NATIONAL BUREAU OF STANDARDS REPORT

1661

NATIONAL BUREAU

OF STANDARDS

, FILE COP

Projects and Publications

of the

NATIONAL APPLIED MATHEMATICS LABORATORIES

A QUARTERLY REPORT

January through March 1952

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
U. S. DEPARTMENT OF COMMERCE

Ghsarles Sawyer, Secretary

NATIONAL BUREAU OF STANDARDS

A. V. As tin. Acting Director

THE NATIONAL BUREAU OF STANDARDS

Hi® scope of activities off the National Bureau of Standard® ie suggested in the following listing of the divisions and sections engaged in technical svork. In general, each section is en- gaged in specialised research, development, end engineering in the field indicated by ita

title. A brief description of the activities, and of the resultant reports and publications,

appears on the inside of the back cover of this report.

I. ELECTRICITY. R«sis tance Measurements. Inductance and Capacitance. Electrical Instruments. Magnetic Measurements. Electrochemistry.

2° OPTICS AHD METROLOGY. Photometry and Colorimetry. Critical Instruments. Photographic

Technology. Length. Gage.

3. HEAT AW POWER. Teoperoture Measurements. Thermodynamics. Cryogenics. Engines and

Lasbricatio® . Engine Fuels. ATOMIC AND RADIATION PHYSICS. Spectroscopy. Kediomsetry . Maas Spectrometry. Physical Electronics. Electron Physics. Atomic Physics. Neutron Meaaureosemts. Nuclear Physics. Radioactivity. X-Rays. Betatron. Nucleonic Instrumentation . Radiological Equipment. Atomic Energy Commission Instruments Branch.

5. CHEMISTRY. Organic Coatiag3- Surface Chemistry. Organic Chemistry. Analytical Chemistry. Inorganic Chemistry. Electrodeposition. Gas Chemistry. Physical Chemistry. TherBBOchemi stry. Spectrochemistry. Pure Substances.

6. MECHANICS. Sound. Mechanical Instruments. Aerodynamics. Engineering Mechanics. Hydraulics. Mass. 'Capacity, Density, ond Fluid Meters. 7. ORGANIC AW FIBROUS MATERIALS. Rubber. Textiles. Paper. Leather. Testing and

Specifications. Organic Plastics. Dental Research.

8* METALLURGY. Thermal Metallurgy. Chemical Metallurgy. Stechanical Metal lurgy . Corrosion.

9.

NUMERAL PRODUCTS. Porcelain and P.ottcry. Glass. Refractories. Enameled Metals. Building Stone. Concreting Materials. Constitution and Microatructure. Chemistry of Mineral

Prsxhsets. 10. BUILDING TECHNOLOGY. Structural Engineering. Fire Protection. Heating and Air Con- ditioning. Exterior and Interior Coverings. Codes and Specifications.

11. APPLIED MATHEMATICS. Numerical Analysis. Confutation . Statistical Engineering. Ma- chine Bevelopaent.

12. ELECTRONICS. Engineering Electronics. Electron Thbes. Electronic Computers. Elecfcramie IsastrwaeBtatiea

.

13. ORDNANCE DEVELOPMENT. Mechanical Research and Development. Electromechanical

Rase®. Technical Service®. Missile Fusing Research. Missile Fusing Development. Pro- jectile Fosses. Ordnance Component*. Ordnance Teats. Ordnance Research.

Ho RADIO PROPAGATION. Upper Atmosphere Research. Ionospheric Research. Regular Propagation Services. Frequency Utilisation Research. Tropospheric Propagation Research. High

Frequency Standards. Microwave Standards.

15. FISSILE DEVELOPMENT. Missile Engineering. Missile Dynamics. Missile Intelligence.

Missile Instrumentation. Technical Services. Combustion.
NBS PROJECT NBS REPORT

Projects and Publications

of the

NATIONAL APPLIED MATHEMATICS LABORATORIES

January through March J9S2

Approved for public release by the

This report is issued j Director of the National Institute of >n, reprinting, or reproduction

?

any form, either in i Standards and Technology (NIST) 'mission in writing is obtained

fro. th. Office cf the I on g

^ 25. D. C.
NATIONAL APPLIED MATHEMATICS LABORATORIES

January 1 through March 31, 1952 ADMINISTRATIVE OFFICE

John H. Curtiss, Ph.D., Chief Edward W. Cannon, Ph.D., Assistant Chief Olga Taussky-Todd, Ph.D. , Mathematics Consultant

W. J. Youden, Ph.D., Mathematics Consultant Ehgene Lukacs, Ph.D., Mathematics Consultant Myrtle R. Kelllngton, M.A. , Technical Aid Luis 0. Rodriguez, M.A. , Chief Clerk John B. Tallerico, B.C.S., Assistant Chief Clerk Margaret W. Dawson, M.A. , Secretary

Elsie L. Busman, Secretary (Calif.) Jacqueline B. Jones, Secretary Violet C. Jordan, Secretary Esther L. Turner, Secretary INSTITUTE FOR NUMERICAL ANALYSIS COMPUTATION LABORATORY

Los Angeles, California Derrick H. Lehmer, Ph.D . Director of Research

Magnus R. riestenes, Ph.D. . . Assistant Director and UCLA Liaison Officer

Harry D. Huskey, Ph.D Assistant Director for Mathematical Services

Gertrude Blanch, Ph.D. . . . Assistant to the Director (Numerical Analysis) Albert S. Cahn, Jr., M.S Assistant to the Director (Adninistration) Research Staff

Forman S, Acton, D.Sc

Charles E. Africa, Jr., B. A

Leonard M. Blunenthal , Ph.D

George E. Forsythe, Ph.D

Jerry W. Gaddum, Ph.D

Cornelius Lanczos, Ph.D

Richard H. Lawson, M.A

Theodore S. Motzkln, Ph.D

Victor A. Oswald, Jr., Ph.D

Lowell J. Paige, Ph.D

David S. Saxon, Ph.D

Isaac J. Schoenberg, Ph.D

Eduard L. Stiefel, Fh.D

Marvin L. Stein, Ph.D

Wolfgang R. Wasow, Fh.D. Graduate Fellows

Stephen G. Gasiorowicz, M.S. (UCLA) Robert M. nayes, M.A. Mathematical Servi

Mathematician UCLA Research Associate Mathematician

Mathematician

Mathenatician

Mathematician UCLA Research Associate UCLA Research Associate

. Consultant (Linguist) Mathematician Physicist UCLA Research Associate UCLA Research Associate Mathematician Mathenatician

Urs D. Hochstrasser, M.A. (UCLA) ces Unit Arnold D. Bestenes, Ph.D Mathematician

Frederick H. Hollander, M.A Mathematician

Marvin Howard, M.S Mathematician

Roselyn S. Lipkis, B.A • Mathenatician

Robert R. Reynolds, M.S Mathenatician

Thomas II. Southard, Fh.D Mathematician

Everett C. Yowell, Fh.D Mathematician

Computing Staff

Thomas D. Lakin, B.S. Norma Logel, B.S. Shirley L. Marks, M.A. Owen R. Mock, B.A. Elmer E. Osborne, M.A. Nan N. Reynolds, M.S. LaVeme E. Rickard, M.A. Albert H. Rosenthal, B.A. Fhilip II. Sayre (UCLA) Jerome A. Schneck, B.A.

M. Winifred Smith Louise M. Straus, M.A. Lindley S. Wilson, M.A.

Warrei A. Baily (UCLA) Leonus T. Batiste, B.S. (UCLA) Patricia B. Bremer, B.A. Patricia L. Childress (UCLA) Dennison A. Curtiss, B.A. (UCLA) Leola Cutler, B.A. Edward D. Fisher, B.S. Alexandra I. Forsythe, M.A. Lillian Forthal, B.A. Gladys P. Franklin, B.S. James W. Gardner (UCLA) Ehma M. Henderson, M.A.

ituth B. Horgan, B.A. Merry M. Kruse, B.S.Machine Engineering and Development Unit Biagio F. Arabrosio, B.S.. Electronic Ehgineer Noman F. Loretz, B.S Electropic. Ehgineer

Ragnar Thorensen, Fh.D. . Electronic Scientist James W. Walsh Electronic Technician

Electronic Laboratory Staff Blanche C. Eidem Edward D. Martinolich John P. E'rancis, B.S. Kenneth J. Millikin (UCLA) Allen E. Garfein (UCLA) Lyle E. Mitchell

Gaylord W. Jones Wallace W. Moore

Louis E. Justice, B.S. John L. Newberger, B.A. Howard K. Marks Eugene M. Rector, B.S. Administrative Unit James D. Allen Clerk (UCLA) Jean 3. Booth , Administrative Clerk Eva Cassirer, B.A. . Translator (UCLA) Albert H. Feldnan Miscellaneous Office Machine Operator (UCLA) Mark H. Ilennes Miscellaneous Office Machine Operator Velma R. Huskey, B.A Information and Editorial Clerk

Earle W. Kimball Fhoperty and Stock Control Clerk Dora K. Madoff Administrative Assistant

M. Evelyn Michaud Administrative Officer

F. Patricia K. Peed Librarian

Gertrude Z. Reider Adninistrative Clerk

Mildred Vu'-nanic Administrative Assistant

James R. Walker, M.A Miscellaneous Office Machine Operator

Madeline V. Youll, B.A Library Assistant

Secretarial Staff Harriett W. Cotton Vendla H. Gordanier Mildred L. Dalton Dorothy M. Hibbard

Betty E. Dean Lucy A. Moore

Mary G. Eskridge Reve J. Vineyard

Jolm Todd, B.S. . . Franz L. Alt, Fh.D

Milton Abraraowitz, Fh.D. . . Ruth K. Anderson, M.A. . . . Frances B. Bauer, Fh.D. . . . William H. Durfee, Fh.D. . , Andre N. Gleyzal, Fh.D. . . . Peter Henrici

Alan J. Ilofflnan, Ph.D. . . . John V. Holberton, M.A. . . . Donald 0. Larson, B.S. . . , Joseph H. Levin, Ph.D . . . . Murray Marinos, Fh.D. . . . , Elliott W. Montroll, Fh.D. . Leon Nemerever, M.S. . . . . Morris Newnan, M.A

Fritz Oberhettinger, Ph.D.

A. M. Ostrowski, Fh.D. . . , Fhilip Rabinowitz, Fh.D. . , Herbert E. Salzer, M.A. . . . Daniel Sokolowsky, M.A. . . Irene A. Stegun, M.A. ...

Charles J. Swift, Fh.D. . . Joseph H. Wegstein, M.S. Norman A. Wiegnann, Fh.D. . . Computing Staff

Hans 0. Braner, B.A. Holla L. Brooks, B.S. Robert B. Bryce, B.A. William F. Cahill, M.S. Ruth E. Capuano

Kathryn Christoph, B.S. Anne R. Cock, B.A. Benjamin F. Cron, B.S. Ira C. Diehm, B.S. Lionell 0. Difford, M.A. Seymour Doranitch Mary M. Duilap, B.S. FYances R. Froberger, B.A. Anne M. Fhtterraan, B.A. Leon Gainen, B.S. Billie R. Gill, M.S. Elizabeth F. Godefroy Allen A. Goldstein, M.A. Lester H. Gordon, Wanda P. Gordon, B.A. Stanley D. Grant, Jr. William G. Ilall, B.S. Helen V. nanmar, B.S. Benjamin H. Handy, Jr., M.S. Genevie Hawkins, B.S. John G. Hershberger, M.S. Frederica R. 'M. Holbert Viola D. Hovsepian, B.S. Renee B. Jasper, B.A. Dorothea H. Jirauch, M.A. Lambert S. Joel, B.A.

Guest Workers Ehzo L. Aparo

M. R. Amette

Dickson H. Call Secretarial Staff

M. Charlotte Case Lillian Sloane ^American University

Chief Assistant Chief ...... Mathenatician Mathematician

, . . AU* Research Associate Mathematician

Mathematician

. . . AU* Research Associate Mathematician

Mathematician

Mathenatician Mathenatician

Mathenatician

Consultant Mathenatician Mathenatician

, . . ALT* Research Associate

. . AU* Research Associate

i Mathematician

. Mathenatician Mathenatician

Mathematician

Mathenatician

F’hysicist Mathenatician

Marlee E. Johnson, B.S. John A. Jordan, B.A. Terry W. Ledley, M.A. Norman Levine, B.S. David S. Liepman

Samuel Mallos, M.A. Ethel C» Marden, B.A. Frank B. Meek, M.A. Theodore R. Mikell, M.S.

Kerrait C. Nelson

Peter J. O'Hara, B.S. Mary Orr Donald S. Park, B.A. Maxine L. Paulsen, B.S. Annabelle W. Fhelps, M.A. Solomon L. Pollack, B.S.

B. Stanley Prusch

Donald I. Rubin, M.S. Ruth C. Shepard, B.S. Wallace R. Soderquist, M.S. Milton Stein Otto T. Steiner, B.A. Frank J. Stockmal, B.A. Gretchen Stolle, M.A. Ellen V. Stubblefield

Sally Tsingou, B.S. George E. Turner, B.S. Bernard Urban, B.S. Edna M. Vi onop, B.A. Mary L. Wadden, B.A. Bertha n. Walter Ruth 7ucker, B.A.

Dino Dalnelli Jerold Mathews

Michael Til son

Ethel Ulitsky Marjorie E. Young

MACHINE DEVELOPMENT LABORATORY

Edward W. Cannon, Fh.D Chief

Karl Goldbarg, M.A Mathematician

Edith T. Norris, B.A Mathematician

Ida Rhodes, M.A Mathematician

STATISTICAL ENGINEERING LABORATORY

Churchill Eisenhart, Ph.D Chief

Joseph M. Cameron, M.S .Assistant Chief

Lola S. Deming, M.A Technical Aid

William S. Connor, Jr. , Fh.D. . Mathenatician

Edgar P. King, F’h.D Mathematician

Julius Lieblein, M.A Mathenatician

Mary G. Natrella, B.A Mathematician

Frank Proschan, M.A Mathematician

I. Richard Savage, M.S Mathematician

Dan Teichroew, M.A UNC* Research Associate

Marlon T. Carson Computer Yvette Bemardon Secretary Shirley C. Schleter, B.A Secretary' Virginia E. Sweeny, B.A Secretary ^University of North Carolina Institute of Statistics
Contents

Pag©

Index. . . . . v Status of Projects as ©f March 31, 1952. ... 1 Institute for Numerical Analysis (NBS Section 11.1)............... i Computation Laboratory (NBS Section 11.2) . 35

Statistical Engineering Laboratory (NBS. Section 11.3). ......... 62

Machine Development Laboratory (NBS Section HA). ............... . 70

Lectures and Symposia ............ yk

Publication Activities. ........... 77

Appendix. . 86
Index of Active Research and Development Projects

NOTE: This index is not intended to cover the numerous special problem

solutions, statistical analyses, and other ad hoc services' to Government

agencies which form an important part of the work of the National Applied

Mathematics Laboratories. These services are, however, fully represented

in the body of the report.

A. Research: Pure Mathematics

Miscellaneous studies in pure mathematics.

B. Research: Numerical Analysis

12 , 36

Classical numerical analysis, Research in. ........... 35

Conformal mapping, Numerical methods in. .... 2

Differential equations, Studies in numerical integration of. . . 3

Diriehlet problem for certain multiply connected domains,

Investigation of Bergman's method for the solution of the. . . 35

Eigenvalues, eigenvectors, and eigenfunctions of linear operators, Calculation of 2

Monte Carlo method, Solution of Laplace equation by 36

Probability methods and sampling techniques. ..... 5

Riemann-zeta-f unc tion , Computation of the imaginary zeros of the l4

Solution of sets of simultaneous algebraic equations and techniques for the inversion and iteration of matrices. ..... 1 Variational methods. ...... .......... 9

C. Research: Applied Mathematics, Physics, Astronomy, and Automatic

Translation

Applied mathematics, Studies in. ................

Compressible flow - method of orthogonal and kernel functions. . Crystal structure, Analysis of ......

Cubic lattices, Distribution of normal modes of vibration of . . Finite matrix theory, Special problems in

Geomagnetic Field, Analysis of.

Heat convection in laminar flow through a tube . Language translation study

Linear programming, Research in . Navier-Stokes equation. . . ...... . Periods and amplitudes of the light variations of the stars

£ Scuti and 12 Lacertae, The determination of the

Program planning, Research in the mathematical theory of. . . . R&dial Mathieu functions

Roots of polynomial equations

Russian mathematical progress, Study of

Scattering functions

Sievert's integral. .....

Spheroidal wave functions . . Struve function of order 3/2

Study of trajectories, Application of the theory of stochastic

processes to the

Theoretical physics, Miscellaneous studies in . . . Three-body problem . Van der Pol. equation

D. Mathematical Statistics

10

39

36

38

37

37

37

16

39

40

27

15

45

38

1

1

47

46

45

47

68

10

37

45

Applications of mathematical statistics to problems of the

Chemical Corps, Research in ...... 68

New projects
Index of Active Research and Development Projects

Committee on Ship Steel, NRC, Statistical services for. . . Extreme-value- theory and applications

Location and scale parameters, Estimation of

Miscellaneous studies in probability and statistics. . . . NBS Administrative Operations, Statistical aspects of. . . NBS Research and testing, Collaboration on statistical as- pects of

Non-pa rametric statistics, Procedures of

Propagation of error, Law of ... . Research ' on application of theory of extreme values to Gust

Load problems

Stochastic processes, Elementary theory of

Theoretical and applied statistics, Investigation of the

applicability of automatic digital electronic computing to

problems in. ......

69

65

63

65

67

66

66

66

67

63

37

E. Mathematical Tables

Antilogarithms, Table of

Arcsin for Complex Arguments, Table of

Bessel-Clif f ord functions, Table of

Bessel functions ¥ (x) , Table of the

Bivariate normal distribution function, Tables of the ....

Chebyshev polynomials, Table of

Collected short mathematical tables of the Computation Laboratory.

Coulomb wave functions, Tables of

E. (z), (z = x + iy) , Tables of

Error function for complex arguments, Table of

Exponential function for negative arguments, Extension of

tables of the . Fermi function II

Gamma function for complex arguments, Table of the

Gases, Table of thermodynamic properties of

Hyperbolic sines and cosines, Extension of the table of. . . . I

n (x,Cy), Tables of (see task 1102-53-1106/49-13) . . Integrals involving higher transcendental functions, Tables of

Lagrangian coefficients for sexagesimal interpolation, Table

of

Mathematical tables, Revision of .... . Mathieu functions II .....

Modified Airy integral, Table of the

Power points of analysis-of-variance tests, Tables of

Probability tables for extreme values

Punched card library.

Random samples, Table to facilitate drawing

Rocket and comet orbits, Tables for ....

Wave function for helium

Wave function for lithium

42

44

18

4i

22

43

4142

4144

44

43

42

48

40

42

4118

44

48

43

19

65

18

45

43

F. Manuals, Bibliographies, Indices, and Technical Information

A.d.c.m., Logical notation and block diagram symbolism for . 17

Fitting straight lines, Manual on 65

Mathematical tables and numerical analysis, Bibliography of . 40

Statistical engineering terminology, Glossary of 62

Statistical literature, Bibliography and guide to 63
Index of Active Research and Development Projects

G. Computing Machine Development

Air Comptroller's computing machine 71 Army Map Service computing machine 72

Bureau of the Census computing machine 70

Navy computing machine 70

Raytheon computer, Coding related to the 72

SEAC : National Bureau of Standards Eastern Automatic Computer . . 71 SEAC: Number- theoretical test problems for 36

Social Security Agency, Investigation of the applicability of

automatic digital electronic computing to problems of the ... 73

SWAC: National Bureau of Standards Western Automatic Computer . . 17

Wright Development Center computing machine . 72
Status of Projects

March 31, 1952

I. Institute for Numerical Analysis

(Section 1L.1)

1 . Fundamental Research

SOLUTION OF SETS OF SIMULTANEOUS ALGEBRAIC EQUATIONS AND

TECHNIQUES FOR THE INVERSION AND ITERATION OF MATRICES

Task 1 101 -1 1 -5100/49-AE2

(formerly 1 1 . 1 /l -49-AE2)

Origin: NBS

Sponsor: Office of Naval Research, USN

Full task description appears in July-Sept 1 9^+9 issue.

Status: CONTINUED. The finite iterative procedure described in

publication (6) has been tried on the SWAC and IBM equipment. These

experiments have been carried out with good success by R. M. Hayes,

U. Hcchs trasser, and L. Wilson for certain small test matrices.

Publication (8) presents a summary of card-programmed calculator

experiments with the accelerated gradient method performed last summer. The results are compared with those obtained with the "almo st-optimum"

methods reported in publication (4).

In publication (7) D. H. Lehmer presents his investigations

of certain character matrices. Let p be an odd prime, oc an integer,

and X (a) Legendre's symbol for the quadratic character of a(mod p).

Simple explicit expressions are derived for the inverse, the characteristic

equation, and any positive power of matrices M^of the form

Moc = (

a ij| = {XC- i + J)} •

These results facilitate the construction of matrices with

elements that can be used for experimentation in numerical

See also the investigation of C. Lanczos on the evaluation

root of certain characteristic equations, publication (8).

Publications: (1) "The extent of n random unit vectors," by

G. E. Forsythe and J. W. Tukey; IN MANUSCRIPT. (2) "A method of computing

exact inverses of matrices with integer coefficients," by J. B. Rosser;

accepted by the NBS Journal of Research. (3) "An extension of Gauss'

transformation for improving the condition of systems of linear equations,"

by G. E. Forsythe and T. S. Motzkin; accepted by Mathematical Tables and

Other Aids to Computation. (4) "Gradient methods in the solution of systems of linear equations," by M„ L. Stein; accepted by the NBS Journal

of Research. (5) "Tentative classification of methods and bibliography

on solving systems of linear equations," by G. E. Forsythe; to appear in

Simultaneous Equations and the Determination of Eigenvalues - Proceedings

of an NBS Symposium held in Los Angeles, Aug. 1951. (8) "Method of consimple rational

analysis

of the smallest
2 Status of Projects

jugate gradients for solving linear systems," by E. Stiefel and M. R. Hestenes; IN MANUSCRIPT. (7) "On certain character matrices," by D. H. Lehraer

submitted to a technical journal. (8) "IBM experiments with accelerated

gradient methods for linear equations," by A. I. Forsythe and G. E. Forsythe;

IN MANUSCRIPT. (9) "Solution of systems of linear equations by minimized

iterations," by C. Lanczos; accepted by the NBS Journal of Research.

NUMERICAL METHODS IN CONFORMAL MAPPING

Task 1101-11-5100A9-CM1

(formerly 1 1 .l/l -5-9-CM1 ) Origin: NBS

Sponsor: Office of Naval Research, USN

Full task description appears in Apr-Jun 195-9 issue.

Status: CONTINUED. The preparation of the report continues. Some

further numerical work has been carried out in the case of an ellipse with

axis ratio 2:1. The results obtained by the Warschawski method are being

checked by differencing and comparison with results obtained on SEAC using

the explicit formulae for the mapping given by G. Szego .E'Conf o rmal mapping

of the interior of an ellipse onto a circle," by G. Szego. Amer. Math. Mo.

iZ, ^7^78 (Aug-Sept 1950)].

CALCULATION OF EIGENVALUES, EIGENVECTORS, AND EIGENFUNCTIONS

OF LINEAR OPERATORS

Task 1101-11-5100/50-3

(formerly 11 . 1 /l —50—3)

Origin: NBS

Sponsor: Office of Naval Research, USN

Full task description appears in July-Sept 195-9 issue.

Status: CONTINUED. A new publication relevant to this task is

publication (2) listed under task 1101 —1 1 —5l 00/52—1 . It appeared in 195-3 in an inaccessible Russian publication and gives background information

about the possible invariant subspaces of the linear transformation defined by a matrix. The paper then goes on to describe specific numerical

methods of extending classical iteration methods to obtain these invariant

subspaces in a general case. It is a useful reference on such practical

problems as separating close eigenvalues, recognizing a defective matrix,

and so on. For other work on eigenvalues, see publication (7) under task

iioi-ii-5ioo/5i-2, p. 5

. In vibration and flutter problems we are frequently interested

in the absolutely smallest root of the characteristic equation. Since the

usual iteration processes boost up the large eigenvalues at the cost of the

small ones, a preliminary inversion of the matrix is required. The present

method dispenses with the preliminary inversion of the matrix. The problem

Ax - Xx = 0 is changed into A*Ax^- Aa*x = 0, and the smallest principal

axis x„ of the Hermitian problem A*Ax = § x ( £ ~ positive and the

o

r O 0 0^0 r smallest of all possible \ ) is considered as the zeroth approximation of x. In first approximation we obtain

x = (1 - XXo

) X

o + Ay
Status of Projects 3

with the quadratic condition

i -r 0 ^ + (

0

2 ~ ri } x2 = 0

where

Ay = x.

„ , x A v. V- _ 1 o o

t — zr --=3— o t x X s o o

_ 1 y A x. Vi — %T 5=r" ' 1 fc^ X X A <-

The solution of the quadratic equation for A approximates the (generally

complex) smallest eigenvalue of the generally nonsymmetric and nonreal

matrix A. This method, if applied to the characteristic equation asso- ciated with a given polynomial, gives an effective method for locating

the absolutely smallest root of an algebraic equation, (or the absolutely

largest root of the inverted equation) without generating the power sums required in Bernoulli's method. (A talk on this subject is to be presented

at the Fresno meeting of the American Mathematical Society, May 3} 1952).

Publications: (1) "Alternative derivations of Fox's escalator

formulas for latent roots," by G. E. Forsythe; submitted to a technical

journal. (2) "Convergence of a method of solving linear problems,"

by W. Karush; submitted to a technical journal. (3) "Sufficient conditions for the convergence of Newton's method in complex Banach spaces,"

by M. L. Stein; submitted to a technical journal. (4) "The determination

of latent roots and invariant manifolds of matrices by means of iterations," by K. A. Semendiaev, translated by C. D. Benster, edited by

G. E. Forsythe, NBS Report No. 1402.

STUDIES IN THE NUMERICAL INTEGRATION OF DIFFERENTIAL EQUATIONS

Task 1101-11-5100/51-1

Origin: NBS

Sponsor: Office of Naval Research, USN

Full task description appears in July-Sept 1950 issue.

Status: CONTINUED. J. W. Green prepared a paper [see publication

( 8 ) below] which describes and studies a new method for the solution of

linear parabolic differential equations. For the equation

L[u] = uxx

- u - g(x,t) u - f(x,t)

subject to the initial conditions u(x,o) = u(o,t) = u(7r,t) = 0, the

method consists in approximating the solution by the sum

n

u

n (x ’ t} = §1 C n,k (t) sin kt >

the C . (t) being determined from the conditions

U ^ K,
b Status of Projects

L[ u 3 — f f sin jx dx = 0, j = 1 , 2, . . .

