alamos, New Mexico
JOHN H. VAN VLECK, Professor of Mathematical Physics, Harvard University, Cambridge
CLEMENT J. VAN VLIET, Statistician, U.S. Navy Electronics Laboratory, San Diego, California
FRANK L. VERDONCK, Computation Laboratory, Harvard University, Cambridge
FRANK M. VERZUH, Research Associate, Massachusetts Institute of Technology, Cambridge
HUGH WAINWRIGHT, Sales Engineer, Sylvania Electric Products, Inc., Boston
J. H. WAKELIN, Associate Director of Research, Textile Research Institute, Princeton, New
Jersey
AN 'VANG, Computation Laboratory, Harvard University, Cambridge
CLIFFORD A. WARREN, Bell Telephone Laboratories, Murray Hill, New Jersey
CHAUNCEY W. WATT, Engineer, Massachusetts Institute of Technology, Cambridge
EDWARD A. WEEKS, JR., Editor, The Atlantic MonthlY, Boston
JAMES R. WEINER, Chief Electrical Engineer, Eckert-Mauchly Computer Corporation, Philadelphia, Pennsylvania
HERBERT G. WEISS, Senior Engineer, Raytheon Manufacturing Company, Waltham, Massachusetts
JOSEPH WEINSTEIN, Mathematician, Signal Corps Engineering Laboratories, Ft. Monmouth,
New Jersey
W. GORDON WELCHMAN, Member of Research Staff, Massachusetts Institute of Technology,
Cambridge
DAVID R. WELLER, Electronic Engineer, Data Utilization Laboratory, Griffiss Air Force Base,
Rome, N ew York
EVERETT T. WELMERS, Chief of the Dynamics Group, Bell Aircraft Corporation, Buffalo,
New York
INA W. WELMERS, Instructor in Mathematics, University of Buffalo, Buffalo, New York
ALBERT WERTHEIMER, Engineer, Bureau of Ordnance, Navy Department, Washington, D.C.
CHARLES F. WEST, Section Head, Raytheon Manufacturing Company, Waltham, Massachusetts
HARALD M. WESTERGAARD, Professor of Civil Engineering, Harvard University, Cambridge
L. D. WHITELOCK, Electronics Engineer, Bureau of Ships, Navy Department, Washington,
D.C.
C. A. WHITTEN, Chief, Section of Triangulation,U.S. Coast and Geodetic Survey, Washington,
D.C.
C. ROBERT WIESER, Research Engineer, Servomechanisms Laboratory, Massachusetts Institute
of Technology, Cambridge
HOLLY B. WILKINS, Computation Laboratory, Harvard University, Cambridge
ROBERT E. WILKINS, Computation Laboratory, Harvard University, Cambridge
DAVID A. WILKINSON, Project Manager, General Electric Company, Syracuse, New York
XXXVll
MEMBERS OF THE SYMPOSIUM
SAMUEL B. WILLIAMS, Consulting Electrical Engineer, Brooklyn, New York
DEAN H. WILSON, Research Associate, Aeronautical Research Center, University ·of Michigan,
Ann Arbor, Michigan
EDWIN B. ''''ILSON, Consultant, Office of Naval Research, Boston
MORRIS WINICK, Chief Engineer, Transducer Corporation, Boston
WILLIAM WOLFSON, Electronic Engineer, Raytheon Manufacturing Company, Waltham,
Massach usetts
RICHARD D. WOLTMAN, U.S. Naval Proving Ground, Dahlgren, Virginia
''''AY DONG Woo, Assistant Professor, Computation Laboratory, Harvard University, Cambridge
H. A. WOOD, Supervisor of Dynamic Analysis, Chance Vought Aircraft, Dallas, Texas
MARSHALL K. WOOD, Assistant Director, Program Standards. and Cost Control, U.S. Air
Force, Washington, D.C.
RALPH V. VVOOD, JR., Air Force Cambridge Research Laboratories, Cambridge
. WILLIAM W. WOODBURY, Northrop Aircraft Company, Hawthorne, California
L. F. WOODRUFF, Chief Scientific Advisor to Director of Intelligence, Washington,.D.C.
R. W. WOODWARD, Underwood Corporation, Hartford, Connecticut
HOWARD WRIGHT, Electronic Scientist, National Bureau of Standards, vVashington, D.C.
ROBERT H. YOUDEN,· Student, Servomechanisms Laboratory, Massachusetts Institute of
Technology, Cambridge
DAVID IV!. YOUNG, JR., Graduate Student, Harvard. University, Cambridge
PATRICK YOUTZ, Research Associate, Massachusetts Institute of Technology, Cambridge
H. 1. ZAGOR, Chief Physicist, Reeves Instrument Corporation, New York
SERGE J .. ZAROODNY, Engineer, Ballistics Research Laboratory, Aberdeen Proving Ground,
Aber~een,l\faryland
G. K. ZIPF, University Lecturer, Harvard University, Cambridge
XXXV111
FIRST SESSION
Tuesday, September 13, 1949
10: 30 A.M. to 12 :00 P.M.
OPENING ADDRESSES
Presiding
Howard H. Aiken
Director of the Computation Laboratory
EDWARD REYNOLDS
HARVARD UNIVERSITY
It is a great privilege and gives me great personal pleasure to have the honor of bringing
the greetings and warm welcome of the President and Fellows of Harvard College to this large
group of distinguished guests visiting the University on· the occasion of this symposium on
large-scale digital calculating machinery at the Harvard Computation Laboratory. We are
all sorry that President Conant could not personally welcome you here and present these
greetings, but he is on the ''''est Coast keeping engagements made more than a year ago.
I regret this necessity for his absence, but express his hope that you will find your visit her~
interesting and productive, and our hospitality cordial.
This is the third such ceremony in connection with Harvard's Computation Laboratory.
The first, about five years ago during wartime, dedicated Mark I, a highly significant development in this new field, which was then generously presented to Harvard by the International
Business Machines Corporation and temporarily located in our Cruft Laboratory, representing
the fruit of several years of collaboration by Professor Aiken of Harvard with the leading
research men in the IBM organization. Mark I, although in some respects overshadowed by
subsequent developments here and elsewhere, is still the reliable old workhorse of the Laboratory which has rendered extremely valuable service to the armed forces. From its initial
operation until the end of last year, it has had the generous support of the United States
Navy, which we gratefully acknowledge. More recently, the United States Air Force and the
Atomic Energy Commission have shared this support. All three of these agencies of the
Government have been most generous and understanding in helping us to broaden the scope
of the problems to which it has been applied and thus to broaden the field of interest" and
usefulness of this type of,machinery.
The second such ceremony, early in 1947, dedicated this new laboratory building and
,offered for inspection the Mark II, then be~ng completed and tested for the Navy. Shortly
thereafter, Mark II was delivered to the Navy and installed at Dahlgren Proving Ground
in Virginia.
Even though we understand that, with its significant advances in speed and capacity over
Mark I, Mark II has proved its usefulness, it has not satisfied our good friends in the Bureau
of Ordnance of the United States Navy, who have continued their generous support of the
research of this Laboratory and are now jointly with the Harvard Computation Laboratory
sponsoring this third symposium at which we have the pleasure of unveiling Mark III, also
destined for delivery in the near future to the Dahlgren Proving Grounds.
We feel that we may properly take some pride in the quality of research being carried
on in this Laboratory. The distinguished character of the talent attending this symposium
3
OPENING ADDRESSES
supports us in this feeling. We feel that rather too much time has been devoted to the development of actual machinery growing out of this research, and have some hope that one of the
by-products of this and other meetings may be to stimulate the interest of others in this phase
of the application of our research and thus to eliminate the need for this activity on the part
of our staff. Certainly the manufacture of parts and even the assembly thereof are not activities
for which we are fitted or which we wish to pursue, except possibly in the production of a
machine for our own use within the University.
As a layman participating actively in the administrative problems which arise out of the
organized group research that has become such an active part of resear~h at universities in
recent years, I am tremendously pleased at the evidences of awakening interest in the usefulness
of these newest appliances in widely scattered fields of research. The inescapable application
of mathematics in practically every field of human endeavor makes it se~m important to us
that the understanding of the availability and usefulness of the developments being made
here and in other mathematical research laboratories, as aids in all other fields of research,
be spread as widely as possible; and we are therefore particularly pleased that the latter
part of the program for this symposium is devoted to discussions of the relation of the work of
this Laboratory to research in a broad range of subjects.
Increased awareness of the usefulness of these new tools in the field to activities in other
branches of science inevitably increases the demand for the already inadequate number of
men and women who are educated not only in the theories of design and operation of such
machinery. but in the understanding of their applicability. This emphasizes the. other great
responsibility of the staff of this Laboratory-meeting t~eir obligations as teachers' to provide
the instruction required for the training and development of personnel interested in these
lines. Here again we gratefully acknowledge the understanding support of our friends in
the Bureau of Ordnance and in the Air Force. While we are endeavoring to develop support
for this program from other sources and to obtain permanent endowment for the Laboratory,
the understanding contractual support from these Government sources has been and continues
to be invaluable.
4
REAR ADMIRAL F. 1. ENTWISTLE, USN
NAVY DEPARTMENT, BUREAU OF ORDNANCE
It is with distinct pleasure that the Bureau of Ordnance joins hands with Harvard University in sponsoring this Symposium on Large-Scale Digital Calculating Machinery. On behalf
of the Bureau of Ordnance I take great pleasure in welcoming you to these meetings. Your
presence-the presence of so many distinguished scientists-assures us that much value will
be gained by us all from the deliberations and discussions which will take place during these
next few days.
During the past few years the relation between the Bureau of Ordnance and Harvard
University has been unusually close and cordial. This happy relation has given the Navy
the benefit of Harvard's talents and facilities and has led to the development of greatly
improved computing machinery and methods. Harvard University is most fortunate in
having on its staff Professor Howard Aiken, under whose unusual leadership, energy, and
ability the Computation Laboratory has grown.
As we in the Bureau of Ordnance look back on the computing problems with which we·
were faced prior to the First World War, we find that large-scale computations arose chiefly
in connection with the problems of ballistics-problems in which we were principally con. cerned with construction of range tables for seagoing gun systems with limited angles of
elevation. In those days, one computer (and by computer I mean a man with a slide rule,
log book, and a set of Engel's Ballistic Tables) handled all such computations. It probably
took the impetus, the acceleration, and the foreboding of World War II to permit conception
of the machine that was originated here and is kno~n as the Mark 1.
In the olden days when we were youngsters, and possibly a bit more impatient, that one
man (that computer) with his slide rule and his tables gave us a series of curves or figures
in a book, and we were to go out to put them to use. Frequently we discovered that we did
not know how to do this or we found that the tables were incorrect.
The availability of accurate tables in time of war is very important indeed. When the
recent war came along with its bombings, rocket firing, and use of heavier guns for antiaircraft
and bombardment, we found our range tables insufficient. In fact, we were about 500 range
tables behind. In the course of some years, that figure was decreased to 350; but still it was
a problem of one man, one slide rule,. and tables. Naturally, 500 tables would equal 500 men
or 500 years; even with 500 men we would still be at least a year behind.
\-\Torld War II indicated by great numbers-in tonnages, people, dollars-the magnitude
of effort required to fight a modern war. I believe it showed us that we can no longer afford
to fight wars of that magnitude. Many of us in the armed services have come to realize that
our job is not to fight wars but to prevent them. If we had realizea. this in the period from
5
OPENING ADDRESSES
1925 to 1930, we might have dissuaded the Japanese in 1939 from exerting the effort that was
subsequently shown. By keeping us prepared to carry through a war, these machines may
help us to prevent wars.
This science of computation, which we have come here to discuss, has grown up from the
association of people requiring such machines and their results with other people willing to
incorporate themselves into the effort to design and to build such machines. The fact that
we can collaborate and coordinate our efforts with a university such as Harvard, and can
arrange for the services of the laboratory here for our mutual benefit, should assure us of the
continuation of our so-called democracy.· In the installation of computers, the time factor
and accuracy of the machine are certainly important. But more important to my mind is
the coordination of the university and the military. This is, in itself, a step forward in the
university's aim of first teaching the individual and then going further to educate the country.
May I again express the continuing deep interest of the Bureau of Ordnance in this important subject of large-scale calculating machinery, and its applications, which may better
equip the Bureau of Ordnance of the Navy Department to carryon its work in national
defense. Both for myself and for the Chi~f of the Bureau of Ordnance, I wish to express our
appreciation of the close and wholehearted cooperation on the part of Harvard University
and the Computation Laboratory, and to acknowledge great and significant contributions
in the development of computi~g machines and methods that they have made through their
skills, their talents, and their facilities. May I add further that the Chief of the Bureau of
Ordnance and I will carryon all we can undertake ,and accomplish to continue this type of
collaboration resulting in this broader effort to prevent wars.
6
HOWARD H. AIKEN
HARVARD UNIVERSITY
As Admiral Entwistle has remarked, for five years the staff of the Comp~tation Laboratory
has been engaged in the construction of automatic computing machinery for the Bureau of
Ordnance. Without the Bureau's constant support, our share of this research could never
have been undertaken. We look upon the completion of Mark III as representing the end
of a phase in the development of this subject.
I have often remarked that if all the computing machines under construction were to
be completed, there would not be staff enough to operate them. Instruction in computing
machinery represents one of the more aggravated aspects of a generally recognized problem
in technical education. We feel that the further development of mathematical methods and
the extended use of computing machines in the various fields represented by speakers here
are those points at which levers should be placed to make the greatest possible advance in
computer research. Only by completing computing machines and then operating them can
the operating experience and experimental results be obtained that are so essential as a point
of departure in passing from one design to another. Therefore, at our laboratory we have
decided not to undertake the construction of any more large-scale computing machines with
the exception of one, which we hope to build for our own use and keep at Harvard.
There is an ever-increasing number of industries interested in constructing computing
machines outside the universities. In applying computing machinery to new and different
fields, many proposals have been made, ranging all the way from devices for an automatic
continuous audit, an automatic continuous inventory, down through an automatically operated
insurance office, public-utility billing department, department-store accounting system, to
more.specific and less general accounting-machine components. Other proposals have included
airline ticket-inventory systems, similar devices for railroad reservations, and automatic
railroad ticket-vending machines. On the technical side, machines have been proposed
involving automatic computers in connection with air-traffic control, airport control and
almost every other manufacturing operation up to and including the automatic factory. But
until our universities are able to offer well-rounded programs in numerical methods and the
application of computing machinery to prepare men to 9perate these machines, the success
of many of the proposed industrial programs will not be realized.
I should like to take this opportunity to express the appreciation of our staff to the Bureau
of Ordnance for its support throughout these years and, more than that, for the privilege of
pleasant associations which we have had with the representatives of that Bureau. It has been a
great pleasure to work with them throughout the construction of both Mark II and Mark III Calculators, and we have built up an association which I have every reason to believe will continue.
7
SECOND SESSION
Tuesday, September 13, 1949
2:00 P.M. to 5:00 P.M.
RECENT DEVELOPMENTS IN COMPUTING MACHINERY
Presiding
Mina Rees
Office of Naval Research
THE MARK
III
CALCULATOR
BENJAMIN L. MOORE
Harvard University
The large-scale digital computing machine known as Mark III has been built by the
Computation Laboratory of Harvard for the Bureau of Ordnance of the Navy Department.
It is to be installed at the Naval Proving Ground, Dahlgren, Virginia. The construction
work has been completed and the machine is now under test.
The decimal number system is used throughout the entire machine. Normal operations
are carried out with 16 digits. Provision is made, however, to use 32, 48, or more digits whe.n
needed. The operating decimal point is manually set by
the operator in one of six positions. In addition, under
control of the sequencing unit, the operator may choose
Coded Decimal
Decimal
Notation
at any time one of three locations of the decimal point.
Digit
2* 4 2 1
In order. to reduce the size of the memory, digits are
'0
0 0 0 0
stored in the coded form where four binary digits are used
0 0 0 1
1
to represent a decimal digit. Figure 1 shows the system
2
0 0 1 0
that has been adopted, where the weights of the four
3
0 0 1 1
4
0 1 0 0
binary digits are 1, 2, 4, and 2, respectively. It should be
5
1 0 1 1
noted that the sum of these weights is 9 and therefore the
6
1 1 0 0
nine's complement of any digit may be obtained by chang7
1 1 0 1
ing zeros to ones and vice versa. This is quite a con8
1 1 1 0
venience electronically. where a positive voltage may be
9
1 1 1 1
used for a one and a negative voltage' for a zero. The
nine's complement can then be obtained merely by in- FIG. 1. Coded decimal number
system used in Mark III.
verting the signal. These complements are used for
subtraction in this machine.
The number-storage system consists of eight rotating drums whose surfaces are' coated
with a thin layer of magnetic material. Information recorded in the form of a small magnetic
dipole during one' revolution may be played back on any succeeding revolution. A zero is
represented by a magnetic dipole oriented in one direction and a one by the opposite orientation. New information is recorded directly over the old making erasure of the surface unnecessary. The played-back voltage signal is double ended, with the positive voltage first
for one orientation and the negative first for the opposite orientation of the dipole. Figure 2
is a photograph of a typical cathode-ray oscillograph pattern having two negative-first pulses
in a group of positive-first pulses. Figure 3 is similar except that the pulses have been reversed
4 X 10 8 times. ~his photograph was taken upon completion of a test to determine whether
I I
BENJAMIN L. MOORE
the noise background increased or the pulses changed shape after many reversals. It is easy
to \see that the two pictures are almost identical.
Using a single magnetic head on a channel or track, the access time to anyone number is
the time for one revolution of the drum. At the expense of more heads the access time can
FIG. 3. Oscillograph trace of a typical
playback-pulse pattern after 4 X 108
reversals.
FIG. 2. Oscillograph trace of a typical
playback-pulse pattern.
be decreased. This machine uses two playback heads per track, so that the access time is
reduced to the time of one half revolution. Since it is electrically more convenient, separate
heads are used for recording and playback. Thus each binary track contains two record
A
A-A
A
fIG. 4. Cross section of a binary magnetic storage channel.
heads as well as two playback heads. Figure 4 shows a typical cross section of a binary channel.
Figure 5 shows the drum storage unit.
As it is convenient to have all components of a decimal digit available simultaneously,
four parallel binary channels are used to represent one decimal channel where the binary
12
FIG. 5. View of the assembled magnetic-drum storage unit.
BENJAMIN L. MOORE
channels will have the weights 1, 2, 4, 2, respectively. The digits of a given number and the
numbers themselves are stored serially around the periphery of the drum. The main. storage
system uses a pulse density of ten pulses per inch and stores ten 16-digit numbers in a decimal
channel. To extend the storage":system capacity beyond ten numbers, parallel channels are
used so that the selection of a given number involves the selection of a channel as well as a
time selection, as the drum moves past the playback head.
The arithmetic unit of this machine is electronic and contains an adder, a multiplier, and
certain sensing units. It contains no divider, since this operation is aCGomplished by an
iteration process. The unit is serial in operation so that it is necessary to have the corresponding
t------....--~
TO
ARITHMETIC
UNIT
SELECTION
GATE
STORA~E
REGENERATION
CHANNEL
CHANNEL
FIG. 6. Block diagram of a regeneration channel.
digits of two numbers to be added available simultaneously. Since, in general, two numbers
will not necessarily be played back from the drum in the same time phase, some provision
must be made to put them in phase. Figure 6 illustrates how this is done. On the left is
the storage channel where a selection is made of the head to be used, depending on where the
number is located and the phase of the drum. At the appropriate number time the gate
opens and records the desired number on the regeneration channel. By the time the last
digit is recorded the first digit is being played back by the regeneration playback so that the
gate lets this number through to be recorded again. Thus the number is recorded 12 times
around the channel and is available at any time thereafter. The two blank spaces on the
storage channel provide time for switching operations and no numbers are stored in this
interval.
MARK III CALCULATOR
With this brief picture of the magnetic storage system, let us examine the organization
of the whole machine, a block diagram of which is shown in Fig. 7.
The storage system is divided into two parts. One, called slow storage, has a capacity
of 4000 16-digit numbers and selection of channels is made by relays that are relatively slow.
However, provision is made for transferring 20 numbers at a time from slow storage to the
other section, which is called fast, where selection of numbers can be done at electronic
R
TAPE INPUT
TRANSFER
CHANNELS
SLOW STORAGE
4000 NUMBERS
8
ARITHMETIC
UNIT
FAST STORAG E
150 CONSTANTS
200 NUMBERS
TRANSFER
CHANNEL
~
TAPE
./TYPEWRITER/
READER
PRI NTER
FIG.
7. Block diagram showing organization of the calculator.
speeds. Ten numbers at a time are transferred from the fast storage to slow. The slow section
is mainly used for storage of functions.
The fast storage has a capacity of 200 numbers in addition to 150 permanent constants.
The constants are used for computing functions such as Ijx, IjVx, cos x, log x, antilog x,
tan- 1 x.
The basic cycle of the machine is the access time to the storage, namely, one half revolution
of the drum. Each cycle the machine delivers two numbers to the arithmetic unit over the
two parallel busses A and B and returns the previous result to the storage. One addition can
be performed each cycle, while multiplication requires 3 cycles. As the speed of the drums
15
FIG. 8. View of the Mark III Calculator.
MARK III CALCULATOR
is slightly less than 7200 rev/min the cycle is about 4.2 msec, that is, addition requires 4.2 msec
and multiplication 12.6 msec.
Numbers are fed into and recorded out of the machine by means of magnetic paper tape.
There are eight mechanisms, anyone or any combination of which can be set to read into
or out of the machine. Recorded tapes are run through a tape reader which in turn operates
o
0000
0000
0000
0000
'-
0000
0000
0000
0000
0000
0000
0000
0000
---
0
c +Rc-xcl+31)c= 31
FIG. 6. Checking identities.
multiplication or division; 2" a transfer operation; 3, a shifting or extraction process; 4, the
two substitution operations; 5, the branch' operations; 6, the floating-point processes; and
7, the codes that pertain to the external memory units.
Space does not permit a complete description of each operation; however, Fig. 8 shows
the manner in which each of six representative operations is programmed. In addition, the
address of the addend is placed in the Address 1 position, that of the augend in the Address 2
position; the addition code 01 is inserted in the Operation Code position. The address to
which the sum is to be transmitted is located at the Address 3 position. If the result is to be
'used immediately in the next operation, and if there is no need to transmit this result to the
memory, then the third address position may be left void, and the result maybe called forth
in the next operation through the use of a special address. However, whether the third address
is void or not, the result of the present operation remains available in a special one-word
register of the arithmetic unit, and is subject to call by employing the above-mentioned special
60
RAYTHEON ELECTRONIC COMPUTER
BINARY
OCTAL
ADDITION
000001
01
ADDITION (DOUBLE - PRECISION - PART I)
000010
ADDITION (DOUBLE - PRECISION - PART 2)
0000 I I
02
03
SUBTRACTION
o0
0·1 00
04
SUBTRACTION (DOUBLE-PRECISION - PARTI)
000101
SUBTRACTION (DOUBLE -PRECISION - PART2)
000110
05
06
MULTIPLICATION (TRANSMIT HIGH ORDER WITH ROUNDOFF)
MULTIPLICATION (TRANSMIT HIGH ORDER NO ROUNDOFF)
MULTIPLICATION (TRANSMIT LOW ORDER)
DIVISION (QUOTIENT NOT ROUNDED - OFF; REMAINDER AVAILABLE)
DIVISION (QUOTIENT ROUNDED - OFF)
001000
001001
0010 10
001 I 10
001 I I I
TRANSFER (NORMAL)
TRANSFER (POSITIVE ABSOLUTE VALUE)
TRANSFER (NEGATIVE ABSOLUTE VALUE)
TRANSFER (SELECTIVE)
I 0000
20
010001
01'0010
o I 001 I
21
22
SHIFT (CONTROLLED)
SHIFT FACTOR (NORMAL)
SHIFT FACTOR (SQUARE ROOT)
EXTRACTION
oI
oI
oI
oI
1000
10 0 I
10 10
I I 10
30
31
32
36
SUBSTITUTION (ADDITIVE)
SUBSTITUTION (SUBTRACTIVE)
100001
I 001 00
44
BRANCH (NORMAL)
BRANCH (EQUALITY SENSING)
I 0 I 000
I 0 I 001
ADDITION (FLOATING)
SUBTRACTION (FLOATING)
MULTIPLICATION (FLOATING)
I 100{)0
110001
61
110010
62
OPERATION
o
TAPE RECORD
TAPE READ
HUNT - PREPARE TO RECORD
HUNT - PREPARE TO READ
I I I 000
I I I 001
I I I 0 I 0
I I I 0 I I
FIG.
7. Operation codes.
61
10
I I
12
16
I 7
23
41
50
51
60
70
71
72
73
RICHARD M. BLOCH
address in the succeeding order. It is clear that by this means useless transmissions to or from
the memory are completely avoided. It should be 'understood, of course, -,that the device just
discuss~d may be applied to any desired operation and is not restricted to the addition process.
To continue, Address 4 will contain the" memory address from which the new order is to be
obtained-or, more exactly, from which the first half of the new order is to be selected. The
programming arrangements for subtraction, multiplication, and division, shown in Fig. 8,
I belieye, are self-explanatory in the light of the above discussion.
The selective transfer order directs the machine to multiply the number in memory position
OPERATION
ADDRESS
#,
ADDRESS OPERATION
#=2
CODE
ADDITION
ADDEND
AUGEND
01
SUM
NEXT
ORDER
SLJBTRACTION
MINUEND
SUBTRAHEND
04
DIFFER ENCE
NEXT
ORDER
MULTIPLICATION
MULTIPLI- MULTIPLiER
CAND
10
PRODUCT
NEXT
ORDER
DIVISION
DIVISOR
17
QUOTIENT
NEXT
ORDER
23
B
NEXT
ORDER
50
C
TRANSFER
(SELECTIVE)
A
BRANCH
(NORMAL)
A
DIVIDEND
C
B
ADDRESS
#3
ADDRESS
#4
D
FIG. 8. Construction of computer orders.
A by + 1 or - 1, according to whethe:f the number in memory position C "is positive or
negative; the result of this process is then to be transmitted to memory position B.
The Branch-Normal operation is somewhat unusual both in its effect on the subsequent
computation and in the treatment of the third address. If the number.in memory position
A is greater than or equal to the number located at memory position B, the machine is directed
to obtain its next order from storage position C as indicated by Address 3; otherwise, the
central control is to obtain the next order from memory position D as specified by the fourth
address.
.
The problem-preparation unit is a manually operated device, independent of the main
computer, that places the programming orders and numerical input information on the
magnetic tape in preparation for entry into the machine. A first Teletype keyboard unit 'is
used to prepare a standard five-hole Teletype paper tape. This tape is used in conjunction
62
RAYTHEON ELECTRONIC COMPUTER
with another keyboard unit to prepare a second paper tape. The operator of the second unit,
reading from the same manuscript that was used in the preparation of the first tape, causes
a second paper tape to be perforated. However, each key that is depressed must establish
an identity with the Teletype code appearing on the preliminary tape; otherwise, the keyboard
is automaticalIy locked and the intended perforation of an erroneous code on the second tape
is intercepted. A printer is also associated with the keyboard so that a printed copy of the
programming is available to the operator in the course of the tape-preparation process.
Programming orders are entered in the keyboard in octal notation, whereas numerical information will generally be entered in decimal notation; this conforms to the notation that is
prescribed for the original manuscript. However, where desired, numbers may also be entered
in the octal system if the information should happen to be available in this form; provision
is made on the keyboard to indicate the particular notation being used.
After the second paper tape is prepared, it is transferred to a magnetic· recording unit
where the Teletype codes ,appearing on the tape are converted to binary or binary-coded
decimal notation in accordance with a coded indication that accompanies each word on the
paper tape; a transfer weighted count of each number and half-order is also automatically
constructed. All of this information is then recorded on the magnetic tape in the required
word-and-block form previously described; this magnetic tape is now transferred to one of
the exterrial memory units from which point the information is automatically available to
the computer. Tliese conversion and weight-counting processes, as well as the magnetic
recording process itself, are all automatically checked.
The printing of the final results of a problem is performed by Teletype printers operating
independently of the main machine. In the course of a computation, numbers to be printed
are shuttled in binary-coded decimal form to one or more of the magnetic tapes associated
with the external memory units. At a later time, these reels are transferred manually to the
output printers where the numerical quantities are typed in final form. Directions to ·the
printer involving considerations such as page format, location of the decimal point, etc., are
supplied by auxiliary control devices.
Certain external memory tapes contain words in binary notation only, these quantities
being intermediate values obtained in the computation and intended for direct feedback to
the machine at a subsequent point in the solution of a problem; therefore, the computer
obviously will not have been instructed to convert these numbers into binary-coded decimal
notation. However, it may be desired on certain occasions to print the binary quantities contained in these intermediate tapes; for this reason, provisions have been made for the output
printers to type numerical quantities in octal notation as well as decimal. Transfer weightedcount checks based upon the actuation of the printer code bars are employed to intercept
printing errors;
A printer directly connected with the computer is provided so that the operator may
monitor intermediate results of the computation while the machine is in operation. The transmittal of information to this printer is prearranged in the original programming of the problem.
63
RICHARD M. BLOCH
Provisions for reading from, as well as into, the various high-speed memory positions of
'the machine under manual control have been made. Furthermore,. the computer is designed
in such a way that when an error occurs, causing the machine to stop, all numerical and
control quantities involved in the execution of the order then governing the machine are
available for immediate read-out by the operator; this feature should be an invaluable aid
in the diagnosis of failures.
The Raytheon computer operates at a mean speed of 1600 complete operations per second.
The machine has a complement of roughly 3500 vacuum tubes and 6500 crystal diodes. It
is expected that construction of the computer will be completed at the end of September 1950
A
GENERAL ELECTRIC ENGINEERING DIGITAL COMPUTER
BURTON R. LESTER
General Electric Company
The General Electric Company feels very grateful for the opportunity to discuss our computer at this symposium. Our present effort is the design and construction of a computer
suitable for the engineering problems that arise within the General Electtic Company-a
computer simple in design, accurate and reliable, easy to. operate, and economical to
maintain.
The General Electric Company has been interested in the computer field for many years.
Our first computers were network analyzers. These were soon followed by the more complex
differential analyzers. ''''e built up small computation groups in our engineering divisions.
In some instances, as the requirements for speed and accuracy increased, we rented IBM
machinery.
Approximately six years ago, the first investigation of the possibility of constructing an
automatic digital computer was made. In 1946 work was started on a small binary machine
for control problems. Our Engineering Council also directed that we investigate the possibility
of constructing another machine for internal use. A careful revie"Y was made of the various
computer projects and a small group of engineers visited these projects to weigh their progress.
These efforts culminated in the decision to construct a computer. Our design and constructiori were based on the experience and ability of our Research Laboratory and our
electronic accomplishments during the past war.
Our purpose in constructing this computer was threefold. The computer would enable
more accurate and rapid computation of our engineering designs. It would provide our
Research Laboratory with a long-needed facility. Last, we would gain extensive knowledge
, and position in this field by constructing and operating this machine.
The purpose of our computer set the major design considerations. Accurate and reliable
operation was foremost. Consequently, only proven principles were utilized. A reasonable
operating speed was set with the idea of increasing it gradually as we become more familiar
with the capabilities of the computer. Operation and maintenance procedures were simplified.
Unitized construction was 'employed to aid design, speed maintenance, and provide means
for adding future improvements.
In discussing the features of any computer, it is well· to break the design down into the
following items for easy assimilation: number base, mode of operation, memory, arithmetic
unit, control unit, input-Olitput mechanism, tape-preparation unit, and printer.
Our .computer operates in the decimal system. All numbers and instructions are expressed
in decimal digits. The 2* coded decimal system is used within the computer. The basic
65
BURTON R. LESTER
length of a number is 8 decimal digits using a fixed decimal point and 6 decimal digits with
a floating point. The range of the latter system is from 10- 9 to 10 9 • The simplicity of the
decimal operation outweighed the loss of capacity as compared with the equivalent binary
machine. Actually, the construction of the machine is binary with the exception of three
vacuum tubes per decimal digit in the accumulator. These tubes and associated circuits sense
and correct forbidden combinations.
Serial and parallel operation are used to advantage." Serial operation occurs between
all major units. Parallel operation is used within the arithmetic and control units. Numbers
and instructions are stored serially in the memory in single binary channels. Serial read-in
and read-out of the memory occurs at a 48-kc/sec repetition rate. Numbers and instructions
are transferred between the arithmetic and control registers at a 200-kc/sec rate. Parallel
operation is utilized for basic operations within the arithmetic and control registers at a
200-kc/sec rate.
Operation of the computer is as follows. An order stored in the memory is read serially
to the control unit. This unit reads each part of the order in parallel by means of sensing
circuits. First it directs the lnemory to transfer the two operands serially to the arithmetic
unit. Then the control unit senses the order to determ~ne the operation code and directs
the arithmetic unit to perfornl this operation in parallel. The answer is transferred serially
to the memory again under the direction of the control unit and to the address specified in
the order. Finally, the control unit senses the present order to determine the address of the
next order and transfers it to the control unit. Thus the basic cycle is repeated until the
"problem is solved.
The computer has a magnetic-drum memory that stores 4000 numbers and instructions
in 100 tracks, 40 numbers per track. Pulses are spaced 20 to the inch in the tracks and the
tracks are spaced 8 to the inch. Forty pulse spaces are required for each number-36 for
the number and four guard spaces. The drum rotates at 1800 rev/min; consequently, the
pulse repetition rate is approximately 48 kc/sec. The drum is constructed" of aluminium with
a magnetic coating and is 24 in. in diameter and 30 in. long.
Separate magnetic heads are provided for playback and recording. They are spaced 3 mils
from the drum surface. Two playback amplifiers are used, one for information and one for
clock pulses. A low-level crystal gating system connects the proper head to the information
playback amplifier. The amplifiers have an automatic gain control to eliminate signal variation
caused by eccentricity of the drum. A high-level gating system delivers recording pulses to
the proper head. Recording and playback heads are spaced 180°. Consequently, a number
recorded may be read back one half revolution later to verify memory operations. This check
is performed after each recording.
Initially the design called for a serial arithmetic unit. However, it soon became apparent
that the shifting registers, utilized to synchronize the numbers received from the memory,
could easily be modified for parallel operation with a small increase in equipment. The
arithmetic unit contains three basic registers: A, B, and C. Register A is an accumulator
66
GENERAL ELECTRIC COMPUTER
and contains additional equipment to sense and correct forbidden binary numbers occurring
as a result of addition. Register B is a shifting register. Register C, in addition to being a
shifting register, can also be connected as a counter for use in multiplication and division.
The arithmetic unit performs the following basic operations: addition, subtraction,
multiplication, division, and choice. Actually, the basic bperation is the addition obtained
in· the accumulator. In subtraction, the nine's complement is used with end-around carry.
Multiplication is performed as a series of additions.
Division utilizes the oscillating overdraft method. Both the multiplication and division
processes ·have been simplified to the extent that they are nominally equivalent to their binary
counterparts.
Table 1. Operation times (ftsec) for basic operations.
Operation
Fixed-Point Operation
Floating-Point Operation
Addition
15
350
Subtraction
15
350
450
450
490
Multiplication
Division
530
The times required for basic operations are listed in Table 1. These times do not include
memory-access time. These times represent a small fraction of the time of one revolution of
the drum. Under average conditions the time for completion of one operation including
access time is equal to the time of one revolution of the memory.
The control unit is the telephone central of the computer. It controls the operations of
the various units of the computer. The control signals are either a 40-v positive 2.5-,usec pulse
or a d-c switching voltage of zero or + 40 v. The cyclic control rate is determined by the
unit under control. Control signals to the memory are based oOn the 48-kcjsec clock pulse
generated on the drum. Control signals to the tape input-output unit are based on synchronizing pulses on the tape. Control signals to the arithmetic unit are based on a 200-kc/sec
. oscillator in the control unit.
This type of control permits a great deal offreedom in introducing modification. Increasing
memory operation only requires that the memory supply the correct clock pulse frequency
to the control unit. No other modifications are needed. This flexibility is highly desirable
to permit future improvements.
The four-address system is used in this computer. The first two addresses of the order
are the locations of the operands. The third address is the location of where the answer is
to be stored and the fourth address is that of the next order. A number consists of nine decimal
digits, numbered from 0 to 8. The zero digit is the sign digit and the remaining digits comprise
the actual number. Since there are 4000 memory positions, four decimal digits are required
67
BURTON R. LESTER
to describe an address and sixteen decimal digits are required to describe four addresses.
Consequently, each order consists of two numbers and is stored in two consecutive addresses
in the memory. The remaining two digits of the order are used to denote the type of operation.
The zero digits of the two numbers comprising an order contain this operation code. These
digits are also used as the means of distinguishing between orders and numbers. Zero and
nine are reserved for positive and negative number sign indication. The remaining values
involving 64 possible combinations are operation codes. Thus a number must have a zero
or a nine in the sign digit and an order may have any value but zero or nine in the sign digit.
Table 2. Operations and corresponding codes.
Operation Code
33
Operation
'Read in
34
Read out
35
Transfer
36
Add using fixed decimal points
43
Add
44
Subtract
45
Multiply
46
Divide
53
Choice
54
Choice zero or not zero
63
Move exponent to No.2 address
65
Change address A
66
Change address B
+ or-
, In Table 2 are listed some of the more important operations and their corresponding codes.
It will be noted that provision has been made to enable the machine to change its orders.
In addition, it is possible to set up a problem in such a way that a single order can modify all
the orders in the problem so as to call in new values of the variables.
Additional operations, such as raising to a power, extracting the root, finding the logarithm,
etc., can be prepared as sub-sequences and stored in the: memory for future use. With the'
addition of extra shifting registers, these operations may be performed entirely within the
arithmetic unit.
The memory storage may be described as a huge matrix, the address being the key to
68
GENERAL ELECTRIC COMPUTER
any elemcnt. The first two digits of the address control the row selcction or the equivalent
number time on the drum. The last two digits control column selection or the track
designation.
The tape input-output component takes the tapes prepared by the tape-preparation unit
and under direction of the control unit reads information into the computer memory. It
also prepares magnetic tapes containing the answers to the problem (the output from the
computer). Input and output tape data are the same; that is, data read out of the 'machine
may be run back into the machine. A unit piece of information on the tape consists of a
number plus its memory location. The tapes are operated at a speed such that approximately
25 pieces of information per second are read into or out of the machine. These tapes are
utilized by the printer to type' out the information. The reels of magnetic tape belonging to
this unit are located in the center of the main console.
Tape preparation and printing are performed by units similar to those of the Harvard
Mark III computer. The tape-preparation unit has a standard keyboard and can be used
to prepare both number and sequence-control tapes.
,
The printer uses an electric typewriter, operating at 10 strokes per second. Provision has
been ~ade to vary the typography of the printed page at the discretion of the operator.
The main components of the computer are housed in five racks 24 in. wide and 8 ft tall.
The Tape-Preparation Unit, Printer, Memory, and Power Supply are housed in separate
units. The Magnetic Memory is contained in a cabinet 3 ft wide, 5 ft long and 6 ft high. It
appears that the power supply will be' contained in a similar cabinet. The Tape-Preparation
and Printer Units will be housed in racks 24 in. wide and 6 ft tall.
Unitized construction is used throughout the computer. At present, there are 15 basic
circuits. These circuits are constructed as plug-in assemblies such that the components are
mounted on turrets between the tube socket and plug. These assemblies ~re approximately
1.5 in. in diameter and 2.5 in. long. The circuits plug into standard panels which contain
the signal and power wiring. The panels are mounted vertically in the cabinets. A vertical
sheet of Plexiglas mounts 2.5 in. in front of these panels. The circuits are plugged into the
panels through the Plexiglas. Air, circulated in the channel formed by these two panels, is
used to cool the circuits. The vacuum tubes mount directly in the sockets and project out
of the Plexiglas panel.
,
One' of the major achievements in the design of this computer has been the reduction of
the tube complement to less than 1000 vacuum tubes. This reduction has been made possible
through the use of germanium Qiodes and careful circuit design. Approximately 4000 diodes
are used in the computer.
In summarizing, I would like to highlight the following points.
First: This computer has been designed to solve problems requiring engineering accuracy.
For problems of this type, it must be reliable and accurate first; speed comes next.
Second: Design, operation, and maintenance have been simplified by the reduction of
tube complement and unitized construction.
6g
BURTQN R. LESTER
Third: The computer design is flexible; that is, individual units such as the Arithmetic Unit, Control Unit, Memory Unit, etc., standby themselves. They can be readily
modified with minor effect on the rest of the ·computer. The resulting building biocks
which comprise this computer can be used to construct a computing machine for almost
any purpose.
BANQUET
Tuesday, September 13, 1949
Toastmaster
Edward A. Weeks,jr.
Editor of The Atlantic Monthly
TOAST BY D. H. LEHMER
There is a man among us here tonight who deserves our special vote of thanks and appreciation. He recognized the necessity for a medium of communication-"a 'standard source
to which one might naturally turn for guidance in connection with all mathematical tables
of importance in contemporary research." Through the National Research Council in 1943
he established the quarterly journal Mathematical Tables and Other Aids to Computation. Now
after seven years of unflagging effort, Raymond C. Archibald is retiring as Editor of MTAC.
I propose that we show him our appreciation for his excellent work.
TOAST BY SAMUEL H. CALDWELL
It is with deep regret that we note the absence from this banquet and from the Sy~posium
of one of the world's great figures in the field of scientific computation. Some of you have
known him as a teacher. Many of you met him and heard him at the Machine Computation
Conference held at the Massachusetts Institute of Technology four years ago. All of us who
have known Doctor L.J. Comrie have been stimulated by his appreciative response, impressed
by his intellectual grasp, and conquered by his wit and charm.
As the founder of the Scientific Computing Service of London, and in his former connection
with His Majesty's Naval Almanac Office, ·Doctor Comrie has been a prolific contributor
to the literature of scientific computation. But history will name him also as one of the pioneers
in the development and application of machine methods to computation problems.
Doctor Comrie is unable to be with us because of serious illness, and this I know is a matter
of profound disappointment to him. It is proposed that we members of this Symposium stand
at the side of Doctor Comrie in his fight for heaith and that we let him kno~ it. I therefore
ask that we request our Toastmaster to send to Doctor Comrie our prayers for quick and full
recovery of his health and vigor, and our earnest hope that he can be with us at our next
Symposium.
73
THE PRESENT POSITION OF AUTOMATIC COMPUTING MACHINE
DEVELOPMENT IN ENGLAND
w. s.
ELLIOTT
Research Laboratories of Elliott Broihers (London) Limited
I have come from a place in England named Borehamwood. Borehamwoodcontributes
both to the arts and to the sciences. A small part of its contribution to the sciences is work
on what our popular press, unfortunately, in my view, calls "Electronic Brains." On the
side of the arts a large part of the British Motion Picture Industry is at Borehamwood. Some
of you may have heard that a certain William Shakespeare has been trying to earn dollars
for his country by writing the screen plays "Henry V" and "Hamlet."
On my desk at Borehamwood, I have a volume which I prize very highly. It is a report
of the proceedings of the first Symposium on large-scale digital calculating machines held
here at Harvard in 1947. This is a book which, I think, contains much weighty and interesting
material-material made no less significant by the advances of the last two and a half years.
To me not the least interesting paper in this volume is that by Richard Babbage dealing with
the work of his English grandfather, Charles Babbage, that first designer of computing machines.
And when I read this paper, my attention focuses on one passage.
"Propose to any Englishman any principle or any instrument, however admirable, and
you will observe that the whole effort of the English mind is directed to find a difficulty, a
defect, or an impossibility in it. If you speak to him of a machine for peeling a potato, he will
pronounce it impossible; if you peel a potato with it before his eyes, he will declare it useless
because it will not slice a pineapple. Impart the same principle or show the same machine
to an American . . . and you will observe that the whole effort of his mind is to find some
new application of the principle, some new use for the instrument."
When Professor Aiken, just ten days ago, asked me to speak at this Symposium, my first
thought was that I might take as a text the differences between English and American COlnputing machines in the light of that passage. But when 1. came to think about it, I decided
I could find no significant difference except perhaps that the groups developing our machines
are a little smaller. Certainly the projects that we have are as diverse as those in this country,
and the ways that the different groups go to work are similarly varied. For instance, the logical
design of one machine was compl~ted well before the team was set up to build it. Another
machine grew as the ideas came. The first machine is more engineered, and the second machine
is breadboard.
I shall mention seven groups in England working on computing-machine projects: three
at Universities-the Universities of Cambridge, Manchester and London; three at Government establishments---the National Physical Laboratory (NPL), which I think corresponds
74
COMPUTING MACHINES IN ENGLAND
to your National Bureau of Standards, the Telecommunications Research Establishment
(TRE) of the Ministry of Supply, and the Royal Aircraft Establishment; and I shall mention
one industrial firm, Elliott Brothers (London) Limited, the Research Laboratories of which
I represent. Of these groups, that at Cambridge has one machine fully operating. . The
Manchester group has a machine fully operating though with restricted input and outpu t
units. Other machines are in various stages of development. I shall describe the Cambridge
FIG. 1. The Electronic Storage Delay Automatic Calculator (EDSAC).
machine more fully and I shall compare other machines with it. I shall not give here precise
figures for the memory capacity, speed, and so on.
The first electronic computer to run in England, the Electronic Delay Storage Automatic
Calculator, or EDSAQ, was designed and built by Mr. M. V. Wilkes, the Director of the
Cambridge University Mathematical Laboratory, assisted by Mr. Renwick. Besides being
a theoretical physicist Mr. Wilkes is a practical electronic engineer.
EDSAC (Fig. I) was projected by Mr. Wilkes during a visit he made to the United States
in 1946 when he attended part of a course on computing machines at the Moore School.
The logical design of EDSAC was influenced by the ideas of Mauchly, Eckert, Goldstine,
and Sharpless of the Moore School. At the outset 'Vilkes stated that he was not interested
in building the best possible machine. He wanted to make a reliable machine and to make it
quickly. He chose mercury delay-line storage as being the only principle which at that time
75
w.
S. ELLIOTT
promised reliable storage. He chose a 500-kc/sec digit rate as being the fastest that, with the
techniques then known, would give a reliable computer. The store capacity is 512 words
(numbers or orders) of 32 binary digits. Input to the machine is by punched paper tapes
prepared on a teleprinter keyboard, and output is directly printed on a typewriter. EDSAC
uses a one-address code for instructions. The storage, control, and arithmetic units were
designed in 1947, and they and the input and output units were built in 1948. Toward the
and of that year parts of the machine were being tested, and the machine was fully operating
and was demonstra ted a t a conference on computing machines held at Cambridge in J une
1949. Today the team there is gaining experience in running problems on the machine, and
I have with me two samples of the work of the machine.
FIG. 2. The M a nchester machine. Control and input circuits are at the left,
the memory in the right center, a nd the arithmetic units at the right.
The first sa mple is a list of the prime numbers up to 1021. The list starts with the number
5- Mr. Wilkes assured me that they know the prime numbers below tha t. The second sample
is a tabulated solution of a second-order differential equa tion.
Before I leave the Ca mbridge group I should like to say tha t Wilkes is very active in holding
fortnightly colloquia during University term time, a nd the different teams in England attend
these colloquia .very well a nd keep closely in touch.
In June of this year the colloquia culmina ted in a four-day conference. Descriptions were
given a t the conference of the various computing-machine projects, not only in Engla nd, but
also in Fra nce, H olland a nd Sweden. A contribution from Doctor Huskey was read, giving
a n account of the p resent state in America. Discussion subj ects included CRT storage,
programming and coding, checking facilities, a nd perma nent a nd semiperma nent storage.
The second University group is a t M a nchester. The machine (Fig. 2) is being built by
the Electrical Engineering Depa rtment under Professor F. C. Willia ms for the use of Doctor
Newman a nd Doc tor Turing of the M a thema tical D epartment. There is close contact between
the engineers and the mathema ticians, but the machine is definitel y being designed by the
engineers. The machine uses the well-known CRT store of F. C. Willia ms and T. Kilburn.
76
COMPUTING MACHINES IN ENGLAND
This store features a standard cathode-ray tube and a physically simple mechanism. Experimental work on the store was completed about March 1948. Having built the store, Williams
and Kilburn wanted to test'it, so they added a second storage tube as an accumulator, and
a third tube as control. They thus had ababy computing machine. The baby machine was
of breadboard construction and today the machine at Manchester consists of these same breadboards and others that have been added. In fact; the machine has grown gradually as ideas
came-unlike some machines, which have been built according to a master plan conceived
at the outset. For this reason any description of the machine is liable to be outdated very
quickly. At the time of the Cambridge conference, there were a fast multiplier CRT and a
special tube for modifying instructions. The machine uses magnetic-drum auxiliary storage
running at the rate of the working store. A feature of this is that the drum is synchronized
from the machine's clock-the drum does not generate the clock. The drum stores true binary
numbers and has access only to and from the main store.
Input to the Manchester machine is on a binary button board, and the output is a CRT
display of the binary content of one of the CRT stores. The digit rate is limited by the CRT
store to a quarte~ of that in EDSAC, but si?ce the store is noncyclic the average access time
in the two machines is similar. One CRT store has the capacity offour long tanks in EDSAC,
that is to say, 32 words.
Because of the restricted input and output units, on the Manchester machine the type of
problem that can be run on it is rather limited but some interesting work has been done on
the' Mersenne numbers.
A second machine for the Manchester group is being built by Ferranti Limited. This is
to be a more engineered version, and it will have 16 main CRT ~tores. The engineers consider
four to eight to be the optimum number of main storage tubes in a computer of this type,
having regard to transfer time from the auxiliary store to the main stores. The mat,hematicians
would be content with eight tubes but in some cases would like 16, so to be on the safe side
16 main storage tubes have been decided on, in addition to an accumulating store, a store for
control numbers and a "B" tube where instructions are modified. There is no proper name
for the Manchester machine, though I understand from Professor Williams that it has a variety
of improper names. The Manchester machine recently gave rise to some correspondence in
the London Times on whether a machine could rival the brain of man. In an interview with
the paper Doctor Turing said he did not exclude the possibility that the machine could write
a sonnet. He added, however, that only another machine could appreciate the sonnet fully.
The third university group, at London, directed by pro A. D. Booth, is working on three
machines. The first is called "Automatic Relay Calculator" or ARC. This is a binary relay
machine which, in logical design, is somewhat similar to EDSAC and follows some of the
ideas of the Moore School, in that, for example, numbers and orders are lumped together in
the store, and orders can be modified. It is a parallel computer with a small magnetic-drum
store. Input and output and semipermanent storage are all on puriched paper tape. The
machine was made by Doctor Booth for experiments in logical design. It has 800 relays and
77
w.
S. ELLIOTT
cost about £2,500. The machine was being tested in June 1949~ The magnetic drum is now
being changed to another storage system, an electromechanical store, which Doctor Booth
is developing and which is of some interest. In essence this is a very concentrated collection
of small relays. It packs 256 numbers, each of 21 binary digits, into about 12 by 8 by 16 in.
Doctor Booth is also designing an electronic machine and is to make two models of it in
parallel, for different uses. It is to have magnetic-drum storage, magnetic-tape input and
output, and magnetic-tape auxiliary storage. It is to have multiplier and divider units, and
Doctor Booth thinks it will have fewer than 1000 tubes. He gives no date for its completion.
I t may be one or two years.
Doctor Booth's third machine is a "Simple Electronic Computer" or SEC. This he proposes as the smallest electronic computer that will have all· the main facilities of a generalpurpose machine but will be as small-as 181 tubes, and he hopes that University departments
will be able to build it for themselves.
Turning now to the Government establishments, a considerable amount of work was done
at the National Physical Laboratory in 1946 on the Automatic Computing Engine or ACE.
This work was done under Doctor Turing and by the end of that year the quite complicated
and sophisticated logical design was completed and several problems had been coded. In
September 1947 an Electronics section was set up at the National Physical Laboratory to
work on electronic computing machines, ~ut before this team had started on the actual construction of the ACE Doctor Turing left. About the middle of 1948, it was decided that the
theoretical team of the Mathematics Division, which was now under Mr. J. H. Wilkinson,
should join the electronics group under Mr. Colebrook, and the two teams should work
togethe~ on the construction, not of the full-scale machine, but of a Test Assembly. This Test
Assembly represents an attempt to construct the smallest machine that will serve as an adequate
testing ground for the concepts involved in the full-scale machine, but that will nevertheless
be large enough to be a useful computer.
The TA is somewhat similar to EDSAC. It has, for example, delay-line storage. It works
at twice the digit rate of the EDSAC. It has some logical orders other than those needed for
arithmetic operations and uses the two-address code for instructions. Input and output are
on Hollerith cards.
In EDSAC, instructions are stored serially in a long tank. This means that after obeying
one instruction the machine has to wait for the remainder of a major cycle before the next
instruction is available. In theTA this is overcome by facilities for putting instructions in
nonserially and in such a way that when one is obeyed the next instruction is immediately
available. The T A is being carefully engineered. About one-half or two-thirds of the chassis
for it is completed, and Doctor Wilkinson hopes it will all-be completed by the end of 1949
so that testing will start in 1950. It is not likely that the full machine of the 1946 design will
be built now. Any further machine will probably have a much smaller number of mercury
delay-line stores and auxiliary magnetic-drum storage.
At the Telecommunications Research Establishment of the Ministry of Supply, Dr. A. M.
78
COMPUTING MACHINES IN ENGLAND
V ttley is working on a parallel electronic machine for the use of mathematical physiCists in
the Ministry of Supply. His decision to make a parallel machine was taken after a visit to
the V nited States in 1948 and was influenced by the fact that no one else in England at that
time was building a parallel electronic machine. He uses storage tubes similar to those of
F. C. Williams, working in the same way, and uses as many storage tubes as there are digits
in his words. The tubes are scanned in parallel, and a word is represented by taking one digit
from the corresponding position in each of the tubes. Like the Manchester machine, it uses
magnetic-drum auxiliary storage, but unlike the Manchester machine the numbers on the drum
are in binary-coded decimals, and there is direct access between the drum and the outside
world. Doctor Vttley's idea is that the drum will be prepared at leisure by mathematicians;
it will then be taken to the machine, and its contents transferred as a whole into the working
store of the machine. In an at,tempt to make the machine completely self-checking, Doctor
Vttley has developed a complete series of three-state circuits for the arithmetic and control
units of the machine. The whole machine works in three states apart from the store, the states
being nought, one, and fault. There is now a one-digit working model of the store and of
all the three-state circuits, and the magnetic drum is completed, together with the tape
puncher, and transfer from tape to drum and from drum to electromatic typewriter. Doctor
V ttley thinks the whole machine will be working in one or two years.
Another small relay machine is being made at the Royal Aircraft Establishment by Doctor
Hollingdale for the use of people in that establishment.
I come now to Elliott Brothers Research Laboratories. Our jnterest is in the development
of reliable components such as storage, arithmetic, input and output units for high-speed
machines~ We are working on a CRT storage method similar to but not the same as that
used by F. C. Williams at Manchester. In his paper F. C. Williams called the method of his
that we use, "anticipation pulse storage." We find that we can use a higher writing speed
than in the dot-dash method that is actually used in the Manchester machine though we
have not decided the maximum number of digits that can be stored reliably on one tube.
We are working on small logical units for serial operation at up to I-Me/sec digit rate and
we have working a series-parallel multiplying unit, using these logical elements, that forms
the rounded-off product of two numbers entering the unit simultaneously, the rounded-off
product appearing in the following number time. In its final form this multiplier will feed
the output straight back to one of the inputs so that n numbers can be multiplied together in
n number times.
We are working on photographic methods of feeding input data and function tables into
a high-speed computer and of recording the output from a computer. The input data and
function tables are prepared by photographing a lamp display controlled from the register
of a desk calculating machine working in binary scale, which we have made especially for
this purpose. The film can be read at /1 megadigit per second into a computer.
No description of the English automatic computing machine projects is complete without
mentioning the name of Professor Hartree, Plummer Professor of Mathematical Physics in
79
w.
S. ELLIOTT
the University of Cambridge. The early work of Hartree and Porter on the differential
analyzer at Manchester is well known, and today Professor Hartree plays a leading part in
encouraging work in England on digital machines. He is a regular attendant at the Cambridge
colloquia and is regarded as our chief contact with work in the United States.
In conclusion, I would say that the greatest diversity of opinion in England at the moment
is on the best method of storage to use. In the Cambridge machine the component that gives
the least trouble in the whole equipment is the mercury delay line. The F. C. Williams' store
is welcomed enthusiastically by some groups in England, though others are unhappy about the
noise level. Doctor Booth's electromechanical store is interesting in its simplicity and digit
density, though it is limited in speed. There is general agreement in England on the use of
magnetic-drum auxiliary storage. I think, however, that the greatest possibility of tec~nical
improvement or simplification is in storage systems~
The tendency in England at the moment. is to gain experience with the small machines·
that have been built or are being built, and I think that after one or two years of gaining this
experience, some furthermachines may be built. It is likely that when this happens the move
will be in the direction of logically simpler rather than of larger machines.
Finally, I should like to return to the subject of human and mechanical brains. Professor
Sir Geoffrey Jefferson of the department of neurosurgery of the University of Manchester
gave the annual Lister Oration to the Royal College of Surgeons of London on this subject.
He referred to the fact that some workers believe that by embodying in a machine the electrical
principles underlying neural activity, light can be thrown on the way we think and. remember.
He did not think, however, that the day would dawn when the gracious rooms of the Royal
Society of London would have to be converted into garages to house the new fellows.
80
THIRD SESSION
Wednesday, September 14, 1949
9: 00 A.M. to 12: 00 P.M.
RECENT DEVELOPMENTS IN COMPUTING MACHINERY
Presiding
. E. Leon Chaffee
Harvard University
SEMIAUTOMATIC INSTRUCTION ON THE ZEPHYR
HARRY D. HUSKEY·
National Bureau oj Standards, Institute for Numerical Anabsis, UCLA
Presently designed calculators cannot be entirely automatic with respect to coding;· they
may do problem after problem automatically without human attention, hut somebody must
initially tell the machine what it is to do. We will develop in this paper a method of operation
of such a computer in which the user need not tell in explicit detail everything that the calculator must do in the course of carrying out the computation. This concept of semiautomatic
instruction has been called abbreviated-code instruction. l
, To illustrate by example, assuming we wish to invert a matrix having m rows and n columns,
the only essential information is (1) where the coefficients of the matrix are, (2) how many
rows and columns there are, (3) what process is to be carried out, and (4) where the answers
are to be placed. We expect to be able to obtain sheets of paper upon which appear in the
appropriate order the coefficients of the inverted matrix without doing more than sending
the initial coefficients into the calculator and·a single coded instruction specifying the three
items mentioned above.
The Zephyr, the electronic digital calculator under construction at the· Institute for
Numerical Analysis, will he used as the model in this paper to illustrate how these coded
instructions will operate. Thus, before explaining the abbreviated code in detail a brief
description of the Zephyr will be given.
The Zephyr consists of: (1) an arithmetic unit where the information is processed or
modified; (2) a high-speed memory which remembers both the numerical and the instructional
words needed during the computation; (3) a low-speed memory, which we shall referto as
the store, 2 inasmuch as it serves as a warehouse wherein numerical information, main routines
of code words, and subroutines of commands or code words are stored; (4) a control unit
which scans the memory for its commands, and executes them by sending out the appropriate
signals to the other units; (5) input-output equipment which we will not discuss in this paper.
Information is stored arid' processed in the Zephyr in units that are 41 binary digits long.
Such a unit is called a word. Words may be interpreted as numerical information or as
instructions.
Numbers can be subclassified as follows. A word may represent a signed binary number
lying somewhere between - 2 40 and
240. Or, it may represent a signed ten-decimal-digit
number where each decimal digit is represented as a four-digit binary number. A floating
binary representation may be used where one is dealing with numhers of the form ± a X 2b.
For example, the first digit represents the sign, the next ten binary digits may represent the
exponent b, and the remaining 30 digits may represent the significant figures of the number
+
8·3
HARRY D. HUSKEY
in binary form. In this manner, with some loss of relative accuracy, a word can represent
numbers in the range between ± 230 X 2± 29.
In a similar manner instructions are subclassified into three classes. First, there are
command words of which there are 13 in the Zephyr. A command word isa 41-binary-digit
word, a portion of which determines one of the 13 operations, and the remainder of which,
in general, specifies four addresses in the memory. A second class of instructions are the
control words. Control words may serve as parameters that determine the number of repetitions
of certain routines;. they may be the bounds used to stop certain computational processes;
or they may serve as factors in logical products or extraction operations. The third class of
i?structions are called code words. A code word is a compact representation in one word of
several parameters that are needed to specify the operation of subroutines. Each subroutine
extracts its various parameters from this one code word. Thus, one may specify a scalar
multiplication with only one code word. The appropriate subroutine in the calculator extracts
and properly places the various parameters from the code word. These parameters must
specify the common factor (that is, specify its address in the high-speed memory), the location
of the elements of the vector (say by specifying the address of the first element and the number
of terms in the vector), and where the result is to be placed .
. We can summarize· the various types of instruction as follows. A command word is a 41binary-digit word which the calculator explicitly understands and obeys. A control word is
not directly obeyed by the calculator nor is it a direct part of the calculation; it in some way
co~trols the course of the computation or enters into the, arithmetic-like operations that are
performed upon command words. A code word is an abbreviated instruction that specifies in
one word a whole sequence of events for the calculator.
The high-speed memory will consist of a bank of cathode-ray tubes used in a manner
devised by F. C. Williams, of Manchester University, England.
This memory will have a capacity of 512 41-binary-digit words and will be able to deliver
anyone of its words to the other units of the machine in about 20 flsec.
The high-speed memory will be divided i~to three parts: first, a part that stores the numerical information temporarily; . second, a part that stores the subroutines which are to be used
in the problem; third, a part that stores that portion of the main routine which must be
stored in the high-speed memory. As the main routine is carried out new segments must be
read in, and in the course of doing the problem numerical information must be transferred
to and frOlTI the magnetic drum. If all the necessary subroutines cannot be stored in the
high-speed memory at once these, too, must be read in and out during the course of the
computation.
The store, or low-speed memory, will consist of a magnetic drum with a capacity of 10,000
words of standard 41-binary-digit length. It will have a multiplicity of reading and recording
heads so arranged that all the 41 digits of a particular word will appear simultaneously at
41 different magnetic heads. Thus, the access time for a word on the drum depends upon
the orientation of the drum when the number is called for, and will vary from a few
84
INSTRUCTION ON THE ZEPHYR
micro-seconds to a maximum of 16,000 p.sec (the time it will take for the drum. to make a
complete revolution).
In similar fashion to the high-speed memory the magnetic-drum storage will be divided
into three parts. One part will store the numerical information needed to do the respective
problems. A secon~ part will store all the standard routines, such as division, floating operations, etc. A third part will store the commands or coded instructions of the main routine.
In our present experience the number of commands per routine seems to average around 30.
Thus, we could store 100 different standard routines on the drum and only take up 3000
words of storage. Most problems should involve only a few hundred instructions, say not
more than 1000. This leaves approximately 6000 words, which, for example, is ample room
for storing all the numerical information involved in solving 70 simultaneous linear equations.
A command word may be represented in the form cx., {J~ y, c5, F; cx., (J, y, and c5, generally,
represent addresses or the position of words in the memory, while F determines which one
of the 13 commands is involved.
In normal situations the next command is specified by a fifth address, called 8, which is
remembered by a binary counter in the control unit. Each time a command is obeyed the
number in 8 is increased by unity; thus, the machine normally obeys a sequence of commands
coming from successive addresses in the memory.
There are three special commands wherein the next command is determined by the fourth
address, c5, of the present command. By the use of these special commands the machine may
transfer source of control with each command. When operating in this manner the machine
may obey.any arbitrary pattern of commands in the memory. Naturally, the special commands
may be interspersed in any desired manner among the other commands.
