NATIONAL BUREAU OF STANDARDS REPORT

4274

PROJECTS and PUBLICATIONS

of the

APPLIED MATHEMATICS DIVISION

A Quarterly Report

April through June 1955

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

FOR OFFICIAL DISTRIBUTION
U. S. DEPARTMENT OF COMMERCE

Sinclair Weeks, Secretary

NATIONAL BUREAU OF STANDARDS

A. V. Astin, Director

THE NATIONAL BUREAU OF STANDARDS

The scope of activities of the National Bureau of Standards is suggested in the following listing

of the divisions and sections engaged in technical work. In general, each section is engaged in

specialized research, development, and engineering in the field indicated by its 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.

Electricity and Electronics. Resistance and Reactance. Electron Tubes. Electrical Instru- ments. Magnetic Measurements. Process Technology. Engineering Electronics. Electronic

Instrumentation. Electrochemistry.

Optics and Metrology. Photometry and Colorimetry. Optical Instruments. Photographic

Technology. Length. Engineering Metrology.

Heat and Power. Temperature Measurements. Thermodynamics. Cryogenic Physics.

Engines and Lubrication. Engine Fuels.

Atomic and Radiation Physics. Spectroscopy. Radiometry. Mass Spectrometry. Solid

State Physics. Electron Physics. Atomic Physics. Nuclear Physics. Radioactivity. X-rays.

Betatron. Nucleonic Instrumentation. Radiological Equipment. AEC Radiation Instruments.

Chemistry. Organic Coatings. Surface Chemistry. Organic Chemistry. Analytical Chemistry.

Inorganic Chemistry. Electrodeposition. Gas Chemistry. Physical Chemistry. Thermochemistry. Spectrochemistry. Pure Substances.

Mechanics. Sound. Mechanical Instruments. Fluid Mechanics. Engineering Mechanics.

Mass and Scale. Capacity, Density, and Fluid Meters. Combustion Controls.

Organic and Fibrous Materials. Rubber. Textiles. Paper. Leather. Testing and Specifications. Polymer Stricture. Organic Plastics. Dental Research.

Metallurgy. Thermal Metallurgy. Chemical Metallurgy. Mechanical Metallurgy. Corrosion.

Mineral Products. Porcelain and Pottery. Glass. Refractories. Enameled Metals. Concreting Materials. Constitution and Microstructure.

Building Technology. Structural Engineering. Fire Protection. Heating and Air Conditioning. Floor, Roof, and Wall Coverings. Codes and Specifications.

Applied Mathematics. Numerical Analysis. Computation. Statistical Engineering. Mathematical Physics.

Data Processing Systems. Components and Techniques. Digital Circuitry. Digital Systems.

Analogue Systems.

Cryogenic Engineering. Cryogenic Equipment. Cryogenic Processes. Properties of Materials.

Gas Liquefaction.

Radio Propagation Physics. Upper Atmosphere Research. Ionospheric Research. Regular

Propagation Services.

Radio Propagation Engineering. Frequency Utilization Research. Tropospheric Propagation

Research.

Radio Standards. High Frequency Standards. Microwave Standards. • Office of Basic Inatrumentation • Office of Weights and Measure.
NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT NBS REPORT

NAML 4274

PROJECTS and PUBLICATIONS

of the

APPLIED MATHEMATICS DIVISION

April through June 1 955

<NBS>

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

The publication, reprli

unless permission isob 25, D, C, Such permla

cal ly prepared If that

Approved for public release by the

director of the National Institute of

Standards and Technology (NIST) on October 9, 2015

part, Is prohibited

lards, Washington

t has been speclflrt for Its own use,
APPLIED MATHEMATICS DIVISION

April 1 through June 30, 1955

TECHNICAL ADVISORY COMMITTEE

David Blackwell, Howard University Pnilip M. Morse, Massachusetts

Edward U. Condon Institute of Technology

Mark Kac, Cornell University Mina S. Rees, Hunter College

A. H. Taub, University of Illinois

DIVISION OFFICE

Edward W. Cannon, Ph.D., Chief

Franz L, Alt, Ph.D., Assistant Chief

Olga Taussky-Todd , Ph.D., Consultant

W. J. Youden, Ph.D., Consultant

Myrtle R. Kellington, M.A. , Technical Aid

Luis 0. Rodriguez, M.A. , Chief Clerk

Yates S. Sladen, Assistant Chief Clerk

Mary E. McKinley, Secretary

Esther L. Turner, Secretary

NUMERICAL ANALYSIS SECTION John Todd, B.S., Chief

Henry A. Antosiewicz, Ph.D.

Leila D. Bram, Ph.D.

Philip Davis, Ph.D.

Ky Fan, Ph.D.

Karl Goldberg, M.A.

Evelyn A. Grigg, Sec'y

Alan J, Hoffman, Ph.D.

Morris

Philip

Eduard

Newman, Ph.D.

Rabinowitz, Ph.D.

Stiefel, Ph.D.

COMPUTATION LABORATORY Milton Abramowitz, Ph.D., Chief (Acting)

Irene A. Stegun, M.A. , Assistant Chief

***Ruth K. Anderson, M.A.

Joseph Brain, Ph.D.

Hans 0. Bremer, B.A. William F. Cahill, M.S.

Ruth E. Capuano

Mary M. Dunlap, B.S.

Anne F. Futterman, B.A. Billie R. Gill, M.S.

Elizabeth F. Godefroy

William G. Hall, B.S.

Genevie E. Hawkins, B.S.

Urs W. Hochstrasser , Ph.D.

Gloria F. Holmes, B.S., Sec'y

Dorothea H. Jirauch, M.A.

Lambert S. Joel, B.A. David S. Liepman

Ethel C. Marden, B.A. Kermit C. Nelson

Peter J. O'Hara, B.S.

Mary Orr

Maxine L. Paulsen, B.S.

B. Stanley Prusch

Ida Rhodes , M.A,

Sally Tsingou, B.S.

Bertha H. Walter

Philip J. Walsh, B.S.

Joseph H. Wegstein, M.S.

Donald F. Wehn, B.A. Ruth Zucker, B.A.

STATISTICAL ENGINEERING LABORATORY Churchill Eisenhart, Ph.D., Chief

Joseph M. Cameron, M.S., Assistant Chief

Robert C. Burton

Marion T. Carson

Willard H. Clatworthy, Ph.D.

Lola S. Deming, M.A.

Mary L. Epling, B.S., Sec'y

MATHEMATICAL PHYSICS SECTION

Peter Chiarulli, Ph.D.

Anne R. Cock, B.A. Leon Feldman, B.A.

Seymour Geisser, Ph.D.

Lela J. Hamilton, Sec'y

Forest L. Miller, Jr. Julius Lieblein, Ph.D.

Peter Henrici, Ph.D.

John G. Hershberger, M.S.

Fritz Oberhettinger , Ph.D.

Mary G. Natrella, B.A.

Hugh M. Pettigrew

Paul N. Somerville, Ph.D.

Marvin Zelen, M.A.

Edith N. Reese, B.A.

Lillian Sloane, Sec'y

Edward W. Cannon, Ph.D., Chief

Robert F. Dressier, Ph.D., Assistant Chief

Under contract with The American University Guest worker On leave of absence
CONTENTS

Page

Status of Projects* as of June 30, 1955 . . 1

Numerical Analysis Section ....... 1

(NBS Section 11.1)

Computation Laboratory 7

(NBS Section 11.2)

Statistical Engineering Laboratory ... 22

(NBS Section 11.3)

Mathematical Physics Section 27

(NBS Section 11. k)

SEAC 30

Lectures and Symposia 32

Publication Activities 3^

*Only unclassified projects are included in this

report.
.

: :

•

, • "

•

- t

— - '

• .

• • .. \ . ..

.

.
Status of Projects

June 30, 19S5

I. NUMERICAL ANALYSIS SECTION

(Section 11.1)

RESEARCH IN NUMERICAL ANALYSIS AND RELATED FIELDS

Task 1101-10-1104/55-55

Origin: NBS Authorized 8/13/54

Managers: J. Todd, P. Davis Revised 8/29/54

Full task description: July-Sept 1954 issue, p. 1 Status: CONTINUED. A library of matrix codes has been completed

by M. Newman. These are general purpose matrix routines which perform any

of the arithmetical operations AB, A+B, A

-

^ in any desired order. All

the routines are for the internal memory of SEAC , the data being taken as single precision floated decimal numbers. The floating subroutines for

these codes were written by K. Goldberg.

The routine for A

-

^

, for example, produces the inverse of a 28 by 28 matrix in approximately 15 minutes. In addition, there are available

routines for the solution of equations, A'A, A'BA, and other routines of

a special nature.

The routines all have the characteristic that the data input and

data output are in the same standard form (either decimal or binary) so that the output of one routine may be used as the input of another. It is

planned to add to this library other basic matrix codes, such as the finding

of characteristic values and characteristic vectors.

P. Henrici continued his program of automatizing certain complicated

arithmetic operations, in particular, computations with power series with

rational coefficients, on SEAC. Two reports have been prepared which

describe the results obtained and discuss future applications.

P. Henrici derived a new proof of the asymptotic expansion of

the function

J

r oo u^ j r e 3 0 - e xu

) ^ . 0

(to be included in the introduction of the Tables of integrals related to

the Airy functions, to be submitted for publication in the NBS Applied

Mathematics Series.)

J. Todd completed a paper giving a direct approach to the problem

of the numerical stability of discrete schemes for the solution of partial

differential equations, by various schemes, implicit and explicit, for such

equations as uxx=u

t

, uxxxx

+u

tt=0, uxx+

u

yJr=u

t

, u

XJ =utt . The main tools

used are (1) the. facts that the characteristic roots of a polynomial P in
2 Status of Projects

a matrix C are the same polynomials P(A) of the characteristic roots A

of C and (2) certain facts about partitioned matrices.

Publications:

(1) Stable systems of differential equations with integrable perturbation

term. H. Antosiewicz. To appear in the Journal of the London

Mathematical Society.

(2) Computation of vibration modes and frequencies on SEAC. W. Cahill

and S. Levy (NBS 6,^-). Accepted, Journal of the Institute of Aeronautical

Sciences

(3) On a problem in the theory of mechanical quadratures. P. Davis. To

appear in the Pacific Journal of Mathematics.

(b) A muJ.ti-purpose or thonormalizing code and its uses. P. Davis and

P. Rabinowitz. J. Assoc. Comp. Mach. 1_, 183-191 (195*+). 1 (5) On the Lerch zeta function. F. Oberhe ttinger . To appear in the

Pacific Journal of Mathematics.

(6) Generation and testing of random numbers on SEAC. 0. Taussky, J. Todd,

M. Newman, and J. Cameron. In manuscript.

(7) On nonlinear differential equations of the second order with integrable

forcing term. H. A. Antosiewicz. J. London Math. Soc. .30? 6^-67 (1955).

(8) Evaluation of a definite integral. P. Henrici. In manuscript.

(9) Asymptotic location of the zero of the Bessel polynomials. P. Henrici.

In manuscript.

(10) The generation of Coulomb wave functions by means of recurrence relations.

I. Stegun and M. Abramowitz. Phys. Rev. ^8, 1851-1852 (June 1 5 ? 1955).

(11) A subroutine for computations with rational numbers. P. Henrici. In

manuscript.

(12) Automatic computations with power series. P. Henrici. In manuscript.

(13) A direct approach to the problem of stability in the numerical solution

of partial differential equations. J. Todd. In manuscript.

(1 b) Some Monte Carlo experiments for computing multiple integrals, P. Davis

and P. Rabinowitz. Submitted to a technical journal.

(15) Abscissas and weights for Gaussian quadratures of high order. P. Davis

and P. Rabinowitz. In manuscript.

(16) Numerical experiments in potential theory using orthonormal functions.

P. Davis and P. Rabinowitz. In manuscript.

(17) Be^griindung fur die Beschaf tigung mit numerischer analysis. J. Todd.

To appear in the Jahresbericht d. D.M.V.

BASIC RESEARCH IN LINEAR PROGRAMMING

Task 1101-10-5116/50-2

Origin: NBS Authorized 3/31/50

Sponsor: Office of Scientific Research, ARDC, USAF,

and Directorate of Management Analysis, USAF

Managers: J. Todd, A. Hoffman

Full task description: Jan-Mar 1950 issue, p. 2

Status: COMPLETED. H. Antosiewicz completed the editing of the

Proceedings of the Linear Programming Symposium held in January. The exact

date of publication is not yet known.