, n.

W. Wasow studied the asymptotic behavior, for large X, of the

differential equation

&

j=i

it ,(x) + X

2 2 b. (x)u (2”k) =

k=l

with b (0) - 0, b^(0) ^ 0. The variable x is complex, and the coefficients

are assumed to be analytic. This problem occurs in the theory of the

stability of laminar viscous flows. It can be solved completely - as far

as the asymptotically leading terms are concerned - by expressing the

solutions asymp to tically in terms of solutions of the special differential

equation

y (4) + X

2 (xy " + y) = 0 .

The asymptotic properties of the latter can be derived by means of contour

integration [see publication (7) below].

E. Stiefel [see publication (9) below] shows that the numerical

work involved in solving a boundary value or eigenvalue problem by finite

difference methods in a domain with many symmetries often can be reduced

by applying the theory of group characters to the group of symmetries of

the domain. He also considers the problem of solving Au = 0 in a cube

when the prescribed boundary values are invariant under the group of rotations of the cube. In the series representation of the solution in terms

of harmonic polynomials only a subset of these polynomials actually occurs,

and the theory of group characters facilitates considerably the determination of this subset.

G. Blanch has been studying a class of second order differential

operators with constant coefficients. The representation of the solution

by an integral equation is well known; but it does not seem to have been

observed before that the numerical evaluation of the solution can be so arranged as not to involve directly the upper limit of integration. The

troublesome problem of iterating approximate solutions is therefore

avoided. Moreover, for certain parameters the solution by means of the

integral equation has distinct advantages over the corresponding method

of stepwise integration, from the viewpoint of the size of the permissible

integration interval, for a given upper bound of error, A paper on the

subject is in process of preparation.

Publications: (1) "On integration of parabolic equations by difference methods. I: Linear and quasi-iinear equations for the infinite

interval," by F. John; accepted by Communications on Pure and Applied

Mathematics. (2) "On the numerical solution of parabolic partial differential equations," by G. Blanch; accepted by MBS Journal of Research.

(3) "On mildly nonlinear partial difference equations of elliptic type,"

by L Bers; submitted to a technical journal. (4) "The expansion

theorem for pseudo-analytic functions," by L. Bers and S. Agmon; submitted to a technical journal. (5) "On the truncation error in the

solution of Laplace's equation by finite differences," by W. Wasow; ac- cepted by the NBS Journal of Research. (6) "On the approximation of

linear elliptic differential equations by difference equations with

positive coefficients" by T. S. Motzkin and W. Wasow; submitted to a technical journal. (7) "Asymptotic solution of differential equations of

hydrodynamic stability in a domain containing a transition point," by

W. Wasow; IN MANUSCRIPT. (8) "An expansion method for parabolic partial
Status of Projects 5

differential equations," by J. W. Green; IN MANUSCRIPT. (9) "Two applications of group characters to the solution of boundary-value problems,"

by E. Stiefel; accepted by the N3S Journal of Research.

PROBABILITY METHODS AM) SAMPLING TECHNIQUES

Task 1101-11-5100/51-2

Origin; NBS

Sponsor: Office of Naval Research, USN

Full task description appears in July-Sept 1950 issue.

Status: CONTINUED. With the aim of deriving useful new results

on error analysis for the Monte Carlo method, within the framework of a unified theory, a study was made by J. H. Curtiss of chain functions of

the type

"N = f(sN)tr

n rz ( S k—1 ,S

k^ N

N

+ 2 zk

,

1 L

where S , S. , . . . , is a simple Markov chain. In the discrete case, suppose that the possible states of the process are x^, i = , .

.

,

2 ,-1 , 0 ,1

, 2 ,

.

and that the transition probabilities PrCS^^ y j S^ = x) = p(x,y) are stationary. Then if we let

vN

(x) = E(Zn | S

q= x), wn

(x) = E(£n

| S

q= x), a(x,y) z(x,y)p(x,y)

it is easily shown that

(*) v^Cx) = H a(x,y) vN

(y), v

Q

(x) = ^(x);

(**) w

N

(x) = f

(X) + w

n_i (y)

w

0 (x) = f'M

If A denotes the matrix [a(x^,Xj)J = [a — ], then the formal solutions

of these recursion relations are in matrix notation respectively

vN

= A

N

f, w

N-1 = (I-AN

)(I-A)

_1

f

If A has the following partitioned form:

where s

1J

A1 >rv>

i a ij a . . ij — 0 1 X“ 1 j

• • © 5 n X 1 }

• • • 9 D

u*. II

• 0 j=n+1 . .

0 S . .

Kronecker’s symbol, and if

-J
6 Status of Projects

then (*) has the form

( **>i=

)

with formal solution

N+1 = A, V* r + N A

2

"

VM

= A

1 N

V"'

+ ( I-A

1 N

) ( I-A

1 )

1 A

2

If the same matrix A is used in (**) but the initial-value vector y is replaced by * - (^" | 0) , then (**) becomes

f

" ' (**** ,

)

“N

= A1 W

N-1

+

and its solution is

*N-1 = ( I -A. Nmi-a^

which is exactly what (***) and its solution become when ‘jjr ' = 0 , A9 = I.

oo ‘ The well-known necessary and sufficient condition for A1 = 0 is that all

eigenvalues of A1 be in absolute value less than unity;

is satisfied, then obviously limiting vectors and w^ exist. If a

2 =

if this condition

I,

the limiting vector in each case satisfies the same system of linear

algebraic equations. This is true whether or not -\j/' = 0. The coex- istence of these two stochastic models for linear equations was pointed

out by Wasow in a forthcoming paper in Mathematical Tables and Other Aids

to Computation (see publication ( 5 ) below). The above discussion, together with its analogue for nondiscrete Markov processes, provides a convenient unified background for most of the applications of the Monte

Carlo method which have so far been made, including two summation and

quadrature problems (the case N = 1), time-dependent transport and diffusion problems, and boundary value problems associated with elliptic

differential equations (the case N = °o) . In problems involving differential and difference equations, the mean values w^(x) , w^x) , play the

role of Green's functions. From the point of view of stochastic processes, the elements of the submatrix A

2 correspond to absorbing or trap

s tales

.

The study of error analysis centered about the variances of

Z^ and 2^ and the reduction of these variances by "importance sampling."

In what follows, the underlying viewpoint is that the matrix A is given,

and the problem is to make a wise choice of z(x,y) and p(x,y) (with

z(x,y) p(x,y) = a(x,y), p(x,y) - 0, Z p(x,y) = 1). Let B = [ z (x

i ,x • ) a (x

i ,x . ) ]

,

and let B 1 and B 9 be the subraatrices of B corresponding to A. and A0

. Let 1 ^ 1 ^

i N(x) - E(Z^ 1 S

Q = x) ,/* N

(x) = E(2^ I S

Q = x) •

Then

= + (I-B^d-Bj -1 M /»? B 9B

Y
Status of Projects 7

also

= BND^+ (X-BN

)(I-B) 1 £

2D( I-A)

-1

'xjr -of-}

- 2Hn

(I-A)

-1 A^yjr ,

where D is a diagonal matrix whose diagonal elements are the components

of Y i an^ where

H = £' B

kDAN_1 - k

. w

0

If inS^, the initial vector is replaced by as in deriving (****)

above, then in/4 ^, the B's and A's can be replaced by Bis and A1 s , and

'jf by \j/" . If the eigenvalues of (as well as those of A^ ) are all

less than unity in absolute value, then it can be shown that limiting

second moments exist for both and and are given respectively by

m 00 = (I-B

1 )“' B ? D"

/^oo= (I-B

1 ) -1 ^2D( I-A

1 )

(The first of these two formulas was derived in the case A^ = I by

Forsythe and Leibler in Mathematical Tables and Other Aids to Computation

IV 127-120 (1950), and the second was studied from a different point of

view in the special case in which A is a stochastic matrix and z(x,y) = 1 by Wasow in his forthcoming MTAC paper referred to previously. Both

papers apply the restriction to that the eigenvalues of the corres- ponding matrix of absolute values are less than unity in modulus, but

this restriction seems to be academic in view of the fact that Z^andHoo

will always be approximated in practice by and 23^, N large but finite.)

These results permit the discussion of a number of dispersion

problems related to the Monte Carlo method. For example, if the duration

of the random walk associated with the partitioned matrix A is defined

as the number of visits to the states corresponding to the elements in

columns numbered j = 1,...,n before reaching the trap states j = n+1,...,

and if E. is the mean duration for initial position x., and finally, if

A -S- 1 E is the maximum mean duration for any i, then the varianc

o£ the duration for starting at x. can be shown to be not greater than 2E^ - E?.

2.

^ /-) i

(The exact formula for the variance is 2^ G(x^,x^) - El, where

G(x^,x^) is the mean number of visits to x^ if the random walk starts

at x^.) Another application is to the comparison of the two methods of

matrix inversion mentioned above (one by means of Z^aod the other by means of S^,) . This application generalizes the comparison of the two

methods made by Wasow for stochastic matrices in the forthcoming MTAC

paper referred to previously.

Turning to importance sampling, a useful selection of the

stochastic model in the case of simple summation or quadrature problems

(these correspond to the case N = 1, the problem being to evaluate

v (x) = Ha(x,y) \j/ (y) or the analogous integral) seems to be as follows:
Status of Projects

Suppose that a(x,y) = 0. Choose 0(y) = 0 in such a way that

w(x) = 2a(x,y) 0(y) can be more easily evaluated than v^ (x) , and let

€ (y) =

[ Y

- 0(y) V0(y) ' Let £ = f€ ( y )(* Now choose

P(x,y> - z(x,y) = 0(y) 3$

Then of course

as was desired, and

E(Z

1

| S

Q = x) = 2Ia(x,y) ^(-y)

Var(Z

1

| S

o = x) = w(x) 2 S £

2

(y) p(x,y) - (v

1 (x)-w(x) )‘

* c 2 ( >2 -£ w(x)

This simple appraisal involves only the given data, but will generally

be rather too high, inasmuch as the neglected term (v^(x) - w(x))^ is

of the order of £2W ( X )2 a i so . If €= 0, Z,'1 becomes a zero-variance

For N>1, it turns out that Z. T will be a zero variance estiN

- i-N es timator

,

mator if and only if z(x,y) = K ~jr (x)/ ijr (y) . In this case, Z^ - K

N ^(x);

and E(Z^) can satisfy the recursion relation (*) for N = 1,2,..., if and

only if K is an eigenvalue of A and f is a corresponding eigenvector.

In the case of the partitioned matrix A described above, this is not a serious restriction because with K = 1

, ^ " can be assigned arbitrarily

as before, and although cannot be assigned, arbitrarily

if 'j'' is to be an eigenvector, nevertheless the effect of ijr' fades out as N—»<x>. In fact, in the limit the vector (voolib") is obviously an eigenvector of A corresponding to the eigenvalue K - 1, so it is reasonable

to suppose that the problem of estimating the solution of (***) in the case N = oo might lend itself easily to a practical importance sampling

technique such as that used in the case N = 1 above. The appropriate

practical arrangement for the case N = oo

, suggested^by the case N = 1,

seems^to be as follows: Suppose again that a(x,y) = 0. Choose any

,0(y) = 0 and define

£(x

a

) = 0(x

i

) - Za(x^,y) 0(y i ), i = 1,...,n,

C(x

i

) = 0 ,

0(x

±

) = 'jf ( x

^ ) , i > n .

(Thus 0 plays the role of a trial vector in the deterministic theory

of linear equations.) Let € = \£ ( y )\. Choose

p( x v ) = «(x, y )0 (y) , v) = 0(xl-€Ul pU,yJ

0(x) - £ (x) ’ zU,yJ 0(y)

Set up Z^ with the trial vector 0 replacing ~j/

r

. Then ECZ^l S

q = x) = v^Cx) , as desired, and

VaKZ^l S

Q = x) = E [ (Z^- 0(\)f | S

Q = x] - (v^Cx) - 0(x)) 2

= E [ (Zqq- 0(x)

)

2

I S

q = x] = £

2 C(x) ,
Status of Projects 9

where C is a complicated vector whose components involve only the

elements of A and components of 0. In the (artificial) case 6 = 0, Z®

becomes a zero-variance estimator. The existence of zero-variance chain

functions seems to obviate the necessity of going into a more elaborate

stochastic process for solving steady-state problens, as advocated by

G. E. Albert in a recent series of internal memoranda prepared at the

Oak Ridge National Laboratory.

The extensions of the above results to continuous operators

instead of matrices are fairly immediate . Similar results have also

been obtained for estimating v^T (x) by importance sampling for just one preassigned value of N. The new results contained above are being incorporated ina research paper. The long-range objective of the study

is a comprehensive exposition of Monte Carlo techniques for essentially

non-stochastic problems, including the estimation of eigenvalues.

In addition to the Monte Carlo work by J. H. Curtiss, M. Cohen

and M. Kac completed the manuscript of their report on experiments with

the Kac-Donsker method for the determination of the lowest eigenvalue

of Schrbdinger’ s equation.

Publications : (1) "Uniformly best constant risk and minimax point

estimates," by R. T. Peterson, Jr.; NBS J. Res. 48, 4-9-53 (Jan. 1952);

RP2282. (2) "On the estimation of an eigenvalue by an additive functional

of a stochastic process with special reference to the Kac-Donsker method,"

by R. Fortet; NBS J. Res. 48, 68-75 (Jan. 1952); RP2286. (3) "Additive

functionals of & Markoff process," by R. Fortet; submitted to a technical

journal. (4) "On some functionals of Laplacian processes," by R. Fortet;

NBS J. Res. 48, 32-39 (Jan. 1952); RP2280. (5) "On the inversion of

matrices by random walks," by W. Wasow; accepted by Mathematical Tables

and Other Aids to Computation. (6) "Metodi probabilis tici per la

soluzione numerica di aicuni problem! di analyisi", by W. Wasow; to be

published in the Proceedings of the Fourth Congress of the Italian Mathematical Union in Messina. (7) "A statistical method for finding the

lowest eigenvalue of Schrodinger 1 s equation," by M. Kac and M. Cohen;

IN MANUSCRIPT.

VARIATIONAL METHODS

Task 1101-11-5100/51-3

Origin: NBS

Sponsors Office of Naval Research, USN

Full task description appears in Juiy-Sept 1950 issue.

Status: CONTINUED. Publication (4), an internal report, was completed by M. L, Stein. R. M. Hayes is continuing his study of general

methods for solving linear boundary value problems involving ordinary

and partial differential equations. He has obtained general convergence

theorems which are applicable to the Rayleigh-Ritz method and also to a generalization of the method given by E. Stiefel and M. R. Hestenes for

the finite case. The proof of these results is based on those given

in publication (1).

Publications: (1) "Applications of the theory of quadratic forms

in Hilbert space to the calculus of variations," by M. R. Hestenes;

Pac. J. Math. I, 525-581 (Dec. 1951)- (2) "On methods for obtaining

solutions of fixed end point problems in the calculus of variations,"

by M. L. Stein; accepted by the NBS Journal of Research. (3) "The

solution of linear equations by minimization," by M. R. Hestenes and

M. L. Stein; NAML Report 52-45. (4) "A Rayleigh-Ritz-like procedure

for minimizing integrals," by M. L. Stein; IN MANUSCRIPT.
10 Status of Projects

STUDIES IN APPLIED MATHEMATICS

Task 1 101 -1 1 -5100/51 -5

Origin: NBS

Sponsor: Office of Naval Research, USN

Full task description appears in July-Sept 1950 issue.

Status: CONTINUED. The computations for Tables I, II and III of

publication (2) were completed as a part of task 1 1 01 -53-1 1 01 /5l -36 ,p . 23-

Publications : (1) An introduction to the tables of the Chebyshev

Polynomials has been prepared by C. Lanezos, to be included in the NBS

Applied Mathematics Series 9? "Tables of the Chebyshev polynomials S

n (x)

and C

Q (x)"; now in press. (2) "Numerical computation of low moments of

order statistics from a normal population," by J. Barkley Rosser; submitted to a technical journal. (3) "Analytical and practical curve fitting of equidistant data," by C. Lanezos; IN MANUSCRIPT.

MISCELLANEOUS STUDIES IN THEORETICAL PHYSICS

Task 1 101-1 1-5100/51-5

Origin: Office of Naval Research, USN

Sponsor: " " Full task description appears in July-Sept 1950 issue.

Status: CONTINUED. The study of the application of variational

methods to quantum mechanical scattering problems is being continued (see

Oc t-Dec 1951 issue, p. 7). Explicit closed form results have been obtained for the differential cross-section for scattering by a Yukawa

potential, including exchange, using a plane wave trial function with

an adjustable wave number. Computations using these expressions are under way. The results will be compared with "exact" differential crosssections obtained from a numerical integration of Sch rodinger 1 s equation

which is now being carried out with the aid of W. Futterman of the

University of Southern California. The extension to the case in which

tensor forces are included in the interaction potential is being studied.

A. Banos, Jr., University of California at Los Angeles, and

R. K. Golden have examined a problem involving electromagnetic effects

associated with moving bodies (see July-Sept 1951 issue, p. 9). They

considered the electric and magnetic fields which a stationary observer

attributes to a rotating uniformly magnetized sphere, first when the

sphere is in empty space and then when it is surrounded by a stationary

concentric conducting shield. Neglecting terms of order v2/c2, they

found that in both cases the magnetic field is the same as that for the

stationary sphere. In addition both internal and external electric

fields are present in the first case while only an internal electric

field is present in the second. This study is a first step in the con- sideration of magneto-hydrodynamical effects in the theory of geomagnetism.

Publications: (1) "Modes of vibration of a suspended chain,"

by D. S. Saxon and A. S. Cahn; accepted by the Quarterly Journal of

Mechanics and Applied Mathematics. (2) "Distribution of electrical

conduction currents in the vicinity of thunderstorms," by R. E. Holzer

and D. S. Saxon; accepted by the Journal of Geophysical Research.

(3) "Radiation characteristics of a turnstile antenna shielded by a sec- tion of a metallic tubs closed at one end," by A. Banos, Jr., D.S. Saxon,
Status of Prohects 1

and L. Bailin; accepted by the Journal of Applied Physics. (4-) "The

torsion of anisotropic elastic cylinders by forces applied on the lateral

surface," by H. Luxenberg; accepted by the NBS Journal of Research,

(5) "An optical model for nucleon-nuclei scattering," by R. E. LeLevier

and D. S. Saxon; accepted by the Physical Review. (6) "The electromagnetic

field^of a rotating uniformly magnetized sphere," by R. K. Golden and

A. Banos, Jr., IN MANUSCRIPT. (7) "Variational calculation of scattering

cross-section," by E. Gerjuoy and D. S. Saxon; accepted as a letter to the editor by the Physical Review.

STUDY OF RUSSIAN MATHEMATICAL PROGRESS

Task 1101-11-5100/52-1

Origin: NBS Authorized 3/15/52

Sponsor: Office of Naval Research, USN

Manager: G. E. Forsythe

• Objective: To prepare, and keep up to date in readily available

form, information regarding a) mathematical monographs in Russian, b)

papers in Russian on numerical mathematical analysis. Also, to make this

literature accessible to U. S. mathematicians by c) obtaining important

monographs and journals for the Institute for Numerical Analysis library,

d) publishing bibliographies of Russian material, e) translating selected

important material on numerical analysis.

Background: Like other European mathematicians, those of. Russia

have always devoted considerable attention to publishing good expository

monographs on advanced mathematical subjects. The output of such monographs has recently increased to an astounding degree. There have appeared within the past few years a number of first class books, several

of which are unparalleled in the Western literature. The existence of

these books is not well known in the United States. Obtaining copies of

them, publishing lists of them, and translating important excerpts will

contribute a great deal towards making them known. It also has the desirable effect of stimulating the study of Russian by mathematicians.

With respect to journal articles the situation is a little different. Formerly a Russian article was ordinarily followed by a summary

in French, English, or German. This practice was stopped abruptly about

195-7? and moreover, Russian mathematicians then closed the practice of

writing some articles in Western languages. Mathematical Reviews is doing

excellent work in reviewing Russian articles, and the American Mathematical

Society (under an O.N.R. contract) is translating important ones - mostly

in pure mathematics. With respect to journal articles, the present task

is intended only to supplement these efforts, especially in numerical

analysis, which has not benefited much from the translation project.

Comments: The sponsor feels that it is a matter of vital importance

to our national self interest that scientists become familiar with the

Russian language and with Soviet scientific work. It is perhaps not so important in pure mathematics, where ideas seem to leap freely over barriers of language and censorship. But in applied work it is often es- sential to follow other men's work and reasoning laboriously from line to

line; reading a brief abstract is not sufficient. Unless the reader knows

Russian fairly well, he will gain little from the original article.

Status: NEW, The Institute for Numerical Analysis library has ac- quired copies of over sixty Russian monographs in mathematics (with a
Status of Projects

1 2

few in physics). It subscribes to about six Russian journals. Publication (1) is being typed for distribution. It lists over 350 monographs

of interest to the practising mathematician, arranged a) by author and

b) by subject. Publication (2) is a translation of an important article

on calculating matrix eigenvalues by classical iterative methods. It

is available for distribution. Publication (3) is a long and interesting

one - over 100 pages in Russian - in which the tools of analysis in Banach

spaces are applied to various problems of numerical analysis, - solving

differential and integral equations, and determining the inverse and

spectrum of a finite matrix.

An informal, but important, result of the task has been the

initiation of a Russian class for mathematicians at the University of

California at Los Angeles.

Publications; (1) "Bibliographical survey of Russian mathematical

monographs 1930—1951 , n by G. E. Forsythe; IN MANUSCRIPT. (2) 'The determination of latent roots and invariant manifolds of matrices by_means

of iteration," by K. A. Semendiaev translated by C. D. Benster, edited

by G. E. Forsythe; NBS Report No. 1402. (3) Translation of "Functional

analysis and applied ma thema tic s

, " by L. V. Kantorovich; IN MANUSCRIPT.

STUDIES IN PURE MATHEMATICS

Task 1 101-1 1-5101/50-4

(formerly 11.1/1-50-4)

Origin; NBS

Sponsor; Office of Naval Research, USN

Full task description appears in July-Sept 1949 issue.

Status: CONTINUED. D. H. Lehmer has continued the investigation

into the properties of Kloosterman sums

P-1 _

S (k) = exP [27ri(h + kh)/p] (hh = 1 (mod p))

p k=1

where p is an odd prime. The inequality

Is (k)| < 2n/F

holds uniformly for all p < 100. The sums of powers

R =

r ? S n(k) N =

+1 p S n (k)

(p> = "I "

extending over the quadratic residues and nonresidues respectively, are integers and have been computed for r = 0(1 )6, 8, 10,1 6 by using the approximate values of S^(k) obtained previously (see Oct-Dec 1951 issue, p. 8),

with the help of certain congruence properties. A further study of these

integers is in progress. . A series of tests for primality of Mersenne numbers 2

P - 1 has

been run on the SWAC. The coding for this calculation was sent in by

R. M. Robinson of the University of California, Berkeley. The results

have been interesting from both a theoretical and a practical viewpoint.

The tests have shown a high degree of reliability in SWAC operation, and

they have given some unexpected information on the distribution of primes
13 Status of Projects

of the form 2^-1. Tests have been run once for p ^ 1 867 and twice

for p ^ 1303 with exact agreement. Only two new primes have been discovered, namely those for p = 521 and p = 607. These are by far the

largest known primes. The test consists of computing p-1 terms of the

sequence 4, l4, 195-,..., [U

n+ .j = - 2] modulo 2P - 1 . Divisibility of

U , by 2^-1 is necessary and sufficient for the primality of 2^-1 . The

P 1 ^ test requires approximately p^ microseconds.

L. M. Blumenthal continues his study of the applications of

metric methods in abstract algebra (see publication 6 ). A Boolean metric

space <3 is obtained by associating with each pair of elements p,q of a set an element d(p,q) of a Boolean algebra as distance, so that d(p,q)=0

if and only if p-q^ d(p,q) = d(q ,p) , and d(p,q)+d(q,r)Dd(p,r) (+ ,Ddenote

Boolean addition and inclusion, in the wide sense, respectively). Boolean

geometry studies the invariants of (B under the group of congruences (i.e.,

distance-preserving mappings). Any Boolean algebra is a space CB with

d (p ,q )-pq

+ P ' q (juxtaposition and 1 denoting Boolean product and complementation, respectively), and this paper begins the systematic study of

these Boolean geometries by developing linearity notions and the theory

of metric segments. Defining these in a suitable manner, they are shown

to possess many properties of segments in ordinary metric spaces. Thus,

(1) the length of a segment is the distance d(a,b) of its end-elements,

(2) if S^, S£ are segments and b is metrically between a,c then their

set-union is a segment S

c

, and (3) for m > 4, any m-tuple of <3 is on some segment if and only if each triple of the m-tuple has that property.

E. Stiefel (see publication 7) succeeded in finding a complete

answer to the question concerning the possible dimensions of spaces in

which generalised systems of Cauchy-Riemann equations can exist. Here n partial differential equations

0 (j — 1,2,...,n)

with constant complex coefficients are said to form a system of generalised

Cauchy-Riemann equations, if there exist constants b®jj k such that

A ,h k Au, = H bk g-

J t^k 0x h

In 1939, 0- Taussky-Todd had proved that n must be a power of 2. Using

methods from the theory of the representation of algebras, E. Stiefel

proved that such systems can exist for n = 1,2,4, 8 only and gave an enumeration of all possible cases. The paper by I. J. Schoenberg and A. Whitney (see publication 8)

is based on previous results concerning the behavior of Polya frequency

functions. The authors obtain necessary and sufficient conditions for

the positivity of any minor of the matrix A (x - y) . The problem separates into a number of subcases, depending on the analytical nature of

the Laplace transform of A(x). Each case is treated exhaustively and

the problem solved. The results obtained give a general and elegant

discussion of the necessary and sufficient conditions of interpolating

uniquely with the help of spline curves of any order.

I. J. Schoenberg discusses a question recently raised by

L. M. Blumenthal (see publication 9). It is known that a real innerproduct space is ptolemaic, i.e., among the distances between any four

points of the space Ptolemy's inequality holds. The question is as follows

Let the real normed linear space S be ptolemaic; does it follow that its norm springs from an inner product? This question is answered in the affirmative. The proof is derived from a new, slightly improved, version of
1 ^ Status of Projects

a characterization of inner-product spaces due to M-. M. Day.