In addition and subtraction operations the capacity of the calculator may be exceeded.
In case this happens the normal commands behave exactly like the special commands; that
is, the next command is determined by the fourth address c5.
In order to explain efficiently the 13 commands, let us introduce the following notation.
Let w(cx.) denote the word stored in address cx.. Let the symbol ~ be read as "replaces." Let
NC = w(c5) me~n that the next command the calculator is to obey is the word in address c5
of the memory. The 13 commands, their symbols and effects, and the' next command are
given in Table 1.
Two principles have been followed in deciding upon the system of commands. The first
. is that there should be as few commands as possible so as to simplify the electronic circuitry.
(Actually, the electronic function table which interprets these commands has only eight
positions.) The second principle is that the commands should be as general as possible. For
example, the Extract Command (logical product) allows the use of any factor whatsoever,
and the elections in case of overflow are completely general.
One should notice that there is no Transfer of Control Command; the special commands
do this automatically. Also, there is no Halt Command; the Input Command with c5 specifying
a nonexistent input device causes the machine to stop. Division is accomplished by a routine.
85
HARRY D. HUSKEY
Table 1. Commands, symbols, effects, and next commands.
Command
Symbol
Next Command
Effect
Addition
ex, (3, y, <5, A . w( ex)
+ w({3)
--+ w(y)
w(s) ;
w( <5) if overflow
Special Addition
ex, (3, y, <5, Al
w(ex)
+ w({3)
--+ w(y)
w( <5)
Subtraction
ex, (3, y, <5, S
w(ex) - w({3) .--+"w(y)
w(s) ;
w( <5) if overflow
Special Subtradion
ex, {3, y, <5, SI
w(ex) - w({3) --+ w(y)
w( <5)
Multiplication with ex, {3, y, <5, M
Round-Off
W( ex) . w({J) rounded off to 40 digits w(s)
and sign --+ w(y)
Special Multiplication ex, {3, y, <5, Ml w( ex) . w({3) rounded off to 40 digits w(<5)
with Round-off
and sign --+ w(y)
Exact Multiplication
ex, {3, y, <5, P
w(ex)· w({3) --+w(y) and w(<5)
Compare
ex, {3, y, <5, C
Causes change in source of com- w(s) if w(ex)
mand
w( <5) if w( ex)
Special Compare
ex, (3, y, <5, C1
.Causes change in source of com.;. w(s) if Iw(ex)1
mand
w(<5) if Iw(ex)1
Extract
ex,{3,y, <5,E
w({3) is blanked (made into zeros) w(s)
wherever there are ones in w( ex),
<
w(fJ);
> w(P)
< Iw({3) I;
> Iw({3) I
the result is shifted right or left
a certain amount as determined
by <5, the final result --+ w(y)
Input
ex, (3, y, <5, I
Information is transferred from an w(s)
input device determined by <5 to
the address ex in the memory
Special Input*
ex, (3, y, <5, II
Incoming information goes to w(s)
instead of w(ex)
Output
ex, (3, y, <5, 0
Information is transferred from w(s)
address (Yo of the memory to the
appropriate piece of output
equipment as determined by <5
w(s)
In the case of the input and output commands <5 may specify that the
transfer is between the memory and the magnetic drum. In this event
y and part of {3 determine the address on the drum.
* This command is particularly useful in the process of initial input (that is, the process of reading-in information when there are no commands in the high-speed memory).
.
It takes nine digits to specify an address in the .memory. Thus, in the standard 4l-binarydigit words there are five digits left after one accounts for the four addresses. One of the five
86
INSTRUCTION ON THE ZEPHYR
digits is used for checking purposes to hasten "the detection of any error caused by the calculator
trying to obey ordinary numbers as commands. Three of the digits define the eight distinct
commands described earlier. The remaining digit defines modifications of five of the eight
commands, referred to in the table as the special commands.
Operations more complicated or more elaborate than those described in the discussion
of the 13 commands must be done by a sequence of commands called a routine or subroutine.
For example, division can be done by repeated subtraction if the appropriate routine is used.
The whole process of division, which may amount to 100 operations, can be completely
determined by approximately 15 commands. Furthermore, various commands used in the
routine are repeated over and over again, operating each time on different numbers.
We cannot go into details of routines at this time. Suffice it to say that the subject is a
very interesting one and that t~ere are many pitfalls for the unwary; for example, has "division
by zero" been taken into account?
In a general-purpose computer there are many relatively simple operations that we want
the calculator to carry out. For example, we want the calculator to perform division, floating
addition 3 and subtraction, floating multiplication and division, store-to-memory sequence
transfers; and many other routines. Each such routine can be represented by a single word.
Table 2. Storage of control words and interpretation routine.
Number
Address
ex
1
{J . 'Y
Remarks
<5
F
0
(= 000 ... 00)
2
1's, .1 's, 1's, O's, 1's
Used to extract the £5 portion of a word
3
1's, 1's, 1's, 1's, O's
Used to extract F portion
4
O's, 1, O's, O's, O's
5
O's, O's, O's, 1's, 1's
/3 addresses by unity
Used to extract cx., /3, and 'Y portions
6
1, O's, 1, O's,O's
Used to increase
Used to simultaneously increase cx. and 'Y addresses by unity
200
Address 200 shall be used to store the present coded instruction
upon which the calculator is operating. This address plays a
role analogous to the control register which registers a command
word while the machine is obeying it.
201
The five addresses following 200 contain an interpretation routine
that keeps track of the coded instruction we are presently
obeying, and provides a method of entering the proper sub. routine. In this system there are no general "links" (transfer
of control instructions) to tell the machine what to do when it
finishes the present subroutine; when each subroutine is finished
the control always returns to this interpretation routine
to
205
HARRY D. HUSKEY
If one were to try to build in circuitry to enable the calculator to perform all these tasks
it is clear that the machine would become so cumbersome from the circuitry point of view that
it would be almost impossible to construct, and, very likely, impossible to maintain in operation.
Howev:er, we have seen that a routine of standard instructions can be set up in the memory
whose effect will be to carry out such operations as those described above.
Before considering in more detail an example of an abbreviated code instruction we '
will look 'into the storage of control words and examine the interpretation routine. ' We
shall assume that control words and the interpretation routine are stored as indicated In
Table'2.
Each abbreviated command will be stored in address 200 while it is being obeyed. A
portion of it (analogous to the function in the command word) is extracted and added'to
a dummy command to arrange for an entry to the subroutine. The first step, in the interpretation routine is to read the code word from a general place in the memory into a fixed
place, address 200. Next the, extraction and addition with the dummy command takes place.
This dummy cmnmand must be carried along to allow the command to be used over and over
(the old command which provided the last entry remains in the subroutine until such time
as it is replaced).
The new entry shifts the source of commands into the subroutine. Each subroutine begins
with certain extraction arid addition commands that split the parameters off the code word
in address 200 and add them into the appropriate blanks in the routine. The last command
to be obeyed in the subroutine refers the control to address 201 for its next instruction. This
address in turn specifies what coded instruction is to be transferred to address 200.
Ifwe assume that fifty such abbreviated commands can be stored in the high-speed memory
and still leave sufficient room for the arithmetic data and the appropriate subroutines, then
either we must arrange that the interpretation routine counts and causes segments of the main
routine to be read in, or every fiftieth cOlnmand must read in fifty new coded commands from
the main routine of the problem.
Consider the coded instruction
20, 18, 60, VC, 19.
This means that a constant in address 18 is to be multiplied by a vector of order 19 stored'
in addresses 20, 21, .. :', 38 and the resultant vector is to be stored in addresses 60, 61, . . . ,
78. Let us also assume that the above vector-constant multiplication coded instruction is
stored in address 225. The sequence of commands is given in Table 3.
The vector-constant multiplication routine is chosen as an example since it clearly can
be consid~red as a unit in a higher-level program (for example, solution of simultaneous linear
equations) and it may itself control subroutines. For instance, the numbers may be stored in
floating-binary form and the command in address 306 would have to be replaced by a coded
instruction calling for a floating-multiplication routine.
We can imagine much more elaborate situations in which the main routine is given as a
sequence of coded instructions. Each of these coded instructions calls for a routine that is in
88
INSTRUCTION ON THE ZEP.HYR
Table 3. Sequence of commands for multiplication of a vector by a constant.
Next Command is
Determined By
w(201)
Last command of
preceding routine
w(202)
8
w(203)
The Command Is
Remarks
1,225,200" O,A ,
Takes (20,18,60,VC,19) to address 200
2,200,204, O,E
The "VC" [= 300] is extracted into address
204 to use as an entry to the vector-constant
routine
205,204,204, O,A
A "dummy" command in 205 has the
extracted VC added to it [w(205)
= 201,4,201,0,Al]
w(204)
8
201,4,201, 300,A 1
Adds'" 1" to the "225" in address b of w(20 1)
to provide for obtaining the next coded
instruction
(The interpretation routine is now finished and we are about to
enter the vector-constant routine at address 300)
w(300)
0(204)
5,200,204, O,E
Extract the "20,18,60" of the VC instruction
into address 204. Note that using 204 does
not harm the interpretation routine
204,302,306, 303,Al
Extractee is added to dummy in 302 to
produce the first multiplication command
[the 0 address
of w(204)]
w(301)
8
'[w(302)
=
O,O,O,O,M]
w(303)
~(301)
3,200,204,1-3,2,E
"19" extracted from w(200) and shifted left
into the (x; position in address 204
w(304)
8
204,306,204, O,A
A bound is produced in address 204· to tell
when to stop this multiplication process
[w(204) = 39, 18,60,0,M]
306,,204, 0, 201,C
The process will be complete when w(306)
= 39,18,79,0,M and the source of command will shift to' address 20 I
w(305)
w(306)
w(307)
8
20, 18, 60, O,M
306,6,306, 305,AI
First product is done
Certain addresses of w(306) are increased by
unity and the calculator turns to the
command in address 305
turn. made up of coded instructions, and so forth, until finally one reaches subroutines whose
elements are commands that the calculator explicitly obeys.
One approach to the problem of keeping track of position as one drops' from one hierarchy
of routine to an~ther is by a process called reversion storage. 4 In this method a so-called queue
89
HARRY D. HUSKEY
is established which stores in reverse order the addresses one must return to after 'completion
of the respective subroutines in order to proceed with the problem.
Our approach to this problem has been to classify all routines into various orders. Firstorder routines are made of units which are explicit commands that. the calculator obeys.
There is no need for an interpretation routine for these routines since the e counter keeps
track of position here. Second-order routines are those whose elements are first-order routines.
Third and higher orders are similarly defined. For each level a different interpretation routine
must be used. Not only this, but one may need to use a certain coded instruction representing
a particular first-order routine as part of, say, a second-order routine and a third-order routine.
Therefore, a record must be kept of the level from which the entry was made to each subroutine. When that routine is finished the source of command will revert to the appropriate
place in the correct level. Thus, we see that the interpretation routine divides into several
parts (one for each order of routines used after the first) with a record section that stores the level
from which the entry to subroutines is made.
In a sense we have made the situation two-dimensional. There are discrete levels on which
we may operate with the record portion of the general interpretation routine controlling the
choice of levels.
Success of a system like this simplifies coding by putting more of the responsibility for
routine operations upon the calculator~
NOTES
1. The term "abbreviated-code instruction" was developed aCthe National Physical Laboratory,
. England, in a group headed by Dr. A. M. Turing of which the author was a member.
2. The term "sto~e" was used by Charles Babbage of England, who is credited with being the
first to design an automatic calculator. In England the term "store" is commonly used when referring
to the "memory." Note that in this paper its use is restricted to the low-speed memory.
3. When two numbers of the form a X 2b are to be added, one must be shifted until they have
the same exponent.
4. This approach was developed at NPL. See note 1.
go
STATIC MAGNETIC DELAY LINES
WAY DONG WOO
Harvard University
The magnetic delay line is a storage device which is built of rectifiers and transformers with
cores made of ferromagnetic material that has nearly rectangular hysteresis curves. As shown
in Fig. 1, a binary "1" is stored in a magnetic core as a residual magnetism in one dir~ct~on,
while a binary "0" is stored as a residual magnetism in the opposite direction. The difference
B
--~----~-----+--------------------~-----H
I'
I
FIG. 1.. Paths of operating point.
between this storage device and the conventional rotating magnetic drum or tape is that the
storage medium is not moving. The information is recorded in discrete cores instead of on
small spots in a continuous medium.
To recorda binary "1" a positive magnetizing current is applied and to record a binary
"0" a negative magnetizing current is applied. After the magnetizing-current pulse is over,
the information will be preserved until another magnetizing force passes through the arc.
In order to read out the information without mechanical motion, it is necessary to apply
a probing magnetizing force H', which is obtained when I' is applied. If the digit is a binary
"1," then a large flux change occurs and a large induced voltage eo is obtained at the output
WAY DONG WOO
winding. If the digit is a binary "0," little voltage is induced. Thus the digit stored is indicated
by the magnitude of the induced voltage when a probing current is applied.
The residual magnetism remains essentially the same before and after a sufficiently small
probing current. However, application of another probing current of the same magnitude
INPUT
(0)
--I--I---+-H I
a
a
(b) PATHS OF OPERATION WHEN DIGIT STORED IS O.
b
- - f - - t - - - + - HI
a
a
(c) PATHS OF OPERATION WHEN DIGIT STORED IS I.
FIG.
2. Diagrams to show basic operation.
will produce a very small change of flux no matter whether the original state of the core is
"1" or "0." After a small H', repeated application of H' will only describe the minor hysteresis
loop shown at 1'; Thus, after the information is read out once, it can be considered destroyed
unless one resorts to increasingly large probing currents.
If a probing current large enough t6 reverse the saturation is applied, a very large induced
voltage results. It is so large as to be able to reverse saturation of another core of identical
92
STATIC MAGNETIC DELAY LINES
construction. Referring to Fig. 2(a), if both cores were saturated in the negative direction
originally, repeated application of II and 12 will not change the saturation of either core, as
shown in Fig . .2(b), and little voltage is induced at the output winding. One can consider
this as a "0" stored in this pair of cores. If, however, core number 1 is positively saturated
originally, application of II will cause flux 1 to change from positive saturation to negative
saturation. The voltage induced in the link winding will produce a current that opposes
the effect of II. This current causes the flux in the second core to go to positive saturation
even if it was originally at negative saturation. Now, if 12 is applied to the second core, the
flux in this core will go back to the state of "0" while that of the first core will go to the state
of "I." Alternate application of II and 12 will result in an exchange of "1" from one core
ADVANCING
FIG. 3. Circuit diagram of magnetic delay line.
to the other, and there is an induced voltage at the output winding on every II and 12 pulse.
Thus a digit" 1" is stored in this pair of cores.
From the pasic mode of operation, a number of cores are connected as shown in Fig. 3.
The. coils are wound so that the advancing current pulses produce a negative saturation
corresponding to "0." The series rectifiers in the link winding ate such as to stop any current
in the link windings that would produce a negative flux. Consider now the case of the cores
C1 and C2 having negative saturation. Then application of advancing current pulse II will
have no effect at all, and both cores retain their "0." One can also consider this as a "0"
having been passed on from C1 to C2 • On the other hand, if C1 is positively saturated but C2
is negatively saturated, when advancing current pulse II is applied, C1 will be saturated negatively, and the current in the. circuit linking C1 and C2 will saturate the latter positively. Core
C1 returns to "0," while C2 takes on the "1" that was originally in C1 •
The rectifiers in the circuit prevent the effect of changes of flux in other cores on the two
cores considered. The shunt rectifiers will prevent positive linking current in core Co when C1
93
WAY DONG WOO
reverses saturation from "1" to "0" and produces the driving voltage so that the" 1" does not,
go in the backward direction. However, it will have no effect when the driving voltage is
from the Co, because in this case the point a is at higher potential than b, while in the former
case, a is at a lower potential than b.
The series rectifiers prevent the effect on C3 when a "1" is advanced from C1 into C2 • As
(a)
( b)
FIG. 4. Flux in a given core as a function of time:
(a) information rate, 3 kc/sec; information = 0111;
(b) information rate, 30 kc/sec; information = 1000.
the flux in C2 goes positive, the voltage induced in the link winding to C3 is such as to produce
current causing negative flux in both C2 and C3 • This current is prevented by the series rectifier.
Aside from isolating C3 from C1 this rectifier also makes change of flux in C2 from negative
to positive easier.
Since each core when pulsed advances its stored digit only t() the next core and has no
effect on any other core, it is possible to advance a digit f~om every other core at the same
time. Thus the advancing current windings of every other core are connected in series. The
advancing current pulse 11 will step the digits in all odd cores to the even cores, and the
94
STATIC MAGNETIC DELAY LINES
advancing current pulse 12 will step all the digits in the even cores to the odd cores. A pair
of the alternate pulses will cause the digit to step two cores, which are considered as one unit
of storage.
It is obvious that material having a nearly rectangular hysteresis loop, high retentivity,
and low coercive force is required. The cores are made of wound strips of Deltamax (manu-
FIG. 5. High-speed magnetic delay line using
selenium rectifiers.
factured by the Allegheny-Ludlum Steel Corporation) of about fQur convolutions. The
diameter is t in. and the strip is t in. wide and 0.001 in. thick.
At present the maximum speed is 30,000 digits (i.e., 30,000 digits can be stepped through
each unit of storage) per second. There is no lower limit of speed. The system acts like a
system of trigger pairs, where digits are stepped from one trigger pair to the next. The fact
that the speed is entirely controlled by the rate of advancing current pulses makes it a very
useful intermediate storage system between two systems of widely different information rates.
~'
FIG. 6. Five-digit magnetic delay line.
Figure 4 shows the flux in a given core as a function of time when the information rate
(a) 3000 and (b) 30,000 digits per second. Figure 5 shows a ten-digit magnetic delay line
on a breadboard. Figure 6 shows a five-digit line mounted on an octal plug.
Professor Howard H. Aiken, the director of the Harvard Computation Laboratory, proposed
this form of storage device, and Dr. An Wang has done most of the work to make it successful.
Special acknowledgment is due the Allegheny-Ludlum Steel Corporation, which has cooperated
actively with the Computation Laboratory and has supplied all the core material.
IS
95
COORDINATE TUBES FOR USE WITH ELECTROSTATIC
STORAGE TUBES
R. S. JULIAN
and
A. L. SAMUEL
University of Illinois
One of the basic problems in connection with high-speed digital computers is that of
storing the necessary amount of information which must be available at high speed as needed
in the course of computation. As the speed of computing systems increases and as the amount
of storage is aiso increased, the problem of locating any desired information becomes more
acute. With this in mind, a research program was instituted,at the University of Illinois to
develop precise and rapid methods of locating stored information. Recent developments in
storage systems in which continual memory refreshing is employed have somewhat reduced
the long-term stability requirements, at least for these case's, so that the system to be described,
may not be needed. However, the system does possess a number of unique features which
were thought to be of general interest and to warrant description. When and if storage systems
progress to the point that a very much larger number of digits-say 106-can be stored in
one electrostatic storage tube, then the need for precise locating equipment will again be
urgent regardless of the type of refreshing used and the present scherne may warrant investigation.
We will assume that the information is stored in a binary code on the surface of target
plates in tubes of the cathode-ray type and that the system envisions the requirement that
anyone bit of information should be obtainable on demand without regard to its particular
location on the screen. To make the matter still more definite, we will assume that a bank of
40 such tubes are to be operated in parallel and that individual digits of a 40-digit code are
stored in corresponding locations in the forty different tubes; we would then like to be able
to locate these 40 digits simultaneously.
To make this possible we propose to combine these 40 cathode-ray tubes together with
two special coordinate tubes into a master-slave relationship in which all of the tubes are
connected to the same power supply with their corresponding deflection plates all tied together.
There will then be a one-to-one co~respondence between spot positions in the different tubes,
although distortions may occur in the mapping from one tube to another as the result of
minor differences and imperfections in the tube structures. If now some independent means
is provided for precisely, identifying specific spots on the screen of one tube, which then acts
as the master, such that the beam of this tube can be returned to these spots with certainty
when desired, the beams of all the other tubes will be returned to the corresponding spots
in these tubes quite independently of any distortions that may occur in the different tubes.
This will be true for a group of tubes that are structurally quite different as long as the other
96
COORDINATE TUBES
voltages on the tubes are maintained constant. If the tubes are reasonably similar in their
geometrical construction, it will still he true to a high degree of accuracy when these other
voltages are allowed to vary within the usual engineering limits. It is only necessary, therefore,
to introduce auxiliary beam-locating equipment into the master tube in order to control the
motion of the spots in all the slave tubes to the desired accuracy. Furthermore, it is quite
feasible, to control the horizontal motions of the beams in the slave tubes with one master tube
while at the same time controlling the vertical ,motion of the beams with a second master tube.
The master tubes need, not contain provisions for storage and they can 'be specialized to the
necessary extent to perform their control functions, all the while preserving their essential
similarity to the slave tubes in regard to one of their deflecting systems so as to retain the
desired mapping characteristics.
Models of two quite different types of master tube have been constructed. Tests made
on these tubes will be described later. Both types of tube are similar to the extent that they
provide a definite number of stable beam positions (in this case 32) by means of mechanical
positions on target plates contained within the master tubes. The beam position is maintained
by means of servo amplifiers which obtain their input signals from the beam currents associated
with the target plates.
, The basic principle underlying the control system can be illustrated by reference to Fig. 1,
which shows a syst~m in which there is but one stable position. A single-stage amplifier is
used in the illustration to simplify the discussion. Assume that the beam of the tube has been
deflected so as to strike the top portion of the second p~ate. All the current of the beam will
then be to this electrode. The voltage produced by this current in the grid resistor of the
amplifier tube (augmented by a d.c. grid bias) will cause the control tube to be biased nearly
to cutoff. As the result the plate current will be small, and the tap on the plate power supply
will be set at a value that will cause the beam to be deflected downward. Alternatively, if the
spot had been deflected downward so, as to strike the interceptor plate only, there would be
no current in the grid resistor, with the result that there would exist an appreciable plate
current. Wit\1 proper adjustments the resulting negative voltage across the plate resistor will
exceed the positive bias on the vertical deflecting vane and the beam will be deflected upward.
Obviously there exists but one stable equilibrium position in which the division of beam
current between the target and the interceptor is such as to produce no net deflecting effect.
If the amplifier circuit is properly designed to prevent hunting, any deviation of the beam
from this equilibrium position will bring into play the necessary restoring forces to return
the beam to the desired location. On the basis of this scheme, it is a relatively simple matter
to visualize the interceptor electrode, in anyone of several forms such that there may exist a
multiplicity of stable positions separated by regions of instability. Given a scheme for stepping
. the beam from one stable location to another, the necessary elements for the master tubes
are evident.
W}1ile it would be possible to step the spot from a starting position to any desired ultimate
97
R. S. JULIAN AND A. L. SAMUEL
position by a series of equal steps, economy-of-time considerations suggest the desirability of
utilizing steps of different sizes corresponding to the particular number system used in specifying
the address (which in this case is binary). If information defining a desired memory location
is supp1ied to the tubes in positional-notation form, then two possibilities exist: either this
FIG. 1. Principle of stabilization.
information can be supplied in timed sequence or it can be supplied simultaneously. These
two alternatives have resulted in the development of two quite different types of coordinate·
tube.
A simplified form of the serial coordinate tube is shown diagrammatically in Fig. 2 (in
this case for only 8 stable positions). A comb-shaped interceptor is used in which the slots
between the teeth are cut to different depths. The vertical position of the beam in this tube
(which will be assumed to be acting as the master governing the vertical motion of the beams
98
I
I
/
./
,.. ..... -- .......
--
.......
I
I
I
" "-"-
",
\
I
FROM
COLUMN
CONTROL
"
I
I
I
I
\
\
\
\
\
\
I
,
I
I
"'"
./
/
/
I
I
,I
I
I
/
-.
FIG.
2. The serial coordinate tube.
99
----_/
./
/
I
I
I
I
I
COLUMN
CONTROL
/
.""..---...
-
..... ....... ,
/
',VERTICAL
, MASTER
I
/
/.
/
\
\
\
\
/
\
\
\
,I
I
I
,
I
I
I
............,_ /I
,
\ HORI ZONTAL
\ MASTER
\
\
,I
\
\
I
I
I
-----
/
........... _, /'
I
,,
,
\ SLAVE
\
\
\
\
,
I
I
,~- ...... ~
./
.,..""
FROM
CONTROL
FIG.
I
--.
---_
WRITING
READING
I
I
I
I
-_/
AND
PULSES
3. The storage locating system shown with a single slave tube.
100
COORDINATE TUBES
in the storage tubes) is controlled by the servo amplifier, the polarity being such that the beam
is stable when partially intercepted by the top of anyone tooth. The horizontal position of
the beam in the master tube (i.e. the one controlling the vertical motion of the beam in the
storage tubes) is, however, subject to independent control, there being three column positions
corresponding to different columns in the binary-notational number system. The beam will
be assumed to be initially deflected to the position labeled 1 in the figure. This corresponds
to the starting position before an address has been located. We will assume that the desired
address is the fifth slot, which in binary notation is 101, and that digits corresponding to this
binary number are transmitted to the tube in timed sequence with provision for stepping
the beam from column to column between these periods. With the beam in position 1, the
first digit of the address is supplied through the circuit designated at the twitcher. This supplies
FIG. 4. Photograph of the serial tube.
to the deflecting plates a step voltage which transports the spot upward beyond the first slot
so suddenly that the servo amplifier is practically not effective. The amplifier then continues
to deflect the beam upward until it comes to rest at the next stable position, labeled 2 in the
diagram. The column-control circuit then deflects the beam to position 3, where it is ready
to receive the second digit of the address; in the present case this digit is zero, so that no signal
is supplied from the twitcher circuit and the beam remains in position 3. The horizontal
position of the beam is then moved to the next column and the final digit of the address is
supplied, causing the beam to step to position 5. This then is the desired location.
If we assume that the tube just described was controlling the vertical position of the spots
in the slave tubes, a similar master tube could be at the same time controlling the horizontal
positions of the beams in these same slave tubes, with the result that the desired address would
be located in a time required to transmit the three-digit address code for either the horizontal
or vertical positions. This is shown in Fig. 3, where, for simplicity, only one slave tube is drawn.
A simple method of returning the beam to the starting position is also illustrated in Fig. 2.
It is only necessary to move the beam to a restoring column on the right to cause it to be
101
R. S. JULIAN AND A. L. SAMUEL
returned to the desired horizontal position; it can then be deflected to the left at position 1
as shown . A number of quite different stepping arrangements have been proposed and
investigated, but since the one shown in Fig. 2 proved to be the simplest and most reliable,
these other schemes will not be discussed in detail.
Several characteristics of this scheme warrant special attenti0n. It
will be observed that the address must be supplied in time sequence
and in the normal forward binary notation rather than in the reverse
binary notation which is frequently employed in serial machines for
the numbers on which arithmetic operations involving carry are performed. This must be carefully noted but should cause no trouble
except in those cases where arithmetic operations are performed on
addresses. The use of a timed-sequence address allows a simple mastertube construction having only a single output lead. W hile the tube
is
essentially a digital device, the beam is stepped across the slots in
FIG. 5. Actual
the target plate by means of an applied voltage. It is important to
sequence of steps to
locate the position
observe that the tolerances on this stepping voltage are so great as not
11101 on the serial
to nullify the digital principle of operation. The most severe requiretube.
ments on this stepping voltage occur when the spot is in the last
column, in which position the amplitude of step must vary by about ± 50 percent to cause
failure. This can be seen by studying the form of the coordinate tube output current as a
function of beam position in Fig. 12. A photograph of an experimental coordinate tube of the
serial-address system is shown in Fig. 4. Figure 5 is a photograph of the screen of this tube
showing the sequence of positions occupied by the beam in going through a complete cycle to
locate the address specified by the binary number 111Ol.
FIG. 6. Photograph of the parallel coordinate tube.
A distinction between the use of a serial or parallel address code and the operation of the
complete computer on a parallel basis should be noted. In the system just described the
address is supplied to the coordinate tubes in time sequence, but since the master tubes control
40 slave tubes each containing one digit, the stored information is available for use in a parallel
adder if this is desired.
If a parallel system is envisioned, it would be more logical to supply the address to the
102
COORDINATE TUBES·
coordinate tubes in parallel rather than in tiined sequence. For this reason a second type of
beam-position tube, shown in Fig. 6, has been constructed. Structurally this tube is similar
to the one just described in that it consists of a cathode-ray tube with a special target replacing
the fluorescent screen. However, this target now consists of a metal plate containing vertical
columns of windows, the vertical position of the beam being stabilized at one of eight locations
determined by each of these windows.
The principle of operation is as follows. The electron beam is swept horizontally by a
sine-wave generator at a speed that is high compared to the vertical operating speed. We
have found it convenient to use a 30-Mc/sec oscillator for this purpose. The trace of the
sweep spans all of the vertical columns of windows, as shown in Fig. 7. Wherever the beam
encounters a window it enters and impinges upon o~e of the curved metal surfaces which
may be seen behind each column. The secondary electrons ejected from a given one of these
. surfaces will either arrive or not arrive at a collector C, according as the bias upon the corresponding grid G is positive or negative. The current to the collectors produces a voltage drop
which is then amplified by a high-frequency amplifier, and rectified, and the output voltage
is supplied to the vertical deflecting plates of the tube in such a way that the vertical position
of the beam rises as long as the secondaries in one or more columns reach a collector.
With this mechanism in mind we can now see how the beam finds the proper vertical
position. The binary digits of the vertical address are applied as biases to the grids G, positive
. bias if a digit is one and negative if it is zero. The most significant digit is applied to Gl,
the next to G2, and so forth. As may be seen now by studying the positions of the windows in
Fig. 7, the beam will rise to a unique position for each three-digit number applied to the grids
if the initial position is near the bottom where the beam encounters windows in each column.
For example, in the figure the beam is shown in the 010 position; only column 2 is open so
the beam rises to the upper end of the lower window in this column and stops. Had the address
been 110, the long window in column 1 would have bridged the gap between windows in
column 2 and the trace would have risen to the top of the upper window in column 2. The
zero position of the beam is established by cutting away the lower portions of the secondary
emitting surfaces so that the primary beam can strike the collectors. This can be seen in
columns 2 and 3 of the figure.
The over-all speed of this type of coordinate tube is somewhat better than that of a serially
operated tube because the beam need stop only once in its search for an address. The tube
also avoids the need of a direct-coupled external amplifier and column-stepping equipment.
On the other hand, the high-frequency amplifier req uired by the parallel tube needs more
gain than does the direct-coupled amplifier of the serial tube. The parallel tube itself is fairly
complicated, and the over-all complexity of the two· systems seems to be about the same.
In any application of coordinate tubes, the speed of operation is likely to be a matter of
primary interest. \Vhile the over-all speed may depend upon the computer as a whole, certain
limitations inherent in the coordinate tube itself may appropriately be discussed here. These
103
R. S. JULIAN AND A. L. SAMUEL
limitations are essentially those that are encountered in any feedback amplifier because of
parasitic capacitance and finite tube transconductance, and are associated with feedback
stability.
60 MC AM Pli FI ER
AND DETECTOR
/""-------.--.--.....,
...............
/
/
.
""
/
/
'""- '\.
"
/
\
I
30 MC SINE
WAVE TRACE
/
\
\
/
/
/
\
\
\
\
\
I
/
/
I
/
'j
I
/
I
I
I
I
/
~~
,/
/
/
./"
/'
./"
/-------~
PARALLEL
ADDRESS
-- -- ---
-:----
_.-/
/
/
/
/
~
FIG. 7. Principle of the parallel tube.
In either of the two typesof tube, the beam is expected to rise during switching until some
edge of the locating comb is encountered and there to remain. For example, referring to Fig.
2, suppose that the beam is given a vertical twitch so that the feedback system causes it to
rise from position 1 toward position 2. When the spot reaches the open slot it must not cross
4
10
COORDINATE TUBES
the slot because the feedback system would then cause it to rise still farther to some undesired
position. This places a restriction upon the speed with which the spot may be allowed to
move relative to the speed of response of the stabilizing amplifier. Moreover, when the spot
is resting i~ some position the feedback system should cause it to remain quiescent; that is,
the feedback loop should be stable. These two requirements are somewhat similar, the second
being ordinarily the more stringent.
The basic factors upon which these two types of stability depend may be understood by
analyzing in detail a specific typical system, first for overshoot and second for static stability.
The system considered is that shown in Fig. 8.
The following nomenclature will be used:
Co, total amplifier output capacitance including the collector electrode of the coordinate
tube;
d, equivalent spot diameter;
D, deflection sensitivity of coordinate tube;
gm, mutual conductance of each amplifier tube;
G, zero-frequency gain per stage [= gmR];
io, net beam current to collector;
L(w), total gain ratio of feedback loop;
N, number of amplifier stages;
R, interstage shunt resistance;
S, vertical position of spot;
.t, time;
Vn) output voltage of nth amplifier stage;
W, slot width;
w, angular frequency.
With the electron beam in the position shown in Fig. 8, the beam current £0 charges Co
at a certain rate. The rising voltage across Co, when amplified, causes the beam 'position to
change at a rate given by
4!. _ io GnD
(1)
dt - Co
.
When the spot position reaches the lower edge of the slot, the current in Co abruptly stops (in
this part of the analysis the beam focus is ass~med to be infinitely sharp). The requirement
we wish to impo~e, then, is that the spot cease rising before it has travelled one additional
slot width.
To calculate this we must know the response of the amplifier to an abruptly starting or
stopping linearly rising voltage. For the amplifier shown in Fig. 5 we find that
(2)
R. S. JULIAN AND A. L. SAMUEL
irall vn's are zero for x
<
tiRe.
O. In this expression x =
If we take
Vo = 0 for t < 0
and
Vo
=
at for t
>
0, .
(3)
where a is constant, then Eq. (2) gives
~n-k ]
vn=ARCGn [x-n+e-X~-k-x7c.
(4)
o
If the deflecting voltage of the coordinate tube is the output Vn of the last amplifier stage,
Eq. (4) says that the ultimate (t~ RC) motion of the spot is given by
S = atGND-aNRCGND.
(5)
The second term of this expression is the ultimate lag of the spot behind where it would have
been with a perfect amplifier and the input given by Eq~ (3). Since this is a linear analysis,
this lag is the same as the overshoot when the beam reaches an edge after long travel. Comparing the two terms in the right-hand member of Eq. (5), we see that the overshoot is
ds
NRC dt'
where dsfdt is the speed with which the spot enters the slot. The condition that the overshoot
be less than one slot width, therefore, is
which may also be written
(6)
The condition that the spot in a coordinate tube shall not overshoot its mark too far must
certainly be met if the tube is to operate at all. Beyond this one might demand that the spot
ultimately come to complete equilibrium and not dance about' on the edge of the slit. Whether
or not the spot- will do this evidently depends upon the behavior of the complete feedback
path, including the effect of finite spot focus. The feedback loop may be characterized by'
means of the complex loop gain, which for Fig. 8 is
WDioRoGN
L( w) = d( 1 + jwRoCo) (I + jwRC)N
(7)
when expressed as a ratio. In this expression d is an equivalent spot diameter defined as the
diameter the spot would have if the rate of variation of collector current with the elevation
of the spot on the slot edge were constant over the sl'0t diameter and equal to the actual rate
at the equilibrium position. This gives a dimension that is proportional to diameter for similar
spots and is of the same order as the visual spot diameter on a fluorescent screen.
The condition that the spot come to equilibrium can now be obtained from Nyquist's
criterion for the stability of a feedback loop. This criterion is that the complex variable L
106
N IDENTICAL STAGES
r---------------------r--r---------------------------,
/
\
\
//
------------""
/
I
,,
\
I
I
,I
I
I
I
I
I
I
_lQo
--.--
R
C
R
C
I
I
/
I
I
I
I
FIG.
~
L-______ _______________
~
I
_
I
I
I
_____ _
_ ___________________ J
8. System used to analyze stability.
HORIZONTAL
STEP
GENERATOR
F
COORDINATE TUBE
-------------------------,
I
I
OSCILLATOR
ADDRESS
GENERATOR
I
I
D.C.
STABILIZING
- AMPLIFIER
I
I
I
I
I
I
I
I
___ ...JI
TWITCHER
FIG.
D
9. Block diagram of test equipment for the serial tube.
I
R. S. JULIAN AND A. L. 'SAMUEL
not encircle the point - 1 as co traverses the rea,l axis from Eq. (7) shows that this requires that
•
io GND
Co
since L(O)
~
<
00
to
+ 00.
Examination of
'IT
Sln-
2N dg m
'IT
GC'
cos 2N
2
1. Making use ofEq. (1) this inequality becomes
ds
dt
<
K dg m
NGC'
.
(8)
where K is a number of the order of unity.
. Comparing (8) with (6) we see that the allowable spot speed for absolute stability is less
than that' for which the spot simply. will not overshoot a slot in about the ratio of the spot
diameter to the slot width. It is interesting to note that all factors describing properties of
the amplifier enter both (6) and (8) in the same way.
The facts that must be considered in choosing the parameters of the amplifier system to
operate a coordinate tube are relations (1) and (8), and a statement (9) of the maximum
deflection which the amplifier must produce. Collected, these are
ds= io GND
dt Co
'
(1)
ds
dg m
dtnstruction of the
experimental tubes, was done by Messrs. Robertson, Peiffer, and Haynes, graduate students
in Electrical Engineering at the University of Illinois.
APPENDIX
HARMONIC OUTPUT OF PARALLEL-TYPE TUBE
As the electron beam scans .sinusoidally across the windows in the parallel-type coordinate
tube of Fig. 6, it produces current pulses to the collectors C of any columns that are not shut
SINUSOIDAL
TRACE
x =Xo cos "'ot
i(t)
~
FIG. 14. Analysis of the output from'
the parallel tube.
off by the grids G. In order that the tube may function properly, these various pulses must
all add in phase, and each pulse should contribute about equally to the output signal. The
followipg analysis discloses the conditions under which this will occur.
The complex output current may be taken as the Fourier series in wot, that is"
00
i(t)
= 2: aneinwot,
(IA)
-00
where the complex constants an are given by
1
an ---: -2 f1T i(t)e-jnwotd(wot).
7T
J-1T
113
(2B)
R. S. JULIAN AND A. L. SAMUEL
Putting into Eq. (2A) the output ~urrent i(t) due to a single open column located as shown
in Fig; 14, we get
.
an =
~ [sin (n cos- ;~)
l
sin
-
(n cos-
l
::)].
Hence, the rmscollector current in the nth harmonic may be written
.
tn =
V2Io 6. [.SIn ( n cos- I
-n-
X)] '
where 6. [] means the value of the function in brackets at
1
r
/
o
07
-I
(,)
c
iii
Xl
minus its value at
\
\
/
'\
\ V
/
X 2•
Al\
I
\
I
t
l:!.
/ 1~1
D ~
><1><
1
."
(3A)
Xo
T
_ 2nd HARMONIC CURRENT
-
.225 10
1
/
/
-I
x
Xo
FIG. 15. Second-harmonic output curve;
6. = (second-harmonic current)fO.225Io·
Since an comes out real regardless of the values of Xl and X 2, we see that the currents fi'om
any number of open slots will be either in phase or 180 0 out of phase. The total current in
a given harmonic may readily be obtain~d for any placement of open windows from a graph
of the function in brackets in Eq. (3A). This is illustrated in Fig. 15, where the function is
plotted for n :!::: 2. From Fig. 15 it is apparent that: if the peak-to-peak sine-wave sweep is
about twice the total width occupied by the windows then all pulses will add in phase and
will contribute about equally.
The curve corresponding to Fig. 15 for the fundamental (n = 1) is a semicircle centered
at zero. From this or direct physical reasoning, one sees that a window at the center of the
sweep produces no fundamental output, and windows on opposite sides ·of the center tend to
cancel. For this reason and because of shielding problems, the second harmonic rather than
the fundamental output is used.
114
BASIC ASPECTS OF SPECIAL COMPUTATIONAL PROBLEMS
HOWARD T. ENGSTROM
Engineering Research Associates, Inc.
This Symposium is impressive, both because of the large number of members and guests
present and also because of the fearful and wonderful developments which have been and
will be revealed in the papers presented here. The preceding papers on this program have
been concerned largely with specific engineering developments; I should like to digress briefly
and discuss some important factors of the general probl~m of procuring effective computing
machinery.
I should like to emphasize, first, that the objective of all work on calculating machinery
is to produce computational results. In spite of the large amount of activity in connection
with the development of digital calculating machinery, I think it is a fact that nearly all of
the computational results so important to our national defense and industrial economy are
still produced by traditional methods. Most of the computational results are still, obtained
by machines of the desk calculator type, supplemented by the excellent machinery of the
International Business Machines Corporation, Remington Rand, Burroughs, and others,
including, particularly, such notable individual contributors as Professor Aiken, who has
been responsible for the development of both numerical techniques and machinery for carrying
. them out.
Not very long ago, a war for which this country was unprepared found us equally unprepared to carry out many necessary computational problems. The procurement of adequate
equipment to carry out these problems was a matter of great difficulty. In the light of the
critical international situation in these postwar days, deep consideration should be given to
the basic problem of procurement of this computing equipment.
In any discussion of a computational problem an immediate question is "Shall we use
general purpose equipment, or design special equipment for this particular need?" It is
axiomatic that any given computational process can be carried out more efficiently (i.e.,
either more rapidly or with less extensive machinery, or both) by equipment d~signed especially
for the purpose. However, the decision between special-purpose and general-purpose equipment is difficult to make. It depends upon a number of factors related to each other in a
complicated way. It depends not only upon the technical character of the problem but also
on economic factors, the work load, and S::l on.,
If solution time is not the most important factor, it is quite possible that general-purpose
equipment may be preferable because of its versatility. If general-purpose equipment is
obtained it can be applied later to .other problems, as they arise. One activity equipped with
general-purpose machinery which comes to mind immediately is, of course, the Computation
I
15
HOWARD T. ENGSTROM
Laboratory at Harvard. Although this laboratory is devoted primarily to research into
methods of computation, rather than to the actual performance of computing services,it
has done much of the latter. Other gener':ll computation facilities are centered at the National
Bureau of Standards. Some such services, notably in England, have been set up under private
enterprise. The International Business Machines Corporation provides services of this character, particularly in connection with their large-scale computer in New York, although this
service is again primarily scientific in objective. Some bureaus and divisions of the government
have found the volume of individual computational problems sufficiently great to warrant
setting up laboratories of general-purpose computing equipment to carry out services of this
character which arise within their particular divisions.
At the other extreme are the computational problems requiring a large volume of specialized
. workof a repetitious nature where the load is kept constant. In these situations special-purpose
equipment is the obvious choice. I shall sketch briefly some exam~les of specialized problems
requiring extensive repetitive computation at nearly constant work load.
The airport facilities and the airways of this country are being subjected to increasing
congestion, particularly under adverse weather conditions. The problems of air-traffic control
and airport time utilization are essentially computational in character. They are problems .
of automatic continuous inventory. 'Vith respect to the airport time-utilization problem, the
basic preliminary design plans for equipment that will solve it have already been prepared.
This equipment will store information on airport runway assignments by hour and minute,
classified as to class of aircraft and arrival and departure times. The proposed equipment will
supply this information upon inquiry and will change the stored information to conform to
changing situations occasioned by weather conditions. Supplementary information, such as
the identity of the plane and its route, likewise will be made part of the record ..
Present control equipment in general use performs no computations, and even the most
routine decisions are presently made by human controllers. Eventually it is planned that the
input and output to airport time-utilization equipment will come from communication
channels, and that the proposed equipment, which I have described very briefly, will be used
at all large airports.
This is a typical problem in which the use of special computational equipment is necessary.
The development of such equipment must be pursued strenuously, and its installation encouraged. Operational control of large numbers of aircraft is of vital importance; the nation
may be faced with a 'need for a practical solution of this problem on short notice.
Although of limited sophistication, the problem of reservation control also is one of impOrt:I.nce. Those of you who spent valuable time during the war sitting in airports in far-off
places waiting for air transportation, or in railroad stations attempting to get railroad transportation, realize this only too well. Technical methods of a computational nature for the
solution of these reservation problems have been proposed. The basic reasons why this type
of service is not yet available are nontechnical in character. They depend upon operational
and financial conditions. For example, these are the questions which arise: Is it better for
116
SPECIAL COMPUTATIONAL PROBLEMS
each airline to maintain a single reservation control, centrally tied in by communication lines
to its outlying offices, or to maintain separate centers in the major cities from which it operates?
Is it preferable for the airlines to combine their reservation control on an intercompany basis
in each major center? To what types of transaction must computational equipment provide
the reply? Answers to these questions are being sought by the Air Transport Association and
committees consisting of representatives of commercial airlines.
Another field in which large-scale computing is required and in which the arithmetic is
straightforward can be designated under the heading of inventory control. Many important
problems in this field are being handled adequately now, but the earlier years of the last war
may be characterized by the statement "too little and too late," largely because of inadequate
inventory control. The later years of the war were marked by the rise of priority systems and
the resulting controversies. One basic assumption which may be made is that any future
wars of these United States will be fought in the economy of limited scarcity. This means that·
improvement in the methods of the control of inventories must continually be carried out
and that plans should be made. to speed up even those methods that are satisfactory now.
Applications to these problems of techniques such as those discussed at this Symposium are
seriously lagging.
There are numerous other fields in which the application of special-purpose computing
equipment is obvious. These are situations in which specific data-reduction problems exist;
problems of control in which the req~ired degree of precision is so high that digital rather than
analogue techniques must be used. In all these fields the question remains: Why have not
the successful results of researches been brought to bear on these problems? I believe the
basic answer to this question lies essentially in nontechnical fields. The following reasons I
believe. are basic:
(1) Lack qfreliability. The reliability of electronic equipment involving l~rge numbers of
vacuum tubes is still questionable. Reliability is of paramount importance in connection
with any problems involving automatic control or inventory. In putting together a digital
computer, whether special or gener~l purpose, a great deal of time is spent in removing the
bugs. Although components operate well individually, the int~rconnecting and matching
problems assume large proportions. The maintenance. of special computing installations,
however soundly engineered, is a problem of the first magnitude.
(2) Economic factors. The economy of this nation is such that sources for procurement of
computational equipment must be found in private industry. The researches on digital' computing equip~ent have been carried out to a large extent at universities under government
sponsorship. Large-scale computing devices are expensive. Private enterprise, which must
make a profit, is naturally reluctant t6 invest the large sums necessary to establish procurement
sources on an industrial basis. Rapid advances in the art are, paradoxically enough, a
hindrance to industrial development because no one wants to spend money on equipment
that may shortly become obsolete. Also, the industrialist requires some competitive protection
in the form of patents or exclusive rights to equipment and techniques. The patent structure
117
HOWARD T. ENGSTROM
with respect to the large-scale computing devices is complicated by the fact that so much of
the work has been carried out either within the government itself or in nonprofit institutions.
Moreover, there arc no accurate data on the cost of producing such equipment because the
methods of accounting employed in universities, government laboratories, and industry are
so different. Hence, the question "What is a reasonable price for a computer?" is difficult
to answe:r;.
I have devoted much of this paper to generalities. In order. to come within the compass
of the title of this session, "Recent Developments in Computing Machinery," I must mention
the contribution of the company which I represent .. I believe our basic contribution to the
practical solution of many special computational problems is our work with magnetic-drum
storage and the ancillary electronic techniques. We have placed considerable emphasis on
the perfection of the magnetic drum, both as a scientific instrument and as a c~mpetitiv~
commercial component. I am happy to report that we have had magnetic-drum equipment
operating satisfactorily for a period of two years. Our efforts to develop a~d'design components
and to evolve manufacturing techniques and processes have attained a degree of success such
that magnetic-drum storage can be considered an industrial component. We have developed
reliable magnetic heads, drum-surface materials and techniques, and. mechanical-design
principles. These will be the subject of papers presented elsewhere. I wish to point out,
simply, that the magnetic drum, as a component of special digital computing machinery,
is now available.
In closing, I should like to state again that the needs for special computing equipment in
many aspects of our. national defense have not been met. Large gaps exist in the fields of
operational control and in highly speciali~ed computing. Components to solve many of these
problems have been developed, but ~re not industrially available. Increased attention must
be given to these problems or the program on large-scale digital calculating machinery may
be given the label "too many words, too few numbers."
118
ELECTROCHEMICAL COMPUTING ELEMENTS
JOHN R. BOWMAN
Mellon Institute
Several fundamental and general electrochemical effects are potentially useful in the design
of digital computing-machinery components. These include chemical deposition, electrolyte
polarization, hydrogen-electrode polarization, anodic-film polarization, and alteration of
surface tension. These can be combined in various types of cell to provide the functions of
storage and selection. Such components have the advantages, over most of their equivalents,
of small size and low cost. In speed, they fall in the millisecond, or more rarely in the microsecond, range. Their main disadvantage is that they are essentially low-voltage direct-current
units, and hence are not particularly well suited to electronic coupling.
Electrochemical devices have found little application in communication engineering.
This is largely because the effects are essentially qualitative and not reproducible to better
than several percent. In digital networks, however, such reproducibility is not required, and
electrochemical devices can be designed that will give good dependability in discriminating
between two discrete states. This applies to all of the cells to be described. All of the effects
discussed are reversible, not, in the thermodynamic sense, but in the sense that input of a
suitable sig~al will bring the device back to an original state after having received an intermediate signal.
From the qualitative character of these effects, detailed development of any unit must be
closely associated with the development of the entire network. Forthis reason, the information
presente~ here is essentially a theoretical discussion of principles supported by a minimum
amount of experimental results. Further experimentation would be useless without a definite
object of computer design as a whole.
As is well known in the electroplating arts, the passage of charge through a cell containing
metal ions may cause deposition of metal on an electrode. This effect is a reversible one, and
the presence of the metal film on the electrode gives .the cell an output voltage. When the
circuit is closed, a current is established in it and the metal returns .to the electrolyte as ions.
This effect can, potentially at least, be utilized to make a storage device. Consider, for
example, a cell composed of similar electrodes and an electrolyte containing ions of a metal
that plates out well. An electric impuise to su~h a cell will cause it to have an emf of sign
opposing the input pulse. Application of a second input pulse of opposite polarity to this
charged cell will cause anodic removal of the metal originally plated out and simultaneous
deposition of metal on the other electrode, the emf of the cell thereby being reversed in polarity.
Such a cell has properties similar to those of a capacitor.
119
JOHN R. BOWMAN
Selection of the electrode and electrolyte materials for a memory device depending on
this principle requires certain obvious considerations. Perhaps the most important is that
the metal to be deposited be not subject to corrosion by the electrolyte. Further, it is desirable
that the cell potential be as high as practical. These requirements are essentially contradictory
because the most active metals present the largest electrode potentials.. The best compromise
appears to be silver. This metal is unique in that it is the most active material that is unaffected
by aqueous solutions of its own salts. It plates out well and develops usefully high potentials.
The electrodes to be used should be inert chemically and should polarize readily with
respect to hydrogen. These requirelnents lead almost uniquely to gold as the electrode
material.
The electrolyte should be stabilized to constant concentration of metal ions, a condition
most readily met by providing for use of a saturated solution and supplying an excess of the
solid-phase salt. For the silver-gold system, silver sulfate fulfills these conditions conveniently,
being stable and soluble to a useful extent in water.
Numerous experiments have been conducted on the cell
Au/Ag2 S0 4 , sat. aq./Au.
It is readily reversible, stable on standing for several months, and gives a steady output emf
when charged to O~l to 0.2 v.
The actual value of' the emf in the charged state is not reproducible, and appears to
depend greatly on the nature of the gold surface on which the silver was plated out during
the charging cycle, the rate at which the silver was deposited, and the amount of silver de-,
posited. In general, high voltages are obtained for rough electrodes where small amounts of
silver are deposited rapidly. In no case, however, was there ambiguity as to the sign of the
polarity.
As will be discussed under hydrogen polarization, the emf of this type of cell may be high,
i.e., I v or more immediately after charging, but this value decreases to that of a normal
silver electrode in a few minutes.
This simple cell has the disadvantage that successive charging pulses or a long-continued
one will deposit additional silver linearly with it, and reversal may require a large charge.
This may be overcome by introduction of acid in the electrolyte to cause concentration
polariza tion.
Since the mobilities of the ions in ail electrolyte, are in general different, a· current in it
gives rise to concentration gradients'; In particular, the hydrogen ion is highly mobile p,nd
will carry a large part of the current relative to its concentration. If the electrolyte bearing
silver ions is initially acidified and uniform, a substantial part of ~he current will initially
be carried by the silver ion, but as the action proceeds the ratio of hydrogen- to silver-ion
concentration near the cathode will decrease sharply. Continued current will then deposit
relatively small amounts of metallic silver, and a saturation effect exists. The charge-retention
characteristic of a typical cell is illustrated in Fig. l. Charging curves have been obtained
. 120
ELECTROCHEMICAL COMPUTING ELEMENTS
on numerous cells of this type. The initial portion of the curve is nearly linear in deposition
of silver. As the charging proceeds, the electrolyte becomes exhausted of silver ion in the
vicinity of the cathode, and the principal reaction at that electrode is release of hydrogen,
which is quickly lost and does not contribute to the charge retained by the cell.
On applying a charging pulse of reversed polarity
to an already charged cell, the limited amount of
silver originally deposited is promptly removed
o
because there is an abundance of sulfate ion in the w
neighborhood of the electrode originally serving z
2 and is
an integer. The exact nature of these characters is immaterial; they. may be holes' in paper
tape, marks on paper, el~ctrical pulses, distinct identifiable positions of rotating elements,
light signals. All we require is that we can recognize any character, and unambiguously
determine to which of the (3 classes it belongs.
12 7
GEORGE W. PATTERSON
It is further assumed that these characters can be arranged in strings with a definite
beginning and end, and that each character except those at the ends has a unique immediate
predecessor and a unique immediate successor. This constitutes a linear syntactical system
or language. Each string is an expression of the language. Multidimensional exp~essions are
also used, for example, in matrices and punch cards, but they will not be considered here.
An expression, a string of characters, can be symbolized by
6.
Xi'
The variables of the meta-
language are restrieted to range over fJ > 2 possible values, i.e., to the characters. The
integer fJ is a parameter that constitutes the, radix or base of the system. If fJ = 10, and we
are dealing with the ordinary written numerals,
6
Xi
= '365' would signify that
Xo
= '5',
=
'6', X 2 = '3'. Note the numbering of characters from "right to left." The single quotation
marks are used to denote the fact· that we are considering the marks themselves and not the
numbers they denote.
Certain other properties are assigned to the fJ distinct kinds of characters. First of all we
require that a discrete cyclic order be established among the fJ characters. The red characters
on ,the telephone dial exemplify this for the' ordinary written system, fJ = 10. The cyclic '
order progresses counterclockwise aro~nd the dial. The cyclic successor of X will be denoted
by O'(x) • . Thus, on the dial, 0'('0') = '1', 0'('9') = '0', etc.' Iterations of the cyclic successor
operator will be denoted by exponents; for example, in the ordinary system, 0'5('0') = '5',
O'IO(X) = O'O(x) = x. In general, O'T(X) = O's(x) if and only if r == s (mod fJ). In addition to
the cyclic order, we single out a particular character and call it 'Nu'. On the telephone dial,
Nu appears immediately below the hook, i.e., in the ordinary system Nu = '0'. In modern
written Arabic, Nu = ' . '; as transmitted by the telephone dial, Nu is a closely spaced time
sequence of ten pulses. The telephone dial is a siIl?-ple, inexpensive, syntactical machine;
when properly manipulated, it transforms the '0' appearing in the directory into the requisite
pulses.
The idea of the.fJ > 2 characters, the cyclic order imposed on them, and the fiducial Nu,
are the elements of the development. A numerical expression is defined as any expression
Xl
I
with at least one character having the property that the first character is not Nu, i.e.,
Ax.
'/, = m
t
is a numerical expression provided m < n, and Xn =1= Nu. It will be useful to have a symbol
for the expression with no characters at all, and 'A' is selected for this purpose. The arch
'~' means 'is followeq by,' and capital letters will be used for expression variables, since
small letters are reserved for character variables. Note that a single character is always an
expression, but not conversely. To illustrate this notation:
A ..-. X = X = X"-' A.
We have defined the numerical expressions as the totality of expressions that begin with
a character distinct from Nu. This is a formation rule. In order to proceed further we define
a transformation rule, which enables the determination of the successor of a numerical expression. in the sense of Peano's axioms, that is to say, the operation of counting. We do this
I28
LOGICAL SYNTAX AND TRANSFORMATION RULES
with the aid of the auxiliary notion of quasi successor which is defined for any arbitrary
expression.
Definition l. Quasi successor (quasinachfolger), qnf (X).
(1) qnf (A)
=
a(Nu).
(2a) If a(z) =j::. Nu, then qnf (Y --- z)
=
Y --- a(z) ;
(2b) If a(z) = Nu, then qnf (Y --- z) = qnf (Y) --- a(z).
This is the syntactical formulation of the operation performed by a counter. The last digit
continually progresses through the cyclic order-note that (2a) and (2b) both terminate in
'a(z)-but if the last digit becomes Nu, it is necessary to perform the counting operation on
the expression formed by discarding the last digit; this is the carry. If when the last digit
is discarded nothing ,remains, a new a(Nu) (corresponding to '1') is prefixed. This assumes,
of course, that the counter has unlimited capacity. This transformation rule is a recursive
operation across the digits, from right to left.
On the basis of the above definition, we can prove
Theorem 1.
IfAxi
=j::. A a- 1 (Nu),
t=m
t=m
.
then t=m
AYi = qnfAxi
if and only if
t=m
rem < k < i) => (Xk = a- 1(Nu))],
(2) [Yi = Xi] <=> (3k) [em < k < i) . (Xk =j::. a- 1 (Nu))].
IffJ = 2, then Xi =j::. a- 1 (Nu) ifand only ifxk = Nu, and X k = a- 1 (Nu) ifand only ifxk = a(Nu)
(1) [Yi = a(x i )] <=> (k)
and hence lines (1) and (2) of Theorem (1) specialize to:
[em < k < i) => (Xk = a(Nu))],
[em < k < i) . (Xk = Nu)].
(I a) [Yi = a(xi)] <=> (k)
(2a) [Yi = Xi] <=> (3k)
The successor of a numerical expression. X is simply qnf (X), and it can be shown that the
qnf operation, thus restricted, satisfies all of Peano's axioms. The theory of operations on
natural numbers can be constructed from this transformation rule.
So far we have not considered the physical nature of the characters, nor how they are
physically strung together to form expressions; we have been considering questions of axiomatic
syntax. Suppose we have ,a language in which fJ = 2, and the chara~ters are represented by
two conditions of potential at a point ~ in an electric circuit. Suppose, ~(t) is the proposition:
the point ~ is at the higher of the two possible potentials at time t. We define X t = Nu and
X t = a(Nu) (there are only two char~cters) as follows:
Definition 2.
EXt = Nu] <=> rv
EXt = a(Nu)] <=>
12 9
~(t),
~(t).
GEORGE W. PATTERSON
vVe are now dealing with physical syntax, since the physical nature of the characters comes
into the picture. Since the characters must have unique immediate predecessors and successors, it will be necessary to quantize our time scale. In a synchronous machine, we assume
that t increases by constant increments. Suppose we have a black box with two inputs and
two outputs (Fig. I). The box has the property that 1] assumes the high potential if and only
if ~ and (J. are at different potentials, and A assumes the high potential ifand only if ~ and (J.
are both at the high potential. Such a device can be synthesized by methods developed by
Burkhart and Kalin, to be described in a forthcoming Harvard Computation Laboratory
publication. 4 As soon as we identify the states of the calculating mechanism with the characters
of the object language, analysis and synthesis of its behavior merges with the discipline of
physical syntax.
FIG. 2. Half adder adapted to
perform the qnf transformation.
FIG. 1. A half adder.
To return to the "black box," which is known as a half adder, its physical behavior is
described by:
(A)
fJ(t) <=> (\j [~(t) <=> (J.(t)],
A(t) <=>
[~(t)
. (J.(t)].
(B)
We define the character Y t by
Definition 3.
[Yt
[Yt
=
=
Nu] <=>
("\.J
1](t),
a(Nu)] <=> 'YJ(t).
Definitions 2 and 3 and statements (A) and (B) give:
(J.(t) <=> [Yt = d(x t )],
(\) (J.(t) <=> [Yt = x t ],
A(t) <=> [(x t = a(Nu)) . (J.(t)].
(C)
(D)
(E)
In other words, if (J. is at the high potential (usually a positive pulse), then the box carries
out the cyclic-successor transformation on each character, but if (J. is at the low potential,
then the box performs the identity transformation.
A delay is now inserted in the circuit (Fig. 2) and the inputs to (J. are so connected that
(J.(t) <=> [y(t) V K(t) 1. y(t) <=> t = (0), and K(t) <=> A(t - 1). Now,
if
26
then
AYi
= qnf0xi.
~ = 0
1= 0
Xi
=1=
26 a(Nu),
LOGICAL SYNTAX AND TRANSFORMATION RllLES
First, we note that for t
>
0, A(t) <=> (tk) [(0
< tk :::; t) =>
~(tk)]·
This is demonstrable by an inductive argument. Hence:
IX(t) <=> (t k) [(0 < tk < t) => (Xt" = a(Nu))]
and
<=> (3t k) [(0 < tk < t) • (Xt" = Nu)].
Replacing IX(t) and (\) IX(t) in (C) and (D) by their equivalents given above, we see by Theorem
rv IX(t)
1 (for f3 = 2) that this circuit performs the qnf transformation.
Another transformation rule will be described. All systems for expressing integers, even
Roman num~rals, must have a qnf transformation rule, but complementation is characteristic
of algorithmic systems, and has more syntactical than mathematical significance.
Definition 4.
f3 comp (aT(Nu))
= 0-<11- 1 )- T(Nu).
This defines the complement for a single digit. This is analogous (for
complement"; e.g., 10, comp ('0') = '9', 10 comp ('7') = '2'.
fJ =
10) to the "nine's
n
n
f3 comp ( ~
Xi') = ~
fJ comp (Xi)·
This definition extends f3 comp to expressions; fJ comp is a "linear operator". with respect to
"~" and is thus more simply mechanized than the so-called ten's complement; there is no
in teraction between characters.
The black box just described can also perform the complement transformation. The
peculiar property of binary systems that makes this possible is given. by
Theorem 2.
If f3 = 2, a- 1 (x) = a(x) = 2 comp (x).
From Theorem 2 and statements (C) and (D) we obtain:
IX(t) <=> [Yt = 2 comp (x t)],
("\) IX(t) <=> [Yt = Xt].
(F)
(G)
If we supply clock pulses to' IX, the circuit complements; if not, then it gives the identity
transformation.
Now consider the problem of forming the "ten's" complement. No special notation is
needed, since it is simply qnf (fJ comp (x)). The qnf indicates the necessity of carry mechanisms
when a ten's complement is formed; qnfis not a "linear operator." From the previous analysis
a complementer can be constructed by connecting two half adders in tandem (Fig. 3). All
that is required is to supply clock pulses at c5 and a starting pulse at y(t) simultaneous with the
appearance of the least significant digit at~. There is a superi~r method, and the basis for
it will now be derived.
The following theorem is easily proved from the preceding definitions and properties of
cyclic order:
GEORGE W. PATTERSON
Theorem 3.
Ifn > m, then
C61 Xi) ~ a(fJ comp (xm)),
If Xm = Nu, qnf (fJ comp ( 6 Xi)) = qnf (fJ comp C61 Xi)) ~ Xm•
(1) Ifx m =1= Nu, qnf (fJ comp C:~~Xi)) = fJ comp
(2)
From Theorem 1 follows
Theorem 4.