A. J. Hoffman continued the study of combinatorial applications

of linear programming. An incidence matrix A with m rows and n columns

is said to have the "unimodular property" if every square submatric of A

has a determinant with absolute value 0 or 1 . Extending an earlier result,

he has shown that A has the unimodular property if

a . . -a . • =1

1J 1 1J 2

for all k>j, either a. . =1

K J -j a kj ,=1 or ^r1 -
Status of Projects 3

This lemma can be applied to the study of partially ordered sets in which

x > z, y / z implies x and y are comparable. Further this lemma seems to

be the underlying reason for the integrality of solutions to linear

programming problems (in which special properties of the "right-hand side"

are not employed) for all known cases not covered by the lemma (used in

the transportation problem and its generalizations) that the incidence

matrix for points and arcs of a linear graph with no odd cycles has the

unimodular property.

Additional work in the field of linear programming will be

reported under task 1 1 01 -1 0-51 1 6/55-56 , Research in Mathematical Topics

Applicable to Numerical Analysis.

Publications:

(1) A theorem on alternatives for pairs of matrices. H. A. Antosiewicz.

To appear in the Pacific Journal of Mathematics.

(2) Discrete analogs of inequalities of Wirtinger. Ky Fan, 0. Taussky, and

J. Todd. Monatsh. Math. £2, 73~90 (19 55).

(3) On distinct systems of representatives. A. J, Hoffman and H. W. Kuhn.

To appear in the Annals of Mathematical Study 38.

(5-) On "overshoot" in the "furthest hyperplane" method. R. Bryce. In

manuscript

(5) An algorithm for solving the transportation problem, A. Gleyzal.

J. Res. NBS 54, 213-216 (1955).

( 6 ) Notes on some scheduling programs. A. J. Hoffman, In manuscript.

(7) On block relaxation. L. S. Joel, In manuscript.

(8) Systems of linear inequalities. Ky Fan. To appear in an Annals of

Mathematics Study.

(9) Systems of distinct representatives and linear programming. A.J. Hoffman,.

To appear in Annals of Mathematics Study 38.

(10) On a theorem of Dilworth. A. J. Hoffman and G. B. Dantzig (RAND

Corporation). To appear in Annals of Mathematics Study 38.

(11) On the minimization of concave and convex functionals. G. B. Dantzig,

A. J. Hoffman, W. Hirsch. In manuscript.

(12) How to solve a linear programming problem. A. J. Hoffman. To appear

in the Proceedings of a Symposium on Linear Programming held in

Washington, D. C., January 1955.

(13) Linear programming in bid evaluation. L. Gainen. To appear in the

Proceedings of a Symposium on Linear Programming held in Washington,

D. C., in January 1955.

RESEARCH IN MATHEMATICAL TOPICS APPLICABLE TO

NUMERICAL ANALYSIS

Task 1 101-10-51 16/55-56

Origin: NBS

Sponsor: Office of Naval_Re se arch

Managers: 0. Taussky-Todd , M. Newman

Full task description: July-Sept 195*+ issue, p, 5 Status: CONTINUED. M. Newman completed a paper entitled "An inclusion

theorem for modular subgroups," which generalizes certain results previously

obtained. Let G be the multiplicative group of matrices (® ^) , where a,b,c,d

are rational integers and ad-bc=1 . Let G(m,n) be the subgroup of G

characterized by b=0(mod m) and c=0(mod n). Then if (m,n)=1 and H is a subgroup of G containing G(m,n), H=G(m^,n^) where m^

| m and n^J n. If

(m,n) > 1, the theorem is false and explicit counterexamples are given.

A further generalization is given to the case where the underlying ring

is any ring of algebraic integers.

Authorized 8/13/55-

Revised 8/29/55-
Matrices with a property which is shared by each of the principal

minors were studied by K. Goldberg. If there is a transformation P of

such a matrix A such that each principal minor’ of A is transformed into the

corresponding principal minor of A** and A and A^ have equal determinants, then

A and A^ have equal roots. This procedure was used in a case where the

transformation matrix A^ was symmetric so that the roots of A were shown to

be real.

A problem in graph theory was studied and partially solved by

K. Goldberg together with E. Dade. The problem is to find the number of

ways that a certain set of graphs can be combined in order to give graphs

with a prescribed number of polygons.

A study of defining combinatorial operators to simplify the

procedures used in the method of generating functions is in progress. These

operators have been used to generalize and give a more direct proof of

previous work done in finding the coefficients of products of formal power

series in non-commutative variables.

Olga T. Todd, jointly with E. G. Straus of the University of

California at Los Angeles, studied algebraic equations satisfied by roots

of natural numbers. A manuscript was prepared. The paper generalizes some

of the work of paper (11) below.

Dr. Todd studied the connection between certain theorems which

hold both for symmetric and for non-negative non-symme trie matrices.

Working jointly with I. L. Glicksberg of the RAND Corporation,

Ky Fan continued the study of fully k-convex normed linear spaces.

Publications:

(1) On a generalization of the normal basis in abelian algebraic number

fields. M. Newman and 0. Taussky-Todd . Accepted, Communications on Pure and Applied Mathematics (New York University).

(2) A note on group matrices. 0. Taussky-Todd. To appear in the Proceedings

of the American Mathematical Society.

(3) The normalizer of certain modular subgroups. M. Newman. To appear in

the Canadian Mathematical Journal.

(4) Generalizations of identities for the coefficients of certain modular

forms. M. Newman. To appear in Journal of the London Mathematical

Society

(5) Compac tif ication of completely regular spaces. Ky Fan and F. Wagner

(University of Notre Dame). In manuscript.

( 6 ) Some inequalities concerning positive-definite Hermitian matrices.

Ky Fan. To appear in the Proceedings of the Cambridge Philosophical

Society

(7) A comparison theorem for eigenvalues of normal matrices. Ky Fan.

To appear in the Pacific Journal of Mathematics.

(8) Power series for loq(e x .ey). K. Goldberg, To appear in the Duke

Journal of Mathematics.

(9) On certain series expansions involving Whittaker functions and Jacobi

polynomials. P. Henrici. To appear in the Pacific Journal of Mathematics,

(10) On generating functions of the Jacobi polynomials. P. Henrici. To

appear in the Pacific Journal of Mathematics.

(11) The number of absolute points of a correlation. A. Hoffman, M. Newman,

E. Straus, and 0. Taussky-Todd. To appear in the Pacific Journal of

Mathematics

(12) Structure theorems for modular subqroups. M. Newman. Duke Math. J.

22, 25-32 (1955).

(13) The coefficients of certain modular forms. M. Newman. To appear in

the Journal of the London Mathematical Society.

(1^) Some computational problems in algebraic number theory. 0. Taussky.

To appear in the Proceedings of the American Mathematical Society

Sixth Symposium on Applied Mathematics, held at Santa Monica, California,

August 1953.

(15) Error bounds for eigenvalues of symmetric integral equations. H. Wielandt.

To appear in the Proceedings of the American Mathematical Society Sixth

Symposium on Applied Mathematics, held at Santa Monica, California,

August 1953.
Status of Projects 5

(16) On eigenvalues of sums of normal matrices. H. Wielandt. Submitted

to a technical journal,

(17) Automatic computations of nerve fiber excitation. K. S. Cole (Naval

Medical Research Institute), H. A. Antosiewicz, and P. Rabinowitz.

In manuscript.

(18) On asymptotic stability. H. A. Antosiewicz. In manuscript.

(19) An inclusion theorem for modular subgroups. M. Newman. In manuscript,

(20) Unimodular circulants. 0. Taussky-Todd . Accepted for the Schur

Memorial Volume, Mathematische Zeitschrift.

(21) Algebraic equations satisfied by roots of natural numbers, E. G. Straus

(U.C.L.A.) and 0. Taussky. In manuscript.

ANALYTIC STUDY OF WAR GAMES

Task 1 101-10-51 16/55-83

Origin and Sponsor: Armament Branch, ARDC, USAF Authorized 12/29/5^

Manager: H. A. Antosiewicz

Full task description: Oct-Dec 195^ issue, p. 7

Status: CONTINUED. H. A. Antosiewicz continued his investigation

of the war games model. As originally proposed, this model included as one of the strategic variables the rate of production of each side in such

a manner that, in a game of attack of offensive forces upon offensive forces

only, the rates of production of the two opposing sides would tend ultimately

to the same goal. This was felt to .be unrealistic and unnecessarily

restrictive. To achieve more realism, H. A. Antosiewicz proposed a modification of the original model which, for the same situation, permits

two different limiting values for the rates of production. At the same time

it is possible to include what might be called overshoot of rate of

production; that is, the rates of production are allowed to increase above

their preassigned goals before ultimately approaching these (from above).

This representation is based upon the reasoning that either side, in case

of an attack upon its offensive forces, would attempt to increase its rate

of production to a set goal as rapidly as possible and that, in doing so,

the rate of production, when first reaching its goal, would tend to exceed

it for some time thereafter. A report on this modification has been

prepared and submitted to the sponsor.

Publication:

(1) Analytic study of war games, H. A. Antosiewicz. Submitted to a technical journal.

STUDIES IN THE THEORY OF ASYMPTOTIC EXPANSIONS

Task 1101-10-5116/55-116

Origin: NBS Authorized 3/31/55

Sponsor: Office of Scientific Research, ARDC, USAF

Manager: P. Davis

Full task description: Jan-Mar 1955 issue, p. 8

Status: CONTINUED. Let z=1 lie on the boundary of a simply connected

region D. For f(z) regular in D, write

oo f (z) /-w 21 a

n

( z_ 1 ) n
6 Status of Projects

to indicate that the right-hand series is the asymptotic expansion (in

the sense of Poincare) for f(z) at z=1 . Write,

R

n (z) = f (z)-a

Q-a

1 (z-1

.-a

R_ 1 (z-1 ) n

f n

(z) = (z-1) nR

n (z).

For a given sequence of positive quantities {

n } we consider the class

A(m

n

) of functions analytic in D for which

2 2 < M k m (n=0,1 ,...).

Here y (I designates some conveniently chosen norm. The norm I f (z) I ds

has been studied in detail. Carleman's idea of introducing a minimum

problem has been combined with the theory of conformal kernel functions to

arrive at a necessary and sufficient condition on (m

n^ in order that A(m ) be a uniqueness class for asymptotic expansions.
II. COMPUTATION LABORATORY

(Section 11.2)

1 . Mathematical Tables

1102-40-1 11 OA3-3 TABLES OF E

1 (z), (z=x+iy)

Manager : I. Stegun

Full task description : Apr-June 1949 issue, p. 4l

Status : Continued. The manuscript of the tables is completed. Preparation

of the introduction for publication continued.

1 1 02-40-1 110A7-2 TABLES OF COULOMB WAVE FUNCTIONS

Origin : NBS

Manager : P. Rabinowitz

Full task description : Apr-June 1949 issue, p. 45

Status : Continued. Computations of F

Q

, F

Q

' , G

Q

, G

Q

1 were made for y=-4>|,

/’ =r

(? /

>=I\/2 ? with 0.5< P<50. These tables will provide initial

values to obtain results for the values of viand P by integration of the

differential equation. Values for L>0 can be generated from the recurrence

relations for the Coulomb wave functions, cf., I. A. Stegun and M. Abramowitz

"The generation of Coulomb wave functions by means of recurrence relations^"

The Physical Review, June 15? 1955.

1 102-40-1110/50-7 WAVE FUNCTION FOR LITHIUM

Origin : NBS

Sponso r: Bureau of Ordnance, USN

Manager : W. F. Cahill

Full task description : Apr-June 1950 issue, p. 36

Status : Completed. The objective of this task was the calculation of the wave function for ionized lithium in the 1s2s ~>S state, and of the

corresponding eigenvalue. The function 0=r..r 2

\Jr, where ^

is the wave function desired and r, and r 2 are the respective distances of the two

electrons from the nucleus, was expressed as a series:

(1) 0(r

1 ,r 2 ,0)= ZZ J2I +^i $^P^(cos 9)

where P« are Legendre functions and © is the angle subtended at the nucleus

by the two electrons. The coefficients $^are functions of r* and r 2 satisfying a set of coupled partial differential equations which are obtained by subs tituting_f rom (1) into the Schroedinger equation for the

problem. The fact that £ 0dominates 5^owas ma(^ e use °* in the computation.

The functions c£|_ 0as we ii as a small correction to <£ 0were computed from

difference equations obtained from the differential equations, using an iteration process. The functions 5)., and we re computed on the SEAC and

used in computing corrections to the eigenvalue.

The results are substantially in agreement with computations

carried out by other methods at Yale University, and they demonstrate the

suitability of the method used here. The SEAC codes developed for this

problem are usable for similar computations for other atoms.