T. S. Motzkin, in a joint paper with E. G. Straus and F. A.

Valentine (see publication 10), studies compact point-sets S in the plane

for which, for each x £ S, the set of points of 5 with maximum distance

from x always, or never, consists of one single point. In the results

special convex curves and various kinds of evolutes of the latter play

a role.

Publications: (1) "On subharmonic, harmonic, and linear functions

of two variables," by E. F. Beekenbach; Revista de Matematica y Fisica

Teorica (Argentina). 8, 7-13 (Nov. 1951). (2) "On relative extrema of

Bessel functions," by 0. Szasz; accepted by the Bolletino della Unione

Matematica Italiana (Firenze). (3) "On the relative extrema of the

Hermite orthogonal functions," by 0. Szasz; accepted by the Journal of

the Indian Mathematical Society. (4) "On a recursion formula and on some Tauberian theorems," by N. G. de Bruijn and P. Erdos; accepted by the

NB3 Journal of Research. (5) "Metric methods in integral and differential

geometry," by J. W. Gaddum; accepted by the American Journal of Mathematics.

(6') "Boolean Geometry I," by L. M. Blumenthal; IN MANUSCRIPT. (7) "On

Gauchy-Riemann equations in higher dimensions," by E. Stiefel; accepted

by the NBS Journal of Research. (8) "On Polya frequency functions III.

The positivity of translation determinants with an application to the

interpolation problem by spline curves," by I. J. Schoenberg and A- Whitney; submitted to a technical journal. (9) "A remark on M. M. Day's

characterization of inner-product spaces and a conjecture of L. M.

Blumenthal," by I. J. Schoenberg; submitted to a technical journal.

(10) "The number of farthest points, by T. S. Motzkin, E.G. Straus, and

F. A. Valentine; submitted to a technical journal.

COMPUTATION OF THE IMAGINARY ZEROS OF THE RIEMANN ZETA FUNCTION

Task 1101-11-5101/50-13

(formerly 11 .1/1 —50—1 3)

Origins NBS

Sponsors Office of Naval Research, USN

Managers D. H. Lehmer

Full task description appears in Apr-Jun 1950 issue.

Status s CONTINUED. The calculation of the first 5000 "Gram points"

has been run twice on the SWAC. The calculation of the Zeta-f unc tion

at these points has been coded and is being run on the SWAC. Among the

first 1 500 Gram points there were found 80 failures of Gram's Law. A

special run to take care of these cases is being coded.
Status of Projects

2. Applied Research

i 5

RESEARCH IN THE MATHEMATICAL THEORY OF PROGRAM PLANNING

Task 1101-21-5102/50-11

(formerly 1 1 . 1 /l —50—1 1 ) Origin: Office of Air Comptroller, U3AF

Sponsor: " " Full task description appears in Apr-Jun 1950 issue.

Status: CONTINUED. The theory of the transporation problem

(F. L. Hitchcock, T. C. Koopmans , G. B. Dantzig, and M. Flood) was furthered by T. S. Motzkin in two ways. A succinct representation of all

basic solutions (before minimization) of the system

z

J x . . ij

was found (see publication^ ) below). Each basic solution is obtained

by superimposing contiguous intervals of lengths b

^ on contiguous intervals of lengths a^ and putting equal to the common part of a^

and b j . Hence (as is known) the coefficients of and bj in x^. are integers (+ 1). The same holds for the multi-index problem (x. s

^ s^ > 1 , r-fold sums) if and only if r-1 and the number of s^> 2 is at

most 2 (see publication (2) below). An example with fractional coefficients and an additional behavior discrepancy for the multi-index problem

was found by A. J. Hoffman.

Besides further hand computing under the supervision of M.

Howard to test relaxation methods, the double description method was coded for SWAC by L. Joel.

Continuing his study of metric methods in linear inequalities,

L. M. Blumenthal defined a homogeneous system of inequalities to be ir- reducibly consistent provided (1 ) the system has a non-trivial solution,

but (2) each subsystem obtained by suppressing a column of the unknowns

has only the trivial solution. The subset C of the unit n-sphere

associated with the coefficients in such a system is non-global, but the

projections of C on each of the great (n-1 ) spheres in which the coordinate planes intersect S are global. It is easily seen that a system

is irreducibly consis tent

n if and only if the solution set H(C) / 0 and

5Z (C) is contained in the interior of one orthant. This orthant is the one that is in the interior of C*, the convex extension of C.

As a modus operand! the above considerations suggest that a system of inequalities be studied by examining first the one-column subsystems, then the two-column systems, etc., with the knowledge that if none of the k-column subsystems has a nontrivial solution, any consistent

(k+1)-column system has all of its solutions in one orthant of the cor- responding k-sphere. This method, together with a device for "casting out"

orthants was applied to a system of ten inequalities in four unknowns

(which turned out to be irreducibly consistent) and the solution orthant,

together with a solution, was found.

J. W. Gaddum proved (see publication (5-) below) that a convex

cone and its polar cone have a nonnull vector in common, unless they are orthogonal linear varieties. This is applied to show, among other things,
1 6 Status of Projects

that if A x ^ 0 has a solution, it has one which is in the convex cone generated by A, and hence AA'y^O has a nonnegative solution.

For other investigations on convex curves compare with task

1101-11 - 5 ioi/5o-4 , p. i

a

T. S. Motzkin participated in the Third Annual Meeting of the

Logistics Conference, sponsored by The George Washington University and

Office of Naval Research, and held on Jan. 2, 3? 7, and 8 ht Washington,

D.C; in a Round Table Conference on the STATAC-SCOOP machine, held on

Jan. 4 at the National Bureau of Standards; and in coordinating conferences

with the Linear Programming Group of the Computation Laboratory of the

National Applied Mathematics Laboratories.

Publications; (1) "Basic solutions of the transportation problem,"

by T. S. Motzkin; IN MANUSCRIPT. (2) "The multi-index transportation

problem," by T. S. Motzkin; IN MANUSCRIPT. (3) "Two existence theorems

for systems of linear inequalities," by L. M. Blumenthal; submitted to a technical journal. ( 4 ) "A theorem on convex cones with application to linear inequalities," by J. W. Gaddum; submitted to a technical journal.

LANGUAGE TRANSLATION STUDY

Task 1 101-21-5104/52-1

Origin: NBS

Sponsor: The Rockefeller Foundation

Full task description appears in Oct-Dec 1951 issue.

Status: CONTINUED. Work continued on the investigation of word

frequency both in Spanish and in German, under the direction of W. E. Bull

and V. A. Oswald, respectively. The Spanish project is now in its

final stages. The material is being written for publication and charts

are being prepared which express the linguistic facts graphically. Present

indications are that the data will be significant in many fields of investigation. In German, a microglossary has been cons true ted - for the field

of brain surgery by scanning sixteen different articles on the subject.

The glossary has been extended well beyond the scope originally planned,

since it was found that the recurrence of nontechnical items could be

predicated to almost as high a degree as the recurrence of technical items.

The statistical import of these findings is now being investigated. The

semantic import can be established just as soon as the work of alphabetizing

the glossary has been completed.

The Research Laboratory of Electronics of the Massachusetts Institute of Technology, aided by a grant of the Rockefeller Foundation, is

planning a Conference on Mechanical Translation to be held June 18-20 at

the Massachusetts Institute of Technology, at which conference reports

vill be presented on the above task.
Status of Projects

3 • Development

17

NATIONAL BUREAU OF STANDARDS WESTERN AUTOMATIC COMPUTER (SWAC)

(previously listed as Air Materiel Command Computing Machine)

Task 1 101 -34-51 04/49-1

(formerly 1 1 .J_/ 22-4-9-1 j Origin: Flight Research Laboratory, Wright Air Development Center, USAF

Sponsor: " " " " Full task description appears in Apr-Jun 1949 issue.

Status: CONTINUED. There are three items which have recently

produced a substantial increase in operating time on the SWAC. These

three items are: 1) decreasing the accelerating voltage to minimize the

effect of flaws, 2) installing a delay circuit to minimize the effects of

spillover, primarily caused by lowering the accelerating voltage, and

3) installing the motor generator set as of last December to stabilize

the voltages and minimize the need for adjustments relative to the

Williams' tube memory system. Improvements have been made in the Williams'

tube amplifiers and a more thorough routine of systematic maintenance has

been established. A new method of input to the SWAC making use of an IBM

collator is almost ready to go into operation. The rate of input with

this device is approximately 80 times that of the Flexowriter equipment

now in use.

LOGICAL NOTATION AND BLOCK DIAGRAM SYMBOLISM FOR A.D.C.M.

Task 1 101 -3^-51 03A9-2

(formerly 11.1/22-49-2)

Origin: NBS

Sponsor: Flight Research Laboratory, Wright Air Development Center, USAF

Full task description appears in Apr-Jun 1949 issue.

Status: CONTINUED. Distribution continued of the preliminary

lists of terminology and block diagram symbols to be used in connection

with automatic computing machines. The lists are now being revised on the basis of the many comments and suggestions which have been offered

in response to our request.
Status of Projects

4. Mathematical Services

MATHIEU FUNCTIONS II

Task 1 101 -53-1 101 A5-1

(formerly 11.1/2-45-1)

Origin; Applied Mathematics Panel MDRC

Sponsor: Flight Research Laboratory, Wright Air Development Center, USAF

Full task description appears in Apr-Jun 1949 issue.

Status: INACTIVE. For status to date see Oct-Dec 1950 issue.

TABLE OF BESSEL-CLIFFORD FUNCTIONS

(formerly listed as SPECIAL TABLE OF BESSEL FUNCTIONS)

Task 1 101-53-1 101 A8-2

(formerly 11.1/2—48—2)

Origin: NBS

Sponsor: Flight Research Laboratory, Wright Air Development Center, USAF

Full task description appears in Jan-Mar 1949 issue.

Status: CONTINUED. The final manuscript has been fully checked.

TABLES FOR ROCKET AND COMET ORBITS

Task 1101-53-1101/48-3

(formerly 11.1/2-48-3)

Origin: NBS

Sponsor: Flight Research Laboratory, Wright Air Development Center, USAF

Full task description appears in Jan-Mar 1949 issue.

Status: CONTINUED. The table is being retyped, on account of a flaw

in the manuscript.

Publication: Approved for publication as NBS Applied Mathematics

Series 20; to be printed by the Government Printing Office.

COMPUTING SERVICES FOR RESEARCH STAFF OF

THE INSTITUTE FOR NUMERICAL ANALYSIS

Task 1 101 -53-1 101/49-1 a (formerly 1 101 -53-1 100A9-1

)

Origin: NBS

Sponsor: Office of Naval Research, USN

Manager: M. Howard

Full task description appears in July-Sept 1949 issue, (see task

1 1 .1/32-49-1 )

.

Status: CONTINUED. Computations on desk calculators were made for

various members of the staff. Among such calculations were: (1) For
Status of Projects 19

F. Acton: Computations were made to test approximations for the probability

integral useful in numerical analysis. (2) For G. Blanch and D. H. Lehmer:

Investigation was made of the zeros of a class of polynomials arising in

connection with a certain asymptotic expansion. (3) For C. Lanczos: Computations were made for further investigation of the Chebyshev iteration

technique, and the method was extended for finding the smallest eigenvalue

of a non-syrametric matrix. (4) For D. H. Lehmer: Brief computations were made involving the evaluation of a matrix and the determination of the

roots of several polynomials. (5) For D. Saxon: Further investigation

was made involving the evaluation of series and functions for the determination of scattering cross-sections, which are attained by using

variational methods.

Problems computed on IBM machines for the research staff included the following: (1 ) For D. Saxon: Numerical solution was made of

a second order differential equation containing two parameters. Solutions were made for thirty sets of parameters.

Problems for the research staff involving the SWAC were:

(1) For F. Acton: A code was prepared for the computation of probability

distribution leading to confidence limits which will include all of a future sample of m from a normal distribution with probability 0^. (2) For

A. Cahn: The routine for the Goldbach problem was recoded to use Alway's

method to test for primality. (3) For M. Cohen: Further work was done

on the Monte Carlo solution of Schrodinger 1 s equation. (4) For G. Forsythe:

A Monte Carlo solution of a boundary-value problem for Laplace’s differential equation was run on SWAC. (5) For M. R. Hestenes: Several sets of

5 linear equations were solved on SWAC, and coding was done to solve 8

equations. (6) For D. H. Lehmer: (a) Mersenne numbers were tested on SWAC for primality, and (2?21 _ -]) and (2^07 - i) were found to be prime.

See task 1 1 01 -1 1 -5l 01 /50-4, p.12? (b) Values of certain character sums were obtained on SWAC, and (c) A routine was coded and checked out for

the continued fraction expansions of square roots of integers. (7) For

W. E. Milne: A solution obtained on SWAC during the previous quarter for

a set of 3 simultaneous differential equations was checked and converted.

(8) For J. B. Rosser: Additional coding and computing has been done for

task 1101-11-5101/50-13, p. I

1!

PUNCHED CARD LIBRARY

Task 1 101 -53-1 101 A9-2

(formerly 11.1/2-49-2)

Origin: NBS

Full task description appears in Apr-Jun 1949 issue.

Comments: A catalog of tables on punched cards which are on file at

the Institute may be obtained by addressing the Institute for Numerical

Analysis, 405 Hilgard Avenue, Los Angeles 24, California. Within the

limits of the program of the computation unit of the Institute, tables

will be duplicated upon request, provided the requester furnishes the

blank cards. Request should be addressed directly to the Institute.

Status: CONTINUED. Pearson's table of the Bivariate Normal Distribution Function has been checked by differences up to and including

the sixth order. Values of the function for arguments k = 0.1, h = 0,0.1;

and k = 0.2, h = 0,0.1, 0.2, for r = + .05(.05).95, have been computed

on the CPC as a further check. The table of random digits (20,000 cards,

50 digits per card), and the table of random normal deviates (10,000 cards) were duplicated and the copies sent to H. Schutzberger, of Sandia Corporation .
20 Status of Projects

METEOROLOGICAL MEANS

Task 1 101-53-1 101/50-17

(formerly 11.1/31-50-1 7)

Origin: Department of Meteorology, UCLA Authorized 3/31/50

Sponsor: Flight Research Laboratory, Wright Completed 3/31/52

Air Development Center, USAF

Manager: E. Yowell

Objective: To calculate the time averages (daily readings for one month) and the space averages (readings, for a given day, around an entire

latitude circle) of various meteorological elements ( i . e . , wind velocity,

temperature, height) of the 700 mb, 500 mb, and 300 mb pressure levels.

Background: This computation arises in connection with research

in the general circulation of the atmosphere being performed by the U.C.L.A.

Meteorology Department for the Watson Laboratories of the AMC.

Comments; Raw data for this computation are furnished to INA on punched cards by the U.C.L.A. Meteorology Department.

Status: COMPLETED. The results were transmitted to the originator.

EARTH TIDES

Task 1 101-53-1 101/51-1

Origin: Geophysics Department U.CL.A.

Sponsor: Office of Naval Research, USN

Manager: T. H. Southard

Full task description appears in July-Sept 1950 issue.

Status: INACTIVE. For status to date see July-Sept 1950 issue.

EVAPORATION COMPUTATIONS

Task 1 101-53-1101/51-3

Origin: Naval Electronics Laboratory

Sponsor: Bureau of Reclamation, Department of the Interior

Full task description appears in July-Sept 1950 issue.

Status: CONTINUED. Data for months of November 1950 through August

1951 have been processed and results transmitted to originator. Daily

averages of all data are being prepared. These are 90 percent complete.

RAYDIST DATA ANALYSIS

Task 1 101-53-1 101/51-10

Origin: Naval Air Missile Test Center (Point Mugu) Authorized 12/1/50

Sponsor: Bureau of Aeronautics, USN Terminated 3/31/52

Manager: E. Yowell

Objective: To calculate the coordinates of an object being followed by

a Raydist tracking system.

Status: TERMINATED.
Status of Projects 21 COMPUTATIONS IN CONNECTION WITH PROGRAM ANALYSIS

Task 1101-53-1101/51-18

Origin: Operations Analysis Office Authorized l/l/5l

Sponsor: Office of Air Comptroller, USAF Terminated 3/31^52

Manager: R. Lipkis

Objective: To perform certain computations, as requested, in con- nection with this work.

Comme.nts: Most of the computing submitted to be done on this

subject will be programmed for the SWAC.

Status: TERMINATED.

STATISTICAL SMOOTHING

Task 1 101-53-1 101/51-19

Origin: Stanford Research Institute, Stanford University

Sponsor: Office of Research Operations, U. S. Army

Full task description appears in Jan-Mar 1951 issue.

Status: CONTINUED. One additional case was completed.

ROCKET GRAIN BURNING

Task 1101-53-1101/51-21

Origin: Naval Ordnance Test Station

Sponsor: Bureau of Ordnance, USN

Managers: F. John, G. Blanch, R. R. Objective: Part I: To solve

u

t = uxx

+ exP(~x )

subject to the boundary conditions

u(x,0) = 0; u(oo,t)

Part II: To solve

"t = uxx

+ exP (x

b

(Inyokern), USN Authorized 2/28/51

Completed 3/31/52

Reynolds

the equation

+ ex P (A ~

= 0; ux (0,t) = 0.

B

- x) + exp (A - —) ,

where x^(t) is that value of x for which u = u^( cons tant) . In part II,

x^ defines a moving boundary and replaces the condition u

x (0,t) = 0.

The other two boundary conditions of Part I hold.

Background: The solution of this equation is desired by the Naval

Ordnance Test Station in connection with research being performed at that

base
22 Status of Projects

Comment; A preliminary investigation of this problem has been

authorized. Decision as to whether or not a complete study will be made

will depend on the preliminary results.

Status: COMPLETED.

COMPUTATION IN CONNECTION WITH A STUDY OF POLARIZATION OF LIGHT

Task 1 101 -53-1 101/51 -25

Origin: Department of Meteorology, UCLA

Sponsor: Flight Research Laboratory, Wright Air Development Center, USAF

Full task description appears in Jan-Mar 1951 issue.

Status: CONTINUED. Completion of the final report is awaiting

introductory material and information concerning format to be furnished

by the originator.

CONVERSION OF HEXIDECIMAL NUMBERS

Task 1 101-53-1 101/51-28

Origin: Naval Air Missile Test Center (Point Mugu)

Sponsor: Bureau of Aeronautics, USN

Full task description appears in Jan-Mar 1951 issue.

Status: CONTINUED. One run of data was reduced this quarter.

TABLES OF THE BIVARIATE NORMAL DISTRIBUTION FUNCTION

Task 1101-53-1101/51-32

Origin: Division 13? NBS

Sponsor: Office of Chief of Ordnance, U. S. Army

Full task description appears in Apr-Jun 1951 issue.

Status: CONTINUED. Differences of Pearson's values and those of

E. Fix have been checked. In certain regions entries will have to be

recomputed to determine their accuracy. This recomputation will be done

on IBM equipment.

PRESSURE FIELDS OF POTENTIAL FLOW PAST A BODY OF REVOLUTION

Task 1 101-53-1 101/51-33

Origin: Naval Ordnance Test Station (Pasadena)

Sponsor: Bureau of Ordnance, USN

Full task description appears in -Apr-Jun 1951 issue.

Status: CONTINUED, An investigation of an integral equation

method of solution was started.
Status of Projects 23

SIMPLIFIED ROLLING PULLOUT EQUATIONS

Task 1 101-53-1 101/51 -3^+ Origin: Cornell Aeronautical Laboratory

Sponsor: Flight Research Laboratory, Wright Air Development Center, USAF

Full task description appears in Apr-Jun 1951 issue.

Status: CONTINUED. Experimental data have been received for eight

test flights. Coding of these flights for the SWAC is under way.

LOW MOMENTS OF ORDER STATISTICS

Task 1 101-53-1 101/51-36

Origin: University of Oregon

Sponsor: Office of Naval Research, USN

Full task description appears in Apr-Jun 1951 issue.

Status: CONTINUED. The computations for the first and second

moments are complete, as are certain quantities required in the computation

of the cross moments. This task, although temporarily halted pending

further financial support, is now being continued under a limited budget.

REDUCTION OF HYDROGRAPHIC DATA

Task 1 101-53-1 101/51-39

Origin: Scripps Oceanographic Institute Authorized 6/22/51

Sponsor: University of California Completed 3/31/52

Manager: R. R. Reynolds

Objective: Given sets of observations of salinity S, temperature T,

and pressure P, of samples of sea-water:

1. Compute Cl, the chlorinity, by the formula

Cl S-0.030

i"B65o 5

2. Compute

^s © o

= -0*069 + 1.4708 Cl - 0.001570 Cl

2

+ .0000398 Cl 3

.

3. Compute the auxiliary quantities Aj, B^., where

Ax = 10 3 T(4.7867 - 0.0981 85T + .0010843T2

)

Bt

= 10"6

T(1 8.030 - 0. 81 64t + 0 . 01667T

2

)

S = _ (T - 3.98) 2

(T + 283)

T 503.570 (T + 67 . 26 )
24 4. Status of Projects

Compute ^g T Q

by the formula

,T,0

~ “^T + ( ®S,0,0 +

1

3

2l+ )[ 1 - A

T+BT

<;(rS,0,0

- 0.1324)] .

5.

Compute °Cg T p by the formula

^S^P 10 ^ PoC‘ 4886S,T, ° 1 1 + 0.0000183P

- [227 + 28.33T - 0.55 IT2

+ 0.004T3

]

+ 10

~4

p [105.5 + 9 . 50 T - o.i 58t2

] - io

-8

(i.5p 2 t)

(50. ~ ~-28 9 S ? OjO,__ [i47.3 - 2. 72T + 0.04T^

- 10

4

p (32.4 - 0.87T + 0.002T2 )]

g

' S ,0,Q- 28

)

2 [4.5 + 0 . 1 T - 10 _4

p (1.8 - 0.06T)]J.

1 ^ The quantities O' and oc are related by the equations «. = ^

- 10 j

(/j -1 )

6.

Compute the anomaly in specific volume

^ = *S,T,P

~ °35, 0,P.

isobaric surfaces P^ and P

^

7.

Compute the anomaly in geopotential distance between

AD - S DP.

Background; The data consists of measurements of temperature T

(in degrees centigrade), and salinity S (in parts per mille) , of a sample

of water taken at a depth corresponding to a sea-pressure P(in decibars).

The density/5

, or the specific volume is a function of all three parameters.

It is desired to determine the specific volume for all points of observation by the method given by V. Bjerknes and J. W. Sandstrom, Dynamic Meteor- oloqy and Hydrography, part 1, Carneqie Institute of Washington, Pub. No.

88, 1910 . Comments: These data are collected by Scripps Oceanographic Institute in connection with a study of Pacific currents.

Status: COMPLETED. The last phase of this task, the training of

Scripps personnel in the methods used to solve the hydrographic data

problem, was completed.
Status of Projects 25

AUTOCORRELATION COEFFICIENTS

Task 1101 -53-1 101/52-2

Origin; Naval Air Missile Test Center (Point Mugu)

Sponsor; Bureau of Aeronautics, USN

Full task description appears in July-Sept 1951 issue.

Status; CONTINUED. Computations are performed os data are received.

One run was computed in connection with phase one during this quarter.

PRESSURE DISTRIBUTION ON A BODY OF REVOLUTION IN SUPERSONIC FLIGHT

Task H01-53-1101/52-3

Origin; Hughes Aircraft Company Authorized 8/20/51

Sponsor; Flight Research Laboratory Completed 3/31/52

Wright Air Development Center, USAF

Manager; R. R. Reynolds

Objective; To devise an I.B.M. procedure for determining the velocity distribution on the surface of a bluff body of revolution moving at

supersonic speed in a steady inviscid, rotational, compressible medium.

Background; The coordinates x,r in a meridional plane and the speed

q and direction 0 of the flow are given along an initial curve, as well as the direction <r of a detached shock front. The equations of the left and

right characteristics are

dr = tan (© + oc)dx and dr = lan(© -oc)dx,

respectively, where oc is the Mach angle. Tollmien ' s equations of com- patibility

d© cot OC q dq sin © sin oc

r sin (© + pc)

dr + sin oc cos pc y( y - 1ds

where Y

1 .405

d© = - CO t oc q dq + sin 6 sin pc

r sin (© ~od

dr sinoccosoc

r(r -i) ds

also hold on the left and right characteristics, respectively. These four

equations are sufficient for the calculation of x,r,q,© and the entropy

s at interior points. On the shock front these quantities are determined

from a shock polar diagram.

Status; COMPLETED. Computations were started on a seventh body,

but the initial conditions proved to be incompatible. Further work will

be done by the originator. The description of the procedure used in this

task is reported in the appendix of this issue.
26 Status of Projects

RAMJET CONTROL

Task 1 101-53-1 101/52-4-

Origin: Marquardt Aircraft Company

Sponsor: Air Materiel Command, USAF

Full task description appears in July-Sept 1951 issue.

Status: CONTINUED. Responses are being determined for another

y(t).

MAGNETO- IONIC EQUATIONS

Task 1 101-53-1 101/52-5

Origin: Air Force Cambridge Research Center, USAF

Sponsor: " " " Full task description appears in Oct-Dec 1951 issue.

Status: CONTINUED. Possible computation methods were examined

further; no computation has been performed as yet.

REDUCTION OF THEODOLITE DATA

Task 1 101 -53-1 101/52-6

Origin: Naval Air Missile Test Center (Point Mugu)

Sponsor: Bureau of Aeronautics, USN

Manager: F. H. Hollander

Full task description appears in July-Sept 1951 issue.

Status: CONTINUED. One theodolite run was computed this quarter.

RANGE ERROR COMPUTATION

Task 1 101-53-1 101/52-7

Origin: Naval Air Missile Test Center (Point Mugu) Authorized 9/19/51

Sponsor: Bureau of Ordnance, USN Completed 3/31/52

Manager: F. H. Hollander

Objective: Given the coordinates of a transmitter and of three

receivers, to calculate the following quantities for each of 1 1 1 specified

points. r Let X , m , n be the direction cosines from the transmitter to

o o

7 o

a point in space, and ,mu , n^ (i - 1,2,3) be the direction cosines from

the receivers to the point.

raa trix

If the matrix (a.^) is the inverse of the

X

1 KB +

O m n +

o + X0

in +

2 o

+ X-, m +

3 o

m n

2 o

e +

o
Status of Projects 27

the quantities desired are Ax _ <4 2 2

Ap

+ a

1 2 + a

l 3

Av _ <4 2 2

Ap

+ a 22+ a 23

Az _ <4 2 2

Ap

+ a 32+ a 33

The computation is to be made for each of 4

Status; COMPLETED.

sets of three receivers.