If
6
Xi =1=
6
Nu, n
>
m, then6Yi = qnf (fJ comp
(6 Xi)) if and only if:
< k < i) => (Xk = Nu)],
fJ comp Xi] <=> (3k) [(m < k < i) => (Xk =1= Nu)].
(1) [Yi = a(fJ comp Xi)] <=> (k) [(m
(2) [Yi =
If fJ = 2 these two lines specialize to:
(la) [Yi = Xi] <::=> (k) [(m
(2a) [Yi
(Xk = Nu)],
= 2 comp Xi] <=> (3k) [(m
<~
< i) . (Xk
= a(Nu))].
This theorem justifies the following ingenious circuit, invented by T. C. Chen.
£~~----------------~~-r-+
.~~
y~
FIG. 4. Circuit for transformation corresponding to the two's complement.
FIG. 3. A complementer.
.
The half adder is connected to a flip-flop as in Fig. 4. The gate-inverter combination
serves to transform the static output of the flip-flop to a string of positiv.e pulses, in order to'
permit a.c. coupling in the half adder. It will be noted that the positive pulses appear at
oc if and only if the flip-flop is set. The flip-flop is assumed to require one unit of time to change
its state, and the condition of the circuit immediately preceding the arrival of the character
Xo is I\) ~(- 1) and y(- 1) and we require that y(t) < . > t = - l . The circuit condition at
oc is then, for t > 0:
oc(t) <=> (3tk) [(0 < tk < t) . ~(tk)]'
r0 oc(t) <=> (tk) [(0 ~ tk < t) => I\)~(tk)] ..
Replacing
~(t)
by its syntactical equivalent from Definition 2:
I\)
oc(t) <=> (3t k) [(0 < tk < t) . Xt k = a(Nu)],
oc(t) <=> (t k) [(0 < tk < t) => (Xtk = Nu)].
Combining this with statements (F) and (G) we obtain
[Yt
[Yt
= 2 comp (x t )]
=
<=> (3tk) [(0 < tk < t) .
Xt] <=> (t k) [(0 < tk < t) =>
13 2
(Xtk
(Xtk =
= Nu)].
a(Nu))],
. LOGICAL SYNTAX AND TRANSFORMATION RULES
Thus Theorem 4 applies and we have proved that the circuit carries out. the transformation
rule (for {J
=
2) corresponding to the two's complement, provided that
6
Xi
=1=
6
Nu.
The circuits analyzed are rather elementary, but these methods provide a link between
the physical properties of the equipment components and the object language, as well as a
method of describing and analyzing the interrelations between the transformation rules of the
object language. This method is capable of being extended to cover more complex situations
which are at present difficult to investigate except by our sometimes fallible intuition.
Special Notation
X"-"Y,
n
the expression formed by adjoining the expression Y to the expression X.
~ Xi'
the expression formed by adjoining the characters
that order).
a(x),
the cyclic successor of the character x.
ar(x),
the rth iteration of the cyclic successor operator.
Xn) X n - 1, ••• ,Xm + 1, Xm
together (in
<=>, if and only i£
(k),
for every (integer) kl
=>,
if, then.
(3k),
for some (integer) kl
rv
not· ..•.
. and·
v
. or .
(or both).
Lower-case letters are character variables; upper-case letters are expression variables;
lower-case Greek letters are statement variables referring to voltage levels.
REFERENCES
1. R. Carnap, Introduction to semantics (Harvard University Press, 1942), p. 1.
2. F. Klein, Elementary mathematics from an advanced standpoint, vol. 1, Arithmetic, algebra, ana{ysis,
tr. by E. R. Hedrick and C. A. Noble, pp. 21-22.
3. R. Carnap, The logical syntax of language (Harcourt, Brace, 1937), p. 1.
4. Staff of the Computation Laboratory, Synthesis of electronic computing and control circuits
(Harvard University Press, ·1951).
133
FOURTH SESSION
Wednesday, September 14, 1949
2:00 P~M. to 5:00 P.M.
NUMERICAL METHODS
Presiding
Raymond C. Archibald
Brown University
NOTES ON THE SOLUTION OF LINEAR SYSTEMS
INVOLVING INEQUALITIES
GEORGE W. BROWN
Rand Corporation
Consider the problem of minimizing a linear function
~AiiX,
j
>
~biXi
subject to the conditions l
i
C1 ,
1, 2, .. " ml
=
j = 1, 2, .. " m2
Notice at the outset that equalities may be admitted in this form by writing each equality as
two inequalities with reversal of signs. Furthermore, the problem may be reformulated so
that only inequalities of the form Xi > 0 are present, by defining appropriate new variables.
Thus it is evident that the above form is simply one standard version of a general problem
involving both inequalities and equalities.
In principle the solution of the problem stated is trivial. Observe that the set of inequalities
defines in m2-space a convex polyhedron (possibly empty) with at most m1 + m2 faces of
dimension m2 - 1, and that the minimum problem is that of finding an extreme point of the
polyhedron in some direction. In general, the extremum will be taken on at a vertex, so the
problem is that of evaluating ~biXi at the vertices and choosing that vertex which yields the
smallest value. A vertex is of course a point at which a subsystem (of rank m2 ) of the inequalities
is satisfied exactly as equalities, with the remaining inequalities satisfied. In principle, then,
one could invert all subsystems of rank m2 , throwing out those whose solutions fail to satisfy
the remaining inequalities, and then evaluate ~biX;, It is clear that this is not a practical·
method beyond the smallest values of m1 and m2 • The. practical difficulties stem from the
fact that the convex polyhedron is specified ~y its faces, whereas the vertices are at the root
of the problem.
.
In passing, it should be noted that the problem stated above has a very simple dual problem, obtained by transposing the matrix A, and making a few other obvious changes. The
dual is the problem of maximizing ~CiYi subject to the conditions
i
=
1, 2, .. " m1
j = 1,2, .. " m2
The two dual problems have the property that if either problem has a solution so has the
other, and the minimum valuein one is the maximum value in the other. In certain economic
applications the solutions of both problems are required.
Consider now the problem of maximizing min ~~iAii subject to ~i > 0, ~~i = 1, i = 1,
j
2, . . " ml ; and the dual problem of minimizing max
i
137
~Aii1];
subject to
1]1
>
0,
~1]; =
1,
GEORGE W. BROWN
i =
1, . . ., m2 • This problem provides optimum mixed strategies for the zero-sum game
with matrix A, where Aii represents the payment from player I to player 2, if player 1 plays
his ith strategy and player 2 plays his ith strategy. The celebrated minimax theorem of von
Neumann says that under the conditions stated
Max Min ~~iAii
;
j
=
Min Max ~Aii1'Ji.
~
i
The common value is referred to as the value of the game and the {~i} and {1'Ji} of the solutions
are the optimum mixtures for players I and 2, respectively. As in the first problem stated in
this paper, geometrical considerations of convex bodies contribute to an understanding of
the problem, and it turns out that in general the problem is practically solved if it is known
which submatrix of A to invert.
There is of course an intimate relation between the theory-of-games problem and the
problem first stated, although they are not quite identical problems, since the game problem
always has a solution, while the first problem does not necessarily. To summarize briefly,
the game problem is directly a special case of the first problem, while the first problem can
always be embedded in a game problem, whose solutions yield solutions to the original problem
if it has a solution. Thus, if problems of one type can be solved, so can problems of the other
type.
Various iterative methods for solution of one or the other of these problems have been
given by von Neumann, Dantzig, and others. While some of these methods may be practical
over a certain range of problems, all of them have an apparent dependence, in required number
of steps, of higher order than the first power of the linear dimensions of the problem. For very
huge matrices not possessing simplifying special properties, such a dependence can be a very
serious obstacle in the way of getting numerical solutions. We will describe briefly, for the
game solution, an iterative scheme which is quite different from those previously suggested,
in that the amount of calculation required at each iterative step is directly proportional to
the linear dimensions of the problem, so that the method has, a priori, some chance of beating
the high-order dependence.
The procedure to be described can most easily be comprehended by considering the
psychology of, let us say, a statistician unfamiliar with the theory of games. Such a person,
faced with repeated choices of play of a certain game, might reasonably be expected to play,
at each opportunity, that one of his strategies 'which is best against past history, that is, against
the mixture constituted by his opponent's plays to date. Such a decision utilizes informati0n
of the past in the most obvious manner. The iterative scheme referred to here is based on a
picture of two such statisticians playing repeatedly together. For purposes of calculation a
slight modification is introduced which has the effect that the two players choose alternately,
rather than simultaneously.
Restating the method algebraically, let A be the game matrix, let in and in be the nth
choices of strategy for the two sides, ~nd let ~i(n) and 1'J/n ) be the relative frequencies of strategies
i and j in (iI' i2, . . ., in) and (j1' i2' ., .. , in), respectively; then in minimizes ~~/n)Aii and
i
'
~
I
2
3
I
3
1.1
1.2
2
1.3
2
0
3
0
I
4
2
1.5'
FIG.
0
10
j=l
j=2
1
2
1.3
2
3
1.3
3
1
4.3
3.1 .
t.1
1. A 4 X 3 matrix.
j=3
Yo
2
0
0
3.1
3
1.2
3
3.1
.65
2.1
1
4.2
4.1
4.3
1.37
1. '1'1
2
5.3
Vo
jo
1=1
1=2
1=3
1=4
0
3.1
1.1
1.3
3.1
3.1
3.3
4.1
4.6
6.1
4
1
7.3
~
5.5
1.30
1.60
2
~
5.3
5.1
5
1
10.3
6.3
6~7
1.26
1.52
2
'1.5
'1.3
6.1
7.6
6
4
12.3
1&
7.8
1.30
1.55
2
8.6
9.3
'1.1
9.1
7
2
13.6
9.8
!&
1.11
1.46
3
9.8
9.3
~
10.2
8
3
13.6
10.8
10.9
1.35
1.46
2
10.9
11.3
11.2
11.'1
9
4
15.6
12.3
12.0
1.33
1.59
3
12.1
11.3
14.3
12.8
10
3
15.6
13.3
15.1
1.33
1.53
2
13.2
13.3
15.3
14.3
11
3
15.6
!!J
18.2
1.30
1.48
2
14.3
15.3
16.3
15.8
12
3
15.6
15.3
21.3
1.28
1.44
2
15.4
1'1.3
17.3
17.3
13
2
16.9
17.3
21.3
1.30
1.48
1
18.4
18.6
1'1.3
19.3,
14
4
18.9
!Y
22.4
1.34
1.49
2
19.5
20.6
18.3
20.8
15
4
20.9
gQJ
23.5
1.35
1.51
2
20.6
22.6
19.3
22.3
18
2
22.2
22.3
23.5
1.39
1.52
1
23.6
23.9
19.3
24.3
24.7
~
20.3
25.8
23.4
26.9
1'1
4
24.2
~
24.6
1.40
1.52
2
18
2
25.5
25.8
~
1.37
1.49
3
25.9
25.9
19
4
2'1.5
2'1.3
25.'1
1.35
1.47
3
27.1
25.9
26.5
28.0
28.3
25.9
29.6
29.1
20
4
29.5
28.8
~
1.34
1.48
3
21
3
29.5
29.8
29.9
1.40
1.49
1
~
2'1.2
.29.6
31.1
22
1
32.5
30.9
31.1
1.40
1.48
2
32.4
29.2
30.6
32.6
33.6
29.2
33.7
33.7
23
4
34.5
32.4
32.2
1.40
1.47
3
24
3
34.5
33.4
35.3
1.39
1.47
2
34.'1
31.2
34.7
35.2
25
4
36.5
34.9
36.4
1.40
1.47
2
35.8
33.2
35.7
36.7
FIG.
2. Cumulative payoffs.
139
GEORGE W. BROWN
in+I maximizes
~Aii1]/n>.
3
This process defines a sequence iI' J~, i 2, j2' . . ., once i l is chosen
(perhaps arbitrarily), except for possible ambiguities of the extrema. A~y convenient rule
will do for handling ambiguities. If En = min ~;iAii and Vn = max ~Aii1]/n>, it is easily
.
.
j
i
seen that En < V < Vm where V is the value of the game. The mixtures {;i} and {1]/n>}
are mixed strategies, and the corresponding f~ and Vn are the most favorable outcomes ensured
to each player if he uses the corresponding mixture.
,
At this moment not much is rigorously established about the properties of this iteration,
except that ifit converges at all it converges to a solution of the game for each side. Of course
it would be sufficient if lim· sup En = lim inf Vn. There is considerable support, however,
based on experience with the method, and also on the study of a related system of differential
equations, for the conjecture that convergence is of the order of lin and does not depend
essentially on the size of the matrix. If this is so, it is extremely important for the solution of
large matrices, by virtue of the fact that each iterative steps requires only a number of operations proportional to the linear size of the matrix. Convergence of order 1In is of course painful
if high accuracy is needed. In such cases it may be possible, however, to uSe a method like
this to get close to the sblution, .finishing 'with one step of another iteration.
Figure 2 is a worksheet showing 25 steps carried out for the 4 X 3 matrix given in Fig. 1.
Note that each line is obtained by adding to the previous line, compone~t by component, the
corresponding row or 'column of the matrix, without troubling todivide by n. The En and
Vn were calculated at each step, by division of the extrema by n, to· show the progress of the
calculation. In case of ties the lowest index was taken. Note particularly that Vn - fn is
decreasing just about like lin, in spite of the excursions which Vn and En make. The initial
choice of il = 2 was made deliberately as an unfavorable choice, with respect to minimum
guaranteed payoff.
It is appropriate to report to this Symposium that preliminary discussions with' Messrs.
Harr and Singer, of the staff of the Harvard Computation Laboratory, indicate that Mark III
could carry out 1000 of these iterative lines for a 40 X 40 matrix in comfortably under an
hour. Of course the problem has not beel?- completely programmed, but the estimate is believed
to be conservative.
REFERENCE
1. The theoretical background of this. paper is based on work of H. Weyl, von Neumann,
Ville, Tucker,G. Dantzig, and others, on convex polyhedra and on the theory of games.
MATHEMATICAL METHODS IN LARGE-SCALE COMPUTING UNITS
D. H. LEHMER
University
of California
The title of this paper covers such a vast subject that it will be impossible to do it justice.
In fact, this title might well have been chosen as that of the whole session. My aim is merely
to discuss in a general way certain features of the mathematics that is characteristic of the
. large-scale computing unit. In pointing up these general remarks I shall discuss in considerable
detail only one problem. Further illustrations will be contained, no doubt, in the other papers
of this session.
The mathematical methods available to a computing unit depend of course on the versatility of the unit. Nearly all units can perform addition, subtraction, ·multiplication, and division. The advent of large-scale digital cOrhputers has added a fifth operation of considerable
importance, namely, discrimination. This, in general terms, is the operation of making a
choice of one of several branches of a program (or course of procedure), depending on the
outcome of a previous calculation. This operation is peculiar to discrete-variable machines,
since its outcome is not continuous. The purely analogue machine cannot distinguish the
larger of two sufficiently small numbers, or determine the sign of either. This fact was recognized early in ,the construction of roulette wheels. By converting the wheel into a discretevariable device countless arguments were avoided.
The ability of a discrete-variable machine to discriminate and thus to. decide for itself
what course of action to take has led to the popular misconception that such machines think
or even have the ability to learn from experience.
Various criteria are employed in discrimination. Decisions are made according to whether:
(1)
(2)
(3)
(4)
(5)
(6)
A given number is > 0 or < 0 (Harvard Mark I) ;
A given number is 0 or not (ENIAC);
A given number is odd or even (ENIAC);
A given sequence of numbers has one sign pattern or another (Bell Telephone);
The sum of two numbers exceeds the capacity of the machine or not (Zephyr);
A given number belongs to one ofa set of residue classes with respect to a given modulus
(Electronic sieve).
These criteria are not independent and others can be constructed from them. All digital
machines are capable of some form of discrimination and those named above are given only
as examples.
The mathematical methods that call for much discrimination are very frequently iterative
141
D. H. LEHMER
ones. .Here discrimination is used to decide whether or not to continue to iterate. Another
simple use of discrimination is in forming the nonanalytic function Ixl. More elaborate uses
arise in the step-by-step solution of differential equations and of course still more in problems
of combinatorial analysis and number theory. Incidentally, the electronic sieve is designed
to make 10 million discriminations per second.
Another feature of mathematical methods that are being used in large-scale· computing
is that they tend to eliminate the elaborate formulas and to introduce instead what might·
be called combinatorial complexities. This is due for the most part to the high speed of operation. For instance, in using a quadrature formula for numerical integration it does not pay
to use the accurate Weddle's rule; it is often simpler and even faster to employ the crude
. trapezoidal rule. As far as I know the· superb method of Gauss for mechanical quadrature
has never been used in large-scale work. The method of Heun is used much more frequently
than the more accurate and complex Runge-Kutta method for the step-by-step solution of
ordinary differential equations. Minima of functions are found by extensive numerical trialand-error methods, rather than by the somewhat more sophisticated and traditional method
of setting derivatives equal to zero and solving. Systems of many first-order differential
equations are solved in lieu of single differential equations of high order. A large number of
trial solutions of differential equations with one-point boundary conditions may be made in
order to obtain a single solution of a two-point boundary problem. Solving problems in terms
of special functions is passe; finite-difference methods are used instead. The power-series
expansions of analytic functions are being ·used to a large number of terms and to a great
accuracy in order to avoid the use of alternative asymptotic expansions.
All these examples show how mathematical subtleties are being replaced by stepped-up
numerical activiti'es. To make this replacement possible the operator naturally must surrender
.much of his control to the machine itself. He simply cannot follow the course of the numerical
work with sufficient rapidity to make on-the-spot decisions as to what to do next. This means
that the programmer may have to incorporate a large number of discriminations or branches
in the program of the problem. Much has been said, but little written, about the logic or
even the topology of programming. Logicians and topologists are not coming to the rescue
of the desperate programmer. General rules for programming have been discovered. Most
of them have been used in the Kansas City freight yards fora long time. This is the combinatorial complexity to which I have referred. Flow diagrams showing the routines, subroutines,
and other wheels within wheels are hardly distinguishable from the block diagrams of the
machine itself; the latter, however, are made once and for all. This then is the white man's
burden of large-scale computing.
The third characteristic feature of discrete-variable methods is the possibility of introducing
number theory into what at the outset appears to be a· problem in continuous functions. By
way of illustration, let me call attention to a method which is the subject of the last paper
142
METHODS IN LARGE-SCALE UNITS
of this session-the Monte Carlo method. In this method it is necessary to produce random
variables. The problem here is not one of producing a table of random digits to be published
and used by others. On the contrary, one can think ideally of a perfect stream of these random
numbers produced at high speed by the machine and passing by a "gate." Whenever the
computer needs a number it opens the gate and takes one. More explicitly, we might list
the following desiderata:
(1) An unlimited sequence of randomly arranged eight-digit numbers;
(2) A simple process by which the machine may produce the sequence:
(3) Immediate access by the machine to the current number of the sequence whenever
necessary; of course, the whole sequence need not be retained in the machine.
Ifwe examine these desiderata we see at once that they are inconsistent. In the first place,
the number of eight-digit numbers is not unlimited. There are in fact only 100 -million of
them. Secondly, condition (2) forces the sequence to be ultimately periodic and therefore.
not random. We therefore scale down our demands and modify (1) and (2) to read:
(1) Millions of pseudo-random numbers
(2)
Un
Un;
= !(Un-l),! a simple function.
A pseudo-random sequence is a vague notion embodying the idea of a -sequence in which
each term is unpredictable to the uninitiated and whose digits pass a certain number of tests
traditional with statisticians and depending somewhat on the uses to which the sequence is
to be put. The worst possible departure from randomness is to have the period of the sequence
small or equal to one. In constructing the functionj, therefore, it is of the utmost importance
to obtain one that produces a guaranteed proper period of immense length.
A method already in use on the ENIAC, due to von Neumann and Metropolis, is the
following. Let Uo be an arbitrary initial eight-digit number. Then U1 is defined as the central
-block of eight digits in the square of uo, and U 2 is defined as the same function of U1 that U1 is
ofuo, etc. -At first sight this would appear to give an ideal source of random numbers. Certainly _is produces an unpredictable sequence of numbers. However, as has been pointed out already
by several writers, this process cannot be expected to give random numbers. In fact, one must
expect. to obtain numbers Un of the form xyzwOOOO before many more than 10,000 numbers
Un are generated. When this happens, either w = 0 and Un + 1 = 00000000 and all succeeding
u's vanish, or w =1= 0 and all succeeding u's are of the form xy'z'w'OOOO, where w' = 1, 5, or 6.
Hence periodicity will set in in fewer than 3000 more steps. Also, one must expect to obtain
numbers of the form Un = OOOOxyzw, in which case Un + 17' Un + 18, • • • , all vanish. _Thus it is
seen that this process cannot be recommended as a source of random digits. It has an additional drawback in that it ties up the multiplier, which is a fairly busy component of any
machine.
If we look at the problem from the standpoint of the theory of numbers, it is not difficult
143
D. H. LEHMER
to find a more satisfactory solution. We may proceed as follows. We begin as before with an .
arbitrary nonzero initial eight-digit number Uo. Next we compute 23uo• In general this will
be a ten-digit number. The ninth and tenth digits (counting from the right) are now removed
and subtracted from the remaining eight-digit number. This produces U1 • The next number
U 2 is produced from U1 in the same way. To illustrate in detail,suppose that the initial Uo is
47594118 (chosen at random from a wastepaper basket of punched cards). Then
20u o = 9
51882360
42782354
23u o = 10
subtract 10
94664714
- 10
U1
=
94664704
As in the first method, this process is necessarily ultimately periodic. In fact, it is actually
periodic of period 5882352. This fact makes all the difference; the reason for it is simple.
In computing Un from Un -1 we are computing the remainder of 23u n _Ion division by 10 8
1.
Hence, in congruence notation
n
Un == Uo 23
(mod 108
1).
+
+
+
1
By the theory of the binomial congruence, Un is periodic of period 5882352 since 10 8
= 17·5882353. The number 23 is the best possible choice in the sense that no other number
produces a longer period, and no smaller number produces a period more than half as long.
As set .up for the ENIAC, for example, the ·process would tie up only two accumulators
and would produce 5000 pseudo-random digits per second. The process would have a period
of 2 hr 36 min 52 sec.
Whether such a set of digits or a reasonable subset satisfies the statisticians' tests for randomness is of course another question. To investigate this matter I have secured the kind
cooperation of Professor L. E. Cunningham of the Astronomy Department of the 'University
of California, who set up the calculation on the IBM calculating punch 602A. This produced
the first 5000 u's (that is, 40,000 digits in all) in about 4 hr (the ENIAC would be faster by
a factor of 1800). One of the secondary .reasons for making the calculation was to test the
accuracy of the 602A. It may distress some and surprise others to know that any isolated
Un can be computed on a desk calculator in 3 min~ Thus U sooo was known in advance. The
fact that this value agreed with the result obtained in 5000 steps is a rigorous check of the
arithmetic unit of the 602A.
Once produced, the results were subjected to four standard tests with the assistance of
Dr. Evelyn Fix and other members of the Statistical Laboratory of the Univers~ty of California.
All four tests were passed successfully. In case anyone is convinced that the numbers Un are
really random, I should like to call his attention to the fact that each .number Un is a multiple
of 17.
For a binary machine a similar process can be set up with respect to a modulus of the form·
144
METHODS IN LARGE-SCALE UNITS
2n ± 1. For example, with the Mersenne prime 231 - 1 as modulus,' more than 66 billion
pseudo-random binary digits can be generated.
The method is based on the function f(x) = ax. A very little extra complication would
be produced by using the general linear functionf(x) . ax + b. However, nothing is gained
by this generalization, since the period is independent of b, as one can see from the theory of
difference equations. With machines capable of parallel operation like the ENIAC and SSEC,
the above process is especially advantageous for two reasons: (a) it can be incorporated into
the program with very little expense, and (b) the "gating" of this routine at irregular time
intervals serves further to randomize the sequence Un. The serial-type machine would simply
use the numbers U m one after another.
I have gone into such great detail on this problem just to indicate how a problem in, let
us say, nuclear physics, when attacked by a large-scale computing unit, can involve a mathematical method taken from the impractical theory of numbers.
It is only fair to point out that, conversely, large-scale digital equipment can be used to
study certain problems in the theory of numbers. In fact, it is sometimes a little exasperating
for the number theorist to assist the applied mathematician in juggling round-off errors,
truncating errors and a flitting decimal point in order to adapt a problem in fluid mechanics
to a discrete-variable machine when all the time the machine, being digital, is· all ready to
work onclean.. cut problems involving whole numbers. However, I realize that this exasperation is shared by very few present. Most of you will be relieved to know that, to the best of
my knowledge, very little valuable time on large-scale computing units has been spent on
such unprofitable problems.
In fact, to date, only one small problem of this sort has been solved and published, and
another is making slow progress. However, I hear that the University of Manchester's new
computing machine is being used on such problems and doubtless there will be some interesting
results published before long.
It maynot be out of place to mention certain kinds of problems for which no mathematical
method would seem to be available in order to apply large-scale computing units in a practical
way. By an impractical application we mean one thatproduces results no more rapidly than
a few hand computers using desk calculators. Since the large-scale machines are based on the
four rational operations they have good control over functions that are defined by algebraic
expressions. However, mathematics abounds with functions that are defined verbally, often
in some negative way. Such functions are apt· to give trouble if they cannot be expressed
directly in terms of operations with which the machine is familiar. Simple examples of such
functions occur in the theory of numbers, algebra; topology, statistics, organi~ chemistry,
genetics, and elsewhere. Often these functions are of the enumerative sort. For .example,
one can ask for the number of none qui valent maps of 135 countries, or the number of different
ways that each map can. be colored in five colors. If ten permutations of the digits 0, 1, . . . , 9
145
D. H. LEHMER
are selected at random, what is the probability that they form a Latin square? If a(n) denotes
N
the number of prime factors of n, is the sum L; (- l)a(n) negative for 1000 O? Suppose that o~e obtains, still obeying the rules of the game, ina noticeable
proportion ex. of tries, a situation A where only, say, ten cards are left uncovered; after that,
however, we meet with "failure." It might be justifiable to restore the ten cards to their
positions in a different permutation and try from the situation A again. By examining a large
number of the 10! permutations we might obtain the number {J expressing the chance that.
starting with A we "win" B.A reasonable g~ess for the chance of success from the beginning
without "cheating" would then be greater than ex.{J. Of course A should be really a class of
positions, not a possible or a very special one. It seems, however, that if the playing of the
. whole game is decomposed into two brmore stages, there will be a saving in the number of
experiments compared with the number necessary to play to the end each time and beginning
anew after each failure from the start, that is, a new permutation of the 52 cards.
The validity of such a procedure can be established in some cases. One has to prove
independence, or estimate from above the correlation between the classes of events A and
success B.
It is of .course obvious that one can study. "experimentally" the behavior of solutions of
equations which themselves describe a random process, by using the digital computer as an
analogy machine,. as it were. l This experimental-that is, statistical-approach by Monte
+
208
MONTE CARLO METHOD
Carlo techniques has been applied .by various authors to linear partial differential equations. 2
In the case of equations that are quadratic or of higher order in the unknown functions and
their derivatives, the obvious Monte Carlo procedure would be much more cumbersome, but
may still have heuristic value. As an example, let us take a bilinear system of two partial
differential equations
where UI and U2 are unknown functions of coordinates x,y, z, and t; OCI and OC 2 are given constants; f31 and f32 are given functions, for simplicity linear in UI and 112 and also involving the
independent variables x, y, z. One would like to know the asymptotic form of UI and U2 (for
large values of t). This problem may be looked upon as a straightforward generalization of
the diffusion model (Fermi) of the Schrodinger equation.! It would correspond to a model
of a system of two particles with the potential function for U i replaced by the corresponding U
function of the other particle~ This linked &ystem, treated then. somewhat in the spirit of a
field theory, is nonlinear. There will not be in general eigenfunctions-the separation Into
a time-independent equation will not be possible; yet for large values of the parameter t the
space part of U may have an almost periodic or summable (by the first mean) behavior. A
numerical approach to the study of such systems could again be a Monte Carlo procedure.
One would diffuse and multiply the (fictitious) particles corresponding to UI ane. U2 according
to their numbers, instead of a given function V of coordinates. Since these numbers change
in t, it will be necessary to make frequent censuses-as it were, to interrupt the calculation
periodically-in order to ascertain the values to be used for "potentials."
The problem of transforming first purely formally, an equation not of a diffusion or Boltzmann type into one of the above type thus becomes of practical importance. Let us indicate
some possibilities in this direction. The equation of Hamilton-Jacobi in one dimension has
the form
1
(1)
dX = 2 (x) .
(dS)2 v
On the other hand, consider the equation
dW _ ~ [. d(VW)]
dt. - dX v dX
.
(2)
This latter equation will describe the probability behavior of a particle starting, say, from the
origin, and performing a random walk on the line, steps being equally probable to the right
or to the left. However, the length of the steps in the position x is proportional to the value
of v(x). If we perform the passage to the limit with the length of the step tending to zero the
resulting continuous process gives a distribution of position in time t obeying Eq. (2). It can
be proved 3 that the crest of the distribution,. that is, the place x where d Wldt = 0, will satisfy
a relation S(x) = ti, where S is the solution of Eq. (1).
20 9
S. ULAM
It is of course quite unnecessary to take recourse to such methods for a one-dimensional
equation that is easily solved explicitly bY' quadratures. The example here given is meant
merely to indicate the possibility of relations between two seemingly very different processes.
One is a strictly deterministic one, described by the equation of geometric optics (or the
equations of mechanics), the characteristic equation of Hamilton. The other is a continuous
random-walk process with the length of the elementary step a given function of position. It
turns out that at least in one dimension the locus of the points where the first derivative with
respect to time of the probability distribution is equal to zero coincides with the locus of the
points where the value of the Hamilton function S = vi. In two or more dimensions, the
two loci are probably at least asymptotically equal, that is, for large values of t.
In the first examples of application of the Monte C?arlo method to empirical evaluation
of properties of solutions of differential equations, one studied the density of the diffusing and
branching, that is, multiplying and transmuting, particles. This density as a function of the
independent variables obeyed a linear partial differential equation of a parabolic or elliptic
type. It is clear thatJor nonlinear equations one will have to examine, not this density directly,
"but appropriately chosenfunctionals of this function.
The diffusion process can be described, of course, as a Markoff chain, and this in turn by
a study of the interaction of matrices with nonnegative coefficients. Let us indicate a way to
study "experimentally" the behavior of powers of matrices with arbitrary real terms. This
possibility rests on the fact that real numbers cap. be considered as matrices, with positive
terms; for example, - 2 corresponds to
(~~) ..This correspondence obviously preserves both
addition and multiplication. Anysystem described by any n-byo:-n matrix giving the transition
moments as real numbers can be interpreted probabilistically by using 2n-by-2n matrices with
nonnegative terms. The diffusion a"nd branching or multiplication are performed by two
kinds of particles-black and red-with the transformation rates given by the matrices
above~
. In having four kinds of particles one can then realize stochastic models for matrices with
complex terms; more generally, with an appropriate number of kinds of particles, one can
realize 'stochastic models for more general algebras over real numbers. 4
The possibility of a statistical or probabilistic evaluation of definite integrals in n-dimensional space affords merely one example of an attempt to gain insight into a situation involving
a system of n particles. Let us think here of n as having a va]ue of the order of 10 or 20. The
"appearance" of a set of points in a euclidean space of this dimension, if the set is defined as
above by many inequalities, cannot of course be studied on graphs directly, or very well by
projections of the set into three-dimensional component spaces. Now, in physical chemistry,
for instance, the occurrence of this situation and its importance are well known. The properties
of a molecule with a large number of atoms depends on characteristics of configurations of
certain n-dimensional sets. The evaluation of various functionals of these configurations can,
probably, be done best by a Monte Carlo procedure, that is, by testing a large number of
210
MONTE CARLO METHOD
n-tuples, chosen at random with appropriate distribution, for the values of these given
functionals.
It is rather curious that one meets with an analogous situation in pure mathematics itself.
Let us describe it very briefly: a formal system in mathematics involves in addition to the
Boolean operations of elementary logic or set theory (the addition and intersection of sets of
points), the so-called quantifiers, the two symbols ~riod.
Extensive computations are often needed, not only in finding theoretical results to check
with experiment, but also in finding out what interpretation to give to the experimental
results themselves. Elaborate calculations on the functioning of an instrument may be required
to 'obtain data that throw light on important questions of fundamental theorY. Professor
Vallarta'spaper discusses the interpretation of observations from a huge ,instrument, the earth
itself acting as a magnetic spectrometer.
The subject of nuclear physics, on which Prqfessor Feshbach reports, is one in which the
wish of the physicist for simple explanation has suffered repeated rebuffs. Nuclear str~cture
differs from atomic structure in such ways that much less can be expected from semiqualitative
arguments. Some of the high-energy scattering data indicate that the basic phenomenon of
the so-called saturation of nuclear forces will have to be explained in a rather complicated
way. In this snqjec( the number of different hypotheses that may need to be tested, as well
as the number of separate problems, is so great that only automatic computation seems capable
of progressing fast enough.
Besides the problems of testing theories, interpreting observations, and deciding between
various hypotheses, there are questions how far existing theories are capable of accounting
for certain kinds of phenomena. For example, it seems to be generally agreed that nonrelativistic quantum mechanics accounts for atomic and molecular structure and for many facts
in the structure of solids, but the question may be raised how far it suffices to cover all of this
last field. Are the striking phenomena of superconductivity to be accounted for as statistical
effects of the ordinary electri(:al and quantum laws, or do they require the introduction of
some perhaps radically new though pleasingly simple assumption? Probably authorities in
this field have definite opinions on such questions, but they can scarcely be really certain
21 7
WENDELL H. FURRY
about them. The difficulty of such statistical calculations is so great that automatic calculation,
for finite but rather extensive lattice structures, may eventually be called into use.
In summary it can be said that there are questions ~n theoretical physics with which automatic computation has nothing to do, and on the other hand there is ,a potentially very great
service in the provision of function tables, which would be universally welcomed. There are
also many questions in the testing of. theories, in the interpretation 0[' observations, in the
choice between hypotheses, and in establishing the range of adequacy of theories, for which
automatic computation could be extremely useful. The settling of such questions would
generally be helpful, and in some cases probably indispensable, in the advancement of
theoretical physics.
218
DOUBLE REFRACTION OF FLOW AND THE. DIMENSIONS OF
LARGE ASYMMETRIC MOLECULES
HAROLD A. SCHERAGA
Cornell University
JOHN T. EDSALL
Harvard Medical School
and
J. ORTEN GADD, JR.
Computation Laboratory of Harvard University
Optical measurements of the double refraction produced when a solution oflarge asymmetric
molecules is subject to a shearing force can be used to determine the dimensions of the dissolved
molecules. I - 4 The method has already been extensively applied,5-10 but· many of t~e data.
obtained or obtainable could not heretofore be interpreted because the theory had been
developed to give numerical values only under certain limiting conditions. The present work
was undertaken in order to extend the applicability of the theory to a much wider range of·
experimental conditions, thus greatly increasing the usefulness of flow-birefringence measurements as a tool in the determination of particle sizes and the characterization of polydisperse
systems of macromolecules ..
Double refraction is produced, in a liquid contammg large asymmetric molecules or
colloidal particles, when a velocity gradient is set up in the liquid. This is most readily achieved
by forcing the liquid through a capillary tube, or by subjecting it to shear between two concentric cylinders, one of which rotates while the other is held fixed. The latter procedure is
best for quantitative measurements, and was employed in 1870 by J. Clerk Maxwell, who
was apparently the first to describe the phenomenon, using Canada balsam ~s the liquid for
study. This is also the method that has been adopted in most studies on double refraction
of flow. 1-16
In the concentric-cylinder type of system, the liquid is placed in the annular space between
the cylinders, the suspended particles assuming random orientation when both cylinders are
at rest, as shown in Fig. 1 (a). When one of the cylinders, say the outer one, is set in rotation,
laminar flow is produced in the liquid and a velocity gradient is set up across the gap.17 The
resulting shearing forces produce an orientation of the. suspended particles, which are here
assumed to be rigid ellipsoids of revolution. 1s This orientation is represented schematically
in Fig. 1 (b). If the cylinders are mounted between crossed Nicol prisms, where AA and PP
represent the planes of transmission of the analyzing and polarizing Nicol, respectively, then
SCHERAGA, EDSALL, AND GADD
the field appears dark when the cylinders are at rest and, when one cylinder is rotating,
becomes light in all regions except for a dark cross [Fig. 1(b)], the "cross of isocline."
To characterize the observed phenomena, ~here are two quantities that must be measured:
(1) the extinction angle X, the smaller of the two angles between the cross of isocline and the
planes of transmission of the Nicols (this is also the angle between the optic axis in the flowing
liquid and the direction of the streamlines); (2) the magnitude of the double refraction f).n,
that is, the difference in refractive index between light transmitted with its electric vector.
parallel, and light with its electric yector perpendicular, to the optic axis. The problem is
to measure' X and f).n ~s functions of the velocity
p
gradient and relate them to the dimensions of the
suspended particles.
Empirically it is found, for solutions containing
iu\-r'i~-A
a
single
type of large molecule, that X approaches
A45° as the velocity gradient G approaches zero, and
approaches 0° asymptotically as G increases to very
large values, provided the flow is laminar; f).n is
p
zero when G = 0, is a linear function of G at low
FIG. 1..Orientation of particles, each
values of G, and gradually approaches a constant
schematically represented by a line
limiting
saturation value at very high values of G.
indieating its optic axis, in a doubly
refracting liquid between concentric Very elongated particles, like those of myosin (the
cylinders, when the outer cylinder is structural protein of muscle) or tobacco-mosaic virus,
(a) at rest, (b) in motion., The lines AA which are seve~al thousand Angstrom units in length,
and PP are the axes of crossed Nicol
prisms. [Adapted from von Muralt and give low. values of X and high values of f).n even at
low velocity.gradients (G = 10 to 300 sec- I in water
Edsall, J. Biol. Chem.89, 315 (1930).]
which has a viscosity of 0.0 1 poise at 20° C). Molecules, like those of human serum albumin and gamma globulin, which are near 200 A in
length, require much higher velocity gradients and solvent media of high visc.o<)ity as well
to .attain a significant degree of orientation (G = 1000 to 10,000 sec-I, or more; viscosity
50 to 100 times that of water).
Colloidal particles or large asymmetric molecules in the flowing solution are subject to
shearing forces due to the velocity gradient G, which tends to orient their major axes. I9 In
addition to the hydrodynamic forces, the particles are suqject to rotary Brownian move~ent
which causes a random fluctuation of the orientation. The Brownian movement is characterized by a rotary diffusion constant 0 .. The relation of X and f).n to the molecular dimensions
has been developed chiefly by Boeder,20 Peterlin and Stuart,3, 21, 22 and Snell man and Bjornstahl. 4 This is expressed as a function of the parameter rt. (or (J in the notation of Peterlin and
Stuart), which is equal to G/0. If 0 is known, the .length of the semimajor axis a of the
molecule can be evaluated.1- 4, 23, 24 The crux of the problem, therefore, is the determination
of 0 from the experimental measurements of X and G.
220
DOUBLE REFRACTION OF FLOW
The double refraction !::..n is the product of an optical factor which is evaluated independcntly 22 and an orientation factor J. Like' the extinction angle X,! is a function of (f.. and '
the axial ratio p [= ajb].
We shall further define the quantity 25 R,
R
p2 - 1
= p2 + 1;
(1)
R is thus equal to unity for an infinitely thin rod (ajb = 00); to zero for a sphere (ajb = 1);
and to - 1 for a flat disk without thickness (ajb = 0).
Peterlin and Stuart obtained expressions for X and! in terms of slowly converging infinite
series in (f.. and p. At very low values of, (X « 1.5), corresponding to X values between 45°
and 38°, these series converge sufficiently rapidly to enable one to evaluate the rotary diffusion
constant from a simple limiting equation. However, the errors in the experimental data are
generally greatest at low velocity gradients, that is, at low values of (X. The data are more
accurate at somewhat higher gradients, but it has not been possible hitherto to evaluate from
theory the numerical relation between X and (X under these conditions. Moreover, it is very
important experimentally to determine whether a given solution under study contains only
one or more than one constituent capable of orientation by the velocity gradients employed.
The only way to be sure of this is to make measurements over a wide range of velocity gradients
and compare the measured X values with values calculated from the appropriate theoretical
curve. 26 However, since only a small portion of the theoretical curve is given by the limiting
formulas of Peterlin and Stuart, this method of analysis could not be satisfactorily carried out.
A semiempirical method has been tried 9 but it was considered essential to have the complete
theoretical curves using a rigid ellipsoid of revolution as a molecular model. 18 If-these were
available it would be possible not only to infer whether only a single type of elongated molecule
is present but also, ifseveral such components are present, to draw some important inferences
concerning their relative sizes and concentrations in the solution. An observed X value, in
such a multicomponent system, is a function of all the values of both X andf that would be
found for each of the components, if it were present in the solution alone, at the same velocity
gradient. The ability to analyze such complex systems would greatly increase the range and
power of the method of double refraction of flow.
We shall, therefore, present the Peterlin and Stuart theory 3, 21, 22 wherein ~e have evaluated
the quantities required to obtain X and f values over a wide range of (X values by the use of
the Mark I computer of the Harvard Computation Laboratory.
If rigid ellipsoidal particles are suspended in a continuous medium under conditions of
laminar flow, a steady-state distribution will be established very rapidly. This distribution
will dep~nd on (X and R and is characterized by a distribution function F which, in the steady
state, is given by the differential equationja7
.
!::..F
1 + R cos 2cp (JF
2
(Jcp
+
R sin 0 cos 0 sin2cp (JF _ 3R sin 2 0 sin 2cp F
2
(JO
2
.
221
(2)
SCHERAGA, EDSALL, AND GADD
The meaning of 0 and cp may be understood by reference to Fig. 2, which is a section of
the gap between the concentric cylinders, the inner one rotating. Here X is the direction of
the streamlines at 0; Z is the direction of the velocity gradient between the concentric'
cylinders; Y is parallel to the cylinder axis and is normal to the streaming plane; x, y, z
are the directions of the principal axes of the index-of-refraction ellipsoid of the birefringent
system; and X is the angle between the z- and X-axes, x and z being coplanar with X and Z.
An individual particle at 0 has its major axis in the ~ direction, where ~, 'YJ, ~ are a set of axes
fixed in the particle. Then 0 is the angle between Y and ~,. while cp is the angle between the
YZ- and Y~-planes. This is the usual notation of spherical coordinates with volume element
dO. = sin OdOdcp.
z
Fig. 2. Part of the coordinate system in the Couette
cylinder apparatus. The X, Y, Z axes are fixed in the
fluid and the x, y, z axes are the principal axes of the
index-of-refraction ellipsoid in the birefringent system.
[From Peterlin and Stuart, Z. ,Physik 112, 1 (1939).]
As will be· shown later, the determination of X and f involves the evaluation of certain
mean values. The distribution function F is required for this purpose and is evaluated as
follows.
Express F as a power series in R,
00
F=
2: RiF
j=O
(3)
j•
Each Fj then satisfies an inhomogeneous equation of the type
! J1Fi _
ex
J
=!
1.
2
dFj
dq;
[cos 2cp d~~_l+ sin 0 cos 0 sin 2cp
d~il -
3 sin 2 0 sin 29.' . F j _ 1] .
(4)
Now F j rJay be expressed in terms of series of spherical harmonics as
00
Fj =
00
n
! 2:anO,jP2n + 2: 2:(anm,j cos 2mcp + hnm,j sin 2mcp)P2n2m,
n=O
n=l m=l
222
(5)
DOUBLE REFRACTION OF FLOW
where
is a Legendre polynomial of the first kind, and
P2n
P2n
2
' •
m
2
= sm
'0
m
d2mP2n
•
(d cos 0) 2m
Since FJ is a function of (X. and is independent of R, the annl,j and bnm,j coefficients will also
,have this dependency.
Substituting from Eq. (5) in Eq. (4) and making use of the orthogonality and recurrence
relations for 'these polynomials,28 one obtains the following recurrence formulas for anm,j and
bnm,j:
n(2n
+ 1) a
,= _ 1 [ _
nO,)
(X.
'4
+
+
(2n - 3)(2n - 2)(2n - 1)2n(2n
(4n - 3)(4n _ 1 ) ,
3(2n - 1)2n(2n + 1)(2n
(4n - 1)(4n + 3)
+ 1) b
'
11-1,1;)-1
+ 2) b
'
n,I;)-I
+ I) (2n + 2) (2n + 3) (2n + 4) b
(4n + 3)(4n + 5)
n+
2n(2n
,],
1,1;)-1
(6)
,
(7)
2n(2n
+ 1)
(X.
'_
1 [
anm,j, + mbnm,j - -
3
4
2n + 1
(4n- 3)(4n- 1) bn- 1,m-1;j-1
~
(4n - -I)(4n
+ 3) bn,m-l;j-1 -(4n + 3)(4n +5) bn+l ,m-1;j-1
_ (2n- 2m- 3)(2n- 2m- 2)(2n- 2m- 1)(2n- 2m)(2n
(4n - 3)(4n - 1)
3(2n - 2m - 1)(2n - 2m)(2n + 2m + 1)(2n
(4n - 1)(4n:+ 3)
+
+,
2n(2n
- manm,j
+
+-
+ 2m +
1)(2n
2m
'(4n
2n(2n
+ 1)
(X.
,3
- (4n- 1)(4n
_
bnm,j -
'4
"
n-l,m+1,)-1
+ 2m + 2) b
•
n,m+l;j-l
+ 2)(2n + 2m + 3)(2n +,2m + 4) b
+ 3)(4n + 5)
1 [
+ 1)b
.
2n + 1
(4n _ 3)(4n _ 1) an-1,m-1;j-1
+ 3) an,m-l;j-l ~ (4n +
2n
3)(4n
]
n+ 1,m+ 1;j-1
(m
* 0),
+ 5) an +1,m-1;j-1
-
(2n - 2m - 3)(2n - 2m - 2)(2n - 2m - 1)(2n - 2m)(2n + 1)
(4n - 3)(4n - 1)
an -1,m+1;j-1
+
3(2n - 2m - 1)(2n - 2m)(2n + 2m + 1)(2n
,
(4n - 1)(4n + 3)
2n(2n
+ 2m +
+,
1)(2n
(8)
+ 2m + 2)
, an ,m+1;j-1
+ 2m + 2)(2n +2m +- 3)(2n + 2m + 4) a
,'_ ]
(4n + 3)(4n + 5)
,
n+1,m+1,) 1
(m
223
* 0).
(9)
SCHERAGA, EDSALL, AND GADD
Normalization, that is, putting SFdQ ~ l,gives aoo,o = Ij27T and all other aOO,j = O. The
complete distribution function is given by Eq. (3) after the F/s are thus evaluated .
The evaluation of the coefficients anm,j and bnm,j as solutions of the simultaneous Eqs. (6-9)
is a formidable task and would have been hopeless without the aid of the Mark I computer.
It should be pointed out that not all the coefficients are required for the problem of double
refraction of flow, but, as has been shown by Peterlin and Stuart, only the all,j and bll,j terms.
However, many other terms are required in order to evaluate these particular ones.Th~ task
is. somewhat eased by the vanishing of many of these terms for certain values of the indices.
Making use of the distribution function F of the particles it is possible to calculate the
effect of the interaction of this oriented system with a beam of polarized light, that is, the
double refraction. Results of such 'a computation are
tan 2X =
2 cos (~X) cos (~Z)
cos (~X) - cos (~Z)
=====--'---========
2
2
(10)
and
/).n
=
27TC (gl - g2)'V[COS 2 (~Z) - cos 2
n
(~X)J2 + [4 cos (~Z) cos.(~X)J2
(11 )
where C is the concentration of the particles, n is theindex of refraction of the' isotropic solution
at rest, and (gl - g2) is an optical factor depending on the axial ratio and indices of refraction
of the particles. 22 Since cos (~X) = sin () sin cp, and cos (~Z) = sin () cos cp, it follows that
(~X)
cos
(~Z) =
(~Z)-
cos 2
(~X)
2 cos
and
cos 2
-l sin 2cp . P22
= 1 cos 2cp . P22.
, (12)
The mean values of these functions are evaluated by multiplying by F and, integrating, giving 29
00
1
""
L. Ri- b11,J'
- tan 2X
;=1
(13)
and
(14)
Heretofore, these equations have been useful for valid computation only in the following
limiting forms, which hold for rJ. < 1.5,
X=
f(rJ.,R)
24R2)
"4 - 12 1 - 108 1 + -35
7T
rJ.
rJ.R [ 1 = 15
By machine computation of the
all,j
[
rJ.2
(
rJ.2 (6R2)
72
1
35
+
+.. . .] ,.
]
+ ....
( 15)
(16)
and bll ,; terms, the sums that appear in Eqs. (13)
224
DOUBLE REFRACTION OF FLOW
and (14) have been evaluated for values of ex up to ex = 200. At these high ex values X has
fallen practically to zero from the initial value of 45° at ex = O. However,Jis still significantly
far from its saturation value Joo which it would have at ex = 00. The convergence of these
series is much more dependent upon the rate of decrease of the values of the all,; and bu ,}
terms, as j increases, than upon the decreasi!lg values of RJ-l, especially since R approaches
0.5
45
40
.=0.4
2o
o
U. 0.3
c:
25
c:
o
.~ 20
U
c:
2
15
W
0 •2
c:
1<
Q)
10
~
00.1
10
15
25
20
30
35
10
Pa ram e t e r, a .
FIG. 3. Extinction angle X as a function
of the parameter ex, together with its
depe~~nce on the axial ratio p.
~
30
35
FIG. 4. Orientation factor f as a function.
of the parameter ex, together with its
dependence on the axial ratio}.
o
~ 0.20
26.0
0
U
c:
20
E o... ~
26.5
Q)
01
c:
1). However, the curves
for an oblate ellipsoid of axial ratio lip are identical with those of a prolate ellipsoid of axial
225
Table 1. Extinction angle X as a function of Cl for various axial ratios p.
to
to
0'>
~
1.00
1.25
1.50
1.75
2.00
2.25
2.50
3.00
3.50
4.00
0.00
0.25
0.50
0.75
45.00
43.81
42.62
41.44
45.00
43.81
42.62
41.44
45.00
43.81
42.62
41.44
45.00
43.81
42.62
41.44
45.00
43.81
42.62
41.44
45.00
43.81
42.62
41.44
45.00
43.81
42.62
41.44
45.00
43.81
42.62
41.44
45.00
43.81
42.62
41.44
45.00
43.81
42.62
41.44
45.00
43.81
42.62
41.45
1.00
1.25
1.50
1.75
40.27
39.12
37.98
36.87
40.27
39.12
37.99
36.as
40.27
39.12
38.00
36.S9
40.28
39.13
38.01
36.91
40.28
39.14
38.02
36.92
40.28
39.14
38.02
36.93
40.28
39.14
38.03
36.94
40.29
39.15
38.04
36.96
40.29
39.16
38.05
36.97
40.29
39.16
38.05
36.98
40.29
39.16
38.06
36.99
2.00
2.25
2.SO
3.00
35.78
34.72
33.69
31.72
35.79
34.74
33.71
31.74
35.81
34.76
33.74
31.00
35.84
34.80
33.79
31.87
35.86
34.82
33.82
31.93
35.88
34.85
33.86
31.98
35.89
34.87
33.88
32.03
35.91
34.CX)
33.93
32.09
35.93
34.92
33.96
32.14
35.94
34.94
33.98
32.17
35.95
34.95
33.99
32.19
3.50
4.00
4.SO
5.00
29.87
28.16
26.57
25.10
29.91
28.21
26.64
25.18
30.00
28.32
26.78
25.36
30.09
28.45
26.94
25·55
30.18
28.56
27.08
25.73
30.25
28.66
25.88
30.31
28.75
27.31
26.01
30.41
28.87
27.47
26.20
30.47
28.95
27.58
26.33
.30.52
29.02
27.66
26.42
30.55
29.06
27.71
26.49
6.00
7.00
8.00
9.00
22.50
20.30
18.43
16.84
22.61
20.44
18.60
17.03
22.85
20.74
18.94
17.40
23.ll
21.05
19.31
17.81
23.35
21.35
19.65
18.20
23.55
21.60
19.94
18.53
23.72
21.81
20.20
18.81
23.98
22.13
20.57
19.25
24.16
22.36
20.84
19.54
24.29
22.51
21.02
19.75
24.38
22.63
21.16
19.90
10.00
12.50
15.00
17.50
15.48
12.82
10.CX)
9.46
15.67
13.03
11.11
16.53
13.97
9~67
16.08
13.47
11.57
10.12
12.09
10.64
16.95
14.45
12.60
11.16
17.31
14.87
13.05
ll.63
17.62
15.23
13.45
12.05
18.09
15.80
14.07
12.72
18.42
16.19
14.52
13.20
18.66
16.48
14.84
13.55
18.83
16.68
15.07
13.81
20.00
22.50
25.00
30.00 .
8.35
7.46
6.75
5.66
8.55
7.65
6.92
5.81
8.98
8.06
7.32
6.17
9.49
8.57
7.00
6.61
10.02
9.08
8.30
7.08
10.50
9.56
8.78
7.54
10.93
10.00
9.22
7.98
11.62
10.71
8.71
12.13
11.25
10.50
9.28
12.51
ll.64
10.91
9.72
12.79
11.94
11.22
10.06
35.00
40.00
45.00
SO.oo
4.86
4.27
3.00
3".42
5.00
4.39
3.91
3.53
5.32
4.68
4.17
3.76
5.73
5.05
4.51
4.08
6.17
5.46
4.90
4.43
6.61
5.as
5.29
4.80
7.03
6.28
5.67
5.17
7.76
7.00
6.37
5.84
8.34
7.58
6.95
6.41
8.80
8.04
7.41
6.87
9.15
8.41
7.78
7.25
60.00
80.00
100.00
200.00
2.86
2.14
1.72
0.86
2.94
2.21
1.77
0.89
3.15
2.37
1.90
0.95
3.42
2.58
2.07
1.04
3.73
2.82
2.27
1.14
4.06
3.09
2.49
1.26
4.39
3.36
2.71
1.38
5.00
3.88
3.15
1.62
5.54
4.34
3.56
1.84
5.99
4.75
3.90
2.04
6.36
5.08
4.20
2.21
27~21
9~95
4.50 .
~
~
~
~
5.00
6.00
7.00
8.00
9.00
10.00
12.00
16.00
25.00
50.00
CD
0.00
0.25
0.50
0.75
45.00
43.81
42.62
41.45
45.00
43.81
42.62
41.45
45.00
43.81
42.62
41.45
45.00
43.81
42.62
41.45
45.00
43.81
42.62
41.45
45.00
43.81
42.62
41.45
45.00
43.81
42.62
41.45
45.00
43.81
42.62
41.45
45.00
• 42.62
41.45
45.00
43.81
42.62
41.45
45.00
43.81
42.62
41.45
1.00
1.25
1.50
1.75
40.29
39.16
38.06
36.99
40.29
39.16
38.06
37.00
40.30
39.17
38.07
37.00
40.30
39.17
38.07
37.00
40.30
39.17
38.07
37.00
40.30
39.17
38.07
37.01
40.30
39.17
38.07
37.01
40.30
39.17
38.07
37.01
40.30
39.17
38.07
37.01
40.30
39.17
38.07
37.01
40.30
39.17
38.08
37.01
2.00
2.25
2.50
3.00
35.96
34.96
34.01
32.21
35.97
34.97
34.02
32.23
35.97
34.98
34.03
32.25
35.98
34.98
34.04
32.26
35.98
34.98
34.04
32.ZT
35.98
34.99
34.04
32.ZT
35.98
34.99
34.05
32.28
35.98
35.00
34.05
32.28
35.99
35.00
34.05
32.29
35.99
35.00
34.06
32.29
35.99
35.00
34.06
32.29
4.00
4.50
5.00
3.~
30.58
29.09
ZT.75
26.54
30.61
29.14
ZT.81
26.61
30.63
29.17
27.85
26.65
30.64
29.19
27.f!l
26.68
30.66
29.20
27.88
26.70
30.66
29.21
ZT.89
26.71
30.67
29.22
27.91
26.73
30.68
29.23
27.93
26.75
30.69
29.24
27.94
26.71
30.69
29.25
27.95
26.71
30.69
29.25
27.95
26.78
6.00
7.00
8.00
9.00
24.45
22.71
21.26
20.02
24.54
22.8)
21.39
20.18
24. ill
22.90
21.48
20.27
24.63
22.94
21.53
20.34
24.66
22.98
21.57
20.38
24.68
23.00
21. ill
20.41
24.70
23.03
21.64
20.46
24.73
23.06
21.68
20.50
24.75
23.09
21.70
20.53
24.76
23.10
21.72
20.55
24.76
23.11
21.73
20.55
10.00
12.50
15.00
17.50
18.96
16.84
15.25
14.00
19.13
-17.05
15.49
14.27
19.24
17.18
15.64
14.J¥.
19.31
17.ZT
15.74
14.55
19.36
17.33
15.81
14.63
19.39
17.37
15.86
14.68
19.44
17043
15.93
14.76
19.49
17.49
16.00
14.84
19.53
17.54
16.05
14.89
19.54
17.56
16.08
14.92
19.55
17.56
16.09
14.93
2O.OQ
22.50
25.00
30.00
]J.OO
12.16
1l.46
10.32
13.29
12.48
11.79
10.68
13.47
12.67
12.00
10.91
13.59
12.00
12.14
11.07
13.68
12.90
12.24
1l.18
13.74
12.97
12.31
1l.26
13.82
13.05
12.41
11.37
13.90
13.14
12.50
11.48
13.97
]J.21
12.58
11.57
14.00
13.24
12.62
11.61
14.01
13.26
12.63
11.62
35.00
40.00
45.00
50.00
9.43
8.70
8.08
7.54
9.81
9.11
8.50
7.98
10.07
9.38
8.79
8.28
10.24
9.57
8.98
8.48
10.36
9.70
9.12
8.63
10.45
9.79
9.Z3
8.74
10.57
9.92
9.36
8.88
10.69
10.06
9.51
9.03
10.78
10.16
9.62
9.15
10.83
10.21
9.68
9.22
10.85
10.23
9.69
9.23
60.00
80.00
100.00
200.00
6.66
5.36
4.45
2.35
7.10
5.78
4.8)
2.58
7.41
6.08
5.09
2.74
7.62
6.28
5.28
2.85
7.78
6.43
5.41
2.93
7.89
6.55
5.52
2.99
8.05
6.70
5.66
3.08
8.20
6.86
5.80
3.16
8.33
- 6.98
5.92
3.Z3
8.40
7.05
5.98
3.27
8.42
7.08
6.00
3.28
43~81
Table 2. Orientation factor f as a function of IX for various axial ratios p.
K)
IV
co
~
1.00
1.25
1.50
1.75
2.00
2.25
2.50
3.00
3.50
4.00
4.50
0.00
0.25
0.50
0.75
0.00000
0.00000
0.00000
0.00000
0.00000
0.00366
0.0C!729
0.01089
0.0000
0.0064
0.0128
0.0191
0.0000
0.0085
0.0169
0.0252
0.0000
0.0100
0.0199
0.0298
0.0000
0.0112
0.0223
0.0332
0.0000
0.0121
0.024l
0.0359
0.0000
0.0133
0.0266
0.0397
0.0000
0.0141
0.0282
0.0421
0.0000
0.0147
0.0293
0.0437
0.0000
0.0151
0.0301
0.0449
1.00
1.25
1.50
1.75
0.00000
0.00000
0.00000
0.00000
0.01-443
0.01791
0.02129
0.02458
0.0253
0.0314
0.0373
0.0430
0.0334
0.0414
0.0492
0.0568
0.0394
0.0489
0.0581
0.0671
0.0440
0.0546
0.0649
0.0749
0.0476
0.0590 0.0701
0.0809
0.0525
0.0651
0.0774
0.0893
0.0557
0.0691
0.0821
0.0947
0.0!)19
0.0718
0.0853
0.0984
0.0595
0.0737
0.0876
0.10lD
2.00
2.25
2.50
3.00
0.00000
0.00000
0.00000
0.00000
0.02776
0.03082
0.03376
0.03925
0.0486
0.0540
0.0591
0.0687
0.0641
0.0712
0.0779
0.0906
0.07!)1
0.0840
0.0920
0.1069
0.0845
0.0938
0.1027
O.ll93
0.0913
0.lD13
O.llO9
0.1288
0.1007
0.ill8·
0.1223
0.1421
0.1069
0.1185
0.1297
0.1507
0.1110
0.1231
0.1348
0.1565
0.1139
0.1264
0.1383
0.1(l)6
3.50
4.00
4.50
5.00
0.00000
0.00000
0.00000
0.00000
0.04422
0.04868
0.05266
0.05620
0.0774
0.0852
0.0922
0.0984
0.1020
0.1123
0.1216
0.1299
0.1204
0.1326
0.1436
0.1534
0.13440.1480
0.1(l)2
0.1712
0.1451
0.1598
0.1730
0.1849
0.1(l)1
0.1763
0.1909
0.2041
0.1697
0.1869
0.2024
0.2165
0.1762
0.1941
0.2103
0.2249
0.1809
0.1992
0.2158
0.2308
6.00
7.00
8.00
9.00
0.00000
0.00000
0.00000
0.00000
0.06211
0.06674
0.07038
0.07326
0.1089
0.ll72
0.1238 .
0.1291
0.1438
C.1550
0.1640
0.1714
0.1700
0.1835
0.1945
0.2035
0.1900
0.2052
0.2178
0.2282
0.2053
0.2220
0.2358
'0.2474
0.2268
0.2456
0.2613
0.2746
0.2407
0.2609
0.m8
0.2923
0.2502
0.2712
0.2891
0.3044
0.2568
0.2786
0.2970
0.3129
10.00
12.50
15.00
17.50
0.00000
0.00000 .
0.00000
0.00000
0.07557
0.07961
0.08213
0.08378
0.1334
0.1411
0.1461
0.1495
0.1774
0.1886
0.19(l)
0.2012
0.2111
0.2253
0.2351
0.242l
0.2370
0.2539
0.2658
0.2746
0.2573
0.2764
0.2902
0.3005
0.28(l)
0.3086
0.3254
0.3383
0.3048
0.3299
0.3487
0.3635
0.3176
0.3444
0.3649
0.3810
0.3267
0.3548
0.3764
0.3936
20.00
22.50
25.00
30.00
0.00000
0.00000
0.00000
0.00000
0.08492
0.08573
0.08634
0.08714
0.1518
0.1536
0.1549
0.1567
0.2050
0.2078
0.2099
0.2130
0.2473
0.2812
0.2864
0.2905
0.2966
0.3085
0.3148
0.3199
0.3276
0.3485
0.3568
0.3637
0.3744
0.3755
0.3853
0.3936
0.4069
0.3942
0.4052
0.4147
0.4299
0.4077
0.4196
0.4299
0.4467
35.00
40.00
45.00
50.00
0.00000
0.00000
0.00000
0.00000
0.08764
0.08797
0.08820
0.08836
0.1!)18
0.1585
0.1591
0.1595
0.2149
0.2163
0.2172
0.2179
0.2619
0.2640
0.2656
0.2667
0.3~
0.3Dro
O.~
0.3331
0.3371
0.3400
.0.3424
0.3823
0.3883
0.3930
0.3967
0.4169
0.4248
0.4311
0.4362
0.4418
0.4513
0.4589
0.4653
0.4599
0.4707
0.4796
0.4871
60 •.00
80.00
100.00
200.00
0.00000
0.00000
0.00000
0.00000
0.08858
0.08880
0.08891
0.08904
0.1600
0.1(l)5
0.1(l)7
0.1611
0.2189
0.2199
0.2203
0.2209
0.2683
0.2699
0.2707
0.2718
0.3100
0.3125
0.3138
0.3155
0.3456
0.3492
0.3509
0.3535
0.4021
0.4082
0.4l14
0.4l61
0.4438 .
