NATIONAL BUREAU OF STANDARDS REPORT

4658

PROJECTS and PUBLICATIONS

of the

APPLIED MATHEMATICS DIVISION

A Quarterly Report

January through March, 1956

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 Structure. 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 Instrumentation # Office of Weights and Measures
CONTENTS

Page

Status of Projects* as of March 31, 1956 1

1. Numerical analysis 1

2. Mathematical tables and programming research. 6

3. Probability and mathematical statistics ... 10

4. Mathematical Physics 14

5. Mathematical and computational services ... 18

6. Statistical Engineering services 31

Application of SEAC 34

Lectures and Symposia 36

Publication activities 40

Only unclassified projects are included in this report.
Wm

.
NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT NBS REPORT

1 1 .0 4658

PROJECTS and PUBLICATIONS

of the

APPLIED MATHEMATICS DIVISION

January through March, 1956




U. $. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

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aniens panslaaloa is obtained la wiltlagfra* tbeOlaa af tba Director, National Bureau of Btaadarda, Washington

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-U-
APPLIED MATHEMATICS DIVISION

January 1 through March 31 > 1956

TECHNICAL ADVISORY COMMITTEE

David Blackwell, Howard University

Edward U. Condon

Mark Kac , Cornell University

Philip M. Morse, Massachusetts

Institute of Technology

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., Administrative Officer

Yates S. Sladen, Administrative Assistant

Marguerite C. Hepburn, Secretary

Mary E. McKinley, Secretary

NUMERICAL ANALYSIS SECTION

Henry A. Antosiewicz, Ph.D.

N. Aronszajn, D.Sc.

Philip Davis, Ph.D.

John Todd, B.S., Chief

Karl Goldberg, M.A.

Evelyn A. Grigg, Sec'y

Alan J. Hoffman, Ph.D.

Morris Newman, Ph.D.

COMPUTATION LABORATORY Milton Abramowitz, Ph.D., Chief

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

Joseph Bram, Ph.D.

Hans 0. Bremer, B.A.

William F. Cahill, M.S.

Ruth E. Capuano

John W. Cooper, B.S.

Mary M. Dunlap, B.S.

Robert L. Durrah, M.A.

Elizabeth F. Godefroy

William G. Hall, B.S.

Genevie E. Hawkins, B.S.

Emilie V. Haynsworth, Ph.D.

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.

Sally T. Peavy, B.S.

B. Stanley Prusch

Elizabeth A. Radcliffe

Ida Rhodes, M.A.

Billie G. Urban, M.S.

Philip J. Walsh, B.S.

Bertha H. Walter

Joseph H. Wegstein, M.S.

Richard G. Williams, M.S

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

William S. Connor, Ph.D.

Mary C, Dannemiller

Lola S. Deming, M.A.

Mary L. Epling, B.S.

Lela J. Hamilton, Sec'y

Julius Lieblein, Ph.D.

Mary G. Natrella, B.A.

Joan R. Rosenblatt, Ph.D.

Norman C. Severn, Ph.D.

Paul N. Somerville, Ph.D

Carolyn Yick, Sec'y

Marvin Zelen, M.A.

MATHEMATICAL PHYSICS SECTION Robert F. Dressier, Ph.D., Chief

Peter Chiarulli, Ph.D.

Leon Feldman, B.A.

Peter Henrici, Ph.D.

Fritz Oberhettinger , Ph.D. **Robert Stoneley, Sc*D.

Edith N. Reese, B.A. *John L. Synge, Sc.D,

Lillian Sloane, Sec'y

Under contract with The American University **Guest Worker ***0n leave of absence
Status of Projects

March 31 ? 1956

1 . NUMERICAL ANALYSIS

RESEARCH IN THEORIES OF DISCRETE STRUCTURES

Task 1100-10-5116/56-159

Origin: NBS Authorized 9/30/55

Sponsor: Office of Naval Research

Manager: 0. Taussky-Todd

Full task description: July-Sept 1955 issue, p. 1 Status: CONTINUED. 0. Taussky continued her investigations

concerning the higher commutators of a matrix A and its transposed and

conjugate A*. For 3x3 matrices examples were constructed which show

that the second, fourth, fifth, or sixth order commutator (A, A, . . . ,A,A*)

of A and A* can be the first one to vanish. If none of these vanishes,

then no commutator of any order will vanish. Group matrices with vanishing

fourth order commutators were also studied.