7
8 Status of Projects

1102-40-1110/51-8 TABLES OF POWER POINTS OF ANALYSIS OF VARIANCE TESTS

Origin ; Section 11.3? NBS

Managers : A. J. Hoffman, S. Tsingou

Full task description ; Apr-June 1951 issue, p. 49

Status ; Inactive. For status to date, see Jan-Mar 1955 issue, p. 10.

1 102-40-1 1 10/52-7 REVISION OF MATHEMATICAL TABLES

Origin ; NBS

Managers ; W. F. Cahill, I. Stegun

Full task description ; July-Sept 1951 issue, p, 4l

Status ; Continued. Revision of "Tables of natural logarithms," vol. IV,

Mathematical Tables MT1 2 (1941), for reissue in the Applied Mathematics

Series continued.

1102-40-1110/52-14 TABLE OF ARCSIN FOR COMPLEX ARGUMENTS

Origin ; NBS

Manager ; B. Gill

Full task description : July-Sept 1951 issue, p. 4l

Status ; Inactive. For status to date, see Jan-Mar 1955 issue, p. 11.

1102-40-1110/52-18 EXTENSION OF THE TABLE OF HYPERBOLIC SINES AND COSINES

Origin ; NBS

Manager : K. Nelson

Full task description ; July-Sept 1951 issue, p. 4l

Status : Continued. The manuscript has been transmitted to the Government

Printing Office for publication.

Publication ; "Table of hyperbolic sines and cosines," (x=2 to x=10, 9S)

Applied Mathematics Series 45; in press.

1102-40-1110/52-23 TABLE OF THE MODIFIED AIRY INTEGRAL

Origin ; NBS

Manager ; P. Rabinowitz

Full task description ; July-Sept 1951 issue, p. 42

Status ; Continued. The tables and the introduction have been completed

and are being prepared for publication.

1102-40-1110/52-25 TABLE OF ERROR FUNCTION FOR COMPLEX ARGUMENTS

Origin ; NBS

Manager ; W. -Hall

Full task description : July-Sept 1951 issue, p. 42

Status ; Inactive. For status to date, see Oct-Dec 1954 issue, p. 11.

1102-40-1110/52-31 EXTENSION OF TABLES OF THE EXPONENTIAL FUNCTION FOR

NEGATIVE ARGUMENTS

Origin ; NBS

Managers ; E. Marden, S. Prusch

Full task description ; July-Sept 1951 issue, p. 43

Status ; Continued. The table is in press at the U. S. Government Printing

Office

Publication ; "Table of the descending exponential," (x=2.5 to x=10, 20D)

Applied Mathematics Series 46; in press.

1102-40-1110/52-37 SPHEROIDAL WAVE FUNCTIONS

Origin ; NBS

Manager ; D. Liepman

Full task description ; Oct-Dec 1951 issue, p, 38

Status ; Inactive.

1102-40-1110/52-49 RADIAL MATHIEU FUNCTIONS

Origin ; NBS

Manager ; I. Rhodes

Full task description ; Jan-Mar 1952 issue, p. 45

Status ; Inactive. For status to date, see July-Sept 1954 issue, p. 10.
Status of Projects 9

1102-40-1110/52-57 SIEVERT'S INTEGRAL

Origin : NBS

Managers : M. Paulsen, P. O’Hara

Full task description : Jan-Mar 1952 issue, p. 46

Status : Continued. Preparation of the manuscript in a torm suitable for

publication has been started on the IBM card-controlled typewriter.

1 102-40- 1 1 10/ 53-52 L-SHELL CONVERSION COEFFICIENTS

Origin : Oak Ridge National Laboratory

Managers : W. Hall

Full task description : Apr-June 1953 issue, p. 45

Status : Continued. Computations for the LT shell have been completed.

Computations for the LTT shell were completed except for z=l5. The

following eigenvalues were run for the L^.^ shell: z=65, z=75.

371 1-60-0009/55-65 AUTOMATIC CODING

Origin : NBS

Manager : J. Wegstein

Full task description : July-Sept 1954 issue, p. 11 Status : Continued. J. Cooper investigated a method for modifying the order

for conversion from binary to decimal notation. The aim of the modification

is to use the order for editing output data automatically,

A routine which will automatically "scan" any elementary function

over a given range of a single variable was written by J. Cooper and is

being checked. This routine is intended primarily to serve as a tool in

planning integration procedures for integrals involving complicated

elementary functions.

Experiments are being made by J. Wegstein with a routine for

automatically determining the minimum point of a function S (x^ ,X£ > . • .x

n

) where S>0 . Such a routine would be useful, among other things, for

fitting curves to arbitrarily chosen functions and for solving sets of

simultaneous (not necessarily linear) equations.

A compiler routine has been written by J. Bram. It takes a program

written in a pseudo code and prepares a standard three-address SEAC code.

The pseudo code consists of symbols which correspond almost exactly with

mathematical and flow diagram symbolism. In particular, any mathematical

expression suitably furnished with parentheses to avoid ambiguity and

consisting only of the four rational arithmetical operations and the

elementary functions can be written by a symbol-f or-symbol transliteration

of the mathematical symbols into the pseudo-code symbols. The final

compiled code is a fixed-point three-address code.

A technical memorandum on "The Base 00 computational system" was completed during this quarter.

2 . Mathematical Services

Note: The tasks under Mathematical Services are arranged serially according

to the digits following the slant lines in the task number. The first two

digits following the slant line designate the fiscal year in which the task was authorized.

1 102-40-5126/50-13 RAY TRACING

Origin and Sponsor : NBS, Section 2.2

Manager : E. Marden

Full task description : Jan-Mar 1950 issue, p. 33

Status : Inactive.
10 Status of Projects

1 102-40-5126/51-37 MOLECULAR STRUCTURE, III

Origin and Sponsor ; Naval Research Laboratory, USN

Manager ; P. O’Hara

Full task description ; July-Sept 1951 issue, p. 50

Status ; Continued. Refinement procedures were continued for improving

the structure of the spurite crystal.

1 102-^0

- 5126/52-20 SPHERICAL BLAST

Origin and Sponsor ; Naval Ordnance Laboratory

Manager ; D. Jirauch

Full task description ; July-Sept 1951 issue, p. 56

Status ; Completed. A report entitled "Dynamic model of a spherically

symmetric explosion" is being prepared. The report is concerned with the

numerical solution of the partial differential equation of hyperbolic

type describing gas flow behind a shock wave. It describes a new method

of programming the numerical integration of such equations for automatic

computors. The method is adapted to cases in which memory space in the

machine is a limiting factor; it is applicable to all partial differential

equations for which stepwise integration is appropriate.

1 102-40- 51 26/52-44 CALCULATIONS FOR d SPACINGS

Origin and Sponsor ; NBS , Division 9

Manager ; A. Futterman

Full task description ; Oct-Dec 1951 issue, p. 47

Status ; Continued. Calculations for d-spacings were performed for the

following cases; (1) orthorhombic crystals: Cs^rO^ , NgSiO^, NH^CIO^;

(2) hexagonal crystals*. (NHi

+ )GeF^ , K^GeF^, Ne^PtF^, NaFeO^; (3) tetragonal

crystal CaWO^. Redetermination of unit cell constants by least squares

fitting to measured d-spacings was carried out for the following crystals;

(1) orthorhombic; Cs-pCrO^., NH^CIO^; (2) hexagonal: Asl^ , (NHj

+ )GeF^

K

2GeF£) ; (3) tetragonal: CaWO^,

1102-^0-5126/53-27 COMPUTATION OF THERMODYNAMIC FUNCTIONS

Origin and Sponsor ; NBS, Section 3.2

Manager ; E. Marden

Full task description ; Jan-Mar 1953 issue, p. 57

Status ; Continued. The SEAC code for computing thermodynamic functions to

include anharmonicity , rotation-vibration coupling, centrifugal stretching,

and non-classical rotational terms has been modified to accommodate

molecules with a greater number of frequencies. Calculations of thermodynamic

functions from 50°K to 15>000°K were performed for the following molecules:

HCN , DCN , TCN , and CF 2C12

. In addition, calculations were made for (C 2C1^) and (C

2

Br)

+

) using

the original code for this problem.

1 1 02-40-51 26/73-51 RADIATION DIFFUSION

Origin : NBS, Section 4.8

Sponsor ; Atomic Energy Commission

Manager ; A. Futterman

Full task description ; Apr-June 1953 issue, p. 57 (Neutron Diffusion III)

Status ; Continued. At present attention is fixed on a study of the diffusion

of radiation from a source in the plane interface between two different

homogeneous media. A calculation of elastic scattering cross sections for

electrons is being carried out. A code for the first phase of this

problem, viz., the evaluation of the gamma function for complex arguments,

has been completed.
Status of Projects 11

1 102-40-51 26/54-13 AWARD OF PROCUREMENT CONTRACTS BY LINEAR PROGRAMMING

Origin and Sponsor : New York Quartermaster Procurement Agency

Manager : H. Bremer

Full task description: Oct-Dec 1953 issue, p. 5-3 Status : Continued. Analyses of contract award problems were carried out on SEAC at the rate of about one per week. The contracts covered items such

as overcoats, trousers, shirts, pillow cases, tents, and insignia.

371 1 -60-0009/54-1 7 DE POLYMERIZATION

Origin : NBS , Section 7 . 6

Manager : J. Bram

Full task description : Oct-Dec 1953 issue, p. 44

Status : Inactive. For status to date, see Jan-Mar 1955 issue, p. 15

371 1-60-0009/54-19 ENERGY LEVELS OF COMPLEX ATOMS

Origin : NBS, Section 4.1

Manager : W. Hall

Full task description : Jan-Mar 1954 issue, p. 4l

Status : Continued. A code to form matrices of electrostatic and magnetic

interactions for four electron spectra with d- and s-electrons

(d Lf +d3s+d2s2) , given 20 parameters which are the Slater integrals of

electrostatic interaction and the integrals of magnetic interaction,

has been checked out. This code has been used to generate matrices

for the spectrum of Tantalum II and Ruthenium I.

1102-40-5126/54-21 EXPERIMENTAL PROGRAM FOR MULTIPLE INPUT AND OUTPUT

Origin : NBS, Section 12.3

Sponsor : Bureau of Supplies and Accounts, Department of the Navy

Manager : I. Rhodes

Full task description : Jan-Mar 1954 issue, p. 42

Status : Terminated. No further studies of a general nature are contemplated

at this time. Specific applications growing out of this task are being

reported under other task headings, especially 1 102-40-5126/55-102, p. 16 . 371 1-60-0009/54-22 ENERGY DISTRIBUTIONS ON OPTICAL IMAGE

Origin : NBS, Section 2.2

Manager : L. S. Joel

Full task description : Jan-Mar 1954 issue, p. 43

Status : Inactive.

371 1-60-0009/54-30 SPECTRUM ANALYSIS

Origin : NBS, Division 4

Manager : S. Prusch

Full task description : Jan-Mar 1954 issue, p. 46

Status : Continued, A SEAC code was prepared for computing wave numbers (<r) from wave lengths (\) . Also a code is being prepared for the searching

of pairs of lines.

1 102-40-5126/54-33 AIR CONDITIONING IN UNDERGROUND STRUCTURES

Origin and Sponsor : NBS, Section 10.3

Manager : S. Tsingou

Full task description : Jan-Mar 1954 issue, p. 47

Status : Completed. The desired integral (cf. Jan-Mar 1955 issue, p. 16)

has been evaluated for about 20 values of the parameter ett/a2 # 1102-40-5126/54-34 EQUILIBRIUM CALCULATIONS FOR WATER

Origin and Sponsor : Naval Ordnance Laboratory

Manager : J. Wegstein

Full task description : Apr-June 1954 issue, p. 38

Status : Inactive.
12 Status of Projects

1 102-40-51 26/54-35 THERMAL STRESSES IN STRUCTURES

Origin and Sponsor : NBS, Section 6,4

Manaqer : W. Hall

Full task description : Apr-June 1954 issue, p. 38

Status ; Inactive. For status to date, see Jan-Mar 1955 issue, p. 16.

371 1-60-0009/55—36 VELOCITY OF LIGHT

Origin : NBS , Section 11.3

Manager : P. Rabinowitz, J. M. Cameron

Full task description : Jan-Mar 1955- issue, p. 47

Status : Continued. Additional third degree polynomials were fitted to data

submitted by the sponsor to determine rotational constants of HCN and DCN

The method of least squares was used.