0.4528
0.4576
0.4649
0.4750
0.4868
0.4933
0.5034
0.4987
0.5132
0.5213
0.5342
0~2513
0.2544
0.2589
0.3038
X
5.00
0.00
0.25
0.50
0.75
0.0000
0.0154
0.0)C17
0.0458
·1.00
1.25
1.50
1.75
0.Dro6 .
2.00
2.25
0.ll61
0.1287
3.00
3.SO
7.c:x,>
8.00
9.00
10.00
12.00
16.00
25.00
SO.OO
CX)
0.0000
0.0158
0.0314
0.0469
0.0000
0.01.60
0.0319
0.0476
0.0000
0.0161
0.0322
0.0480
0.0000
0.0162
0.0324
0.0483
0.0000
0.0l63
0.0325
0.0486
0.0000
0.0164
0.0327
0.0489
0.0000
0.0165
0.0)29
0.0492
0.0000
0.0166
0.0331
0.0494
0.0000
0.0166
0.0332
0.0495
0.0000
0.0167
0.0332
0.0496
0.0621
O.C1769
0.0914
0.1053
0.0630
0.C1781.
0.0927
0.1069
0.0636
0.0788
0.0936
0.1000
0.0640
0.0793
0.0942
0.1087
0.0643
0.C1797
0.0947
0.1092
0.0647
0.0802
0.0953
0.1098
0.0651
0.0807
0.0958
0.ll05
0.0654
0.0811
0.0963
O.lllO
0.0656
0.0813
0.0965
0.lll3
0.0656
0.0813
0.0966
0.1114
O.ll~
0.1636
0.~9
0.1319
0.1443
0.1676.
0.1206
0.1338
0.1464
0.1700
0.1218
0.1351
0.1478
0.1716
0.1226
0.1359
0.1488
0.1727
0.1231
0.1366
0.1494
0.1735
0.1239
0.1374
0.1503
0.1745
0.1246
0.1382
0.1512
0.1756
0.1252
0.1388
0.1519
0.1764
0.1255
0.1392
0.1523
0.1768
0.1256
0.1393
0.1524
0.1769
5.00
0.1842
0.2029
0.2198
0.2:351
0.1887
0.2078
0.2251
0.2408
0.1914
0.2108
0.2284
0.2444
0.1932
0.2128
0.2,306
0.2467
0.1945
0.2l42
0.2321
0.2483
0.1954
0.2152
0.2331
0.2494
0.1965
0.2165
0.2.345
0.2509
0.1977
0.2177
0.2359
0.2524
0.1986
0.2l87
0.2370
0.2536
0.1991
0.2192
0.2376
0.2542
0.1992
0.2194
0.2377
0.2544
6.00
7.00
8.00
9.00
0.2617
0.2839
0.3028
0.3191
0.2681
0.2910
0.3106
0.3275
0.2721
0.2954
0.3153
0-3:326
0.2747
0.2983
0.3185
0.3360
0.2765
0.3003
0.3206
0.3383
0.2778
0.3017
0.3222
0.3399
0.2795
0.3036
0.3242
0.3421
0.2812
0.3054
0.3262
0.3443
0.2825
0.3069
0.3278
0.3460
0.2832
0.3076
0.3286
0.3469
0.2834
0.3079
0.3289
0.3472
10.00
12.50
15.00
0.3334
0.3624
0.3848
0.4028
0.3423
0.3726
0.3962
0.4152
0.3477
0.3788
0.4032
0.4229
0.3513
0.3830
0.4C178
0.4279
0.3538
0.3858
0.4110
0.4314
0.3556
0.3879
0.IJ.33
0.4340
0.3579
0.3906
0.4163
0.4373
0.3603
0.3933
0.4193
0.4406
0.3621
0.3953
0.4216
0.4431
0.3630
0.3964
0.4228
0.4444
0.3633
0.3968
0.4232
0.4449
0.4177
2.SO
4.00
4.50
t~
~
c.o
6.00 .
17.50
0.C1751
0.0892
0.1029
20.00
22.50
25.00
30.00
0.4302
0.4412
0.4592
0.4311
0.4445
0.4564
0.4761
0.4393
0.4533
0.4659
0.4867
0.4448
0.4591
0.4721
0.4937
0.4486
0.4632
0.4765
0.4985
0.4513
0.4661
0.4796
0.5020
0.4549
0.4699
0.4837
0.5066
0.4585
0.4737
0.4878
0.5113
0.4612
0.4766
0.4910
0.5148
0.4626
0.4782
0.4926
0.5166
0.4631
0.4787
0.4932
0.5173
35.00
40.00
45.00
50.00
0.4736
0.4854
0.4952
0.5037
0.4921
0.5055
0.5168
0.5267
O.9J37
0.5181
0.5304
0.5413
0.5114
0.5265
0.5395
0.5511
0.5167
0.5324.
0.5459
0.5580
0.5206
0.5366
0.5505
0.5630
0.5257
0.5422
0.5565
0.5696
0.5308
0.5478
0.5626
0.5763
0.5347
0.5521
0.5673
0.5814
0.5367
0.5543
0.5698
0.5841 .
0.5374
0.5551
0.5706
0.5850
60.00
80.00
100.00
200.00
0.5169
0.5338
0.5434
0.5588
0.5425
0.5632
0.5753
0.5952
0.5590
0.5826
0.5966
0.6199
0.5701
0.59(D
0.6ll3
0.6372
0.5780
0.6055
0.6219
0.6498
0.5838
0.6125
0.6298
0.6592
0.5914
0.6218
0.6403
0.6718
005.991
0.6314
0.6511
0.6051
0.6389
0.6596
0.6954
0.6083
0.6429
0.6642
0.7010
0.6094
0.6442
0.6657
0.7029
0.6850
SCHERAGA, EDSALL, AND GADD
ratio p. This ratio enters into Eqs. (13) and (14) only through the function R. From Eq. (I)
it is clear that when p changes to lip, R changes sign: R(p) -:- - R(llp). However, in the
summations I;Ri-1 an ,j and I;Ri- 1bu ,j, only the a and b coefficients for whichj is odd have values
different from zero. Hence, only even powers of R enter into the summations, from which
it follows that X(rx,p) = x(rx,l/p) and f(rx,p) = - f(rx,l/p).30 These relations were clearly
recognized by Peterlin· and Stuart22 (see their figures for X andf as functions of rx), but were
not explicitly stated by them. From Eq. (13),31
.
- bUl
6
hm tan 2X = - - ' =-.
R~O
The equation lim tan 2X
a~O
=
=
all,!
rx
61rx, which had already been derived by Boeder20 for the case of
.
1), is the same as Eq. (15) as rx approaches zero.
Thus in all cases, when rx approaches zero and
45
the degree of o!ientation beco~es very small, f
40
approaches
zero and tan 2X = 61rx~ This holds for
CD 35
'0
all
p
values
and, in the limiting case where p'
.a 30
approaches I-that is, the ellipsoid approaches a
.~ 25
U 20
sphere-these relations hold also for all values of rx.
.~
;B 15
If experimental values of X are plotted as a
10
function of ~n, the data may be fitted by a theoretical
5
curve of X as a function of ~f, where k is an adjust0.5
0.6
0.1
0.2
0.3
0.4
able constant used to fit the data to the curve.
Orientation Factor
When determined, it gives the optical factor
FIG. 7. Extinction angle as a function (gl - g2)' since
of orientation factor, together with its
.dependence on the axial ratio p.
thin rods (R
If this procedure is adopted, it is unnecessary to extrapolate to low values of rx where the
experimental errors are large. 9, 10 The nature of the curve for X as a function of f and its
dependence on p are shown in Fig. 7.
Some question may arise as to the validity of the Peterlin and Stuart solution, especially
since the viscosity problem, treated by Peterlin with the same distribution function, is at
variance with the results of Simha,32, 33 whose treatment is considered to be the valid one. 34
The Simha treatment of viscosity is also identical with that of Kuhn and Kuhn 35 for low
gradient. However, the disagreement in the viscosity theories does not arise from the use of
an incorrect distripution function,36 but rather from Peterlin's omission .of certain terms in
the hydrodynamic equations, which were taken into account by Simha. Thus, the inadequacy
of the viscosity theory in no way affects the valid use of this distribution function for the
treatment of double refraction of flow.
If the omitted terms were taken into account, Peterlin's viscosity treatment would presumably be valid. During the course of the computation in connection with the present problem
DOUBLE REFRACTION OF FLOW
numerous other coefficients besides the aU,j and bU,j terms were evaluated. The availability
of these additional terms may be of use for a viscosity theory formulated on a similar basis.
The first and crucial step in the organization of this problem for machine computation
was an analysis of the recurrence formulas (6-9) and the following generalizations applying
to them:
1. All terms with negative indices are zero;
2. anm,O = 0, and aOO,j = 0, except aoo,o = Ij2r.;
3,. bnm,o = bnO,j = bOm,j = 0.
A rigorous analytical study seemed to be out of the question. Therefore several decisions
were made arbitrarily. Since the limited internal storage capacity of the Mark I computer
was one of the chief obstacles to be over- m
come, it was decided to consider each
o
o
o
o
o
o
o
24-column storage counter as two counters 8
of 12 columns each so that the a and b
o
o
o
o
o
o
o
7
terms for any nm,j coul,d be contained iq
o
o
o
o
o
o
o
one counter, all terms being carried to'ten 6
places of decimals.
o
o
o
o
o
o
o
It may be observed that the values of 5
all terms with an index ofji depend entirely
o
o
o
o
o
o
o
4
upon terms with the index ji-I. Thus the
o
o
o
o
o
o
o
decision was made to assign a storage 3
counter to each nm combination, to put in
o
o
o
o
o
o
o
these counters the anm,O and bnm,o terms, to 2
compute fronl these the anm,l and bnm,l terms
o
o
o
o
o
o
o
and transfer these results into the same set
of counters, then to compute the anm ,2 and 0 ~--o---o----o----o-_~-o----o---o--o
234
5
6
7
8
bnm~2 terms, and so on. This led to the
adoption of the viewpoint that values of
FIG. 8. Array of points in nm-plane.
points in the nm-plane (Fig. 8) were being computed at time "zero," "one," . . . , that is,
j = 0, 1, . . . There is, of course, a pair of values at each point, one value for the a term
and one for the b term.
It was hoped that a newly computed pair of values anm,j and bnm,j might immediately
replace the pair anm,j-l and bnm,j-l in the storage counter assigned to that particular nm combination. Unfortunately, this seemed impossible because the previous values are needed to
compute values at neighboring points, and it appeared that a second set of storage counters
would be required. However, further study of the recurrence formulas shows that the values
of all terms with the index m i depend only upon terms with m i - I or m i + 1 indices. Since it is
known that, for j = 0, the orily point at which either the a or the b term has a value is the
point 0,0 in the nm-plane, it is obvious that at time "one" (j = 1) values occur only in the
SCHERAGA, EDSALL, AND GADD
°
rowm = 1 and at time "two" only in the rows m =
and m = 2. Thus it is seen that for
evenj only the rows of even m, which depend only upon the rows of odd m, need be calculated,
and that for oddj it is necessary to compute only the odd rows in the m direction. This meant
that only one set of storage counters was needed, since a newly computed pair of values for
a point nm,j could immediately replace the values in storage for that point in the nm-plane,
which were values for nm,j - 2.
An inspection of the n indices shows that values can extend only one place farther in the
n direction with each increase of 1 in j. Since 0,0 is the only point with a value for j = 0,
it is clear that all anm,j and bnm,j are zero for n > j.
n -I
n
n +1
A parallel assertion holds for m > j, but this
om+1
om+1
m+I,
j -I'
j-I'
j-I
becomes inconsequential in view of the fact that,
in the present problem, it has been found unnecessary to compute the value of any point where
m > n. This is the same as saying that the points
to the left of the diagonal (Fig. 8) have no effect
o
o
upon the values at the point 1,1 in the nm-plane,
which is the only point at which results are required.
By the recurrence relations, the a and b terms at
any point in the nm-plane depend upon the previous
dme's terms at the three points centered directly
n-I
n
o~!1
above
it and the three centered directly below it
m-I
orn-I
/
j -I'
j-I'
j -I'
(Fig. 9). But, for any point on the diagonal (n = m),
FIG. 9. Illustration of recurrence
the coefficients applied to the first two points in
relations for a given term represented
in the nm-plane.
the upper row vanish. Likewise, for any point
just off the diagonal to the right (n = m
1), the
coefficient of the first term in the upper row becomes zero. Therefore, values to· the left of
the diagonal have no effect at any time upon any point on, or to the right of, the diagonal.
As a summary· of the results of this analysis, the following may be added to the initial
generalizations:
4. If m is even and j is odd, anm,j = 0, and bnm,j = 0;
5. If m is odd andj is even, anm,j= 0, and bnm,j = 0;
6. If n > j, or if m > j, anm,j = 0, and bnm,j· 0;
7. In this problem, terms where m > n need not be computed as they do not affect the
values of alI,jOr bn,j.
Considering these findings as well as the storage capacity of Mark 1, it was decided to
conlpute the a and bterms at all the points within the block n -< 8, nz -< 8, j -< 15 that contributed anything to the final results. This meant that 40 points in the nm-plane were to be
computed and that 40 storage counters would be required. Since terms with n or m indices
of 9 werc nbt being computed when j =.9, terms with n or m indices of 8 would be in error
when j = 10, so these terms were not computed. Likewise, terms having n or m indices of 7
/
+
DOUBLE REFRACTION OF FLOW
woul~l be incorrect when j == 11 and so were not computed. Thus, when j = 15, the terms
al,I;15 and b1,1;15 were the only terms calculated. It remained' then for the computer to determine whether 15 terms, or 8 nonzero terms, would suffice for the convergence of the series,
or whether some change in organization would have to be made. All in all, there were now
the following 136 n,m;j points to be computed:
],1;1
5,5; 11
6,2;8
6,1;9
2,2; 10
2,1;5
5,2;6
7,3;7
7,1;9
3,2; 10
1,0; 12
3,1;5
5,5;7
7,2;8
1,0;2
6,2;6
8,2;8
4,2;10
2,0;12
4,1;5
4,4;6
6,5;7
3,3;9
2,0;2
4,4;8
4,3;9
5,2; 10
3;0;12
5,4;6
7,5;7
5,1;5
2,2;2
6,2;
10
5,3;9
4,0;12
6,4;6
1,1;3
3,3;5
7,7;7 , 5,4;8
4,4;10
6,4;8
2,2; 12
1,0;8
6,3;9
2,1;3
4,3;5
6,6;,6
5,4;10
7,4;8
7,3;9
3,2; 12
1,1;7
3,1;3
5,3;5
2,0;8
6,4;10
8,4;8
5,5;9
4,2;12
5,5;5
2,1;7
3,0;8
3,3;3
4,4;12
6,6;
1
0
6,6;8
6,5;9
3,1;7
4,0;8
1,0;4
1,0;6
,1,1;11
1,1; 13
7,6;8
7,5;9
4,1;7
5,0;8
2,0;4
2,0;6
2,1; 11
8,6;8
7,7;9
2,1;13
5,1;7
6,0;8
3,0;4
3,0;6
3,1; 13
3,1; 11
8,8;8
1,0;10
4,0;6
6,1;7
7,0;8
4,0;4
4,1;11
3,3;13
1,1;9
2,0;10
7,1;7
8,0;8
2,2;4
5,0;6
5,1;11
1,0;14
2,1;9
3,0;10
6,0;6
3,3;7
2,2;8
3,2;4
2,0; 14
3,3; 11
3,1;9
4,0;10
4,3;7
3,2;8
'4,2;4
2,2;6
4,2;8,
4,1;9
5,0;10 '4,3;11
2,2;14
5,3;7
4,4;4
3,2;6
1,1;15
],.1 ;5
5,3;11
5,1;9
6,0;10
6,3;7
5,2;8
4,2;6
Next, Eqs. (8) and (9) were solved and written in the form
a
+
+ +
+ +
+
.=
_
nm,]
where k '= 2n(2n
1),
A = alcl
a2c2
a3c3
a4c4
B = blcl
b2c2
b3c3
b4c4
2n
1
c1 = 4 (4n - 3) (4n - 1) ,
3
c2 = - 4 (4n - 1) (4n
3) ,
+
+
+ kB
+ k2/ex'
exmA
exm2
b . = kA - 'XmB
nm,]
exm2
k 2/ex '
+
(17)
+ ar,cs + ascs,
+ bscs -to bscs,
+
n
2 (4n + 3)( 4n + 5) ,
C4 == (2n -,2m - 3)(2n - 2m - 2)(2n - 2m - 1)(2n - 2m)cl ,
Cs = - (2n - 2m - 1)(2n - 2m)(2n
2m
1)(2n
2m
2)c 2 ,
c6 = - (2n + 2m
1)(2n
2ni +2)(2n +2m
3)(2n
2m
4)c3 ,
c3
=-
+
+
+
+
+
+
and
a1
=
an -1,m-1;j- b
a4 = an-I,m + l;j- h
bl = bn - 1,m-l;j-l,
b4 = bn - 1,m + l;j- b
a2 = an ,m-1;j- b
as = an,m + l;j- b
b2 = bn ,m-1;j-.,
bs = bn,m + 1;j- b
a3 = an + l.m-l;j- b
a6 = an +1,m+1;j-b
b3 = bn + 1,m-l;j- b
b6 = bn + 1,m+l;j-l'
233
+
+
+
SCHERAGA, EDSALL, AND GADD
In order that one computation routine might be used throughout, it was decided to compute the terms for m -.:.. 0 exactly like the others and then to wipe out the bno,j term and double·
the anO,j term, in accordance with Eqs. (6) and (7).
The first control tape placed upon the calculator simply computed Cl~ C2 , Ca, C4 , Cs, cs, k, k2, m2,
and the sum of these nine quantities for each of the 136 points, and punched the results on
cards. These calculations were not checked internally. Next was run another s~mple tape
which fed these punched cards into the machine and printed all quantities. These printed
sheets were checked for accuracy. Since all these quantities are functions of nand m only,
most of them were comput~d more than once. In fact, the case of n = m = 1 occurs eight
times. A check that the results in these duplicate cases agreed was a part of the visual check.
However, the real check at this point was in the fact that all these quantities had been hastily
hand computed in advance, and the tape was run on Mark I simply· to verify these results
and to produce them on punched cards. This portion of the job consumed 6 hours of machine
time.
The next step was the calculation of the terms anm,j and bnm,j. This was by far the major
part of the problem and consumed some two weeks of machine time. A separate run was
made for each value of~, the quantities ~ and l/~ being placed in constant switches. The main
control tape read into six working counte~s the values in the six counters containing the
particular al andbl , a2 and b2 , • • • , as and bs applying to the point being computed. A subsequence routine then computed anm,j and bnm,j, whereupon the main control tape dire~ted.
these results into the counter assigned to that nm combination and. set up the six 'working
counters for the next point. The main tape thus comprised 136 of these small sections. When
it had finished its ·run, which required about 6t hours, it was started over again with new
values of ~ and 1/~.
The subsequence routine, which consumed about 3 minutes of running time for the computation of the a and b terms at each point, comprised the following operations. First, the
cards containing the nine constants for that point were fed into the calculator, and the sum
of these constants was checked against the punched card containing their sum. Then the
A and B of Eqs. (17) were computed by 12 multiplications and checked by six additional
nlultiplications, using the relation
A
B = (a l
bt)cl
(as
bs)cs•
Then anm,j and bnm,j were computed directly by means of Eqs. (17). Since these equations
involve both m and m2, both k and k 2, and both ~ and 1/~, no intermediate checks were necessary. The only check at this point was the substitution of the computed. anm,j and bnm,j into
Eqs. (8) and (9). Next was applied the .test whether m was zero, in which case the a term
was doubled and the b term erased, in accordance with Eqs. (6) and (7). Then anm,j and
bnm,j were printed and, if they were for the point 1,I;j, they were punched on cards.
For very small values of ~ the convergence was extremely rapid, the terms al j l;5 and bl ,I;5
being zero to ten decimal places. Not until ~ reached 6 did the terms al,1;15 and bl ,I;15 exceed
10- 1°, and for all ~ under 25 it was felt that the error being committed in the dropping of the
+
+
+ ... +
234
+
DOUBLE REFRACTION OF FLOW
and b1,lj17 terms was insignificant. However, it could be seen that more terms would
be needed for accuracy when rt. was large. So the case rt. = 100 was then run in order that
the results might indicate that many unnecessary terms were being computed. It was found
that, even in this case oflarge rt., the values of terms with an m index of5 or more never exceeded
10-10, and that there were several other points where this was the casco Thus it was possible
to code a new control tape which would compute the terms through al,1;23 and b1 ,1;23 with the
use of only 36 counters for the storage of 36 points in the nm-plane. This required the computation of the following 244 n,m;j points:
al,lj17
1,1;1
1,0;2
2,0;2
2,2;2
1,1;3
2,1;3
3,1;3
3,3;3
1,0;4
2,0;4
3,0;4
4,0;4
2,2;4
3,2;4
4,2;4
4,4;4
1,1;5
2,1;5
3,1;5
4,1;5
5,1;5
3,3;5
4,3;5
5,3;5
1,0;6
2,0;6
3,0;6
4,0;6
5,0;6
6,0;6
6,2;8
2,2;6
7,2;8
3,2;6
4,2;6
8,2;8
4,4;8
5,2;6
1,1;9
6,2;6
4,4;6
2,1;9
1,1;7
3,1;9
4,1;9
2,1;7
5,1;9
3,1;7
4,1;7
6,1;9
5,1;7
7,1;9
6,1;7
8,1;9
7,1;7
9,1;9
3,3;9
3,3;7
4,3;7
4,3;9
5,3;7
5,3;9
6,3;9
6,3;7
7,3;9
7,3;7
1,0; 10
1,0;8
2,0;8
2,0;10
3,0;8
3,0;10
4,0;10
4,0;8
5,0;10
5,0;8
6,0;10
6,0;8
. 7,0;10
7,0;8
8,0;8 . 8,0;10
9,0;10
2,2;8
10,0;10
3,2;8
4,2;8
2,2;10
3,2;10
5,2;8
4,2;10
5,2; 10 . 11,0;12
6,2;10 12,0;12
2,2;12
7,2;10
8,2;10
3,2;12
4,4;10 4,2;12
1,1;11
5,2;12
2,1;11
6,2;12
3,1; 11
7,2;12
4,1;11
8,2;12
4,4;12
5,1; 11
1,1;13
6,1;11
7,1;11
2,1;13
3,1;13
8,1;11
4,1;] 3
9,1; 11
10,1;11
5,1; 13
6,1; 13
11,1;11
7,1;13
3,3; 11
8,1; 13
4,3; 11
5,3; 11
9,1; 13
6,3; 11 10,1;13
7,3;11 11,1;13
1,0;12
3,3;13
4,3;13
2,0;12
5,3;13
3,0; 12
6,3;13
4,0; 12
7,3;13
5,0;12
6,0;12
1,0;14
7,0;12
2,0;14
3,0; 14
8,0;12
4,0;14
9,0;12
10,0;12
5,0;14
235
6,0;14
7,0;14
8,0;14·
9,0;14
10,0;14
2,2;14
3,2;14
4,2;14
5,2;14
6,2;14
7,2;14
8,2; 14
4,4;14
1,1; 15
2,1; 15
3,1; 15
4,1;15
5,1; 15
6,1; 15
7,1;15
8,1;15
9,1; 15
3,3;15
4,3;15
5,3;15
6,3;15
7,3;15
1,0; 16
2,0;16
3,0;16
4,0;16
5,0;16
2,2;18
6,0;16
3,2;18
7,0;16
4,2;18
5,2;18
8,0;16
2,2;16
6,2;18
3,2;16 4,4;18
4,2;16
1,1;19
5,2;16
2,1;19
6,2; 16 3,1;19
7,2; 16 . 4,1;19
8,2;16
5,1;19
4,4;16
3,3;19
1,1;17 4,3;19
2,1; 17 5,3;19
3,1; 17
1,0;20
4,1;17
2,0;20
5,1;17. 3,0;20
6,1; 17
4,0;20
7,1;17
2,2;20
3,3;17
3,2;20
4,2;20
4,3;17
5,3;17 4,4;20
1,1;21
6,3; 17
7,3; 17
2,1;21
1,0; 18 3,1;21
2,0;18
3,3;21
3,0; 18
1,0;22
4,0;18
2,0;22
5,0;18
2,2;22
1,1;23
6,0; 18
SCHERAGA, EDSALL, AND GADD
This tape was run for all values of (X from 25 up, each run consuming 12 hours of machine
time. The four extra terms in the series improved the convergence considerably. Although
no accurate estimate of the size of the remainder term could be made, some idea of the rate
of convergence of these series at high (X values may be obtained from the data of Tables 3
Table 3. Values of -
~all,j
as a function of) for various values of (x.
1'Z
25
40
1
3
5
7
9
0.037622
0.045133
0.046291
0.045537
0.044715
0.044315
0.044244
0.044290
0.044338
0.044361
0.044367
0.044366
0.038913
0.048785
0.052465
0.053452
0.053261
0.052705
0.052205
0.051896
0.051754
0.051713
0.051715
0.051729
11.
13
15
17
19 .
21
23
200
100
&J
0.039395
0.050268
0.055325
0.057790
0.058821
0.059032
·0.058871
0.058608
0.058368
0.058191
0.058079
0.058013 .
0.039646
0.051071
0.056967
0.060498
0.062630
0.063820
0.064424
0.064699
0.039753
0.051419
0.057700
0.061765
0.064509
0.066313
0.067487
0.068260
0.068783
0.069127
0.069338
0.069460
0.064~
0.064821
0.064804
0.064711
Table 4. Values of ~bll,j as a function of) for various values of (x.
r-z
1
3
5
7
9
11
13
15
17
19
21
23
25
40
0.009029
0.015774
0.019813
0.021399
0.021545
0.021256
0.020927
0.020828
0.020818
0.020833
0.020844
0.020849
0.005837
0.010806
0.014767
0.017518
0.019029
0.019628
0.019723
0.019616
0.019482
0.019383
0.019328
0.019303
200
100
60
0.001193
0.002298
0.003359
0.004385
0.005295
0.006051
0.006651
0.007110
0.007451
0.007699
0.007874
0.007997
0.002379
0.004560
0.006605
0.008500
0.010092
0.011325
0.012225
0.012848
0.013258
0.013519
0.013677
0.013768
0.003939
0.007460
0.010583
0.013207
0.015137
0.016389
0.017107
0.017461
0.017577
0~017620
0.017595
0.017563
and 4, where the values of the - ~all,j and ~bll.j series are given for several values of (x. These
data are also plotted in Figs. 10 and 11. It will be noted that the b terms converge much more
r-------______ __
a = 25
a = 200
~
a = 100
a=60
a=40
a= 25
361~~3~~5~~7--~9--~1I--~13--~15--~17--~19--~21~23
FIG. 10. Values of - ~all,j as a function
of) for various values of (x.
01
3
5
7
9
II
13
15
17
19
21
23
FIG. 11.. Values of ~bll,j as a function
ofI for various values of (x.
DOUBLE REFRACTION OF FLOW
slowly than do the a terms. It will also be noted that the series for (X = 200 and (X = 100
clearly have not converged. It is felt that, for (X <; 60, the error that has been committed is
well under 1 percent, but that, for (X = 80, 100, or 200, the results listed in Tables 1 and 2
are significantly in error for all but the very small values of p. These "bad" results have been
included in the tables in the hope that they may shed some light on the question of convergence.
If they had been included in Fig. 7, for the case p = 00, they would lie significantly lower
than the curve shown, so that the extrapolated value offat X = 0 would be about 0.75 instead
of the value 1 as it should be and as,in fact, it appears to be when the vaIues at (X = 80, 100,
and 200 are omitted ..
It is unfortunate that the series of b terms converges slowly when the sum of the series is
. small, thus making the proportional error committed in the truncation of the series that much
greater. It should be noted that, as (X approaches 00, all terms in the series ~bl1,j approach
zero and thus approach each other. If the sum of this series is desired to within a certain
"percentage error," the number of terms required becomes infinite as (X approaches 00.
In order to study the propagation of error~, the case (X = 100 was run with an intentional
error introduced in the' value of aoo,o. As the successive terms were computed, the effects of
this error became less and less. Therefore it is believed that roundoff errors do not accumulate
and need not be considered.
The final control tape in the problem governed the computation of tan 2X andf2. Since
, only three or four decimal places of accuracy were required in the results, in the interests of
economy it was decided to obtain X andffrom tables by hand methods. Thus it was possible
to complete this stage of the job in 1 day of machine use.
The deck of cards containing the terms aU,j and bll,j for all the values of (X was fed into the
calculator along with another deck of cards containing R and ~2 [Eq. (1)] for each of several
values of p. Then, by means of Eqs. (13) and (14), tan 2X andf2 were obtained directly by
multiplications and a division. The multiplications in the computation of tan 2X were checked
by the distributive law, those in the computation of12 by the associative law, and the division
by multiplying the quotient by the divisor. All results were printed in duplicate and were
also punched on cards in order that they might be available for future usc if desired.
Using the two sets of printed results, the values of X andfwere computed by two people
working independently and later comparing the two sets of computations. 37
It is believed that no error has been made at any stage of the computation except in the
truncation' of the, series of aU,j and b11,j terms. Th~s it is felt that the results contained in
Tables 1 and 2 are entirely accurate when either (X or p is very small, are significantly in error'
when (X is very large (unless p is very small), and are probably accurate to well within 1 percent
in all cases except when (X = 80, 100, or 200.
The total machine time consumed by this problem was slightly over two weeks. It is felt
that here is a perfect example of the situation where a large-scale automatic calculator has a
tremendous advantage over hand or desk computers, perhaps not so much in the matter of
speed as in the problem of organization.
237
SCHERAGA, EDSALL, AND GADD
The method of double refraction of flow in systems containing large asymmetric molecules
gives experimental data which,. when interpreted in light of the theory of Peterlin and Stuart,
enable one to calculate nlOlecular lengths; information about the polydispersity of the system
and about the optical properties of the solute particles may also be obtained from such data .
.Heretofore, this theory had been developed so that. the data could be interpreted only
under the limiting condition of low velocity gradient where the degree of orientation of the
solute particles is very small. With the aid of the Mark I computer;· the necessary equations
have been solved to give numerical values over a much wider range of velocity gradients, .
thus greatly increasing the usefulness of flow-birefringence measurements for the study of
macromolecular systems.
REFERENCES
1. Edsall, "Streaming birefringence and its relation to particle size and shape," Advances in Colloid
Sci. 1, 269 (1942).
2. Cohn and Edsall, Proteins, amino acids, and peptides (Reinhold, New York, 1943), p. 506.
3. Peterlin and Stuart, Handbuch und J ahrbuch der Chemische Physik, Bd. 8, Abt. IB (1943).
4. Snellman and Bjornstah1,
Kolloid~BeiheJte
52, 403 (1941).
5. Signer and Gross, Z. physik. Chern. [A.] 165, 161 (1933).
6. Meh1, Cold Spring Harbor Symposia in Quantitative Biology, vol. 6, (1938) p. 218.
7. Lauffer and Stanley, J. Biol.
Chem~
123, 507 (1938).
8. Foster and Edsall, J. Am. Chern. Soc. 67,617 (1945).
9. Edsall, Foster, and Scheinberg, J. Am. Chern. Soc. 69, 273 (1947).
10. Edsall and Foster, J. Am. Chern. Soc. 70, 1860 (1948)~
II. Edsall, Gordon, Meh1, Scheinberg, and l'vlann, Rev. Sci. Instruments 15, 243 (1944).
12. von Muralt and Edsall, J. Biol. Chern. 89, 315, 351 (1930) .
.13. Edsall and Mehl, J. Biol. Chern. 133, 409 (1940).
14. Sadron, J. phys. radium [7] 7, 263 (1936).
IS. de Rosset, J. Chern. Phys. 9, 766 (1941).
16. Lawrence, Needham, and Shen, J. Gen. Physiol. 27, 201 (1944).
17. Above a critical speed of rotation, the flow becomes turbulent (see reference I), but we are
here concerned only with conditions in which the flow islaminar.
18. A rigid ellipsoid of revolution is considered a moderately good approximation to the shape
of a protein molecule, but not to the shapes of flexible coiling molecules such as those of many synthetic
polymers.
19. Jeffery, Proc. Roy. Soc. (London) [A] 102, 161 (1922-23).
20. Boeder, Z. Ph)'sik 75, 258 (1932).
21. Peterlin, Z. Physik 111,232 (1938).
22. Peterlin and Stuart, Z. Physik 112, 1, 129 (1939).
23. P,errin,
~/.
phys. radium [7] 5,497 (1934) ..
23 8
n.oUBLE REFRACTION OF FLOW
24. For example, for prolate ellipsoids (a > b), rotary Brownian movement of the a-axis about
the b-axis is characterized 23 by the rotary diffusion constant 0 b and a relaxation time T a , where
3q2(2 - q2)
1
vI _ q2
---===- In
- 3 q2
+
v1- q2
q
2(1 - q4)
. where q = liP = bfa, and the zero subscript refers to quantities for a sphere of the same volume.
If a > Sb, the following approximation is valid within 1 percent:
1
3kT
0 b = - = - -3 (2 In 2p-l),
27"a
161T1]a
where 1] is the viscosity of the solvent.
It is thus easily seen that 0 b is not a very sensitive function of the axial ratio p as compared to
the length of the semimajor axis a. Therefore p can be determined with sufficient accuracy for present
purposes from viscosity measurements, and may be taken as a known quantity. A determination of
0 b from flow-birefringence data thus gives the molecular length 2a.
2S. This quantity is denoted by b by Peterlin and Stuart.
26. The problem of calculations based on double refraction of flow measurements in polydisperse
systems will be considered in a later publication'.
27. The expression for the distribution fl,lnction for the steady state may also be given in the
equivalent form 4
011F = div Fw,
where w is the angular velocity of the particle and is a function of G and R; 0 is the rotary diffusion
constant showing the analogy between the rotational problem and the similarly expressed problem
of translativnal diffusion embodied in Fick's laws. The substitution of Jeffery's expressions 19 for
wand performance of the indicated vector operations leads to Eq. (2).
28. Jahnke and Emde, Tables offil~ctions (Dover, New York, 1945),p: 107.
29. The series appearing in Eqs. (13) and (14) can be shown to be equal to
~
Sf
~ . 1
= - - ' - cos 2X, and L.. RJ- b11
Sf.
j = - - sm 2X.
161TR
j=I'
161TR
These terms appear explicitly in the treatment offlow birefringence in polydisperse systems. 26 Tables
of these functions would be of significant help for such computations.
.
RJ-
L..
j=1
1
all j
'
30. The change of sign in the equation for J, when
the parentheses in' Eq. (14).
a
31.
11,1
=_~,
87T 1
1 .. b
+ 36/a2 '
11,1
p changes
to l/P, is due to the factor R before
=_3_
1
.
47ra 1 + 36/a2
32. Simha, J. Phys. Chern. 44, 2S (1940).
33. Eirich, Reports on progress in physics, vol. 7 (1940), p.329.
34. Personal communications with Drs. Onsager, Peterlin, and Simha.
3S. Kuhn and Kuhn, Helv. Chim. Acta 28, 97 (194S); see especially Eqs. (73) and (74).
36. This distribution function appears to be correct although its convergence has not' been estab'lished. An indication' of the possibility of the convergence was obtained during the computation
procedure described in the next section.
37. We should like to express our thanks to Dr. Eric Ellenbogen for aid in computing one of
these sets.
239
L-SHELL INTERNAL CONVERSION
MORRIS E. ROSE
Oak Ridge National Laboratory
While the problem of internal conversion is of considerable interest to nuclear physicists,
and may be of some interest in connection with the proceedings of this Symposium as an
example of an intricate and imposing calculation, part of which has already been carried out
on the Automatic Sequence Relay Calculator (Mark I), I wish to use it merely as a jumping-off
point to discuss a more general problem. This more general problem, to which I may give
the title "Interaction of electrons and electromagnetic radiation," is one that is ripe, so to
speak, for the utilization of modern computing machinery. My principal thesis is that as a
by-product of the internal-conversion work we obtain a very important contribution to the
problem of numerical solution for the description of other processes which are of prime interest
to' the physicist .. These processes are:
(a) Bremstrahlung, or the emission of light by an electron in the neighborhood of an atom ..
(b) Pair formatiori, or the transition of an electron: from a negative energy state to a
positive energy state under the influence of an external electromagnetic field. Again, this
process takes, place when there is an atom nearby. The'result qf the transition is to create an
electron-positron pair.
(c) One-quantum pair annihilation, the reverse of process (b).
(d) Photoelectric effect, the absorption of light by a bound electron.
(e) .Compton scattering, the scattering of light by bound electrons.
To this list we may add the internal-conversion process that is to be described. One
considers a system of nucleus plus atomic electrons. As a result of a nuclear transmutation
the nu'cleus is often left in an excited state. It can get rid of its excitation energy by one of
several mechanisms, of which the most important are: (I) emission of high-frequency Ilght
(y-ray) or (2) transfer of the energy to one of the atomic electrons, say one of those in the
K-shell, which is usually the most probable event of this type. This electron is ejected from
the atom and appears as a sharp line in an energy spectrum measured with an electron spectro, meter. Processes (1) and (2) are alternative modes of decay and from a measurement of the
ratio of the rate of process (2) to that of process (I )--that is, the internal-conversion coefficient
-one obtains the following vital statistics concerning nuclear structures. ''''hile there is a
great deal we do not know about a nucleus we do know that in each state, in addition to the
energy, the angular momentum of the nucleus is a constant of the motion. We also know that
the parity is a constant of the motion ; this is a two-valued (even,-odd) quantity describing
the behavior of nuclear wave functions under space inversion. Now the fact that these two
quantities and the energy are constants of the motion is the only nuclear information inserted
L-SHELL INTERNAL CONVERSION
into the problem .. \Vhat comes out of an experimental and theoretical study of the internalconversion coefficient, together with other data of nuclear spectroscopy, is a quantitative
knowledge of all the constants of the motion, energy, angular momentum, and parity for the
pertinen~ nuclear states. There is no doubt in the minds of many physicists that at the moment
the most promising approach to an understanding of nuclear structure is through the accumulation ot information on the quantum numbers of nuclear states. It is also a well-known fact
that completely detailed information about the structure of quantum systems is not always
necessary in order. to make some useful applications. An example is the role of the quantum
mechanics of molecular structure in elucidating the empirical rules of chemical valence. It
is to be expected that the study of nuclear spectroscopy will be equally useful.
In order to make clear the thesis as originally stated, it is necessary to say a few words
about the details of the internal-conve"rsion calculations. The rate of electron ejection appears
as a sum over all possible final states of the electron, selected according to conservation laws
of energy, angular momentum, and parity, of squares of matrix elements referring to each
final state. These matrix elements, as always, involve certain averages over the configuration
space of the electron of an equivalent electromagnetic field corresponding to the nuclear
transition. Thus one deals with integrals of the form
r
F( r;Z,p,J) XI (kr )J(r;Z,j)dr
(1)
in which there appear certain physical parameters Z, p, k, J, I, j. Of these one is interested
in a particular j,. and the conservations laws are such that once j is chosen there are only
three free parameters-Z, p, I, for example. The Xl is a spherical Hankel function of the
first kind of half-integral order. The functions F and f will be referred to as wave functions,
and, in a relativistic treatment of the problem, they appear as solutions of coupled linear
homogeneous differential equations to which certain boundary conditions are applied. 1 These
boundary conditions constitute an eigenvalue problem which has a discrete part and a continuous part; F belongs to the continuous part, f to the discrete part .
. \!\Then the electron is in the K-shell, which is closer to the nucleus than any other shell,
it is permissible to neglect the effect of all the other electrons in the atom, and then analytic
representations of the wave functions are available. The integrals (I), of which there are seven
. for each Z, p, I, can be represented in terms of functions that have known properties but are
un tabulated, namely, hypergeometric functions with complex parameters. The complications
involved in the computations for the number of points in Z, p, I space which was required
was sufficiently imposing to bring up the question of an alternative procedure, namely,
numerical integration of the differential equations for at least the F function, which brings
in most of the complication, and then evaluation of matrix elements, typified by (1), by
numerical quadrature. For the purpose in hand this procedure turned out to involve a considerably greater number of operations and the first-mentioned. procedure, computation of
hypergeometric series, was adopted. As mentioned, this work was done at the Computation
241
MORRIS E. ROSE
Laboratory under Professor H. H. Aiken. In the light of later developments and a broadening
of our point of view, it can be said in afterthought that the second procedure of computing
wave functions would have been more desirable.
For the L-shell internal conversion it is not at all legitimate to forget the presence of other
electrons. They will produce a net field which must be added to the nuclear field and this
will modify the motion of the ejected electron in an important way. The modification of the
potential field in which the electron moves can be determined and this itself is a problem of
no mean proportions. Fortunately, this problenl has been solved by J~ Reitz, assisted considerably by the ENIAC. The potential field, which we call a screened field, must be inserted
in the differential equations, which then determine the wave functions, and, since the field
is known only numerically, it is necessary to integrate the wave equation numerically. Other
possibilities, such as analytic representation of the potential function, or perturbation OI
variational techniques, for finding the wave functions turn out to be highly impractical.
I will omit any discussion of the many interesting problems that arise in connection with
the numerical solution of the wave equation, except to remark that one of the most difficult
parts of this problem is to obtain well-behaved solutions in the discrete spectrum. For
this purpose it is well to remember that a highly accurate eigenvalue may give a poor
wave function and that some interpolation procedure for applying the boundary conditions
is required. This and a number of other problems of methodology have been solved and it is
hoped to put the L-shell internal-conversion computations on the Mark III shortly.
In the process of calculating the internal conversion it will be necessary to tabulate 3N
discrete-spectrum wave-function pairs where N is the number of atoms for which the calculations will be made; N will be about 10. In addition, for radiation fields of angular momenta
l = 1 through l = 5 and for six values of the energy (parameter p or k) and N = 10 it will
be necessary to tabulate 840 wave-function pairs in the continuum. These will·comprise
final-state electron waves of all angular momenta J up to 13/2. 2 With these wave functions
about 10 4 matrix elements (quadratures) will have to be computed. This in briefis the program
for the L-shell internal' conversion.
The 840 continuous-spectrum wave functions thus obtained, which are solutions' for the
relativistic Dirac electron in a screened field, would represent a compendium of the most
accurate set of wave functions available. In fact, no wave functions, even without screening,
have been available in tabulated form. These same wave functions are involved in the quanti~
tative description of all the processes involving interaction of electrons and light which were
mentioned in the preceding. Hitherto all these processes have been calculated by approximate
methods involving wave functions describing electrons subject to no atomic or nuclear fields
at all. These approximate calculations suffice where high energies of the electrons are involved
(say 10 Mev or more) although even here 10- to 15-percent discrepancies between theory and
experiment have been observed. At low energies very large errors may be incurred by the
use of these approximate wave functions. Thus for pair production at 1.5 Mev in lead the
error in the calculated cross section is 100 percent.
242
L-SHELL INTERNAL CONVERSION-
The wave functions used in the internal conversion work correspond to a representation
in which the angular momentum and one of its components are constants of the motion.
For the more general class of interaction problems one needs a representation which corresponds
to an outgoing current in a definite direction and this implies a linear combination of angularmomentum wave equations. In order that component functions with J > 13/2 shall contribute inappreciably we are restricted to'low electron energies (up to about 2 Mev). This
restriction could be removed, of course, by a more extended program of computation of wave
functions, although even with the restricted number of angular-momentum values some
interesting work could be done in connection with all the processes that do not involve pairs.
In the case of the pair phenomena there is another difficulty. The description of these processes
requires wave functions belonging to the negative-energy continuum. Essentially, these are
obtained from the positive-energy continuum by changing the sign of the parameter Z. While
this is a more or less trivial change in an analytic representation of the wave functions, it is
far from trivial when the wave functions must be obtaine.d by computational methods and
are to be given in numerical form. The negative-energy wave functions can be obtained in
exactly the same manner as were the positive-energy functions. This again calls for an even
more extended program for computing these wave functions.
It seems to me that such a program would be very much worth while and would constitute
an invaluable contribution to physics.
NOTES
1., The function F is coupled to another function G, f to a function g. Both G and g appear in
other matrix elements of the form (1).
2. The unit of angular momentum is 1i (Planck's constant divided by 27T); 1i = 1.05 X 10- 27
erg sec.
THE USE OF FAST COMPUTING MACHINES IN THE THEORY
OF PRIMARY COSMIC 'RADIATION
MANUEL S•. VALLARTA
University of Mexico*
Computing machines ,have been used in the theory of primary cosmic radiation in two
cases: (a) in the theory of the geomagnetic effects, that is, to solve the 'equations of motion
of a, charged particle in the field of a magnetic dipole; (b) in connection with problems involved
in the ,theory of the emission of cosmic rays from the sun, that is, to solye the equations of
motion of a charged particle in the variable magnetic- field of a sunspot, and the equations of
motion of a charged particle in the field of two crossed magnetic dipoles.
It is well ~nown from Gauss's analysis of the magnetic field of the earth as observed at the
earth's surface that the potential component of this field is the sup~rposition of a dipole and
a quadripole field, of which the former is by far the more important. Further, the former
varies as the inverse cube of the distaI?-ce from the dipole, while the latter varies as the inverse
fourth power of the distance from the quadripole. It follows that at distances from the earth
of the order of magnitude ofa few earth's radii the dipole field controls the motion ofa charged
particle, while the quadripole field plays the role of a small perturbation. Since primary
cosmic rays are charged particles coming to the earth from distances large compared with the
earth's radius, only the dipole component of the geomagnetic field has to be taken into account.
The equations of motion are readily set up from the classical laws of motion. The force
acting on the particle is simply the Lorentz force which, since it always acts perpendicular
to the path, does no work and 'hence the kinetic energy of the particle is a constant of the
motion. As a consequence the particle's mass is not its rest mass but its relativistic mass which
remains constant throughout the motion.
The equations of motion ofa charged particle in the field ofa magnetic dipole are integrable
in terms of known (elliptic) functions only in the case of motion in the plane perpendicular
to the dipole, that is, the plane of the geomagnetic equator. Elsewhere they must be integrated
by making use of methods of numerical or mechanical integration. Methods of numerical
point-by-point integration have been extensively used by StOrmer and his assistants at the
University of Oslo, Norway. As the geomagnetic field varies rapidly, particularly in the region
,close tothe earth, the interval of integration must be chosen correspondingly short to reach
adequate precision, and this means that a very large amount of labor is required. Depending
on the kind of trajectory, a numerical integration by standard methods requires from a day
to more than a week, and this circumstance rules out the possibility of solving problems where
* Read at the Symposium by J. C. Street, Harvard University.,
244
FAST COMPUTING MACHINES IN THEORY OF PRIMARY COSMIC RADIATION
a large number of complicated trajectories are needed. Hence the necessity of using fast
modern computing machines.
When the field has axial symmetry, as in the case of a dipole, a second integral of the
motion exists in addition to the kinetic-energy integral already mentioned. This second
integral is the projection of the 'moment of momentum of the particle at infinity, relative to
the dipole axis, on this axis. As a consequence the motion can be split up into two motions:
(a) a mot jon in the meridian plane, that is, in a plane containing the dipole axis; (b) a rotation
of the meridian plane. Most of the important problems related to the theory of primary
cosmic radiation, in particular the geomagnetic effects,' do not require the knowledge of
the motion of the meridian plane.
A particle of given energy and angular momentum can move only within certain regions
of space known as the Stormer regions. Outside of these regions the kinetic energy would
become negative, or, what amounts to the same thing, the time would become imaginary.
The shape of these regions is determined from the value of the energy and the angular
momentum.
The main result of the analysis of the motion ofa charged particle in the field ofa magnetic
dipole is that all particles of a ,given energy and sign must arrive at any point on the earth
from directions within a certain cone, known as the allowed cone. This cone has its vertex
at the observer on the earth and its generators are further described below. The allowed
cone is a cone of many sheets: the first sheet determines the so-called main cone, which has
the property that any direction within is an allowed direction; the last sheet determines the
shadow cone, which has the property that any direction outside is a forbidden direction.
Between the main cone and the shadow cone the sheets of the allowed cone determine alter'nately bands of allowed and bonds of forbidden directions of very complex structure. This
has been called the region of penumbra.
The generators of the main cone are trajectories, which are asymptotic to members of.
the family of unstable periodic orbits that are farthest removed from the earth, and which
do not form loops, known as trajectories of the first kind; the generators of the shadow cone
are trajectories which do not form loops and are tangent to the earth before reaching the
observer. These trajectories are known as trajectories of the, second kind. The generators
of the penumbr~ bands are trajectories of either the first or second kinds which form one or
more loops before reaching the observer. An asymptotic trajectory may at the same time
be tangent to the ea~th and thus mark a direction along which two sheets of the allowed
cone, for instance the main cone and the shadow cone, touch each other. Such trajectories
are known as trajectories of the third kind.
The equations of motion of the particle in the meridian plane, in appropriate coordinates,
are
~:~ = (~~) 4 e2,1: _ e~X + e-2x cos 2 }~,
d2 }.
-
da?
=
e-
•
2,1:
sm }. cos }.-
245
sin ).
cos 3 }.'
MANUEL S. VALLARTA
and the equation of motion of the' meridian plane is
dcp _ _1_ _ e- X
da - cos 2 A
•
.'
No use of this last equation will be made in what follows.
The determination of the main cone requires (a) the knowledge of the members of the
family of periodic orbits that lie farthest from the dipole; (b) the knowledge of the asymptotic
orbits to each member of the family of periodic orbits. To find the shadow cone, all trajectories
that do not form loops and are tangent to the earth at some point apart from the point of
observation must be known. To determine the penumbra bands all trajectories, either
asympt~tic or tangent, making one or more loops must be available. All these problems
depend for their solution on the integration of the equations of motion.
The integration of the equations of motion was carried out by means ofthe first differential
analyzer at the Massachusetts Institute of Technology during the years from 1933 to 1939.
The first integrals may be written
dA -_
da
J(e
2x
1).
- cos 4 A SIn
1
II.
cos II.1d.a.
In this form the system of differential equations may be immediately set up and solved in the
differential analyzer. Five, input tables are needed to introduce the functions '(e Xj2Yl)4 - eX, .
e- 2x, cos 2 A, cos- 4 A, and sin A cos A. Four integrators are required.to integrate ,the product
of the two functions u~der the integral signs and two more to integrate dxjda and dAjda. The
output is plotted with x as abscissa and A as ordinate.
The initial conditions are introduced in the machine from a knowledge of x and the initial.
slope. The knowledge of x and A is sufficient to set the starting points on the input tables.
To s~t the starting points on the six integrators the values of the functions e- 2x , (eXj2Yl) 4 - eX,
cos 2 A, -cos -4 A, sin Acos A, dx/da, and dAjdcr must be calculated. The first four are computed
from the known values of x and A, the last two from x, A, and the initial slope.
One way to find the family of periodic orbits is to start the trajectory at right angles to
the boundary of the StOrmer forbidden region and continue the integration until the point
where the trajectory has a tangent parallel to the }.-axis is reached. If the point of "vertical"
tangent is on the equator, the required periodic orbit has been found; if it is not, then the
starting point is moved along the boundary of the forbidden region until the required trajectory
has been discovered. In this way it was found that periodic orbits exist only for a limited range
of values of the 'angular momentum, and .for any value of angular momentum within this
interval they exist in pairs, of which one is stable and the other unstable.
To find the asymptotic trajectories a point is chosen on the equator between the earth
and the outer (unstable) periodic orbit and a trajectory is started in the direction toward
this orbit; the initial slope is then adjusted by trial until the trajectory neither falls short of
246
FAST COMPUTING MACHINES IN THEORY OF PRIMARY COSMIC RADIATION
nor intersects the periodic orbit. Five to ten trials are necessary, and the critical initial slope
is determined by this method with a precision of a few thousandths of a radian, as shown by
independent calculation. I , 2
To determine shadow orbits one may choose a given value of the energy and angular
momentum and start trajectories at fixed intervals of a few degrees of latitude, tangent to the·
earth. These trajectories are then continued until they st;ike the earth at some other point.
Two limits of latitude are determined in this way. The lower limit is characterized by the
fact that the orbit through this point is a self-reversing orbit, that is, an orbit that reaches the
boundary of the forbidden region of StOrmer and reverses along itself. The upper limit
corresponds to an orbit that has an inflection at the point of tangency and is therefore a
transition orbit between simple orbits of the second kind and orbits having maxima at the
point of tangency and minima at points within the earth .. All orbits of the latter class are.
clearly in "shadow" and are not generators of the shadow cone.
The determination of penumbra orbits is very much more complicated but follows in
general along the same lines. 4, 5, 6
The precision of the trajectories determined with the help of the differential analyzer is
far from constant and depends on two factors-the number and sharpness of turns and the
length of a trajectory. For short runs without sharp turns the precision reached may be of
the order of a few thousandths of a radian; for long runs with many sharp turns the error
may be as high as half a radian.
Another problem requiring the use of high-speed modern computing machines is the
emission of cosmic rays fi'om the sun. A few years ago it was found that the appearance of
certain solar flares was followed by a sudden large increase in the intensity of the cosmic
radiation as observed everywhere on the earth except at equatorial latitudes. It seems certain
that charged particles present in the neighborhood of sunspots can be accelerated up to
cosmic-ray energies by the action of the variable magnetic fields of sunspots and that they
can escape only when the proper conditions are satisfied between the permanent dipole
magnetic moment of the sun and the transient dipole moment of the pair of sunspots associated
with the flare. To find out the actual trajectory followed by such charged particles from the
sun to the earth requires the integration of the equations of motion in the combined field
of the permanent and transient dipoles; This problem has no axial symmetry; consequently
the angular-momentum integral is lost and only the kinetic-energy integral remains. In
phase space the trajectories are therefore subject to the condition that they must remain on
the surface vx 2
v1/
vz 2 = 1 (in appropriate units). This condition, translated into configuration space, yields the result that, provided the ratio between the permanent and the
transient dipoles and their relative orientation is within cet;tain limits, and provided also the
ratio between the field ~alues is above a certain constant, a tunnel is drilled through the
Stormer forbidden region of the permanent dipole, and through this tunnel the particles
acceleI,'ated by the variable sunspot field can then escape. This condition is necessary but not
+
+
247
MANUEL S. VALLARTA
sufficient; in other words, it is not known in advance whether trajectories exist that start
from the sun and come out of the tunnel.
The equations of motion in the field of the two dipoles are
2
~ d x = dy (k[3(Z - zo)p· p _ cos
K2 ds 2 . ds
r5
r3
y] +
[3(Z - zo)Q·
R5
_ dz (k[3(Y - Yo)p· p _ cos fJ]
ds
r5
r3 .
+
p_
C])
3
CROS
[3(Y - Yo)Q . p' ~ cos b])
R5
R3'
and two other equations obtained by cyclic interchange, where K2 = 9Ms/mv, k = Mss/Ms,
Ms is the magnetic moment of the permanent dipole, Mss is that of the transient dipole;
p, P, p', Q arethe vectors whose components are, respectively,
(cos cx., cos (J, cos y), the direction cosines of Mss;
(x - Xo,Y - Yo, z - zo), Xo,Yo, Zo being the position coordinates of Mss;
(cos a, cos b, cos c), the direction cosines of Ms;
(x - Xo,Y - Yo, z - Zo), X o, Yo, Zo being the position coordinates of Ms;
r2 = (x - X O)2 + (y - YO)2+ (z - ZO)2, R2 = (x - XO)2 + (y - YO)2 + (z - ZO)2;
and the origin of coordinates is at the mouth of th~ tunnel nearest the sun.
In the region close to the sun the field is large and changes rapidly. A preliminary integration has shown that in this region the trajectory starts out as a tight spiral of a few hundred
kilometers radius and a few thousand kilometers pitch. As a consequence the velocity vector
changes rapidly but the displacement is smalL Further, the interval of integration must be
taken very small to keep within the required precision.
The U.S. Army has very kindly made available the ENIAC machine at Aberdeen, Marylan~ for the purpose of carrying out the integration of the equations of motion. For the reason
mentioned above, even this fast rriachine is unable to carry through the integration in cartesian
coorqinates without prohibitive labor. In order to circumvent this difficulty we have made
use of the fact that both the radius and the pitch of the spiral' trajectory are slowly varying
functions of the distance along the trajectory, and only the angle turned through is a rapidly
varying function. We have therefore introduced helical· coordinates' defined by the transformation x =M cos (), Y = M sin (), z = N(), where M, N, and () are functions of the arc
length s measured along the trajectory. The transformed 'equations are
2
2
. -dM -d() sin () - M cos () (d())
d2 ()
'd-M
cos
.()
2
M
sin
()
-ds 2
ds ds
ds
ds 2
=
(d:! sin () + }vI cos () ~:) (- . .) _ (d: () + N~:) (- . .),
and two more equations obtained from these as in the previous case. It is hoped to start the.
integrations with the help of the EN lAC this coming fall. The interval of integration will
be a few thousand kilometers at the start, and. will be doubled every other step.. From fifty
to a hundred trajectories will be required, and in all some ten million operations will be needed.
248
FAST COMPUTING MACHINES IN THEORY OF PRIMARY COSMIC RADIATION
Without taking account of the time required for necessary machine repairs, the time required
for the actual integration on the machine will be about a month.
vVe are indebted to Professor John von Neumann of the Institute for Advanced Study,
Princeton, New Jersey for his interest and help with' the problem of the emission of cosmic
rays from the sun.
REFERENCES
1. G. Lemaitre and M. S. Vallarta,
P~)'s.
Rev. 49, 719 (1936).
2. G. Lemaitre and M. S. Vallarta,
P/~ys.
Rev. 50,493 (1936).
3. G. Lemaitre and M. S. Vallarta, Ann. soc. sci. Bruxelles 56, 1.02 (1936).
4. E.J. Schremp, Phys. Rev. 54,158 (1938).
5. R. A. Hutner, Phys. Rev. 55, 15 (1939).
6. R. A. Hutner,
P~ys.
Rev. 55, 614 (1939).
7. M. S. Vallarta, An outline of the theory of the allowed cone ql cosmic radiation (University of Toronto
Press, Toronto, 1938).
8. S. E. Forbush, P. S. Gill, and M. S. Vallarta, Rev. Mod. Phys. 21,44 (1949).
249
COMPUTATIONAL PROBLEMS IN NUCLEAR PHYSICS
HERMAN FESHBACH
Massachusetts Institute of Technology
At present the computing machine is employed by theoretical nuclear physicists as a tool
of research. This is perhaps a more exciting role than such an instrument commonly plays;
on the other hand, there is the concomitant danger that many of the results record an experiment, an attempt to explain the properties of the atomic nucleus and its constituents as following
from some initially chosen hypothesis. This is particularly true now when our knowledge of
the forces that hold nuclear particles together is vague and fragmentary. It is usual that such
an attempt will be a failure and the numerical results obtained will be ·replaced in the future
by ones of greater validity. It is thus of the greatest importance to choose problems that are
presumed to be most fruitful in exposing the inner workings of the nucleus.
As a consequence of this view of the type of calculations likely to occur, it is highly desirable
that the computing machine be flexible. Thus, it should be possible to make rapid changes
in the input of the computer-for example, to change the numerical values of some of the
parameters; or one might need to change some of the computational details, for in many
problems where the final results cannot· be envisaged easily a priori it is' difficult to have all
such details in order for every choice of parameters. Note also the human element here for
someone must judge how the parameters are t6 be changed. Thus, for some of the very fast
computing machines under construction, it seems likely that the nuclear problem cannot be
run continuously. Rather it must be possible, particularly for economic reasons, to switch
.
to some other problem while the decision is being made.
Before discussing some of the computational problems that occur in nuclear physics, it is
useful to tabulate the types of data which nuclear theory must be expected to correlate and
explain. Some notation is necessary: A is the total number of particles in the nucleus, Z is
the nuclear charge and therefore the number of protons, A-Z is the number of neutrons.
(I) The mass of the nucleus may be measured. Its deviation from the sum of the masses of
the individual neutrons and protons that make up the nucleus is called the binding energy,
constituting the first body of nuclear data to be explained. (2) Under the influence of external
forces such as those produced by y-rays, electrons, or nuclear projectiles, the nucleus can gain
energy and exist for a short while in an excited state .. The energies required to excite each of
.these various excited states form a second set of data. (3) In comparing two nuclei with the
same number of nuclear particles, that is, the same value of A, it is found that one of them is
more stable. The optimum value ofZ for each A, and the differences in binding energy
between the stable nucleus and the unstable nuclei, must be explained. (4) Another set of
important data is found by applying electric or magnetic fields to the nucleus. From these
25°
COMPUTATIONAL PROBLEMS IN NUCLEAR PHYSICS
measurements we may obtain the magnetic moment and the electriC quadripole moment of
the nucleus. A related datum is the total angular momentum of the nucleus about its center
of mass.
As an example, consider the case A = 2. The most stable system with this value of A is
the deuteron, consisting of a neutron and proton. The other systems with this value of A,
two neutrons or two protons, are less stable; actually, they are unstable. The deuteron has
a binding energy of 2.23 Mev. It has an electric quadripole moment of 2.73 X 10- 27 cm. 2 ,
and a magnetic moment of 0.8565 nuclear magnetons. "The angular momentum is 'Ii (Planck's
constant divided by 27T). There are no excited states for the deuteron in which the neutro~ "
and proton remain bound to each other.
The scattering and absorption of particles by" nuclei, together with the resultant transmutations of the target nuclei, provide another important source of data. In this type of
experiment, the incident particle-neutron, proton or alpha particle-upon striking the nuclea"r
surface may be reflected, giving rise to scatterh~g, or it may be absorbed. Upon absorption,
the resultant nucleus will be unstable emitting the~ a particle (it can be the same type as the
~ncident particle). The residual nucleus is generally not the same as the target nucleus. A
simple example is
BIO
+n
-7
Li7
+ oc.
Deuterons, and late1y tritium (H3), are also employed as projectiles. Deuteron reactions
are of a rather different nature inasmuch as the deuteron is a rather loosely bound combination
of neutron and proton. When the deuteron approaches the nucleus, the electrostatic field
of the nucleus repels the proton and thus polarizes the deuteron, so that whenever the neutron
lies between the nuclear surface and the proton, the deuteron is stretched. If the neutron
should strike the nucleus with the proton outside, the neutron is absorbed or reflected by the
nucleus. In either event, the bond that held neutron and proton together in the deuteron
combin~tion is not strong enough to keep them together under this impact, the proton going
off independently of the neutron. Upon some occasions, of course, the complete deuteron
may strike the.nucleus and be absorbed but, at least for small energies, the electrostatic field
of the nucleus tends to prevent the proton from reaching the nucleus, making thi~ process
relatively improbable.
Considerably more fundamental experiments occur when the elementary nuclear particles-neutrons and protons-scatter from each other as in neutron-proton and protonproton scattering, for then we are dealing directly with the nuclear forces between
particles.
In this field" of nuclear reactions and scattering a number of functions occur which should
be and indeed are in part tabulated. This circumstance arises from the fact that one factor
in describing the probability of an event, for example, absorption by a nucleus, is the probability
that the particle will strike the nucleus. Since this depends on the motion of the incident
particle while it is outside the nucleus where the forces acting are known, it becomes possible
to tabulate this probability for various energies and charges of the incident particle. If the
25 1
HERMAN FESHBACH
incident particle is a neutron,then the probability may be 3tated in terms of the solutions of
the equation
2
d uZ
[k2_1(l + 1)] u = 0
(I an integer) ,
(1)
dr2 +
r2
Z
,
subject to the boundary condition
U t -7
eiler. Here k
= V2MEjli 2, where E
is the energy of
r~OCJ
the incident particle and M is its mas,s. The required solutions are well known to be kr times
the spherical Hankel function krhl(kr) , where hl(kr) = VTr/2kr Hz+ l(l)(kr). The important
physical quantities are' the phase and amplitude of this function. These have been tabulated.