M. Newman and 0. Taussky had proved earlier that a unimodular

circulant of integers of the form AA* is also of the form CC 1 when C

is again a circulant of integers. They now have shown that the weaker

condition of positive definiteness does not suffice for circulants of

order n ^ 8. A manuscript has been prepared.

M. Newman and 0. Taussky are studying hermitian matrices and

their decomposition into the form AA* where A may be rectangular.

Publications:

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

fields. M. Newman and 0. Taussky. Comm. Pure Appl. Math. (New York

University) .2, 85-91 (1956).

(2) A note on group matrices. 0. Taussky. Proc. Amer. Math. Soc. 6,

98^-986 (1955).

(3) The formal power series for log (ex »ey). K. Goldberg. To appear in

the Duke Mathematical Journal.

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

E. Straus, 0. Taussky. To appear in the Pacific Journal of Mathematics.

(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, California, 1953

(6) Bounds and asymptotic behavior of the coefficients in the series

log (e x ey). K. Goldberg. In manuscript.

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

(University of California at Los Angeles) and 0. Taussky. To appear

in Pacific Journal of Mathematics.

(8) Pairs of matrices with property L, II. T. S. Motzkin and 0. Taussky.

Trans. Amer. Math. Soc. 80, 387_1+01 (1955).

(9) Generation and testing of pseudo-random numbers. 0. Taussky and J. Todd

To appear in the Proceedings of a Symposium on Monte Carlo Methods, held

at Gainesville, Florida, 1 9 5^+,

1
2 Status of Projects

( 10 ) On the Hilbert matrix, T, Kato. Submitted to a technical journal.

( 11 ) Commutators of A and A*. T. Kato and 0. Taussky. J. Wash. Acad.

Sci. 1+6, 38-i+O (1956).

( 12 ) Some free subgroups of the modular group. K. Goldberg and M„ Newman.

In manuscript.

( 13 ) Classes of positive definite unimodular circulants. M. Newman and

0. Taussky. In manuscript.

RESEARCH IN NUMERICAL ANALYSIS AND RELATED FIELDS

Task 1 101 -10-1 10V55-55

Origin: NBS Authorized 8/1 3/5*+

Managers: J. Todd , P. Davis Revised 8/29/5*+

Full task description: July-Sept 195*+ issue, p. 1 Status: CONTINUED* H. A. Antosiewicz has continued his work on the

survey of conditions for the stability of solutions of ordinary differential

equations which are based on Lyapunov's second method. Particular emphasis

has been paid to those conditions that guarantee stability "in the large."

Among them and related to them are criteria for the ultimate boundedness

of every solution. The concept of ultimate boundedness is important in

many cases of practical interest: For ' non-linear differential equations

of the second order it has been investigated previously by G. E. Reuter

and for more general equations by T. Yoshizawa. The latter's method of

attack is closely related to Lyapunov's method.

P. Davis and Ky Fan have completed the present phase of their

study on inequality and approximation in function spaces. It will probably

be resumed when Profes-sor Ky Fan returns to the Bureau in the summer.

J. Todd has continued his survey of various topics in the theory

of approximation that are applicable in numerical analysis. Arising out

of this, J. Todd has planned a new tabulation of the Cauer parameters,

with an independent variable different from that used by Glowatski. These

parameters arise in the design of electrical circuits and are expressed

in terms of elliptic functions, the modular parameter of which depends,

in the case of filters for example, on the permissible width of the frequency

band in the pass-and-stop interval.

K. Goldberg continued his study of iterated convolutions.

Publications:

(1) Stable systems of differential equations with integrable perturbation

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

Society.

(2) On the Lerch zeta function. F. Oberhettinger , To appear in the Pacific

Journal of Mathematics.

(3) A subroutine for computations with rational numbers. P. Henrici. J.

Assoc. Comp. Mach. 3? 6-9 (1956).