THE DETERMINATION OF THE PERIODS AND AMPLITUDES

OF THE LIGHT VARIATIONS OF THE STARS

S SCUTI AND 1 2 LACERTAE

Task 1101-53-1101/52-9

(formerly 1101-53-1100/49-4)

Origin: MSS

Sponsor: Flight Research Laboratory, Wright Air Development Center, USAF

Full task description appears in Apr-Jun 1949 issue (see task 1 1 .1/32-49-4)

Status: INACTIVE. For status to date see Oct-Dec 1950 issue.

BUBBLE DISTANCES

Task 1 101-53-1 101/52-1

Origin: Meteorology Department, University of California at Los Angeles

Sponsor: Flight Research Laboratory, Wright Air Development Center, USAF

Full task description appears in Oct-Dec 1951 issue.

Status: CONTINUED. Listings of original data were checked by the

Meteorology Department. After corrections were made, the corrected data were listed and differenced. Bubble pairs have been selected on approximately half of the data.

SUBSONIC AIR FORCES

Task 1101-53-1101/52-14

Origin: Flight Research Laboratory, Authorized 12/15/51

Wright Air Development Center, USAF Terminated 3/31/52

Sponsor: " " " Manager: R. Reynolds

Objective: To determine lift and moment coefficients for 65 wingaileron combinations oscillating with reduced frequency cu = ,04( .04) . 52

in a stream at Mach number 0.7* The wing section is assumed to be a straight line extending from x = -1 to x = 1 with the control surface

hinge at x = ~.1(.2).'9 (x / .5)*
28 Status of Projects

Comments: The formulas used, based principally on the investigation

of Poisson's integral equation by H. G. Kiissner and L. Schwarz, are derived in H. E. Fettis' "Calculation of non-s ta tiona ry air forces at

subsonic speeds," OAR Technical Report No. 5 0 951), Flight Research

Laboratory, Wright Air Development Center, USAF

.

Status: TERMINATED. The sponsor withdrew the problem.

EVALUATION OF TRANSCENDENTAL EXPRESSIONS

Task 1 101 -53-1 101/52-1 7

Origin: North American Aviation, Inc.

Sponsor: Flight Research Laboratory, Wright Air

Development Center, USAF

Manager: Roselyn Lipkis

Objective: For each 765 combinations of the parameters A, b, ©, and

x, determine the integral values of D and W which produce a minimum of

AW + D

W+D

1 - bx

subject to the restrictions 0 = D = 30 and 1 = W = 5. Background: These computations arise in connection with the work at

North American Aviation, Inc., in connection with an Air Force Contract.

Status: COMPLETED. The computations were performed on the SWAC, and

the results were submitted to the contractor.

Authorized 12/15/51

Completed 3/31/52

INHERENT ERROR ANALYSIS FIRE CONTROL EVALUATION PROGRAM

Task 11 01 -53-1 1 01/52-1

9

Origin: Naval Ordnance Test Station (Inyokern) Authorized 4/l/52

Sponsor: Bureau of Ordnance, USN

Manager: A. D. Hestenes

Objective: This study is concerned with the inherent error analysis

of the air to ground rocket fire control evaluation program of the client

and consists of four phases as follows:

(A) Review of evaluation program and determination of the mathematical expressions for the inherent errors.

(B) Recommendation of statistical experiments by which NOTS can determine measurement errors, reading errors, etc., required for a numerical evaluation of the expressions obtained in (A).

(C) Study of the permissable measurement errors for a desired

accuracy in the resulting data.

(D) Numerical evaluation of the mathematical expressions obtained

in (A) using numerical values of errors obtained as a result of (B).

Background: This study is needed to determine the validity of the

results of fire control evaluation programs.
Status of Projects 29

Status: NEW. Phase (A) is proceeding as well as anticipated.

No time has been spent on Phases (B) , (C) , and (D)

TABLE OF BIVARIATE NORMAL DISTRIBUTION FUNCTION FOR SPECIAL

VALUE OF THE PARAMETERS

Task 1101-53-1101/52-23

Origin: Engineering Department, Columbia University

Sponsor: Office of Naval Research, USN

Manager: G. Blanch

Objective: To compute

Authorized 2/l 5/52

Completed 3/31/52

L(-h, -h,r) for r = + i

L(-h, -2h,r) for r = i

h = 0(.01 ) 2. 6; 5D

where by definition

L(h,k,r) =

and

0(x,y,r) = p==L===— exp [-£(x2 + y

2 - 2rxy)/( 1 -r

2

) ] . 277-7(1 - r 2

)

00 (

dx

{

dy 0(x,y , r)

Background: A table of L(h,k,r) for h, k - 2.6 has been published

by K. Pearson. A supplementary table of the same function for h,k up to 4,

computed by E.Fix of the University of California* Berkeley, is

available in manuscript form. (The National Bureau of Standards is now collating the tables of Pearson and Fix, to be published in one volume.)

Interpolation in the tables of E. Fix and Pearson is not linear, however,

and there is a need for the table at the smaller interval, in connection

with certain research now in progress.

The computations will involve, essentially, 5-point interpolation

in available tables, and use of known relations between L(h,k,r) and

L(-h,-k,r). A few values of L(-h,-2h,r) for h beyond 2 will have to be

computed especially, since they are outside the range of either table.

Publication; "Tables of L(-h,-h,r) for r = + i and L(-h,-2h,r) for

r = i for h = 0(.01)2.6; 5D," by G. Blanch; NAML report 52-53-

Status: COMPLETED (NEW).

6th ORDER DIFFERENTIAL SYSTEM

Task 1 101-53-1 101/52-26

Origin: Hughes Aircraft Co. Authorized 4/1/52

Sponsor: Flight Research Laboratory, Wright Air Development Center, USAF

Manager: T. H. Southard

Objective: To solve a system of three second order non-linear ordinary differential equations with prescribed initial conditions. The equations
Status of Projects

30

involved are similar to those of task 1101 —53—1 1 01 /5l —37 those of

task 1101—53—1101 /5l —27 from the same originator.

Status: NEW. Approximately half of the task has been computed

and turned over to the originator for analysis. Further computations

await the originator's decision.

SYSTEMATIC AND RANDOM ERRORS

Task 1101-53-1101/52-29

Origin: North American Aviation Co. Authorized Vi/52

Sponsor: Flight Research Laboratory, Wright Air Development Center, USAF

Manager: A. D. Hestenes

Objective: Compute the salvo kill probability of a square target

of side 2a as a function of the parameters: aiming (systematic) error,

ammunition dispersion (random error), and salvo size. The expression to be evaluated is

P 5K 2

00 CO E y Q(i,j,N) P.(i,j)

i--0 p0

A

where

E

h (i ? j) =

PA(

i

’ j ) =

Q(i,j,N) = 1 - [1 - P

k Ph (i,j)J

y(W2(1 _ iii»+ + ili))

<Tr n & R

"

- ¥>) + - 55

5

1

) - s)’ A

5#51 - k))

- - k >)

X u

i r _ 2 ^

(x)

The distributions are assumed to be circular normal distributions with

standard deviations o' for the random error and o' tor the aiming error. N is the salvo size and yJ = ka, x =0 represents the center of the

o o

r aiming point distribution. is a probability that a target will be

destroyed if hit by a single shot independent of the effect of other shots,

a/n represents the mesh size. n is to be chosen such as P

S(

, is correct

to three decimals.
Status of Projects 31

SK is to be evaluated for the following values of the parameters

0.

2, 0.4, 0.7, and 1.0

1 , 5 , io, 25, 50, ioo, 150, 200

0, a, 2a, 3a, 5a, 10a, 15a, 20a

a, 2a, 3a, 5a, 10a, 1 5 a , 20a

a, 2a, 3a, 5a, 10a, 15a, 20a.

Background: These computations are needed to predict performance

of new (as well as old) weapons.

Comments: Although much work of this nature has been done piecemeal

in the past, the program outlined above goes beyond the previous work and

represents a comprehensive program which will be useful to many organizations interested in defense programs.

Status: NEW. A preliminary study of the mesh size has been made.

The main portion of the task is now being coded for computation on the

SWAC.

CONTROL SYSTEM EQUATIONS

Task 1 101-53-1 101/52-30

Origin: Summers Gyroscope Company Authorized 4/1/52

Sponsor: Flight Research Laboratory, Wright Air Development Center, USAF

Manager: R. R. Reynolds

Objective: To compute the positive quantities

Rq ~ a^(3a - 2x)/l2xz©

o

,

R

e = (-3a^ + lOa^x - 6a

2x

2 - 6a

2x + 6ax 2

-1 2ax2

y + 1 2a 2

xy + 6x 2

y - 6x2

y 2

)/l 2xz©

0 ,

Rg = 2(a - x)/x ,

R| = x(6y2 - 2z 2 + 6y + 3z)/3a 2

(1 - z) , where

x = (1 - c)z , ©

o = R(R + 1 )/2 ,

R = (6y

2 + 6yz + z2 )/6(1 - z) ,

a is a root of
32

Status of Projects

a”

1 - (.75x + °Oa 2 + (^ x +/^)a -Y - 0 ,

<* = .75(1 - 2y) , /& = i . 5y ( 1 - y) , r = .25y

2 (3 - 2y) ,

"3 p p and c . = 1 - [zJ - c* z +/3z - r )/z ( . 75z - oc) ] , for the parameters mm

c = c . , and the values . 25( .05) .4( . 1 ) . 9( .025) . 975 greater than C . , min imi m

z — .005, .01, .02, .05, .1, .2, .5, y = .001,. 002, .005, .01, .02, .05.

Background: This problem arises in the roll-axis stability analysis

of an airframe using a flicker type control system.

Status: NEW. After some preliminary analysis, the problem is being

coded for the SWAG.

SMOG ANALYSIS

Task 1 101-53-1 101/52-31

Origin: Consolidated Engineering Corporation Authorized 4/1/52

Sponsor: NBS

Manager: E. C. Yowell

Objective: To perform the following computations on 38 sets of mass spectrograph data: let x^j be the mass spectrograph reading of the

ith line (counted from an arbitrary m/e value) in the j th sample. Normalize each measurement producing an x^j by the equation

Then compute the ratios,

— i ] y x . . 1

X J

k. . , i, J,1

x •

i ,1

Background: The client believes that an examination of the k. . 1 ratios will enable him to identify chemical compounds present in one sample but not in another.

Comments: During the summer of 1951, 38 samples of atmospheric

constituents were collected in liquid oxygen traps set up in the Los

Angeles area. Many of these samples were taken in contrasting pairs of

observing conditions. Each sample was then subjected to a mass spectrograph analysis, and the results of these analyses are being studied for

clues as to the primary components contained in the local smog.

Status: NEW. These computations have been 50 percent completed.
Status of Projects 33

CORRELATION COEFFICIENTS

Task 1101-53-1101/52-3*+

Origin: Sheridan Supply Company Authorized 5-/1/52

Sponsor: Flight Research Laboratory, Wright Air Development Center, USAF

Manager: F. H. Hollander

Objective: To compute all two by two tables from a set of 102 scores obtained on an opinion questionnaire given to 600 Basic Airmen. Then

51 51 tetrachoric correlations are computed from these tables by the approximate formula given below. The first step involves obtaining the

cell frequencies for each possible pair of questions as in the table:

Variable y

0

Variable x

1 0

a b

c d

The second step involves the , computation of the tetrachoric coefficients

by the formula

Background: Originally the score for each question could be any

integer from zero to eleven, inclusive. These were converted into a score of zero or one by separating the original scale into two parts

approximately at the median of the scores of all airmen for that question

so that all four marginal totals for each table would be approximately

300. The client feels that this justifies the use of the above approximate formula for tetrachoric correlation.

Status: NEW. These computations have been 90 percent completed.

SIERRA WAVE PROJECT

Task 1101-53-1101/52-36

Origin: Department of Meteorology, U.C.L.A. Authorized *+/l/52

Sponsor: Flight Research Laboratory, Wright Air Development Center, USAF

Manager: T. H. Southard

Objective: To determine the space position of a sailplane as a function of time, being given data as to the difference between the distances of the sailplane from each of several pairs of fixed (Raydist)

s tations

Background: The sponsor is interested in investigating the meteorological phenomenon of orographic lee waves in the atmosphere, and is

sending sailplanes to probe the wind field existing in the lee of the

Sierra Nevada mountains of California.

Status: NEW. First data have been received and key-punched, but

no computations have been performed as yet.
35 Status of Projects

CHARACTERISTICS OF MOMENT ESTIMATORS OF EXTREME VALUES

Task 1 101 -53-1 101/52-38

Origin: Section 11.3, NBS Authorized Vi/52

Sponsor: Dynamic Loads Division, NACA

Manager: D. Teichroew

Objective: To compute, by empirical sampling, the following

moments of the mean, y, and standard deviation, s, in random samples of

size n - 10, 20, 30 from the extreme-value distribution with c.d.f.

F(y) = exp (-e~y )

p 2

<r (y) , cov(y,s) , ^(s) , cr (s) .

Background: These computations are needed in the study of the twoparameter asymptotic distribution of extreme values Z(x) = exp(-e~y),

where y = (x-u )//3 , being carried on under task 1 1 03-21 -5l 06/51 -1 . If a sample of observations is assumed to come from this distribution then the

parameter = u + / yp indicates the (smallest) value which will not be

exceeded by the fraction ( 1 — P ) of the observations in the long run. yp

is a tabulated value depending only on the probability P. E. J. Gumbel

has given an estimator for - which depends upon the sample mean x and

standard deviation s . This estimator is given by

^

A A v

= U + /3 y p ,

where u = x — — Cs

x

, /$ - —^ s^, and C (=0.5772. . . ) is Euler' s constant.

The above computations will make it possible for the first time to determine the bias and efficiency of Gumbel ' s moment-estimator for several

samples of moderate size.

Comments: This is a companion project to tas k 1 102-53-1 106/52-67,

which has for its aim the construction of order-statistics estimators of

the parameter

p which are unbiased and have minimum variance. The re- sults of these two tasks will make possible a comparison between the two

types of estimators needed in connection with research on estimation of

extreme values being carried on under task 1 1 03— 21— 51 06/51 —1 (Research on Application of Theory of Extreme Values to Gust-Load Problems).

Status: NEW.
II. Computation Laboratory

(Section 11.2)

1 . Research

RESEARCH IN CLASSICAL NUMERICAL ANALYSIS

Task 1102-21 - 1104/50-1

(formerly 11.2/11-50-1)

Origin: NBS

Sponsor: Flight Research Laboratory, Wright Air Development Center, USAF

Full task description appears in Jan-Mar 1950 issue.

Status: CONTINUED. In connection with his study of a new set of

orthogonal polynomials H. E. Salzer completed the computation of the zeros

and the corresponding Christoffel numbers for the first eight orthogonal

polynomials. With the aid of these complex numbers integrals of the type

1 27Ti c-ioo

P(i-)x dx

which arise in the inversion of Laplace transforms, can be evaluated

exactly for P, any polynomial of degree not exceeding 16 and with no constant term.

An article dealing with the numerical calculation of Laplace

transforms of a function given at equal intervals has been completed by

Mr. Salzer. Also he has completed an article discussing equally weighted

quadrature formulas over infinite intervals.

RESEARCH IN MODERN NUMERICAL ANALYSIS: INVESTIGATION

OF BERGMAN'S METHOD FOR THE SOLUTION OF THE DIRICHLET

PROBLEM FOR CERTAIN MULTIPLY CONNECTED DOMAINS

Task 1102-21 - 1104/50-2

(formerly 11 . 2/11 - 50-2 ) Origin: NBS

Sponsor: Flight Research Laboratory, Wright Air Development Center, USAF

Full task description appears in Jan-Mar 1950 issue.

Status: CONTINUED. F. Alt has prepared a paper to be presented to the Association for Computing Machinery describing the results of this

investigation. It is expected that this paper and the tabular material

will be submitted for publication by the NBS.

35
36

Status of Projects

MISCELLANEOUS STUDIES IN PURE MATHEMATICS

Task 11 02-21 -1 1 04/50-4

(formerly 11.2/11-50-4)

Origin: NBS

Manager: 0. Taussky-Todd

Full task description appears in Jan-Mar 1950 issue.

Status: CONTINUED. M. Newman is continuing his studies of function

analogous to the Ramanujan function (n) which arise as coefficients in

the expansions of certain elliptic modular functions.

oo r Publications: (1) "The coefficients of 1 l (l-x n

) by M. Newman;

IN MANUSCRIPT. (2) "Remarks on some modular identities," by M. Newman;

submitted to a technical journal. (3) "On Cauehy-Riemann equations in

higher dimensions," by E. Stiefel; accepted by the NBS Journal of Research

NUMBER THEORETICAL TEST PROBLEMS FOR SEAC

Task 1 102-21 -1 104/50-5a

(formerly 11. 2/1

1 —50—5 3 Origin: NBS

Managers: 0. Taussky-Todd and K. Goldberg

Full task description appears in Apr-Jun 1950 issue.

Status: INACTIVE. For status to date see Oct-Dec 1951 issue.

ANALYSIS OF CRYSTAL STRUCTURE

Task 1102-21-1104/51-3

Origin: NBS

Full task description appears in July-Sept 1950 issue.

Status: INACTIVE. For status to date see July-Sept 1951 issue.

SOLUTION OF LAPLACE EQUATION BY MONTE CARLO METHOD

Task 1102-21-1104/51-6

Origin: NBS

Managers: C. J. Swift and J. Todd

Full task description appears in July-Sept 1950 issue.

Status: INACTIVE. For status to date see Oct-Dec 1951 issue.
37 Status of Projects

HEAT CONVECTION IN LAMINAR FLOW THROUGH A TUBE

Task 1 102-21-1 104/51-9

Origin: NBS

Full task description appears in Jan-Mar 1951 issue.

Status: CONTINUED. The report has been written and is now being

reviewed.

THREE-BODY PROBLEM

Task 1102-21-1104/52-4

Origin: NBS

Full task description appears in July-Sept 1951 issue.

Status: INACTIVE. For status to date see Oct-Dec 1951 issue.

ANALYSIS OF GEOMAGNETIC FIELD

Task 1102-21-1104/52-8

Origin: NBS

Full task description appears in July-Sept 1951 issue.

Status: INACTIVE. For status to date see Oct-Dec 1951 issue.

INVESTIGATION OF THE APPLICABILITY OF AUTOMATIC DIGITAL

ELECTRONIC COMPUTING TO PROBLEMS IN THEORETICAL AND APPLIED STATISTICS

Task 1102-21-1104/52-28

Origin: NBS and Institute of Statistics, University of North Carolina

Managers: J. Levin and R. B. Bryce

Full task description appears in July-Sept 1951 issue.

Status: CONTINUED. Research was conducted by D. Teichroew on

(i) continued fractions, (ii) history of empirical sampling, and (iii)

sampling on high-speed machines. The results regarding continued fractions have been set forth in a paper "Use of continued fractions in hispeed computing," by D. Teichroew, which is to be submitted for publication.

Also a report on "importance" sampling has been prepared.

SPECIAL PROBLEMS IN FINITE MATRIX THEORY

Task 1102-21-1104/52-34

Origin: NBS

Full task description appears in Oct-Dec 1951 issue.

Status: CONTINUED. (1) 0. Taussky-Todd continued work on matrices

with property L and, in particular, on different characterizations of

this property. She also worked on inequalities for characteristic roots
38 Status of Projects

of products of matrices in connection with recent results of H. Weyl and

K. Fan. ( 2 ) A. J. Hoffman obtained a new proof of some results of

J. von Neumann on the location of the eigenvalues of the second kind of

the sum of two matrices. Q) Id. Mannos obtained results of "Eigenvectors

of matrix polynomials." (4) 0. Taus sky-Todd and J. Todd completed an ex- tensive manuscript on "Systems of ’ simul taneous equations, matrices and

determinants". This was prepared at the request of the editors of a technical journal.

Publications: (1) "Normal matrices with property L," by N.Wiegmann;

submitted to a technical journal. ( 2 ) "Pairs of normal matrices with

property L," by H. Wielandt; submitted to a technical journal. (3) "On

the eigenvalues of A + B and AB , " by H. Wielandt; submitted to a technical

journal

.

ROOTS OF POLYNOMIAL EQUATIONS

Task 1102-21-1104/52-51

Origin: NBS Authorized 2/1/52

Manager: I. C. Diehm

Objective: To produce a SEAL routine which will determine the

roots, real and complex, of any given polynomial equation, provided its

degree is not so large that computation time becomes excessive.

f (z)

Then

Background: Let C be a closed curve in the complex plane, and

a function which is meromorphic within, and has no singularities

1 27/1

f

1 (z)

f (z) dz =

on C.

where r^ ,

r

Q ,

, . . .

a

re the orders of the zeros and s^,S

2 ?s^... the orders

of the poles of f(z) within C. Since a polynomial has no poles, it can

be determined whether the polynomial has zeros in a given region, and, if

so, the region can be subdivided and the process repeated until the roots are known to any desired accuracy.

Status: NEW. A code has been completed and is being tested,

DISTRIBUTION OF NORMAL MODES OF VIBRATION OF CUBIC LATTICES

Task 1102-21-1104/52-62

Origin: NBS Authorized 2/25/52

Sponsor: " Manager: I. C. Diehm

Objective: To compute the distribution of normal modes of vibration

in cubic lattices, on which depend the vibrational contribution to the

thermodynamic properties of polyatomic molecules and crystals.

Exact distribution functions have been found for two-dimensional

lattices, but the amount of computation necessary in the three-dimensional

case has hitherto been prohibitive. This program will investigate application of high-speed computing devices to the problem.
Status of Projects 39

Background; The characteristic frequencies of normal modes of

vibration of a cubic lattice are roots of cubic equations. Since the

number of equations is of the order of the number of particles in the

lattice (0(10 -1 )), the time required for the calculation of the frequencies

is tremendous. It is expected that by taking small crystals (0(104 ) lattice points) one would obtain a considerable amount of information

concerning the distribution.

Status; NEW. A flow chart has been prepared, and coding is in

progress.

RESEARCH IN LINEAR PROGRAMMING

Task 1102-21-5115/50-2

(formerly 1 1 02-53-1 1 06/50-2)

Origin: Air Comptroller's Office, USAF

Sponsor: " " Managers: J. Todd and A. J. Hoffman

Full tas& description appears in Jan-Mar 1950 issue,, see 11.2/12-50-1,

Status: CONTINUED. Twenty small scale problems have been solved

by the Simplex method, the Brown Game method, and the relaxation method.

The results which have been obtained are being investigated so as to compare the relative efficiency of these methods in solving problems of

linear programming.

The Simplex technique was successfully employed in solving

the "Aircraft Deployment" problem 5 that of finding the most efficient

assignment of crews and aircraft to combat and training so that a maximum

number of sorties can be flown. The solution time for the handling of

a 48 x 71 matrix was 30 hours. The same problem attempted by the Brown

Game method showed the latter considerably slower.

Publications: (1) "Preliminary report on the ?>EAC experience with

the relaxation method," by N. Wiegmann; IN MANUSCRIPT. (2) "Preliminary

report on the SEAC experience with Brown's method of fictitious play,"

by M. Mannos; IN MANUSCRIPT. (3) "The order of degeneracy of the solution

of the Simplex problem," by A. Hoffman; IN MANUSCRIPT. (4) "Hie 'closeness

of a vector to a solution of a system of linear inequalities," by A. Hoffman

IN MANUSCRIPT. (5) "Bibliography on bounds for characteristic roots of

finite matrices," by 0. Taussky-Todd, NBS Report 1162 (Sept 1951); limited

number of copies available upon request.

COMPRESSIBLE FLOW-METHOD OF ORTHOGONAL AND KERNEL FUNCTIONS

Task 1102-21-5116/52-16

Origin: Flight Research Laboratory, Wright Air Development Center, USAF

and Harvard University

Full task description appears in July-Sept 1951 issue.

Status: CONTINUED. A coarse tabulation was made on SEAC of the X

and © values as functions of M. The codes have been completed and are being checked on SEAC, for: (1) the tabulation of M as a function of X

and © and (2) the evaluation of F corresponding to the M obtained in (1).

Further theoretical in\'es tigatioras are being made to determine the appropriate grid for A and 9 preparatory to the coding of the multiple integrals

mentioned in the last issue.
40

Status of Projects

TABLES OF INTEGRALS INVOLVING THE HIGHER TRANSCENDENTAL FUNCTIONS

Task 1 102-21-51 1 7/52-33

Origins NBS

Sponsor: Office of Naval Research

Full task description appears in July-Sept 1951 issue.

Status: CONTINUED. The preparation of the tables was c'ontinued.

NAVIER-STOKES EQUATION

Task 1 102-21-51 1 7/52-50

Origin: NBS

Sponsor: Flight Research Laboratory, Wright Air Development Center, USAF

Full task description appears in Oct-Dec 1951 issue.

Status: CONTINUED. The numerical evaluation of the Oseen linearization. of the Navier-S tokes equations has been continued. The following

parts of the problem have been coded: (1) the determination of the

Fourier coefficients for the boundary values of the first seven functions

of the complete systems and their normal derivatives, and (2) the evaluation of the coefficients of the systems of linear equations which determine

approximate solutions for both the classical problems (all velocity components, zero, and the Ackeret problem (normal velocity zero; tangential stress,

zero). A trial run on SEAC has been made and was successful.

2. Applied Research: Tables and Experimental Computations

BIBLIOGRAPHY OF MATHEMATICAL TABLES AND NUMERICAL ANALYSIS

Task 1102-21-1110/50-5

(formerly 1 1 02-21 -1 1 04/50-5)

Origins NBS

Managers: J. Todd and H. E. Salzer

Full task description appears in Jan-Mar 1950 issue, see 11.2/2-50-5.

Status: CONTINUED. W. H. Durfee has prepared a mimeographed

list of standard mathematical periodicals available in the NBS Washington

library.
Status of Projects 4l COLLECTED SHORT MATHEMATICAL TABLES OF THE COMPUTATION LABORATORY

Task 1102-21-1110/51-4-

( formerly 1 1 02-21 -1 1 04/51 -4-)

Origin: NBS

Manager: H. E„ Salzer

Full task description appears in July-Sept 1950 issue.

Status: CONTINUED.

Publication: "Collected Tables of the Computation Laboratory:

Volume I„ Tables of functions and of zeros of functions"; accepted for

publication in the NBS Applied Mathematics Series.

TABLE OF THE BESSEL FUNCTION Y (x)

Ji Task 11 02-21 -ill 0/51-10

(formerly 1 1 02—21 — 1 104/51-10)

Origin; NBS

Full task description appears in Jan-Mar 1951 issue.

Status: INACTIVE. For status to date see July-Sept 1951 issue.

REVISION OF MATHEMATICAL TABLES

Task 1102-21-1110/52-7

Origin: NBS

Full task description appears in July-Sept 1951 issue.