371 1 -60-0009/55--38 COMPRESSIBILITY FACTORS OF DRY AIR

Origin : NBS, Section 3.2

Manager : M. Paulsen

Full task description : Jan-Mar 1954 issue, p. 48

Status : Continued. The code was completed and a test run was made. This

code, using floating-point routines, computes a table of compressibility

factors z and densities^ where z=A+B^ +C^ 2+D^8+ # #

and £

= P/RzT. For

a given table the temperature T remains constant and the pressure P

varies

The virial coefficients A,B,C,D,... for a given temperature are furnished by the sponsor. The test runs carried out thus far have been

concerned with a third degree polynomial with A=1 . The present code can handle polynomials up to the eleventh degree with A arbitrary.

1 102-40-5126/55-39 MOLECULAR vibrations

Origin and Sponsor : NBS, Section 3.2

Manager : K. Goldberg

Full task description : July-Sept 1954 issue, p. 16

Status : Continued, Nineteen symmetric matrices of the form A’BA were computed, given the rectangular matrix A and the symmetric matrix B.

The eigenvalues and eigenvectors of these matrices were then calculated.

1 102-40-5126/54-43 CHARACTERISTICS OF CONDUCTING RESISTORS

Origin and Sponsor : NBS, Section 12.1

Manager : B. Gill

Full task description : Apr-June 1954 issue, p. 40

Status : Continued. The code was adjusted to process smaller blocks of data

20 sets were run and the results were transmitted to the sponsor.

371 1-60-0009/55-53 ELECTRONIC FUNCTIONS

Origin : NBS, Section 4.5

Manager : W. G. Hall

Objective : To prepare eight tables of the quantities

V

V* = V + —

2

2mQC^

•= -tv?

} =

h 1 42mQ

e 4V*

P* = H

e =/3W*
Status of Projects 13

e 2 V = W*-1

W* = E*+1

/3 = £* = w*

where each of the eight variables V, V*, H^>,/1, p* , E*, W*,,3is taken in

turn as independent variable. Then to merge the tables 0 !' these functions

into one table arranged according to increasing values of V. The functions

are to be tabulated so that each of the variables takes on the values

axlOP, for a=1 (

. 1 )3( .2) 5( . 5)9. 5 for integral values of P ranging from 1 to as high as 19 for some of the variables.

Bac kqround : These tables of electronic functions will be a revision of those

included in NBS Mathematical Table MT17) "Miscellaneous Physical Tables,"

(1941; now out of print). The present computations are being carried out

in accordance with theoretical investigations of L. Marton of the NBS

Electron Physics Section (4-. 5).

Status : New. Coding of the problem was started,

1102-40-5126/55-58 ELECTROMAGNETIC RADIATION FROM LIGHTNING

Origin and Sponsor : NBS, Section 82.1

Manager : B. Gill

Full task description : Jan-Mar 1955 issue, p. 17

Status : Inactive. For status to date, see Jan-Mar 1955 issue, p. 17.

1 102-40-5126/55-61 ELASTIC CROSS SECTION FOR NEUTRON SCATTERING

Origin and Sponsor : Naval Research Laboratory

Manager : I. Stegun

Full task description : Oct-Dec 1954 issue, p. 18

Status : Continued. Computations were carried out on SEAC for 335 sets*of

parameters. Comparison of theoretical results with experimental results

for lead, copper, carbon, tin, and aluminum continued,

1102-40-5126/55-62 INTEGRALS INVOLVED IN SUPERSONIC FLUTTER

Origin and Sponsor : National Advisory Committee for Aeronautics

Manager : S. Tsingou

Full task description : July-Sept 1954 issue, p. 17

Status : Completed. A code to check the tables was completed, and the tables were checked out. A manuscript was prepared and results were sent to

the sponsor.

1 102-40-5126/55-64 HIGH-TEMPERATURE THERMODYNAMIC TABLES (FMT)

Origin and Sponsor : NBS, Section 3.2

Manager : J. Wegstein

Full task description : July-Sept 1954 issue, p. 18

Status : Inactive.

371 1-60-0009/55-66 RECONSTITUTION OF MONOCHROMATIC LIGHT INTENSITIES

Origin : NBS, Division 30

Manager : H. Bremer

Full task description : July-Sept 1954 issue, p. 18

Status : Inactive. For status to date, see Oct-Dec 1954 issue, p. 19.

371 1-60-0009/55-68 CRYSTAL STRUCTURE CALCULATIONS

Origin : NBS, Division 9

Manager : A. Futterman

Full task description : Jan-Mar 1955 issue, p. 18
14 Status of Projects

Status : Continued. Checking of the code for the calculations of d-spacings

continued. This code is related to the one used in task 1 102-40-51 26/52-44

(p. 10) but is more general in that it includes the monoclinic and triclinic

systems to which the older code does not apply.

371 1-60-0009/55-69 THEORY OF DIELECTRIC RELAXATION

Origin ; NBS, Section 7.6

Manager ; K. Goldberg

Full task description ; July-Sept 195^ issue, p. 19

Status ; Continued. A paper explaining the method used to prove the reality

of the roots of certain matrices and applying this method to the matrices

arising in the theory of dielectric relaxation is in preparation.

1 102-40-5126/55-72 MARYLAND IN TER-INDUS TRY STUDY

Origin and Sponsor ; pirectorate of Intelligence, USAF*

Manager ; L. S. Joel

Full task description ; July-Sept 1954 issue, p. 20

Status ; Completed. All desired computations, consisting of the inversion

of certain Leontieff matrices, were carried out and the results were

transmitted to the originator.

1 102-40-5126/55-73 HEAT CONVECTION

Origin ; University of Minnesota

Sponsor ; Atomic Energy Commission

Manager ; G. Hawkins

Full task description ; Oct-Dec 195^ issue, p. 20

Status ; Completed. Several programs were written and solutions were obtained

corresponding to minor variants in the mathematical formulation of the

problem. Results have been communicated to the University of Minnesota

where a decision on publication will be made.

1 102-40-5126/55-7^- LIQUID-VAPOR TRANSITION, II

Origin and Sponsor ; Naval Medical Research Institute

Manager ; S. Prusch

Full task description ; Jan-Mar 1955 issue, p. 19

Status ; Continued. The integral equation was solved on punched-card

machines for two values of the parameter and two given sets of functional

values

371 1-60-0009/55-75 PARAMETER OF THE DISPERSION EQUATION FOR

Origin ; NBS, Section 1.6

Manager ; R. Zucker

Full task description ; Jan-Mar 1955 issue, p. 20

Status ; Continued. The constants K. of the equation

2 _L K± \

2

n -1 = 21 3

i4 A

2-*/

OPTICAL GLASS

were evaluated by the method of least squares for given values of n,Aand

A^. Then a table of n was calculated for the given values of A and and

the obtained values of K. . A table of the absolute differences of the

observed and calculated values of n was also obtained. Finally, a table

of n was submitted to the sponsor f or A =0( .02) 1 ( .04) 2( . 2) 3 ( . 5) 1 0( .2) 50.

*The origin and sponsor of this task was incorrectly reported in previous

issues.
Status of Projects 15

1 102-40-5126/55-78 NEUTRON TRANSPORT

Origin and Sponsor : Atomic Energy Commission, New York

Manager : S. Tsingou

Full task description : Oct-Dec 1955- issue, p. 21

Status : Completed. No SEAC computations beyond those already reported

appear to be required at this time.

1 102-40-5126/55-79 COMPLETE DEGRADATION IN THE NEUTRON

Origin and Sponsor : Atomic Energy Commission, New York

Manager : A. Futterman

Full task description : Oct-Dec 1954 issue, p. 21

Status : Completed. No further computations are required at this time.

1 102-40-5126/55-80 ALPHA-ANALYSIS

Origin and Sponsor : Atomic Energy Commission, New York

Manager : W. G. Hall

Full task description : Oct-Dec 1955- issue, p. 22

Status : Completed. No further computations are required at this time.

3711-60-5126/55-81 COMBINING TESTS FOR SIGNIFICANCE

Origin : NBS , Section 11.3

Manager : L. S. Joel

Full task description : Oct-Dec 1954, p. 23

Status : Continued. Checking of the code continued.

371 1-60-0009/55-82 THERMOMETER CALIBRATIONS

Origin : NBS, Section 3.1

Manager : S. Prusch

Full task description : Jan-Mar 1955 issue, p. 20

Status : Inactive. For status to date, see Jan-Mar 1955 issue, p. 21.

371 1-60-0009/55-86 FLOW COEFFICIENTS FOR FLUIDS

Origin : NBS, Section 6.7

Manager : P. J. Walsh

Full task description : Jan-Mar 1955 issue, p. 21

Status : Continued. Using empirical formulae similar to (1) in the previous

report (Jan-Mar 1955 issue, p. 21), a code was prepared to evaluate flow

coefficients for 10 values of D ranging between 2.067 and 15.25 and for

approximately 40 values of 3 where ,1<y3<.82. The code is now being checked.

1 102-40-5126/55-87 "ZERO" METHOD DETERMINATION OF CRYSTAL STRUCTURE

Origin and Sponsor : NBS, Section 9.7

Manager : J. Bram

Full task description : Jan-Mar 1955 issue, p. 22

Status : Inactive. For status to date, see Jan-Mar 1955 issue, p.' 22.

1 102-40-5126/55-88 STRESSES IN a WALL FOUNDATION

Origin and Sponsor : NBS, Section 10.1

Managers : E. Marden, G. Hawkins

Full task description : Jan-Mar 1955 issue, p. 22

Status : Continued. Stresses in a wall foundation were computed for a given

set of data corresponding to a single wall.

371 1-60-0009/55-89 TEMPERATURE DISTRIBUTION IN SOLID WINGS HEATED

AERODYNAMICALLY

Origin : NBS, Section 6.4

Manager : P. Davis, W. F. Cahill

Full task description : Oct-Dec 1954 issue, p. 22

Status : Inactive. For status to date, see Jan-Mar 1955 issue, p, 23.
16 Status of Projects

371 1-60-0009/55-90 STUDY OF A DIFFERENCE EQUATION ARISING IN STATISTICAL

MECHANICS

Origin : NBS, Section 3.2

Manager : M. Newman

Full task description : Oct-Dec 195*+ issue, p. 23

Status : Terminated. It appears to be unnecessary at this time to pursue

the study of this problem beyond the solution previously reported (cf.

Oct-Dec 1 95*+ issue, p. 23).

3711-60-0009/55-91 COMPUTATION OF VIRIAL COEFFICIENTS

Origin : NBS, Section 3.2

Manager : P. Davis

Full task description : Oct-Dec 1955- issue, p. 23

Status : Terminated.

3711-60-0009/55-92 CYLINDRICAL ELECTRON LENS CALCULATIONS

Origin : NBS, Section 4.5

Manager : W. G. Hall

Full task description : Jan-Mar 1955 issue, p. 24

Status : Continued. Approximately 200 trajectories were computed. Two

values of A were used corresponding to two systems. Except for prints

the same routine was used for all cases. Only x and the direction and

magnitude of the initial velocity were changed.

1 102-^0-5126/55-96 DISTRIBUTION OF LIGHT OUTPUT FROM NEUTRON SPECTROMETERS

Origin and Sponsor : NBS, Section 4.11

Manager : I. Rhodes

Full task description : Jan-Mar 1955 issue, p. 25

Status : Completed. The results were -transmitted to the sponsor. They show

that it is feasible to construct a spectrometer 100 times as powerful

as those previously available.

1102-40-5126/55-98 THERMAL DIFFUSIVITY

Origin and Sponsor : NBS, Section 3.2

Manager : P. Davis

Full task description : Jan-Mar 1955 issue, p. 25

Status : Terminated. Present needs have been taken care of with standard

solutions

1 102-40-5126/55-102 COST ACCOUNTING OPERATIONS FOR HIGH SPEED COMPUTERS

Origin and Sponsor : Bureau of Supplies and Accounts, Dept, of the Navy

Managers : G. Hawkins, E. Marden

Full task description : Jan-Mar 1955 issue, p. 25

Status : Continued. Three main routines were written and checked on UNIVAC.

In addition, several input data handling routines and an edit routine were written and checked.