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

of partial differential equations. J. Todd. To appear in th« Proceedings

of a Symposium on Partial Differential Equations, held at Berkeley,

California, 1955.

(5) Some Monte Carlo experiments for computing multiple integrals. P. Davis

and P, Rabinowitz. To appear in Mathematical Tables and Other Aids to

Computation

.

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

and P. Rabinowitz, J. Res. NBS 56, 3 5~37 (1956), RP26^5.

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

P. Davis and P. Rabinowitz. J. Wash. Acad. Sci. i+6? 12-17 (1956).

(8) Complete sequences and approximations in normed linear spaces. P. Davis

and K. Fan. Submitted to a technical journal.

f 9) A matrix with real characteristic roots. K. Goldberg. To appear in

Journal of Research, NBS.
Status of Projects 3

(10) Note on bounds for determinants. E. V. Haynsworth. In manuscript.

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

RESEARCH IN MATHEMATICAL TOPICS APPLICABLE TO

NUMERICAL ANALYSIS

Task 1101-10-5116/55-56

Origin: NBS Authorized 8/1 3/5^+

Sponsor: Office of Naval Research Revised 8/29/5*+

Managers: J. Todd', M. Newman

Full task description: July-Sept 195*+ issue, p. 5 Status: CONTINUED. Some experiments in inverting matrices and

solving linear systems (using the floating point routines for these

purposes, described previously) were undertaken by M. Newman and I. Stegun.

It was found that even for matrices of orders as high as 60, round-off

error remained negligible. Comparing these results with those obtained

from fixed point routines, the conclusion seems to be that both time-wise

and accuracy-wise floating point routines are to be preferred. A more or less detailed description and evaluation of various codes for matrix

operations is being prepared.

Some new floating point routines for computing such functions as log x, e , have been prepared by M. Newman.

A. J. Hoffman has completed an expository article (see publication

(22) below) on linear programming, summarizing the highlights of the theory,

computing procedures and applications of the subject,

Ky Fan, I. Glicksberg (University of Notre Dame), and A. J. Hoffman,

have completed a manuscript (publication (23) below) on systems of inequalities

involving convex functions. Questions relating to the consistency, independence

and so forth, of a system of inequalities f^(x) < 0, where the f^ are convex

functions defined on a convex set in a real linear space, are shown to be

resolvable almost as simply as the corresponding questions for systems of

linear inequalities that they generalize,

A. J. Hoffman has shown (in publication (2*+)) that a simple

combinatorial lemma yields a generalization of a well-known theorem of Konig

on systems "representing" two partitions of a given set. The lemma states

that if the incidence matrix of points vs. subsets for a given set R and

family of subsets [S-,...,S } has the unimodular property that every square

sub-matrix is singula r or u nimodular , and 0 < < 1, then there exists a set C£ R such that |c<\S.- olJi I < 1. (A denotes the cardinality of the

set A.) 1 1

Publications:

(1) The normalizer of certain modular subgroups. M. Newman. Canadian Math.

J. 8, 29-31 (1956).

(2) Generalizations of identities for coefficients of certain modular forms.

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

(3) Compactif ication of completely regular spaces. Ky Fan and F. Wagner

(University of Notre Dame). In manuscript.

(*+) On dominant eigenvalues of positive matrices. T. Kato. In manuscript.

(5) A comparison theorem for eigenvalues of normal matrices, Ky Fan. Pac e J. Math, i, 911-913 (1955).

(6) On generating functions of the Jacobi polynomials. P. Henrici* Pac. J.

Math, i, 923-931 (1955).

(7) 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, 1953.
Status of Projects

4

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

(Bryn Mawr College). To appear in Annals of Mathematics Study 38.

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

Mathematics Study 38.

(10) Systems of distinct representatives and linear programming. A. J.

Hoffman and H. W. Kuhn (Bryn Mawr College). To appear in the American

Mathematical Monthly.

(11) Dilworth's theorem on partially ordered sets. G. B. Dantzig (RAND

Corporation) and A. J. Hoffman. To appear in Annals of Mathematics

Study 38.

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

manuscript.

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

( 1 ^+) On block relaxation. L. S. Joel. In manuscript.