Status: CONTINUED. The revision of AMS1 , "Tables of the Bessel

Functions Y^Cx), (x) , K

q (x), K, (x) , 0^x6 1," is completed; this will

be reissued as AMS 25. Also the revision of the following tables is in

progress; (1) "Tables of Circular and Hyperbolic Sines and Cosines for

Radian Arguments," (MT3); (2) "Table of Sine and Cosine Integrals for

Arguments from 10 to 100," (MT13); (3) "Tables of Probability Function,"

volume II (MT14-); and (4) "Table of Arc Tan x," (MTl6). These tables

will be reissued in the NBS Applied Mathematics Series.

TABLES OF E^z), (z = x + iy)

Task 1102-21-5120/4-3-3)

(formerly 1 102-21 -1 1 10A3-3)

Origins Canadian National Research Council

Sponsor: Flight Research Laboratory, Wright Air Development Center, USAF

Full task description appears in Apr-Jun 194-9 issue, see task 11.2/2-4-3-3.

Status; CONTINUED. Checking of the final manuscript is underway.
42 Status of Projects

TABLE OF THE GAMMA FUNCTION FOR COMPLEX ARGUMENTS

Task 1102-21-5120/46-1

(formerly 1102-21-1110/46-1)

Origin; NBS

Sponsor; Flight Research Laboratory, Wright Air Development Center, USAF

Managers; H. E. Salzer and M. Stein

Full task description appears in Apr-Jun 1949 issue, see task 11.2/2-46-1.

Status; CONTINUED. The checking and preparation of interpolation

schedules was continued. An extensive bibliography of tables in this

field and of their applications has been compiled by H. E. Salzer.

TABLES OF COULOMB WAVE FUNCTIONS

Task 1102-21-5120/47-2

(formerly 1102-21-1110/47-2)

Origin: NBS

Sponsor; Flight Research Laboratory, Wright Air Development Center, USAF

Full task description appears in Apr-Jun 1949 issue, see task 11.2/2-47-2,

Status: CONTINUED. The checking of the manuscript for volume II

is being continued.

A program for the computation of the functions on a production

basis Is now being studied.

f^iblicatiao: "Tables of Coulomb wave functions, volume I," NBS

Applied Mathematics Series 17? in press, Government Printing Office.

TABLE OF ANTILOGARITHMS

Task 1 1 02-21 -51 20/47-1

(formerly 1102-31_-d.no/47-3)

Origins NBS

Sponsor: Flight Research Laboratory, Wright Air Development Center, USAF

Full task description appears in Apr-Jun 1949 issue, see 11.2/2-47-3*

Status: CONTINUED. This volume is to be submitted for publication

in the NBS Applied Mathematics Series.

TABLE OF LAGRANGIAN COEFFICIENTS

FOR SEXAGESIMAL INTERPOLATION

Task 11 02-21 -5l 20/48-2

.

(formerly 1102-21-1110/48-2)

Origin: NBS

Sponsor: Flight Research Laboratory, Wright Air Development Center, USAF

Full task description appears in Apr-Jun 1949 issue, see 11.2/2-48-2.

Status: CONTINUED. The manuscript for the three- and four-point

interpolants has been typed. Arrangements are being made for the writing

of an introduction for the volume.
Status of Projects 43

FERMI FUNCTION, II

Task 1 102-21-51 20A9-10

(formerly 1102-21-1110/49-10)

Origin: NBS , Section 4.4

Sponsor: Flight Research Laboratory, Wright Air Development Center, USAF

Full task description appears in Apr-Jun 1949 issue, see 11.2/33-49-10.

Status: CONTINUED. The second page proofs were corrected.

Publication: The table is being printed by the Government Printing

Office and will be issued as "Table for the analysis of ^-spectra," NBS

Applied Mathematics Series 13-

TABLE OF CHEBYSHEV POLYNOMIALS

Task 1 102-21-51 20/50-3a

(formerly 1 1 02-21 —111 0/50—3«)

Origin: NBS

Sponsor: Flight Research Laboratory, Wright Air Development Center, USAF

Full task description appears in July-Sept 1949 issue, see 11.2/2-50-3.

Status: CONTINUED. The final page proofs were corrected.

Publication: "Table of Chebyshev polynomials," NBS Applied Mathematics Series 9j in press, Government Printing Office.

PROBABILITY TABLES FOR EXTREME VALUES

Task 11 02-21 -5l20/50-4a

(formerly 1 1 02-21 -1 1 1 0/50-4a)

Origin: NBS, Section 11.3

Sponsor: Flight Research Laboratory, Wright Air Development Center, USAF

Full task description appears in Oct-Dec 1949 issue, see 11.2/2-50-4.

Status: CONTINUED. Galley proofs have been received and corrected.

Publication: "Probability tables for analysis of extreme-value

data", NBS Applied Mathematics Series 22; in press, Government Printing

Office.

WAVE FUNCTION FOR LITHIUM

Task 1102-21-5120/50-7

(formerly 1 1 02-21 -1 1 04/50-7)

Origin: NBS

Manager: J. Todd

Full task description appears in Apr-Jun 1950 issue.

Status: CONTINUED. A code has been prepared and is being checked.
1+

1+ Status of Projects

TABLE OF ARCS IN FOR COMPLEX ARGUMENTS

Task 1 102-53-5120/52-1^

(formerly 1 102-21 -1 1 1 0/52-1 4)

Origin: NBS

Manager: J. Levin

Full task description appears in July-Sept 1951 issue.

Status: CONTINUED. A code has been completed and is being checked.

EXTENSION OF THE TABLE OF HYPERBOLIC SINES AND COSINES

Task 1102-21-5120/52-18

(formerly 1102-21-1110/52-18)

Origin: NBS

Full task description appears in July-Sept 1951 issue.

Status: CONTINUED. The code has been revised for high-speed

output. The completion of the table is pending until machine time

becomes available.

TABLE OF THE MODIFIED AIRY INTEGRAL

Task 1102-21-5120/52-23

(formerly 1102-21-1110/52-33)

Origin: NBS

Full task description appears in July-Sept 1951 issue.

Status: CONTINUED. The truncation error occurring in the numerical

integration of the integral is being studied.

TABLE OF ERROR FUNCTION FOR COMPLEX ARGUMENTS

Task 1102-21-5120/52-25

(formerly 1 1 02-21 -1 1 1 0/52-25)

Origin: NBS

Managers: H. E. Salzer and I. C. Diehm

Full task description appears in July-Sept 1951 issue.

Status: CONTINUED. The code has been completed and checked.

EXTENSION OF TABLES OF THE EXPONENTIAL FUNCTION FOR

NEGATIVE ARGUMENTS

Task 1102-21-5120/52-31

(formerly 1102-21-1110/52-31)

Origin: NBS

Full task description appears in July-Sept 1951 issue.

Status: CONTINUED. The code has been completed and checked and

will be run when machine time is available.'-"'

</
Status of Projects 45

WAVE FUNCTION FOR HELIUM

Task. 1102-21-5120/52-32

(formerly 1102-21-1104/52-32)

Origins NBS

Full task description appears in Oct-Dec 1951 issue.

Status: CONTINUED. The difference equation corresponding to the helium wave equation using a coarse mesh has been solved. Plans

have been made to solve the equation using a finer mesh.

SPHEROIDAL WAVE FUNCTIONS

Task 1102-21-5120/52-37

(formerly 1 102-21 -1 1 1 0/52-37)

Origin: NBS

Full task description appears in Oct-Dec 1951 issue.

Status: CONTINUED. A code has been completed and checked and the

first part of the problem solved by its aid. A code for the second part

of the problem has been completed and partly checked.

VAN DER POL EQUATION

'Task 1102-21-5120/52-43

(formerly 1102-21-1110/52-43)

Origin: NBS

Full task description appears in Oct-Dec 1951 issue.

Status: CONTINUED. A code has been completed and checked and results

have been obtained for small values of the parameter.

RADIAL MATHIEU FUNCTIONS

Task 1 102-21-5120/52-49

Origin: NBS Authorized 2/1/52

Sponsor: Atomic Energy Commission

Managers: I. Rhodes , J, Todd

Objective: To tabulate the radial solutions of Mathieu's differential equation

(1) y" - (b - s cosJi

2x)y = 0 . Range of tabulations s from 0 to 10, at those points for which the

characteristic values and trigonometric coefficients are available;

x - 0( .02)2( .01 )3 j 5D, for orders r = 0,1 , ... ,1 5. Functions to be tabulated : AJe

p (s,x), AJe'p (s,x), Ne

p (s,x)/A, Ne'

p (s,x)/A

AJ q

p

( s , x ) , AJo'

r (s,x), No

p

( s , x

)

/A , No'

r (s,x)/A

( 2 )
46 Status of Projects

where A = 4

r

r

, g ~ ir.

> Background: The above functions are defined in the NBS publication

Tables Relating to Mathieu Functions (published by Columbia University

Press) . Associated with equation (l’) is the following equation:

(3) y" + (b - s cos

2x)y = 0

When the parameter b assumes a value be (s) which belongs to a countably

infinite set of characteristic values the solutions of (3) are even and

periodic, or period v or 2 w. Similarly there is another countably infinite set. of characteristic values bo

p (s) corresponding to which the

periodic solutions (of the same periods) are odd. The characteristic

values of orders up to and including 15? together with the trigonometric

coefficients associated with them, have been adequately tabulated in the

NBS publication mentioned above. A pioneering table of the periodic

solutions of the first five or six orders have been published by Ince,

and a more extensive table for orders up to 15 is in process of computation at the NBS Computation Laboratory. No tables at all are available

for the "radial" solutions —that is, the solutions of (2), although in

many problems the periodic solutions cannot be used without the corres- ponding solutions of (2). The values of the radial solutions at x = 0 can be obtained easily from the "joining factors" tabulated in the NBS

publication and the information supplied in the Introduction of the same volume. Moreover, the magnitude of the functions, with varying s, can now be inferred from the properties of the tabulated joining factors,

and it is hoped that the scaling factor,. A, introduced in (2), is such

that the resulting functions will be of order of magnitude of unity

throughout their range.

Ideally it would be desirable to tabulate the functions up to a point where the known asymptotic expansions yield approximations to about

three significant figures. The range suggested here may not meet this

criterion, especially for high orders r. However, the initial tabulation

suggested here, or any part of it, will provide some means of judging

how much more remains to be done.

The range suggested here comprises about 408,000 entries. If 8

functions are printed across a page, the space needed will be 1,020 pages,

or 2 volumes.

Status: NEW. Several programs for the computation of the Mathieu

functions have been coded for SEAC. A study of the relative efficiency

of these codes is underway.

SIEVERT'S INTEGRAL

Task 1102-21-5120/52-57

Origin: NBS

Managers: 0. Steiner and W. Soderquist

Authorized 2/12/52

S, where

Objective: To construct a table of values of Sievert's integral

= JV*-x sec ©

d©

for the range of values 0 = 0(1°)90°, x = 0 ( .01 ) 1 ( .02) 5( *05) 1 0.

Background: Among the uses of Siebert's integral is the calculation
Status of Projects 47

of intensity of radiation from line sources of radioactive material.

Status: NEW. A code is in preparation.

STRUVE FUNCTION OF ORDER 3/2

Task 1102-21-5120/52-60

Origins NIBS Authorized 2/12/52

Managers E.G1 Marden

Objectives To compute

/27? „ / _. N „ 2 sin x , 2 \ /5T H3/2 (x) - 1 X ^2 (1 ~ cos x)

for x - 0(. 02)l5j where H-^0 (x) is the Struve Function of order 3/2.

Background: The Struve functions of half integral order appear

in radiation problems wi th sp herical symmetry. One important quantity

that is proportional to \/2tt/x H^^Cx) is the fraction of energy scattered

from a plane wave by a spherical obstacle when its wave transmission

characteristics are not too different from those of the external medium.

Status: NEW. A code has been completed and checked.

SCATTERING FUNCTIONS

Task 1102-21-5120/52-63

Origin: NBS Authorized 3/10/52

Manager s A. N. Gleyzal

Objective: To develop practical numerical and analytical methods

for determining the scattering of plane waves by obstacles (or force

centers in quantum mechanical cases). Mathematically this requires the

solution of the wave equation

x + k^(r)x = 0

with proper behavior at large distances from the scatterer. The possibility of using high-speed digital computers for this problem will be

inves tigated

.

Background: There is considerable technological interest in the

scattering of electromagnetic waves by large molecules, fogs, dust particles, etc. The interpretation of experimental results in the scattering

of high energy particles by nuclei is one of the central problems in

nuclear physics.

Status: NEW. A study of the literature is being made preparatory

to formulating an extensive computation program.
48 Status of Projects

TABLES OF POWER POINTS OF ANALYSIS OF VARIANCE TESTS

Task 1304-34-6351/51-8

Origin; Section 11.3? NBS

Full task description appears in Apr-Jun 1951 issue.

Status: CONTINUED. A code for the computation of x, where p, q,oc

are given and X (p,q) =oC , has been completed.

3 . Mathematical Services

LINEAR PROGRAMMING ON STANDARD PUNCHED CARD MACHINES

Task 1102-53-1106/49-3

(formerly 11.2/36-49-3)

Origin: Air Comptroller's Office, USAF

Sponsor: " " ” Full task description appears in Apr-Jun 195-9 issue.

Status: CONTINUED. The computation of Air Force triangular models

has been progressing as required by the Office of the Air Comptroller.

The 192 x 192 inter-industry matrix has been prepared for inversion on

the UNIVAC.

SHOCK WAVE PARAMETERS

Task 11 02-53-1 106A9-1 3

(formerly 11. 2/3 3 —4-9 —13)

Origin: Bureau of Ordnance, USN

Sponsor: " " Full task description appears in Apr-Jun 195-9 issue.

.Status: INACTIVE. For status to date see Apr-Jun 1951 issue.

WAVE RESISTANCE OF SHIPS, III

Task 1 102-53-1 106/50-1 1 (formerly 11. 2/33—50—1 1 ) Origin: David Taylor Model Basin, USN

Sponsor: " " " Full task description appears in Oet-Dec 195-9 issue.

Status: CONTINUED. The integral Mn = *T x

n sinc?c x dx has been^^

computed for the new values of**. In addition the integral R = J^x^e dx

has been computed for values ofcc.
Status of Projects 49

MOLECULAR STRUCTURE CALCULATIONS, II

Task 1102-53-1106/50-16

(formerly 11.2/33-50-16)

Origin: Naval Research Laboratory, USN

Sponsor: " " Full task description appears in Jan-Mar 1950 issue.

Status: INACTIVE. For status to date see July-Sept 1951 issue.

PROGRAM COMPUTATION ON THE SEAC

Task 1102-53-1106/51-7

Origin: Office of the Air Comptroller, USAF

Sponsor: " Managers: A. J. Hoffman and F. B. Meek

Full task description appears in July-Sept 1950 issue.

Status: CONTINUED. A new code for the solution of aircraft

deployment pattern models has been devised. The code has been checked

and several test problems have been run off successfully. The new code

eliminates the need for hand computation of patterns and in addition,

furnishes information which was also previously computed by hand about

allocations of planes to combat ahd trailing.

FLOW IN SUPERSONIC NOZZLES

Task 1102-53-1 106/51-13

Origin: Naval Ordnance Laboratory

Sponsor: " " Managers: p. Alt and L. Nemerever

Full task description appears in Oct-Dec 1950 issue.

Status: CONTINUED. One of the finite difference schemes investigated has been coded and run on SEAC; it is apparently stable. Results are being transmitted to the sponsor.

INTERNAL CONVERSION COEFFICIENTS FOR L-SHELL

Task 1102-53-1106/51-19

Origin: Atomic Energy Commission, Oak Ridge National Laboratories

Sponsor: 11 " " Manager: J. H. Wegstein

Full task description appears in Jan-Mar 1951 issue.

Status: CONTINUED. Coding of the main routine continues.
50

Status of Projects

X-RAY PENETRATION

Task 1 102-53-1 106/51 -20

Origin: Atomic Energy Commission, New York Office

Sponsor: " " " Full task description appears in Jan-Mar 1951 issue.

Status: CONTINUED. Low energy cases have been run, and several

results have been obtained which were transmitted to sponsor. The

completion of the problem is pending until machine time becomes available.

A code for a point-monodirectional radiation source is in preparation.

Origin: Naval Medical Research Institute

Sponsor: " " Full task description appears in Jan-Mar 1951 issue.

Status: CONTINUED. The integral equation was solved for additional

values of the parameter.

Origin: Naval Research Laboratory, USN

Managers: M. Abramowitz and P. J. O'Hara

full task description appears in July-Sept 1951 issue.

Status: CONTINUED. Calculation of the most prominent maxima of

the series

LIQUID-VAPOR TRANSITION

Task 1 102-53-1 106/51-22

MOLECULAR STRUCTURE, III

Task 1102-53-1106/51-37

liKl was continued.

SHOCK WAVE PARAMETERS

Task 1 102-53-1 106/51 -38

Origin: Bureau of Ordnance, USN

Sponsor: " " Full task description appears in Apr-Jun 1951 issue.

II

Status: INACTIVE. For status to date see Oct-Dec 1951 issue.
Status of Projects 51 STANDARD LORAN TABLES

Task 1 1 02-53-1 106/51 -39s Alaskan Chain

Task 1 1 02-53-1 1 06/52-38 : Loran Radux Stations II

Origin: U. S. Navy Hydrographic Office

Sponsor: " " Full task description appears in July-Sept 1949 issue, see 11.2/34-50-1.

Status: CONTINUED. The manuscript for the Alaskan chain was completed and submitted to the sponsor. The calculations for the first

two Radux pairs of Loran Radux Stations II have been completed. and the

results have been transmitted to the sponsor.

PRESSURE DISTRIBUTION ON BODIES OF REVOLUTION

Task 1102-53-1106/52-3

Origin; David Taylor Model Basin, USN

Full task description appears in July-Sept 1951 issue.

Status: CONTINUED. All cases but one have been completed.

POWDER DIFFRACTION

Task 1102-53-1106/52-6

Origin: NBS , Section 9*7

Full task description appears in July-Sept 1951 issue.

Status: INACTIVE. For status to date see July-Sept 1951 issue.

PLASTIC DEFLECTION OF A COLUMN

Task 1 102-53-1 106/52-1 1 Origin: NACA, Langley Aeronautical Laboratory

Sponsor: " " " Full task description appears in July-Sept 1951 issue.

Status: CONTINUED. The results are being transmitted to the

sponsor as completed.

ROUNDING ERRORS IN STEPWISE INTEGRATION OF A PARTIAL DIFFERENTIAL

EQUATION

Task 1102-53-1 106/52-13

Origin: Naval Ordnance Laboratory, USN

Sponsor: " " Full task description appears in July-Sept 1951 issue.

Status: CONTINUED. Results are being transmitted to the sponsor

as completed.
52 Status of Projects

NEUTRON DIFFUSION

Task 1 102-53-1 106/52-1 5 Origin: Rand Corporation

Sponsor: Air Materiel Command, USAF

Full task description appears in July-Sept 1951 issue.

Status: CONTINUED. The completion of the computations is pending

until data becomes available.

PRECISE DETERMINATION OF THE PARAMETER OF DISPERSION

EQUATION FOR SEVERAL TYPES OF OPTICAL GLASS

Task 1 102-53-1 106/52-1 7

Origin: NBS , Division 2

Sponsor: " Full task description appears in July-Sept 1951 issue.

Status: CONTINUED. The computations are being performed as reques ted

ROLLING MOMENT DUE TO SIDESLIP

Task 1 102-53-1 106/52-19

Origin: National Advisory Committee for Aeronautics

Sponsor: " " Full task description appears in July-Sept 1951 issue.

Status: CONTINUED. The code is completed and checked. Preparation of data by the IBM unit is almost completed.

SPHERICAL BLAST

Task 1 102-53-1 106/52-20

Origin: Naval Ordnance Laboratory, USN

Sponsor: " " Managers: D. H. Jirauch, L. Nemerever

Status: CONTINUED. New equations were submitted by the sponsors

after further analysis. A flow chart has been completed.
Status of Projects

53

MAGNETIC FIELD EXTRAPOLATION

Task 11 02-53-1 1 06/52-22

Origin: Naval Ordnance Laboratory, USN

Sponsor: " " Full task description appears in July-Sept 1951 issue.

Status: CONTINUED. A code has been completed and checked. The

continuation of this task is pending until data for the next problem

is available.

CHECKING OF NAVIGATION TABLES

Task 1 102-53-1 106/52-26

Origin: USN Hydrographic Office

Sponsor: " " Full task description appears in July-Sept 1951 issue.

Status: CONTINUED. The checking of these tables is 60 percent

completed.

CABLE IN A UNIFORM STREAM

Task 1102-53-1106/52-27

Origin: David Taylor Model Basin

Sponsor: " " Full task description appears in July-Sept 1951 issue.

Status: INACTIVE. For status to date see Oct-Dec 1951 issue.

LORAN GRID TABLES

Task 1102-53-1106/52-39

Origin: Hydrographic Office, USN

Sponsor: " " Full task description appears in Oct-Dec 1951 issue.

Status: CONTINUED. Programming for SEAC has been completed.

CALCULATIONS FOR d SPACINGS

Task 1 102-53-1 106/52-44

Origin: NBS , Division 9

Sponsor: " Managers: J. A. Jordan and M. Abramowitz

Full task description appears in Oct-Dec 1951 issue.

Status: CONTINUED. The calculations were completed and transmitted to the sponsor.
54 Status of Projects

SMOOTHING AND REDUCTION OF THEODOLITE DATA

Task 1 102-53-1 106/52-45

Origins Long Range Proving Grounds, Florida

Sponsor; " " Full task description appears in Oct-Dec 1951 issue.

Status: CONTINUED. A preliminary study of the form of the

approximating polynomial and the number of points to be used is being

made

.

INTENSITIES OF SPECTRAL LINES II

Task 1 102-53-1 106/52-46

Origin; NBS , Division l4

Sponsors " Full task description appears in Oct-Dec 1951 issue.

Status: CONTINUED. Computations are being perfoined as requested.

INTER-INDUSTRY ECONOMICS

Task 1 102-53-1 106/52-47

Origins The George Washington University

Sponsors U. S. Navy

Full task description appears in Oct-Dec 1951 issue.

Status: CONTINUED. The calculations are about 60 percent completed.

KINETICS EQUATIONS FOR CONCURRENT CHEMICAL REACTIONS

Task 1102-53-1106/52-48

Origins NBS, Section 7*3

Sponsors " Managers A. Gleyzal

Authorized l/l/52

Completed 3/31/52/

Ob j ec live s To solve the

dx

dt

= V

3

equations

(3AC - x)(a - x - y)

djL = dt

KDh (a - x - y)

with x( 0 ) = y( 0 ) = 0 , and to express the solution functions

x(t, K^, Kp, A

q

, h,a),y(t, K^, K

Q

, A

q

, h, a) in terms of functions of

two variables.

Background: The kinetics of the oxidation of aldoses of chlorous

acid is complicated by the concurrent decomposition of chlorous acid
Status of Projects 55

having independent evaluation of orders and rate constants of the decomposition and oxidation reaction; a mathematical solution of the rate

equation would establish the stoichiometry of the oxidation of aldoses

with chlorous acid.

Status: COMPLETED (NEW). The results were transmitted to the

originators

.

CONSUMER PRICE INDEX SURVEY

Task 11 02-53-1 1 06/52-53

Origin: Bureau of Labor Statistics

Sponsors " " Managers J. A. Jordan

Authorized l/l/52

Completed 3/31/52

Objectives To obtain the percentage of reported expenditures, and

the amount of average expenditure, for each type of commodity, in 91 selected cities. Resulting figures to be used for revision of consumer

price index to a 1950 base.

Background: Supplied data consists of numbers of consumer units,

amount reporting, and aggregate amount reported within each item number,

for each city.

Status: COMPLETED (NEW). The results were transmitted to the

sponsor.

BETATRON SCHXFF SPECTRUM CALCULATION

Task 1 102-53-1 106/52-5^

Authorized^ l/l/52

Completed 3/31/52

, where

E (electronic energy) =

e

k = T =

€ =

1 =c2

and z Background: The quantity arises in the formula for a spectrum

in a straight forward direction. The problem was proposed by Dr. H.W. Koch.

Status: COMPLETED (NEW ) . The results were transmitted to the sponsor.

0(2)30.51 (5)50.51 mer.

,

0(2) [

E

e

]

,

2(1 -€) (Inoc -1 )+ € 2

(1 noc -i) , k_E 2

z

1/73

2E ( i - e ) 0

1 1

1

= . 51 1 mer. , = 74 .

Origin: NBS , Section 4.11

Sponsor: " Manager: I. Stegun ^2

e Objective: Computation of the quantity ^
56 Status of Projects

HEAT LOSS FROM CAVITIES

Task 1 102-53-1 106/52-56

Origin: NBS Section 10.3 Authorized 2/12/52

Sponsor: " " Manager: L. Nemerever

Objective: To obtain an analytical expression for the heat loss

from a spherical cavity.

Background: This task arose in connection with research on thermal

properties of underground cavities, carried on in the Heating and AirConditioning Section of NBS. Dr. Harold Woolley had previously derived a formula applicable to spherical cavities. The present task consisted

mainly in verifying Dr. Woolley's work.

Status: NEW. Preliminary checking of material has been done.

WAVE LENGTH OF SPECTRAL LINES

Task 1 102-53-1 106/52-59

Origin: NBS Section 5.2 Authorized 2/15/52

Sponsor: " Completed 3/31/52

Manager: I. Stegun

Objective: To fit a quadratic polynomial, by the method of least

squares, to a set of experimental data.

Background: The computation arose in connection with experimental

work of the NBS Radiometry Laboratory. The wave numbers (frequencies)

of lines in the rotational-vibrational absorption spectrum of CS 9 are expressed as a function of quantum number. This problem was requested

by E. K. Plyer and N. M. Gailar of Section 5.2, NBS.

Status: COMPLETED (NEW). The results were transmitted to the

sponsor.

MULTIPLE COMPTON SCATTERING OF LOW ENERGY GAMMA RADIATION

Task 1 102-53-1 106/52-65

Origin: Naval Research Laboratory Authorized 5/1/52

Sponsor: " " Manager: I. Stegun

Objective: To evaluate Fourier integrals for a selected set of

pa rameters

.

Background: The integrals arise in the theoretical studies being

carried out at the Naval Research Laboratory. These computations were specifically requested by Mr. O'Rourke.

Status: NEW.
57 Status of Projects

LONG PATH USABLE FREQUENCY PREDICTIONS

Task 1 102-53-1 106/52-66

Origin: NBS Section l4.4 Authorized 4/l/52

Sponsor: " Manager: M. Stein

Objective: To analyze available radio traffic and field strength

data to determine observed maximum usable frequencies over long paths.