1102-40-5126/55-103 MISSOURI RIVER PROBLEM

Origin and Sponsor : Missouri River Division, Corps of Engineers, U.S.Army

Manager : H. Bremer

Full task description : Jan-Mar 1955 issue, p. 26

Status : Continued. Codes have been written and checked to obtain simple and

multiple regression coefficients. These codes have been written for

UNIVAC

1 102-40-5126/55-105 AERONOMIC TIDAL WINDS OF THERMAL ORIGIN

Origin and Sponsor : NBS, Section 82.1

Managers : B. Gill, H. Howe

Full task description : Jan-Mar 1955 issue, p. 26

Status : Inactive. For status to date, see Jan-Mar 1955 issue, p. 26.
Status of Projects 17

i 102-40- 51 26/55-106 reflex klystron

Origin and Sponsor : Diamond Ordnance Fuze Laboratory, Department of the Army Manager J. Bram

Full task description ; Jan-Mar 1955 issue, p. 2

Status : Completed. The desired integral n, representing the efficiency 0 f

a reflex klystron, was evaluated for about 300 combinations of the parameters

c, coTq and V-j/Vq (for definitions, see previous report, p. 27). Specifically,

for each of the values c =100, 10,1 , ,5> .2, .1 , values of ouTq were chosen so that the expression 9q=342 goTq(4c+1 -Vc

)

has one of the values 27r(n+3/4) , n=3 ?5 9 7?9?11; for each of these combinations of c and uTq, n was computed

for about ten values of V-j/Vq lying between 0 and (V-j/Vq)*, the latter

being approximately the value at which n=0. (For fixed c and uffn , n increases monotonically from 0 at V.j/Vq=0 to a maximum, then decreases

and crosses the V^/Vq axis.)

1 102-40-51 26/55-1 09 ELECTRON PULSE HEIGHT DISTRIBUTION

Origin and Sponsor : NBS , Section 4.11

Manager : R. Zucker

Full task description : Jan-Mar 1955 issue, p. 27

Status : Completed. The results were transmitted to the sponsor.

1 102-40-5126/55-1 13 REACTOR DESIGN

Origin : Westinghouse Atomic Power Division

Sponsor : Atomic Energy Commission Mager : U. Hochstrasser

Full task description : Jan-Mar 1955 issue, p. 28

Status : Continued. Methods for the numerical solution of this problem have

been investigated. A simple test problem has been set up and a set of

difference equations approximating the system of differential equations

has been derived. A code is being prepared for finding the solution of

this set of 600 difference equations using D. Young's overrelaxation

method. This test problem will be used to investigate certain variations

of the overrelaxation method and to work out procedures and codes for

problems more general in the geometry of the reactor and in the type of

system of differential equations.

1102-40-5126/55-115 ADSORPTION INTEGRALS

Origin and Sponsor : NBS > Section 3.1

Manager : J. Cooper

Objective : To compute the integrals

for a given range of values of V and for various values of S2» and A*>. For the initial phase of the problem

Background : The function T represents the adsorption, of radiant energy

as it passes through a slit of width AVwhose geometry is specified by

the <T(AV,x) function. The results are to be compared with experimentally

observed values to judge the accuracy of such computations.

Comment : The problem was requested specifically by A. Bass of the NBS

Temperature Measurements Section (3.1).

Status : New. A code to compute the integrals was written using 16-point

Gaussian quadrature. Computations were carried out for values of Ai/=0.04,

0.2, 0.4, 0.8, 4.0, with s/ttcc =0.36 and oC=.4, The integrals were plotted

for j/=0(. 04)2.0.

CT(A^,x) = where y =

y

(if —x ) tt A*>
18 Status of Projects

1 102-40-51 26/55-1 1 7 ATTENUATION OF PRESSURE PULSES OF FINITE AMPLITUDE

Origin ; NBS, Section 3.2

Sponsor : Bureau of Aeronautics , U. S. Navy

Manager : M. Paulsen

Objective : To investigate the attenuation of finite-amplitude pressure

pulses in a simple pneumatic system with an enlarged section in the

transmission line. This involves solutions to the second-order partial

differential equation of the hyperbolic type,

(a2_u2) £

_

d* axaz az^

with u = by the method of characteristics for this type of equation.

Background : In many pneumatic ^sterns , attenuators are often inserted in

the system in order to reduce the transmission of pressure pulses from

the actuator upstream to the pressure regulator. The present investigation

studies the characteristics of such an attenuator as affected by the size

and geometry of the attenuator. To simplify the computation, the flow is

assumed to be one-dimensional and isentropic,

Comment : This problem was requested by D. H. Tsai of the Pneumatics

Laboratory (NBS, Section 3.2).

Status : New, Routines are in the process of preparation for computing the

particle velocity /

and sound velocity a at the boundaries between various

sections of the attenuator in a simulated pneumatic system. A study is

being made which will select the appropriate scheme for computing these

quantities at the boundaries of the various sections of the attenuator,

1102-40-5126/55-118 THERMOMETER CALIBRATION

Origin and Sponsor : NBS, Section 3.1

Manager : B. Gill

Objective : To fit polynomials to temperature-resistance measurements for

platinum-resistance thermometers and to use the polynomials obtained to

compute calibration tables.

Background : In 1939. E. W. Hogue and F. Brickwedde, of the NBS Temperature

Measurements Section (3.1), established a temperature scale in the region

from 10 to 90°K. This scale has been maintained at the NBS by means of a series of capsule-type platinum resistance thermometers. The scale is

generally accepted as the unofficial standard in this country, and it has

been distributed throughout this country by calibrating similar platinum

thermometers against our standards. This calibration involves comparison

of resistances at 15 different temperatures. Previously the data from

these comparisons have been reduced to a smooth table of resistance vs. temperature entirely by hand, involving about three weeks of computation

for a group of three thermometers.

Comment : This problem was requested by J. L. Riddle (NBS Section 3.1).

Status : New. The code for computing the table has been prepared. Experimentation

to determine a minimum-degree polynomial of satisfactory fit is in progress.

1102-40-5126/55-120 SOLUTIONS TO MATRICES

Origin and Sponsor : NBS, Section 5.9

Manager : K, Goldberg

Objective : To find the characteristic roots of several small (n<13) symmetric

matrices

Background : The problem originates in the calculation of the electronic

energy levels of the methane molecule.

Comment : The problem was requested by M. B. Wallenstein of the NBS

Thermochemistry Section (5.9).

Status : Completed (New). The results were transmitted to the sponsor.
Status of Projects -19

1102-40-5126/55-121 ELECTRON PENETRATION

Origin : NBS, Section 4.8

Sponsor : Atomic Energy Commission

Managers : P. Walsh, A. Futterman

Objective : To calculate a number of electron depth distributions by solving

an integro-dif f erential diffusion equation which describes fairly

accurately the penetration of electrons.

Background : The diffusion equation governing electron penetration and

diffusion in plane geometry contains three independent variables describing

an electron's energy, location, and direction. Integral methods have

recently been developed and tested for calculating spatial moments of

fairly high order of the electron flux. These integral methods, which

reduce the calculation of spatial moments to a recursive system, are ideally suited to the capabilities of high speed computers. Spatial

moments up to the 9th, 10th, or 12th order will be calculated with the

SEAC. A functional representation of the desired distribution which

agrees with these spatial moments will then be constructed. The latter

requires a numerical solution of a set of simultaneous equations, which

can also be easily accomplished with the SEAC. The whole procedure is

to be applied to enough situations, i.e., scattering media, source geometries, and source energies, so that interpolation will be feasible

to obtain almost any situation of interest to experimentalists up to

electron source energies of 10 Mev or so. Perhaps as many as 1 50

individual problems will be solved in the course of this program, i.e.,

involving 8 to 10 source energies, 6 to 8 scattering materials, and 2 to

3 geometries. Publication of standard tables of the results is planned.

Comment : This task is an extension of task 1 102-40-51 26/54-24. It is

requested by L, V. Spencer of the Nuclear Physics Section (4.8).

Status : New. The problem was analyzed and coded. The code is now being

checked on SEAC.

1 102-40-5126/55-1 22 SOLUTION OF NORMAL EQUATIONS

Origin : Advisory Committee on Weather Control

Sponsor : U. S. Weather Bureau

Manager : M. Newman

Objective : To compute (S'S - v

2

^ > where S, v^,V2 are rectangular

matrices, for several sets of data.

Background : The problem arises in weather prediction. The data given

represents annual precipitation averages throughout various parts of

Canada. The results are used to determine regression coefficients

which are employed in predicting the weather for those areas. Comment : The problem was specifically requested by H.C.S.Thom, Chief

Climatologist of the Advisory Committee,

Status : New. Eight runs consisting of four 40x6 and four 30x6 S-matrices

were completed, and the results were transmitted to the sponsor.

1102-40-5126/55-123 TEMPERATURE DISTRIBUTION

Origin : NBS , Section 6.4

Sponsor : NBS, Section 6.4

Manager : W. G. Hall

Ob jective : To compute non-steady temperature distributions for an I-beam,

taking into consideration heat convection and radiation. The structure

of any cross section is divided into 32 or fewer rectangular elements,

and the temperature of each element is given by the finite-difference

analog of the heat transfer equation.

The transfer of heat through any side is expressed by one of

the following relations
20 Status of Projects

(D q = [c+dC^2

)]

^

(t -o

mn

(2) q = [(»Ht„)A.(tX)'.]

(3) q =<rAi (GfHt

n ) Jl F (t

k

-t

Bk

) m- l

where t is the temperature in the m-th element adjoining the n-th

element with temperature t n

. The remaining parameters are constants

depending on the geometry of the structure and the material.

Background : This task is a generalization of a problem that was done

earlier on SEAC; see task 1102-40-5126/54—3. The problem was requested

by S. Goodman of the NBS Engineering Mechanics Section (6,4).

Status : New. Coding of the problem was started.

1102-40-5126/55-124 EVALUATION OF INTEGRALS RELATED TO GAS FLOW

Origin : NBS, Section 3.2

Sponso r: Bureau of Aeronautics, U.S. Navy

Manage r: J. Cooper

Objective: To evaluate the integrals

„ P

3 dr

Rl

~l (1 + ar)

fe /7 7l-r 2

"

2 =L

for various values of /3 and r. oC = 0.528, and a>0,0C</3^1«

Background : These integrals represent the time taken to fill a reservoir

of gas from a container at a higher pressure. The variable of integration

r represents the ratio of pressure in the two vessels, and integration

is performed from a lower limit where these equations apply to an upper

limit which is one or less. An upper limit of unity represents a complete

filling of the container, whereas a limit lower than one represents only

a partial filling of the container.

The two integrals R^ and R

2 are needed to represent both turbulent

and non-turbulent flow between the two containers.

The factor a in the equations depends on the relative sizes of the

reservoirs and on the deviation of the flow from the adiabatic case due

to heat losses.

Status : New. The code has been written and checked using 1 6-point Gaussian

quadrature. Values of R. and R

2 were computed for a=0.491 and /S ranging

between 0.528 and approximately^. 900 in steps of .005, and from 0.900

to 1 in steps of .002. Additional values at the same range of /3 were

computed for a=.1(.05)1.3 for R^ only. '

1 1 02-40-51 26/55-1 29 PROCESSING OF PUBLIC HOUSING DATA

Origin : NBS, Section 12.1

Sponsor : U. S. Public Housing Authority

Manager : I. Rhodes

Objective : To attempt to program the tabulation of certain statistical data

from reports on families in low rent housing, with special attention to

the handling of errors due to inconsistencies in the information reported.

Bac kground : A report on the reexamination of families in low rent housing

must be drawn up regularly at the U. S. Public Housing Authority. This
Status of Projects 21

is a compilation of data from detailed individual family reports, consisting

of answers to approximately 35 questions for each person in the family and

a comment column. Heretofore, mechanization of the compilation of the data

has been considered to be an impossible task because of errors and inconsistencies

in the data reported and because of the non-numerical comments that contain

pertinent statistical information.

Status : New. A few sample reports involving almost all possible types of errors were drawn up for study. All types were listed and each type was assigned a number. A code for tabulating inconsistent data was worked

out whereby the machine would be instructed to detect inconsistencies and

to indicate each by its number. At present, the code has been written for

the SEAC , but codes for other available machines will be considered. A

demonstration was held for the benefit of the staff of the Public Housing

Authority and other interested parties.

1102-40- 5126/55-128 GROUND REFLECTION COEFFICIENTS

Origin and Sponsor : NBS, Section 82.1

Managers : B. Gill, H. H. Howe ( 82 . 1 ) Objective : To prepare tables showing the sum of the intensities of the

directly incoming wave and its reflection for each of six components of

electromagnetic field, for various values of dielectric constant, angle of

incidence, and ratio of conductivity to frequency.