If the incident particle is charged (for example, a proton, a triton, or an (X-particle), then
the solutions of the following differential equation are required:
q~~ + [k2 _ 2'Tj _ 1(1 + 1)] u = 0
dr2
r
r2
l,
(I an integer)
'
where 'Tj = MZZ' e2jli2, Z is the charge on the target nucleus, and Z'
incident particle.
It is more convenient to use a dimensionless independent variable
p
Then Eq. (2) becomes
d2u Z
dp2
+
= kr.
IS
(2)
the charge on the
(3)
2
[1 _2ZZ'e
~ _ 1(1 + 1)]
=0
liv P
p2
,z
,
u'
(4) "
where v is the velocity of the incident particle. The solutions ofEq. (4) are known as Coulomb
wave functions. The solutions of interest must satisfy boundary conditions similar to those
•
given in Eq. (2):
ZZ'e 2
(5)
(X = 1iV
p~OCJ
Eq. (4) may be reduced to the equation for the .confluent hypergeometric function. Thus,
power-series expansions of this function with an infinite radius of convergence exist as well
as expansions in terms of the spherical and cylindrical Bessel functions. Extensive tabulations
have been made. of the imaginary part of the solution lto by the Computation Laboratory
in New 'York City. The higher I values may be obtained by recUrrence formulas, although
successive application of some will result in loss of accuracy. Note that by making appropriate
changes in the parameters (X and k the solutions may' be utilized in the discussion of protonproton scattering.
The problem of the nlotion of a deuteron in the electrostatic field of a nucleus has not yet
been solved. In this case one does not expect to obtain exact analytic solutions; rather the
attempt is made to reduce the partial differential equation to a form that would be suitable
for machine calculation. The equation is
(6)
Here \l n 2 and \l p2 are the Laplacians in the coordinates
rn
and
rp
respectively. The function
COMPUTATIONAL PROBLEMS IN NUCLEAR PHYSICS
V is the neutron-proton potential. It is appreciable only when the neutron and proton are
separated by less than about 2 X 10- 13 cm. The function 1p(rn' rv) should b~have as given
by Eq. (5) as the center of mass of the deuteron goes to infinity. At the surface of the nucleus,
according to recent models of nuclear reactions, the logarithmic radial derivative with respect
to rn and also to rv must satisfy specified boundary conditions. Actually this statement of,
the boundary conditions already contains considerable simplifications. It would, however,
take us too far afield to discuss these here.
Coulomb wave functions for electrons also are needed in many strategic places. Here we
are again dealing with known forces and consequently known equations of motion. For low
electron velocities and "bare" nuclei (that is, disregarding the extranuclear electrons) the
electronic Coulomh wave functions satisfy Eq. (4) with Z' = - 1. The effect of the extra.nuclear electrons, particularly for low velocities, complicates the calculation considerably, as
is clear from the earlier discussion by M. E. Rose. However, when the electrons are moving
rapidly, it is necessary to employ the Dirac equation which satisfies the requirements of relativity. Again the wave functions satisfy Eq. (4), except that d is no longer an integer. There
is the additional complication that in the Dirac case there are two such solutions which must
be properly combined to give the final solution. Wave functions of this type would be useful
in a large number of problems, all of which are more or less concerned with nuclear structure.
Besides the problems of internal conversion (see the paper by M. E. Rose) and internal pair
production, there are the problems of electron excitation and disintegration of nuclei, electron
production of mesons, and, of great importance in electrodynamics, the production of x-rays,
the production of electron-positron pairs, and the scattering of electrons by nuclei, where it
is necessary to take into account the distrihution of charge within the nucleus itself.
vVe have now exhausted the catalogue offunctions whose tabulation would be of permanent
value, particularly in nuclear physics. It should be emphasized, however, that these functions
are not descriptions in any way of the nuclei but rather are tools by means of which the analyst
may extract the salient features of such a description from the experimental data. For example,
in the scattering and absorption of particles by nuclei the energy at which resonance occurs
and the width of the resonance may, by means of the functions discussed above, be translated
into the value of the logarithmic derivative at the surface of the nucleus at or near the resonance.
This is a property of the interior of the nucleus. The problem of understanding these facts
in terms of a theory of nuclear. structure remains.
Let us now turn to the problem. of determining nuclear structure itself. . Here we attempt
the calculation of such properties as binding energy, the energies of the excited· states and the
associated widths, stability questions, electromagnetic properties of nuclei, relative yields in
nuclear reactions, and so on. The general plan of action consists in utilizing the properties
of the simpler nuclear systems to test and finally choose the law of force between nucleons.
Then this law of force is to be employed to predict the properties of more complicated systems.
Is it possible for such a progranl to succeed? It is not altogether clear that it is; for example,
the recent discovery of "shell structure" in nuclei indicates at least the possibility that the
253
HERMAN FESHBACH
heavier nuclei are in some ways less complex than the very light nuclei. Or it may be that
the notion fundamental to this program, that there are laws of force which depend only upon
the coordinates of the nucleons, may be incorrect. We begin to see that some calculations
may prove to possess only an ephemeral value.
Of course we cannot discuss this entire program in detail here. It will suffice to point out
the mathematical questions involved and some suggested methods of solution. It will be seen
that much machine ,computation is involved, barring some revolutionary d.iscovery that would
succeed in reducing the present complication to simplicity much as the Copernican theory
of the motion of the planets reduced the older Ptolemaic theory.
The mathematical problem to be solved is that of the Schroeding~r equation:
- 1i2
[ 2M ~Vl
~
+ ?Vii] 1p =
t,J
E1p,
(7)
i>j
where 1p is a function of all the coordinates of each particle; the subscript i denotes the partiGle
involved, so that Vii is the potential energy between the ith and jth particles. The word
coordinate as employed here includes not only the space coordinates but the charge (lor 0)
and spin (intrinsic angular momentum of each nucleon) coordinates as well. The energy of
the system is denoted by E; it has its lowest value for the ground state of the system where
its value is the negative of the binding energy. The excited states of the system all have larger
energies. The outstanding characteristic of these forces is their short range and their consequent
rapid variation as the distance between the particles changes. These features of the nuclear
forces make calculations difficult to perform.
Equation (7) can be reduced to, a system oflinear second-order ordinary differential equations
only for the two-particle nuclear systems, for which A =2. These equations may be integrated
by either numerical or analytic methods if possible. However, even, for the two-body case it
is often more economical to adopt approximate procedures which lead to the desired results
more rapidly and easily. For all other 'nuclear systems, three-body and more, approximate
methods must be employed.
There are three such nlethods which have been employed in the past and upon which
we may expect to rely in the future. These are (1) the perturbation method, (2) the RayleighRitz method, and (3) the variational-iterational method."
In the first' of these, the solution 1p is expanded in terms of the eigenfunctions of some
approximate problem. We rewrite Eq. (7) symbolically in the form
H1p = E1p,
where H is an operator. Suppose that
H= HO
H',
+
and that the approximate problem with eigenfunctions Xn. and energies En is
HOXn = EnXn.
The energy E is then the solution of a secular determinant
IH'nm - (E - En)Nnml = 0,
254
(8)
COMPUTATIONAL PROBLEMS IN NUCLEAR PHYSICS
where
N nm = (Xm Xn)~lIm' Hnm' = (Xm H'Xm)·
We are employing Hilbert space notation; ~nm is the Kronecker~. Approximate formulas
which assume that the difference between Hand HO is small have been developed 2 and are
customarily employed rather than the full secular determinant. Calculations of this type
have been made for the lighter nuclei, employing as the unperturbed wave functions the
harmonic-oscillator wave functions 3 (Hermite functions) for each particle, assuming particle
independence. For the heavy nuclei, plane-wave approximations for each particle have been
assumed. 4 Perturbation methods have also been employed in the resonating-group method
where the nucleus is presumed to exist for· a time in certain subgroups; it is decided a priori
which groups are most likely..This essentially provides a scheme wherein it becomes possible
to base the properties of a nucleus A on those of A-I. Because of the length of the calculations, the perturbation method has never been pushed far enough to obtain convergence;
usually it has stopped early with but a few terms permitting a qualitative understanding of
some nuclear properties, and on the other hand leading to some very grave misconceptions.
It may not be the ,most appropriate method but certainly it is one that is readily adapted to
machine calculation.
The Rayleigh-Ritz method enjoyed a very great success in the theory of the atom and
consequently has been employed in nuclear problems. The method as it is customarily
employed consists of two parts. First (note that one may use either the independent particle
picture or the resonating-group method) one assumes a form for the function "p involving
nonlinear parameters. These nonlinear parameters are then determined by the variational
principle .. However, as is well known, this procedure yields only an upper bound to the eigenvalue. The second step is an attempt to determine the eigenvalue itself. The initial c40ice
for "I) is made the first term in an infinite series offunctions with linear undetermined coefficients,
the functions involved forming a complete set for the problem under discussion. Introducing
this series into the variational principle yields a secular determinant similar to Eq. (8). By
considering the successive values of E as the number of terms i~ the series is increased, one
may estimate the convergence and thus the final value of the eigenvalue E. Unfortunately,
this procedure is not foolproof, 5 for sometimes the convergence obtained may be false. This
is caused in part by the faulty choice of the type of unperturbed problem. However, a considerable fraction of the difficulty lies in the rapid variation of nuclear potentials with
interparticle distance, implying the need to employ a considerable number of eigenfunctions
with fairly large quantum number. Hence the lack of convergence.
In the discussion of both the perturbation and the variational methods we have concentrated
on the calculation of binding energies and the energy levels of nuclei. However, it should be
noted that both of these methods apply as well to scattering and nuclear-reaction problems.
Their application to these problems has been made for only the very light nuclei.
The variation-iteration method adds to the variation method (a) a systematic method of
improving the initiaJ trial function and (b) a method of obtaining a lower bound which,
255
HERMAN FESHBACH
combined with the upper bound given by the Rayleigh principle, deter~ines the required
eigenvalue to a certain accuracy. The method has been applied successfully by several authors.
Convergence is rapid and security in the results is available because of the, existence of a
. lower bound.
The problems under discussion may all be written in the form
A1p = AB1p,
(9)
where A and B are Hermitian operators and A is an eigenvalue. For example, in Eq. (7),
'2 Vij) B = 1, A = E. There is another possibility, however. Let
A = - (1i2 j2M) '2\1 i 2
+
ii,j
i>j
Vii = qfij, where fii is just the form of the dependence of the internucleon potential energy,
and q is the measure of the strength of that potential. The number q may also be considered
as the eigenvalue A, A = E + (1i2j2M) '2\1 i 2, B = '2hi. In this formulation E is assumed to
i
i,j
i>j
be known, and the necessary strength of potential needed to obtain this value of E is computed.
At the present stage in the history of nuclear physics, this is actually a inore convenient order,
for we are now interested in determining what Vii will yield the known experimental binding
energies. It is this formulation which has been employed in the calculations that so far have
been made with this method.
The technique goes as follows. By some means or other, either by the Rayleigh principle
or by knowing a reasonably good approximation to the correct eigenfunction, an initial wave
function CPo is chosen. The iteration method is employed to improve the initial trial function.
It is important to employ in the iteration an operator that essentially involves an integration
rather than a differentiation. In the problem under consideration, therefore, the successive
iterates CfJn are generated as follows:
CfJl
= A -IBCfJo,
Cf2
= A -IBepl,
(10)
epn+l = A -IBepn·
It is easy to see how the successive applications of A -IB improve epo. Since the solutions of
Eq. (9) form a complete ortho~ormal set {1pn} we may expand epo in terms of the set:
co
p1pp.
'2a
o
epo =
Then
co
Pn
~ap
=6
Apn 1pp,
p=o
where Ap is the eigenvalue associated with 1pp. Inasmuch as there is an eigenvalue in the set
Ap , say Ao, which has the lowest absolute value, ep", -+ (constant) 1po.
",~co
COMPUTATIONAL PROBLEMS IN NUCLEAR PHYSICS
From these successive iterates it is possible to form successive approximations ,to the eigenvalue by employing the iterates as trial functions in the two variational principles:
and
(1p, A1p)
Ao = stat. value of (1p, B1p)
(11)
.
(1p, B1p)
Ao = stat. value of (1p, BA -1B1p)'
(12)
Introduce into these expressions CfJ1, CfJ2, ',' • for 1p. The resultant values of the ratio ,are
A (1)
_
(1,0)
o
-
(1:1)'
A 3/2
,_
(1, 1)
-' (1,2)'
o
'A (n-l/2) _ (n- 1, n- 1)
o
-
(n,n - 1)
,
(13)
A (n) ~ (n,n - 1)
o (n,n) ,
A (n +"1/2) _
o
-
(n,n)
(n,n ~ 1)'
where (n, m) = (CfJm BCfJm) = (CfJm ACfJm+l)' We now tabulate some theorems with regard to
the quantities )'o(n) and i Ao(n+l/2). Two cases are to be distinguished. The inequalities are
special cases of a more general inequality which may be readily found.
Case 1: A, and B are positive definite operators.
(b) Ao(n-l/2)
> Ao(n) >
'] ~ A (n+l) 1
()
( C
ILo:P" 0
Ao
>
(n+l/2) (
Ao·
. A1p
(d) mIn B1p
.
1-
< Ao <
Ao(n+l/2) > Ao;
A (n+l/2) _ A (n+l»)
0
0
)'1 _ Ao(n+l)
,
Aon - AOn+l/2) .
A1 _ At+l/2 ,
A1p
max B1j);
(e) the error decreases in the ratio Ao/Al'in going from Ao(n to Ao(n+l/2).
Case 2: B is positive definite; A is not positive definite.
(a) Ao(n-l/2)
> Ao,
Ao(n- q-l/2)
(b) Ao(n+l/2)
>
>
Ao(n+ p );
P > 0, q >
Ao(n+3/2) • ••
>
Ao if A12
°
>
Ao(n+l)A o(n,+3/2);
257
HERMAN FESHBACH
A (n+l/2) - A (n+l))
0 A21- Ao(no+l)
if Al < 0,
1
where the problem has been adjusted so that Ao is > 0. There is one substantial difference
between the two cases that should be mentioned here. When A and B are positive definite,
the successive approximations to Ao approach Ao monotonically. This is not true in the other
case discussed.
Here Al is the next eigenvalue above Ao in absolute value, A2 the one above that. It is
generally necessary to have a lower-bound estimate of AI. This may be obtained in several
ways. In one we employ the relation
Ao
> Ao(n+l) ( 1 -
Spur (A-IB)2 = 2:(I/Ap2).
(14)
P
Hence
(15)
The variation-iteration method may be improved by several simple methods, of which
we shall give two here. One involves using the functions CfJn generated by the iteration as
base functions for. the Ritz method discussed earlier. This leads to a secular determinant·
whose elements may be expressed in terms of Ao(n) and Ao(n+l/2) and have therefore been already
computed. The secular determinant is
AO
I-A (n-l)
1 ( 1 - A (n-l/2)
AO)
Ao(n-O
o
Ao
1 - A (n-l/2)
o
Ao
1- A (n)
1
(
Ao )
A (n-l/2) 1 - Ao(n)
o
1 (n) ( 1 - A (n+l/2)
Ao) .
A (n-l/2)A
o
o
o
1 ( 1 -A (n+l/2)
Ao)
Ao(n)
o
Ao)
Ao(n) A1(n+l/2) ( 1 - Ao(n+l)
o
0
0
Ao ) •
A (n-l)A1 (n-l/2) ( 1 - Ao(n)
o
o
..
. (16)
From the solution of this equation one obtains an upper bound not only to Ao but also to Al
andA 2, etc., depending upon the size of the determinant. Employing an upper bound for
AI' it is possible from the spur in Eq. (14) to obtain a lower bound for A2 required in one of
25 8
COMPUTATIONAL PROBLEMS IN NUCLEAR PHYSICS
the inequalities above. Finally, it also becomes_ possible to give another set of lower bounds
based on the new approximations to "Po obtained by solving the secular determinant.
The variation-iteration method may also be combined with the relaxation method to
yield another procedure for automatically improving the initial trial function. These results
are equivalent to those incorporated in Eq. (16), except that the quantities involved occur
in a somewhat different order.
Applications of this method to problems in nuclear physics have been made by Thomas 6
and Svartholm. 7 In recent months a series of extensive calculations on the properties of the
deuteron have been made in which the method was utilized with great success in a rather
difficult problem. This case was, as a matter of fact, rather interesting, for it involved a nonpositive definite operator, with the consequence that the eigenvalues extend from - 00 to
. -1- 00. The convergence of the method in this case depends upon the ratio IAo/ All and in
some cases ~1 was very close to - Ao, corresponding to a degeneracy in the iterated eigenvalue,
so that convergence would be slow if the method was applied without modification. One
may improve the convergence by employing the secular determinant (16) after the first two
iterations, or as it turned out it was easy to derive an expression which extrapolates to the
final answer and which is particularly applicable to the nearly degenerate case.
The variation-iteration method has only been employed for the lightest nuclei, A < 4.
The problem of extending this type of calculation, or indeed any of the others, to heavier
nuclei lies in the large number of coordinates involved and the consequent large number of
multiple integrals of many dimensions that would be required. Probably some approximate
technique such as that given by the Monte Carlo method would be necessary. In any event,
it would seem foolhardy to extend the calculations much above A = 4, in vi~w of the present
uncertainty in nuclear forces and the imminent possibility that some simplifying notions in
the physics may turn up in the near future.
In conclusion, we would like to compare the ease with which computing machines could
be utilized in each of the three methods mentioned. It is rather clear from the outset that the
variation-iteration" scheme is much more easily adapted to machine computation than either
the perturbation or the variation method. This is primarily because of the repetitive nature
of the operations involved, which simplifies considerably the number of directions-the number
of stored functions-that need to be fed into the input side of the device. When we combine
this considerable advantage with those already mentioned, it seems to be not too risky to
predict the increasing use of the variation-iteration method in nuclear problems.
REFERENCES
1. The real and imaginary parts of the spherical Hankel functions have been tabulated by the
Mathematical Tables Project, Tables of spherical Bessel functions (Columbia University Press), vols.
1 and 2. The required phases and amplitudes have been computed by Morse, Lowan, Feshbach,
and Lax in a report issued by NDRC, Division 6, entitled "Scattering and Radiation from Circular
Cylinders and Spheres."
259
HERMAN FESHBACH
2. For the most recent version of these perturbation formulas see E. Feenberg, Phys. Rev. 74, 206
(1948); H. Feshbach, Phys. Rev. 74, 1548 (1948).
3. H. Margenau and W. A. Tyrrell, Jr., Phys. Rev. 54, 422 (1938); H. Margenau and D. T.
Warren, Phys. Rev. 52, 790 (1937); H. Margenau and H. Carroll, Phys. Rev. 54, 705 (1938); H.
Nlargenau, Phys. Rev. 55, 1173 (1939); D. T. Warren.and H. Margenau, Plzys. Rev. 52, 1027 (1937).
4. H. Eliler, Z. Physik 105, 353 (1937·); S. Watanak, Z. Physik 113, 482 (1939).
5. D. T. Warren and H. Margenau, Phys. Rev. 52, 1027 (1937).
6. L. H. Thomas, Plzys. Rev. 47, 903 (1935).
7. N. Svartholm, Thesis (Hagon Ohlssons Boktryckeri, Lund, 1945).
SIXTH SESSION
Thursday, September 15, 1949
2:00 P.M. to 5:00 P.M.
AERONAUTICS AND APPLIED MECHANICS
Presiding
Harald M. Westergaard
Harvard University
COMPUTING NIACHINES IN AERONAUTICAL RESEARCH
R. D. O'NEAL
University of Michigan
I shall attempt to outline the possible application of computing machines, both digital
and analog, to some of the principal fields of aeronautical research. Although this symposium
is mainly concerned with high-speed digital machines, a discussion of the application of computing machines to aeronaQtical research could not be complete without considering analog
machines, because they have already proved themselves extremely useful. The fields of research
that I shall consider and the order in which I shall consider them are as follows:
1.
2.
3.
4.
5.
6.
Over-all flight-path problems for aircraft,
Stability of aircraft in flight,
Airflow studies to determine aerodynamic coefficients,
Structural analysis of aircraft,
Dynamic simulation of aircraft,
Traffic handling.
Although the use of computers in the last field will likely be more as a part of a system rather
than for aeronautical research, I shall want to discuss it briefly because I believe it is one of
the problems which most requires high-speed digital computing machines.
Let us first consider the general flight-path problem for an aircraft. I shall use the term
aircraft to mean both airplanes and guided missiles. The equations of motion can easily be
derived from consideri~g the forces acting on the craft. If we neglect external forces such. as
that of ,~ind, which mayor may not be small depending upon the ratio of the velocity of the
aircraft to that of the wind, the Coriolis force, which is a small effect even for very fast super~
sonic aircraft, and variations in the acceleration due to gravity, which is also definitely a small
effect, then the equations of motion for flight in a plane are (see Fig. 1):
= Tsin cp - Cn Sp(x 2 + j2)
My = T cos cp - C n Sp(X2 + j2)
Mx
sin
f) -
cos
f)
CL Sp(X2
+ j2) cos f),
+ CL Sp(X2 + j2)
sin
f) -
(1)
Mg,
(2)
where T is the thrust exerted on the aircraft by whatever propulsive system is used to drive
it, which will, in general, vary with altitude y; Cn is the drag coefficient, which is a function
of the Mach number and angle of attack, becoming quite high in the transonic range; C L is
the lift coefficient, which is also a function of the Mach number and angle of attack;. S is a
~haracteristiccross section of the aircraft; p is the air density; 0 is the angle that the velocity
vector makes with the vertical; cp is the angle that the thrust vector makes with the vertical;
M is the mass of the aircraft; !{ is the acceleration due to gravity.
26 3
R. D. O'NEAL
Since no a~alytic solution has been found for' the above set of non1inear differential equations, it is necessary to use numerical or analog methods to obtain the flight path. Obviously,
hand methods of numerical' solution can be very long and laborious for some flight paths.
The automatic digital machine can be extremely useful in solving these problems. The l\1ark I,
or a similar ,machine, caI?- handle the problem quite well. The use of computing machines
should make parametric studies of a 'preliminary aircraft design much easier and more economical. For instance, the effects of various climb programs" of various thrusts, and of other
parameters can be studied by varying each parameter within reasonable limits and studying
the effects of these variations on the over-all flight path. These parametric studies can be
carried out with analog equipment, providing a high degree of accuracy is not required and
L
Mg
1y
x
..
FIG. 1. Forces on airplane during flight in a plane.
providing--ruitable equipment is used. Special function equipment is required for representing
variation of CD and CL with Mach number, variation of air density with altitude, and possibly
variation of thrust with altitude. However, if high accuracy is required, digital machines
must be used.
In the foregoing discussion of flight-path problems, a stable aircraft was assumed. Some
of the major problems in aeronautical research are those involving stability. The problems
of stability and control are closely related. For instance, in order to keep the control forces
small, the static stability should be low. In fact, highly maneuverable planes such as fighters
may actually be statically unstable. The problems of stability and control may initially be
studied separately, particularly in case the aircraft is statically stable, and later tied together
when the over-all aircraft system, including pilot (either a human pilot or an autopilot) is '
studied. I shall not discuss the stability problem in detail because it is discussed f~lly by E. T.
Welmers in the next paper. However, I would like to say that for ·some stability problems,
264
COMPUTING MACHINES IN AERONAUTICAL RESEARCH
such as the linearized pitch-plane stability problem, electronic analog computing equipment
can be used. If one makes the assumptions that deviations from given flight conditions are
small, that acceleration has negligible effect upon the aerodynamic parameters, and that the
coefficients are constant for the duration of the analysis, then the stability equations are
ordinary linear differential equations that can be solved easily by electronic analog computing
equipment. Parametric studies to indicate the effects of variations over wide limits of each
of the parameters can be made fairly easily, and these are very useful in design. The accuracies
achieved with analog equipment are usually sufficient. It may be that even problems such
as these, which can be adequately handled by analog equipment, will be handled by digital
computing centers when such centers are more plentiful and when the programming time
can be ~educed. 'Vhether the problems will continue to be solved by analog equipment or
whether most of them will besolved by the digital centers will probably be determined mainly
by economic factors.
In order to make studies· of the performance and flight characteristics of an aircraft, we
have already seen that it is necessary to know certain aerodynamic coefficients, such as the
drag and lift coefficients. The solutions of the airflow problems are aimed at furnishing these
coefficients. In the past the computations for airflow problems have been largely performed
with the aid of standard desk machines and at a considerable expenditure of time and effort.
Digital computers can and will be very useful in solving airflow problems.
Let us consider, as an example .of an airflow problem, the p~rtial differential equations
describing in cartesian coordinates the flow about a body for an axially symmetric case:
(3)
(4)
In these equations, x andy are the rectangular coordinates ofa point in a fixed meridian plane
of the body, the direction of the x-axis being along the·axis of symmetry; u and v are x- and
y-components of the velocity of the air relative to the body; and a denotes the local speed of
sound. Of course, the appropriate boundary conditions· must be satisfied for any particular
configuration being considered .. These boundary conditions are (a) that the component of
air velocity normal to the aerodynamic body be zero and (b) that there shall be conservation
of mass, conservation of energy) and conservation of momentum across the shock.
Eq. (3) is hyperbolic for supersonic flow and elliptic for subsonic flow. Eq. (4) is the condition for -irrotational flow. Aerodynamicists have usually used one of two methods for solving
these equations for the condition of supersonic flow: (a) the method of linearization; or
(b) the method of characteristics, by which the foregoing system of partial differential equations
is reduced to an equivalent but simpler system of ordinary differential equations. The method
of linearization is not exact enough for many cases but often the results obtained with the
26 5
R. D. O'NEAL
method of characteristics have been no more accurate, because the labor of getting many
points in the step-by-step numerical solution, required when the method of characteristics
is used, has been too great, and so insufficient points wt:re obtained to assure good accuracy.
However, if high-speed digital computing machines are used, this difficulty will be overcome
and· the greater exactness of the method of characteristics can be realized. Indeed, at the
meeting of the Associationfor Computing Machinery, held at the Ballistic Research Laboratory
at Aberdeen in December 1947, the computation of the airflow about a cone-cylinder for the
axially symmetric case and for irrotational flow, that is, the flow governed by the equations
above, was discussed. At that time it was planned that a solution to this problem would be
carried out on the ENIAC, and I believe that this has been done. The general three-dimensional problem for supersonic flow has not yet been worked out, although Ferri has recently
. published a technical note on "The method of characteristics for the determination of supersonic flow over bodies of revolution at small angles of attack."!
For, airflow studies, large-scale digital computing machines will probably be most useful
in two general types of problems: (a) those involving a large set of similar problems such as
the work done at the M.LT. Center of Analysis on tables of supersonic flow about cones,2,3
and (b) as a research tool in helping to obtain a better understanding of the nature of supersonic
flow, particularly in studying the effects of viscosity and interference effects in aircraft. It
must be emphasized that the previous discussions of airflow have been for the nonviscous case.
Actually, the effects of viscosity may be rather large in some cases. For this reason exact
solutions of the above exp~essions would certainly not replace the wind tunnel for obtaining
aerodynamic coefficients of lift and drag. Rather, the high-speed digital computer can be a
valuable tool in conjunction with the wind tunnel for basic research. The effects of various
parameters can be more easily isolated by a method of numerical experimentation with a
digital machine than by physical experimentation with the wind tunnel. The aerodynamicists
with whom I have discussed these problems believe that the digital computing machine will
be a vahiable tool not only in working out the theory of visco-compressible flow, but also in
studying the interference effects in supersonic flow.
It is axiomatic that aircraft must be built structurally shung, but still as light as possible.
For this reason a large amount of effort has gone into structural analysis and the stress-analysis
problem on a modern aircraft is an extremely long and time-consuming task. The method
which is principally 1:lsed is the "unit method" described first by F. R. Shanley and F. P.
Cozzone. 4 Their paper presented not only improvements in the methods of analysis for
determining axial and shear stresses in box beams, but also a tabular method whicli permitted
a considerable saving of time and which can be adapted to machine methods. This method
consists essentially of dividing the beam structure, fot instance, an aircraft wing, into a number
of parts, taking cross sections normal to the length or longitudinal axis of the plane and further
subdividing the structure by longitudinal planes. To facilitate computations, flange material
is assumed to be concentrated into effective units that coincide with these spanwise divisions.
266
COMPUTING MACHINES IN AERONAUTICAL RESEARCH
The analysis begins at the outer or free end of the structure and works stepwise in to the fixed
end, each step using the results of the previous step. Only simple algebraic expressions have
to be solved for each step but the total amount of computation is large. IBM equipment is
being used to very good advantage by aircraft companies in this analysis. In fact, before the
IBM equipment was used, I understand that it was often difficult to do a complete structural
analysis for an airplane, especially taking into account all of the various loadings that might
occur for different flight conditions. It is quite an advantage to be able to do so, because
weak spots can then be found before planes are built and so a considerable saving in cost
can be effected.
Let us consider briefly the problem of simulating an aircraft system-that is, the aircraft,
and the pilot or autopilot that exerts the control on the aircraft. Such a system is a closed
servo-loop and may be treated as such. We may consider as an example the control of an
airplane in elevation or pitch by means of an autopilot. This case is considered in a reportS
by Hagelbarger, Howe, and Howe. First the equations of motion of (1) the airplane, (2) the
autopilot, and (3) the elevator are determined. Then each of these equations of motion is
8
I
8
1 ELEVATOR 11--__-t~ AIR PLAN E
1
I
L
AUTO- PILOT 1 Hm
' -_ _ _---oj
:1-.....--8
FIG. 2. Block diagram of aircraft system.
simulated by means of analog computers. Finally, the three components are tied together
so as to represent the complete system. This system is shown in block-diagram form in Fig. 2.
. The equ~tion of motion about the center of gravity for the airplane is
I··
MtJ ()
C
+ MtJ
() 1
C
M;;
() = 0 2
Zoo
J
odt,
(5)
where () is the angle of pitch, or the angle that the airplane makes with the horizontal; 0 is
the angle of the elevator with respect to the stabilizer, M tJ , C1, C2, and Zoo are constants, W
being the forcing frequency applied to the elevator. This single equation does not, of course,
completely represent the motion of the airplane, but is illustrative of the type of equation
that is used.
The constants used in the work mentioned above were taken from a report covering the
steady-state response of a B-25] airplane to sinusoidal oscillation. Eq. (5) is derived on the
assumption that the angles 0 and () are small so that the forward' velocity of the airplane
remains constant.
When the calculated steady-state response of the airplane was compared with points
obtained from the computer, there was '.lgreement to within the limits of error of the recorder
used in conjunction with the computer.
A circuit was designed that would give about the same gain and phase characteristics
26 7
R. D. O'NEAL
as for a B-24 autopilot amplifier, whose response curves were known, except for a frequency
ratio of approximately two, since it was assumed that the frequencies for a B-25 would be,
for the same response, about twice those for a B-24.'
It was assumed that the equation of motion of the elevator is of the form
(6)
where To(t) is the net torque applied to the elevator, I is the moment of inertia of the elevator,
C is the aerodynamic damping coefficient, and K is the aerodynamic restoring torque for a
unit deflection of <5.
RESPONSE 9
TO A STEP
INPUT 90 '
FOR B = 3.
urn (~( (( t( t( ~Hf( !ttl tt ("(-tt HH
IIIII
11\'I\\\II\I\\ll\II\\\\\\\\
ff'IIIIUNRH (t'
I'N'TIIII 111111111
I I I I I
iT II
I
r
I
1-+-
FIG.
1111111 ( I ( II ( I
1111111111111111
ro(,t I
I I
RESPON SE 9 TO A
STEP I N PUT 90 FOR
AUTOPILOT GAINFACTOR B = 15. .
I (I I ( f f I f 1 I I 1 I
'I
\ \
B -,
5 di'f lie
3. Effect of change in gain on stability;
I ~ was further assumed tha t
To = KIHm
+ K2 (J,
in which Hm is the hinge moment applied to the elevator and is equal to Bco, where eo is the
output of the autopilot circuit and B is the autopilot gain factor.
Now these three elements were tied together to form the complete,aircraft system. Again
checks were made of the steady-state frequency response to compare the calculated curves
with the curves determined by the computer. Again the agreement was good, as would be
expected. The resonant frequency of the system was measured, and the degree of stability
was studied as a function of the autopilot gain factor by using a step input signal. The effect
of this change in gain can 1;>e seen in Fig. 3. It is to be noted that for the higher gain, (J follows
(Jo much more closely and the static erroris almost zero.
With such a simulator it is quite easy to make parametric studies to find how changes in
various parameters affect the over-all stability. Various arbitr~ry disturbances can be put
into the system and the response of the system to these disturbances studied. A human pilot
might be substituted for the autopilot provided the forces that he would undergo in flight
could be put upon him, and so his reactions to various design changes could be studied.
268
COMPUTING MACHINES IN AERONAUTICAL RESEARCH
Although the analog computing equipment is limited in accuracy, there seem to be many
simulation problems that can be handled adequately by it. Again, as in my comments on
stability, 1 should like to say that digital computing machines may be used for these simulation
problems as high-speed digital computing centers,become plentiful and provided it is economically advantageous.
I believe that one of the most important uses of high-speed digital computing machines in
aeronautics' of the future may well be in the handling of air traffic. As the speed of airplanes
increases, as the number of flights operating into and out of each major airport increases by
a large factor, and as all-weather flying becQmes a reality, the present handling systems,
which depend upon human reaction times, will be inadequate. It is reasonable to believe
that the vastly shorter reaction time of a high-speed digital computer will be required to take
the positional information on each plane in the neighborhood of an airport and per~orm the
necessary computations to determine where it should fit into a complicated and rapidly moving
landing pattern. Airplanes with very different flying and holding speeds '(holding speeds of
the airplanes already vary by about a factor of two) and landing characteristics will have to
be handled and so a fairly large number of rules and airplane-performance data will have to
be in the machine in order that decisions can be made automatically by the machine, such as
the altitude at which the airplane will enter the landing pattern, the speed at which it shall
fly and rate of descent, how close it will be allowed to ~ome to other planes, and the turning
program to be followed. I do not mean to imply that the system of landing aircraft would
necessarily be entirely automatic. Actually, there would still be, a pilot in the airplane, but
he would be receiving his instructions from a high speed digital computer instead of from a
human controller. A standby human controller would have to be available for emergencies.
Such an automatic system for handling aircraft probably cannot be realized for several
years to come, not only because of lack of high-speed computi~g machinery, but also because
terminal equipment is not available. By terminal equipment I mean the devices for converting
the positional information, obtained from radars or other devices, into digital data that can
be handled by a computer, and for performing the reverse function of converting the digital
commands into intelligence that can be used by a pilot.
I have not discussed the use of computers for such problems as the reduction of flight dat~,
or for the preparation of design tables such as Professor Aiken's work done at the Harvard
Computation Laboratory entitled "Tables for the design of missiles,"6 but I believe that I
have discussed eno.ugh examples of the use of computing machines in aeronautical research
to indicate clearly that the aeronautical industries and the aeronautical research centers have
been using available computing equipment as it is developed. I believe that nearly every
aeronautical research center and industry now has analog computing· equipluent which has
either been purchased or built. Many of them have IBM installations, and I believe that
nearly all the large digital computing machines that are now working have already solved
problems in aeronautical research. I am very certain that the field of aeronautics will continue
26 9
R. D. O'NEAL
to use new computing aids as q~ickly ·as they are available. Problems such as the traffichandling problem may well tax the handling capacity of even the fastest and biggest machines
now contemplated.
REFERENCES
l. A. Ferri, NACA Technical Note .No. 1809
(Februa~y
1949).
2. M.LT. center of Analysis, Technical Report No. I, Tables of supersonic flow around cones; work
performed under the direction of Z. Kopal under NOrd contract No. 9169.
3. M.LT. Center of Analysis Technical Report, Tables of supersonic flow around yawing cones.
4. F. R. Shanley and F. P. Cozzone, "Unit method of beam analysis," J. Aeronaut.. Sci. 8, 246:....255
'(1941).
.
5. D. W. Hagelberger, C. E. Howe, and R. M. Howe, "'Investigation of the utility of an electronic
analog computer in engineering problems," University of Michigan Report, UMM-28, Project
MX-794, USAF Contract W33-038-ac-142~2 (Apri~ 1, 1949).
6. Staff of the Computation Laboratory, Tables for the design of missiles (Harvard University Press,
Cambridge, 1948).
27 0
PROBLEMS OF AIRCRAFT DYNAMICS
EVERETT T. WELMERS
Bell Aircraft Corporation
The upsurge of interest in computational devices and techniques, so well exemplified in
the Harvard Computation Laboratory, has been welcomed in all fields of aeronautics, but
perhaps nowhere so warmly as in the field of dynamics. The analysis of the flutter or stability
characteristics of a modern airplane involves so much computational work that in many aircraft
companies an equality sign is understood to exist between the words "flutter calculations."
A detailed discussion here of the derivation of the equations for the problems of dynamics
would be too lengthy, and actually unnecessary. However, the relation of these problems to
modern computing methods may be made more clear if some indications are given of their
sources. Three major problems will be considered, namt;ly, flutter, aerodynamic stability,
and servomechanisms. The order in which they are discussed is chosen only to permit certain
comments about their interdependence. Although generalizations concerning standard
methods of solving these problems cannot be made, the flutter problem has usually been solved
by digital computation, while the problems of servomechanisms have frequently been studied,
by analog methods.
The experience obtained from continued analysis and flight testing has established certain
broad policies for the guaranteeing of stability. To avoid excessive weight or configuration
penalties, to allow for unconventional designs or speeds, and to permit consideration of the
aircraft as a whole, all these require a careful analysis. The dangers involved in testing an
airplane, especially for flutter, are such that the time spent on analytical work can usually be
justified. The result has been that a rather complete theory exists for idealized problems and,
in many instances, experimental verification has been sought for a developed theory, rather
than theories devised for the explanation' of the physical phenomenon.
It must be emphasized that the problem statements and met,hods of solution discussed
here are not unique. In particular, adaptation to modern, digital .techniques may suggest
more convenient methods of attack.
Flutter can be described as a self-induced oscillation involving aerodynamic, inertial, and
elastic forces; at least two degrees of freedom are usually required, for example, wing bending
and torsion, or wing bending-torsion-aileron rotation. Above a critical velocity (or in a
certain velocity range) any slight disturbance of the airfoil results in an oscillation of increasing
amplitude, frequently sufficient to cause structural failure; below this critical velocity (or
outside the range) such a disturbance causes a damped oscillation. Symmetric and unsymmetric motions are usually considered separately.
EVERETT T. WELMERS
The determination of the inertial and elastic forces involved in the problem is relatively
straightforward. First attempts at determining the aerodynamic forces, in which they 'were
taken to be proportional to the instantaneous position of the airfoil, were not satisfactory.
The gradual development of a theory that included the oscillation frequency and the phase
relations culminated in a complete solution for the forces on an oscillating airfoi~ in incom- ,
pressible flow by Theodorsen, l Kussner, 2 Cicala, 3 and Kassner and Fingad0 4 about 1935.
The parameter on which these forces depend is bw/v, usually called the reduced frequency,
where v is the forward velocity of the' airfoil, b the semichord, and w the frequency of oscillation.
In extending oscillating-airfoil theory to compressible fluids, at each Mach number aero";
dynamic forces must be determined for various values of the reduced frequency.
The mathematical structure of the problem can be illustrated by the simple example of
wing bending-torsion flutter, and more involved cases can be discussed in terms of the notation
used there. Applying the differential equations for vibrating beams to the weight, inertia,
and stiffness distributions of the actual airfoil, or using an influence coefficient' method or a
Rayleigh-Ritz variational method, the fundamental bending and torsion frequencies Wh and
(i)cx, and normalized mode shapes h(x) an~ lX,(x), can be determined. The mode shapes h(x)
and lX,(x) are used as generalized coordinates qi' Because the airfoil properties are not simple
mathematical functions, the solution for qi is usually a step-by-step digital or a matrix process;
however, it is not an obvious problem for very large-scale digital computers, since no family
of solutions is required for a given structure nor is the same coding likely to be convenient
for different structures.
In matrix form, the differential equations of motion for a strip of unit width can be written
(Aq
+ Iq)q + Aq q + (Aq+ E~)q =
(1)
0,
where the matrices have the following interpretations: q is the column matrix of generalized
coordinates; dots denote differentiation with respect to time; the A's are square ma~rices
of aerodynamic terms whose subscripts indicate association with acceleration, velocity, or
displacement in the generalized coordinates; the elements depend on the reduced frequency
and the geometry of the airfoil, and will usually be complex to include phase differences;
lij is a square matrix of inertia terms; the main diagonal involves weight o~ moment-of-inertia
terms; the nondiagonal (or coupling) elements involve unbalances or products of inertia;
E is a diagonal matrix of elastic terms, usually expressed as frequehcies in the funda!llental,
modes. If a three-dimensional theory is considered, integrations over the span of the airfoil
are necessary. Eq. (1) is not changed in form, but the matrices A and Iij then involve the
assumed deflection mode shapes.
Using a method standardized by the Air Forces,5 consider the amount of structural damping
required to maintain simple harmonic motion. To do this, let q = qoe iwt and add another
term to the coefficient of q, namely iGq • For convenience, damping coefficients in all modes
are assumed equal, and thus
'
q
=
qoe iwt , iGq = igI; last term of Eq. (1) becomes (Aq
27 2
+ Eq + igI)q.
(2)
PROBLEMS OF AIRCRAFT DYNAMICS
If g > 0, damping must be added to give a 'steady oscillation, implying instability under
actual circumstances. Substituting Eqs. (2) into Eq. (1) and combining the aerodynamic
matrices gives
(3)
(A + Iq + Eo.)q~ = 0,
. where 0. is a complex eigenvalue,
(4)
involving a known reference frequency w a , and the required unknowns, frequency wand
damping g. Vanishing of the determinant of the matrix (A + Iq + EO.) is required for a
solution and determines values of o.. Substitution of an o. i allows solution for n - 1 of the
qo elements in terms of the other.
Change in the reduced frequency bw/v will be reflected only in the matrix A. For an
assumed value of reduced frequency, the eigenvalues 0i and the associated eigenvectors qOi
FIG. l. Damping g and velocity v for various values of bw/o.
can be determined. The status of stability is indicated by the sign of the imaginary part of
Oi' a positive sign denoting instability. Substituting the known Wa into the computed complex
eigenvalue, the frequency wand the damping g can both be determined. For the reduced
frequency chosen, substitution of the computed frequencywand the semichord b, gives
bw = k, v = (1)
-;
k bw.
(5)
A graph (Fig. 1) of the damping g and associated velocity v 'for a range of values of bw/v is
sufficient to determine the critical flutter velocity.
The determination of complex eigenvalues and eigenvectors in a problem of this type is
an excellent example of digital calculation; the expansion of the 2 X 2 determinant, the
solution of the resulting quadratic, and the determination of relative amplitudes (or the eigenvector) in this illustration is only a $mall problem for a desk calculator. I would like to indicate
directions in which the problem expands sufficiently to make its consideration at this symposium
justified. Complications of three types are obvious-enlargement of the matrices by introduction of more degrees of freedom, modifications of the aerodynamic matrix A, and modification
of the structural matrices I and E. These will be considered in inverse order.
Two possible reasons for modification of the matrices I andE can be proposed. First, it
273
EVERETT T. WELMERS
is not always possible to estimate with the desired accuracy the elements of the matrices.
Uncertainties as to root fixities, or even manufacturing tolerances in a wing attachment, will
cause a frequency variation; it is almost impossible to calculate the frequency of control-surface
rotation considering the linkages and supports involved; elastic axes of structures are difficult
to determine and, for structures with discontinuities, somewhat meaningless; effective weights
and inertias for concentrated masses are frequently difficult to evaluate .. As a result, it is
often desirable to solve the flutter problem not only for the best estimate of values for elements
of E and I, but also for variations of these which may lead to more critical conditions. If
such conditions appear, changes can be made in the final design or in fabrication that will
reduce the probability of their occurrence.
Rather than stemming from ignorance like the first, the other reason arises from the variety
of configurations under which the aircraft may fly, some of which will change its flutter
characteristics. Wing-tip external fuel tanks may be full, empty, or may have been jettisoned;
rockets or bombs may be loaded under the wings; different control-surface actuation methods
may be used. 'These modifications may also change the aerodynamic matrix, as in the case
of external wing-tip tanks extending far ahead of the wing. Thus a family of solutions may be
required for a single airplane.
Modification of the aerodynamic matrix A can be likewise justified on the basis of ignorance;
this has frequently been done in studying the flutter of tabs attached to movable control
surfaces. However, the chief cause for modification is to introduce compressibility or Machnumber effects. The aerodynamic forces determined by Theodorsen l were for an incompressible fluid, or M = 0; corrections to the M = 0 flutter speed based on the Glauert factor
(1 - M2) -1 were used for high subsonic speeds. 5 ,6 More exactly, the actual aerodynamic
forces in compressible flow have been determined by Dietze 7 and others, for values of M below
the critical value at which shock waves form. Above a Mach number of about 0.8, no theory
exists until definitely supersonic speeds are reached. The papers of Garrick and Rubinow, 8
Temple and Jahn,9 and others, based on the fundamental work of Possio,lo permit a consideration of the aerodynamic forces from M > 1 to hypersonic speeds.
One method of analyzing flutter in compressible fluid is to construct a stability graph
similar to Fig. 1 for each value of M. Only one velocity on each graph will correspond
to the Mach number for which the graph was drawn. The ~ocus of these points gives
the complete stability graph. (Fig. 2). A gap wil( exist in the locus f~om approximately
0.8 < AI < l.2.
The extension to more degrees of freedom perhaps is most quickly suggested by the existence
of high-speed digital calculators. Hand calculation passes rapidly from the difficult to the
inefficient to the impossible beyond four degrees of freedom.
Recent analytic and experimental work has demonstrated the importance of including the
rigid-body degrees of freedom, vertical translation and pitch in symmetric motion, and roll
for unsymmetric motion. Since there are no elastic restraints in these modes, the square
matrix E is of higher order than rank; thus, for symmetric motion the number of eigenvalues
a
274
PROBLEMS OF AIRCRAFT DYNAMICS
will be two less than the number of degrees of freedom, and one less for unsymmetric
motions.
The inclusion of additional portions of the airplane in the analysis is a second source of
additional degrees of freedom. To the simple wing' bending-torsion problem can be added
ai1eron rotation and tab rotation. Or the bending-torsion of a st~bilizer becomes a minor
in the problem of fuselage vertical bending, stabilizer bending, stabilizer torsion, elevator
rotation, tab rotation, and the rigid-body motions. After noting how a vibrator on the wing
cart excite the most remote parts of the airplane when at the correct 'frequency, it becomes
rather questionable to consider a wing-flutter analysis as separate from a tail-flutter analysis.
A unified analysis of a complete airplane may be unnecessarily involved, even for our best
computers, particularly if it is possible to reorder the degrees of freedom in such a way that
Q;
M=O
M= 1.2
M=O
FIG.
2. Stability graph for a compressible fluid.
elements of the matrices far off the main diagonal tend to vanish. There may be some advantage in coding a problem for perhaps ten degrees of freedom and forcing all flutter calculations
into that pattern.
In connection with an analysis of flutter in swept wings, Spielberg, Fettis, and T oney l1.
have adopted the following method of introducing the boundary conditions at the wing roots.
The various bending and torsion modes and frequencies are computed for the actual wing of
the airplane cantilevered without sweep at its root. Several of the lower bending and torsion
modes are used as generalized coordinates; the boundary conditions inherent in sweep are
introduced into the equations, and the eigenvalue problem is solved as before. Unless the
wing is complicated by concentrated masses, two or three bending and one or two torsion
modes are usually sufficient, making five to seven degrees of freedom for a simple bendingtorsion symmetric flutter including rigid-body motions. The same method can be used in the
solution of the flutter problem for unswept wings.
Another desirable increase in the number of degrees offreedom is suggested by the preceding.
A possible way of determining the vibration modes of a nonuniform beam or wing is to assume
modes fora uniform beam that satisfy the boundary conditions and combine them, by variational principles, into the modes of the actual wing. Similarly, modes for a uniform beam
can be considered as generalized coordinates in the flutter equation. Accuracy of the same
275
EVERETT T. WELMERS
order as that obtained in the previous method will likely require two to four more degrees of
freedom, a very costly penalty to pay without high-speed digital computers. Today, however,
it is not unlikely that uniform deflection modes for all problems will sufficiently simplify the
preparation of the problem for a computer to justify the added computation.
The problem of stability of an· aircraft differs somewhat from that of flutter in several
respects. Whereas we have been interested in determining a critical speed for flutter instability,
we are here primarily interested in the frequency and damping of oscillations at specific
velocities. The aircraft is considered to be a rigid body and only rigid-body motions are
studied. Finally; the air forces involved are those of steady-state aerodY!lamics, as contrasted
with forces for oscillating airfoils studied in flutter. These differences,' however, are more
traditional than essential. In fact, they are rapidly breaking down:
,
The basic problem may be described as the determination of the dynamic response of an
aircraft due to the introduction of forces or moments, either externally or by control-surface
deflections. The forces to be considered are inertial and aerodynaplic. Since quasi-steady
aerodynamics is assumed, there will be no phafie lags and the aerodynamic terms will.be real.
The complete system of equations of motion consists of six simultaneous differential equations
if control surfaces are assumed fixed, and nine such equations if the three control surfaces are
assumed free, that is, themselves movable under the action of aerodynamic and inertial forces.
In many instances subsections of the whole problem are of interest; for illustration, the
equations for motions of an aircraft in its plane of symmetry with free controls will be written. l2
The dependent variables are D..VjVo, the velocity change divided by original velocity; D..y, the
change in flight path angle; D..(), the· change in angle of pitch. Unprimed terms of the form
ai; involve aerodynamic· coefficients and perhaps certain initial conditions; primed terms
involve both aerodynamic and inertial forces. The differential operator djdt is represented
by D; the fi(t), frequently step functions, serve to introduce the disturbance instigating the
dynamic response. We have
(6),
Terms ail which multiply the velocity variable are dependent on the Mach number M, and
thus, solutions must be found for variou3 values of M. As in the case of oscillating air forces,
the transonic region, 0.8 < M < 1.2, remains questionable.
'Since only real elements in the matrices are involved, classical criteria such as Routh's
discriminant can be used for the indication of stable or unstable solutions. Laplace-transform
methods are ideally suited for determining the actual analytic solution. Analog computers
can usually be applied directly, and are especially useful for surveys involving numerous
parameter changes. Solution by a digital computer requires establishment of a sufficient
27 6
PROBLEMS OF AIRCRAFT DYNAMICS
number of cycles to permit determination of the frequency and damping. The degree to
which digital computers can furnish a solution to the more involved problems of this nature,
for example, those requiring nonlinear aerodynamic terms, is primarily a function of coding
difficulties. Standardization to a few basi.c and inclusive types may be the best method of
attack. Any efforts being made to narrow the gap between digital and analog machin~s will
be particularly' useful here.
'
The tr:aditional assumptions on the stability equations are no longer always valid. Aeroelastic effects introduced by the relatively flexible airplanes of today frequently dominate the
stability problem. In the case of a missile, this may only require introducing one additional
degree of freedom-fuselage bending ; the consideration of aeroelastic stability may require
doubling a fuselage skin thickness that was satisfactory. in all other respects. For an entire
airplane, such as a swept-wing bomber, the complete problem has probably never been stated
mathematically. The availability of large calculators bears directly on the interest shown and
effort expended on such a problem.
.
I t should not be assumed that the only difficulties in the problem are calculational. The
aeronautical engineer has considerable info·rmation concerning aerodynamics "in the large,"
total lifts and moments for rigid members. The effects of local deflections, or aerodynamics
"in the small," are on a much less satisfactory basis. Even though the equation coefficients
are not sufficiently well determined to permit a reliable solution, variations of the coefficients
ma~ lead to certain relatively invariant properties which may be of interest. The attempt to
obtain results with inaccurate data frequently points out the important errors, or comparison
of the unreliable calculated results with experiment may lead to better estimates of the
coefficients.
Another of the assumptions associated with stability-that aerodynamic forc~s depended
only on the instantaneous position of the airfoil-is now being questioned. German research
during the war indicated that, under certain circumstances, the use of the oscillating air forces
from flutter theory gave different results, even for the long-period phugoid oscillations.
Although the oscillations in the Theodorsen theoryl are assumed to be harmonic, other unsteady air-force theories' can be used for highly damped motions. l3 A proper evaluation of
the conditions under which unsteady theories are· of importance in stability must await
additional research activity.
Various electrical and mechanical' analogs to inertia, damping, and elastic forces are well
known. The basic components of servosystems lend themselves to the same analogs. As a
result, the appearance of systems of equations similar to Eqs. (3) and (6) in studying the
internal dynamics of servos is to be expected. The use of digital computers in solving problems
relating .to a servosystem itself is not immediately likely. The similiuity between the components of analog computers and servomechanisms is ,so pro~ounced that analog methods seem
to be simpler. Also, it is possible to combine servo units with analog computers to test systems
without knowing all the analytic details of the servo components.
277
EVERETT T. WELMERS
Rather· than illustrate the servomechanism a'spect by considering its internal stability, it
may be more informative to mention the stability or dynamic equations for a servo-controlled
airframe. Again the system lends itself to analog solution in many cases; availability, coding
simplicity, and unification are factors that will influence the use of digital machines. Mention
was made that stick-free stability involved additional equations, one for each control surface,
which described the inertia and aerodynamic forces on it. The control surface angle is here
determined by autopilot intelligence and is applied by a servomotor, rather than by inertial
and aerodynamic forces. Two additional equations are required for each control surface ..
Let E indicate the autopilot ~utput voltage and ~b the change in elevator angle. Only
motionS'in the plane of symmetry of the aircraft are considered. In one instance, Eq. (6) was
augmented in the following way:
D+all
a l2
a21
aat'
ala
0
a15
D+a 22
a 2a
·0
aa2
D2 + aaa'D + aaa"
0
a25
,
,
aa5
~V-
Vo .
~y
~O
au +.!t(t)
a24 + f2(t)
, (7)
aa4 +fa(t)
D2+KI D+K2 D2+Ka D + K 4 0
E
a44 + f4(t)
0
. Ks
0
~b_
0
_aM + f5(t)D+Kswhere the Ki are autopilot and servo constants. In many instances rapid'changes of altitude or
of l\1ach number will require variation of the coefficients au'
The general comments of the preceding section can be repeated for this case. In. one
respect the problem here is more serious; the desirable high natural frequencies of a servosystem are closer to the frequencies associated with flutter than to stability oscillation frequencies~ The flutter matrix will be augmented in much the same way as Eq. (7). ~he
effect of a high-frequency control-surface oscillation on, the autopilot intelligence is difficult
to evaluate. The addition of only one degree of freedom involving servo natural frequency
may present a sufficiently accurate picture. Servo flutter has attracted some analytic attention
recently,14 but the importance of the problem cannot now be properly evaluated.
0
0
If the "general analysis" philosophy of E. H. Moore is applied to the dynamic problems
being discussed, the mere association of the word "stability" with these three cases implies
the existence of a unified theory. This is being realized in practice, partly because efficiency
requires that methods of attack on similar problems not be contradictory, and partly because
a satisfactory airplane design depends on their interrelation. Realizing that sheer bulk of the
problem prevented any logical unified attack, the dynamicist has been unable to influence
design to the proper. extent. The jumps in the order of magnitude of possible problems that
have 'been brought about in the last few years now will permit him to attempt the unification
and contribute more directly to the design.
Perhaps one way of describing this unified dynamic problem is to consider frequency
responses. If the response of the complete airplane is known at .all frequencies from zero to
27 8
PROBLEMS OF AIRCRAFT DYNAMICS
beyond the highest structural natural frequency present, all the problems individually discussed
will be solved. The response of the airplane to slow control-surface deflections will be found
near zero frequency; abrupt deflections can be analyzed by considering various frequency
. components. Aerodynamic stability oscillations, with periods from a minute down to a second,
,vill supply the next peaks in the frequency spectrum. Flutter will contribute resonances with
frequencies as high as 40 to 60 c/sec.
Although the flutter problem discussed here is shown as a system of simultaneous algebraic
equations [Eq. (3)], and the stability and servo aspects as simultaneous differential equations
[Eqs. (6) and (7)], the unity still exists. The differences in statements reflect traditional ways
of handling. problems previously considered independent. In comparing calculation with
flight test, it is desirable to know solutions of the basic flutter equation [Eq. (1)] rather than
to introduce harmonic motions. The particular trends in solution methods for the unified
dynamic problem will depend on the computing machines used; thus the problem and its
method of solution are related.
As has been indicated, the problems of flutter have been solved digitally from the beginning,
while problems involving servos have been frequently treated by analog devices. Hand calculators have been replaced by punch-card methods as the number of degrees of flutter freedom
has increased. In several instances as many as eight degrees of freedom are consistently studied
with IBM equipment. In some cases, direct' expansion is carried out; in others, iterative
methods are used. Within the last few months a flutter problem involving five degrees of
freedom and four eigenvalues has been solved on the :rvIark I; in the solution, the 5 X 5
determinant with complex elements was expanded directly and the resulting quartic in Q
solved by approximation methods. The equations listed for stability have been used on various
digital machines in the calculation of flight paths for missiles and general dynamic-response
pro blems of aircraft.
Two factors influence the use of digital computers in aircraft dynamics. The first is
primarily educational. Tl:J.e type of problem considered, the degree of complication, and the
interpretation of results should all be influenced by the tremendously increased calculational
capacity. "Shotgun" methods are possible, that is, a variety of problems can be solved which
surround the somewhat uncertain location of the actual. Research activity should tend to fill
in the gaps of a unified theory. Only by a proper realization of the power of methods of
solution now ayailable can worthwhile problems be proposed.
The second factor relates to coding difficulties. Many problems which! could be done
quickly on large-scale digital machines require a large amount of coding and analysis· time
for trivial machine time. This immediately tends to discourage attempts at. machine solution
and allows tedious and inaccurate hand calculation to compete in efficiency. I have attempted
to indicate in this paper that numerous parameter changes require repeated solutions of similar
problems; also that it is possible to consider a large, inclusive problem as the basis, and extract
from it pertinent sections. Thus, if a flutter problem involving ten degrees of freedom and
eight eigenvalues were already coded, it would be possible to force a large variety of flutter
EVERETT T. WELMERS
problems into this form; in some cases many of the matrix elements would be zero, but use
of the same code would still be justified by the resulting standardization. Similar standardization of the other. dynamic problems is possible. And as e.£ficient high-~peed computers ~ecome
more available, a single coding tape for the c.omplete dynamic problem may be possible.
In conclusion, the problems of aircraft dynamics do have much in common. The increasing
complexity of aircraft and missiles has required extension of various aspects of dynamic
prob~ems beyond the possibilities of hand calculators. Utilization of large-scale computing
machines permits this necessary extension and also allows, for the first time, a study of the
, coupling between these problems, which have previously' been separated for simplicity in
analysis.
REFERENCES
1. T. Theodorsen, "General theory of aerodynamic instability and the mechanism of flutter,"
NACA Report No. 496 (1935).
2. H. G. Kussner, "Zusammenfassender ijericht iiber den Instationaren Auftrieb von Fliigeln,"
Ll!fifahrt-Forschung 13, 410-424' (1936).
3. P. Cicala, "Le azione aerodinamiche sui profili di' a]a oscillanti inpresenza di corrente
uniforme," Mem. reale accad. sci. Torino [2, pt. I] 68, 73-98 (1934-35).
4. R. Kassner and H. Fingado, "Das ebene Problem der Flugelschwingung," Luftfahrt-Forschung
13, 374-387 (1936).
5. B. Smilg and L. Wasserman, "Application of three-dimensional flutter theory t6 aircraft
structures," Air Corps Tech. Report No. 4798 (9 July 1942).
6. T. Theodorsen and I. E. Garrick, "Mechanism of flutter," NACA Report No. 685 (1940).
7. F. Dietz, "Die Luftkrafte des harmonisch Schwingenden Fliigels im kompressiblen Medium
beiUnterschaIl-geschwendigkeit;" DVL (Jan. 1943); AAF Translation F-TS-506-RE (Nov. 1946).
8. I. E. Garrick and S. I. Rubinow, "Flutter. and oscillating air.:.force calculations for an airfoil
in a two-dimensional supersonic flow," NACA Report No. 846 (1946).
9. G. Temple and H. A. Jahn, "Flutter at supersonic speeds," R.A.E. Report No. S.M.E. 3314
(April 1945).
10. C. Possio, "L'azione aerodinamica suI profilo oscillante aIle velocidt ultrasonora," Pontif.
Acad. SCi. Acta 1, 93-105 (No. 11, 1937).
11. Spielberg, Fettis, and Toney, "Methods for calculating' the flutter and vibration characteristics of swept wings," Air Materiel Command Eng. Div. Memorandum Report MCREXA5-4595-8'C""4
(3 Aug. 1948).
'
12. W. L. Mitchell, "Dynamics of aircraft 'flight," Bell Aircraft Report 02-981-006 (1948).
13. H. Wagner, "Uber die Entstehung des Dynamischen Auftriebes von Tragflugeln," Z.A.M.M.,
5,17-35 (1925).
14.
J.
Winson, "The flutter of servo-controlled aircraft," J. Aeronaut. Sci. 16, No. 7 (July 1949).
A
STATISTICAL METHOD FOR' CERTAIN NONLINEAR
. DYNAMICAL SYSTEMS
GEORGE R. STIBITZ
Consultant in Applied Mathematics, Burlington, Vermont
I would like to describe a way of treating certain nonlinear dynamical systems that
reverses the usual trend of expertness in scientific study. Everybody knows that the trend of
expertness is in the direction of learning more and more about less and less until one knows
everything about nothing. I do not want to suggest carrying the reverse process to a limit,
but I do want to propose, quite seriously, a method that gives us no information at all about
the behavior ora dynamical system in any particular cases, but tells us quite a bit about its
behavior in a lot of cases. In other words, the results are statistical.
This paper is divided into three parts; it discusses first ·Why statistical·results may be useful;
second, the theory of a statistical method that has been found useful in some problems; and
third, the application of the method, particularly showing how automatic computers will be
needed in this. application.
The parameters of the physical systems that we treat in applied mathematics are never
known exactly. Sometimes this ignorance is the kind we are born with and sometimes it is
acquired. In other words, 'there are problems in which parameters are not known exactly
because it is impossible to make perfect measurements, and there are problems in which we
do not want to bother about fixing their values. For instance, when we set manufacturing
tolerances, we are in effect saying that we could measure certain dimensions quite accurately,
but for- convenience we prefer to give them some latitude.
Sometimes small errors or variations are unimportant, and sometimes they are very
important. For instance, if I calculate my ~ime of departure for the railroad statio~ and adjust
my speed of travel aiming to get to the station just as the train is about to leave, a small
variation one way or the other determines whether I catch the train or miss it. I find, of
course, that the time of my arrival at my final destination depends very critically on minute
variations in the values of the parameters I choose. In this dynamical system there is a discontinuity in my response to the initial values, and in the thermal energy dissipated at the
station.
On the other hand, if I were to drive all the way to my final destination, then small variations in the speed and time of starting would not be so important.
Now, suppose I were commuting daily. Then the data that are most important to me are
not what will happen on any particular trip, but the number of times- per year that I miss the
train. I could make a statistical study of this factor as a function of my probable error in
estimating speed, of the time I use in eating another piece of toast, and sn on.