To develop empirical methods of calculating long path maximum usable

frequencies to bring them into agreement with observations.

Background: Experience has shown that the maximum usable frequencies predicated by methods currently in use at the Central Radio

Propagation Laboratory are discrepant with and in general lower than

those actually observed. It is believed that empirical methods can be

devised to improve this situation.

Status: NEW.

ORDER-STATISTICS ESTIMATORS OF EXTREME VALUES

Task 1 102-53-1 106/52-67

Origin: NBS Section 11.3 Authorized 4/1/52

Sponsor: National Advisory Committee for Aeronautics

Manager: I. A. Stegun

Objective: 1. To express the solution w^ n ^ 0 f the

following linear system of n+2 equations explicitly in terms of the two

arbitrary quantities a and b:

EL <r (n) w.

(n) + x

j=1 ij + c . (n) ^ = 0 , i - 1 ,2,...,n.

n ( n)

E

w

j j=i

" C.

(n) w.(n)

E j j j=1

= b

Here c.

1 (n) is the first moment of the i-th order statistic (in ascending

order) in random samples of size n from the extreme-value distribution

with c.d.f. exp (-e ^) , and (T. is the covariance between the i-th

and j-th order statistics, The coefficients c^ n

), d e P en d upon

certain sums to be evaluated and upon the sample size n. The n+2 solutions will all be of the form r^

a + s^ b.
58 Status of Projects

2. To evaluate the quadratic form

V Q (w

i (n) (n) (n)

n = E

j=1

nE

i=1

(n) (n) (n)

ij

w.

(n) ( n )*

for w. = w- , i = 1,2,...,n. The result will be of the form

* 1 2

1 2

Qn = C

n

a + D

n ab + En b ’ where c

n

’ D

n

’ E

n are the numerical quantities

whose values are desired.

Calculations are to be made for at most n = 3(1)7? and accuracy

to at least 5-D is desired in the coefficients C , D , E , at least for

n

7 n

7 n

7 the smaller values of n. Background: In connection with the study of methods of analyzing

extreme-value data under task 1 1 03-21 -5l 06/51 -1 , it is desired to obtain

order-statistics estimators of the parameters of the form T = ST w.( n )x.

n i=

1

i i

which are unbiased and have minimum variance. This requires the minimiza

tion of the quadratic form Q defined above subject to linear restrictions

of the form

M ( n ) E w i = «> j=i J

n. E

j=i c.

(n) w.(n)

J J = b

The linear equations under (1) express the conditions for minimization.

Comments: See Comments under task 1 1 01 -53-1 1 01/52-38, to which

this task is a companion project. The results of both these tasks are needed for research in progress under task 1 1 03-21 -51 06/51 -1 (Research

in Application of Theory of Extreme Values to Gust Load Problems). Results for each value of n will be transmitted to originator for analysis

to determine whether to proceed with calculations for the next succeeding

value of n. Status: NEW.

ATMOSPHERIC REFRACTION

Task 1 102-53-1 106/52-68

Origin: NBS Section lh-,9 Authorized 4/1/52

Sponsor: " Completed 3/31/52

Manager: R. Brooks

Objective: To obtain the autocorrelation function of time series

consisting of measurements of the atmospheric index of refraction for

radio waves. Background: The measurements are made by the NBS Microwave Standards Section (NBS 1 *+

. 9 ) • The autocorrelation analysis is required in

studies of the scattering of microwaves in the atmosphere and has application to investigations of atmospheric turbulence. This analysis

was requested specifically.

Status: COMPLETED (NEW). The results have been transmitted

by Dr. H. E. Bussey, lA.y to the sponsor.
Status of Projects 59

GAS ADSORPTION BY HIGH POLYMERS

Task 1 102-53-1 106/52-70

Origin: Befhesda Naval Medical Center Authorized 4/1/52

Sponsor: ,s " Manager: I. A. Stegun

Objective: To obtain numerical solutions of a certain transcendental equation for a large number of values of the parameters occurring

in it.

Background: The equation under consideration occurs in theoretical

investigations of the adsorption of gases by high-polymer molecules.

This was specifically requested by Dr. T. Mill, Bethesda Naval Medical

Center.

Status; NEW.

TABLES OF THERMODYNAMIC PROPERTIES OF GASES

Task 0302-51 - 2606/49-5

( f o rae rl y 11.2/33-49-5

)

Origin: NBS , Section 3-2

Sponsor: National Advisory Committee on Aeronautics

Managers: I. A. Stegun and M. Stein

Full task description appears in Apr-Jun 1949 issue.

Status: CONTINUED. Calculations are being performed as requested.

A table for helium (No. 6.10) has been published in the NBS-NACA tables

series of thermal properties of gases.

BASIC IONOSPHERIC DATA

Task 1 401 -34-1 473A9-1 4

(formerly 11.2/33-49-14)

Origin: NBS, Section 1 4 .

3

Sponsor; " Managers; M. Stein and W. Gordon

Full task description appears in Apr-Jun 1949 issue.

Status: CONTINUED, The calculations are being performed as requested.

IONOSPHERIC WINDS

Task 1401-1 1-1401/50-7

(formerly 1 1 . 2/33-50-7)

Origin: NBS, Section 14.1 Authorized 12/1/49

Sponsor: » Completed 3/41/52

Manager: B. S. Prusch

Objective: To analyze certain records of fast-fading ionospheric

reflections of radio beams by transcribing them from the original paper
60 Status of Projects

T% tape to punched cards and performing correlation analyses by means of

punched card machines.

Background: Measurements of intensity of reflection are recorded

at three stations located at the vertices of a right triangle. As an ionospheric disturbance travels over the stations, it causes successive

fading of the three records. By cross-correlating the records with various time lags and maximizing the correlation coefficients one obtains

the components of the average velocity of travel of such disturbances.

The problem was suggested by Mr. C. D. Salsberg of the Ionospheric Research Laboratory.

Status: COMPLETED.

Ta

( f 0 Orig in : NBS

,

Sec tion 2.2

Spon sor: t! Full task description app

S ta tus

:

CONTINUED.

and numerical analyses of

RAY TRACING

sk 0202-21-2308/50-13

rmerly 11.2/33-50-13)

ears in Jan-Mar 1950 issue

Several routine problems

image errors of the third

on skew ray tracing

order were done.

COMPUTATION OF SINGLE AND

Task

DOUBLE WALL SOUND TRANSMISSION INTEGRALS

0601-31-3527/51-7

Origin: NBS

Manager: N. Levine and J. Levin

Authorized 6/25/51

Terminated 3/31/52

Objective: To compute the integral

x

= 2 ft

vdv

2

where

A^ = 1 + 5 [ R(R+ 1 ) + p^v^ ] + 5- sin^bv R( R+ 1 ) + p^ -p^v"|'

- 4pv sin 2bv [ R(R+1 )+p^vft

R is a damping parameter, b is a parameter which depends on the ratio of an airspace thickness to the wave length of sound, and p is a parameter

depending on the mass reactance of the wall and the stiffness reactance

resulting from the existence of flexural waves. Background: This integral represents the ratio of transmitted to incident energy in the case of a random sound field striking a wall con- sisting of two partitions separated by an airspace. The variable v is

the cosine of the angle of incidence.
Status of Projects 61

Comments: This problem was formulated by the Sound Section,

National Bureau of Standards. The occurrence of flexural waves in the

wall is taken into account.

Status: TERMINATED.

CONDUCTION OF HEAT INWARD FROM A PLANE SURFACE EXPOSED TO HIGH

INTENSITY RADIATION

Task 1002-1 1-4720/51 -I 1 * Origin: NBS , 10.0

Sponsor: " Managers: M. Abramowitz and W. F. Cahill

Full task description appears in Apr-Jun 1951 issue.

Status: CONTINUED. Several new cases have been run on the SEAC.
III. Statistical Engineering Laboratory

(Section 11.3)

1 . Fundamental Research in Mathematical Statistics

GLOSSARY OF STATISTICAL ENGINEERING TERMINOLOGY

Task 1 103-1 1-1 107A8-3

(formerly 1 1 .3/2-9-8-3)

Origin: Section 11.0, NBS

Manager: I. R. Savage

Authorized 1 /23/1+8

Terminated 3/31/52

Objective: To prepare a glossary of the statistical terminology

associated with acceptance sampling and process control, statistical

analysis and interpretation of experimental and test data, and statistical

design of experiments and tests.

Background: The application of statistical concepts and techniques

to acceptance sampling and process control has given rise to new terms,

and many everyday terms are used with very specific connotations. The

relatively new art of the statistical design of experiments and tests

also has a special vocabulary. Finally, the concepts, principles, and

techniques of statistical inference as applied to the analysis and

interpretation of experimental and test data have been revised and ex- panded considerably during the past two decades with consequent changes

in the meanings of terras and the introduction of new terms.

It is highly desirable, therefore, that a glossary of statistical engineering terminology be prepared to eliminate some of the

present confusion in this field and to facilitate wider understanding

of the subject.

Status: TERMINATED. Since the inception of this task, a Committee

on Educational Aids (chairman,M. G. Kendall, and manager, W. R. Buckland,

both of the University of London) of the International Statistical Institute has come into being. One of its stated purposes is the preparation of a list of statistical terms and symbols in the field of

statistical methodology; another is the preparation of a chapter on statistical terminology for inclusion in the handbook of statistics to be issued by the American Statistical Association. The continuation of

the present task as such would only rival this similar effort; therefore,

it has been decided to terminate this task and to cooperate with and

assist the above-mentioned committee as needed. As a start in this direction, at Professor Kimball's request, we are making the glossary in

its present form available to the committee and we plan to send additions

as these grow out of the work on other tasks. A number of worthwhile

additions to our glossary have been made this quarter by members of the

technical staff.

Dr. Eisenhart has continued the weekly conferences with

G. A. Bicking, Ordnance Research and Development Division, in connection

with Task Group 9 (Precision and Accuracy) of Committee E— 1 1 of the

American Society for Testing Materials. Continued cooperation in this

regard should in no way be affected by the termination of this present

task.

62
Status of Projects 63

BIBLIOGRAPHY AND GUIDE TO STATISTICAL LITERATURE

Task 1 103-1 1 -1 107A9-1 a (formerly 11.3/2-49-1)

Origin: NBS

Full task description appears in Jan-Mar 1949 issue.

Status: CONTINUED. The present file of abstract cards consists of

all abstracts in Mathematical Reviews (1940 through 1950) under the two

general headings "Theory of Probability" and "Mathematical Statistics".

Classification by a coded subject index of these abstracts has been com- pleted. The emphasis this quarter has been on extending the bibliography

to include selected abstracts of statistical significance from other

fields, such as actuarial mathematics, astronomy, biological problems,

combinatorial analysis, economics, theory of errors, ergodic theory,

games, design of experiments, moments, and numerical and graphical methods.

Back issues of Mathematical Reviews have been reviewed again to make these

further selections. The work of clipping, pasting onto cards, and classifying these additional abstracts continues. Thus, except for this re- latively small back-log of abstracts, the preparation of the bibliography

is now on a current basis. Decisions about the final mechanical handling

of the abstracts and the manner in which information from them shall be

made available have yet to be made. This card index has proved useful

in connection with the work on non-pa rametric statistics being carried

out under task 1 1 03-1 1 -1 1 07/52-2.

ELEMENTARY THEORY OF STOCHASTIC PROCESSES

Task 1103-11-1107/49-3

Origin: NBS

Full task description appears in July-Sept 1950 issue.

Status: CONTINUED. The manuscript of H. B. Mann has been accepted

for publication and is in press at the U. S. Government Printing Office.

Publication: "Introduction to the theory of stochastic processes

depending on a continuous parameter," by H. B. Mann; to appear as NBS

Applied Mathematics Series 24.

ESTIMATION OF LOCATION AND SCALE PARAMETERS

Task 1103-11-1107/50-1

(formerly 11.3/1-50-1)

Origin: NBS Authorized 3/1/50

Manager: Churchill Eisenhart Terminated 3/31/52

Objective: To compare statistical properties of alternative es- timators of the location and scale parameters of particular probability

distributions, from the viewpoints of (1) practical applications and (2)

clarification of the aims and principles of statistical estimation.

Background: Statistical estimation is a field of research in which

there is considerable activity at the present time. One line of research

is directed toward determining the "best" estimators, that is, toward the

development of principles and techniques for determining such estimators
6/ Status of Projects

when they exist; and another line is directed toward the development of

easier-to-compute estimators that sacrifice as little as possible of the

desirable properties of the "best". Even in the definition of "best"

there are some difficulties - thus, for purposes of combination of es- timates from several independent samples, unbiased estimators are desired;

whereas, for use in single instances, there is much to be said for closes t estimators; but unbiasedness and closeness are not always compatible, e.g.,

in estimating the standard deviation of a normal distribution. Much

of the work to date along these lines has concerned itself with the determination of asymptotically "best" estimators, which have certain optimum

properties for infinitely large samples. The present project, on the

other hand, aims to concentrate primarily on estimators of value in the case of finite samples, particularly "small" samples of ten values or less,

such as are frequently met with in physical science and engineering

measurement.

Comments: This task, conceived as a continuing study, unifies

and broadens the scope of work initiated under projects 11.3/1 -^7-1?

1 1 . 3/1 -*+7-2 , and 11.3/1-5-9-2. The first of these was completed in March

19/9 (for a summary of results obtained, see Projects and Publications

Jan-Mar 19/9)* The other two are now terminated.

The numerical results of project 11. 2/33—^+9—1 8 (full project

description appears in Ap r- June 195-9 issue) bear on the study of the

arithmetic mean as an estimator of location being continued under the

present task.

Likewise, the numerical results on moments of order statistics

in samples from a normal distribution evaluated under project 11.1/32-50-5

(full project description appears in Oct-Dec 195-9 issue) promise to be

of value in connection with the present task.

Status: TERMINATED. Hereafter work of the type summarized below,

and in recent status reports under this task, will be reported under

task 1 1 03-1 1 -1 1 07/51 -2, Miscellaneous Studies in Probability and Statistics . C. Eisenhart and L. S. Deming resumed work on the table of

probability points of order statistics in small samples previously described under this task (see Jan-Mar 1950 issue, p. 5-0). For all of

the probability levels concerned, the table has now been completed and

checked from n = 2 to n = 6; and it has been about 75 percent completed

from n = 7 to n = 10. A substantial portion of the entries for the .001

level came from L. Joel, as a by-product of SEAC computation relating to task 1 304-3Z-6351 /5l -8. The table is being extended to n - 20, and

the entries for n - 11 to n - 20 are about 50 percent completed for the

probability levels from .005 to .995 inclusive.

In response to a request from industry, E. P. King is preparing

a memorandum on the efficiency of the average range as an estimate of

the standard deviation of a normal population which partitions the total

loss of efficiency into two portions; one is attributable to the grouping

into sub-samples, and the other is attributable to the use of a linear

estimate rather than a root mean square estimate.
Status of Projects

65

EXTREME-VALUE THEORY AND APPLICATIONS

Task 1 103-1 1-1107/50-1a

(formerly 11 <,3/2-50-1)

Origin: NBS

Full task description appears in Jan-Mar 1950 issue.

Status: CONTINUED. The review of the draft of the monograph by

E. J. Gumbel by 0 technical reader continued.

MANUAL ON FITTING STRAIGHT LINES

Task 1 1 03-1 1-11 07/50—

2

(formerly 11 . 3/2-50- 2 ) Origin: NBS

Full task description appears in Jan-Mar 1950 issue.

Status: CONTINUED. Several aspects of the two-variables-in-error

problems were worked on by F. S. Acton in Princeton in conjunction with

J. Tukey, as were also several of the confidence region problems that

arise in these chapters. Illustrative examples were collected both in

Washington and Princeton.

TABLE TO FACILITATE DRAWING RANDOM SAMPLES

Task 1103-11-1107/51-1

Origin: NBS

Full task description appears in July-Sept 1950 issue.

Status: INACTIVE. For status to date see July-Sept 1951 issue.

MISCELLANEOUS STUDIES IN PROBABILITY AND STATISTICS

Task 1103-11-1107/51-2

Origin: NBS

Full task description appears in July-Sept 1950 issue.

Status: CONTINUED. Cl) Two short working papers were prepared by

I. R. Savage on the limits of ES/'o" and on a problem in evaluating

definite integrals using various Monte Carlo methods. (2 ) 0. Szasz and

E. Lukacs continued their study of non-negative trigonometric polynomials

and are preparing a second paper on the subject. (For details see SeptDec 1951 issue, p. 59) C3) 0. Szasz and E. Lukacs studied properties of

characteristic functions which are analytic in a neighborhood of the

origin.

Publications: Cl) "On the derivation and accuracy of certain

formulas for sample sizes and operating characteristics of non-seqnential

sampling procedures," by U. Cfaaod; NBS J. Res. 42, 491-502(Dec 1951).

C2) "Properties of statistics involving the closest pair in a sample of

three observations," by J. Lieblein; accepted by the NBS Journal of Research. (3) "An essential property of the Fourier transforms of distribu-
66 Status of Projects

tion functions," by E. Lukacs; accepted for publication , in the Proceedings of the American Mathematical Society. (4) "The stochastic independence

of symmetric and homogeneous linear and quadratic statistics," by

E, Lukacs; submitted to a technical journal. (5) "Some non-negative trigonometric polynomials connected with a problem in probability," by

E. Lukacs and 0. Szasz; accepted by the MBS Journal of Research.

LAW OF PROPAGATION OF ERROR

Task 1103-11-1107/52-1

Origin; NBS

Full task description appears in Joly-Sept 1951 issue.

Status: INACTIVE. For status to date see July-Sept 1951 issue.

PROCEDURES OF NON-PARAMETRIC STATISTICS

Task 1 103-1 1-1 107/52-2

Origin: NBS

Full task description appears in July-Sept 1951 issue.

Status: CONTINUED. (1 ) The bibliography of non-parametric methods

is nearing completion; a final check for accuracy will be made before its

preparation as a report. (2) The final draft of the reports on Tchebycheff

inequalities has been prepared. (3) A working paper on the asymptotic

power of non-parametric procedures was prepared. C*+) A working paper

on a useful transformation of multivariate data was prepared. (5) Preliminary work on a paper showing the independence of tests of randomness and

on other tests of hypotheses was begun.

2. Applied Research in Mathematical Statistics

COLLABORATION ON STATISTICAL ASPECTS OF

NBS RESEARCH AND TESTING

Task 3011-60-0002/51-1

(formerly 3000-21-0002/51-1 ) Origin: NBS

Full task description appears in July-Sept 1950 issue.

Status: CONTINUED. Activity under this project fell into two main

categories

:

A. Design of Experiments: A quintuple lattice with 25 treatments,

separable into 5 replications, was constructed for comparing 25 enameled

metals in 25 tests pf 5 specimens each. A position effect in the test was eliminated by running a complete replication at each test position.

B. Development or Selection of the Appropriate Methods for

Analysis and Interpretation of Data: A method was devised for testing
Status of Projects 67

whether two curves of unknown functional form have the same functional

form.

Publications: (l ) "Tables for constructing and for computing the

operating characteristics of single sampling plans," by J. M. Cameron;

accepted by Industrial Quality Control. (2) "The operating characteristic

of the control chart for sample means," by E. P. King; submitted to a technical journal. (3) "Confidence and tolerance intervals for the normal

distribution," by F. Proschan; submitted to a technical journal. (4) "The use of random numbers," by F. Proschan; accepted by Industrial Quality

Control. (5) "Control charts may be all right - but ..., by F. Proschan;

accepted by Industrial Quality Control. (6 ) "Statistics and planning

tests at elevated temperatures," by W. J. Youden; accepted by Experimental

Stress Analysis. (7) "The interpretation of chemical data," by W.J. Youden;

accepted by Industrial Quality Control. (8) "Statistical units of measurements," by W. J. Youden; submitted to a technical journal. (9) "Experimental statistics - a review," by W. J. Youden and R. J. Hader; Anal.

Chem. 24, 120-124 (Jan 1952).

STATISTICAL ASPECTS OF NBS ADMINISTRATIVE OPERATIONS

Task 3011-60-0002/52-1

Origin: NBS

Full task description appears in Oet-Dec 1951 issue.

Status: CONTINUED. A study of sick and annual leave practices

of NBS employees for the calendar year 1950 was completed. At the re- quest of the Personnel Division, a quick estimate of the percentage of

NBS employees in a particular age category was obtained from a small

sample of employee records to serve as a basis for determining the

feasibility of a periodic health examination of employees in this group.

A memorandum was submitted to the Chief of the Administrative Services

Division on a combinatorial problem arising in connection with the preparation of keys for tumbler locks.

RESEARCH ON APPLICATION OF THEORY OF EXTREME

VALUES TO GUST LOAD PROBLEMS

Task 1 103-21-5106/51-1

Origin: NACA, Dynamic Loads Division

Full task description appears in July-Sept 1950 issue.

Status: CONTINUED. Work during the present quarter has been marked

by a discovery that radically alters the basic computation techniques

for this task. The new method involves a feasible means of determining

the variance of extremal estimators based on order statistics for samples

of any size. With this tool it is expected to obtain unbiased estimators

of good efficiency, and the scope of the results can thereby be greatly

extended without much, if any, increase in cost over what had been planned.

These matters were outlined in an administrative progress report submitted

to the National Advisory Committee for Aeronautics (NBS Report 1521, March

1 4, 1952).

A meeting with NACA representatives was held on March 20, 1952,

at which time plans for completion of the work were taken up. Afterwards,

two computation tasks were initiated: (a) 1 102-53-1 106/52-67? (see p.57)
68 Status of Projects

for hand computation of "minimunii l! -va fiance order-statistics estimators

for small samples; and (b) 1 1 01 -53-1 1 01/52-38, (see p.3i-)'£or bias and

variance of an estimator (based on sample moments) for several moderate

sample sizes.

At the close of the quarter, a technical article was in preparation presenting the mathematical basis of the new method.

RESEARCH IN APPLICATIONS OF MATHEMATICAL STATISTICS TO

PROBLEMS OF THE CHEMICAL CORPS

Task 1 103-21-51 1 8/52-1

Origin: Biological Laboratories, Chemical Corps, Dept, of the Army

Sponsor: " " " Full task description appears in Oct-Oec 1951 issue.

Status: CONTINUED. An operations analysis approach to extensive

bivariate data led to the development of a simple performance criterion.

A certain set of data has been resolved into its principal components

by the fitting of decay type curves. Estimates of error have been obtained

which may be used in evaluating the fitted curves. Special analysis of

covariance techniques are being devised to determine the effects of

several treatments on some bivariate observational data with an inherently

large experimental error.

APPLICATION OF THE THEORY OF STOCHASTIC PROCESSES TO

THE STUDY OF TRAJECTORIES

Task 1103-21-5119/52-1

Origin: U. S. Naval Ordnance Test Station, Inyokern Authorized l/l/52

Sponsor; " " " Manager: E. Lukacs

Objective: To determine whether mathematical-statistical tools

associated with the theory of stochastic processes can be profitably

applied to the analysis of trajectory data of the type gathered at the

Naval Ordnance Test Station, Inyokern.

Background: The theory of stochastic processes deals with timedependent phenomena in which there is a probability relationship between

a state at a given instant and one or more states at preceding instants.

Some of the difficulties encountered in analyzing ordnance data can be overcome by considering a trajectory to be a stochastic process, thus

eliminating the difficulties created by the fact that only one observation is available for the position of the missile at each instant.
Status of Projects 69

Status: NEW The first step in this work was the adoption of a stochastic

model. As always in such a situation one has to find a compromise between

the wish to establish a realistic model and the necessity to use a model

which does not lead to a forbiddingly complicated mathematical analysis.

A fundamental random process is the Wiener process, and methods for the

estimation of its parameters have been developed under a previous Bureau

task (Elementary Theory of Stochastic Processes, Task 1 1 03-11 -1 1 07A-9-3) • This is the main reason why it seems desirable to attempt to use in this

study the Wie.ner process as the stochastic model. Clearly, this involves

a certain amount of idealization. It is therefore desirable to safeguard

against too radical departure from reality by testing first whether the

observed trajectory data can be assumed to come from a Wiener random process.

Since no such test is known at present the first problem under

this tas\ is the development of a test for the hypothesis that a sequence of observations comes from a Wiener process.

The following theorem was derived in this connection mainly

by using a result of Khinchine:

Let y(t) be a stochastic process and assume that

(i) y(t) is a process with independent increments.

(ii) y(t) is strongly continuous in the interval [a,b].

Then y(b) - y(a) is normally distributed.

This theorem shows that the normality of the increments follows

from a certain continuity property of the process. In view of the physical

situation encountered in the study of trajectories a continuity assumption

is certainly reasonable. Therefore the test to be developed should not

emphasize the normality of the increments; the independence of the increments seems to be the property deserving greater attention.

A technical report containing the proof of this theorem was completed and will be issued at the beginning of the next quarter.

E. Lukacs visited Professor H. B. Mann in February 1952 and

discussed with him various aspects of this task.

STATISTICAL SERVICES FOR COMMITTEE ON SHIP STEEL, NRC

Task 1 103-53-5105/52-1

Origin: Ship Structure Committee, NRC

Sponsor: " " Full task description appears in Oct-Dec 1952 issue.

Status: INACTIVE. For status to date see Oct-Dec 1952 issue.
IV. Machine Development Laboratory

(Section 11.4)

in cooperation with Electronic Computer Section

(Section 12.3)

1 . Development: Design and Construction of Automatic Digital

Computing Machines

THE BUREAU OF THE CENSUS COMPUTING MACHINE

Task 1 1 04-34-5107/47-1

(formerly 11.4/21-4-7-1)

Origin: The Bureau of the Census

Sponsor: " " Full task description appears in Apr-Jun 1949 issue.

Status: CONTINUED. At the end of the quarter, the first UNIVAC

System had been in operation for a full year. This equipment has remained

in the Philadelphia plant of the Eckert-Mauchly Division of Remington Rand

Inc., and has been maintained by the company under contract with the

National Bureau of Standards. The Bureau of the Census has employed the

eouipment predominantly for the tabulation of a portion of the 17 th decennial census. The operating experience obtained during this period has

been reported in considerable detail in a recent paper, "Review of electronic digital computers by J. L. McPherson (Bureau of the Census)

and S. N. Alexander (National Bureau of Standards), presented at the

Joint AIEE-IRE Computer Conference, February 1952.

Alterations to improve the reliability of auxiliary equipment

are being provided on the second and third UNIVAC Systems and will be

incorporated into the Bureau of the Census UNIVAC during the coming

quarter. The present plans are to leave the installation in Philadelphia

during most of 1952, with the Bureau of the Census assuming complete

responsibility for its operation and maintenance.