Background : At very low frequencies such as predominate in Sf erics, the

earth is usually a good reflector. It is impossible to measure directly

the intensity of an incoming radio wave because a receiver receives both

the directly incoming wave and the wave reflected from the ground. Under

reasonable simplifying assumptions the formula for the ground reflection

coefficient, which is the ratio of the intensities of the direct and

reflected waves, is known: it is given as a complex number, representing

both attentuation and change of phase. The coefficient depends upon (1)

angle of incidence, (2) dielectric constant of ground, (3) conductivity

of ground, and ( 4 ) frequency of wave. Items ( 3 ) and ( 4 ) enter the formula

for the coefficient only by their ratio, so that only three essential

parameters are involved. The formula is rather intricate, nevertheless;

moreover, there are two reflection coefficients, according to the plane

of polarization of the incoming wave. The tables, giving the sum of the

intensities of the direct and reflected waves will be useful in the

interpretation and analysis of observations in Sferics.

Status : New. The code was written and is now being checked.
III. STATISTICAL ENGINEERING LABORATORY

(Section 1 1 .3)

1 . Fundamental Research in Mathematical Statistics

BIBLIOGRAPHY AND GUIDE TO STATISTICAL LITERATURE

Task 1 103-10-1 107A9-1 a Origin: NBS Authorized 1/9/49

Manager: L. S. Deming

Full task description: Apr-June 19^9 issue, p. 75

Status: CONTINUED. For a description of the continuing activity

on this task, see the Jan-Mar 195*+ issue, p. 49.

MANUAL ON FITTING STRAIGHT LINES

Task 1103-10-1107/50-2

Origin: NBS Authorized 3/1/50

Manager: F. S. Acton

Full task description: Jan-Mar 1950 issue, p. 42

Status: CONTINUED. The manuscript is currently in the hands of

technical advisors, for comment.

TABLES TO FACILITATE DRAWING RANDOM SAMPLES

Task 1103-10-1107/51-1

Origin: NBS Authorized 7/1/50

Managers: C. Eisenhart, L. S. Deming

Full task description: July-Sept 1950 issue, p. 57

Status: INACTIVE. For status to date see July-Sept 1952 issue,

P. 64.

22
Status of Projects

23

MISCELLANEOUS STUDIES IN PROBABILITY AND STATISTICS

Task 1103-10-1107/51-2

Origin: NBS Authorized 7/1/50

Manager: C. Eisenhart

Full task description: July-Sept 1950 issue, p. 58

Status: CONTINUED. Dr. Geisser prepared a manuscript giving

necessary and sufficient conditions for the independence of the mean and mean square succession difference (see ( 3 ) below).

Publications:

(1) On moments of order statistics from the Weibull distribution.

J. Lieblein. Ann. Math. Stat. 26, 330-333 (June 1955).

(2) Time-discrete stochastic processes in arbitrary sets, with applications

to processes with absorbing regions and to the problem oi loops in

Markoff chains. D. van Dantzig. Accepted for publication (in French)

in Annales de l’Institut Henri Poincare (Paris).

(3) A note on the normal distribution. S. Geisser. Submitted to a technical journal.

(b) Inequalities for probabilities associated with the multivariate normal

distribution. I. R. Savage. Submitted to a technical journal.

PROCEDURES OF NON- PARAMETRIC STATISTICS

Task 1103-10-1107/52-2

Origin: NBS Authorized 9/17/51

Manager: I. R. Savage

Full task description: July-Sept 1951 issue, p. 66

Status: INACTIVE. For the latest report on this task see July-Sept

195^ issue, p. 22.

Publications

(1) Easily used simultaneous confidence limits for a line. W. S. Connor.

Submitted to a technical journal.

(2) Contributions to the theory of rank order statistics. I. R. Savage.

Submitted to a technical journal.

STUDIES IN THE MATHEMATICS OF EXPERIMENT DESIGN

Task 1103-10-1107/53-1

Origin: NBS Authorized 10/15/52

Manager: W. H. Clatworthy

Full task description: Oct-Dec 1952 issue, p. 60

Status: CONTINUED. M. Zelen's work on the analysis of covariance

as applied to incomplete block designs is now in manuscript (see •(*+) below).

Mr. Zelen has recently been able to show that when one replaces

the treatments in a partially balanced design (two associate classes)

with a balanced incomplete block design, the resulting design is a partially

balanced design with not more than five associate classes.

The reworking of publication (3) listed below to include two-way

elimination of heterogeneity is in progress. The revised manuscript is

intended for publication in the Applied Mathematics Series.

Publications:

(1) Partially balanced incomplete block designs with two associate classes

and two treatments per block. W. H. Clatworthy. J. Res. NBS

177-190 (Apr. 1955).

(2) Partially replicated latin squares. W. J. Youden and J. S. Hunter

(American Cyanmid Co.) Accepted for publicatibn in Biometrics.
2k Status of Projects

(3) Some fractional factorial arrangements for factors at two levels.

W. H. Clatworthy, W. S. Connor, and M. Zelen. In manuscript.

C

1*) On the analysis of covariance. M. Zelen. In manuscript.

RESEARCH ON MATHEMATICAL ASPECTS OF ORDER

STATISTICS METHODS

Task 1103-10-1107/55-110

Origin: NBS Authorized 3/3/55

Manager: J. Lieblein

Full task description: Jan-Mar 1955 issue, p. 31

Status: CONTINUED. A talk was presented on the theory and application

of extreme values before the U. S. Department of Agriculture Departmental

Committee on statistical design and analysis. A talk on the subject was

also presented at a Building Technology staff meeting (NBS), and discussions

were carried on with members of the Division regarding the application of

extreme-value methods to wind velocity data in connection with building

codes. In connection with these discussions, methods were worked out for

appropriate use of return period information in conjunction with probability

data. Advice was furnished in connection with a request from the David

Taylor Model Basin concerning the most suitable method of studying data

on extreme stress variations observed on a vessel during its winter voyages.

2 . Applied Research in Mathematical Statistics

COLLABORATION ON STATISTICAL ASPECTS OF NBS

RESEARCH AND TESTING

Task 3737-60-0002/51-1

Origin: NBS Authorized 7/1/50

Managers: W. J. Youden, J, Cameron

Full task description: July-Sept 1950 issue, p, 60

Status: CONTINUED. During this quarter members of the Section

provided statistical assistance and advice to a number of Bureau personnel.

The following are representative examples:

(1) Fitting of polynomials to data on atomic spectra for E. K. Plyler

and H. C. Allen, Jr., of section k,2 . ( 2 ) Analysis of data on a test of fire extinguishers for

A. F. Robertson and H. Shoub of 10 . 2 . (3) Presentation of material to personnel of Division 10 on the

applicability and uses of extreme value methods.

(k) Analysis of data to determine effects of size and shape on noise of composition type resistors for G. T. Conrad, Jr., of section 1.6.

(5) Planning and analysis of experiment to determine the radiant

energy absorption by water vapor on the output of total radiation pyrometers

for S. T. Lonberger, 3.1.

( 6 ) Analysis of data to determine precision of thermocouple

calibrations for B. E. Squires of section 3.1.

(7) Planning of fractional factorial experiment to determine

effects of various factors on fatigue characteristics of steel for H.E. Frankel

of 8.3.

(8) An analysis of data from an interlaboratory study of the

effect of cathode melts on transconductance of vacuum tubes for C.P. Marsden,

of 1.2.
Status of Projects 25

STATISTICAL ASPECTS OF NBS ADMINISTRATIVE OPERATIONS

Task 3737-60-0002/52-1

Origin: NBS Authorized 10/1/5

Manager: C. Eisenhart

Full task description: Oct-Dec 1951 issue, p. 56

Status: TERMINATED. Activities under this task will be reported

together with the activities of the major task under this project,

3737-60-0002/51-1

STATISTICAL SERVICES FOR COMMITTEE ON SHIP STEEL, NRC

Task 1103-^0-5105/52-1

Origin and Sponsor: Ship Structure Committee, NRC Authorized 12/1/51

Manager: W. J. Youden

Full task description: Oct-Dec 1951 issue, p. 58

Status: INACTIVE. For status to date see Oct-Dec 1954 issue, p. 27.

RESEARCH IN APPLICATIONS OF MATHEMATICAL STATISTICS TO

PROBLEMS OF THE CHEMICAL CORPS

Task 1103-40-5118/52-1

Origin and Sponsor: Biological Laboratories, Authorized 10/1/51

Chemical Corps, Dept, of the Army

Manager: C. Eisenhart

Full task description: Oct-Dec 1951 issue, p. 57

Status: CONTINUED. Computations are proceeding to find the operating

characteristics of a test involving the combination of two tests of

significance in the analysis of variance. Work on fractional factorial

designs with two way elimination of error continues.

STATISTICAL ANALYSIS AND DESIGN OF EXPERIMENTS

FOR THE U. S. GEOLOGICAL SURVEY

Task 1103-40-5140/54-1

>rigin and Sponsor: U. S. Geological Survey, Authorized 10/9/53

Department of Interior

Managers: C. Eisenhart, W. J. Youden

"ull task description: Oct-Dec 1953 issue, p. 50

Status: CONTINUED. (1) A statistically based sampling plan was worked out for the sampling of fine-grained deposits in streams for use

in a study of mineral content for Dr. Dorothy Carroll of the Sedimentary

Petrology Project of the Geochemistry and Petrology Branch.

(2) An investigation was made of the applicability of statistical

methods in establishing relationships between location, gradient, and

other factors with size distribution of pebbles in studies in soil hydrology

and erosion for J. T. Hack of the General Geology Branch.

(3) A mixed model analysis of variance was worked out for the

analysis of data from a study of the effect of solid state phenomena on

the fluorescence of uranium flux. This study is being conducted by

the Geochemistry and Petrology Branch in Denver.
26 Status of Projects

(b) A sampling plan based on the theory of extreme values was worked out in collaboration with W. C. Krumbein for use in an exploratory

study of particle sizes in sedimentary deposits, such as certain gravel

pits in Maryland. The data were analyzed with considerable success by means of extreme value methods and a joint paper on the outcome of the

study was prepared for publication by J, Lieblein and Dr. Krumbein as noted below.

Publication:

(1) Application of extreme value theory to cobbles and boulders in gravel

deposits. W. C. Krumbein (Northwestern University) and J. Lieblein,

In manuscript.

MANUAL ON EXPERIMENTAL STATISTICS

FOR ORDNANCE ENGINEERS

Task 1103-^+0-5l l+6/55-93

Origin and Sponsor: Office of Ordnance Research Authorized 1 2/29/5*+

Manager: C. Eisenhart

Full task description: Oct-Dec 195*+ issue, p. 28

Status: Paul N. Somerville, who arrived April 1, has begun active

work on the manual. Conferences were held at several Ordnance establishments

on various aspects of the manual.
IV. MATHEMATICAL PHYSICS SECTION

(Section 1 1 .*+)

RESEARCH IN MATHEMATICAL PHYSICS AND RELATED FIELDS

Task 110*+-1 0-1 11 5/55-57

Origin: NBS Authorized 9/1/5*+

Manager: E. W. Cannon

Full task description: July-Sept 195*+ issue, p. 27

Status: CONTINUED. P. Henrici revised and enlarged a manuscript

on Legendre and Gegenbauer functions. He has also completed a report on the evaluation of the integral

(J

1 (t/2)) 2

J

2 (ty) dt,

connected with problems in structural analysis, A network analysis has

also been completed, which demonstrates the averaging property for electrical

controls connected with NBS combustion experiments.

R. F. Dressier is continuing an investigation on local curvature

effects on edge stresses in bent plates.

Publications:

(1) Addition theorems for Legendre and Gegenbauer functions. P. Henrici.

Accepted for publication in Journal of Rational Mechanics and Analysis.

(2) On the vibration of a square clamped plate. M. Abramowitz and W. Cahill.

Accepted for publication in Journal of the Association for Computing

Machinery.

(3) Kleine Bemerkung in asymptotischen Entwicklung des Fehlerintegrals

P. Henrici. Accepted for publication in ZAMP.

(*+) Evaluation of a definite integral. P. Henrici. An NBS report (April

195*0.

RESEARCH IN ELECTROMAGNETIC THEORY

Task 11 04-1 0-51 60/5*+-*+7

Origin and Sponsor: Diamond Ordnance Fuze Authorized 6/29/5*+

Laboratory, Department of Revised 9/29/5*+

the Army

Manager: F. Oberhettinger

Full task description: July-Sept 195*+ issue, p. 28

Status: CONTINUED. F. Oberhettinger continued a study of problems

for electromagnetic fields. Expressions for the two Green's functions

of the equation Au+k^u=0^ different from the representations by modes,

have been obtained for an infinitely long cylinder with the rectangular

cross section. This study is connected with the radiation problem for a slotted wave guide, A report on this task is in preparation.