281
GEORGE R. STIBITZ
The subject of commuting has been treated rather fully by Streeter and Williams, so I
will leave that subject to them', and turn to a somewhat more dignified problem. The problem
that started the present study of a statistical method actually has many features in common
with that of commuting, but arose during the war when we tried to make a device called the
dynamic tester. The dynamic tester included a servo that made 60 attempts every second to
catch one of two trains in opposite directions. We did not care whether it caught the right
train on any particular trip, provided it did not get too far off schedule on the average.
Figure 1 is a rough schematic diagram of the servo. The tester is intended to put some
,equipment through its paces at normal speeds, following precalculated course. A motor M
drives the equipment being tested; every sixtieth of a second a control device C reads from a
punched tape what the motor position ought to be, and compares this position with the actual
position of the motor at that instant. If the motor is lagging, the control device fires a thyratron
tube and kicks the motor forward, while if it is leading the
required position, the control kicks it backward. Between
kicks, the motor coasts at practically constant speed.
Theoretically, the kicks are all of the same size, and the
motor catches one or the other every time. In practice,
the system was stabilized by supplying extra kicks whenever
FIG. 1. Schematic diagram
the sign of the error changed, but we will ignore this detail
of a control servo.
for the moment.
Several things are quite clear about this system. By no stretch of the imagination could
it be solved as a linear system. The impulses to the motor are not at all proportional to the
error. They are either full-sized positive or full-sized negative kicks. They cannot be linearized
by averaging over long periods of time, because the sampling interval is not short enough to
be negligible. On the other hand, we do not care about the exact history of the servo motor's
position, provided its probable error in following the course is small enougJ:!, and provided
the error never (or hardly ever) gets so large that the motor gets out of step.
Dynamically, this servo is very simple, if all the parameters, such as the initial position
and velocity of the motor, the course, and the size of the kicks are known exactly. Knowing'
the position at any sampling instant, we can compare it with the position required by the
course data, and determine whether the motor will be accelerated or decelerated. Adding or
subtracting the resulting increment of velocity, we can easily calculate the motor position
and velocity at the ll(~xt sampling instant, 1/60 sec later. This completes one cycle of the
operation, and we can repeat it as often as we like.
However, like the time of arrival at the end of a railroad trip, the motor position and
velocity are discontinuous functions of the initial conditions, and these are never known exactly.
Tiny variations in the initial position or speed would make the servo lead or lag at the next
sampling instant, and hence cha~ge the sign of the kick; and the entire future path would be
thereby altered, not infinitesimally, but by a finite amount. Luckily, this is one of the cases
where we are not interested in the details of individual runs, but in the statistics of many runs.,
a
A STATISTICAL METHOD
One of the vital bits of statistical information we need is the probability that the error will
exceed one half of a revolution, for when this happens, the motor' gets out of step and is lost.
With the servo taking 60 chances every second, this probability must be very small indeed,
if the number of failures is to be admissible. The probability of failure should not exceed
10- 5 at most. If we want to estimate such a probability by the usual method, we would need
to calculate something of the order of a million steps, for each' proposed design. Such a
program is staggeringly inefficient.
Evidently a new approach was needed. The servo I have been talking about is one of the
simplest cases of this kind of problem, but it is not the only one. In general, a dynamical
system that is subject to random influences of some kind, and that has severe nonlinearity or
actual discontinuity so that simple linear theory cannot be applied, may be suitable subject
matter for the statistical method.
With this brief statement of the reasons for wanting a method of treating the effects of
random influences in highly nonlinear dynamical systems, I shall outline the scheme that
finally solved the dynamic-tester servo problem, and suggest its extension to similar but more
complicated problems. It is the extension to larger problems that calls for automatic computers.
As background for this outline, let us recall some ideas that are probably quite familiar,
but that we shall use in a rather different way. The notion of a '''phase space" for a dynamical
system is very old, as such things go. The phase space for the dynamic-tester servo is very
simple. It can be represented on paper by making a graph with the position and velocity
(or momentum) of the motor as coordinates.
This is true because the state of the servo at any instant is completely determined when we
know its position and its velocity, and we can represent th~se quantities by a single point on
the graph. So we can calculate the future behavi·or. of the servo under any set of forces when
the location of its representative point in phase space is given.
As timt; goes on, the servo motor is kicked back and forth, and its position and velocity
change. Its representative point therefore moves about in phase space, and we could trace
its trajectory if we like; or, if we prefer, we can imagine that we take a series of motion-picture
frames of the phase space, each frame showing 'therepresentative point in a slight1y different
position.
The usual metho~ of dynamics traces the path of the representative point through phase
space, starting at an arbitrarily chosen initial position. The tra~ing may be done by an
analytic solution in some few cases, or it may be done step by step, as we said the dynamictester servo could be solved. The essential thing is that a rep!,esentativepoint is tagged, so
to speak, and, followed as long as its history is of interest .
. We have noted that the values of the variables and parameters (including the initial values)
cannot be measured or established with absolute exactness. Even if we tried to start the
system from a given point in phase space, we could not do so. All we can do is to start it so
that its representative point is somewhere near a given spot in phase space. If we made a
28 3
GEORGE R. STIBITZ
great many tries and plotted the representative point in each case, a microscopic examination
of the phase space near this spot would .look as if a shotgun had sprayed it, or as if a swarm
of bees had settled there. As time goes on, 'each representative point moves through phase
space, so the swarm of bees drifts along.
This swarm (to use the bee analogy) may stay in a compact mass, actually condensing
more and more, or it may disperse. In a stable dynamic system, the swarm keeps together
more or less, but in an unstable system the swarm expands without limit, either actually or
practically.· If the system is linear or mildly nonlinear, the members of the swarm are wellbehaved bees, s~ldom crossing each other's paths, and in general going along nicely side by ,
side. If the system is discontinuous, however; every bee makes sudden decisions, changing·
its mind, and darting through tJ:1e swarm. Despite this erratic behavior, the swar-m may keep
wi thin fairly definite limits.
In terms of this picture of a swarm moving through phase space, it is easy to point out the
distinction between the usual dynamic methods and the statistical approach. The usual
methods follow the flight of one particular bee, whereas in the statistical method we shall
study the motion of the swarm as a whole, observing its density and its tendency to disperse,
and noting what proportion of the swarm gets lost.
Now that we have decided upon a point of view we can see whether it shows us a means
for the practical solution of problems that arise.
Suppose that at any instant of time we have a picture of the phase space of a dynamical
system, showing a swarm of representative points. For brevity, I shall call them simply dots.
We can arbitrarily partition off the phase space into small cells, and we can count the·number
of dots in each cell. Then we can record these counts in the cells· of our graph.
As time passes, the dots move about from cell to cell, according to the dynamical laws of
the system, For the moment we can concentrate on two cells, A and B, that lie not too far
apart in phase space. ·Suppose that at time t = 0 there are DA dots in cell A and none elsewhere. The first frame of our imaginary movie then shows a density D.A in cell A, and zero
density everywhere else. The next frame, which we shall ~ay is 1/60 sec later, shows these dots
somewhat scatter;ed. A certain fraction of them, say N(A,B), has moved from cell A into
cell B, so that at this time there are N(A,B)D A dots in cell B.
Using the dynamical relations of the system, we can calculate the "transfer ratio" N(A,B)
for every pair of cells in phase space. When we have found the function N(A,B), we can put
our imaginary camera out of focus, so it no longer shows individual dots, but merely records
the densities of dots in each cell.
That is, we are no longer interested in individual runs of the system, but only in the density
of dots in a given cell at a given time. Obviously, we interpret this density as a probability.
The density in a given cell at a given time is proportional to the probability that the system
will be found at that instant to have the velocity and position corresponding to that ·cell.
I. have passed· very lightly over the construction of the transfer ratio N(A,B). Evaluating
this ratio is analogous to setting up the computing routine for the numerical solution of the
284
A STATISTICAL METHOD
dynamical system, in the usual treatment. It varies from problem to problem, and I will
explain later how it is set up for the particular example of the dynamic-tester servo.
I wish first, however, to review the general features of the statistical method. It will be
recalled that the whole scheme depends upon plotting the state of a dynamical system in a
phase space, so that when a representative point or dot is given, the future behavior of the
system can be calculated fronl a knowledge of the forces acting on it. In the case of the simple
dynamic-tester servo, the phase space is two-dimensional, with position and velocity of the
motor as coordinates.
The dynamic specifications for the system tell where any dot moves to in an interval of
time, but instead of following one dot throughout its path, we propose to deal with the density
of these dots. Using the same dynamic specificatio'ns as in the usual method, we construct a
transfer ratio that tells us how the density in each cell at the end of a short time interval is
related to the density distribution throughout the phase space at the start of that interval.
Instead of starting with an assumed initial point in phase space and following its path,
then, we start with an assumed initial density of dots representing the initial probability
distribution for the system, and apply the transfer operation to see how this distribution changes
with time. Vve frequently find that we can choose the coordinates so that the distribution
quickly settles down into a static pattern, unless the system itself is unstable. In the latter
case, of course, the density distribution spreads out more and more, approaching zero every~here in any finite region.
We have seen that the simple servo has a phase space of two dimensions, but ifit had more
degrees of mechanical freedom, the phase space would have more dimensions~ I think it is
clear why such a problem would require an automatic computer.
Having mentioned some problems in which a statistical method seems desirable, and having
outlined the scheme that was found useful in solving one of those problems, I want to give
a summary of the numerical methods and results of one simple example.
We shall reduce the dynamic-tester servo to its simplest form, and shall suppose that the
"course" called for by the control mechanism is identically zero. Let p be the angular position
of a brush carried on the motor shaft, measured in arbitrary units clockwise from the zero
position. For convenience in computation, these units may coincide with the size of the cell;
they might, for instance, be 10° of actual motor rotation. In the same way, the velocity q may
be measured in units of one position unit per unit of time. It is convenient to use the interval
between samplings as the unit of time. Then 1 sec,is 60 units of time.
We draw a picture (Fig. 2) of the phase space for this system, with coordinates p and q.
A dot at Po represents the servo at an instant when the motor passes through a position 3 units
clockwise from its zero position, with a velocity of 2 units. Ideally, th€ control would deliver
, a negative impulse to the motor whenever the dot is to the right of p = a in the graph. We
make the simple assumption that the kick is of unit magnitude, so that it instantaneously
changes the velocity by 1 unit, and we assume that the motor moves with constant speed
between kicks.
GEORGE R. STIBITZ
In the picture, we see two possible paths for the dot. If the motor were not kicked, its
dot would move to the right 2 units of position; because its velocity is 2 units. Since the motor
does get a kick, the dot moves down one space, because its velocity is reduced from 2 units to
1 unit. The dot therefore moves to the right one space, since the motor travels for 1 unit of
time with 1 unit of velocity. Similar paths can be found for dots in each cell.
q
3
p=-2 p=-I p=O p=1
~
1
-2
-I
p=2
Po
2
I
0
2
~-+~~~+--4--~--;-~
~~~~~~---~~-+----T---~
P2
4
3
q=2
r
5 p
q =1
q=O
q =-1
-I
q=-2
l
-2
FIG. 3. Motions of contents of some of
the cells of the phase space.
FIG. 2. Phase space for the simplest
form of dynamic-tester servo.
It will be convenient to express the change in position of the dot during one unit of time
by writing equations for the changes /)..p and /)..q in the coordinate values p and q. Then the
simple assumptions we have made are equivalent to saying that
/)..q
=-
sgn p,
/)..p
= q
+ /)..q,
where p and q are the coordinates at the start of the interval.
p
p
50
=0
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
t
=0
q
=0
50
50
=0
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
q
=0
50
t
=1
FIG. 5. The distribution of Fig. 4 after
one application of the transfer scheme.
FIG. 4. Uniform distribution of
1000 dots in 20 cells.
In this trivial case, it is easy to see that the entire contents of each cell moves into another
cell whose position is defined by the conditions on 6..q and /)..p just stated. Figure 3 shows how
the contents of some of the cells will move. We will carry the trivial case one step further by
distributing a thousand dots uniformly over the 20 cells shown in Fig. 4. Each cell· has 50
286
A STATISTICAL METHOD
dots, but since we are not interested in their individualities, we simply mark each cell with
the number of dots it contains. We apply the transfer scheme already discussed to this distribution, and find that the densities have shifted t6 the pattern of Fig. 5. One more application
of the transfer (Fig. 6) will be enough of this rather uninteresting example. We have already
p =0
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
q
=0
50
FIG. 6. The distribution of Fig. 4 after two applications
of the transfer scheme.
gone far· enough to see that the distribution shows signs of dissipating and, as a matter of fact,
the system will be found to be unstable.
To make an interesting and useful example, we need, first, to take account of the imperfections of the system, and second, to introduce a stabilizing mechanism.
It is clear that in practice the control cannot decide without error whether p is positive
or negative. Because of the finite width of the control brush, as well as backlash, vibration,
errors in timing the sample, and so on, there will be occasions when the motor gets a kick of
Table 1. Distribution of probability X 1000 at t
= O.
q
0
0
0
0
0
0
0
50 50
50 50
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
50 50
50 50
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
50 50
50 50
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
50
50
50 50
0
0
0
0
0
0
0
-1
0
0
0
0
0
0
0
50 50
50 50
0
0
0
0
0
0
0
-2
-9
-8
-7 -6
-5
-4 -3 -2 -1 0
2
3
4
5
6
7
8
9
P
1
the wrong sign. Of course; the farther the motor is from zero position. the less chance there
is for a mistake of this kind. In an actual application of the method, an estimate of the probability of such mistakes would be made by examining the mechanism. For our example, we
shall simply say that 30 percent of the dots in the blocks between + 1 and - 1 are subject
to error. Then 30 percent of the contents of the first column of cells to the right of p = 0
will move as if they received positive kicks, and 70 percent as if they received negative kicks,
with corresponding conditions in the left-hand side. All cells further removed will receive
proper kicks.
GEORGE R. STIBITZ
There are many factors that affect the size of the impulses~ as well as their time of application. Again, if ·this were an actual problem, we would need to examine the mechanism to
estimate the probabilities involved, but as this is merely an example, we arbitradly choose a
Table 2. Distribution of probability X 1000 at t
;:=::
1.
q
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Q
3
2
5
0
0
0
0
0
0
0
0
0
0
0
0
0
~O
35
15
2
0
4
0
0
0
0
0
0
0
0
0
0
0
5
43
35
15
2
0
0
3
0
0
0
0
0
0
0
0
0
0
5
43
35
15
2
0
0
0
2
0
0
0
0
0
0
0
0
2
8
48
35
15
2
0
0
0
0
1
0
0
0
0
0
0
12
30
48
48
30
12
0
0
0
0
0
0
0
0
0
0
0
2
15
35
48
8
2
0
0
0
0
0
0
Q
0
-1
0
0
0
2
15
35
43
5
0
0
0
0
0
0
0
0
0
0
-2
0
0
2
15
35 43
5
0
0
0
0
0
0
0
0
0
0
0
-3
0
2
15
33
40
0
0
0
0
0
0
0
0
0
0
0
0
0
-4
2
3
5
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-5
2
5
22
52
92
93
95
83
58
58
83
95
93 .92
52
22
5
2
Tota.l
-9
-8
-7
-6
-,5
-4
-3
-2
-1
2
3
6
7
8
9
p
0
1
4
5
Table 3. Distribution of probability X 1000 at t = 2.
q
0
0
0
0
0
0
0
0
0
0
0
o.
0
0
11
0
1
4
5
5
7
4
2
3
0
0
0
0
0
0
0
0
0
28 32 48 31
14
2
0
0
2
o. 0
0
0
0
0
0
0
2
15
35 51
49 40
15
2
0
0
0
0
1
0
0
o' 0
0
0
2
15 35 49
49 35
15
2
0
0
0
0
0
0
0
2
0
0
0
0
0
0
2
14 31
2
4
7
2
4
-9
-10
15 40 49
15
2
0
0
0
0
0
0
0
0
-1
48 32 28 11
0
0
0
0
0
0
0
0
0
0
0
-2
4
0
0
0
0
0
0
0
0
0
0
0
0
-3
9
19 38 67 75 92 99
99
99 99 92 75 67 38 19
9
4
2
Total
-8
-7
4
8
9
10
P
5
5
-6
-5
1
0
-4 -3
51 35
-2
-10 1
2
3
5
6
7
·simple law, and say that 10 percent of the contents of any cell will get kicks that are smaller
than normal by 1 unit, and 10 percent kicks that are larger. We shall select as .the normal
impulse, for this example, one that makes t1q equal to 2 units of velpcity.
288
A STATISTICAL METHOD
Next, we consider the question of stability. Actually, the dynamic-tester servo was stabilized
by making the impulses much greater whenever the error changed sign, but the analysis will
be a little simpler if we calculate the distribution when a friction flywheel is used. This flywheel
Table 4. Distribution of probability X 1000 at t = 3.
q
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
0
0
4
.0
0
0
0
0
0
0
0
0
0
0
0
0
3
12
12
2
0
0
0
3
0
0
0
0
0
0
0
0
0
0
3
16
34
34
14
1
1
0
1
1
2
0
0
0
0
0
0
0
3
15
38
46
49
28
11
9
9
5
2
0
0
1
0
0
0
0
2
12
29
43
31
39
39
31
43
27
12
2
0
0
0
0
0
0
0
2
5
9
9
11
28
49
46
38
15
3
0
0
0
0
0
0
0
-1
1
1
0
1
1
14
34
34
16
3
0
0
0
0
0
0
0
0
0
0
-2
0
0
0
2
12
12
3
0
0
0
0
0
0
0
0
0
0
0
o·
0
-3
0
0
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-4
1
1
3
9
25
47
75 108 111 126
126 111 108 75
47
25
9
3
1
1
Total
-10 -9
-8
-7
-6
-5
-4
5
6
7
8
9
10
P
-3
-2
-1 0
1
2
3
4
Table 5. Distribution of probability. X 1000 at t= 7.
q
0
0
0
0
0
0
0
0
0
0
0
0
0
1
2
3
1
0
4
0
0
o.
0
0
0
0
0
0
0
0
1
11
19
13
2
0
0
3
0
0
0
0
0
0
0
1
3
5
16
3.3
34
16
4
3
1
0
2
0
0
0
0
0
0
2
9
19
36
56
37
23
13
2
1
1
1
1
0
0
0
1
6
7
23
43
54
54
43
23
7
6
1
0
0
0
0
1
1
1
2
13
23
37
56
36
19
9
2
0
0
0
0
0
0
.. 1
0
1
3
4
16
34
33
16
5
3
1
0
0
0
0
0
0
0
-2
0
0
2
13
19
11
1
0
0
0
0
0
0
0
0
0
0
0
-3
0
1
3
2
1
0
0
0
0
0
0
0
0
0
0
0
0
0
-4
1
3
9
22
55
75
96
117 125 96
75
55
22
9
3
1
Total
-9
-8
-7
-6
-5
-4
-3
3
4
5
6
7
8
9
p
125 117
-2
-1 0
1
2
is loosely coupled to the motor, and slips for a short period after the application of each impulse.
Therefore the change in position is greater than it would be if no such slippage occurred.·
It can be shown that this is a stabilizing influence. Let ~y be equal to p
2~q, the figure 2
representing a stabilizing factor. Again, we ignore effects that would be taken into account
+
28 9
GEORGE R. STIBITZ
in an actual problem. The friction is not constant, the timing is not exact, the velocity of the
motor is not strictly constant, and so on. These things are easily taken into account, but we
omit them for simplicity.
Table 6. Distribution of probability X 1000 at t
=
11.
q
0
0
0
0
0
0
0
0
o"0
0
0
1
3
3
1
4
0
0
0
0
0
0
0
0
0
Z 11
16
13
1
0
3
0
0
0
0
0
0
0
1
3
5
17
33
34
16
5
4
1
2
0
0
0
0
0
2
11
22
38
54
37
23
12
2
0
0
1
Q
0
1
3
6
19
40
53
53
40
19
6
3
1
0
0
0
0
0
2
12
23
37
54
38
22
11
2-
0
0
0
0
0
-1 "
1
4
3
16
34
33
17
5
3
1
0
0
0
0
0
0
-2
0
1
13
16
11
2
0
0
0
0
0
0
0
0
0
0
-3
1
3
3
1
0
0
0
0
0
0
0
0
0
0
0
0
-4
2
8
24
48
74
93 123 121
121 123 93
74
48
24
8
2
Total
-8
-7
-6
-5
-4
-3
2
4
5
6
7
8
p
-2
-1 0
1
3
Table 7. Distribution of probability X 1000 at t = 12.
q
0
0
0
0
0
0
0
0
0
0
0
0
1
3
3
1
4
0
0
0
0
0
0
0
0
0
0
2
11
16
13
2
0
3
0
0
1
0
0
0
0
0
0
0
0
0
2
0
0
1
3
5
17
0
0
1
13
20
1
4
5
17
36
0
2
13
16
11
1
3
3
1
0
2
9
22
50
72
95 121 119
-8
-7
-6
-5
-4
-3
3
6
19
37
36
17
5
4
1
2
11 . 22
38
54
37
20
13
1
0
0
1
36
50
50
36
17
5
3
1
0
0
0
37
54
38
22
11
2
0
0
0
0
0
-1
37
19
6
3
1
0
0
0
0
0
0
-2
2
0
0
0
0
0
0
0
0
0
0
-3
0
0
0
0
0
0
0
0
0
0
0
-4
119 121 95
72
50
22
9
2
Total
4
5
6
7
8
p
-2
-1
0 1
2
3
Often it is simpler to tell where the contents of a cell go than to tell where the cell gets its
contribution from. In other" words, it is easier to express the distribution of density at the
end of an interval as the sum of a number of partial densities,each consisting" of the flow of
probability due to one of the contributory causes. This formulation seems to make for easier
A STATISTICAL METHOD
application of machine computation, also, since it reduces the amount of sorting required.
In the present example, for instance, we see that, except for the first column of cells, 10 percent
of the contents of each cell on the right move down one cell. If qo is the value of q from which
it moves, then this density moves to the right q - 2 cells. Only 70 percent of the first column
to, the right of zero is affected by ,this move, and only 30 percent of the first column on the left
is affected. This is a simple transfer, which is easily mechanized. It gives us a partial distribution. Similar partial distributions can be formed of the 80 percent that moves down two
i
~ ~ :;.~ :. ;~i:.· i!~.: :.~·~+.·;~.: i.~ t.:~.i ~ .~ .~1;f:~t; =\3·
.." ..:.:..... :......:.:.......:.:.:.:.....:..........:::...
...
..
j
.w·..
. . ..
:'4·
J ...:.:...
. ~ :. :.~ ;: ~: :.-·~j;:,~!.·t.):~'.!t:~;.·:l+.~ i.· :.~ r~ t~ :=~
--t..
.:•:..:..:.:•:.•. •:. . . ;. :. .:
::-
....
.~>;;lt~;~t,~~i,P:
t'~ .-
. ;"..(:J~W(ffr'ft~';'·:
----l
~~~~~~====~====~.
. .:./~ i~}!~f .i;'I-:;:_:'~
..:::/!0ff,1f1-f!r'.r:·: . 6 .'.
t----i'
t :
:.~ ;:l_ .:~r.;: I~.:jT.·~;:;~.~m~litf:=(~7·: 1--.-:::.~'.::.:·:.:.~:; :.·:;.:_. ;.:~ .:~ .: T.: i·.!~·.~ ;t~.g~M;t)·=:f-:;8:-·:_-1
- - 1 1 -...
---l::
••: l.:::..:..:.::. ::.:.:.::,.:
. ..::.::.......
...:..
..
..
..
..
. -
:. .
i.::.i.·.:...;:.......
..( . ;
-
FIG. 7. Results of applying the transfer scheme to an initial
rectangular distribution of density (see Tables 1 to 7).
rows and right q - 4 columns, and so on. Summing all such partial transfers, we have the
total transfer of probability or density.
Tables I to 7 give the results of applying the transfer for the simplified dynamic-tester
serv~' to the same rectangular distribution of density that we used as the initial distribution
for the trivial example. The same results are shown graphically in Fig. 7.
In this paper we have di,scussed dynamical problems, notably those about systems having
severe nonlinearities or actual discontinuities, for which we want statistical information. One
of these is the dynamic-tester servo.
Next,
statistical method has been worked out for following a distribution of runs of
a
29 1
GEORGE R. STIBITZ
such a system instead of dealing- with individual runs. The method consists of plotting' densities
or probabilities in a phase space, and of calculating transformations for such densities. In
practice the transformation can be carried out by automatic computers, and if the number
of degrees of freedom for the dynamical system is large, there will be a great many cells, and
automatic computers will really be needed.
Finally, we have glanced at a simple example, and have seen how the probability flows
around in phase space and gradually 'settles down to a steady flow pattern.
COMBUSTION AERODYNAMICS
HOWARD W. EMMONS
Harvard University
Fluid-mechanics problems have been developed to date with restrictive assumptions based
in part upon the problems whose solutions were sought and in part on mathematical conveni~
ence. By far the largest amount of work has been done on questions involving the flow of
incompressible ideal fluids, that is,· fluids of constant density and zero viscosity". Since the
beginning of this century considerable progress has been made in the extension of our knowledge of fluid mec~anics through the addition of studies of the effects of viscosity-thus
abandoning the ideal fluid-and, mo·re recently, of problems in the flow of compressible fluids
--thus abandoning the assumption of constant density. Most work on compressible fluids
has involved a specific type of compressibility, namely, that of the ideal gas.
In the study of compressible fluids the interesting phenomenon of shock waves appears.
These dis·turbances are studied directly in order to determine their fundamental nature, that
is, the variation of temperature, pressure, velocity, etc., through the shock wave itself. They
are studied indirectly by considering ideal nonsteady flow to determine when and how shock
waves first develop, and by studying the flow in regions between shock waves, the shock waves
themselves being treated as discontinuities.
In a few cases authors have treated problems that involve, besides the various phenomena
already mentioned, heat transfer between various parts of the fluid and between the fluid
and the walls. There is one very important phenomenon, however, that is excluded from
most of these treatments, namely, combustion.. Fluid me~hanics has progressed to the point
where this phenomenon, or at least the simpler aspects of it, can be added. Take, for example,
a phenomenon with which everyone is familiar, the flame· on a Bunsen burner. When this
flame is small, there is a very steady, sharp, central cone surrounded by a stream of hot,
somewhat luminous gas which fans out above it upon the top of the burner. When the flame
is large, everyone is familiar with its rather random oscillations. This phenomenon, which
involves the stability of a jet and the stability of the combustion process simultaneously, would
undoubtedly .be difficult to analyze. However, a small quiescent flame appears innocent
enough and suggests itself as an object for study. Curiously enough, this apparently simple
phenomenon has not to my knowledge been computed.
I need not spend any appreciable time enumerating practical problems in which a knowledge of combustion aerodynamics is important. Besides the simple gas flames such as that
already mentioned, which are used not only for laboratory work but also for cutting torche'.l,
welding torches, and other devices, there are all the various furnaces and combustion chambers
used throughout industry. In every case the aerodynami.c phenomena are responsible for
293
HOWARD W. EMMONS
accomplishing the distribution of heat in the desired way. At prescnt our knowledge of this
field is almost entirely empirical. Certain general principles are yet unknown. This is brought.·
out most clearly when one observes the difficulty encountered in attempting to change the
scale of a piece of equipment involving a flame.
Perhaps the most important approach available to tIle engineer on problems that are
inherently too complex for present-day computation is the possibility of testing on a model
scale and then using the resulting information for the design of the prototype. With combustion this is impossible, since the development of a small furnace gives few clues to the performance of another, say twice as large. In the present paper, the relatively simple problem of
the Bunsen burner will be considered.
It is .clear that the most important addition to present-day aerodynamics reqUIrIng
consideration in order' to include combustion is the interaction of chemical reactions with the
motion of the fluids. Consideration here will be limited to the combustion of premixed gases.
'Ve might note that this does not include all of the phenomena of importance, since very
frequently the fuel and air are not premixed., but, for example, a liquid fuel is sprayed in fine
droplets from a nozzle. These droplets must evaporate and the resulting vapor must mix
with air before it can burn.
Since we are here interested only in the aerodynamics of combustion, we need consider
only the chemistry of the problem to the extent to which this influences the fluid motions.
At this point we encounter a setback since, while the aerodynamics of combustion has an
almost nonexistent literature, the chemistry of combustion has a most extensive literature.
In spite of this, however, the setback is serious' since an examination of this literature shows
that in spite of valiant attempts the chemistry of combustion, which in large part is the chemistry
of reaction rates, is by no means well understood. In f<).ct, for even the' simplest of reactions
the lack of understanding is still tremendous. From our point of view a review of what is
known brings out the' following as important phenomena. The reaction which obviously
propagates into the unburned material from the burned gase~, thus continuing the reaction
in the neighborhood of the relatively stationary flame front, is propagated through a combination of effects. The. diffusion of chemical species from the burned into the unburned material
may act as chain carriers and thus initiate further reaction. Heat from the hot, burned gases
may propagate forward by thermal conduction (and perhaps in some· cases by radiation)
into the unburned gases and thus bring about, through the dependence of reaction rate on
temperature, the reaction in the unburned gas. The reactions themselves depend of course
upon the precise chemical nature of the combustible mixture. Ions and free radicals and
various molecular species are present in concentrations that vary from place to place through
the region of combustion. In fact, probably the best definition of the region of combustion
~ould be that region in which the composition of the gases differs significantly from the
reactants and combustion products. The chemistry of the reaction region is not only exceedingly complex but is ~till covered by an· extremely dense veil of ignorance.
294
COMBUSTION AERODYNAMICS
Thus there appear open to us in the study of the aerodynamics of combustion the same
two possible approaches as are available and are used in connection with shock-wave phenomena. On the one hand, we might focus our attention upon the aerodynamics of the flame
itself and ask for the variation of temperature, pressure, velocity, and, what is more important,
composition, and so forth, through the reaction region. Numerous such studies have been
attempted with more or less success in'the past in order to clarify the chemistry of the problem.
A great deal remains to be done in this direction and it is to be hoped that the study of combustion aerodynamics may indeed lead ultimately to the understanding of flame propagation.
On the other hand, we can focus our attention not on the reaction region but on the flow
before and after that region. Our success in this approach, like our success in the corresponding
approach to flow with shock waves, depends very largely upon the physical dimensions of the
com~ustion region. "If, like shock waves, the combustion region is indeed a small fraction of
an inch in thickness, then the assumption that it can be replaced by a mathematical discontinuity, will not invalidate our results. However, if the reaction region is wide compared
with other pertinent dimensions of the apparatus, our results would be without significance.
We will here make the assumption that the reaction region is thin and can be replaced by a
discontinuity. Thus we are considering those cases of combustion in' which the reaction is
essentially completed over a very short distance. At present it appears that for most premixed
gases this assumption is correct. Even in the case of luminous flames in which carbon particles
are obviously burning over a large region, the primary combustion which liberates these
excess 'carbon particles takes place very rapidly, the luminous region being present only
because of the mixing of the "burned" gases with additional air, thus permitting secondary
combustion of the carbon in a diffusion flame.
vVe will thus assume that a mixture of combustible gases in an equilibrium mixture arrives
at the flame front. These gases then- instantaneously react to an equilibrium mixture of
combustion products. Flame-front relations can be derived by the application of the continuity,
momentum, and energy laws to an element of the flame front. Such analyses have been made
many times under fairly general circumstances.
Since we are here intending to set up for solution the entire flow field, it is desirable to
simplify the problem as completely as possible. Thus it is to be observed that for a Bunsen
burner and other low-velocity (laminar) combustion processes, the pressure variations throughout the flowing gases are relatively small, as are also the temperature variations except across
the flame front itself. We may assume, therefore, that the unburned gases flow as. an incompressible fluid, and in addition that the burned gases also flow as an incompressible fluid.
We must, however, take into account the discontinuous change of density across the flame
front. We will denote the density ratio by n. Then
PI
qllt
- = '-,
(1)
qn.
where P is the density of the gas, qn is the component" of velocity of the fluid normal to the
flame front, the subscript 1 refers to the unburned mixture and the subscript 2 to the products
n
=
P2
295
HOWARD W. EMMONS
of combustion. For incompressib1e fluids, the value of n completely determines all of the
flame pr;operties except the flame-propag~tion rate St. The flame-propagation rate, which is
the velocity of propagation of the flame front normal to itself into the unburned gases, will
be assumed· to be cons tan t. Then
qri = qi sin (Jw = St,
l
qn
2
I
=
q2 sin:
(Jw. =
nSt,
(2)
where q is the velocity at the flame front and (Jw is the angle between the velocity vector and
the flame front.
It is clear that both nand St must be determined for the particular combustible being used.
For a given combustible, n can be computed with considerable precision from the equilibrium
properties of the burned and unburned fluids. The transformation rate St, however, cannot
. yet be computed; moreover, ~urrent experimental data, as interpreted, do not show St to
be an absolute constant. In fact, one of the principal' uses to which the present theory would
be put is to take accurately into account those aerodynamic aspects of flames which must be
understood in order to determine accurately whether or not St is constant for a given flame.
, 'Viih these two constants the flame-front relations can be completely determined. We
note from Eqs. (2) that the velocity normal to the flame front changes discontinuously from
St to n times this value'. The momentum equation written for an axis along the flame front
shows that the tangential velocity componentqt does not change;
(3)
Thus the resultant velocity changes discontinuously from ql to a larger value q2 which deviates
in direction from qi by an angle o. All these relations are shown in Fig. 1. The difference in
velocity components in a given direction making an, angle l' with the flame front is shown by
the geometry of Fig. l. Thus
(4)
At this point we shall restrict further consideration to two-dimensional flow. Thus, in
Fig. 1 we show x- andy-axes with a flame front making an angle (f. with the x-axis. For this
case, with v = (f., we get
(5)
,and with v = ( f . - 90°,
(6)
.
where u and v are the components of q along the coordinate axes.
- As will' be seen in the following, the only other flame-front relation we need is the discontinuous change in total pressure Po. To derive this relation we start with the momentum
equation written for an axis normal to the flame front. This gives
P2 -
PI =
PI St 2 (1 - n),
(7)
where P is the static pressure. The' total pressure Po, which is constant on streamlines between
discontinuities, -is now found from the 'Bernoulli equation,
P + }pq2 = Po.
29 6
(8)
COMBUSTION AERODYNAMICS
Thus we get, for the difference in total pressure,
POI
_
2 ( _ 1) (1
POI -. 2 n
_ P1St
+
cot
2
n
OWl) .
(9)
The flow of an incompressible fluid in two dimensions is described by the continuity and
irrotationality relations (using the usual hydrodynamic notation),
au
av _
0
ax + ay - ,
(10)
av _ au __ 2w
ax
ay -
We introduce the volume-flow stream function
'IjJ
by
- d'IjJ
U
= -
(11 )
.
ay' v =
a'IjJ
(12)
dX'
PRODUCTS OF
COMBUSTION
2w
'IjJ
ar
r
FIG. 2. Relation between rotation and
velocity variation.
FIG. 1. Aerodynamic flame-front relations.
Thus by Eq. (11)
=aq +.s..
is given by
( 13)
For an incompressible fluid the rate of rotation is related to the total pressure of the fluid;
the relation is most easily derived frorn Fig. 2. We have
2w =
aq
q
ar -to ;;
(14)
-but, by Eq. (8),
aq
ap
dr+ pq
apo
ar = Jr'
( 15)
and by the radial-momentum equation,
(16)
Thus
2w =
J.
pq
297
apo.
ar
(] 7)
HOWARD W. EMMONS
But
dtp
q = dr
(18)
Thus finaIJy the desired relation is obtained:
l,dpo
2w =
pd--;P.
( 19)
Since Po, the Bernoulli constant, is a function of tp, the rate of rotation is fixed on streamlines,
w(tp) . . We are now in a position to write the equations that have to be solved in order to
understand the aerodynamics of a ,simple two-dimensional combustion problem.
For the unburned mixture (subscript 1),
d 2'1PI
d 2tpl
(20)
dX2 + dy2 = 2w1 ;
for the products of combustion (subscript 2)
d2tp2
d 2tp2
dX2
~y2
+
=
2W2·
(21 )
For boundary conditions we must rely upon our: physical knowledge td assure ourselves that
we have a sufficient set to obtain a solution. For the unburned mixture we specify the flow
passage, by specifying, for example, the channel walls. In addition' the velocity distribution
must be given at some upstream point (perhaps x = - (0).
Thus· setting tpl = 0 on one wall, we compute the value of tpl on the upstream section
from the given velocity distribution. The value of tp on the other channel \\-all.is then found
and is equal to the total volume flow of combustible gas. These boundary conditions are not
sufficient to determine tpl, since as yet no conditions closing the domain on the flame side
have been given. We note that the vorticity distribution in the inlet stream is given by the
given velocity distribution by use of Eqs. (8) and (19). Since the rate of rotation w is constant
on streamlines, the vorticity distribution WI is determined simultaneously with tpl.
For the products of combustion, we again can specify a priori boundary conditions on;
three sides. Channel walls may be given if the combustion takes place within a passage, or
free streamlines may be specified as for a Bunsen flame. In either case, the stream function
is known by continuity:
(22)
For free streamlines the additional fact of constant pressure, hence constant velocity, is needed.
The free-streamline location will be given by the solution. At some downstream section
(perhaps x = (0), the pressure is taken as constant. This is a sufficient condition, since Eq. (8)
is a relation bet:ween P0 2 and q2 and hence between a function of tp, P0 2( tp) and its first derivative
q2 = dtp/dn, where n is normal to the as yet undetermined streamlines.
To complete the specification of the problem we must add sufficient conditions connecting
tpl and tp2 along the flame front so that tpl' tp2 an~ the flame-front location can be found. A
sufficient set of conditions is provided by Eqs. (5), (6), and (22) in the form
(22}
COMBUSTION AERODYNAMICS
(23)
(24)
To make the problem soluble, we yet require a method of finding CO 2' This is supplied
by Eqs. (9) and (19). From these we find at the flame front the relation
(25)
The only analytical solutIOn so far found to the above system of equations is the plane
oblique flame separating two uniform parallel streams.
If the combustible IS flowing along the x-axis with constant velocity U, the unburned
stream function is
(26)
"PI = - Uy.
If a plane flame at angle
(X
=
Ow passes through the z-axis its equation is
y' = x tan
(X
= mx,
(27)
and the resulting stream function of the products of combustion is
1 + nm2
(n - 1)mU
"P2 = -
1
+ m2
x-
1
+ m2 Uy.
(28)
Equations (26), (27), and (28) are the analytic expressions for the flame of Fig. 1 (with ql = U,
parallel to the x-axis).
The channel problem shown in Fig. 3 has not yet been solved but will be set up for ·solution
by two different methods.
The first is an integral-equation method in which the equations can be solved numerically
by an iteration procedure. It is based upon the observation that the flame front can be
considered a line source of strength density
d"P'
)
(--_.
'dl
- (n - l)S
If -
t,
(29)
where I is distance along the flame front measured to the right when crossing with the fluid,
and the subscript indicates differentiation along the flame front. In addition to the constant
source strength, the flame front position is determined by the condition that it propagates at
the constant rate St;
(30)
We now have the entire flow specified by Eqs. (12) and (13). The boundary conditions
on four sides~inlet, outlet, and channel-are the same as before, no distinction being made
between' burned and unburned fluid. The separation of burned and unburned fluid is accomplished by setting a line source of strength given by Eq. (29) in such a location as to satisfy
Eq. (30).
299
HOWARD W. EMMONS
The channel of Fig. 3 imposes the specific boundary conditions
=
°
1jJ
at x =0,
= 1 at x = 1,
1jJ
=
1Jl
d1jJ
dY
=
x aty -->-- -
°aty
(31 )
(32)
00,
-->--00.
(uniform parallel stream of combustible at inlet with
velocity v = 1)
(33)
(asymptotically constant velocity on each streamline-this
is equivalent to the constant-pressure condition)
(34)
y
STREAMLINE
I
I
I
I
I
I
I
I
I
I
--L.REFLECTED
I PASSAGE
I
I
I
I
I
I
\
I
" \ 1/
I I
~
I
I
I
2
x
INLET
VELOCITY
DISTRIBUTION
FIG.
3. Combustion in a channel.
The foregoing variables may be considered dimensionless if we use as the unit dimensions
the width of the channel, the fluid velocity at the inlet, and the total inlet volume flow.
The velocities induced at any point (x, y) by a source, of strength Q at the point (xo, Yo)
(including (- xo,Yo) and aU image points) to satisfy the boundary conditions are
Q[
uQ
.sin 7T(X - xo)
cos 7T(X ~ xo)
="4 cosh 7T(Y-Yo) = Qqu (x,y, xo,Yo) ,
+
, sin 7T(X
xo)
Yo) - cos 7T(X
+ cosh 7T(y -
+ xo)
]
(35)
300
COMBUSTION AERODYNAMICS
v
[
sinh 7T(Y - Y o ) ·
4 cosh ir(y - Yo) - cos 7T(X - xo)
= -Q
Q
sinh 7T(y + Yo)
+ _-::----;-__
-:-=_---=--..c~-_
cosh 7T(Y - Yo) - cos 7T(X
]
+ xo)
= Qqv(x,y"xo,yo),
(36)
The corresponding velocities induced by a vortex of strength
the same boundary conditions are
.
r
at the point (xo,Yo) meeting
Ur = - rqv(x,y, xo,Yo),
(37)
= rqu (x,y, xo,Yo)'
(38)
Vr
Let the flame be situated along the liney, = y,(x,). Ifwe suppose this line known we can
write down the velocity components at any point (x, y) for the solution to Eq. (13) using as
the source strength
)211 'dx,
Q = d1J/ = (n - I)St dl = (n - . I)St [ 1 + (dy
d~
(39)
and as the vortex strength
(40)
The velocity components are
u(x,y)
=
(n - I)S,
r (:;rr
[I
+
qu[x,y, x/,y/(x/)]dx/
- J.I dxof_CX)CX)w(xo,yo) qv(x,y, xo,yo)dyo,
v(x,y) = (n- 1)S.J:
(41)
{I + (tJf 1J.(x,y,x/,y/(x/))dx/
+
fol dxof_CX)CX)w(xOyo)qu(x,y, xo,yo)dyo
+
Vo,
(42)
where Vo is a constant to be selected to provide the given inlet velocity. We now use the.
condition that the flame propagates at a fixed rate St-Eq. (30)-in the form
dx,
St = v (f[ -
U
dy,
(f[.
(
v-
U
dy') [
dx, I
+
( dy,)
dx,
2] -
1
(43)
Thus the integral equation to be solved for y,(x,) becomes
U+ (tJr = (n- I) r[, + (tJf[qv(xt,y"x/,y/) -
t; qu(xt,Yt,x/,y/)]dX/
(44)
The functions qw qv have poles at (x/,y/) = (x"y,). The first integral on the right is to
be taken around the pole on the side of the burned fluid, that is J'/ > y, at x/ = x,. If the
301
HOWARD W. EMMONS
principal value of the first right-hand integral is' taken, the value obtained in passing around
the pole must be added. This value is
- (n-; 1) [1 + (XJf.
To determine the constant
strength as
(45)
it is simplest to integrate Eq. (29) to get the total source
Vo
"P' = (n - 1).
(46)
Since aty = - 00 (the inlet) this fluid is uniformly confined between x "',0 and x
velocity induced at the inlet by the flame source is
Vi
=
=
- "P'
- (n - 1)
2~x =
2
1, the
(47)
The vorticity which is confined to the burned gases in the present channel problem induces
no velocity at x = - 00. Hence, since the resultant velocity was given as unity,
1
=
Vo
+ Vi.
(48)
Thus finally the integral equation becomes
n
t 1[1 + (x:rf
= (n -
+ 1)
1) f [1 + (1:,:r1' [q.(Xt>YP/,Y/) -
X~ qu(x"y" x/,y,:)] dx/
.f.l
[
qy,
]
+ ° dxo w(xo,yo)
St
qu(x"y" xo,Yo) + dx, qv(x"y" xo,Yo) dyo,
t
where principal values are to be taken of all integrals.
The solution proceeds by first assuming a straight flame front,
+
(n
2S
(49)
(50)
where the slope is obtained by supposing the inlet velocity to be unaltered up to the flame.
Note that (50) satisfies Eq. (49) if the integrals are ignored.
Now substitute (50) in the first integral on the right ofEq. (49), ignore the second integral,
and solve for dy//dx/. By integration computey,/(x/).
In the next approximation it is necessary· (in a channel) to include the second integral in w.
To do this we note that WI = 0 while W2("P) is not zero but is given by Eq. (25). To find tp
we start with the complex potential for a source and integrate over the flame front and fora
vortex and integrate over the products of combustion.
"P = "Po
+ "PIX -
(n - 1) . .
27T S tZp
l+iY/(l)
f.
+
0
• 7T ( Z - Zo ) • 7T ( Z In SIn
2
SIn
2
1
27T
V •
J
P
.
-
Zo
)
. dzo
(51 )
+
7T(Z - zo) • 7T(Z
z~)
w(x,y) In SIn
2
SIn
2
dxodyo·
rCJ!ion bchind
flame front
The constants ."Po and "Pl. are found from the conditions at the inlet, while w(x,y)
302
=
w2 (x,y)
COMBUSTION AERODYNAMICS
is found from the previous approximation. Taking the real and imaginary parts of Eq. (51),
as indica ted,
'f/l = .'f/lo +
'f/lI X -
tanh~
I)Stfl('
)!
(n-1 tanh; (y- Yo)
-1
(y- YO)) (
dYo
2n
tan n w
+ tan
1 + -d
dxo
~
tan 2 (x - xo)
tan '2 (x + xo)
-
along
flame front
1
+ 2n
J
I cosh n(y - Yo) - cos n(x.- xo) d d
w(xo, Yo) n cosh n(y -,)'0) - cos n(x + xo) Xo 910·
region behind
(52)
flame front
Aty =
-00, 'f/l.=
x, but by Eq. (52)
'f/l = 'f/lo
(n - 1)
+ 'f/lIX + -2-"
(1 - x).
(53)
Thus
'f/lo
~
-
n-l
n+l
(54)
-2- , 'f/ll = -2- ,
in agreement with Eqs. (47) and (48). For the first approximation we find "P2' from Eq. (52)
by inserting the value ofy,' just found and w 20 = o. Now W 2 can be computed fromy,', 'f/l2'
and Eq. (25). An iterative solution now proceeds by replacing Eq. (50) by y!'(x') and repeating
the previous steps. This time, however, wl is used in the integrals.
In view of the involved nature of the integral-equation attack just outlined it seems desirable
to consider also another method based upon difference equations ..
For this purpose it is most convenient to introduce the stream function 'Y based upon
mass flow, This can be conveniently done in terms of the previous stream function
(55)
The basic Eqs. (20) and (21) applied to the flame in a channel yield
'1"11
'1"2
1
+
'1"1
+
'1"22
2
+
'1"1
3
+
+ 'Y 23 +
4
'1"1 -
4'1"1
4
'P'2 -
4'1"2
0
0
= 0,
= 2€5 2W 20,
(56)
where the superscripts refer to values at a point 0 and its surrounding points 1, 2, 3, 4 on a
square net of points of spacing €5.
The boundary conditionsofEq. (31), (32), and (33) are now used directly. The boundary
condition of Eq. (34) is applied by assuming that '1"2 = x at first, and then progressively
correcting this by the equation
(57)
which can be used as soon as an approximation to (02(X,y) is available, that is, as soon as a
first approximate solution to 'I" and the flame front has been obtained.
HOWARD W. EMMONS
In deriving Eq. (57) use was made of the flame internal boundary condition
(58)
'1"2 = '1"1'
which follows from Eq. (22).
Equations (23) and (24) expressed in finite-difference form in terms of'F1 and '1"2 complete
the boundary conditions required to find the location of the flame front and the stream-function
solution.
Any technique for the solution of the finite-difference system is suitable. The following
appears to be a good method of handling the flame-front conditions, for either· a relaxation
calculation or a computing machine. A square net of points is placed to cover the entire
channel. A flame front is placed in the channel by guess. This flame of course falls between
net points almost everywhere. Values of '1"1 and '1"2 are placed at the net points in their
respective regions by guess in the usual way. At every net point adjacent to the flame front
a value of'F1 and '1"2 is placed. Thus in a band of points along the flame front there are values
of the stream function 'I" appropriate to both burned and unburned material.
Equations (23) and (24) expressed for pairs of points between which the flame passes
provide a relation between the boundary values of '1", that is, the value of '1"1 on the (2) side
and vice versa. During the course of solution of the (1) and (2) regi'ons the flame-front boundary
equations become increasingly in error. Periodically, therefore, these equations are used to
readjust the boundary values by starting at the center line of the channel where the flame is
assumed held and recomputing the 'I" boundary values in steps along the flame front. This is
the finite-difference manner of flame-front characteristic propagation.
Finally, the flame front itself is located between net points by use of the equaiity of 'Fl and
'I" 2 at the front, using linear interpolation between nearest points Qn either side.
The theory of flanles herein described is a first attempt to solve the whole aerodynamic
problem of combustion in a form that will permit a careful check of the streamlines and flamefront position with those obtained experimentally.
Although the theory is stripped to its barest essentials, it is still of such complexity as to
be solvable by the relaxation ~ethod only with considerable labor. It is to the computing
machine that those wishing to solve nonlinear boundary-value problems must increasingly
turn if our growing empirical knowledge is to be supported and guided by a real understandi~g
of the phenomena involved.
ApPLICATION OF. COMPUTING MACHINERY TO RESEARCH
OF THE OIL INDUSTRY
MORRIS MUSKAT
Gulf Research & Dellelopment Company
The writer is neither an analyst nor an electronics expert .. Nor is he qualified to speak at
all about computational problems as such. At the most, he can claim to be playing the role
of an interested spectator of the rapidly developing science of computation.
This paper will be unique in this Symposium. In contrast to the others on the program
it will contribute nothing to the problem of computing-machinery development. It will not
discuss the analytic aspects of specific computing problems, nor exhibit any completed solutions
of mathematical equations treated numerically 'or by large-scale computers. If it will serve
any purpos~ at all beyond fulfilling a promise made to Doctor Aiken, it may be that of stimulating the development of computing services from the point of view of the industrial user as
well as that of the computing organization itself.
Having been in the oil industry for 20 years, I might well be expected to be able to discuss
authoritatively the subject of computational problems in all phases of the oil industry, as the
title of this paper suggests. Unfortunately, however, specialization in the oil industry is as
severe as in many other engineering fields, and anyone individual must content himself with
developing complete knowledge in at most very restricted aspects of the whole industry. In
the case of the author, his personal technical activities have been largely confined to the
physics· of oil production. Nevertheless, for completeness, brief reference will be made to the
basic types of mathematical problems arising in other pha.ses of the oil industry,' although no
attempt will be made to do more than' exhibit some of the fundamental equations involved.
It is only with respect to the equations arising in the physics of oil production that the author
has had direct experience involving the 'use of large-scale computing machinery.
To the author's knowledge no serious attempt has yet been made to apply large-scale
computing machinery to solve the basic problems of geophysical prospecting, oil refining,
or lubrication. The reason, of course, is that the fundamental equations underlying these
subjects were formulated long before the development of large-scale computing equipment
(this term is used throughout the paper to mean digital rather than analog computing
machinery) and in those cases where solutions were urgently required dir:ect numerical methods
were applied or such approximations were introduced as to make the equations analytically
tractable. While undoubtedly the availability of the powerful computing facilities currently
being developed will stimulate their application to problems arising in future research, no
active interest in immediate applications in the fields of geophysical prospecting, refining,
MORRIS MUSKAT
or lubrication seems to have yet materialized.' Accordingly, we shall merely list some of the
governing equations pertaining to these fields as indicative of the types of problems that may
be proposed for computational analysis in the future.
,
~he three major types of geophysical prospecting having widespread application in the oil
industry are known as the gravity, magnetic, and seismic methods. In effect, they are all
composed of procedures of making measurements at the surface and inferring from these
the nature and ,geometry of the subsurface rocks which presumably give rise to the surface
data. Gravity and magnetic prospecting are both based on potential-theory principles, and
their aL llytic aspects are quite similar in many respects. It will suffice for our present purposes
l
to not( h.erely that the problem of "gravity interpretation'~ is essentially equivalent to that of
solving' ,the integral equation:
,( ) , k
T)dT
(I )
gz X,Y
r3'
Iza (
where g z(x,y) is the vertical component of the acceleration due to gravity measured at the
surface-the x,y-plane; a(T) is the density "anomaly" at the volume element dT lying at the
depth z below the surface and at the distan~e r from the origin; and k. is a constant. The
quantity g z is to be cons~dered as the "reduced" value of the acceleration due to gravity after
correction for surface terrain and the uniform contribution due to an ideal subsurface of
constant density.
There'is a voluminous literature on the practical solution of Eq. (1) and its analogs for
magnetic prospecting by indirect and approximate procedures,l and the direct solutions of
the integral equations corresponding to simplified forms of Eq. (1), including questions of
their uniqueness, have been investigated' quite thoroughly.2 From a practical standpoint,
therefore, there seems to be but little urgent need for undertaking additional analysis by
large-scale computing equipment.
In seismic prospecting the situation is essentially the same, although there have been very
few fundamental investigations of the mathematical aspects of the seismic method. Virtually
all procedures for interpreting seismic data~re limited to evaluations of the times of travel of
the various reflected
or refracted waves in t'erms of depths and velocities of assumed surfaces
.
of discontinuity in the underground strata. The wave equation, as such, plays no direct role
in the application of seismic data. It seems unlikely that machine computation will be called
on in the study of seismic prospecting, except possibly for long term investigations of phenomena
which may arise in media of continuously variable elastic properties.
The term "refining" encompasses such a vast scope of technical activities that no single
problem can be properly considered as typical. The theories of catalysis, fractionation, solvent
extraction, distillation, and chemical kinetics all provide potential subjects for detailed investigation by machine computational methods. Of these only the last will be exhibited as an
illustration. This may be expressed by the set of equations
,
dN i = '",
N'
~ aii i
dt
J
+ ~'" 'b iikNi N k + .
Jk
306
(2)
APPLICATION OF COMPUTING MACHINERY
describing the homogeneous-phase kinetics of the interactions and transformations between
m molecular species of instantaneous concentrations N l , • • • , N m with reaction-rate coefficients
aij) biik , • • • which are to be considered as empirically determinable constants. The Ni(O)
are also to be assumed as known.
The nonlinearity of these equations evidently makes the development of general analytic
solutions impractical except for extremely. specialized simplified cases. While it is doubtful
whether the oil industry would support large-scale computing programs for solving these
equations as a matter of general interest, it is not inconceivable that special circumstances
may arise where the immediate potential applicability of the solutions would warrant the
computational treatment of specific sets of these equations. The problem of determining the
equilibrium distributions-where dNddt = O-among the Ni for heterogeneous reactions has
been given special analytic study in such a form as to facilitate computational treatment, 3 but
no similar analysis of the transient problems even for gas reactions has yet been reported.
From an analytic standpoint the only phase of lubrication which has been sufficiently well
crystallized to lead to strict mathematical formulation is that known as "thick-film" or
"hydrodynamic" lubrication, in contrast to "boundary" lubrication where surface phenomena
and interactions are superposed to an important if not predominating degree on the strictly
hydrodynamic effects. The basic equation was developed as long ago as 1886 by O. Reynolds.
When generalized to include thermal effects on the density and viscosity of the lubricant,
though neglecting centrifugal forces and heat transfer by thermal conductivity within the film,
two interdependent equations are required, namely, 4
a [yh( 1 ax
(1 -
2
3
h ap)]
a
(yh ap)
61l U dX - ay -gilD ay
= 0,
h2 ap) aT h2 ap aT 2p,U{
h4 [fap)2 (ap)2]}
61l U ax dX - 61l U ~y ay = yCh 2 1 + 12p,2U2 \ax + dy
,
(3a)
(3b)
where p, T are the lubricant-film pressure and temperature at (x,y); y, fl are the density
and viscosity; C is the specific heat; U is the velocity, in the x-direction, of the moving surface;
and h is the lub~icant-film thickness. When the thermal effects are completely neglected and
y, It are taken as constant or functions only of the pressure, Eq. (3a) becomes independent of
Eq. (3b), which can then be solved, in principle, in sequence. However, even then Eq. (3a)
still remains virtually intractable analytically except when the film thickness h is of extremely
simple form, such as the thrust-bearing wedge, or when the bearing is assumed to have infinite
width (a/ay= 0).
The complexity of Eqs. (3) would suggest that only large-scale machine computation could
cope with their solution in a practical manner. However, it has been founds that a numerical
treatment by relaxation methods is quite feasible even when the thermal effects are taken.
into account. There are many special lubrication systems for which the specific solutions of
Eqs. (3) would be of considerable interest. But in view of the power of the relaxation method
it is doubtful whether these will call for the application of large-scale digital computing equipment. The only immediate possibility would appear to lie in the investigation of dynamically
30 7
MORRIS MUSKAT
1oadedjournal bearings when the finite bearing width,film discontinuity, and thermal reactions
are taken into account, although when these latter effects are neglected the equations describing
the gross dynamical features of the journal-bearing motion can be solved 6 by- mechanical
integration.
These brief remarks about geophysical prospecting, refining and lubrication should not
be interpreted as implying that each is in the status of a closed book. Research in these fields
is being vigorously prosecuted. While at the moment computational problems do not constitute major bottlenecks to progress, it may well be that, once computational facilities and
services become generally available on a practical basis, many old problems which were
previously dropped because they did not warrant the laborious and time-consuming hand
calculation and new problems arising in current research will be sllbmitted for machine
computation. These phases of the oil industry therefore should not be written off as having
no interest in large-scale machine computation from a long-term standpoint.
In the field ,of oil production the history of the mathematical developments has been
marked by a sequence of evolutionary steps. The first serious attempt to treat analytically
problems of fluid flow in porous media appears to have been that pertaining to a study of
flow of artesian water into well bores, reported 7 in 1863. While this was based on Darcy's
law expressing the linearity of the relation between the fluid velocity and the hydraulic
gradient, which is the fundamental basis ofall viscous-flow phenomena in porous materials,
no general formulation was developed. The latter first appeared8 in 1897 in the form of
Laplace's equation for the pressure distribution, namely,
\l2p
= O.
(4)
In addition to illustrative solutions exhibited in this original work, many others have since
been reported 9 for systems simulating in some degree those of interest in oil production. These,
involved little more than the application of conventional potential-theory techniques.
The scope of problems in fluid flow through porous media governed by Laplace's equation
is limited to iricompressible liquids fully saturating the porous media. An extension of Darcy's
law10 to gas flow· in 1931 led to a nonlinear differential equation, which can be expressed in
the form
\72 (1+m)/m = (1 +- m)fflYol/m dy
y
(5)
v
.
.
k
dt'
where y is the gas density, t the time, f the porosity of the medium, k its permeability, fl the
gas viscosity, 1'0 the atmospheric' density, and m ,a quantity that defines the thermodynamic
character of the gas expansion.
A further extension l l of the single-phase fluid theory to the flow of compressible liquids,
with constant compressibility, showed that such flow systems could be described by the heatconduction equation with the liquid density y as the. dependent variable, that is,
\7 2y _fflK dy
v
-
k
308
dt'
(6)
APPLICATION OF COMPUTING MACHINERY
being the compressibility of the liquid and 1, /1, k having the same meaning as In
Eq. (5).
Both Eqs. (5) and (6) have been applied to practical problems. 12 The former, being
nonlinear for the transient case, has been treated by approximation methods. And the wellknown procedures for solving the Fourier equation, supplemented by direct electrical-circuit
analogs,13 have sufficed for solving a great variety of problems in flow of compressible liquids.
Mathematical problems of a much higher order of complexity arise when the physical
situation is generalized to the actual practical conditions obtaining in most oil-producing
reservoirs, namely, the simultaneous flow of two or more fluid phases-gas, oil, and water-through the same porous medium. For the case of simultaneous flow of oil and gas in a
producing well bore, representing the operation of a "solution gas drive" reservoir, the
corresponding equations may be written
K
d [ F1 (p)b(p) ddUP]
a(p) dU
d
+ VU
[F2(p)e~p) ddUP]
I
= e2u [f(p) - g(p)p]
~
()U
[F (p)b('P) ddUP] -_ e
1
2u
+ F1 (p)c(p) (dVUP)
~~"
2
(7)
P
[O(
z p ) dP
dt -.JO()
P P ddt ] '
where p is the fluid pressure, p the oil saturation, t the time, and U the logarithm of the radial
coordinate. The functions a(p), b(p), . . . , J(p) are to be considered as knowri-'functions of p,
determined by the thermodynamic properties of the gas and oil, and Fl (p), F 2 (pJ as known
, functions of p, reflecting the dynamical characteristics of the porous medium. The quantity p
itself, which expresses the fraction of the pore space of the rock occupied by oil, may be assumed
to have initially a uniform value, less than I,. and must always remain positive and never
exceed its initial uniform value. The pressure p likewise may be taken to be uniform initially,
and must subsequently always be lower than this value, though positive. At a closed sand
body, dP/dU and dp/dU will vanish. And at the producing well one may impose the history of
either P or of the flux: Fl(P)b(p) dP/dU.
It is this last set of equations that has been the basis of the writer's personal interest in
the subject of computing machinery. The equations, in their essential aspects, had been
formulated 14 in 1936 on the basis of experimental work done 15 at the Gulf Research & Development Company on the fundamental laws of multiphase fluid flow through porous media.
In principle, these hydrodynamic equati~::ms, when suitably generalized:, govern the whole
complex of physical processes underlying the recovery of oil from underground reservoirs.
They are evidently too complicated to permit analytic solution. So in order to show that the
equations were somewhat more than academic curiosities, a numeri~al solution for an equivalent simplified linear system was carried through. ''''hile the results seemed physically
reasonable, they were in no sense precise and had been subjected to considerable smoothing,
guided only by physical intuition. However, while this situation was by no means satisfactory,
30 9
MORRIS MUSKAT
the six months computing labor required even for the simple ideal system completely discouraged undertaking the analysis of more complex and practical systems.
In lieu of practical methods of solving Eqs. (7) directly, approximation~ have been introduced. By neglecting the pressure gradients in Eqs. (7) or by an equivalent derivation from
first principles, one can obtain 16 an ordinary nonlinear equation of the first order relating
p and p which suffices to give the gross production history of the reservoir. This has been
applied extensively by numerical integration in predicting oil recoveries, the pressure versus
oil-recovery history, and the effect of returning the produced gas to the oil-bearing formation. 17
Unfortunately, however, there are a number of important questions relating to oil production
which cannot be answered by such simplified treatments, since they pertain to effects of the
neglected· pressure gradients. Perha ps the two major problems of this type are (1) the effect
of the spacing between the producing wells on the ultimate recovery, and (2) the effect of
the rate of production on the recovery. Attempts to evaluate these effects have all been
beclouded by the uncertainty whether the approximations that have been made have not
au.tomatically predetermined the quantitative aspects of the conclusions. Yet well spacing
and production rates are among the most important parameters that are subject to the choice
of the operator in controlling the ultimate oil recoveries.
Except for the original attempt at hand calculation already referred to, the problem of
solving Eqs. (7) directly remained dormant until July 1946, when an announcement appeared
of the war development of the ENIAC. Negotiations with the government were then entered
into for applying the ENIAC to the solution of these equations. Although several plans were
developcd over a period of mc;>re than a year for carrying out this project, it was not found
feasible, because of legal difficulties, to arrange for the required coqperative effort betwecn
the government and an industrial concern. ''''hi Ie this situation was· subsequently resolved
by one of the governmental agencies becoming interested in the problem and assuming
sponsorship for the work, it was ultimately found, much to the embarrassment of the author,
that the project had to be abandoned anyway because the memory capacity of the ENIAC
would not suffice for handling the large number of operational or,ders required.
This unhappy history is referred to here to serve as an illustration of what can happen
when one unfamiliar with the science of computation and its ramifications is left to the mercy
of his own naIve optimism. The writer has learned the "hard way" that there is more to the
computational solution of, complex equations than the desire to have them solved.
shall
discuss this matter further below.