THE NAVY COMPUTING MACHINE

Task 1 104-34-51 07/47-2

(formerly 11.4/22-47-2)

Origin: Mathematical Sciences Division, ONR

Sponsor: " " " Full task description appears in Apr-Jun 1949 issue.

Status: CONTINUED. The assembly of the Hurricane Computer, on whose design the ONR-NBS machine is to be patterned, has been completed,

and the engineering tests and modifications have been in progress during

the entire past quarter. Certain components of the ONR-NBS machine, which

70
Status of Projects 71

had been authorized earlier, have been completed. However, no additional

work will be authorized until the Hurricane Computer has passed its acceptance test and a satisfactory understanding is reached with the company

regarding both the price and the delivery of the ONR-NBS machine.

THE AIR COMPTROLLER'S COMPUTING MACHINE

Task 1104-34-510 7/47-3

< (formerly 11.4-/24—47-3)

Origin: Office of the Air Comptroller, USAF

Sponsor: " " Full task description appears in Apr-Jun 195-9 issue.

Status: CONTINUED. During February 1952, the fourth part of the

revised acceptance test procedure was passed, and the UNIVAC System was accepted for delivery to Washington. This was the second attempt to pass

the more rigid requirements that were put on the ability of the central

computer to communicate with the magnetic input-output system. In order

to meet these requirements, the company found it necessary to introduce

several design modifications in the Uniservos and the input-output circuitry. These same modifications will be used in System No. 3, and appropriate modifications of System No. 1 will be made in the near future.

The National Bureau of Standards contracted for the transfer

and installation of the UNIVAC System in the Pentagon Building. During

the past quarter, the power and control wiring, the power supply, and

the central computer has been installed at the site. It is anticipated

that all elements of the system will be delivered and put into operation

during the coming quarter.

NATIONAL BUREAU OF STANDARDS EASTERN AUTOMATIC COMPUTER (SEAC)

Task 1 1 04-34-51 07A9-1

(formerly 11.4-/24-5-9-1)

Origin: NBS

Sponsor: Air Comptroller's Office, USAF

Full task description appears in Apr-Jun 194-9 issue.

Status: CONTINUED. During this quarter, the over-all operating

reliability of SEAC was 79 percent for a 1 68-hour week which includes

8 hours of scheduled maintenance. Reliability has been somewhat affected

by the recent change to temporary a-c power lines. Since SEAC has thus

far operated off unregulated power supplies, a-c stabilizers have been

ordered to correct the situation.

Additional circuit refinements have been made in the Williams

electrostatic memory, and its use in problem solution has increased

steadily during the past quarter. The additional circuitry for the 3-

address system has been installed but will not be put into service until

the next quarter. Changes and additions to the control circuitry as- sociated with the input-output equipment have also been installed so that the Raytheon tape unit can be utilized along with the other magnetic

tape units.
72 Status of Projects

WRIGHT DEVELOPMENT CENTER COMPUTING MACHINE

(formerly AIR MATERIEL COMMAND COMPUTING MACHINE)

Task 1 I01f-3I+-5107A9-1 a (formerly 1 1 . 4/23-49-1 ) Origin 1 Flight Research Laboratory, Wright Air Development Center, USAF

Sponsor: " " " " " Full task description appears in Apr-Jun 1 9^+9 issue.

Status: CONTINUED. The assembly of the computer is completed ex- cept for the magnetic drum memory. The completed equipment is undergoing

engineering tests, and individual operations are being initiated with

the aid of a small test drum. The Inscriber and Outscriber equipment are

now assembled , and tests on them have been started. The complete assembly

and the testing of the entire system will be delayed until next quarter

because of mechanical difficulties with the magnetic drum for the main

memory. The drum was coated and assembled for trial operation. However,

at operating speed, the eccentricity was found to be excessive, and the

drum has been returned for adjustment. Present plans are to have all

equipment tested aid proved in so that, when the drum is installed, com- plete system tests can be started immediately.

ARMY MAP SERVICE COMPUTING MACHINE

Task 1 104-34-51 07A9-1 b (formerly 1 1 .4/25-49-1 ) Origin: Army Map Service, USA

Sponsor: " " Full task description appears in Apr-Jun 1 9^9 issue.

Status: CONTINUED. During the past quarter, UNIVAC System No. 3 was completely assembled and has been undergoing engineering tests. The

tests on several of the auxiliaries has been completed and accepted.

The acceptance test for the central computer and the input-output system

has been modified slightly in the light of experience gained in the ac- ceptance tests for the first two UNIVAC Systems. The final tests will

require the UNIVAC to utilize a complete set of ten Uniservos. Since it

is anticipated that the acceptance test will be concluded early in the

coming quarter, arrangements for transferring the system to its site in

the Army Map Service Building have been made.

CODING RELATED TO THE RAYTHEON COMPUTER

Task 1 104-53-51 08A9-3

(formerly 11.4/3-49-3)

Origin: Mathematical Sciences Division, ONR Authorized 12/1/48

Sponsor: Office of Naval Research, USN Terminated 3/31/52

Objective: To evaluate the mathematical sufficiency of the proposed

Raytheon electronic computer.

Background: The Raytheon Company has submitted to the Bureau a series of design modifications intended to increase the flexibility and

power of the proposed computer. By coding basic mathematical routines for
Status of Projects 73

solution on the Raytheon computer, in its various stages of design, the

Bureau has evaluated successive designs.

Status: TERMINATED.

INVESTIGATION OF THE APPLICABILITY OF AUTOMATIC DIGITAL

ELECTRONIC COMPUTING TO PROBLEMS OF THE SOCIAL SECURITY

AGENCY

Task 1 1 04-53-51 08/51 -1

Origin: Social Security Agency

Sponsor: " " Full task description appears in Oct-Dec 1950 issue.

Status: INACTIVE. For status to date see July-Sept 1951 issue.
Lectures and Symposia

Numerical Analysis Colloquium Series

(Los Angeles, Calif o rnia

)

CURTISS, J. H. Some chain functions which are useful in the Monte Carlo

method. January 21, 1952.

FRANKEL , S. P. (California Institute of Technology). Accuracy and

stability considerations in partial difference equations. February 1

8

, 1952.

SCHOENBERG, I. J. (University of Pennsylvania, University of California,

and NBS). Interpolation by spline curves. March 1 7 , 1952.

Statistical Engineering Seminars

ACTON, F. S. Analysis of straight line data. January 1 6 , 1952.

EISENHART, C. On the use of randomization in experimentation. February

6, 1952.

LIEBLEIN, J. Summary of some contributed papers given at the Boston

meeting of the Institute of Mathematical Statistics. January

1

1 , 1952.

MANDEL, J. (NBS Section 7.5) Fitting a straight line to cumulative data.

March 7, 1952.

SAVAGE, I. R. On some portions of the proceedings of the Second Berkeley

Symposium on Mathematical Statistics and Probability. January

25, 1952.

Talks Presented by Members of the Staff

At Statistical Seminars

For NBS Division 13 and Section 12.2

EISENHART, C. The evaluation of the effectiveness of material - statistical aspects. January 31,1952.

KING, E. P. (1) Use of the t-test. March 6, 1952. (2) Use of randomized

blocks. March 20, 1952. (3) Concepts of interactions. March

27, 1952.

PROSCHAN, F. (1) Confidence limits for the population proportion. February 21, 1952. (2) Comparison of two percentages. February

28, 1952. (3) Analysis of variance. March 13? 1952.

YOUDEN, W. J. Statistics at work. February 7, 1952.

75-
Lectures and Symposia 75

Papers and Invited Talks

Presented by Members of the Staff

at Outside Organizations

EISENHART, C. The reliability of measured values —fundamental concepts.

Presented to the meeting of the American Society of Pho tog rammetry

Washington, D. G. , January 10, 1952.

FORSYTHE, G. E. Automatic digital computations. Presented to the Los

Angeles Chapter of the American Statistical Association, Los

Angeles, Cal., January 24, 1952.

HESTENES , M. R. (1) Iterative method for solving linear systems. Presented to Laboratory Personnel of the Naval Ordnance Test Station,

Inyokern, China Lake, Cal., January 30, 1952. (2) Minimax problems on calculus of variations. Presented to the Peripatetic

Seminar at the University of California at Los Angeles, February

4, 1952.

HUSKEY, H. D. (1) Mathematical services available at the Institute for

Numerical Analysis. Presented at the U. S. Navy Electronics

Laboratory, San Diego, California, March 7? 1952. (2) Recent

developments in computer design. Presented at the Industrial

Engineering Conference sponsored by the University of California

at Los Angeles and the American Society of Mechanical Engineering,

Los Angeles, March 26, 1952.

LANCZGS, C. Radiation of cylindrical antenna. Presented at the Physics

Seminar, University of California at Los Angeles.

LEHMER, Dc H. Computing machine development and research in pure mathematics. Presented to the Southern California Section of the

Mathematical Association of America Meeting, Occidental College,

Los Angeles, March 8, 1952.

SCHOENBERG, I. J. On the shape of curves. Presented to the Southern

California section of the Mathematical Association of America

Meeting, Occidental College, Los Angeles, March 8, 1952.

TODD, J. The development of numerical analysis for electronic digital

computers. Presented to the meeting of the Columbia Mathematics

Club, Washington, D. C., , January 7? 1952.

WASOW, W. Approximation of elliptic differential equations by difference

equations with positive coefficients. Presented to the Peripatetic Seminar at the University of California at Los Angeles,

March 3j 1952.

YOUDEN, W. J. (1) Statistics at work. Presented to the Parkersburg,

W. Va. Section of the American Society for Quality Control,

January 2, 1952. (2) The control and measurement of experimental

error. Presented to the personnel of the Naval Powder Factory,

Indian Head, Md. , January 28, 1952. (3) Statistics at work.

Presented to the personnel of the Naval Powder Factory, Indian

Head, Md,, February 7, 1952. (4) —(12) Interpretation of

chemical data. Presented to the following section meetings of

the American Chemical Society; Warren, Pa., February 14; Youngstown, Ohio, February 1 5 5 Columbus, Ohio, February 1o; Painesville,

Ohio, February 19; Cleveland, Ohio, February 20; Erie, Pa.,

February 21. Also presented to the staff of the Diamond Alkali

Co., Painesville, Ohio, February 19* Presented to the research
7 6 Lectures and Symposia

personnel of the Celanese Corporation of America, Bishop, Tex.,

February 25; and to the Border Section of the American Chemical

Society, Corpus Christi, Tex., February 26. (13) Statistical

units of measurement. Presented to the Philadelphia Section

of the American Society for Metals, March 4, 1952. (1*0 Principles of experimental design. Presented to the Regional Conference of the American Society for Quality Control, New York, N. Y.

,

March 28-29, 1952.
Publication Activities

1 . PUBLICATIONS WHICH APPEARED DURING THE QUARTER

1.2 Manuals, Bibliographies, Indices

(1)

A guide to tables on punched cards. G. Blanch and E. C. Yowell.

MTAC V, 185-212 (Oct. 1951). Reprints available.

1 .3 Technical Papers

(1) Fifth order aberration in an optical system. R. K. Anderson.

Proceedings. I.B.M. Computation Seminar. 130-131 (Aug. 1951).

(Published by I.B.M. Corp., 590 Madison Ave. , New York, 1951).

(2) On subha rmonic , harmonic, and linear functions of two variables.

E. F. Beckenbach. Revista de Matematica y Fisica Teorica (Argentina) 8, 7-13 (Nov. 1951).

(3) On the derivation and accuracy of certain formulas for sample

sizes and operating characteristics of non-sequential sampling

procedures. U. Chand. NBS J. Res. 4]7, 491-501 (Dec. 1951)?

RP2277- Available from Superintendent of Documents, U. S. Government Printing Office, Washington 25, D, C. , 10 cents.

(4) On some functionals of Laplacian processes. R. Fortet. NBS J. Res. 48, 32-39 (Jan. 1952); RP2280. Available from Superintendent

of Documents, U. S. Government Printing Office, Washington 25?

D. C . , 10 cents.

(5) On the estimation of an eigenvalue by an additive functional

of a stochastic process with special reference to the Kac-Donsker

method. R. Fortet. NBS J. Res. 48, 68-75 (Jan. 1952); RP2286.

Available from Superintendent of Documents, U. S. Government

Printing Office, Washington 25? D. C. , 10 cents.

(6) Random determinants. R. Fortet. NBS J. Res. 42? 465-470 (Dec.

1951); RP2274. Available from Superintendent of Documents, U. S. Government Printing Office, Washington 25? D. C. ? 10 cents.

(7) Applications of the theory of quadratic forms in Hilbert space

to the calculus of variations. M. R. Hestenes. Pac. J. Math.

I, 525-581 (Dec. 1951). Reprints available.

(8) The solution of Ax = ^ Bx. M. R. Hestenes and W. Karush. NBS

J. Res. 42, 471-478 (Dec. 1951); RP2275* Available from Superintendent of Documents, U. S. Government Printing Office, Washington 25? D. €. , 10 cents.

(9) Semi-automatic instruction on the National Bureau of Standards

Western Automatic Computer. H. D. Huskey. Proceedings of a Second Symposium on Large-Scale Digital Computing Machinery,

Annals of the Harvard Computation Laboratory xxvi, 83-90 (Dec.

1951). No reprints.

(10) Determination of the extreme values of the spectrum of a bounded

self-adjoint operator. W. Karush. Proc. Aai. Math. Soc. 2, 980-

989 (Dec. 1951). Reprints available.

77
78 Publication Activities

(11) On the variation of the determinant of a positive definite matrix

.

A. M. Ostrowski and 0. Taussky. Proceedings Royal Netherlands

Academy of Sciences at Amsterdam Series A, 54, No. 5? and Indag.

Math. Ij,, No. 5, 383-385 (1951 ). Reprints available.

(12) Uniformly best constant risk and minimax point estimates.

ft. T. Peterson, jr. NBS J. Res. 48, 49-53 (Jan. 1952); RP2282.

Available from Superintendent of Documents, U. S. Government

Printing Office, Washington 25, D. C. , 10 cents.

(13) Formulas for finding the argument for which a function has a given

derivative. H. E. Salzer. MTAC V, 213-215 (Oct. 1951). Reprints available.

( 1 4) Some general theorems in iterants. P. Stein. NBS J. Res. 48,

82-83 (Jan. 1952); RP2288. Available from Superintendent of

Documents, U. S. Government Printing Office, Washington 25, D. C .

,

5 cents.

(1 5) A note on the bounds of multiple characteristic roots of a matrix.

P. Stein. NBS J. Res. 48, 59-60 (Jan. 1952); RP2284. Available

from Superintendent of Documents, U. S. Government Printing Office,

Washington 25, D. C. , 5 cents.

(16) The convergence of Seidel iterants of nearly symmetric matrices.

P. Stein. MTAC V, 236-240 (Oct. 1 9 5 1 ) « Reprints available.

1 .4 Reviews

(1), Experimental statistics —a review. R. J. Hader and W. J. Youden.

Anal. Chem. 24, 120-124 (Jan. 1952). Reprints available.

1.5 Miscellaneous Publications

(1) Gauss to Ceding on relaxation. G. E. Forsythe. MTAC V, 255-258

(Oct. 1951)- Reprints available.

(2) New frontiers in business management control are being established

by electronic computers. H. D. Huskey and V. R. Huskey. Journal

of Accountancy 69 -75 (Jan. 1952). Reprints available.

2. MANUSCRIPTS IN THE PROCESS OF PUBLICATION March 31, 1952.

2.1 Mathematical Tables

(1) Tables of the Chebyshev polynomials S (x) and C (x) . NBS April n plied Mathematics Series 9* In press, Government Printing

Office.

(2) Table for the analysis ©f /3 spectra. NBS Applied Mathematics

Series 13- In press, Government Printing Office.

(3) Tables of Coulomb wave functions, Vol . I. NBS Applied Mathematics

Series 17- In press, Government Printing Office.

(4) Tables for rocket and comet orbits. S. Herrick. NBS Applied

Mathematics Series 20. In press, Government Printing Office.

(5) Probability tables for analysis of extreme-value data. NBS Applied Mathematics Series 22. In press, Government Printing

Office.
Publication Activities 79

(6) Tables of normal probability functions, NBS Applied Mathematics Series 23. Supercedes NBS Mathematical Table MTl4,

Tables of probability functions, volume II.) In press, Government Printing Office.

(7) Tables of Bessel functions Y- (x) , Y (x) , Kn

(x) , K (x). O^x^l.

NBS Applied Mathematics Series 25. (Reissue of AMS1 . ) In

press, Government Printing Office.

(8) Table of Arctan x. (Formerly NBS Mathematical Tables MT16.)

To be issued in the Applied Mathematics Series.

(9) Collected tables of the Computation Laboratory: Volume I. Tables of functions and of zeros of functions. To be issued

in the NBS Applied Mathematics Series.

(10) Tables of 10x. Submitted for publication.

(11) The third Wilson prime and an extended table of Wilson quotients.

K. Goldberg. Submitted to a technical journal.

(12) Table of zeros and weight factors of the first twenty Hermite

polynomials. H. E. Salzer, R. Zucker, and R. Capuano. Accepted

for publication in the NBS Journal of Research.

2.2 Manuals, Bibliographies, Indices

(1) A guide to the tables of the normal probability integral. NBS

Applied Mathematics Series 21. In press, Government Printing

Office.

(2) Introduction to the theory of stochastic processes depending

on a continuous parameter. H. B. Mann. NBS Applied Mathematics

Series 2k. In press, Government Printing Office.

(3) The hypergeometric and Legendre functions with applications to integral equations of potential theory. C. Snow. NBS Applied

Mathematics Series 19. (Supersedes NBS MTl5> which is out

of print.) In press, Government Printing Office.

2.3 Technical Papers

(1

)

Analyzing straight line data. F. S. Acton. Submitted to a technical journal for publication,

(2) The relaxation method for linear inequalities. S. Agmon.

Submitted to a technical journal for publication.

(3) The expansion theorem for pseudo-analytic functions. S. Agmon

and L. Bers. Submitted to a technical journal.

(4) On subordination in complex theory. E. F. Beckenbach and

E. W. Graham . To appear in "The construction and applications

of conformal maps: Proceedings of a symposium," to be published

by the National Bureau of Standards.

(5) On mildly nonlinear partial difference equations of elliptic

type. L. Bers. Submitted to a technical journal.

(6) On the numerical solution of parabolic partial differential

equations. G. Blanch. Accepted for publication in the NBS

Journal of Research.
Publication Activities

(7) Two existence theorems for systems of linear inequalities.

L. M. Blumen thal . Submitted to a technical journal.

(3) Tables for constructing and computing the operating characteristics of single sampling plans. J . M. Cameron. Accepted for

publication in Industrial Quality Control.

(9)

Contribution to the theory of Markov chains. Kai Lai Chung.

Accepted for publication in the NBS Journal of Research.

(10) On the recursion formula and on some Tauberian theorems. N.G.

de Bruijn and P. Erdos. Accepted by the NBS Journal of Research.

(11) Alternative derivations of Fox's escalator formulas for latent

roots. G. E. Forsythe. Submitted to a technical journal.

(12) An extension of Gauss' transformation for improving the condition of systems of linear transformations. G. E. Forsythe

and T. S. Motzkin. Accepted by Mathematical Tables and Other

Aids to Computation.

(13) Additive functionals of a Markoff process. R. Fortet. Submitted to a technical journal.

(1M-) Practical solution of linear equations and inversion of

matrices. L. Fox. Accepted for publication in the NBS Journal

of Research.

(15) Metric methods in integral and differential geometry.

J. W. Gaddum. Accepted for publication by the American

Journal of Mathematics.

(

6 ) The sums of the dihedral and trihedral angles in a tetrahedron.

J. W. Gaddum. Submitted to a technical journal.

(17) A theorem on convex cones, with applications to linear inequalities. J. W. Gaddum. Submitted to a technical journal.

(id) A 2-basic set of density zero. K. Goldberg. Submitted to a technical journal.

(19) Bergman's Integraloperator erster Art und Riemannsche Funktion. P. Henrici. Submitted to a technical journal.

(20) On the variation of the spectrum of a normal matrix. A. J. Hoffman and H. W. Wielandt. Submitted to a technical journal.

(21) Distribution of electrical conduction currents in the vicinity

of thunderstorms. R. E. Holzer and D. S. Saxon. Accepted

for publication in the Journal of Geophysical Research.

(22) On integration of parabolic equations by difference methods.

I: Linear and quasi-linear equations for the infinite interval.

F. John. Accepted for publication in Communications on Pure

and Applied Mathematics.

(23) Convergence of a method of solving linear problems. W. Ka rush

.

Submitted to a technical journal.

{2b) The operating characteristic of the control chart for sample

means. E. P. King. Submitted to a technical journal.
Publication Activities 81 (25) Solution of systems of linear equations by minimized iterations. C. Lanczos. Accepted for publication in the NBS

Journal of Research.

(26) On certain character matrices. D. H. Lehmer. Submitted to a technical journal.

(2.7) Properties of statistics involving the closest pair in a sample of three observations. J. Lieblein. Accepted by the

NBS Journal of Research.

(28) A method of summing infinite series. S. Lubkin. Accepted by

the NBS Journal of Research.

(29) The stochastic independence of symmetric and homogeneous

linear and quadratic statistics. E. Lukacs. Submitted to a technical journal.

(30) An essential property of the Fourier transforms of distribution functions. E. Lukacs. Accepted for publication in the

Proceedings of the American Mathematical Society.

(31) Some non-negative trigonometric polynomials connected with a problem in probability. E. Lukacs and 0. Szasz. Accepted for

publication in the NBS Journal of Research.

(32) The torsion of anisotropic elastic cylinders by forces applied on the lateral surface. H. Luxenberg, Accepted for

publication .in the NBS Journal of Research.

(33) Graphing with the 407 accounting machine. S. Malles. Submitted to a technical newsletter.

(34) The estimation of parameters in certain stochastic processes.

H. 3. Mann. To be published in S&nkhya; The Indian Journal

of Statistics.

(35) On a method for the determination of converging factors, applied to the asymptotic expansions for the parabolic cylinder

functions. J.C.P. Miller. Submitted to a technical journal.

( 36 ) The double description method. T. S. Motzkin, H. Raiffa,

G. L. Thompson, R. M. Thrall. To appear in the Annals of

Mathematical Studies Volume on Contributions to the Theory

of Games, II.

(37) The number of farthest points. T. S. Motzkin, E. G. Strauss,

F. A. Valentine. Submitted to a technical journal.

(38) On lineal entire functions of n complex variables. T.S. Motzkin

and I. Schoenberg. Submitted to a technical journal.

(39) Matrices with property L. T. S. Motzkin and 0. Taussky.

Accepted for publication in the Transactions of the American

Mathematical Society.

(40) On representations of finite groups. T. S'. Motzkin and

0. Taussky. Submitted to a technical journal.

(41 ) On the derivative of a polynomial and Chebyshev approximation.

T. S. Motzkin and J. L. Walsh. Submitted to a technical

journal

.
Publication Activities

( 4

- 2 ) On the approximation of linear elliptic differential equations

by difference equations with positive coefficients. T. S.Motzkin

and W. Wasow. Submitted to a technical journal.

( 4

- 3 ) Remarks on some modular identities. M. Newman. Submitted to a technical journal.

( 4

- 4-) On two problems in abstract algebra connected with Horner's

rule. A. M. Os trowski . Submitted to a technical journal.

( 4

- 5 ) On a discontinuous analogue of Theodorsen's and Garrick's

method. A. M. Ostrowski, To be included in "The construction

and applications of conformal maps: Proceedings of a symposium,"

to be published in the National Bureau of Standards Applied

Mathematics Series.

( 4

- 6 ) On the convergence of Theodorsen's and Garrick's method of con- formal mapping. A. M. Ostrowski. To be included in "The con- struction and applications of conformal maps: Proceedings of a symposium," to be published in the National Bureau of Standards

Applied Mathematics Series.

( 4

- 7 ) Bounds for the greatest latent root of a positive matrix.

A. Ostrowski. Accepted for publication in the Journal of the

London Mathematical Society.

( 4

- 8 ) Confidence and tolerance intervals for the normal distribution.

F. Proschan. Submitted to a technical journal.

( 4

- 9 ) A method of computing exact inverses of matrices with integer

coefficients. J. B. Rosser. Accepted by the NBS Journal of

Resea rch

.

( 50 ) Numerical computation of low moments of order statistics from a normal population. J. B. Rosser. Submitted to a technical

journal

.

(51 ) Formulas for numerical differentiation in the complex plane.

H. E. Salzer. Accepted for publication in the-Journal of Mathematics and Physics.

( 52 ) An elementary note on powers of quaternions. H. E. Salzer.

Submitted to a technical journal.

(53) On calculating the zeros of polynomials by the method of Lucas.

H. E. Salzer. Accepted for publication in the NBS Journal of

Piesearch.

(54-) An optical model for nucleon-nuclei scattering. D. S. Saxon.

and R. E. LeLevier. Accepted for publication in Physical Review.

(55) Modes of vibration of a suspended chain. D. S. Saxon and A.S.Cahn.

Accepted for publication in the Quarterly Journal of Mechanics

and Applied Mathematics.

( 56 ) Distribution of electrical conduction currents in the vicinity

of thunderstorms. R. E. Holzer and D. S. Saxon. Submitted to

a technical journal.

( 57 ) A remark on M. M. Day's characterization of inner-product spaces

and a conjecture of L. M. Biumenthal. I. J. Schoenberg. Submitted to a technical journal.
Publication Activities 83

(58) On Polya frequency functions III: The positivity of translation

determinants with an application to the interpolation problem

by spline curves. I. J. Schoenberg and A. Whitney. Submitted

to a technical journal.

(59) A bibliography of numerical methods in conformal mapping. W. Seidel.

To be included in "The construction and applications of conformal

maps: Proceedings of a symposium," to be published in the National

Bureau of Standards Applied Mathematics Series.

(60) Gradient methods in the solution of systems of linear equations.

M. L. Stein. Accepted for publication in the NBS Journal of Research .

(61 ) On methods for obtaining solutions of fixed end point problems

in the calculus of variations. M. L. Stein. Accepted for publication in the NBS Journal of Research.

(62) Sufficient conditions for the convergence of Newton's method in

complex Banach spaces. M. L. Stein. Submitted to a technical

Journal

.

(63) A note on the bounds of the real parts of the characteristic roots

of a matrix. P. Stein. Accepted for publication in the NBS

Journal of Research.

(64) On Cauehy-Riemann equations in higher dimensions. E. Stiefel.

Accepted for publication in the NBS Journal of Research.

(65) Two applications of group characters to the solution of boundaryvalue problems. E. Stiefel. Accepted for publication in the

NBS Journal of Research.