27
28 Status of Projects

RESEARCH IN MATHEMATICAL ELASTICITY

Task 11 0^-10-51 60/55-85

Origin: NBS Authorized 1 2/27/5^+

Sponsor: Office of Scientific Research, ARDC, USAF

Manager: R. F. Dressier

Full task description: Oct-Dec 195^ issue, p. 30

Status: CONTINUED. R. F. Dressier and W. F. Cahill have formulated

an unsymmetric matrix of order *+5 to obtain the lowest eigenvalue and

eigenvector of a triangular vibrating plate with aspect ratio 2 and Poisson's

ratio O" equal to .29. The complete calculation was carried out on SEAC.

The resulting frequency obtained is 19 °/o higher than the corresponding

experimental result reported in Gustafson, Stokey, and Zorowski (J. Aer.

Sci. , May 1953). This is to be expected since imperfect clamping and

aerodynamic loading both have the effect of lowering an experimental result

from the mathematical value.

R. F. Dressier and W. F. Cahill have also computed the lowest

frequency and mode for this shape using an order 28 matrix, to determine

the dependence on mesh size. The order 28 yields a frequency only 7°/

higher than the order *+5 result, and the corresponding vectors differ by a negligible amount. This result is encouraging since the order 28 formulation

has a mesh so sparse that only one interior point can retain its differential

stencil unmodified by the effect of the boundary conditions. It is expected

that the principal results will be based upon matrices of order 78. For

this size a new code had to be developed. This has now been completed by

I. Rhodes, and pilot models to test the code are being run. This code

incorporates a technique with a hypergeome trie polynomial of degree 19 to

separate eigenvalues for faster convergence and uses a power method for

unsymmetric real matrices to obtain higher modes. Experience with the

previous computations showed that convergence was so slow, due to the

proximity of adjacent eigenvalues, that some accelerating technique was necessary.

For the corrugated diaphragm problem, R. F. Dressier modified

linear elastic shell equations with rotational symmetry for arbitrary

meridial shape to include lateral loading (air pressure). In the first

form, the two unknowns are meridial rotation and a modified shear force,

leading to a non-homogeneous system of two ordinary differential equations

of total order four, requiring additional integrations to obtain displacements

Secondly, equations have been derived using displacements and shear as unknowns, giving three equations of total order six. For both formulations,

meridial arc length is taken as independent variable. The first set is not

directly applicable for regions where the shell is perpendicular to its

axis, and it possesses no advantage for electronic computation in requiring

two successive integration steps. Therefore the second formulation has

been chosen as the basis for analyses. These shell equations reduce to

plate bending and plane stress equations with circular symmetry over the

central flat portion of the diaphragm. The known solutions for these have

been used to define initial values at the first corrugation for the computer

calculations

The Runge-Kutta method has been coded for this sixth order system,

and a separate code has also been completed for calculating the many variable

coefficients appearing in the equations. The first diaphragm shape now being

solved is the one used by Grover and Bell (Soc. Exp. Stress Anal, j?, 2, 19^8),

in which the meridian consists of arcs of circles and straight line segments.

Three independent homogeneous solutions and one non-homogeneous solution

are needed to satisfy any boundary value problem at the outer edge. W. Cahill

has now obtained the first homogeneous solution, which is simultaneously

serving as a code-check. Independently of the machine, A. Cock is checking

the code by appropriate hand calculations. Approximate machine time for each

solution is ^5 minutes, using 1 50 integration points over the diaphragm radius
Status of Projects 29

The shell equation coefficients contain first and second

derivatives of the raeridial curvature. At each of the eleven transition

points between circular and straight sections, where curvature is

discontinuous, a suitable smoothing function has been calculated and

inserted in order to take account of the effect on the solutions of these

delta functions and their derivatives.
APPLICATION

of

NATIONAL BUREAU OF STANDARDS AUTOMATIC COMPUTER (SEAC)

The record of SEAC operations for unclassified tasks of the Applied

Mathematics Division for the period April 1 through June 30 is as follows:

Task No . NBS:

T7o4/55-55

1 110A7-2

1110/50-7

1110/53-52

5126/52-34

51 26/52-44

5126/53-15

5126/53-27

5126/53-29

5126/53-51

5126/53-53

0009/5^-1

0009/5^-19

5126/55-33

0009/55-38

51 26/55-5-3

0009/55-39

0009/55-53

0009/55-65

5126/55-75

5126/55-81

5126/55-82

5126/55-86

5126/55-87

5126/55-88

0009/55-92

5126/55-96

5126/55-97

5126/55-115

5126/55-117

5126/55-118

5126/55-121

5126/55-124

0002/51-1

5160/55-85

Misc

Misc

Hours Used:

Code Productive

Title Checkinq Operations

Numerical analysis 5 17

Tables of Coulomb wave functions 1 6

Wave function for lithium 1 L-shell conversion coefficients 90

Molecular vibrations 42

Calculations for d spacings 1 4

Nerve fiber reaction 1 3

Thermodynamics functions 1 3

Dynamic • behavior of aircraft structure 5 Radiation diffusion 7 52

Heterogeneity model 40

Matrix reduction 5

1 Energy levels of complex atoms 5 Air conditioning in underground structures

4

2

Compressibility factors of dry air

Characteristics of conducting resistors 1 Molecular vibrations 51 Electronic functions 5 Automatic coding 18 11 Dispersion equation for optical glass 1 2

Combining tests for significance 3 3

Thermometer calibrations 3 1 Flow* coefficients for fluids 3

"Zero" method determination of crystal

structures

1 2

Stresses in a wall foundation 2 9

Cylindrical electron lens calculations

Distribution of light output from neutron

16

spectrometers 1 25

High temperature properties of air 5 29

Adsorption integrals

Attenuation of pressure pulses of finite

4 2

amplitude 2

Thermometer calibrations 7

Electron penetration 2

Evaluation of integrals related to gas flow

Collaboration on statistical aspects of

2 3

NBS research and testing 3 32

Research in mathematical elasticity 5 61

Roots of polynomials 24 2

Training, subroutines, etc. 22 3

30
31

Task No . Title

Hours Used;

Code Productive

Checking Operation

Other:

5126/51-3 Meteorology 1 7

5126/55—13 Award of procurement contracts by linear

programming 1 16

5126/55—45- Flight performance 1 17

5126/55-61 Elastic neutron scattering 1 12

5126/55-62 Integrals involved in supersonic flutter 2 6

5126/55-72 Maryland inter-industry study 8

5126/55-73 Heat convection 11 4

5126/55-101 Transcendental equations 2 12

5126/55-104 Calculations of various fuzing systems 1 158

5126/55-105 Aeronomic tidal winds of thermal origin 2 3

5126/55-106 Reflex klystron 2 26

5126/55-107 Missile trajectory 10 4

Totals: T56 809
Lectures and Symposia

Note: In general, copies of papers or talks listed in this section are not available from the National Bureau of Standards. If and when a paper

is to be published, it will be listed in the section of this report on Publication Activities.

Numerical Analysis Lectures

(Offered by the American University in cooperation

with the National Bureau of Standards)

STIEFEL, E. (Federal Institute of Technology, Zurich). (1) Strategy in

solving linear equations. Methods using orthogonal polynomials,

conjugate gradients. April 4. (2) Application of the Fredholm

integral equation of conformal mapping. April 11. (3) Determination

of eigenvalues by vector iteration. Method of Lanczos. April 18.

(4) Determination of eigenvalues: Quotient-difference algorithm,

April 25.

Papers and Invited Talks

Presented by Members of the Staff

at Meetings of Outside Organizations

ABRAMOWITZ, M. (1) Computations on mathematical tables, and (2) Pitfalls

in computation. Presented at a Physics Seminar at the Massachusetts

Institute of Technology, Cambridge, Mass., June 23 and June 24.

ALT, F. L. (1) Problems exceeding the capacity of present day computers.

Presented at a Mathematics Seminar at the Massachusetts Institute

of Technology, Cambridge, Mass., May 24. (2) Computing machines:

Organization, functions, and components. A series of six lectures

presented at Wayne University, Detroit, Mich., June 6-11.

EISENHART, C. Some antecedents of modern experiment design. Presented

at a joint meeting of the Statistics Section of the Virginia

Academy of Science and the Virginia Chapter of the American

Statistical Association, Harrisonburg, Va., May 13.

HOFFMAN, A. J. Computing linear programs. Presented before the Mathematics

Division of the American Society for Engineering Education, at

Pennsylvania State College, June 20-24.

LIEBLEIN, J. (1) Theory and application of extreme value methods.

Presented at the U. S. Department of Agriculture, Departmental

Committee on Statistical Design and Analysis, Washington, D. C.,

May 4. (2) Some topics of interest in the application of extreme

values. Presented at the staff meeting of the Building Technology

Division, NBS , May 31.

32
Lectures and Symposia 33

TAUSSKY-TODD , 0. (1) Some practical bounds for eigenvalues of finite

matrices. Presented at the AEC Computing Facility, Institute of

Mathematical Sciences, New York University, New York, N. Y.

,

April 21 . ( 2 ) Matrix methods in algebraic number theory.

Presented at the Conference on Number Theory at California Institute

of Technology, Pasadena, California, June 22-2*+. (3) Integral

group matrices. Presented before the Numerical Analysis Research

Group, University of California at Los Angeles, California, June 27 . TODD, J. (1) The practical significance of stability in the solution of

partial differential equations. Presented at the Berkeley

Conference on Partial Differential Equations, University of California

Berkeley, Calif., June 20-July 1. (2) Controlled experiments in

high-speed computations. Presented to the California Chapter,

Society of Industrial and Applied Mathematics, San Francisco,

Calif., June 23.

YOUDEN, W. J. (1) Statistics for skeptics. Presented . before the Cleveland

Section, American Society for Quality Control, Cleveland, 0.,

April 1 . ( 2 ) Interlaboratory evaluation of analytical methods.

Presented as part of the Camp Detrick Course in Statistics, Camp

Detrick, Md., April 7 . Also presented at the Naval Research

Laboratory, Washington, D. C. , April 12 . (3) The gun problem.

Presented at the Marshall Laboratory, E. I. DuPont de Nemours and

Co., Philadelphia, Pennsylvania, April 15.

(*+) Dice, data, and deductions. Presented at a joint meeting

of the Virginia and Hampton Roads Sections, American Chemical

Society, William and Mary College, Williamsburg, Virginia, April 23.

(5) Statistical applications in the chemical industries. Presented

before a joint meeting of the Buffalo Section, American Society

for Quality Control and Buffalo Chapter of the American Statistical

Association, Buffalo, N. Y., April 25. ( 6 ) Problems arising in

engineering investigations. Delivered at a symposium on Statistical

Engineering, Chemical Corps Engineering Agency, Edgewood, Md.,

April 28.

(7) -Ten talks given before local sections of the American

Chemical Society on: Principles of experimental design; Interpretation

of chemical data; and Statistical aspects of analytical determinations

presented in Kansas City, Mo.; Manhattan, Kan.; Wichita, Kan.;

Ponca City, Okla.; Stillwater, Okla.; Tulsa, Okla.; Bartlesville,

Okla.; Pittsburg, Kan.; Columbia, Mo.; and Macomb, 111.; May 2-13.

ZELEN, M. Some new techniques in the design of experiments. Presented

before the George Washington University Logistics Group, George

Washington University, Washington, D. C., April 28.
Publication Activities

1 . PUBLICATIONS THAT APPEARED DURING THE QUARTER

1.1 Mathematical Tables

(1)

Table of characteristic values of Mathieu's equation for large

values of the parameter. G. Blanch and I. Rhodes. J. Wash. Acad.

Sci. 4£, 166-196 (June 1955’)

1 .3 Technical Papers

(1) On nonlinear differential equations of the second order with

integrable perturbation term. H. Antosiewicz. J. London Math.

Soc. 30 , 65-67 (19 55).

(2) Partially balanced incomplete block designs with two associate

classes and two treatments per block. W, H. Clatworthy. J. Res.

NBS £4, 177-190 (Apr. 1955).

(3) A multi-purpose orthonormalizing code and its uses, P. Davis and

P. Rabinowitz. J. Assoc. Comp. Mach. 1_, 183-191 (Oct. 195*0.

(4) Discrete analogs of inequalities of Wirtinger. K. Fan, 0. Taussky,

and J. Todd. Monatsh. Math. 73“90 (1955).

(5) An algorithm for solvinq the transportation problem. A. Gleyzal.

J. Res. NBS £4, 213-216 (Apr. 1955), RP2583.

( 6 ) Indentation pressure of a smooth punch. E. Levin, Qu. App. Math.

13, 134-137 (July 1955).

(7) On moments of order statistics from the Weibull distribution.

J. Lieblein. Ann. Math. Stat. 26, 330-333 (June 1955).

( 8 ) Structure theorems for modular subgroups. M. Newman. Duke Math. J.