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

(RAND Corporation), A. J. Hoffman, W. Hirsch (New York University).

In manuscript.

(16) The lowest frequency of a free square plate. H. Fujita (University

of Tokyo), T. Kato, Y. Nakata (University of Tokyo), and M. Newman,

In manuscript.

(17) An inclusion theorem for modular subgroups. M. Newman. Submitted

to a technical journal.

(18) A table of the coefficients of the powers of f^('T). M. Newman. To

appear in the Proceedings of the Royal Netherlands Academy of Sciences.

(19) On the existence of identities for the coefficients of certain modular

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

Society.

(20) An alternative proof of a theorem on unimodular groups. M, Newman.

Proc. Amer. Math. Soc. 6, 998-1000 (1955).

(21) Integral boundary points of convex polyhedra. A. J. Hoffman and

J. B. Kruskal (Princeton University). To appear in the Annals of

Mathematics Study 38.

(22) Linear programming. A. J. Hoffman. To appear in Applied Mechanics

Reviews

.

(23) Systems of inequalities involving convex function?, Ky Fan (University

Notre Dame), I. Glicksberg (RAND Corporation), and A. J. Hoffman.

In manuscript.

(2*+) Generalization of a theorem of Konig. A. J. Hoffman. In manuscript.

ANALYTIC STUDY OF WAR GAMES

Task 1101-10-5116/55-83

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: INACTIVE.

STUDY OF DIFFERENTIAL EQUATIONS FOR NERVE EXCITATION

Task 1 101—1 0— 51 1 6/56—1 48

Origin and Sponsor: National Institutes of Health Authorized 9/30/55

Bethesda, Md.

Manager: H. A. Antosiewicz

Full task description: July-Sept 1955 issue, p, 7
Status of Projects 5

Status: CONTINUED. A new method has been adopted for the determination

of the values of the parameter K for which the solutions of the basic system

of differential equations have a prescribed behavior. The previously used

method was based on an approximation to the positive eigenvalue, and its

corresponding eigenvector, of the 5x5 matrix which represents the linear

part of the system of differential equations. Now, this positive eigenvalue

and its eigenvector are computed for each value of the parameter K, and the

initial conditions determining the particular solution are then chosen

for the eigenvector. The change in the computation was felt necessary

because the bounds for the derived values of K found by the old procedure

were so close that further approximations might have yielded spurious

results.

About 30 cases, both for temperatures 6.3°C and 18.5°C, were

run during this quarter and were submitted to the sponsor.
2. MATHEMATICAL TABLES AND PROGRAMMING RESEARCH

TABLES OF E^Cz), (z=x+iy)

Task 11 02-40-1 110A3-3

Manager: I. Stegun Authorized 7/1/47

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

Status: CONTINUED. The introduction to the tables has been prepared

and checking is in progress.

TABLES OF COULOMB WAVE FUNCTIONS

Task 1102-40-1110/47-2

Origin: NBS Authorized 7/1/47

Manager: M. Abramowitz

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

Status: CONTINUED. The code is being revised to carry out computations

of the functions for a range of values of jo corresponding to given and

TABLES OF POWER POINTS OF ANALYSIS OF VARIANCE TESTS

Task 1102-40-1110/51-8

Origin: Section 11.3? NBS Authorized 3/26/51

Manager: S. Peavy

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

Status: INACTIVE. For status to date, see Oct-Dec 1955 issue, p. 8.

REVISION OF MATHEMATICAL TABLES

Task 1102-40-1110/52-7

Origin: NBS Authorized 8/10/51

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 Table MT1 2 (1941), for reissue in the Applied

Mathematics Series was completed, and the manuscript is ready for submittal

for publication.

6
Status of Projects 7

TABLE OF ARCSIN FOR COMPLEX ARGUMENTS

Task 1 102-40-1 11 0/52-14

Origin: NBS Authorized 10/1/51

Manager: B. Urban

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

Status: TERMINATED. It was decided to discontinue preparation of

the table for publication because a more comprehensive table of the

function was recently published elsewhere.