(66) On the relative extrema of the Hermite orthogonal functions.

0. Szasz. Accepted for publication in the Journal of the Indian

Mathematical Society.

( 67 ) On the relative extrema of Bessel functions. 0. Szasz. Accepted

for publication by the Bolletino della Unione Matematica Italiana

(Firenze)

.

(68) On the Gibb's phenomenon for a class of linear transforms.

0. Szasz. Accepted by "Publications de 1'Inst. Math, de l'Acad.

Serbe des Sciences," Vol. IV.

( 69 ) Classes of matrices and quadratic fields, II. 0. Taussky-Todd.

Accepted for publication in the Journal of the London Mathematical

Society

.

(70) On conformal mapping of variable regions. S. E. Warschawski.

To be included in "The construction and applications of conformal

maps: Proceedings of a symposium," to be published in the National

Bureau of Standards Applied Mathematics Series.

(71) On the truncation error in the solution of Laplace's equation by

finite differences. W. Wasow. Accepted for publication in the

NBS Journal of Research.

(7 2) On the inversion of matrices by random walks. W. Wasow. Accepted

by Mathematical Tables and Other Aids to Computation.
84 Publication Activities

(73) On singular perturbation problems in the theory of rion-linear

vibrations. W. Wasow. To be published in the Proceedings of

the Symposium on Non-Linear Vibrations, held at Isle de Porquerolles, France, September 18-22, 1951.

(75-) Metodi probabilistic! per la soluzione numerica di alcuni

problemi di analisi. W. Wasow. To be published in the Proceedings of the 4th Congress of the Italian Mathematical Union in

Messina

.

(75) Normal matrices with property L. N. Wiegmann. Submitted to a technical journal.

(7 6 ) Pairs of normal matrices with property L. H. Wielandt. Submitted to a technical journal.

(77) On the eigenvalues of A+B and AB. H. Wielandt. Submitted to

a technical journal.

(78) Statistical units of measurement. W. J. Youden. Submitted to a technical journal.

2.5 Miscellaneous Publications

(1) The construction and applications of conformal maps: Proceedings

of a symposium held at the NBS Institute for Numerical Analysis

Los Angeles, California, June 195-9. NBS Applied Mathematics

Series l8. In press, U. S. Government Printing Office.

(2) Simultaneous equations and the determination of eigenvalues.

Proceedings of an NBS Symposium held in Los Angeles, August 1951.

To appear in the NBS Applied Mathematics Series.

(3) Scientific teamwork in a Computation Laboratory. J. H. Curtiss.

To appear in the Proceedings of the Third Institute on Administration of Scientific Research and Development, held at American

University, October 13, 1 95l •

(4) The use of random numbers. F. Proschan. Accepted for publication

in Industrial Quality Control.

(5) Control charts may be all right, but—. F. Proschan. Accepted

for publication in Industrial Quality Control.

(6) Statistics and planning tests at elevated temperatures. W. J. Youden. Accepted for publication in Experimental Stress Analysis

3. NBS Reports

Note: The following reports contain material for which formal publication

in scientific journals is not presently planned. A limited number

of copies of each was available at the end of the quarter.

(1) Translation by C. D. Benster (UCLA) of K. A. Semendiaev. "The

determination of latent roots and invariant manifolds of matrices

by means of iterations." Translation edited by G. E. Forsythe.

NBS Report 1402. >

(2) A statistical method for finding the lowest eigenvalue of

Sch roedinger 1 s equation. M. Kac and M. Cohen. NBS Report 1553.
Publication Activities 85

(3) Combining tolerances. E. P. King. MBS Report 1313*

(^) Bibliography on bounds for characteristic roots of finite matrices.

0. Taussky. NBS Report 1162.
Appendix

Pressure Distribution on a Body of Revolution

in Supersonic Flight

(Task 1101-53-1101/52-3)

In August, 1951 ? at the request of the Missile Aerodynamics Department of Hughes Aircraft Company, the Institute for Numerical Analysis

began a series of computations for the purpose of devising a workable

procedure on the Model 1 IBM Card-Programmed Calculator (CPC) for determining the velocity distribution on the surface of the forward portion

of an arbitrary body of revolution moving at supersonic speed in the direction of its axis through a steady, inviscid medium* The flow was as- sumed to be rotational; i.e,, the pressure depended on both density and

entropy , although the latter remained constant along a given streamline.

In addition, the rest enthalpy was assumed constant everywhere. Six

bodies were considered, each of them a cylinder with a spherical cap attached by a transition surface concave to the axis*

If such a body is assumed to extend indefinitely rearward, the

flow field around it is divided into two regions by a detached shock

surface which resembles the body close to the axis and approaches a definite conical shape farther away from the axis. On the side of the

shock surface away from the axis the flow is uniform, the velocity being

parallel to the axis with constant Mach number M > 1. As the flow enters

the region between the shock surface and the body, it is deflected away

from the axis and reduced in speed. Close to the nose the flow is subsonic (M < 1); farther back it becomes supersonic (M > 1 ) and approaches

the speed and direction of the uniform flow in the undisturbed stream

outside the shock surface.

Take the point of intersection of the body and its axis as the

origin of the x-r plane, the positive x-axis pointing along the body axis

toward the rear and the r-axis being any normal to the x-axis at the

origin. This plane, then, is a meridian section and the non-uniform flow

is delimited by the shock curve and the body curve (see insert in Fig. 1).

If the velocity and entropy are known accurately on an initial curve ex- tending from the body through an entirely supersonic region to the shock

curve, then it is possible to determine these flow variables to the right

of the initial curve by the method of characteristics* Initial data for

the six problems computed are presented in Table 1. (Values of r, 0 5 and s have been smoothed from empirical and theoretical data to a uniform number

of decimal places.) The curve is seen to rise from the body with a steep,

increasing, positive slope which soon becomes infinite; thereafter, it veers to the left with increasing curvature. The variable 0 is the angle

86
87

Figure 1
Table 1

Number

of Point X r 0 s

(body) 1 .91 8 1 .813 .-6335 .0476

2 • 9225 00 00 .6085 .0471

3 .927 1 . 95-8 . 5638 .0462

4 .936 2.037 .5272 .0447

5 .9595 2.21 6 .4682 .0430

6 .963 2.351 .4343 .041

1

7 • 9765 2.530 .3985 .0390

8 .99 2.709 .3699 .0364

9 • 995-5 2.888 .3467 .0340

10 1 .008 3.11 2 .3208 ,0314

1

1

1 .01 25 3.291 .3038 .0288

1 2 .999 3.515 .2860 .0264

13 .972 3.695 .2725- .0240

14 .9405 3.874 . 261 8 .021 8

15 .873 4.098 .2493 .0198

16 .7875 4.277 .2413 .01 82

17 .6525 4.501 .2306 .01 68

|( shock) 1 8 .3375

L 4.680 .2234 .0142
89

(in radians) between the velocity vector and the positive x-axis; s is

the dimensionless quantity (s - s 0 )/c v

> where s and s g are the entropies

per unit mass at the point and in the undisturbed stream, and c y

is the

specific heat at constant volume. Along the initial curve the speed

q is assumed constant; q* , the ratio between <?£ and the critical speed of

sound, can be defined by

(i ) q* = M

[

(r + i )/[r - i ) m

2 + 2 ]j *, r= i .4o 5,

and is given = 1.048. Moreover, the Mach number of the undisturbed stream

is given = 1 .62 and the angle between the positive x-axis and the tangent

to the shock curve at point 18 on the initial curve is o' = 1 . 01

8

radians.

The upper meridian section of each of the bodies consists of an arc of

a semicircle tangent to the r-axis, a transition curve which is different

for each body, and a straight line. It will be seen later that if the

velocity distribution on the body between the initial curve and some point

P is to be found, it is necessary first to determine x, r, q* , 0, and s at a sufficient number of points in the region bounded by the body, the

initial curve, the shock curve, and a certain curve (the right characteristic) drawn from the shock curve (or the initial curve) through P. The use of characteristics for solving problems of this type

has been systematized by G. Guderley, R. Sauer, and W. Tollmien and is

described in detail in Technical Report No. F-TR-1 1 73A-ND (GDAM A-9-M ^E/l),

Air Materiel Command, Wright Field, 19*+7J The Method of Characteristics

in Compressible Flow, Part I (Steady Supersonic Flow) , prepared by

J. S. Isenberg under the supervision of C. C. Lin in the Graduate Division of Applied Mathematics, Brown University, for the Analysis Division,

Intelligence Department, under contract W33-038ac1 5004 (16351). A description of Tollmien 's method adapted to CPC computation follows.

In the region of non-uniform supersonic flow, the stream function yj/"(x,F) satisfies a nonlinear, second order, hyperbolic, partial

differential ©"nation, in which the coefficients of the second order de-
90

rivatives are expressible as rational functions of the velocity components

u = q cos 0, v = q sin 0; it is thus possible to compute q(or q*) and 0

directly without using . The equations ox the characteristics of the

differential equation are

(2^) dx - /‘dr , /“• = cot(0 + oc.)

,

(2^) dr = A dx , A = tan(0 -oc) ,

where ec is the first quadrant angle

(3) ec = sin" 1 £

[ ( r + 1 ) - ( r- 1 ) (q*)

2 ]/2(q*) 2}

*

At any point in the field, equations (2) define directions making angles

+ ec and - oc with the velocity vector; the family of non-intersecting

curves representing the solutions of (2^) are called left characteristics,

while the family of non-intersecting curves representing the solutions of

(2p) are called right characteristics. Together they form a network

covering the field. Although they cannot be constructed beforehand be- cause of the dependence of ^ and Aon q* and 0, nevertheless their general

behavior can be predicted from the fact that as the flow moves toward

the rear and away from the body, 0 decreases from a first quadrant angle

to zero, and q* increases from a little more than 1 to its value in the

undisturbed stream. Hence, oC decreases to a value depending on M in

the free stream, and Q + oc decreases to zero from an angle

M 1 1 -5 2 2.5 3 3-5 4 00

q* 1 1 .36 1 .63 1 .82 1 .96 2.06 2.13 2.44

«s( degrees) 90 42 30 2b 19 17 1 b 0
91 which may lie in the second quadrant for points near the intersection

of the initial curve and the body curve. Moreover, 0 - oc is a negative,

acute angle which approaches zero toward the rear. Thus, right characteristics slant from the upper left to the lower right, while left ones slant from the lower left to the upper right, except those starting from

a point on the body close to the initial curve where 0 + oc > 90° - these

slant upwards to the left and, if they do not run into the initial

curve, they become tangent to a parallel t© the r-axis and then slant

to the right. The curvilinear quadrilaterals formed by intersecting

left and right characteristics increase in size away from the front end

of the body. This is partially due to the fact that at points farther

away from the initial curve the characteristics have such little, curvature

that all quantities vary almost linearly and it is accordingly expeditious to drop points s© that each mesh becomes almost square, Near the

initial and shock curves the meshes are smaller and elongated in the

direction of the left characteristics; the actual size of the meshes

depends on the distribution of points where data are given on the initial

curve.

If yu. and A are known at two points P^(x^,r^) and P2^ x

2

,r

?? which

occupy such positions that the left characteristic from the former intersects the right characteristic from the latter at a point F-^Cx^r^) lying

in the interior of the nonuniform flow region to the right of the initial

curve, then a first approximation to the coordinates of this point is

obtained by replacing the differentials in (2) by differences and solving

the system

(\) x

3 ~ x

i =/*! ( r 3

- ^

> »

r^ — 2*' X

3 x

2

^

simultaneously. The subscripts appended to /c and A indicate that these

coefficients are to be evaluated at and Pg.
92

The remaining desired variable q,* , 9, and s are obtained from

tire compatibility equations

(5 j) d© - Adq* - Bdr.+ Hds ,

(5R

) d© = -Adej* + /& ds - Hds ,

where

A - cot oc /q*

( 6 )

B = sin© sin oc /r sin(0 + ) ,

= sin© sin©c/r cos(0 - eC ) , H = since cos ©c/ T {f - 1) .

It is known that (5^) holds on a left characteristic and (5R

) holds on

a right one.. The difference approximations for the triad P

^ , P

p , are

(7 l

) ©3 - ©

1 = A

1 (q3

*

- ^

(r

3 - ri ) + H

1 (S3 -

Sl ) ,

( 7r

) ©-, - ©p = -Ap (

q

3 * -

<J 2 *) +/5 0 (x

3 - Xp) - H9

(s-^ - Sp) .

These are not sufficient for the evaluation of the three unknowns q-,*,

0- , and s 3

; however, substitution of

T= In [r+ 1) - (r- l)(q*) 2

] - in 2

transforms (5) into

(9 l

)

' d© = -H(rdT'- ds) - Bdr ,

( 9r) d©~ H(rdT- ds) + y5dx ,

with the attendant difference equations

(10

L

) ©

3

- ©

1 = -H

1 [ (rt- - s 3

) - (rT, - Sl )] - B

1 (r

3 - r

1 ) ,

(io

R

) ©

3 - ©p = Hp[ ( r r

3

- s 3

) - (rr2 - s 2

) ] +

/

32

(x

3 - x

2

) ,

which can be solved for Q-^ in terms of previously computed quantities

by eliminating *>" T - . The variable s is computed as follows: Let
93

( x

ia ,sW k® tSie point of the streamline through and the

line joining P, and P

2

; then, since 9., is the direction of the first

line

,

OD p3

- r

B, = tan0

3

(' x

3

- *m

) •

Also, if it is assumed that s varies linearly on the second line,

02) (x, - *„)/<x, - x

2

) = (r, - g/Cr, - r2

) = («, - ..)/(«, - # 2

) .

From ( 11 ) and ( 12 ) it is possible t© get s^, which is the same as s-,, since the entropy remains constant on a streamline. Finally, either of

equations ( 7 ) yields

In practice the foregoing formulas are reduced to the following

five

( 13 «) ^3 ~ [^-j ^

f*"

\

( P 2

—

^"i — ^> 2X2"> ^ ~ f

i

‘\ ^”0 3 i

(13b) r, = r2 + ^

2

(‘ x

3 ~ x

2

^ ’

(i3c) e

3 = [^©2 + h

2©

1 + ^h2

- r2) - ( $1 - s 2 )}

~ B

1 H2

(5*3 “ p-j ^ + ^2^1 f

' X

3 ~ + ^

2

^ 5

(13d) = s

1 + (s 2 -s^ ) [ (x^-x. ) tan©

3

- ( r^-r, ) ]/[ (x 2~»

1 ) tan©^ -Cr,,-^)] ,

(I3e) = q2

* + [^ 2

(x

3

_x2') ~ H

2

<‘ S 3

_S 2^ ~ ( Qy

Q2^//

'A2 ’

for the first approximation. A second approximation is obtained by

putting these values of x, r, 9, s, q* in ( 2 ), ( 3 ),' and ( 6 ) to calculate

^“3

? Q^, B^, and , and then recomputing ( 13 ) with the

coef f icientsyw^ , A

2

, , H

2

, A

2 replaced by the averages

+ y*3 )/2, (A 2 +A

3 )/2, (H^ + H7 )/2, (H

2 + H-J/2, etc. This procedure

has the effect of giving the coefficients is (4), ( 7 ), and ( 10 ) mean values

whi make the difference equations approximate the differential equations more accurately. Reiteration is possible; however, in the examples inves-
94

tigated even the first iteration changed the value by much less than one percent.

A method for starting the computation is 'now described. The

directions 0 + ©c and 9 - &C. of characteristics emanating from points on the initial curve are obtained from Table 1 and equation (3) ( Y

1 .405,

q* = 1 .048) . For any pair of successive initial points below the one numbered 11 the backward left characteristic (at an angle + 06

^ + ir) from the upper point (x.,,r, ) and the forward right characteristic (at

an angle 0 p

- oC

2

^ * r0!B t!ke fower point (x ?

,r 0 ) intersect at a point

(X3, 1*3) lying to the right of the initial line. The pentad x 0^,s^ is calculated by iterating (13). Call the right characteristic

from an initial point numbered n, Rn, and the left characteristic from

the same point, La; also, let the intersec tiers point of Rm and Ln be

designated (m,n). Then the first interior point to be calculated is

(2,3)5 obtained as the intersection R2 drawn, from 2 and L3 drawn from 3.

This is the point marked A in the lower left corner of Fig. 1. Points

described subsequently are also marked by a capital letter in the figure.

From 3 and 4 the point 18(3,4) is obtained , and from 4 and 5 5 C(4,5), L5

is extended farther down to intersect with R3 in 0(3,5). Along L6 the

points E(5»6), F(4,6), and G(3,6) are determined; along L7, 11(6,7)5...

1(3,7); along L8, J(7,8) , . . .K(3,8) ; and along L9, L(8,9) , . . . P(3,9)

.

It is not practical to continue in this manner since the left

characteristics drawn down from points 10 and 11 almost coincide with L9*

The data on L9 are used to construct right characteristics leading to

the body curve. The simultaneous solution of (4^) and the equation

( 1 4) r = f(x) x = , r = ,

of the body curve yields the coordinates ©f the point ( x 3? r 3) 0ES the

body from information at the interior point (xp,pp). The angle is given

by

(1?) ®3 = tan_i (df/dx)-^ ;
95 .

§ ^ is eoual to the value of s at point 1 (the intersection of the body curve and the initial curve) since the body itself is a streamline; and

is computed from (I3e). The approximations are improved by replacing

the coef f icieats ^2 in (5-^) and /5 0 , , and A

2 in (l3e) by their averages

with , and A^ , just as in computing at an interior point.

If r = fix) _is not given analytically, it may be fitted by a formula.

In either case, if the function is too complicated, a first approximation

to (x^,r^) can be got by determining the intersection of the right characteristic from (x 2

«,r 0 ) with the tangent drawn from a previously computed

point on the body. For the value of x thus calculated, r on the body

is found. From this body point another tangent is drawn to intersect

with the characteristic and the procedure is repeated as often as desired.

Two such iterations were sufficient to give results converging in the

sixth decimal for the transition curves occurring in the problems considered.

If the point of intersection of R^- with the body curve is

called Q(4, 10), then a left characteristic (which can be designed L10

without ambiguity because no left characteristic was drawn through point

10) is started upward by getting R(5,10) from 0(5,9) and Q( 1+,10) ; R5 is

continued from R(5,10) to S(5,1l) on the body by the procedure of the

last paragraph, and L1 1 is started. To complete R6, T(6,10) is computed

from N(6,9) and R(5,10), U (6 , 1 1 ) from T(6,10) and S(5?11)> and V(6,12)

(on the body) from U ( 6 , 1 1 ) (and possibly S(5,11) if the body curve has

a complicated equation). To complete R7, W(7, 10) is computed from

M(7,9) dd T(6,10), X(7,11) from W(7,10) and U(6,11), Y(7,12) from

X(7 , 1 1 ) and V(6,12), and Z(7,13) (on the body) from Y(7,12) (and possibly V(6,12)). All the remaining interior points are computed in this

same systematic manner. Starting with known values on L9, the initial

curve (between points 9 and 1 6) 5 or the shock curve (see next paragraph),

it is possible to determine the point (m,n) from (m,n - 1 ) on the same

right characteristic Rm and (m - 1 ,n) on the preceding right characteristic
96

R(m - 1). [This uniform numbering scheme holds provided the left, characteristics extending upwards from 12,13?--*1

7

are called L9 , L85 ...L4 .]

When the quantities x,r,«,Q,s at any point are computed on the CPC, they are held in the memory until values are obtained at the next point and

also are punched on a single card. A set of these cards for all the

points on a certain right characteristic is combined with the instruction

deck containing formulas (13) to calculate points on the next right

characteristic. The point being computed is (x^,r^), the one on the same characteristic is («£ , sr,-,) , the one on the preceding characteristic

is (x^,r^). The instructions are to take x^,...s^ from the card, take

X25 *-*S

2

^ roni th© memory, compute the appropriate coefficients, evaluate

the formulas in (13), iterate once, put the results x^,...s^ in the

memory and punch them on a new card. The process requires three minutes

for each point and is repeated for each point on Rm without intervention

of the operator until the body is reached. Here the instruction deck is

replaced by another which determines the point on the body as described

in the last paragraph. If f(x) is given by more than one analytic

expression, the appropriate portion of the deck must be altered accordingly.

Each such point (including two iterations) requires five minutes to

compute.

As the flow crosses the shock surface, the uniform velocity

and. entropy suffer an abrupt change. The following formulas are valid

immediately behind the shock:

( 1

6

) (1 + tan

2Q)(u *) 3 - [ 2M* + (1 + tan

2Q)(M*)

-1 + 2M*tan2Q/(r + l)](u*) 2

+ [M*) 2 + 2] u* - M* = 0 ,

( 17 ) v* = u* tan© ,

(

1

8) q* = [u *) 2 + (v*) 2

] 2 = u*/cosO ,

o' = tan

- "' (19) [ (M* - u*)/v*] 5 o <<r< tt/2,
97

(20) s = -or + 1 )ln(r + 1 ) - 2 rinM + rin[ (r - 1 )M2 + 2]

+ ln[2rM2- (r - 1 )] ,

where M = M sin 0" , M is the Mach number of the undisturbed stream, O' is

the angle between the shock curve and the positive x-axis, u* = q* cos 0,

v* - q* sin 0, and M* is the constant value of q* in the undisturbed

stream (see'(l)). Six quantities (x, r,q,0, s , ) are known at point 18 , the intersection of the shock curve and the initial curve. A first approximation to the next point on the shock curve is obtained from the

intersection of its tangent at 18 with L4, the left characteristic

running up from 17* Better approximations are obtained by the iteration

procedure described subsequently. The right characteristic emanating

from this point is called R 1

7

(since no right characteristic was drawn

through 17) and is extended by getting B'(1 7*5) from A' (1 7 A) (the point

on the shock curve) and 16 , and continuing t© the body. The next shock

point D ’

(

1

8

, 5 ) is computed as the intersection of the shock curve and L5;

R1 8 is started by getting E* (1 8 , 6 ) from D

1 (

1

8 , 5 ) and C'(17,6). More

shock points and net points are calculated in this way until the velocity

distribution is known over as much of the body surface as desired.

Let (x~,r^) be the shock point being computed-, (x

2 ,r2

) the

previously calculated one, and (x^,r^) the first interior point of the

right characteristic through (x-,,^) (or on initial point). Hie intersection of the left characteristic from (x^,r^) with the tangent

(21) r^ - r 2 = tan <r2 (x^ ~ x 2^

to the shock curve at (x

2 ,r 5

) is given by

(22) = [ Xl + /*-l ) r 2 - r

i ~ x o tancr

2 )]/[l tan<f

2

] ,

(23) = r

1 + (x^ - x

i )//*\ (from (4^) ) .

If q* and s are assumed constant on the shock curve segment, then (7^)

yields which can be used in (l 6 ),( 17 )? and ( 18 ) to calculate a second
98

approximation of q^*. Finally, <r^ and an improved sx are given by (19)

and (20). These values of x

^ , , e",, and are used to calculate the coefficients ? etc.; then formulas (22), ( 23 ), (7jO , ( 1

6

) ,

( 17 ), ( 18 ), ( 19 ), and (20) are used with^ replaced by ( /^

1 + f*-))/2, etc.,

to recompute the hexed,. About six such iterations were necessary to produce results good to five significant figures. Most of the time of com- puting for a shock point - twenty minutes in all - is consumed in finding

the root of the cubic ( 16 ). As a first approximation to this root, which

is the intermediate one of three real roots, the value of u* at the previous shock point is used; five iterations by Newton’s metfoot were programmed in the instructions.

Data on the initial curve had been determined empirically except

at the shock curve where the results ©f theoretical investigation were

used. These values must be physically correct to about five per cent

and must be compatible with the shape of the body. If, after some com- putation, everything is known on a certain left characteristic L, then L may be treated as a new initial curve. The values on L depend only on the data on the original initial curve and the shape of the body to the

left of L. Conseouently , in the six cases considered, L was taken as the

rightmost left characteristic that intersected the circular arc common

to the meridian sections of all the bodies. The complete network was constructed for only one body; for the other five it was different only

to the right ©f L„ Calculation time was cut considerably, for instead of

the 16 hours, required to compute the complete field (including 200 interior

points, 15 shock points, and 2.5 body points, at respective rates of 3, 20,

and 5 minutes per poirst) , the field to the right of L was done in only

b hours, although it did require about 5 hours to change the coding for

each body. A glance at the formulas shows that if the Mach number of the

undisturbed stream is altered, the coding of shock points only is affected,

while if the food}'' curve is changed, the coding of body points only must

be altered.
99

Interpolation of points in the network becomes necessary when

(1) a segment between adjacent net points is too curved to be replaced

by a straight line, (2) a pair of adjacent characteristics diverge, (3)

the left characteristic rising to the shock curve is too long, or (4)

the initial data are too widely spaced. It is more desirable to graph

the net points as they are computed and just notice the behavior of the

characteristics than to attempt to include any selection principle in

the coding; however, an all purpose interpolation deck can be prepared

and inserted at will by the operator. This deck is made up to interpolate each variable linearly at a point designated (m,n + rp) between

(m,ra) and (m,n + p) , p < 1. Examples of such points in the problems

solved were A" (4,9 1/2), B"(7,10 1/2) and C"(11, 10 lA). Actually,

only five points were given on the initial curve; the eighteen points in

Table 1 were determined by smoothing the data and taken as initial points.

Since the left characteristic from point 17 intersected the shock curve

at a considerable fdistancS" from 17 , the point 17 1/2 was interpolated

linearly on the initial- The left characteristic from 17 1/2 was called

L3 1/2 and intersected the shock curve at D"(l6 l/2, 3 l/2), whence R1 6 l/2 was started. The point F"(l6 l/2, 3 3/4) was interpolated between

D"(l6 l/2, 3 1/2) and E"(l6 1/2,4) and a left characteristic run up to

the shock curve at G"(l6 3/4, 3 3/4). This cumbrous notation may be

simplified by relabeling the characteristics L3 l/2, L3, R1 6 l/2, R1

6

3/4,

B17, ©nd R18 as L3, L2, -R17, Rl8, Rl9,.and R20.

The procedure used for calculating at most of the interior

points in the six examples may be used for all of the interior points if

the initial curve has such a position that left characteristics run upward from it and right ones run downward from it. Otherwise, as in the

examples, a different scheme has to be used in the neighborhood of certain

segments of the initial curve. Although the bodies considered here all have rounded noses,

the method is not restricted to such forms but may be used for axially
100

symmetric bodies of more general shapeThe details of computation on the CPC were worked out by

E. C. Yowell and carried out by P. Bremer. This report was prepared

by R. R. Reynolds.

Ma rch 26 , 1 951 •
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