To complete the record, following the realization that the ENIAC was not sufficiently
powerful to solve Eqs. (7), the equations were submitted to the International Business Machines
Corporation for their consideration. After a number of preliminary discussions, the IBM
Corporation undertook to place the problem on the Selective Sequence .Electronic Calculator.
This work is still in progress. Needless to note, this project is being given a thorough preliminary
analytic formulation by the IBM staff prior to final machine computation.
''''ith respect to the general field of the physics of oil production, it should be noted that
''e
3 10
APPLICATION OF COMPUTING MACHINERY
Eqs. (7) themselves represent highly simplified systems in which it is assumed that the oilproducing rocks are everywhere uniform. Such reservoirs actually never occur in practice.
'Vhile the development of the implications of Eqs. (7) would in itself he a constructive accomplishment, it will ultimately be of considerable interest to investigate their generalization to
nonuniform systems. Moreover, effects of gravity have been ignored in constructing Eqs. (7). ,
Their inclusion would make the physical problem three-dimensional, which would lead to
an additional order of complexity. And even aside from studies of the gravity effects, the
investigation of the three-dimensional analogs of Eqs. (7) will be of importance in treating
stratified producing systems with mutual cross flow. ~inally, Eqs. (7) take no account of
interfacial capillary phenomena, which may be of importance under special conditions, and
especially when gravity is an important factor in the producing operations.
In fact, the detailed study of Eqs. (7) constitutes only a beginning in the establishment of
the quantitative aspects 'of what is now generally known as "reservoir engineering." Even
without anticipating new developments in this field as research continues, it is clear that
large-scale computing machinery will find wide and important applications in oil production
for many years to come. And it is not inconceivable that while only a passive interest in
extensive digital computation has thus far developed in other branches of the oil industry,
comparable applications in refining and geophysical prospecting may ultimately be found
once the practical availability of these powerful tools becomes disseminated throughout these
other fields of activity.
A single and obviously unique experience is a dangerous basis for generalization. The
following remarks are not to be construed as direct implications of the above outlined personal
contact of the author with problems of computation. On the other hand, the program of
this Symposium itself is evidence that outside of government organizations and academic
institutions the application of computing equipment to the solution of specific problems
apparently has thus far been rather fragmentary. It therefore seems appropriate to explore
the general subject of computing-machinery service for industrial applications, even though
much of the discussion must be of a speculative character.
The computing-machinery service to be considered here is that which woul~ require the
use of large-scale equipment localized at computing ~enters such as the Computation Laboratory at Harvard, the IBM Corporation, the Bureau of Standards, ,and similar organizations
which may provide their facilities, at ,least in part, for the investigation of industrial problems.
The specific question involved is essentially that of defining the term "service."
There are two aspects of the composite problem of application of computing equipm~nt
about which there will be little question. The first is that the one who is primarily interested
in the solution must provide both the analytic and the physical statements of the problem.
Second, the computing-service organization must carry out both the actual machine operation
and the coding of the problem. It is in the intermediate coupling of these two contributions
that the situation remains uncertain. And it is in this link that the efficiency and value of the
31 I
MORRIS MUSKAT
computing project may be ultimately determined. It is here that control may be. applied
on the accuracy of the solution, and often on its physical reality and convergence. It ·is the
writer's belief that this bridge of analytic programming should be made available, when
necessary, by the computational organization.
It may wen appear that the very program' of this Symposium belies the suggestion that
analytic preparation and programming should be a part of computational services. For many
of the papers presented here report on investigations in which those with whom the problems
originated carried through all aspects of the problem short only of the machine operations
themselves. You will note, however, that in almost an cases the authors represent academic
or similar research institutions. The same may be expected with respect to many problen:ts
arising in the aircraft, automobile, shipbuilding, explosives, telephone, and railroad industries,
or in the larger companies in the electrical, steel, radio, and glass industries. By their very
nature the major industrial concerns in these fields require virtually self-contained large
tcchnical staffs capable of handling all phases of their engineering activities. However, in
spite of the great contribution to our total industrial effort made by such organizations, by
far the largcr part of industry as a whole is cqmprised of the composite resultant of the hundreds
of intermediate- and small-sized concerns engaged in some form of technical activity.
In their own specialized fields the engineering problems of these companies are essentially
the same as those encountered by the large corporations. Yet in contrast to 'the latter they
cannot afford to maintain the permanent, c?mplete, and well-rounded research organizations.
which can attack effectively virtually any problem that may arise. In particular, with respect
to mathematical problems, or such where analytic treatment may be required at least to guide
experimental research or design, these smaller firms may be fortunate if their engineers have
enough mathematical background merely to construct the equations to be solved. Of the
members of the American Mathematical Society who ga~e their employment affiliation on
the membership list, fewer than 325, or 9.0 percent, indicated connections with indu~trial
concerns, including those who have a direct interest in the development of computing
machinery.
As the writer. himself has learned by'painful experience, and as any "outsider" attending
this Symposium or meetings of the Association for Computing Machinery would quickly
observe, the science of computation is a highly specialized technical field. In many respects it
is still in its infancy-a war baby~but it is growing with accelerating speed. The practicingengineer or physicist of today literally heard nothing- of it during his academic training. The
terms coding, programming, the binary system, and many others that are commonplace in
the language of the modern computation science are quite foreign to those in the engineering
professions.
Among those on the membership list of the Association for Computing Machinery who
have given their, employment affiliation more than 82 percent are in government agencies,
on academic or research institute staffs, in computing organizations, or are employed by
industrial concerns that are obviously engaged in some phase of computing-machinery
3 12
APPLICATION OF COMPUTING MACHINERY'
development. Ivlore than half of the remainder are employed by aircraft and insurance
concerns, and very probably a number of the 38 "residuals',' also are primarily interested in
the equipment development itself. It is thus clear that to thG extent that membership in the
Association is' an index of interest and contact with the computing profession, such interest
has yet been disseminated but slightly into industry as a whole.
If the engineer or industrial physicist or chemist in the average commerCial firm' must stop
to take a training course in the theory of computational machinery, even if he should be
temperamentally suited to absorb such specialized disciplines, before having his problem
accepted by the computing organization, the probability is great that he will drop or circumvent the problem. And even if he were willing and could make arrangements to "study up"
on the basic elements involved; it is still very unlikely that he would thus develop the required
analytic skill to guide the choice of the mesh to be used, decide on the differencing procedures
that may be required for convergence, carry through the preliminary numerical solutions,
or even prepare functional representations for the empirically variable functions in hisequations, if this should be necessary.
At present most of the applications of computing equipment are being made by members
of governmental agencies or academic institutions. Many of these are well staffed for analytic
wO,rk. Moreover, in spite of the importance of these problems, it is doubtful whether the,
pressure for their speedy solution is comparable to that in industry, where diversions into the
purely computational aspects of the problem may not be accepted without prejudice. Undoubtedly, there is a large backlog of demands by such organizations for the use of presently
operating computing machinery. So there may appear to be no need to cater to and accept,
computation proposals from those who are unprepared or unable to submit a completely
programmed problem. Such, however, it is believed, would be a shortsighted policy and would
lessen the long-term possibilities of growth of the science of computation.
It is not suggested that the argument is one-sided. No doubt the provision of this type of
service by computing organizations will involve difficult personnel problems, though these
same difficulties would be even more serious in most industrial firms. It is also true that such
, service would increase the total charges, which might discourage the interest in them by small
concerns with very limited engineering development budgets. But at the same time it would
make it possible to extend the applications of large-:-scale computing equipment to many
organizations that would otherwise simply have to give up because of lack of qualifications.
Perhaps the strongest reason for centering the intermediate analytic facilities within the
computing organizations lies in the importance of experience in this phase of numerical
computation. In a science as young as this, virtually each problem gives rise to new questions.
of detailed treatment. It is not yet ready for standardization and the p~eparation of tabulated
instructions. The analyst who is continually engaged in programming will no doubt accumulate a wealth of experience which will be of inestimable value~both to the computing organization and to its clients. To have each problem prepared and analyzed for computation by a
beginner will be pitifully inefficient as compared to their handling by personnel for whom
MORRIS MUSKAT
such work is their daily professional business~ In fact, the writer ventures to predict that if
and when the "buyer's market" overtakes the computing industry the burden of selling
computing services will. fall on competitive claims of the experience of the organization and t.he
·completeness of the service rather than on the number of milliseconds the machines take for a
multiplication or whether the price is xor x - /).X dollars per hour.
I t is to be understood, of co'urse, that even if the computing organization provides the
analytic preparation of the problem the sponsor must still' accept the ~esponsibility of interpretation and evaluation of the solutions. Except possibly when solving purely arithmetic
problems, as systems of algebraic equations or function-table p~eparation, it is the sponsor
who must supply the guidance in dropping terms if such should be necessary to make the
problem tractable, in fixing the order of accuracy required,and in evaluating the physical
. significance of the solution. This may well call for visits by the sponsor to the computing
organization during the planning, programming, and coding, and in most cases his continuous
presence there during the time the problem is actually on the machine. Indeed, the experience
and background of the sponsor in the technical field giving rise to the problem may be just
as indispensable in achieving a satisfactory solution as the experience of the computing staff
with respect to its analytic aspects.
There is no easy way to accomplish difficult tasks. Cooperative effort by all parties concerned is required. The ultimate impact of the science of computation on our technology
and industrial life will most certainly be tremendous compared to what"has already materialized
and what can now be envisioned. It has already evoked an absorbing interest from and recruited
into its ranks soine of the outstanding leaders in the fields of engineering and electronic design
and mathematical analysis. Let us therefore plan to guide this important growing effort so
that its fruits may be enjoyed by the maximum number for the greatest benefit of our nation
as a whole.
REFERENCES
1. See, for example, L. L. Nettleton, Geophysical prospecting for oil (McGraw-Hill Book Co., New
York, 1940).
2. H. M. Evjen, Geophysics 1, 127 (1936); H. Bateman, J~ App. Plrys. 17,91 (1946); E. C. Bullard
and R. 1. B. Cooper, Proc. Roy. Soc. (Lon~on) [A] 194, 332 (1948); G. Kreisel,Proc. Ro.y. Soc. (London)
[A] 197, 160 (1949); L. J. Peters, Geophysics 14, 290 (1949).
3. S. R. Brinkley, Jr., J. Chem. P~ys. 14, 563 (1946); 15, 107 (1947).
4. See W. F. Cope, Proc. Roy. Soc. (London) [A] 197, 201 (1949).
5. D. G. Christopherson, Proc. Inst. Mech. Engrs. (London) 146, 126 (1942).
6. J. T. Burwell, J. Applied Mechanics'14, A-231 (1947).
7. J. Dupuit, Etudes thioriques et pratiques sur le mouvement des eaux (1863).
8. C. S. Slichter, U.S.G.S. 19th Annual Report (1897.:...98).
9. M. Muskat, Flow of homogeneous fluids through porous media (McGraw-Hill Book Co., New York,.
1937; J. W. Edwards, 1946).
3'4
APPLICATION OF COMPUTING MACHINERY
10. M. Muskat and H. G. Botset, Physics 1,27 (1931).
11. T. V. Moore, R.J. Schilthuis, and W. Hurst, Oil Week{y69, 19 (May 22, 1933); M. Muskat,
Physics 5, 71 (1934).
12. M. Muskat, reference 9.
13. V. Paschkis and H. D. Baker, A/ME Trans. 64,105 (1942); W. A. Bruce, A/ME Trans. 151,
112 (1943).
14. M. Muskat and M. W. Meres, Physics 7, 346 (1936).
IS. R. D. Wyckoff and H. G. ,Botse,t, PI!ysics 7, 325 (1936).
16. M. Muskat, fl. Applied Physics, 16, 147 (1945).
17. M. Muskat, Physical principles
of oil production' (McGraw-Hill Book Co., New York, 1949).
WILLIAM W. WOODBURY
Northrop Aircraft, Inc.
For the past year and a naIf Northrop Aircraft, Inc. has operated a computing machine
.built by the International Business Machines Corporation, which differs radically in its
treatment of problems from the usual computing installation of IBM accounting machines.
This machine consists of three standard machines-a Model 405 printer, a Model 603 electronic
multiplier, and a ]Model 517 summary punch. They are interconnected to function as a
single unit.
The printer, commonly known as a tabulator, is a machine of parts. Two sets of brushes
which read punched cards are the machine's point of entry. Each card is read consecutively,
first by the "upper" brushes, then by the "lower" brushes. A card has room for 80 decimal
digits,· indicated by punching. A number" is represented on a card by the vertical position
of punches in the 80 columns. It is represented in the machine by the relative time at which
the brushes make contact through these punched holes. All machine elements are synchronized
to the movement of a card past both sets of brushes. The card actually has not ten, but 12
vertical positions. Ten are used for digits, while the remaining two are used principally for
algebraic signs, or for control of switches (selectors) which will be described below.
The machine has a counter capacity of 80 decimal ~igits. These are arranged in groups
-four each of 2, 4, 6, and 8 digits-but the groups may be combined to produce individual
accumulators up to 80 decimal digits. Eighty-seven type bars, through which any information
in the machine may be printed in a single cycle of the machine's operation, are also important.
A detachable plug board may be wired according to the arrangement of counters and type
. bars desired for a particular problem. This is a convenient feature of the machine, since
several of these may be wired for various problems in advance, thus permitting immediate
change-over as $oon as a problem is finished. Auxiliary equipment includes six 10-pole and
16 single-pole double-throw switches or selectors; 20 positions for numerical comparison
which yield impulses if the numbers entered are unequ"al; and two distributors for separating
impulses in a circuit with respect to time.
The items mentioned above represent regularly available accessories to the machine.
Special additions have also been made, such as the multiplier entry, exit, and control connections which will be described in connection with the multiplier. There are 16 8-pole, four
4-pole, and 40 s.ingle-pole double-throw switches, plus five 8-pole quadruple-throw switches
called chain selectors. A second plug board is provided for wiring these additional elements.
The multiplier, an electronic device, develops a l2-digit product from two 6-digit factors.
These factors are entered at the same time that the product from the previous entry is read
3 16
THE
603-405
COMPUTER
out. The multiplication is executed during the time between cards. Since the tabulator
does its adding and subtracting in the "nines complement" system, provision has been made
for reversing the factor entries in time when a counter is standing in complement. For example,
take the number 999998. Counting forward in time to the number yields 999998.. Counting
from the number to 999999, which is equivalent to zero for this machine, gives - I. If 2 is
added and the leftmost position carry entered in the units position, the correct answer, + I,
is obtained.· Since the absolute value of the product is developed, provision is made for a
negative-sign impulse when a negative product is read out. This impulse reverses the add or
subtract instruction given to the receiving counter. Provision is also made for round-off, by
addition of five in. the leftmost position dropped. The multiplier control is logically complete;
numbers may be entered from any source, either as complements or as absolute values with
signs; and t4e product with its sign may be taken to any place, including reentry as a factor of
a succeeding product.
The remaining machine-the summary punch-provides a means of punching the information from the counters on cards.
It is now possible to compare this machine to the present conception of an adequate
all-purpose computer. It has an arithmetic organ-the tabulator counters plus the multiplier.
It has an input and two outputs, one of which is the input medium; An internal memory
is achieved by apportioning a part of the 80 counters to ~emory. The external memory, in
the form of punched cards, is indefinitely large. Control could'be established from the counters.
This, however, is not expedient. The control instructions must be punched into cards for
entry into the machine, and are just as well left there. Besides, the limited internal memory
capacity will hold only a tfivial program. Programs are sometimes wired implicitly, however,
so that only blank cards need be fed after the initial data have been entered.
The general-purpose computer is presently conceived of as being organized around one
or two channels. These may be either serial, in which case words are moved about digit by
digit, or parallel, where the entire word is moved at once. This machine has no channel as
·sucb-, but its array of switches is used to construct the channels best suited to the problem at
hand. This frequently permits a kind of multiple-parallel operation, in which several computations are made simultaneously. A table look-up operation from the control cards may
be channeled into one counter,while higher derivatives are being integrated (Ilt = a power
of ten), and while the multiplier and a counter or two are iterating for a square root. This
kind of operation is commonly performed in actual problems. The machine's speed is determined by the card-feeding rate. When no output is required, it accomplishes 150 cycles/min,
performing one multiplication and one or more additions or transfers in 400 msec. When
transactions must be printed, the speed drops to 75 cycles/min. In this kind of printing cycle,
called a list cycle, the operations mentioned above can be performed in 800 msec. Another
kind of cycle prints the contents of the counters, and mayor may not clear the counters. The
duration of this cycle is equal to that of the list cycle, but no computation can be performed
while it is in progress. This cycle is called a total cycle.
WILLIAM W. WOODBURY
In order to punch cards, the machine must be stopped for approximately I sec. The design
of the machine makes it necessary to take. a totai cycle ,at this time.
One limitation on the speed of the machine is that it can perform only one multiplication
in a cycle. Thus, problems involving many multiplications but few other operations take more
time than problems containing relatively few multiplications.
The direct-printing feature is worthy of emphasis. It involves no subsidiary machines.
It does not await manual operations. It simply prints whatever is in or passing through the
machine when the machine is so instructed. In trouble-shooting and in checking, the results
of previous cycles are there for comparison with the result of the present cycle. When a
problem is complicated, intermediate results may be printed at will, to provide a picture of
the relative magnitude of various factors. When final answers only are required, printing
can be restricted to these answers. In the inching process-that is, printing every cycle-the
. exact nature of errors, not only in the program but frequently in the mathematical formulation;
are immediately apparent., Indeed, we dispense almost completely with checking at the
transcription level, since errors are discovered so easily in this manner.
This brings up an -interesting point. On many problems this machine produces results
about as fast as they can be apprehended. ,That is, fora problem of an investigative nature
wherein various configurations are to be tried, additional speed is not so desirable as another
machine when someone else wishes to work with another problem. It is true that much work
is done on this machine that is of the nature of tabulation of functions. For the moment, no
one is much concerned with the development of the answers. In this work great speed would
be an advantage.
A resume of the kind and. scope of problems with which the writer has had experience
follows, for those concerned with the industrial application of this equipment. The computer
was built to integrate a system of six nonlinear differential equations in a single independent
variable. These equations were of the first order in four of the dependent variables, and of
the second order in the other two dependent variables. In addition, the sine and the cosine
of one of the dependent variables entered four of the equations as coefficients. Since, the
continuity of the solution was good, the sine and cosine were integrated stepwise along with the
equations themselves. A second system offour nonlinear second-order equations was integrated,
with the interesting program variation generated through a relation between two of the
dependent variables: xj(x 2 y2)1. This ,expression was evaluated through a table look-up
operation and, because of the wide variations in x and y, was done with a floating decimal
point. The above equations all represent work in connection with servomechanisms having
several degrees of freedom, and with cross-product terms of considerable magnitude. Stochastic
processes have been part of the bread-and-butter work for the machine and are especially
adapted to it because of the multiple-channel operation and the further possibility of making
several simultaneous discriminations for future choices, even 'as the consequences of the last
choice are being computed. Run-of-the-mill work has involved the reduction of test data
and structural analysis. In an aircraft company, test data mean wind-tunnel and strain-gage
+
3 18
THE
603-405
COMPUTER
information, which may be programmed to completion in one machine passage. Structural
analysis has been limited by the need of the machine for other more pressing problems, but
the machine is quite capable of handling problems in this field.
An investigation of the behavior of the biharmonic difference equation for cantilever plates
has been going on, as time has been available. I would lik:e to be able to give some definitive
results from this investigation, but· possibilities remain to be explored. It seems probable,.
however, that the number of computations required for a reasonable· convergence of an
iterative process is of considerably higher order than the number of computations required
to invert the matrix of the points, the ratio being possibly somewhere in the neighborhood of
n2 , where n is the number of points in the lattice. This is for an unsophisticated pattern which
simply substitutes the new value for a point when it is obtained. Convergence could probably
be improved by second-order corrections, but I doubt whether it could be improved enough
to equal matrix-inversion speed. The inversion of high-order matrices using an elimination
algorithm requires about (nJ20)3 working days. This time is slow and this operation one of
the weakest for the machine be~ause of the preponderance of multiplication. Each multiplication, incidentally, involves 3 cycles to retain sufficient accuracy.
The work with the biharmonic equation indicates a limit of application. Unless results
are of sufficient value to justify the expenditure of several months' time, \J4cp = - q with
three free boundaries is beyond the machine's power. With respect to the simpler harmonic
equation, we hardly feel ready to compete with the relaxation technique described by Southwell.
Checking is accomplished in various ways, according to the problem. In integrating
differential equations, continuity of the solution is often a sufficient check. For final structures
reports, when balances are not available as a check, duplicate runs are made and compared
mechanically. Sample calculations are made on a desk calculator in order to avoid systematic
errors. The machine is operated 24 hours a day, 5 days a week, and has averaged about
10 percent down time.·
In conclusion, I wish to say that this card-controlled computer was thrown together in
about four weeks to meet a need for a powerful computer to do a complicated integration.
It seems to fill a useful place in its ability to integrate differential equations in a single independent variable and to do routine calculations involved in engineering design work with an
over-all efficiency of better than ten times that. of any other generally available equipment.
We have here a machine different in nature from most computers. It can perform a multiplication per cycle and several additions and transfers simultaneously. Thus, it is more'efficient
in use of its rate than other computers for which each operation is exclusive. This machine
never has to wait to find out what to do next. Even if what it is to do next is dependent on the
solution so far, this is readily incorporated in the W'iring through the use of a selector, so that
no time is lost. Olle is led to feel that as the clock rate of an internally programmed machine
is increased it should be easy to increase the input rate of the externally programmed machine,
so that the time loss inherent in program operations is still large. This is to emphasize the
time cost of internal operations upon program instructions. The work with the biharmonic
WILLIAM W. WOODBURY
equation suggests that partial differential equations will require far too much time for the
iterative solution which can be accomplished with a relatively small high-speed memory.
This indicates that the memory capacity should be based on the storage requirement for
inverting large matrices, which is of the order of n2 words where the matrix is n X n. Less
storage than this will require continuing use of the input and output, with the consequent loss
of time. From these considerations we at Northrop who are close to this work feel that· the
internally programmed machine will require perhaps ten times as much high-speed memory
as has been considered to date to our knowledge, the possibility of executing program revisions
simultaneously with explicit computation, .and the elimination of access time through the
use of multiple registers.
320
SEVENTH SESSION
Friday, September 16., 1949
9:00 A.M. to 12:00 P.M.
THE ECONOMIC AND SOCIAL SCIENCES
Presiding
Edwin B. Wilson
Office of Naval Research
ApPLICATION OF COMPUTING MACHINERY TO THE SOLUTION OF
PROBLEMS OF THE SOCIAL SCIENCES
FREDERICK MOSTELLER
Harvard University
This paper discusses some of the applications and limitations of the use of modern computing machinery in the social sciences. Such a discussion could scarcely be expected to be
exhaustive, but merely indicative of the kinds of applications occurring now and in the near
future. Of course, some remarks must be included about the use of computing machines in
social science's principal quantitative tool, statistics. We have not included economics in the
social sciences because the title of the program-the Economic and Social Sciences-indicates"
that these fields are to be considered separately. For our purposes the social sciences might be
regarded as including education, social psychology, and sociology. It would be a tour de
force at this time to include cultural anthropology, history, political science, and similar
largely nonquantitative subjects, in a discussion of modern computing machinery.
Thus far, most direct applications of computing machinery to social-science problems are
associated with routine problems of solving simultaneous linear equations, either homogeneous
or nonhomogeneous. The commonest and most widely applied technique" is multiple regression. Here we have one dependent or criterion variable Y which we desire to predict from
a flock of independent variables Xl' X 2 , • • ., X k • The standard approach is by means of least
squares, where we are required to find weights a i to minimize the function
(1)
In the relation (I) the subscript j refers to the observations. This well-known minimization
produces a set of k simultaneous linear nonhomogeneous equations which we solve for the
weights ai • From this solution we get a linear prediction equation
Y =
k
2:
i=O
aiXi , Xo = I.
(2)
vVe are often asked if it would not he better to try to fit some function of the X's to Y. It
certainly would, but we do not ordinarily know the function. The reasonable thing, therefore,
is to take the plane given in Eq. (2) as a first approximation to this function and to hope that
the range of the variables is sufficiently small that this method will be adeq~ate for predictive
purposes. If we go to th~ quadratic approximation, we will have k(k + I) /2 additional terms
to fit. Even if k is as small as 6~ we will have 7 + 21 = 28 simultaneous equations to solve
for the second approximation. The resulting reduction of the residual sum of squares is seldom
32 3
FREDERICK MOSTELLER
worth the work. A more common device is to adjust the scale on which variables are measured
to make the linearity assumption of Eq. (2) more realistic.
The applications of computing machines in the above examples are rather obvious. First,
of course, we want the equations solved. But second, and perhaps more hnportant, we want
to know how much faith we can put in such an equation. We must remember that the observa, tions ordinarily are good to only one or two significant figures.
A problem closely related to multiple regression is the discriminant function. In its simplest
form we are asked to divide a population into two groups. Whereas inmultiple regression
we might be asked to predict degree of marital success, or degree of adjustment of a paroled
man to the outside world, or degree .of success as a pilot, in the case of the discriminant function
we are asked to produce a function that will separate sheep from goats. Will the postulated
marriage end in divorce; if we parole this man will' he return to jail; will the candidate get
his wings or not? A little further afield, we may even ask whether Alexander Hamilton wrote
this particular essay from the Federalist Papers-or was it James Madison?
The distinction between multiple regression and the discriminant function, then, lies in
the nature of the dependent variable. In the case of the discriminant function it is dichotomous.
\Ve want to construct an index number
(3)
and establish a criterion number C, so that according as Z ~ C we can predict, successful
marriage or divorce, good citizenship or recidivism, Hamilton or Madison, with' a .reasonable
percentage of success.
Another way oflooking at this problem is that we want to find A'S such that we can maximize
G=
2
(Zl - Z2) 2
n.-,---"-'---
:2 :2
-
(4)'
(Zij - Zi)2
i=1 j=1
where Zl and Z2 are the Z means of the success and failure groups, and zu, j = 1, . . . , ni ,
i = 1,2 are the Z values for particular individuals. In other words, we want to maximize the
ratio of the between-groups square to the within-groups sum of squares. The numerator of
, G measures the separation of the groups; the denominator measures the variabilities of the
groups within themselves. This method of looking at the problem is chosen because of a
connection with a later problem. It turns out, after some manipulation due to R. A. Fisher,
. that the A's will be obtained by solving the equation
:2AqSpq = cdp,
q
P=
1, . . . , k
(5)
where
(6)
and, c
i~
an arbitrary nonzero constant. Here the d's are the distances between the means
COMPUTING MACHINES IN THE SOCIAL SCIENCES
of the two groups on the independent variables and the S's are weighted covariances between
pairs of independent variables summed for the two groups. As in multiple regression, we
are left with simultaneous linear nonhomogeneous equations to solve (c is arbitrary). Once
this is done we can compute the Z values for each group and discuss the effects of choosing
various values of the cutoff point C. How we choose this point will depend on the costs
of making wrong decisions of either kind. If plenty of pilot candidates are available, and
training costs are high, we will make the cutoff on the index quite high to reduce the washout
percentage, realizing that we are discarding numero.us candidates who would have made
good pilots.
Up to now, little has been done about discriminant functions when it is desired to split the
population into three or more groups. This lack of progress may be due partly to the very
heavy computational work that would, undoubtedly be associated with a decent formulation
of the problem.· As modern computing machinery becomes available to scientists, it is likely
that they will no longer be so reluctant to formulate problems that require heavy computation.
An example of the application of homogeneous equations is supplied by Guttman's scaling
theory. One such problem is that of scaling attitudes. More detailed expositions of this
. problem are given in Paul Horst, The Prediction of Personal Adjustment (Bulletin 48, Social
Science Research Council, 230 Park Avenue, New York, 1941), and The American Soldier,
vol. IV (Princeton University Press,Princeton, N.J., to be published shortly). We will restrict
ourselves to dichotomous questions (answer yes or no) for the explanation. If we have six
such questions, they would form a perfect Guttman scale if the responses to all the questions
by all respondents could be arranged into one of the six forms shown in Table 1. In this table,
Table 1. Guttman scale for the responses to six questions.
Question
Favorable
Unfavorable
1
2
3
4
5
6
X
X
X
X
X
X
0
X
X
X
X
X
0
0
X
X
X
X
0
0
0
X
X
X
0
0
0
0
X
X
0
0
0
0
0
X
0
0
0
0
0
0
I
X corresponds to Yes and 0 to No. The numbers attached to the questions are dummies.
If we could achieve such a perfect state of affairs we would clearly have formed a scale on the
favorable-unfavorable axis which could be thought of in Steven's classification as ordinaL
FREDERICK MOSTELLER
The direction of the 'scale is determined by the content of the questions. The perfection displayed in Table 1 can scarcely be expected in practice. Therefore we request that scores be
assigned to individual patterns of responses to accomplish this ordering of patterns of responses
as nearly as possible. The criterion used is that we should maximize a certain correlation ratio.
This maximization leads to a set of simultaneous homogeneous equations~ ActuaIIy there is
a perfectly decent and workable approximation scheme (caIIed the scalogram method) that
can be used to get the initial rankings of the people and the questions, and we could usuaIIy
avoid computation in practical applications were it not for some further developments. The
(/)
z
o
z
a..
o
IJ..
o
>to-
(/)
Z
W
toZ
UNFAVORABLE
FAVORABLE
FIG. 1. Diagram showing schematlcaIIy a curve of intensity as
measured by the strength with which opinions expressed are held,
plotted against the score as obtained from a Guttman scale. It is
conjectured that the lowest point on the intensity curve corresponds
to neutrality 'and should be regarded as the psychological zero point.
If the first component is regarded as the score on the attitude scale,
,and the score on the second component is plotted against the first,
similar U-shaped curves appear, with minima quite close to those
of the intensity curve.
scoring that is achieved represents mathematically the principal component of the system.
Since there are more components 'available, and since these have been found to have meanings
in other fields of endeavor, it is not unreasonable for the psychologist to wonder whether these
further components might not have further meaning for him. 'In particular, the second component, has been found in some attitude studies to correlate extremely well with the concept
of intensity, where intensity has a separate definition. More recently, Guttman has worked
on a possible interpretation of the third component. I must admit that I take a rather different
view of these components and that I feel it is rather a fortuitous accident that intensity is closely
related to the second component. Intensity with which an opinion is held" as it is ordinarily
defined, leads to U-shaped functions when graphed against the favorable-unfavorable scale
(see Fig. 1). In so far as the first component is arranged in a roughly linear fashion against
326
COMPUTING MACHINES IN THE SOCIAL SCIENCES
this scale, the second component, which is orthogonal to the first, must be rather U-shaped.
Considering the reliability of the observations, all U-shaped functions look pretty much alike.
'Ve need not try to decide this matter here. The main point is that social scientists are interested
in these further components, but we have no very good practical way to get them except by
direct computation. When the questions are numerous, as they often are, the work requires
heavy computation.
If we agree to identify the second component with intensity, it is possible to get at
a zero point on an attitude scale by agr~eing to take the lowest point on the intensity
scale as determining the score on the first component, which will be regarded as neutral
(see Fig. 1).
It might be useful to indicate a type of scale, not unlike Guttman's, in which it is easier
to explain the criterion for obtaining the scores. We might take k attitude items on a special
topic and ask the subject to endorse the r that come closest to his opinions. Out of such an
experiment we would ideally obtain a set of responses like those in Table 2. In this example
Table 2. A second type of attitude scale.
_..
-~-
Item
Favorable
Unfavorable
1
2
X
3
4.
5
6
7
8
X
X
0
0
0
0
0
0
X
X
X
0
0
0
0
0
0
X
X
X
0
0
0
0
0
0
X
X
X
0
0
0
0
0
0
X
X
X
0
0
0
0
0
0
X
X
X
I
I
I
--- -
k = 8, r = 3. If individuals chose only the response patterns indicated above, we would have
a perfect scale. Actually, there will be response patterns with gaps between the checked
items. We formulate the problem this way. We want to assign weights to the items so that
when an individual t chooses items i, j, k, we can give him the score St = Wi
Wi
Wk.
As our criterion we take the ratio of the variability 'of the scores Sl to the variability of the
weights making up a score, the latter summed over the individuals. This view of the situation
is entirely analogous to the criterion given earlier for Fisher's discriminant function. From the
point of view of analysis of variance, we #ant to maximize the ratio Qf the sum of squares
. between individuals to the total sum of squares (because there is an additive relation between
"between individuals," "within individuals," and "total sum of squares"). This ratio of
"between" to "total" is proportional to the correlation ratio. If we try to maximize this ratio
we are led again to solutions of homogeneous linear equations. If the number of items is
+
+
FREDERICK MOSTELLER
large, we have a long computational problem. The method just suggested is in some ways a
variation of Thurstone's method of equal-appearing intervals.
Similar problems arise in education. For example, we may have letter grades in four
courses for a number of individuals. We would like to pool these letter grades to form a scale
of scholastic achievement. However, the distributions of the grades in the several courses are·
quite different. We want to assign numerical values to the grades in the different courses,
and then add these to get a score for the individual. The problem is not unlike the one just
treated, except that for each course (item) we need several weights.
The problems discussed above are common to m~ny of the fields of social science: sociology,
education, social psychology, and perhaps even economics. We could continue to multiply
these examples from scaling theory without difficulty. We have not touched on the problem
of factor analysis-the attempt to find the meaningful psychological or sociological dimensions
of a space of test scores, while reducing the dimensionality of the space-although these
problems are again concerned largely with matrix manipulation. Nor have we discussed the
analysis of time series. However, time-series problems are so general irr all sciences these days
that social scientists can expect generous contributions on this problem from their more
mathematically minded friends in the natural sciences.
To dream a little, I think that certain social problems may be capable of being formulated
in terms of game theory. Then certainly computing machines will be useful, but this application waits on two developments-first, the ability to describe a social problem in terms of a
game, and second, the development of good methods of finding solutions to games. I have no
doubt that progress on the second problem will be more rapid than on the first. Similarly,
the applica'tion of the computing machine as a model for certain problems in clinical psychology
seems to me extremely speculative at this time.
We move from direct to indirect applications of computing machinery in the social sciences
when we discuss problems in theoretical statistics. I would like to call a few of these to the
attention of computing experts. Both theoretical and practical reasons make the normal
distribution one of the most important of all distributions. Therefore, estimates of its paramet,ers
from samples is a constant problem. For a long time the view was held that efficient statistics
(in a technical sense) for estimating parameters were the best ones to use.' Efficiency (or
relative precision) of two unbiased estimates of the same parameter is measured by the ratio
of the variances ofthe two computing estimates; it is the ratio of the smaller variance to the
larger variance. However, it has turned out that effi~ient statistics are not always the easiest
ones to .compute.
It has been found th.at a few carefully selected observations from a large sample can produce
extremely good estimates of the mean and the standard deviation with little calculation.
Similarly, in very small samples it turns out that little efficiency is lost by estimating the mean
from the average of the largest and smallest values, and that the standard deviation can be
very adequately estimated from the range instead of from the cumbersome root-mean-square.
3 28
COMPUTING MACHINES IN THE SOCIAL SCIENCES
The result of these practical findings has been an interest in order statistics. If we draw a
sample from a distribution and order the n observations from least, to greatest,
x1 -< x2 -< ... -< x' < ... < Xm
t
-
(7)
--
then Xi is called the ith order statistic. Statistics constructed from these order statistics-for
example, range, median-are called systematic statistics when they take cognizance of the
order (the mean does not). In studying the worth of these systematic statistics, it is of the
greatest i~terest to know certain properties of the order statistics. In particular, we wish to
know for the normal distribution the mean, the variance of any .order statistic, and the covariances between pairs.
.
For microstatistics (n < 10) we have good tables of these quantities. The first attempt to
get covariances by numerical integration resulted in two-decimal accuracy in. spite of eightdecimal initial values. The latest attempt ismuch improved (five-decimal accuracy) because
of the discovery of a method of exact integration which works up to n = 10, but does not seem
to want to -go further. For macrostatistics' (say n > IqO) we are in fairly decent shape with
asymptotic theory helping us. However in the middle range (100 > n > 10) we are in trouble.
This is a fairly standard situation in statisti~s, the middle-sized samples causing us considerable
worry because it is not clear when the asymptotic theory will be accurate enough to take over
from the computer.
The probability element of the ith order statistic from a sample of n drawn from a continuous probability-density functionf(x) with cumulative distribution F(x) is
g(xi)dx i = (i _ I) ~ ~n _ i) ! [F(Xi)]i-l[1 - F(Xi)]n-ij(Xi)dxi'
while the probability element of the joint distribution of Xi and Xi; i
< j,
(8)
is given by
n!
h(Xi' X;)dXidxi = (i - I) ! (j - i-I) ! (n - j) !
[F(Xi)]i-l[F(x j ) - F(Xi)y-i-l[1 - F(x;)]n-jf(xi)f(xj)dxidx j •
(9)
The quantities we are particularly interested in are
E(x i) = .[tJooXig(Xi) dx i,
,E(Xi2)
= LOOooXi2g(X i) dx i ,
(10)
E(Xi' x;) = f_oooo f:~XiXih(Xi' Xj) dX i dx j,
for n in the middle range. This would make it possible to construct and discuss the efficiency
of any linear systematic statistic. It would also open the door to improving approximations
,vhich would be useful in noncomputational theoretical investigations.
A statistic used in social sciences, where data are frequently ordinal rather than metric,
is· the rank correlation coefficient (I choose this example rather. than soine others for ease of
32 9
FREDERICK MOSTELLER
exposition). We have objects ranked from greatest to least o~ two characteristics. The rank
correlation depends entirely on the sum of the -squares of the differences of the pairs of ranks
given to the objects. There are n! arrangements of the second ranking when we hold the first
one fixed. These n ! rankings produce a distribution of the sum of squares. We refer an obtained
sum of squares to this distribution to decide whether there is reason to believe that there is
really correlation between the rankings or whether such a sum of squares might have arisen
by chance. For example, with n = 4 the distribution is given by
d2
f('2:d 2 )
0
2
4
3
6
8
10
12
14
16
18
20'
4
2
2
2
4
1
3
Total
24 = 4!
For such a small n we probably would not feel much confidence in any correlation unless the
rankiqgs agreed perfectly. Such distributions have been tabulated by Olds, and independently
by Kendall, by hand up to n = 8. vVe know from work of Hotelling and Pabst that for large n
the' distribution tends to normality. However, even for n = 8, the normality is not close
enough for us to get very good approximations to the percentage points of the distribution
function. Without the help of high-speed computing machinery, we cannot push this simple
calculation much further. The real bother is that n! goes up so rapidly that after n is pushed
a few steps further even modern computing machines are bound to be defeated.
We have numerous problems like this in statistics. l\1any of the attempts to create useful
nonparametric statistics bog down at exactly this computational point. Some of these difficulties can be solved in time by sufficiently clever combinatorial devices. But those of us
who want methods for practical use, rather than the sheer joy of mathematical investigation,
are beginning to wonder whether we might not get more work done in the long run by having
tables made by computing machine. It is often easier to solve combinatorial problems when
the answer is essentially known. And certainly the table is what we often want for practical
work. In other words, why hold up the practical problem for the theoretical investigation
when machines can solve the problem, often more accurately, directly? The result of such a
trend would be to leave more time for thinking about problems and their solutions and reduce
330
COMPUTING MACHINES IN THE SOCIAL SCIENCES
the time required for arithmetic manipulation. At the same time we relieve the statistician
ofa side condition. He ordinarily thinks in terms of solutions that he can compute. Computing
machines should extend his horizon. For example, very extensive work has been done in
multivariate analysis of interest to educators and economists among others, but although the,
theory is in good shape nothing much has been tabled. Computing machinery can get us
these tables and put some of these methods to work.
As a final application I might mention the use of sampling experiments. In some statistical
work we cannot get a very workable formulation of the problem mathematically, or, even if
we do, the computation becomes too nasty for even modern computing machines. In such
cases we are leaning more and more to the sampling experiment. A simple example is supplied
by the problem of the truncated normal distribution. We sample from a normal distribution,
but part of one or both of its tails has been removed., We would like to compare several
methods of estimating the original parameters of the untruncated normal from such truncated
samples. With a sample of 15 or 20 the integrations required seem quite unreasonable.
Instead, we try the various methods a number of times and use the empirical results in place
of the theory. It seems to me that such experiments are admirably suited to computing
machines because they involve many repetitions of the same procedure.
By giving these examples of applications of computing machines to social-science problems,
and to statistical problems and theory which in turn can be applied to social-science problems,
I do not care to give the impression that the uses are really very general, or that modern
computing machines will make very fundamental contributions to social science in the near
future. 1fost studies are not very large and the calculations can be handled with a desk
computer. Some exceptions are psychological investigations as carried out by Thurstone at
the University of Chicago, those done by the Educational Testing Service at Princeton, and
censuses and sample censuses as carried out by the Bureau of the Census .. Social scientists
generally think in nonquantitative terms and, except for economics, there is no large body of
mathematical theory available to make quantitative studies on a grand scale sensible. Only'
in the last ten years has any progress been made in applying mathematics to the social sciences,
and the authors of these attempts are quite agreed that little has been done. There has been
. vague mention of the use of computing machines as logic choppers,the notion being that this
is what the social scientist need~ because he thinks qualitatively. Until some definite use in
the social sciences for a logic machine is suggested, I cannot see how it would apply, however
interested I might be in the development of such a device. Some of the burden of limitation
falls, of course, on the computing-machinery people. The social scientist interested in population and sociology problems would like to be able to play around with the census data. He
would like to make tabulations himself, or to his own order (the Bureau of the Census will
make sample studies for him). One might think this would be a job for high-speed computers,
and no doubt it will be, but just now I do not think we are very good at high-speed scanning
and tabulation of large masses of original data.
33.1
FREDERICK MOSTELLER
At present, most of the direct applications of computing machines in the social sciences
outside economics' fall into the realm of simultaneous linear equations----:-homogeneous or.
nonhomogeneous. Examples are multiple regression, discriminant function, scaling theory,
factor analysis. Computing machines can help statistics, the tool subject of quantitative
social science, in numerous ways. Some of these are by tabulating functions, by computing
properties of statistics, both directly and by' means of sampling experiments. This process
will help in the development of statistical theory, and make possible the practical use of a
large body of theory which is little used because of computational difficulties .. Outside economics there is not yet a large body of mathematical theory in the social sciences. This fact
sets severe limits on the direct applications of, high-speed computing machines except in some
special cases mentioned earlier.
33 2
DYNAMIC ANALYSIS OF ECONOMIC EQPILIBRIUM
WASSIL Y W.LEONTIEF
Harvard University
At a similar occasion three years ago 1 had the opportunity to discuss a computational
problem arising in connection with quantitative analysis of mutual interrelations of the
different sectors of a national economy. Since the subject of this paper represents the next
step along the same path of inquiry, it can best be introduced through a short recapitulation
of the original problem. l
The mutual interdependence of the many different branches of production, transportation,
distribution, etc. (1 will refer to all of them from now -on as the different "industries") is
basically due to the fact that they exist by "taking in each other's wash." The inputs of any
one industry are the outputs of the others! Let Xi represent the annual rate of total output
(measured in appropriate physical units) of industry i, X ik the amount of the product of industry
k absorbed annually by industry i, and Xnk the amount of the same product k made available
for "outside use," that is, for'consumption not by anyone of the m industries explicitly included
in the economic system under consideration. The over-all input-output balance of a whole
national economy comprising m separate industries can be described in terms of m linear
, equations
k=m
Xi -
2:
k=1
X ki =Xni •
i
=
1, 2, .. .,.m.
(1)
Turning to the internal input-output structure of any particular industry, we find that
there exists a definite relation, rather narrowly determined by te~hnological-in the widest
sense of the word-considerations, between the rate of its output and the, quantities of all the
various materials and services required to aC,hieve it. As a first empirically justified approximation, the assumption can be made that the quantity of each kind of input absorbed by an
industry per unit of its output is fixed. Thus the magnitudes included in the balance equations
are subject to the set of structural relations
i = 1, 2, . . ., m; k = 1, 2, . . ., m.
(2)
i = 1, 2, . . ., m
(3)
i = 1, 2, . . . , m
(4)
Substituting Eq. (2) in Eq. (1) we have
k=m
Xi -
2:
k=1
which, solved for the
X/s,
akiXk
=
xni ,
gives
where the Aik'S are elements of the inverse of the structural matrix Iakil.
Given an "outside bill of goods" X nl , X n2 , • • ., X nm ' be it the final' domestic consumers'
333
WASSILY W. LEONTIEF
demand, allocations to the foreign countries under the Marshall Plan aid, or itemized material
requirements of a military mobilization program, the last system enables us to determine the
corresponding level of output in all the individual sectors of the economy .
. Only a few years ago the computational difficulties involved in the inversion of a square
matrix of order 40, 100, or 150 would have been considered practically insurmountable.
Now our main concern is that of collecting sufficiently accurate primary quantitative information on the basis of which such matrices are heing set up.
But new computational problems arise as the thinking on the subject advances.
The theoretical scheme of interindustrial relations as presented above is entirely static.
All variables occurring in it are time rates of input and output flows. The actual economic
process involves, however, not only flows but also stocks of commodities: stocks of machinery,
stocks of buildings, inventories of raw materials or goods-in-process and of finished commodities.
Explicit incorporation of stocks as well as of flows' in the model of the national economy
. leads to formulation of adynamic theory. The change in the magnitude of any particular
kind of stock-if it is at all possible-is achieved through accumulation or decumulation of a
flow over time. Let Sik represent the stock of commodity k used in industry i at the time t;
Sik describes, then, the flow pf additions to (or subtractions from) that particular stock. The
balance equations (1) can now be rewritten
k=rn
Xi -
2:
k=l
k=m
X ki -
2:
k=l
Ski
=
X ni •
i = 1,.2, . . ., m.
(5)
The flow of commodities from industry i to industry k is being split here explicitly into two
components, Xki representing that part of it which is being used "on current account" and
Ski the other part added (if Ski> 0) or subtracted (if Ski < 0) from the stock Ski.
A corresponding ~odification must be introduced also into the description of the internal
structure of the separate industries. The equations of set '(2) as formulated above refer only
to technical input requirements on current account. All additions to stock have, in the original
static f~rmulation, been treated as parts of the independent "outside demand," that is, the
vector X n1 , X n2 , • • ., Xnm; they were not explained but rather treated as known parameters.
The more comprehensive dynamic formulation contains an additional, second set of structural
equations in which each stock or capital requirement ora particular industry is related to its
rate of output,
(6)
or differentiating,
(6a)
The constants bki can be referred to as the capital coefficients ..
Substitution of Eqs. (2) and (6a) in Eq; (5) gives a set of m linea·r differential equations
with constant coefficients,
(7)
334
DYNAMIC ANALYSIS
O~
ECONOMIC EQUILIBRIUM
A general solution ,of this dynamic system can be written
k=m
Xi(t) = 2: Ck(X10, x 20, • • • , XmO)KkieAJ!,
L i (x n1 , x n2 , ••. , Xnm);
k=l
i = 1, 2, . . . , m
+
(8)
where the A'S are the characteristic roots of system (7); the Ck's, linear functions of the initial
conditions (expressed in terms of the rates of output of all industries at some point of time to) ;
the K/s, appropriate functions of the constants, the a's, b's, and A'S; while L i (x n1 , Xn2 ,. • ., xnm)
are linear functions of the outside demand Xn1 , Xn2 , • • ., Xnm~
Once obtained in numerical form, this solution makes it possible to answer various types
of questions arising in conne~tion with the explanation of the behavior of the economic system
over ,time.
Western Europe, for example, is striving to accomplish an investment program which in
a certain number of years would make it independent of the negative "outside demand,"
that is, outside supplies currently being made available to it under the Marshall Plan. Had
the necessary primary i~formation been available one could have determined the m rates of
surplus imports of various commodities (the xn/s) that would be required to raise the domestic
output from a given original level X 1(tO), X 2(tO), •.• , to some prescribed higher level X 1(t1),
X 2(t1), . . . over a stated period of time tl - to'
To obtain the desired answer it would be only necessary to insert in Eq. (8) the original
levds of output as the initial condition, set t in the exponentials equal to the prescribed recovery
period tl - to, and equate the right-hand terms of the equations to the desired final levels of
output XI (t1 ) , X 2 (t1 ), • • • The resulting m linear equations can then be solved for the m ,
unknown quantities Xn1 , Xn2 , •.. , Xnm of surplus imports.
If som'e of the characteristic rqots of the differential equations (7) turn out to be complex,
the outputs of the individual industries will display a typical periodic pattern of motion with
increasing constants of diminishing amplitudes depending upon the magnitude of the real
parts of the roots. The consideration of such periodic solutions constitutes the theoretical
basis of many a contemporary business-cycle theory.
Although very attractive because of its obvious simplicity, this explanation of alternative
booms and depressions has a serious weakness which, ifit is overcome by appropriate theoretical
reformulation, leads to a new and interesting computational problem.
The original static system has been transformed into a dynamic one by the introduction
of stock-flow relations as described by a set of appropriate capital coefficients. Not all capital
stocks can, however, be decumulated, that is, reduced in the same way in which they are
being accumulated. Investments in raw materials, goods-in-process, and finished commodities,
in short, stocks which are associated with the concept of working capital, can indeed move
downward as easily as they can go up; not so with fixed capital, that is, machinery, buildings,
permanent investment in roadbeds, soil conservation, etc. Provided sufIicient sources of
supply exist, these stocks can change in the upward direction ~s readily as working capital.
335
WASSILY W. LEONTIEF
In the case of contracting demand, however, fixed capital cannot be as readily reduced as,
say, inventories of raw materials. For obvious technological reasons, the rate at which
machinery and buildings, not to speak of so-called permanent land improvements, can be
used up is strictly limited and at best is very low. The appearance of unused capaCity of idle
fixed capital (in times of downward production trends and during the initial 'phases of recovery
when output has not yet reached the previous high) constitutes one of the most characteristic
aspects of modern business fluctuations.
In the light of these observations, the use of unchanging stock-flow ratios well suited-at
. least in the first approximation-to the analysis of the dynamics of working capital is inappropriate in application to fixed capital. As soon as the rate of k: 2.2.g;k2
j k
1
1.00
2
3
0.20
0.04
5
4
=
0.3200, W 2 = 16.00.)
6
7
8
- 0.16
-0.08
~·0.08
0.00
0.00
-0.08
·0.00
0.00
0.00
0.08
0.08
0.08
- 0.16
2
0.20
1.00
- 0.l6
0.04
. - 0.08
3
0.04
' - 0.l6
1.00
0.20
0.00
4
- 0.l6
0.04
0.20
1.00
0.00
0.00
0.08
5
- 0.08
-0.08
0.00
0.00
1.00
0.20
0.04
6
- 0.08
-0.08
0.00
0.00
0.20
1.00
-·0.16
0.04
7
0.00
0.00
0.08
0.08
0.04
- 0.16
1.00
0.20
8
0.00
0.00
0.08
0.08
- 0.16
0.04
0.20
1.00
The matrix G of Table 5 is defined by Eq. (6):
G = E-BB'
+ 1.
(6)
Off-diagonal entries in G are interpreted as the residual portion of E not accounted for by
the matrix B. The diagonal entries must be unity as a consequence of Eqs. (1) to (5). The
general size of the off-diagonal entries is of special interest. First, as the number of factors
-that is, columns of matrix A-is increased, the general size of these off-diagonal entries
decreases. A second important relation is with the number of cases on which observations
were made. When observations are made on an extremely large group of cases, the off-diagonal
entries in G are vanishingly small for a given set of variables and a given number of columns
in matrix A. As smaller groups of cases are considered, these off-diagonal entries will usually
increase in size. Consider the index Wr defined in Eq. (7):
Wr = N'L'Lgik 2,
j> k
(7)
j
k
where N is the number of cases in the sample. The double summation is of squares of the
off-diagonal entries of G below the diagonal. When the off-diago~a'l entries differ from zero
only because of chance sampling effects, Wr for a nu·mber of samples of size N will have a
frequency distribution in accordance with the chi-square distribution with appropriate degrees
of freedom. Values of chi-square have been tabulated. It is possible, therefore, to set up a
34 2
COMPUTATIONAL PROBLEMS IN PSYCHOLOGY
value above which we would believe the possibilities of observing Wr owing solely to chance
sampling effects to be negligible. Thus, whenever a
is observed above this critical value,
it will be concluded that more columns of A are required. This reasoning leads to
w,.
Condition II. The number of columns r of matrix A is the smallest number where Wr is less than
chi-square, with i[ (n - r) 2 - n - r] degrees offreedom, for a given level of significance.
When a IO-percent level of confidence is used,a chi-square can be computed that will be
exceeded 10 percent of the time by chance sampling effects. This IO-percent level of confidence
seems, to the author, to be appropriate for the present problem.
By making us~ of an excellent approximation devised by Edwin B. Wilson and Margaret
M. Hilferty, a direct computational procedure can be designated. Here r is to be the smallest
integer for which
'
Wr
< t[(n - r)2 - n - r]{ 1 - 9 [
( ) ; -n-r]
n-r
+ 1.2816. J9( n-r); -n-r }3.
(8)
Solutions have been obtained for the illustrative problem for 0, 1, and 2 columns of matrix A.
Values of Wr , degrees of freedom, and actual chi-squares and approximate chi-squares for
a IO-percent level of confidence are listed in Table 6. ~or no factors and for one factor, that
Table 6. Values of Wr , number of degrees of freedom, and actual and approximate
chi-square for IO-percent level of confidence for illustrative example.
r
Wr
Degrees of
freedom
Chi-square
Actual Approximate
0
228
28
37.92
37.91
1
106
20
28.41
28.40
2
16
13
19.81
19.80
is, r equal to 0 or 1, Wr is greater than the corresponding chi-square. For two factors, Wr
is less than the chi-square. We'therefore conclude that two factors are appropriate for the
illustrative example. The last two columns of the table indicate the excellence of the
approximation.
In review, the computational problem is to begin with a matrix C and to obtain matrices
A and values of Wr for successively increasing numbers of factors until the inequality (8) is
satisfied. The matrix A with the 'smallest number of columns for which the inequality is
satisfied is the solution.
The major difficulty lies in obtaining a matrix A, with any specified number of columns,
from matrix C in order to satisfy Eqs. (1) to (5) and Condition I. Direct solutions seem
impracticable. One possible successive-approximation method is based on a procedure
developed by Harold Hotelling for obtaining principal components. An initial trial matrix
343
LEDYARDR. TUCKER
is obtained by any of several methods. Random numbers may be used, provided the. Ui1 2
of Eq·.(9) is greater than zero:
. Al
Ujl2
=
(9)
Laim12.
m
Cu -
It is possible to have zero-entries for all but two cells in anyone· column of AI' Computations
for one trial for the illustrative example are presented in Table 7. The case illustrated is that
Table 7. One trial of a: successive-approximation method.
Variable
number
I
II
1
-0.28
0
0.27
2
2.44
0
3
- 2.22
4
12.81
5
Al
LI
u2iZ
A2
MI
I
II
- 0.28
0.10
II
I
- 1.04
0
- 3.4
- 0.1
0.85
2.87
0
30.1
2.6
0
6.27
- 0.35
0
- 26.9
0.6
- 2.18
1.41
61.30
0.21
0
159.6
21.2
12.93
2.13
2.18
0
'0
24.50
0.09
0
25.8
- 5.7
6
4.68
4.66
17.63
0.27
.0.26
. 63.3
16.4
5.13
i
0.53
- 0.22
3.4
13.3
0.28 - 5.79
2.04
26.08
40.84 .
0.02
8
- 5.66
'0
.
I
0.05
II
.~
K-I
I
K-2
I
2.14 '
4.23
1.75
KI'
I'
I
II
I
153.66
15.71
I
12.40
0
I
0.081
II
15.80
7.19
II
1.27
2.36
II
0
I'
2.09 - 3.53
6.8 .
26.4
0
2.44 - 0.19
, II'
II'
- 0.043
0.424
in which two factors are under consideration. A similar series of trials was performed for the
case of one factor, and the first column of Al in Table 7 isthe approximate solution obtained.
The values in cells for rows 6 and 7 wer~ obtained by solving Eqs. (10),; (1 i), and (12) ,:
1ik == Cik -
.a62 a72 =
167'
a62 2
!s6
,
-2=r'
a 72 . J 77
ajla kl ,
(10)
(11)
( 12)
The entire matrix F with cell entries 1ik was computed and the largest off-diagonal entry
selected. This was for tests 6 and 7 ~ Equations (11), and (12) yield reasonable values for
a62 and a72 • Obtaining atrial matrix AI, the Uil 2'S are computed by Eq.9.
Equations (13), (14), and (15) indicate the computations for subsequent steps:
(13)
344
COMPUTATIONAL PROBLEMS IN PSYCHOLOGY
(14)
(15)
K I -2 = Ll' A,11 •
The matrix Ll is obtained by dividing each entry in a row of Al by the Uil 2 for that row.
Matrices MI and KI-2 are obtained as matrix products. Matrix KI-2 should be symmetric
except as a result of rounding-off errors in MI' Usually minor errors in symmetry wiil not
materially affect succeeding steps. The matrix KI~I is computed to satisfy Eq. (16) and to
have zero entries above the diagonal:
(16)
(KI-l)(KI-l), = KI-2.
The inverse ofK1-1 and the matrix product of Eq. (17) are obtained:
MIKI'
=
( 17)
A 2•
The matrix A2 is the second approximation.
Usually this system should converge on the desired solution. The rate of convergence is
likely to be low, and many trials may be required for each number of factors. As a result,
the kind of computational assistance offered by large-scale digital computing machines is
essential. Other simpler factorial methods are in use by psychologists mainly to avoid the
computational labor involved in the procedure we have outlined. These other procedures
have not answered the question of the number of factors to be considered, and they suffer
,from the usual effects of approximations. It would be quite desirable to be able to apply
. Lawley's method.
Another computational problem with which many psychologists are concerned can be
mentioned here, but we will not spend much time on it. The matrix A can be considered
to give the coordinates of n vectors, one for each test, in r-dimensional space, one dimension
for each factor. It is possible to restate Eqs. (4) and (5) and Condition I to allow the rotation
of axes vyithin the space defined by matrix A. L. L. Thurstone has stated his principle of
simple structure to solve the problem of where to rotate the axes. It is sufficient to state here
that much computational labor is involved and assistance will be welcomed by psychologists.
Unfortunately, Thurstone's principle is qualitative in nature. Until his principle has been
successfully restflted in precise mathematical relations, the solution will depend on the judgment
of the analyst, and the applicability of large-scale calculating machines will be doubtful.
We shall now turn to our second problem.
Problem 2. Determination of Distribution
Correlation.
of Items
in Difficulty to Yield Maximllm Test-Ability
The computational problem can be stated quite simply. Consider Eq. (18):
V
= 2: rs 1 e- ithl
It
V27T
{2:lthf';() V27T
1 e-ixh2dxh _ (2: fa) 1 e-ixh2dXh)2
Jt V27T
It
+ 2: 2: f.a)f.a) 27T V 11h ii=ll th
ti
It
rs4
h
exp [- 2(1 1 .4) (Xh2
. - 1s
345
+ X 2i
2x h Xi r,s2)] ¢XidXh}-l. (18)
LEDYARD R. TUCKER
This equation arises in a mathematical development related to the assembly of standardized
aptitude examinations. Let us take as an example a test in addition, where it is desired to
differentiate, among people most validly with respect to rapidity and accuracy of adding
columns of numbers. When, however, a large number of problems are tried out on a group
of people, it is found that some problems are easy and others are more difficult in terms of the
percentage of the group that gives the correct answers in the time allowed. These differences
in difficulty could arise from differing lengths of the columns to be added, or from differind
numbers of digits in each number, or from use of numbers differing in ease of addition. It
is easier to add l's and 2's than 6's, 7's, or 8's. Thus the test technician has at hh disposal
a large number of problems of different difficulty. How should the problems be selected as
to .difficulty? If only those problems are used to which half of the group gives the correct
answer in the time allotted, then the people who are very poor in addition will be grouped at
low sco~es and the people who are very good in addition will be grouped at high scores, there
being, therefore, little differentiation among members in either of the extreme groups. It
thus seems that the problems should be spread out in difficulty. Exactly what is the best
distribution of problems in difficulty is not known. Our Eq. (18) is an attempt to obtain the
answer.
We have assumed a scale of ability. The coefficient. Vis the correlation between the test
scores and scores on this ability. The quantity rs is a measure of the relation of each test
problem and the underlying ability. The th's are indices of the problems' difficulties .. The
subscript i is used alternatively with h to designate the problems. We would like, for any
given number of problems and value ofrs, to seiect that ~et ofth's' which 'would yield a maximum
value for V. It will be noted that the independent variables~the th's--.:.appear as exponents
and as limits of definite integrals for which 'only numeric solutioI?-s exist. Tabulated sets of
values of the th's that maximize V for several conditions of numbers of problems between
10 and 500.and of values of rs between zero and unity would be of considerable assistance as
guides in the construction of tests. I have not tried a computational type of solution, but the
reports of capabilities of the large-s~ale digital computers indicate that they should yield the
desired result~.
In this paper r' have outlined two problems on which the large~scale computing machines
should be of assistance to psychologists. These problems were chosen because (a) they were
so stated that computational-type solutions were feasible, (b) the volume and complexity of
the calculations indicated a ne~d for assis~ance from a larg;e-scale machine, and (c) the problems
were of special interest to the author.
REFERENCES
H. Hotelling, "Analysis of a complex of statistical variables into principal components," J. Educational Psychol.24, 417-441, 498-520 (1933).
'
D. N. Lawley, "The estimation of factor loadings by the method of maximum likelihood," Prot:.
(Edinburgh) 60, 64-82 (1940).
R~y. ~oc.
COMPUTATIONAL PROBLEMS IN PSYCHOLOGY
D. N. Lawley, "Further investigations offactor estimation," Proc. Roy. Soc. (Edinburgh) 61, 176-185
(1941 ).
D. N. Lawley, "The application of the maximum likelihood method to factor analysis," Brit. J.
Psycho!. 33, 172-175 (1943).
L. L. Thurstone, Multiple factor analysis (University of Chicago Press, Chicago, 1947).
Edwin B. Wilson, and Margaret M. Hilferty, "The distribution of Chi-square," Proc. Nat. Acad.
Sci. U.S. 17,684-688 (1931).
347
COMPUTATIONAL ASPECTS OF CERTAIN ECONOMETRIC PROBLEMS
.HERMAN CHERNOFF
University of Chicago
Economists are frequently interested in constructing models of economic behavior and
estimating the parameters involved. In many cases the traditional least-squares treatment
fails and it becomes necessary to maximize a likelihood function of many variables. Such
problems frequently involve many matrix operations where it is of great importance for
computers to have high computing speeds, either large· internal memory or rapid transfer
from internal to external memories, or both, and the ability to take advantage of matrices
which are very simple in that most' elements are zero.
As an illustrative· example, suppose that Ytl represents the quantity sold of a certain good
in ye~r t; Yt2' the price of the good inyear t; Ztl, the national income in year t~ and Zt2, the
wage rate in the producing industry in year t. Then the economist hypothesizes the following
model, consisting of the demand and supply equations
Ytl =' IXLYt2
Ytl =
where
eXI ,
Utl
and
IX2' IXa, f31'.
Ut2
f32'
f3LYt2
+ IX 2Z tl
+ IXa + UtI,
+ f32 Z t2 + f3a + Ut2 ,
are random unobserved disturbances. It is desired to estimate the parameters
f3a because they can be used to forecast the effect of an excise tax, say, on the
price and quantity of a good (once the national income and the wage rate in the producing
industry are known). Furtherm~reJ these parameters in themselves have important theoretical
significance.