22 , 25-32 (1955).

(9) Osculatory interpolation in the complex plane. H. E. Salzer (now

Department of the Army). J. Res. NBS £4, 263-266 (May 1955), RP2587.

(10)

The generation of Coulomb wave functions by means of recurrence

relations. I. A. Stegun and M. Abramowitz. Phys. Rev. £8, 1851-1852

(June 15, 1955).

2. MANUSCRIPTS IN THE PROCESS OF PUBLICATION JUNE 30, 1955

2.1 Mathematical Tables

(1) Table of hyperbolic sines and cosines, x=2 to x=10, 9S. Applied

Mathematics Series 45. In press, U. S. Government Printing Office.

(2) Table of the descending exponential, x=2.5 to x=10. Applied

Mathematics Series 46. In press, U. S. Government Printing Office.

34
Publication Activities 35

2.3 Technical Papers

(1) Forced heat convection in laminar flow through a tube. M. Abramowitz.

Submitted to a technical journal.

( 2 ) On the vibration of a square clamped plate. M. Abramowitz and

W. F. Cahill. Accepted, Journal of the Association for Computing

Machinery.

( 3 ) Approximate method for rapid Loran computation. M. Abramowitz,

D. H. Call, and J. C. Mathews. Submitted to a technical journal.

(4) A theorem on alternatives for pairs of matrices, H. Antosiewicz.

Accepted, Pacific Journal of Mathematics.

(5) Analytic study of war games. H. Antosiewicz. Submitted to a technical journal.

(6) Asymptotic solution of linear differential equations with a parameter.

H. Antosiewicz. Submitted to a technical journal,

( 7 ) On a certain integral involving Bessel functions. H. Antosiewicz.

Submitted to a technical journal.

(8) On the differential equation x + k(f(x) + g(x)x)x = ke(t).

H. A. Antosiewicz. Submitted to a technical journal.

(9) Stable systems of differential equations with integrable forcing

term. H. A. Antosiewicz. Accepted, Journal of the London

Mathematical Society.

( 10 ) Computation of vibration modes and frequencies on SEAC. W. F. Cahill

and S. Levy (NBS 6.4-). Accepted j Journal of the Institute of

Aeronautical Sciences.

( 11 ) On the design of two-dimensional nozzles by the method of characteristics.

B. Chaix (Federal Institute of Technology, Zurich) and P. Henrici.

Submitted to a technical journal.

( 12 ) Easily used simultaneous confidence limits for a line. W. S. Connor.

Submitted to a technical journal.

( 13 ) Time-discrete stochastic processes in arbitrary sets, with

applications to processes with absorbing regions and to the problem

of loops in Markoff chains. D. van Dantzig. Accepted for publication

(in French) in Annales de l’lnstitut Henri Poincare (Paris).

(14) On the theorem of Dilworth. G. B. Dantzig (RAND) and A. J. Hoffman.

To appear in Annals of Mathematics Study 38 . (15) On a problem in the theory of mechanical quadratures. P. Davis.

Accepted, Pacific Journal of Mathematics.

(16) Some Monte Carlo experiments in computing multiple integrals.

P. Davis and P. Rabinowitz. Submitted to a technical journal.

( 17 ) Some sampling results on the power of nonparame trie tests against

normal alternatives. W. J. Dixon (University of Oregon) and

D. Teichroew. Submitted to a technical journal.

(18) Comparison of theories and experiments for the hydraulic dam-break

wave. R. F. Dressier. To appear in the Proceedings of the Tenth
Publication Activities

General Assembly, of the International Union of Geodesy and

Geophysics, held in Rome, Italy, September 195*+.

(19) Heat flow in a fluid with eddying flow. W. H. Durfee (now with

Operations Research Office, Johns Hopkins University). Submitted

to a technical journal.

(20) A comparison theorem for eigenvalues of normal matrices. K. Fan.

Accepted, Pacific Journal of Mathematics.

(21) On systems of linear inequalities. K. Fan. Submitted to a technical journal.

(22) Some inequalities concerning positive-definite Hermitian matrices.

K. Fan. Accepted, Proceedings of the Cambridge Philosophical Society

(23) Systems of inequalities involving convex functions. Ky Fan.

Submitted to a technical journal.

(2^) Asymptotic lower bounds for the fundamental frequency of convex membranes. G. E. Forsythe. Accepted, Pacific Journal of Mathematics

(25) A note on the normal distribution, S. Geisser. Submitted to a technical journal.

(26) Power series for log e xe^. K. Goldberg. Accepted, Duke Journal

of Mathematics.

(27) Addition theorems for general Legendre and Gegenbauer functions.

P. Henrici, Accepted, Journal of Rational Mechanics and Analysis.

(28) On certain series expansions involving Whittaker functions and

Jacobi polynomials. P. Henrici. Accepted, Pacific Journal of

Mathematics

,

(29) On generating functions of the Jacobi polynomials. P. Henrici.

Accepted, Pacific Journal of Mathematics.

(30) On systems of distinct representatives. A, J. Hoffman and

H. W. Kuhn (Bryn Mawr College). To appear in Annals of Mathematics

Study 38.

(31) Systems of distinct representatives and linear programming.

A. J. Hoffman and H, W. Kuhn (Bryn Mawr College). To appear in

the American Mathematical Monthly.

(32) Acoustic radiation pressure on a circular disk, H. Levine. To

appear in the Proceedings of the Fifth Symposium on Applied

Mathematics of the American Mathematical Society, held in Pittsburgh,

Pa., June 1952.

(33) A proof of Hilbert's Nullstellensatz . T. S. Motzkin. Submitted

to a technical journal.

(3*+) The assignment problem. M, Motzkin. To appear in the Proceedings

of the American Mathematical Society's Sixth Symposium on Applied

Mathematics, held at Santa Monica City College, August 1953.

(35) On Fejer sets in linear and spherical spaces. T. S. Motzkin and

I. J. Schoenberg. Accepted, Annals of Mathematics,
Publication Activities 37

( 36 ) Pairs of matrices with property L, II. T. S. Motzkin and

0. Taussky. Accepted, Transactions of the American Mathematical

Society.

( 37 ) An alternative proof of a theorem on unimodular groups. M. Newman.

Submitted to a technical journal.

( 38 ) An identity for the coefficients of certain modular forms.

M. Newman. Accepted, Journal of the London Mathematical Society.

( 39 ) Generalizations of identities for the coefficients of certain

modular forms. M. Newman. Accepted, Journal of the London

Mathematical Society.

(40) The diophantine equation ax

ra -by n=c. M. Newman. Submitted to a technical journal.

(41) The normalizer of certain modular subgroups. M. Newman. Accepted,

Canadian Journal of Mathematics.

(^2 ) On a generalization of the normal basis in abelian algebraic

number fields. M. Newman and 0. Taussky. To appear in Communication

on Pure and Applied Mathematics of New York University.

(*+3 ) On asymptotic series for functions occurring in the theory of

diffraction of waves by wedges. F. Oberhe ttinger . Submitted to

a technical journal.

(41!-) On the Lerch zeta function. F. Obe rhe ttinger . Accepted, Pacific

Journal of Mathematics.

(^ 5 ) De terminanten mit ueberwiegender Hauptdiagonale und die absolute

Konvergenz von linearen Iterationsprozessen. A. M. Ostrowski.

Accepted, Comentarii Mathematici Helvetici.

(*+6) On Gauss' speeding up device in the theory of single step

iteration. A. M. Ostrowski. Submitted to a technical journal.

(^7) On the convergence of Gauss' alternating procedure in the method

of the least squares , I. A. M. Ostrowski. Submitted to a technical journal.

(*+8) On spectra of second-order differential operators. D. Ray.

Submitted to a technical journal.

(^9) On the convergence of asymptotic solutions of linear differential

equations. R. M. Redheffer (U.C.L.A.) and W, Wasow. Submitted

to a technical journal.

( 50 ) Contributions to the theory of rank order statistics. I. R. Savage.

Submitted to a technical journal.

( 51 ) Inequalities for probabilities associated with the multivariate

normal distribution, I. R. Savage. Submitted to a technical

journal.

( 52 ) A note on group matrices. 0. Taussky. Accepted, Proceedings of

the American Mathematical Society.

( 53 ) Unimodular circulants. 0. Taussky. To appear in the I. Schur

Memorial Volume to be published by Mathematische Zeitschrift.
38 Publication Activities

( 5*+) Some computational problems in algebraic number theory. 0. Taussky.

To appear in the Proceedings of the American Mathematical Society

Sixth Symposium on Applied Mathematics, held at Santa Monica City

College, August 1953 .

( 55 ) Generation and testing of pseudo-random numbers. 0. Taussky and

J. Todd. Submitted to a technical journal.

( 56) An improved cathode ray tube storage system. R, Thorensen. To

appear in the Proceedings of the Western Computer Conference of

the AIEE-IRE-ACM held in Los Angeles, Calif., February 1953 .

( 57 ) Machine attacks on problems whose variables are permutations.

C. B. Tompkins. To appear in the Proceedings of the American

Mathematical Society Sixth Symposium on Numerical Analysis, held

in Santa Monica, Calif., August 1953 .

( 58 ) An extension of a theorem of Dantzig's. C. Tompkins and I. Heller

(George Washington University). Submitted to a technical journal.

( 59 ) Best approximation polynomials of given degree. J. Walsh. To

appear in the Proceedings of the American Mathematical Society

Sixth Symposium on Applied Mathematics, held in Santa Monica,

Calif., August 1953 .

( 60 ) Discrete approximations to elliptic differential equations,

W. Wasow, Accepted, Zeitschrift fur angewandte Mathematik und Physik,

(61) On the asymptotic transformation of certain distributions into

the normal distribution. W. Wasow. To appear in the Proceedings

of the American Mathematical Society Sixth Symposium Applied

Mathematics, held at Santa Monica, Calif., August 1953 .

( 62 ) Error bounds for eigenvalues of symmetric integral equations.

H. Wielandt. To appear in the Proceedings of the American

Mathematical Society Sixth Symposium on Applied Mathematics, held

at Santa Monica, Calif., August 1953 .

( 63 ) On eigenvalues of sums of normal matrices. H. Wielandt. Submitted

to a technical journal.

( 65-) National physical standards and design of experiment. W. J. Youden.

To appear in Revue de L'Institut International de Statistique

(The Hague).

( 65 ) Partially replicated latin squares. W. J. Youden and J. S. ilunter

(American Cyanmid Co.) Accepted, Biometrics.

(66) The number of absolute points of a correlation. A. J. Hoffman,

M. Newman, E. G. Straus, and 0. Taussky. Accepted, Pacific

Journal of Mathematics.

COMM—DC—I4.9OI42
THE NATIONAL BUREAU OF STANDARDS

Functions and Activities

The functions of the National Bureau of Standards are set forth in the Act of Congress, March

3, 1901, as amended by Congress in Public Law 619, 1950. These include the development and

maintenance of the national standards of measurement and the provision of means and methods

for making measurements consistent with these standards; the determination of physical constants

and properties of materials; the development of methods and instruments for testing materials,

devices, and structures; advisory services to Government Agencies on scientific and technical

problems; invention and development of devices to serve special needs of the Government; and the

development of standard practices, codes, and specifications. The work includes basic and applied

research, development, engineering, instrumentation, testing, evaluation, calibration services, and

various consultation and information services. A major portion of the Bureau’s work is performed

for other Government Agencies, particularly the Department of Defense and the Atomic Energy

Commission. The scope of activities is suggested by the listing of divisions and sections on the

inside of the front cover.

Reports and Publications

The results of the Bureau’s work take the form of either actual equipment and devices or published papers and reports. Reports are issued to the sponsoring agency of a particular project or program. Published papers appear either in the Bureau’s own series of publications or in the

journals of professional and scientific societies. The Bureau itself publishes three monthly periodicals, available from the Government Printing Office: The Journal of Research, which presents

complete papers reporting technical investigations; the Technical News Bulletin, which presents

summary and preliminary reports on work in progress; and Basic Radio Propagation Predictions,

which provides data for determining the best frequencies to use for radio communications throughout

the world. There are also five series of nonperiodical publications: The Applied Mathematics

Series, Circulars, Handbooks, Building Materials and Structures Reports, and Miscellaneous

Publications.

Information on the Bureau’s publications can be found in NBS Circular 460, Publications of

the National Bureau of Standards (SI. 25) and its Supplement (SO. 75), available from the Superintendent of Documents, Government Printing Office. Inquiries regarding the Bureau’s reports and

publications should be addressed to the Office of Scientific Publications, National Bureau of

Standards, Washington 25, D. C.