One s~ould refrain from using the traditional least-squares regression method to estimate
the parameters, for the conditions justifying the use of least squares are not s·atisfied and that
method may give meaningless results even in large samples. (To justify least squares one must
assume that Utl, U t2 are distributed independently ofYt2' Ztl, Zt2; this is not the case sinceYtl
andYt2 are determined by Ztl, Zt2, Utl, u t2 .) In this case the maximum-likelihood· method should
be applied.
In general, our system of equations may be written
+ f312.J't2 +.... + YnZtl + Y2l Z t2 + ... =
f321Ytl + f322Yt2 + ... + Y2l Z tl + Y22 Z t2+ • • • =
f3nYtl
{3gIYtl
+
f3g2Yt2
+ ... + YgIZtl + Yg2Z t2 + ... =
348
Utl,
U t2 ,
lltg,
COMPUTATION IN ECONOMETRIC PROBLEMS
where they's and z's are obse:r;ved variables apd the parameters f3ii' Yik are subject to certain
restrictions derived from economic theory. A case which occurs frequently is that where
linear functions of these parameters are known to vanish. In this case a considerable number
of simplifications may enter. The Uti are unobserved random disturbances which are assumed
to be jointly normally distributed, independently of the Zti'l
The above equations may be written
By/ + rz/, =
or
u/,
Ax/ = u/,
where
A
=.
(B
r), 'x/
=e::).
Then it can be shown' that the maximum-likelihood method involves maximizing
log L(A)
=
-llog det (A W A') -
t
lo.g det (A MA')
as a function of A, where A is subject to the a priori restrictions due to economic theory.
Here
1 T
(Mn .M12 )
]vI = T L X t X t = M' L{ ,
I
21
t=1
1
T
.
Mn = T LY/Yt,
t=1
W
=
(;:'n
M12
.Lv,
22
1
T
= T L Y/Zt, etc.,
t=l
~),
Wn = Mn - M12M22-1M21'
Iterative methods have been applied to maximize L. These are gradient methods where
one starts with a certain approximation P and takes a step in the direction of steepest ascent. 2
In particular, the Newton method is a special case of a gradient method where convergence
per iteration is very rapid but the amount of calculation per iteration is quite large. This
method converges faster as the approximation gets closer to the maximizing value. To apply
gradient methods one must consider first derivatives of L. For the Newton method, second
derivatives are also required. It is found convenient to work with the Taylor expansion
log L(P
+ hD) =
log L(P) h tr{[(PWP/)-l PW - (PMP/)-l PM]D/}
+ O(h2),
where tr represents the trace or the sum of the elements along the main diagonal. From
this expansion it is seen that to compute the first-order derivatives one must essentially compute
(PWP')-l PW - (PMP/)-l PM. In a typical case of an eleven-equation system, the number
of rows and columns corresponding to P, M, Wn , respectively, are (11 X 40), (40 X 40),
(11 X 11).
In the case where each of the restrictions is linear in the coefficients of one equation, one
can reduce the number of computations per iteration considerably at the cost of introducing
a few initial operations. For example, consider the following formula for the direction of
349
HERMAN CHERNOFF
steepest ascent with respect to the linearly· independent parameters a in terms of which P
may be computed:
d = a[(PWP')-l @ W - (PMP')-l @M]cI>'B-l,
where B = cI>[(PMP')"':'l @ (M - W)]cI>', cI> is a constant matrix determined by the restrictions,.
a is in a special sense the vector of the independent or unrestricted variables involv~d in P,
and @ indicates the Kronecker product of two matrices.
Among the circumstances that can be used to. reduce the number of comp~tations per
iteration is the fact that cI> is usually an extremely siInple matrix most of whose elements are
zero or one. In the above-mentioned eleven-equation system, the sizes of the matrices d,
a, cI>, B, respectively, are(I X 60), (1 X 60), [60 X (40 xII)], (60 X 60).
Thus to treat the above system or larger ones it is necessary to have a machine which can
rapidly refer to large matrices which are kept constant throughout the iterations to perform.
over a dozen matrix operations per iteration and preferably to make good use of the simplicity
of certain. rna trices (cI».
REFERENCES
. 1. The condition of normality can' be relaxed without seriously affecting the properties of the
estimates obtained. in the manner to be described below. The possible generalizations of conditions
are considered iIi more detail in a forthcoming monograph of the Cowles Commission.
2. See H. Chernoff and J. Bronfenbrenner, "Gradient methods of maximization," Cowles Commission Discussion Paper 332, and T. C. Koopmans, H. Rubin and R. Leipnik, ."Measuring the
equation systems of dynamic economics," Art. II (esp. Sec. 4) in "Statistical inference in dynamic
economic models," Cowles Commission Monograph 10.
350
PHYSIOLOGY AND COMPUTATION DEVICES
WILLIAM
J.
CROZIER
Harvard University
In discussing briefly the significance of high-speed calculation instruments for inquiry in
the realm of physiology I am putting to Ol1e side, for present purposes, th?se aspects of organic
activity which involve interactions between and among individuals and between groups of
individuals and environmental' forces or conditions. Undoubtedly, as data accumulate and
analytical insight grows (although the two may not march in close company), the service of
such devices can be great in reducing the labor of calculations required for the testing of
hypotheses and the making of predictions. Even here, however, one can doubt whether it
will be possible or even desirable to resort to the push-button selection of hypotheses concerning
ethology, ecology, population genetics, and the like, let alone for the setting up of differential
equations basic to the erection of quantitative conceptions. The problem must be stated
before a solution can, be tested.
It is rather in connection with questions of the nature of the individual organism and its
constituent processes that some less obvious considerations are required. Two foci of interest
are clear. The sheer complexity of these processes requires that as understanding of them
improves it will be more and more necessary to recognize, for theory building, that the organic
property under scrutiny is essentially a multivariate one, in which many parameters are
significant. In the end, now only dimly approached, it is likely that all available mathematical
devices and aids to calculation will be called upon for real progress toward formulation and
pe~hapscomprehension.
This first point of interest finds parallels in, for instance, weather prediction or even in
cosmology, where the basic~ifficulty is rather a mechanical one, arising from the degree of
appropriateness of the mathematical system applied and the onerousness of using it, granted
data that are sufficient. Beyond this is the second focus of interest, essentially more attractive
and more stimulating. It centers upon the possibility ofcreat,ing models of individual biological
processes. Such models, both mathematical and material, have of course played a considerable
role in general physiology-as aids to the clarification and the cqncreteness of thinking, and
as springboards for new experiments. Somewhat crudely put, but not unfairly, the question
has arisen: Do complex, fast, computation devices provide an effective model of mental
processes, or even of one general class of human cerebral operations-even to the extent that
,such machines, with developments of kinds now foreseeable, may be used to serve as surrogates
for human decisions or actions?
Note that there are here two distinct questions. The adequate imitation of a given kind
of end result as achieved by an organism does not at all imply that the mechanism whereby
35 1
WILLIAM
J.
CROZIER
the organism acts or deCides has been duplicated. 'For engineering purposes, as in the "no
hands" operation of a production line, this may be quite immaterial (so long as men keep the
surrogate in good worki~g order). But the physiologist's job is different. What he seeks is
not merely an over-all.model. He really looks for an understanding of the actual mechanisms
whereby the organic, biological machine operates. An even partially successful model may
he enormously helpful in furthering this quest, in a particular case; but its character as an
analogical crutch must not be lost sight of.
The biologist is not necessarily too grumbly about this-or at least I should not like to
have it thought tha.t ,he is. He is perfectly able to accept, for the time being, a system ofkinetic
equa,tions embracing the data of photosynthesis as picturing the known essence of the mechanism of this process. He also wants to learn the molecular inwardness of the matter, the
kinetic mechanism with all its defects serving as a ladder toward specific experimental inquiry.
He is not so crude as to look for a nexus of springs pulling dashpots throughb,aths of hydraulic
oil when he peers at muscle fibers in electron-microscope pictures, though this kind of model
has had brilliant uses for some purposes. On the other hand, he knows in a practical way,
as the logician knows, that reasoning by analogy from dynamical properties of the model is
liable to stumble over imperfections in the analogy. Gross instances are available in which
properties of the mechanical model alone, not recognized at the time as such, have misled
inquiry into blind alleys .
.An illustrative case is in point. It is drawn from the field of visual excitation, which has
many advantages for my present purpose. It is significant as illustrating several principles
important for the method of evaluating analogs for organic events and processes. It happens
that a variety of visual ~ata have been submitted to description in terms of the notion that
a visual (seen) threshold effect is brought about when the incidence of light produces a fixed
amount of freshly formed decomposition product of aphotos.ensitive material in the retina
(investigations by S. Hecht). One type of such data was that presented by the contour of
critical flicker frequency as a function of flash intensity. The equation, derived from an
experimental foundation, used for description (by visual, aesthetic test of fitness) implied
certain consequences as necessary if the temperature of the organism were to be altered or
if the light-dark sequence in the flash cycle were to be changed from that commonly characteristic iIi such experiments, nam~ly, 1: 1. The test of the cogency of this formulation cannot
be made by any proc,ess of curve fitting. It has to be made by the use of what I may be allowed
to call parametric ana[ysis-the deliberate modification of experimental conditions in such ways
as to reveal the number and the (or some) quantitative properties of the .parameters of the
necessary formulation. In this specific case, it was easily shown (Crozier) that the photostationary-state equations were incompetent because they completely failed to predict even
the direction of the modifications of the flicker contours when tests such as those already
referred to were made.
It is interesting to note in outline just what had happened here. It provides a useful
commentary on the dangers of analogy. The visual systems of animals exhibit reversible
35 2
PHYSIOLOGY AND COMPUTATION DEVICES
changes of excitability according to the prevailing illumination to which they are exposed.
As a first approximation, it was inevitable that help should be sought from the roughly parallel
kinetic properties of known reversible-reaction systems in physical chemistry. This involved,
however, the crass assumption that seeing, as a decisive behavioral act, occurs in direct
proportion to retinal.events. It also involved the delusion consequent upon the implicit
assumption that the retina is, in the sense of chemical kinetics, a homogeneous medium. It
likewise avoided recognition of the patent fact that the equations to which the initial assumption
led were not in fact unique, even accepting their apparent descriptive adequacy for the data.
In form the equation is identically a logistic in which the light intensity enters as the logarithm.
This of course cannot be distinguished from a Gaussian integral except by very precise measurements extending close to the asymptotes. Data of this kind, as well as the quantitative properties
of the parameters, help to decide in favor of the logarithmic-Gaussian equation.
The important, central fact is of course that the visual system-even at the retina, and
even in a single receptor unit-is obviously a microheterogeneous system. It comprises
assemblages of semi-isolated reaction chambers. Thereby is generated an 'essentially statistical
situation. Without. enlarging here upon details, it can pretty certainly be shown that here
one really has a situation in which the assumptions used by Gauss are implicit; hence sensory
effect is expressed as a Gaussian integral, which is logarithmic in the abscissa in common
cases because the basic parameter fluctuates spontaneously.
.
There is involved here the conception, for which there is direct observational support,
that at sundry levels in the hierarchy of assemblages of neural units and subunits implicated
in a behavioral decision or act one has to do with populations of units which individually
fluctuate in their contributions to the end result. In the light of this we may examine briefly
several concrete cases. From a study of them there emerge two c'o~siderations, forming the
core of what is here suggested. The first is that any model of a mental process, which to be
analytically significant must include in its dynamical structure the essential nature of the
process, must in this context operate in such a way that decisions are automatically achieved
by thorough statistical comparisons. The second is that it is precisely in this way that elementary "nlental" decisions are arrived at.
One is thinking here of elementary decisions neurally mediated. If one is to have a model
of mental processes rather than merely a mimicking of their results, it is necessary te stored, in the
shortest possible time without consideration of any previous selection. A device with such a
digitalized address system and such direct access to any stored signa~ can be used singly or in
groups in a most flexible manner, since no amplitude-sensitive qualities have to be dealt with
and no specific sequences are intrinsic to the memory.
The Selectron (Fig. 1) is a vacuum tube designed in an attempt to realize such an ideal
memory device. The principle of the tube depends on quantizing both the address of the
stored information and the information itself. The selection of the address is obtained by
means of two orthogonal sets of parallel spaced metallic bars forming a checkerboard of windows. A shower of electrons impinges on this checkerboard. Electrons are stopped in all
windows except in a selected one by applying address-selecting voltages to certain groups of
bars connected into appropriate combinations. The storage is in terms of the two stable
potentials that tiny floating metallic elements, located in register with the windows, assume
under continuous electron bombardment. The reading signals .are sizable electron currents
passing through a hole in the storing elements. The signals produce also a visual monitoring
display.
The basic principle of the Selectron has not been changed. The main improvement is
the use of discrete metallic eyelets as the storing elements. In addition to very reliable storage,
. these eyelets have a "grid-action" effect yielding strong electronic reading signals. ,
The Selectron tube, called SE256, has 256 storing elements, is 3 in. in dIameter arid 7 in.
long, and utilizes a 40-lead stem. The diametral and axial cross sections of the tube are shown
in Figs. 2 and 3. Eight elongated cathodes of rectangular cross section are located in a diametral
plane of the tube. Between and parallel to the cathodes are a set of nine selecting bars of
square cross section. These vertical selecting bars are connected into six groups: VI' V2 , V3 , V4 ,
JAN RAJCHMAN
and Vl', V2 ', as shown in Fig. 4. On either side of the plane of the cathodes and V bars there
is a set of 18 p arallel bars of square cross section at right angles to the V set. These two sets
of horizontal selecting bars sandwich the cathodes and V ba rs as do all subsequent electrodes
FIG. 1. The Selectron.
of the tube, the tube being symmetrical with respect to the cathode plane. The 36 horizontal
selecting bars are connected in 12 groups: Hl to H4 and Hl ' to H s', as shown in Fig. 4. There
are nine vertical bars for eight gates and 36 horizontal bars for 32 gates, the excess bars taking
care of the end effects.
On either side beyond the horizontal bars there is a collector made of two flat plates
perforated with round holes whose centers match the centers of the windows formed by the
366
THE SELECTRON
V and H-bars. Adjacent to the collector plates there are two perforated mica sheets holding
between them 128 metallic eyelets. These eyelets, made on automatic screw machines, have
a conical head, a center hole, a holding collar and a shielding tail. They are nickel-plated
2. Diametral view of Selectron.
steel. On the other side of the two mica plates is another perforated metal plate-the writing
plate. The two collector plates, the two eyele~ mica plates, and the writing plate form a tight
assembly riveted together at the ends and in the center.
Beyond the writing plate is another metal plate-the reading plate-perforated. with holes
in register with the holes of the other plates. Beyond it is a Faraday cage fonned by two
perforated plates spaced some distance apart and closed on all four sides by a metallic wall.
. FIG.
367
FIG.
3. Axial cross section of Selectron.
368
THE SELECTRON
A glass plate coated with a fluorescent material is placed against the outer plate of the cage.
In the central plane of the cage there are nine wires which are spaced so as to be between
the holes of the perforated plates. These reading wires are connected together and the
corresponding lead to the stem is shielded.
In the quiescent state of the tube storing informa- ~,:..---++----,.----t-----r--lion previously written-in, all the selecting bars are
at the potential of the cathodes (0 v) and all other
electrodes at potentials indicated in Fig. 5. In this
condition electrons emitted from the cathodes are
focused into 256 beams by the combined action of
the Vand H bars at zero potential and the collector
,plate at some positive potential, such as 180 v.
These beams are focused through the centers of the
collector holes and are directed on the eyelets.
Since the eyelets are not connected anywhere-are
electrically floating-their potentials ,will adjust
themselves so that the net electron currerit to them
is exactly zero. It turns out that there are two
naturally stable potentials for which this is the case.
This can be understood, by examining the current
to the eyelet as a function of its potential as shown'
in Fig. 6; When the eyelet is more negative than
the cathode, no current reaches it because it repels
any incurring electrons. As the eyelet is made more
positive, some electrons strike it, producing a negative current. At a still more positive, potential,
secondary emission from the surface of the eyelet
starts as a result of the primary bombardment and
tends to cancel the negative current, being. a loss of
negative charge. Eventually, the two are equal at
i= =
m [6 1Tl, [E 3, ~ i4 ' ,0, ~
4' ;
23
I
±
:
I
VERTICAL CONNECTIONS
2 + 4· 6 LEADS
2 X 4 = 8 BARS
(\J
~
x
CD
(/)
IZ
w
';:E
w
..J
W
.
~
~
+
CD
(/)
~-H-+-~
o
Z
o
-"
zO 300
Or
C>u 200
I-w
:::>...J
OW
C>
:::>
0:
J:
o
ttZ
tt-
• Z
w.1
ri
-=~~==~~~~~~::::~::~:;::::~2=5~0~VOLTS
11\
V
w ....
" 0 o:::w
O:::c(
:::>w
UJ:
C)t-
~w
1--1
-w
-
0:::
:::>...J
UO
J:
POTENTIAL OF EYELET
WITH RESPECT TO
CATHODE
en 100
c>Z
ZO
_0:::
C)t- ~t- 200
~w OU
O...J U W
300
c(w
z...J
_w
w>">- o:::w
~w
~
----------------IW
400
FIG.
6. Current to eyelet as a function of its potential.
Writing and reading are done one element at a time (or two if the tube is used as a twochannel device) and require selection.
To write into a particular element, current is interrupted everywhere except to that
element. Then a voltage pulse of the shape shown in Fig. 6 is applied to the writing plate.
Because of the capacitive coupling between the eyelet and the writing plate, the rapid rise
of this pulse will cause the eyelet to jump up in potential by an amount adjusted to be a
substantial proportion of the collector potential or more. If the eyelet was. initially at cathode
potential, it will now have been brought near collector potential and will settle at that potential
during the plateau· of the pulse. If it had initially the col,lector potential, it will acquire
momentarily twice the collector potential and will receive substan·tial negative current (see
Fig. 6) which will also bring it. to the collector potential during the plateau time. Whatever
37 1
JAN RAJCHMAN
the'initial condition, at the end of the plateau time the eyelet will be at collector potential.
At this instant the choice is made between positive and negative writing. For positive writing,
no additional pulses are applied to the selecting bars, and the current remainson the eyelet
during the relatively slow decay of the writing pulse. The decay is slow enough to allow the
electronic locking current to keep the eyelet at the collector potential in spite of the displacement capacitive current tending to drag it to cathode potential. This "slow" decay is in fact
only one to several microseconds. For negative writing, 'an additional pulse is applied to one
or more of the four selecting bars in the groups V, V', H, and H', which cuts off the current
to the selected eyelet during the decay, time of the writing pulse. The capacitive down drag
is therefore not counteracted and the eyelet is brought to cathode potential.
Immediately .after the end of the writing pulse the selection pulses end, and current is
reestablished to all eyelets. Only residual ohmic (on other second-order electron or ionic
currents) affect the unseleded eyelets, during the short selection time, and therefore at the
end of the writing pulse they have almost their original potential. This potential is reached
almost immediately thereafter by virtue of the stabilizing currents.
The reading signal is derived from the current passing through the central hole in the
eyelets. Part of the current directed at the eyelet is directed at that tiny hole. When the eyelet
is positive, at collector potential, the electrons directed at the hole go through it by virtue of
their inertia. When the eyelet is negative, at cathode potential, it exercises "grid action" and
electrons are repelled and do not go through the hole. The electrons' paths are shown in
Fig. 5 for the three cases, while the current characteristics are shown in Fig. 6. The presence
or absence of the current through the eyelet is therefore an indication ofthe state of the eyelet.
In the quiescent state of the tube the reading plate is biased off nega!ively and the reading
current going through all the positive eyelets (any number from 0 to 256) does not reach the
reading circuits. To read, an element is selected by applying negative pulses, to all but four
, bars, as explained above. Immediately thereafter a positive pulse is applied to the reading
plate which allows the current through the selected element, if current there is, to proceed
to the output electrodes. The electrons penetrate into the Faraday cage, strike the fluorescent
screen, producing a ,light signal, and also cause the emission of secondary electrons. These
secondary electrons are collected by the ,reading' wires which are connected in parallel and
constitute the reading output signal. The reading wires have a .low electrostatic capacity
and are well shielded from capacity pick-up by the Faraday cage.
For monitoring purposes it is convenient to bias positively the reading plate. A display,
of the stored pattern appears then on the fluorescent screen.
The main characteristics of the Selectron SE256 may be summarized as follows. The tube
has a, capacity of 256 on-off signals. The storage time is indefinite. 'J'he access time to any
element is approximately 10 psec and is independent of all previous accesses to other elements.
The address selection is by means of combinations of non-amplitude-critical pulses of about
200 v applied to circuits with pure capacitive loading of 10 to 20 ppf. The writing and reading
require also pulses whose amplitude and duration have considerable tolerances and are applied
37 2
THE SELECTRON
to pure capacitive loading, 200 !tflf for writing and 50 flflf for reading. The output isa direct
electronic current of 20 to 40 flamp per element. The tube is its own monitor. The supply
voltages have wide tolerances. The total power dissipation is 40 w.
About a score of tubes have been made to date. These tubes were tested first by d.c. or
simple pulse tests. Uniform characteristics of selection and control have been observed in all
tubes, as these depend on geometric factors. that are easily reproducible. The cathode emissions
and secondary emissions of the eyelets were also found essentially uniform. The period of
quiescent-state storage has, of course, been found to be as long as desired or as there was
pa tience to observe it.
A program has been initiated to test the tubes in conditions as similar as possible to those
of an actual computer straining its memory severely. The system consists of taking two
Selectrons, setting an arbitrary pattern of stored information in one of them, interrogating
the elements of that tube one by one in succession, and registering the answers in the corresponding windows of the other tube. The stored pattern will thus be transferred from tube
No.1 to tube No. 2. The pattern is then transferred in a similar manner from tube No.2
back into tube No.1, but this time the polarity is reversed so that positive elements in one
tube correspond to negative ones in the other. The life test consists of letting this back-andforth transfer proceed automatically at a reasonably high repetition rate and observing whether
the initially set pattern remains unspoiled in the system.
To date, runs 0(20 hr without any failures have been observed. The over-all characteristics
of the pair of tubes in the life-test circuit did not change measurably in 700 hr. We are engaged
at present in improving the testing circuits to be certaih that they are not the cause of the
occasional failures that still occur in long runs. We are also attempting to gain greater safety
factors in the tubes themselves.
The research has reached the stage at which a Selectron of a capacity of 256 elements has
been designed. It is practical and reliable in its operation and reasonably easy to build. While
the life tests are still in progress and· data from them are incomplete, there is every reason to
believe that tubes with fairly long life can be made. The fast access time, the digitalized
operation for address reading and information registering, the relatively intense output signals
and self-monitoring by luminous display make the tube particularly useful for electronic
computing machines and other information-handling machines.
REFERENCE
1. J. Rajchman, "The selectron-a tube for selective electrostatic storage," Proceedings of a Symposiumon Large-Scale Digital Calculating Machinery (Harvard University Press, Cambridge, 1948), p. 133.
373
TRAITS CARACTERISTIQUES DE LA CALCULATRICE DE LA MACHINE
A
CALCULER UNIVERSELLE DE L'INSTITUT BLAISE PASCAL
L. COUFFIGNAL
Institut Blaise Pascal*
I. CONSIDERATIONS GENERALES
La destination meme du laboratoire de calcul mecanique de l'Institut Blaise Pascal est de
poursuivre des recherches relatives a. des materiels de calcul numerique, et specialement a.
des machinesarithmetiques, et aussi des recherches relatives au mode d'utilisation de ces
materiels, c' est-a.-dire aux· methodes de calcuL
C'est l'une des raisons pour lesquelles les caracteres constructifs· de la machine a. calculer
universelle de l'Institut Blaise Pascal n'ont pas ete arretes a priori et de fac;bn definitive.
Meme, apres sa mise en service, cette machine pourra subir des modi~cations, soit par remplacement de certains organes par des organes nouveaux, soitpar adjonction d'autres organes;
elle sera, par elle-meme, une sorte de laboratoire.
'Cette souplesse, cette aisance de transform'ation, est peut-etre Ie plus caracteristique de ses'
traits; c'en est du moins un trait fondamental.
Son role d'instrument de recherche lui impose d'etre veritablement universelle, c'est-a.-dire
de pouvoir etre equipcc de manierea. executer toute sorte de calculs. Vne telle exigence serait
excessive pour la machine a. calculer d'un laboratoire de recherche consacre a. des travaux
deter~ines, et dont les calculs sont d'un nombre limite de types bien definis; il suffit dans
ce cas d'une machine permettant d'effectncr cescalculs dans les meilleures conditions de
rapidite, d'economie, et aussi dans les meilleures conditions de simplicite de manipulations;
puisque les operateurs d'une telle machine ne sont pas en general des specialistes du calcul
mecanique, et qu'une machine a. calculer est pour eux l'un des nombreux appareils de leur
laboratoire dont ils ont a. apprendre la manipulation. II y a aussi grand avantage a. ce qu'une
tclle machine ne soit que d'un faible enc,ombrement. Nous pensonsque la machine-Iaboratoire
de l'IBP servira a. determiner les caracteristiques de machines plus reduites, destinees a. des
laboratoires particuliers, adaptees Ie mieux possible aux besoins de ces laboratoires, et de
manipulation simple. Cette consideration no us a conduit a. etudier avec un soin particulier
la realisation materielle des elements de la machine, en vue d'une fabrication de type industriel
et de l'echange standard des unites sujettes a. usure ou accident, notamment celles qui comportent des tubes a. vide; c'est la, pensons-nous, un second trait caracteristique de nos recherches et des parties de la machine deja construites; on verra dans quelques instants les
resultats obtenus dans cette voie.
*
Read at the Symposium by Leon Brillouin, Harvard University
374
LA CALCULATRICE I.B.P.
Considerant que l'element essentiel d'une machine a calculer universelle est Ie mecani~me
calculateur, nous avons d'abord fait porter nos efforts sur ceUe partie de la machine.
L'experience acquise dans l'utilisation de machines mecaniques nous y incitait deja, et
nous a guide utilement. Nos recherches en ce domaine en sont au point OU nous pensons
avoir obtenu des resultats a peu pres definitifs, du moins si'l'on se borne a utiliser comme
materiel elementaire celui que peuvent actuellement fournir les fabricants de materiel de radio.
C'est donc ceUe partie de la machine sur laquelle je me propose de donner quelques details.
Nous l'appelons la calculatrice. J'espere que les renseignements relatifs a notre calculatrice
donneront une idee neUe de l'orientation de nos recherches.
II est clair, enfin, que la' plupart des travaux mathematiques qu'une machine a calculer
peut etre appelee a faire ont pour origine des recherches concernant la technique ou les sciences
de la nature. Les travaux de mathematiques pures necessitent rarement des calculs numeriques
importants; l'utilite de ces calculs nesemble pas aussi imperieuse. Cette remarque nous a
conduit a etudier de fa<;on approfondie Ie calcul mecanique de la racine carree, operation qui
intervient frequemment dans les calculs techniques.
L'etude, poursuivie sur ces bases, nous' a confirme dans la preference d'une calculatrice
parallele a l'exclusion d'une calculatrice a sequence; une analyse rap ide de l'execution des
operations fondamentales, chiffrage, addition, soustraction, multiplication, racine carree, dans
, une calculatrice parallele, donnera, avec l'explication logique de la structure de la calculatrice
de la machine de l'IBP, la justification de notre choix.
II. LES OPERATIONS FONDAMENTALES
ChijJrage. Le chijJrage, operation consistant a representer materiellement un nombre, exige,
dans Ie systeme de numeration binaire, un organe par ordre binaire capable de prendre deux
etats distincts, et un'second organe capable de maintenir Ie premier dans l'etat qu'on lui a
fait prendre; nous appelons Ie premier organe un inscripteur elimentaire, Ie second un verrou
et l' ensemble des deux, un chiiJreur elimentaire. Les chiffreurs elementaires des divers ordres
. binaires constituent un chiffreur binaire; leur nombre est la capacite du chiffreur, un chiffreur
de capacite k peut represellter tous les entiers de 0 a 2k - 1.
Addition. L'addition necessite, pour etre. automatique, un reporteur, dispositif effectuant
Ie report des retenues de telle sorte qu'apres inscription successive de deux nombres sur Ie
chiffreur, ce dernier represente la somme des deux nombres. Nous appelons totalisateur
l'ensemble d'un chiffreur et d'un reporteur.
Exemple (Fig. 1): x = a + b, a = 11011, b = 1001, k ~ 6.
Dans cet exemple, on suppose conformement a la plupart des realisations mecaniques,
electromccaniques, ou electroniques, que Ie reporteur est constitue par un chiffreur auxiliaire
qui enregistre les reports a faire pendant l'inscription du secon,d terme de la somme et Ie
transmet ensuite au chiffreur. Le reporteur de la machine IBP qui va etre decrit n'est pas
de ce type.
Soustraction. La soustraction peut se ramener. a l'addition par la methode bien connue des
375
L. COUFFIGNAL
complements. Le complement de h pour la capacite k est 2k - h, et l'on sait que, si l'on inscrit
sur un totalisateur de capacite k les nombres a et 2k - h, Ie totalisateur marque a - h, Ie
chiffre I dans l'ordre k ne pouvant pas etre ·represente par la machine.
La methode que nous utilisons derive de la methode des complements (Fig. 2). Le complement du retrait est remplace .par Ie permute, qui s'obtient en permutant les chiffres Oet I
0=11011
b = 1001
.'
k=6
!5 4 3 2 I 0
0=
Chiffreur I I I I II I
Report
I I I I I I
1011111011111
1010101010.1
a+b =
Ie
Avant '101 I 10 101 I 101
report 10111010111
1010101010101
11101011101
Report
{ 1110101110101
10101010101
FIG. 1. Exemple d'addition.
dans la figuration de ce nombre, et Ie reporteur est complete par un element enregistrant les
reports provenant du chiffreur elementaire de l'ordre Ie plus eleve pour les transmettre au
chiffreur elementaire de l'ordre Ie plus faible; ce report, appele report sans fin est l'application
au systeme binaire d'un procede deja en usage dans certaines machines decimales mecaniques.
L'avantage de cette methode est que Ie calcul mecanique du permute est beaucoup plus
aise que celui du complement; ce dernier s'obtient en permutant les chiffres 0 et I, sauf Ie
dernier I a droite et les 0 qui Ie suivent; il exige donc
une commande conditionnelle que n'exige pas Ie calcul du
permute.
Nous ne donnerons pas la demonstration de l'equivalence
FIG. 2. Schema d'une
des deux methodes, qui est tres facile, mais il est utile de noter
soustraction.
deux particularites.
D'abord, l'operation a - h, OU a et h sont positifs, n'est exacte que si a > h, car rien ne
distingue, sur Ie totalisateur, la difference 0 - h, et la somme 0 +- h', en design ant par h' Ie
permute de h qui se presente comme nombre arithmetique; il faut donc que Ie permute du
retrait, qui joue Ie role d'un ajoutenegatij, soit accompagne du signe -; on voit aisement qu'il
suffit pour cela d'ajouter un chiffreur elementaire a la gauche duchiffreur et de lui attribuer
un reporteur, en convenantque, dans ce chifJreur de signe Ie signe + soit represente comme Ie
chiffre 0 et Ie signe - comme Ie chiffre I. Un tel totalisateur peut etre appele totalisateur
algebrique.
LA CALCULATRICE I.B.P.
Exemple (Fig. 3): x = a - b, a = 100100, b = 1001, k = 6.
Notons au passage que Ie systeme binaire est Ie seul OU soit possible l'assimilation des signes
distinctifs des nombres positifs et des nombres negatifs a des chiffres de la numeration; c'est
la un· avantage du systeme binaire qui a deja ete utilise, mais ne parait pas avoir ete souligne
de fa<;on nette.
La seconde particularite de cette methode tient a la nature de la realisation mecanique
de la soustraction. Si l'on applique Ia methode prece.dente au calcul de a - a, on trouve une
figuration formee de I dans tous les chiffreurs elementaires, y compris Ie chiffreur de signe.
a-b
Chiffreur
de signe
Nombre
=
~
Permute
de b: u
Totalisant
a et 0
1001, k
=
~
.
b = 1001
k =6
Report
~~ I I I , ,./Sans
I=IIIII=-
a
FIG.
Exemple: a
a =100100
fin
[2]11 0 10 111 0 10 1
r;"I
L!JIIIloIIIIlol
3.. Exemple de soustraction.
6.
inscription de a
01001001
figuration de - a = ! 110110
somme
= 1111111
II faut considerer cette figuration comme r~presentant O. Comme Ie chiffreur de signe
porte Ie signe -, nous l'appellerons Ie zero negatif, et par opposition, no us appellerons zero
positij, la figuration 101000000 (k = 6).
Multiplication. Pour Ia multiplication, dans notre premier modele de calculatrice, nous nous
sommes arretes a la methode classique d'additions repetees. Le muitiplicande m est inscrit dans
un chiffreur M, Ie multiplicateur x dans un chiffreur X. Le multiplicande est transfere a un
totalisateur P ou non seIon que Ie premier chiffre de x est I ou 0; puis Ie multiplicande m
subit un deplacement d'un pas vers la droite dans Ie chiffreur M, tandis que Ie multiplicateur
x subit un deplacement d'un pas vers Ia gauche dans Ie chiffreurX; Ia meme suite d'operations
se reproduit jusqu'a epuisement des chiffres de x.
377
L. COUFFIGNAL
Notons en particulier, d'une part que, si la capacite de M est k et celIe de X, k', la capacite
de P doit etre k + k'; et d'autre part que la multiplication comporte une commande conditionnelle, dependant de la natur,e 0 ou I du chiffre figure dans un certain chiffreur elementaire
(Ie premier chiffreur elementaire de X).
Divisionet r{lcine carree. Les methodes operatoires precedentes sont deja connues dans leur
ensemble.
contraire, la methode de ladivision et celIe de l'extraction d'une racine carree,
que nous allons exposer, nous paraissent nouvelles. L'expose precedent eclaire dans une
certaine mesure la theorie de la division et de l'extraction d'une racine carree; en outre, il
contribuera a mettre en relief la condensation des mecanismes que permettent les methodes
que nous decrivons.
On peut developper pour la division et l'extraction, d'une racine carree des theories
analogues, qui me me s'etendraient aisement a des racines d'ordre superieur a 2.
Sbit a diviser a par b; Designons par qn Ie nombre forme par les premiers chiffres du
quotient jusqu'au chiffre d'ordre n et par qn-l Ie chiffre suivant.
Par definition:
bqn2n < a < b(qn + 1)2n,
Au
b(2qn
+ qn_l)2 n- 1 < a <
b(2qn
+ qn-l + 1)2
n
-
1
•
(1)
Posons:
(2)
Des relations (1) et (2) on tire:
< rn, + - b2 n- 1 < qn_lb2n-1,
(qn-l - l)b2n-l
CIt
P1
2
3
g
0
CD
8
LI
FIG.
c:
0
~
g
0-
.2
<
W
';:
7
V
.
'C
c:
E
e
0
0
'0
(.)
Q.
'V
V
13
14
A
4. Element de totalisateur binaire.
Permutation. La permutation est obtenue' par l'attaque de' tous les etages du totalisateur
par la borne 11, qui est reliee a un generateur d'impulsion unique, et par Ie blocage simultane
des -reporteurs par la borne 8, qui est reliee a un autre generateur d'impulsions.
Transfert. Le transfert s'effectue d'un etage a tous les etages du me me ordre hinaire des
, totalisateurs auxquels peut etre transfere Ie nombre marque par Ie totalisateur auquel appartient
l'~tage considere; la borne de sortie 9 est reliee a cet effet a toutes les bornes d'entree 7 des
etages du meme ordre binaire de ces totalisateurs, mais ceux qui ne doivent pas recevoir de
nombre sont bloques en 8, comme pour la permutation. Le transfert est realise par une
impulsion positive envoyee en 14, sur tous les etages simultanement; cette impulsion n'est
pas suffisante pour que la triode Lv atteigne Ie cut-off, mais la grille de cette triode peut recevoir
une polarisation positive statique de la plaque de la triode Fv , qui est en tension haute lorsque
380
LA CALCULATRICE I.B.P.
la triode de chiffrage Fe debite, et ainsi marque Ie chiffre I; l'impulsion 14 peut alors mettre
en debit la triode Lv, qui envoie une impulsion negative d'inscription dans Ie circuit 9.
Deplacement. Vne impulsion negative, envoyee en 10 dans tous les etages simultanement,
ramene a 0 les triodes Fe qui marquent I et n'agit pas sur celles qui marquent 0; cette commande produit done Ie meme effet que la commande de l'addition du nombre a lui-meme.
Par l'action du reporteur, l'addition devient effective, or, l'addition du nombre a lui-me me
est identique a la multiplication de ce nombre par 2, c'est-a-dire a son deplacement d'un
pas vers les positions hautes. Pour Ie deplacer vers les positions basses il suffit de monter Ie
reporteur en sens contraire.
FIG. 5. Ligne de retard (organes interieurs).
Effafage. L'impulsion de deplacement en 10 ramenant tous les chiffreurs a 0, il suffit de
bloquer en meme temps les reporteurs par une impulsion en 8 pour obtenir l' effa~age.
On voit que toutes les operations elementaires ont une duree de moins de 2(), sauf Ie report
dont la duree peut atteindre k().
Realisation matirielle d'un etage binaire IBP. Le schema montre qu'il nous suffit pour constituer un etage binaire de deux doubles triodes et d'une ligne de retard. Materiellement la
ligne de retard est constituee par quelques bobines plates enroulees sur un tube de carton
fort, et les autres pieces- sont montees en un ensemble compact porte par un socle a 14 broches
(Fig. 5). Cet ensemble est coiffe par Ie tube support de la ligne de retard qui lui sert de carter
(Fig. 6). L'etage binaire ainsi constitue a 5 pouces de haut et Ii pouces de diametre. Dans
un souci de standardisation, on a pris pour Le et Lv les triodes d'une double triode identique
a celle qui sert au chiffrage, bien que Le et Lv aient des fonctions independantes et ne soient
pas montees en flip-flop. En outre, pour faciliter Ie remplacement des tubes uses, Ie montage
s'ouvre transversalement vers Ie milieu de sa hauteur.
Realisation materielle d'uf!,e calculatrice. Puisqu'un totalisateur porte en lui-meme des moyens
381
L. COUFFIGNAL
de permutation et de deplacement il suffit de remplacer par des totalisateurs les chiffreurs
M et X consideres dans la theorie des operations algebriques, pour constituer les organes cal. culateurs d'une calculatrice paralleIe.
Les etages bin aires qui constituent ces totalisateurs sont engages dans des douilles placees
cote a cote sur une plaque de fondation commune. Les 'connexions entre etages sont realisees
de fac;on fixe sous cette plaque, qui constitue elle-meme Ie couver-de d'une boite dans lag.uelle
souffle un vent suffisant pour refroidir les tubes a vide, en circulant a l'interieur de chacun
des tubes carter de chacun des etages binaires.
FIG.
6. Ligne de retard (vue d'ensemble).
La hauteur totale de cet ensemble est de 8 pouces environ; sa surface, celle de 6k carres
de 1t pouce de cote, k design ant la capacite du multiplicande et du multiplicateur; par
exemple, pour la machine IBP, qui travaille sur 15 chiffres decimaux, k = 50 et Ia surface
des totalisateurs de Ia calculatrice est de moins de 700 pouces carres.
Les dispositifs de commande et Ies generateurs d'impulsions demandent une cinquantaine
de tubes, quelle que soit Ia ca?acite des totalisateurs. Ces tubes sont du meme type que ceux
des totalisateurs a l'exception de quelques pentodes et thyratrons.
Nous croyons pouv I
13.
I I I I I 0 I 0 I I I __~~.. O I 100
14.
0 0 0 0 0 I 0 I 00 0
L...
'-===*=========> I
15.
0 0
__~-+.. O 0 Q I 0
8.
12.
k' =5
p
I
--
--
01111111111
I.
X
k=6
I I I
I I 0 I 0 I I I
,I I I I I I I I I I I J
~::~l_____----JI
__----------------~T
I
Stop {_:..-------------------------l
0 I I 0 0
16.
11.
M:
Multiplicande
I : Chiffreur
"
:
x:
Multiplicateur
du Nombre I
Transfert
FIG.
P:Produit
-- -
l
L...- -_ _ _
Y
Report
Contrale
7. Exemple de division.
W : Contrale
et deplacement
L. COUFFIGNAL
comme Ie zero du chiffreur de signe de P; et d'autre part, que Ia permutation de M depend
de Ia comparaison des signes successifs de P; on voit apparaitre deux nouvelles commandes
conditionnelles speciaIes, tenant a Ia fois a la structure de la calculatrice et a la methode de
calcul utilisee.
IV. UN PRINCIPE DE RECHERCHE. LA QUESTION DE LA MEMO IRE
Nous voudrions, a cette occasion, rappeler un principe que nous formulions des 1933, et
dont les confirmations se sont multipliees. L'observation de l'evolution des machines existant
a cette date nous conduisait a avancer que Ie progres, en calcul mecanique, resultait d'une
adaptation mutuelle des machines a calculer et des methodes de calcul. Un exemple particulierement typique d'adaptation des methodes aux machines est, dans Ies analyseurs differentiels, Ia determination des fonctions elementaires, sin x, L x, etc., par des analyseurs differentiels
auxiliaires, c'est-a-dire, mathematiquement, Ia substitution a une fonction d'une equation
differentielle dont elle est solution. L'expose qui precede offre de nombreux exemples de
detail, de reaction mutuelle des recherches mathematiques et des recherches techniques; en
particulier, la simplicite des methodes de division et d'extraction de ra'cine carree est fort
accrue par la simplicite de la technique du deplacement et de la permutation.
Les confirmations renouvelees de ce principe nous conduisent a considerer comme inefficace, dans l'etat actuel de la technique, une discussion logique a priori de la realisation
materielle d'une machine a calculer universelle.
Par exemple, Ie debit tres eleve d'une calculatrice telle que celIe dont no us venons de
donner une description schematique, met en question la methode de calcul des fonctions
elemen~aires, et nous conduira vraisemblablement a abandonner les tables mecaniques, que
nous conseillions, en 1938, pour une machine ~lectromecanique, sous une forme voisine de
celle que l'on peut admirer dans la machine Mark I du professeur Aiken.
On comprendra aussi, pensons-nous, pourquoi nous avons declare, en plusieurs circonstances, .que nous ne savons pas encore queUe sera Ia nature de la memoire de notre machine.
FonctioneUement, nous considerons comme necessaires une memoire interne et une
memoire externe, etcomme avantageuse la separation de Ja memoire des nombres et de la
memoire des commandes.
La structure de ces diverse~ memoires doit dependre, a notre avis, des calculs a faire et de
Ia methode adoptee. Par exemple, la memoire n'intervient pas dans les memes conditions si
.l'on calcule des trajectoires ou si l'on resout un systeme de 50 equations Iineaires a 50 inconnues; dans Ie second cas, les phases sont de une ou deux operations, dans Ie premier cas,
elles peuvent atteindre la centaine d'operations.
C'est-a-dire que notre machine-Iaboratoire comportera plusieurs types de memoire dont
Ie mode d'emploi aura a etre etudie systematiquement, en liaison avec les problemes traites.
Nous donnerons pour terminer Ie schema d'urie memoire que les essais poursuivis jusqu'a
present nous conduisenta considerer comme avantageuse dans la fonction de memoire interne
d'une calculatrice parallele (Fig. 8).
'fE
HT
A
NC
HT
On
J
HT
HT
I
I
I
~I
NC
I
JI
I
J
HT
I
~I
NC
I
I
)1
I
J
HT
HT
I
~I
A
NC
I
)1
I
~
FIG.
8. Schema d'une memoire.
J
L. COUFFIGNAL
Chaque chiffre est enregistre au moyen d'une diode a gaz NC. Les diodes constituant les
chiffreurs eIementaires d'une me me chiffreur binaire sont figures sur une meme ligne horizontale; les diodesdu meme ordre binaire dans les divers chiffreurs sont figures sur une meme
ligne verticale; Ie schema montre donc une memoire de 4 nombres de 4 chiffres; il faut
comprendre en outre que les plots representes par la meme lettre sont reunis entre eux.
Le fonctionnement du dispositif se fonde sur la remarque que Ie seuil de tension d'allumage
d'une diode est nettement pl~s eleve que son seuil de tension d'extinction.
L'inscription s'effectue en envoyant une impulsion positive par les bornes A, B, C, . . .
dans tousles ordres bin aires OU doit etre representc lechiffre I, et en bloquant les tubes des
chiffreurs OU l'ins~ription ne doit pas etre faite par une impulsion opposee.
La lecture s'effectue en envoyant, par les bornes E, F, G. . . . une impulsion negative
trop faible et trop 'breve pour provoquer l'extinction des diodes. L'effac;age s'obtient en
prolongeant l'impulsion de lecture.
L'impulsion de lecture peut avoir pour duree 0, duree de basculement d'un flip-flop des
totalisateurs; c;est, croyons~nous,la plus faible duree atteinte pour l'extraction d'un nombre
d'une memoire et son transfert a un chiffreur. C'est cetie caracteristique de fonctionnement
de la memoire a diodes qui en fait l'interet; cette memoire ne retarde en rien la calculatrice,
car l'inscription dans la memoire et l'effac;age peuvent se poursuivre pendant que la calculatrice travaille isolement.
Malgr.-e Ie nO,fibre des diodes, qui peut paraitre eleve, ce dispositif reste simple et sur,
parce qu~ les diodes' a gaz sont des tubes robustes, et que l'on peut les utiliser dans des conditions OU leur fon~tionnemenLne produitguere d'usure. En outre, ces tubes sont peu couteux.
THE FUTURE OF COMPUTING MACHINERY
LOUIS N. RIDENOUR
. University of Illinois
The title of these remarks is somewhat misleading, in that one of the things Professor Aiken
has requested of me is to give a very brief critical summary of the proceedings of the present
Symposium; following this, I venture a few speculations regarding the principal directions
in which the research and development on computing machinery seem to be tending.
The central interests and concerns of the more than 700 people in attendance at the present
symposium are extremely diverse; the fields.in which papers have been presented are various
and wide. There have been papers on computing machinery,. on methods of numerical
analysis, on the solution of problems involving numerical analysis in the fields of physics,
engineering, economics, and social science. No doubt, the fact that interest in this ~ymposium
. has been so splendidly sustained in spite of this diversity of subject matter can be explained by
observing that, once a problem has been reduced to a mathematical form, then what proceeds
from that point onward is of common interest "to those concerned with numerical analysis,
almost without regard to the way in which the original equations to be solved arose.
Thus a prominent effect of the development of computing machines is likely to be that
of producing important unifications and sharings of viewpoint among various scientific disciplines which present problems amenable to attack by numerical analysis. The reports
presented at this Symposium encourage the belief that the art of computing machines may be
entering a new phase-a phase of increased maturity. We are assembled here to celebrate
the completion of the Harvard Mark III machine, and ,many of the papers presented here in
the sessions on computing machines have described completed and operating machines, rather
than the plans for constructing machines not yet built. It is clear that powerful methods of
numerical analysis are being developed, and that the new numerical problems posed by the
extreme speeds of modern machines are becoming evident and are beginning to be attacked.
Many of the numerical problems that have been described here-in physics, engineering, and
social science-are not merely proposed for solution, but actually have been attacked and
solved in whole or in part. The keynote of the present meeting thus seems to be achievement,
even if limited achievement, rather than promise.
Let us now consider some of the papers presented here, in the order in which they appear
in the program. It wi1l not be possible to mention each of the some forty papers presented,
but an effort will be made to deal with typical ones in each category.
No comment on the Harvard Mark III machine is offered beyond saying that we have all
had an opportunity to inspect this machine and to learn something of its design and its
properties. I should like to remark upon the very consider~ble debt that the entire high-speed
387
LOUIS N. RIDENOUR
computer art owes to the early, continued, and effective work of Professor Aiken and
his group.
The Bell Telephone Laboratories computer that was described seemed remarkable principally for its complete avoidance of the use of conventional vacuum tubes. There were used
as computing elements mainly electromechanical relays, together with fewer than one hundred
vacuum tubes. Possibly because of this unconventional design, this machine and its relatives
in 'the series of Bell machines have achieved a very remarkable record of continued reliability.
Very interesting progress reports on machines under construction were offered by the
Massachusetts Institute of Technology, the Raytheon- Manufacturing Company, the General
Electric Company, the National Bureau of Standards, Mr. Elliott for the British, and the
Institut Blaise Pascal. Many of the machines described are scheduled for completion in the
year 1950; that year should be a very interesting one for those concerned with computing
machines.
One aspect of the British developments seems worthy of special remark. This is the quite
evident difference in the approach to the problem of constructing a large computing machine
adopted respectively by British workers a~d by American workers. Before launching upon
the construction of large machines, the British prefer to make preliminary experiments, and
to gain experience, with small machines of admittedly limited scope which, however, posses~
sufficient g~nerality to be educational. American practice has been, on the other hand, to
embark from the beginning on the construction of quite ambitious machines, usually without
preliminary experience on small-scale models. To some degree, this may express the greater
availability of research funds from the American' government, but I think that it goes deeper
than that; I think that it expresses a difference in the national character.
During the recent war, I was frequently distressed by what seemed evidences of stupidity
and ineptitude in our Air Force operations as they concerned my area of interest-airborne
radar. On one occasion I was complaining to a general officer about this, and pointing out
to him how much better the Royal Air Force managed its affairs. He said: "Well, you have
to expect that. There arc two ways to fight a war: _you can fight a smart war, or you can
fight an overwhelming war, but you can't do both .. The British are fighting a smart war, but
we aren't. We made our choice a long time ago; we decided to fightan overwhelming war,
and that's what we're doing. Don't expect us to be smart." It seems that this approach has
been carried over to the computer field; we Americans have a tendency to overwhelm our
difficul ties.
Several papers were presented on the subject of components for computing machines.
I t was clear from these that the outstanding component problem still is-as it has been for
some time-that of an adequate high-speed storage device, or inner memory, for a computing
machine. While special methods for reducing demands on an inner memory can usually be
devised for any particular problem, nevertheless the scope of a machine increases and its
operation becomes simpler as the capacity of the inner memory rises. Quite a lot of wo~k is
being done on this problem. The work of F. C. Williams, and that of the Eckert-Mauchly
388
THE FUTURE OF COMPUTING MACHINERY
group, which appears to be derived from it, seems to be very promising; so is the success that
has recently been obtained in the use of mercury delay lines as high-speed storage elements.
Further, two papers given here reported on novel and interesting devices whose further
development seems very promising. These are the magnetic delay lines and memory elements,
on the one hand, and the highly ~uggestive work on electrochemical storage elements and
relays, on the other.
Mr. Engstrom called to our attention the importance of special-purpose machines. Naturally enough, attention has mainly been focused on what are called, "general-purpose" machines;
but it is desirable to remember that for many purposes, notably those of industry and government, special-purpose machines are quite adequate and can often be realized for fewer dollars
per function performed than could a general-purpose machine. An interesting example of
a special-purpose machine is the Northrop assemblage of IBM equipment to make a simple,
rapidly assembled, useful, and quite reliable machine.
In the session on numerical methods, Mr. Brown proposed a scheme for solving certain
types of problems by playing a game. He has consulted with workers here at the Computation
Laboratory of Harvard, and finds that their conservative judgment is that a 40 X 40 matrix
can be dealt with completely in a thousand steps, and with an error of one part in a thousand,
in a total time somewhat less than one hour. The complete program has not been prepared,
and this is only an estimate, but it seems a promising one.
All those concerned with machine design should be grateful for Mr. Lehmer's elegant
scheme for the generation of pseudorandom numbers by machines. Such numbers, and their
production by a simple scheme, will take on increasing importance as lengthy analysis is
replaced by statistical experiments conducted on machines, in the fashion of the Monte Carlo
method described to us by Mr. Ulam.
Other papers in the session on numerical methods dealt with important problems in
numerical analysis. Mr. Milne catalogued the outstanding needs in iterative schemes for the
solution of the Laplace equation and other elliptical partial differential equations. Further
work is needed, first, on the development of ways of programming for machine use such rapid
systems of error removal in iterative solutions as the relaxation methods of Southwell; second,
on ways for dealing with curved boundaries; third, on schemes for selecting good initial
values of the functions being dealt with; and fourth, on better methods for handling mixed
boundary conditions.
,
In the session on applications to physics, Mr. Furry made the general observation that'
the high-speed computing machine permits experimentation in theoretical physics with less
labor and better results than have ever been/accessible before. Such "theoretical experimentation" (if this is a good term) includes the testing of theories, the decision among competing
hypotheses, the determination of ranges of validity of various approximations, and so on.
Examples of the actual use of machines in the solution of problems were offered. Problems
presented included the birefringence produced by viscous flow, the trajectories of cosmic-ray
particles, and the interaction of atomic electrons with electromagnetic radiation. In connection
389
LOUIS N. RIDENOUR
with the last, Mr. Rose remarked that he has assembled the first compendium of wave functions
for Dirac electrons in the screened nuclear field. It is indubitable that this catalogue of wave
functions will be important and useful in many other investigations. It is to be hoped that
Mr. Rose or others will carryon in the direction he has marked out, computing such wave
functions for higher values of angular momentum, and for electron states in the negative-energy
continuum.
In the sessions on aeronautics and applied mechanics, the papers presented made it clear
that the use of computers in these fields will be very extensive. I mention particularly Mr.,
Welmer's prediction that the complete solution of the now separate problems of flutter, aerodynamic· stability, and ser~omechanism performance of an airframe may soon be found in
terms of the complete frequency-response spectrum for a particular airplane. Mr. Emmons
and Mr. Muskat outlined two other practical applications in which computers will be extremely
useful.
Mr. Mosteller set forth, in the session on economics and social science, the types of problems
likely to be dealt with. He asserted that these are mainly solutions of simultaneous linear
equations, both homogeneous and inhomogeneous, giving as specific examples problems in
multiple regression, the finding of discriminant functions, scaling theory, and factor analysis.
He further remarked that' the present lack of adequate mathematical theory outside the field
of economics now gives machines little to do in social science, while at the same time it points
up very clearly the major present job of those interested in a quantitative social science.
Examples of specific problems suitable for attack by machines were given by Mr. Tucker and
Mr. Chernoff, and Mr. Waugh pointed out that many important economic problems do not
require the use of machines. He urged that those present at the Symposium would assist the
economist in preventing the establishment of a fad for using high-speed computing machines
for all purposes, whether justified or not. This having been said, Mr. 'rVaugh remarked that
machines have a very important place in the solution of veryimpoitant problems, notably
those of Government jn these days of increased central control. 'He reminded us that formulation of such problems is difficult and that economists often do not kno:w what questions to
ask, what answers to seek, or how to secure the public acceptance of policies necessary to
implement the answers found.
On the ground, no doubt, that physiology can be regarded as an elementary sort of social
science, Mr. Crozier found himself on the social-science program. He pointed out first of all
that the multivariant character of organic processes almost certainly means that, when a
proper mathematical description of such processes is formulated, it will be so complicated
that machine computation will be demanded. He then addressed himself to the question of
the validity of using the physiology of high-speed computing machines as an analogy for the
physiology of the nervous systems of living organisms. He reminded us of the dangers of
misleading analogies and concluded, from the example concerning vision which he quoted
and from other evidence, that elementary neural decisions in a living organism are reached
statistically. Thus, according to Crozier, a true thinking machine would have to have a very
39 0
THE FUTURE OF COMPUTING MACHINERY
large redundancy in individual elements whose individual performances fluctuate, in order
to imitate in any meaningful way the performance of the neural system of a living organism.
Let us turn now to the questions more directly suggested by the title: The Future of
Computing Machinery. The first such question, in this time of vigorous development, design,
and construction, is perhaps: "Who is likely to possess large high-speed computing machines
in the future?" Some workers in the computing-machine field, and some people interested
in the field, are quite pessimistic about the ultimate wide availability of large high-speed
computing machines, on the grounds that such machines are complic~ted and expensive to
build, expensive to maintain and operate, and therefore cannot ever be afford~d by institutions
such as the normal middle-sized university. This is a point of view with which I disagree
completely. I strongly believe that a competent high-speed computing machine will very
soon be recognized as an important and inevitable part of the research equipment of any
university having even modest research pretentions.
Thus I regard the computing machine as being not in the category of the large astronomical
telescope, which is a pleasant but optional luxury for a university, but rather in the category
of the electronuclear particle accelerator, which is a necessity for any university that desires
to cultivate modern nuclear physics. In the early and middle nineteen-thirties, when Lawrence
. was having his first successes with the cyclotron, I remember many discussions bf whether
this or that institution should build a cyclotron. There were always those who argued that
the cyclotron was expensive to build and run, that it had a limited ficid of usefulness, that
there were already plans to build all of them that the country needed or could support, and
therefore that the institution concerned in the discussion need not and should not build a
cyclotron. Now it is not quite true that the only universities that have made substantial
contributions to nuclear physics are the ones who ignored such skeptical notions and built
cyclotrons, but it is nearly enough true to be significant. And the successful institutions that
do not have cyclotrons do have, in all cases, some competing form of particle accelerator.
By analogy, I suggest that high-speed computing machines will be part of the routine
and necessary research equipment of univers'ities, industrial laboratories, government research
establishments, and indeed any institution where any substantial volume of scientific research,
in any field, is carried on. Possibly this trend will be readily discernible in a year or two, and'
surely its full implementation is less than a decade off.
Of course, the wide availability of high-speed computing machines will be greatly forwarded
by improvements in reliability and reductions in cost, in consequence of continued developments of improved components and better'logical design.
Let us now ask: "How large, how fast, and how' complicated should a large, high-speed,
general-purpose computing machine be?" The ENIAC still holds the record for the total
number of vacuum tubes. More recent designs are considerably more ambitious in terms of
the speed of individual operations, the size of the inner memory, and the general competence
of the device; yet in spite of this they have fewer tubes, which they use harder, so to speak.
39 1
LOUIS N. RIDENOUR
1 should like to propose that the answer to the question of where to draw the line in designing
a general-purpose machine is set entirely by considerations of reliability. That is, a large
. general-purpose machine ought to be as big and fancy and competent as it can be made,
subject to the limitation that it must not commit errors oftener than once in, say, four hours.
There is no other significant limitation on the total complexity of the device; for the machines
that we have now, even those that have not yet been realized, but are in design, are still
inadequate to deal with many problems we should like to put to them.
Professor Aiken has quoted to me a remark of Hartree's. Hartree said that the fastest
computing machine that has yet been designed is still some 10 10 times too slow to solve completely the problem of the wave equation for the copper atom. Mr. O'Neal, in his remarks
before this Symposium, said that the solution of the traffic-handling problem for aircraft on
the airlines of this country would tax the capacity of the biggest and fastest computing machines
now in existence.
Warren McCulloch, a professor 0f psychiatry at my university, has interested himself in
the sort of analogy between computing machines and neurophysiology that Mr.· Crozier
regards as being so dangerous. He has remarked that the over-all complexity of the largest
and most complicated computing machine now in existence or proposed is just about equivalent
to the complexity of the nervous system of the flatworm. You mayor may not regard this
as being a fair comparison. It is based upon drawing a parallel between a single flip-flop in
the machine and a single neuron in the nervous system of the flatworm, and I think that it is
safe and suitable for our present purpose.
There is little question that, so far as the carrying out of n~merical computations is concerned, the computing machine is more useful than the flatworm. There are two obvious
major reasons for this. First, the machine is specialized in its function, while the nervous
system of the worm is not. The machine can deal only with special classes of situations, while
the delimitation of the flatworm's competence is far less narrow. Second, the machine works
about a thousand times faster than any organic nervous system.
Without claim~ng in any way that a computing, machine "thinks" in the sense of origination,
we must admit that it relieves human computers of a tremendous burden of routine mental
effort which is ordinarily classified as thinking. This thinking is special, in the sense that it
is governed by formal logical rules of manipulation, but in the past it has had to be managed
by human nervous systems. With the help of machines, it can be directed by human nervous
systems, but carried out without human intervention or assistance.
Thus, we are not talking about machines possessed of the ability to "think" in the sense
to which Mr. Crozier was objecting, but rather machines which can perform logical processes
in a rapid, uniform, and unerring way. The faster and more competent we can make such a
machine, the bigger will be the burden of routine thought that it can take away from men.
If we can make a machine large enough and competent enough, and if in the meanwhile
.we have learned more than we know now about the logical organization of the nervous systems
of living organisms, we may at last be able to make a machine capable of origination and
39 2
THE FUTURE OF COMPUTING MACHINERY
problem-solving behavior. But this lies in the rather distant future; our present problem is
to make large 'and reliable the machines of unitary function which are designed simply for
the straightforward application of the logical rules built into their design.
We want, therefore, to make computing machines as large and as complicated as we can;
for the fanciest machine that we can realize today is powerless in the face, of problems that
we can readily pose, but not yet solve. The limitation on size and complexity is set by reliability; for a machine will be useless to us if it is not sufficiently reliable to be depended upon
for hours at a time.
This leads us to my final question: "How can a computing machine be made more reliable,
so that its complexity can be increased without increasing the chance of failure?" Of course,
there is no simple or evident answer to this question, or such an answer already would have
been exploited in machine design. There are some promising indications on the horizon.
I suggest that the first thing that should be done is to look toward as complete as possible an
elimination from computing machines of vacuum tubes and electromechanical relays. These
two components are presently the major sources of failure in existing machines, partly because
they are so numerous, and partly because they wear out with continu~d use. 'Vhat we need
is computing elements that can perform the same nonlinear functions as those we now achieve
with tubes or relays, but elements that are far less prone to depreciation in'use.
Another drawback of the vacuum tube, of course, is the ridiculously large amount of energy
that must be expended to boil offfree electrons from its cathode. At the time McCulloch made
his remark about the flatworm, he also observed that if a computer built on present principles
should be made to have the same number of individual elements-let us call them "neurons"
-as there are in the human central nervous system, then all the power of Niagara Falls would
be required to light the tubes, and the complete water flow over the Falls would be required
to keep the device cool. The human nervous system, though slow in electronic terms, is
incomparably efficient in terms of energy expenditure per individual computing element.
What is needed is to replace the present basic nonlinear elements used for computers with
another type of element that does not require enormous quantities 'Of stand-by power, and is
not depreciated by continued operation-an element that, once installed, can be relied on
indefinitely unless it is abused. There are some hints as to the possible nature of such a device;
some of these have been reported upon at this Symposium. The most promising ones visible
today are, first, semiconductor devices of the sort of the recently announced transistor; second,
magnetic devices like those reported here; and third, the electrochemical devices that may
be developed from the pioneer work of which Mr. Bowman has told us. A great deal of intensive
work on promising unconventional dements for computer use will be repaid if the over-all
reliability of computers can thereby be increased. Reliability, as we have seen, brings in its
train larger, more complex, and more competent computing machines. Presumably it also
brings in its train a greater availability and a lower cost for the computers of present size and
scope; inevitably, it will bring wider general use and acceptance of computing machines of
all sorts.
393
Source Exif Data:
File Type : PDF File Type Extension : pdf MIME Type : application/pdf PDF Version : 1.3 Linearized : No XMP Toolkit : Adobe XMP Core 4.2.1-c043 52.372728, 2009/01/18-15:56:37 Producer : Adobe Acrobat 9.3 Paper Capture Plug-in Modify Date : 2010:02:26 18:45:48-08:00 Create Date : 2010:02:26 18:45:48-08:00 Metadata Date : 2010:02:26 18:45:48-08:00 Format : application/pdf Document ID : uuid:46936fca-de69-489f-bcd4-2dd3cd82ba51 Instance ID : uuid:35db5b16-0d26-4bd5-b35d-24691d260345 Page Layout : SinglePage Page Mode : UseOutlines Page Count : 431EXIF Metadata provided by EXIF.tools