TABLE OF THE MODIFIED AIRY INTEGRAL

Task 1102-40-1110/52-23

Origin: NBS Authorized 10/4/51

Manager: I. Stegun

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

Status: CONTINUED. Checking of the completed manuscript is in

progress.

TABLE OF ERROR FUNCTION FOR COMPLEX ARGUMENTS

Task 1102-40-1110/52-25

Origin: NBS Authorized 10/5/51

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.

SPHEROIDAL WAVE FUNCTIONS

Task 1102-40-1 110/52-37

Origin: NBS Authorized 11/28/51

Manager: D. Liepman

Full task description: Oct-Dec 1951 issue, p. 38

Status: INACTIVE.

S IEVERT'S INTEGRAL

Task 1102-40-1110/52-57

Origin: NBS Authorized 2/12/52

Managers: M. Paulsen, P. O'Hara

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

Status: CONTINUED. Checking of the tabular values and preparation

of the introductory material continued.
8 Status of Projects

L-SHELL CONVERSION COEFFICIENTS

Task 1102-40-1110/53-52

Origin: Oak Ridge National Laboratory Authorized 5/20/53

Manager: W. Hall

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

Status: CONTINUED. Computations for the Ljjj shell were completed

for z = 65 for all values of k; for z = 95? k = .05, 0.1; and for the

Lj shell for z = 55? k = .05.

AUTOMATIC CODING

Task 3711-60-0009/55-65

Origin: NBS Authorized 9/29/54

Manager: J. Wegstein

Full task description: July-Sept 1954 issue, p. 11

Status: CONTINUED . A new Base code, BASE 11, has been completed.

It is intended to be used specifically for small problems involving

integrals. In it the 1-, 2- and A-orders of Base 00 are replaced by twoword "integrate" orders completely specifying an integration of the form

f

k f(x)dx except for the coding of f(x). (See July-Sept 1955 issue, p. 13.)

By the use of the three integration orders general expressions

of the form fff f (x ,y ,z )dxdydz can be coded using only two words to

specify each integration and the coding necessary to compute the various

functions. The integration rule is Gaussian quadrature, and 2-, 4-, 8-,

and 16-point rules for any interval may be specified.

Base 11 should be useful in coding small problems quickly where

the behavior of the functions involved is well known and only one integration

is involved or separation of variables cannot be effected. It may also

be used for larger problems provided the necessary numerical analysis is

carried out.

In addition to the integration routines, Base 11 also contains

all of the service routines and subroutines of Base 01 (see Oct-Dec 1955

issue^p. 10.)

Work was begun on the BUIC system for SEAC. It is a generating

routine which accepts problems stated in a notation written directly from

the flow diagram of the problem. Algebraic expressions are accepted

directly by the computer in the same form that they can be understood by

any mathematician. Complete logical details have been worked out for the

algebra encoder portion of this system. The feasibility of building a computer which will operate at high speed directly on algebraic and flow

chart symbols is also being studied.

MATHEMATICAL SUBROUTINES

Task 3711-60-0009/56-160

Origin: NBS Authorized 9/30/55

Managers: Staff

Full task description: July-Sept 1955 issue, p. 13

Status: CONTINUED. The development of a base code for matrices

which may be used in problems requiring several types of operations on small
Status of Projects 9

matrices was started. The code will be similar to the Base 00 and will

use most of the Base 00 operations. It will contain subroutines to add

and multiply matrices, to compute the value of determinants*, to orthonormalize

a set of vectors, and to compute the inverse of a given matrix. The

elements of the matrices will be expressed in floating binary point form.

During this quarter, the addition, multiplication, and determinant evaluation

subroutines were written and code checking of these routines was started.
3. PROBABILITY AND MATHEMATICAL STATISTICS

BIBLIOGRAPHY AND GUIDE TO STATISTICAL LITERATURE

Task 1 1 03-1 0-1 1 07/49-1 a Origin: NBS Authorized 1/9/49

Manager: L. S. Doming

Full task description: Apr-June 1949 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: Jai-Mar 1950 issue, p, 42

Status: COMPLETED. The manual on the fitting of straight lines has

been completed and is being edited for publication by a commercial house.

Some material from it is being abstracted for use by the ASTM in their

handbook on the "Presentation of Data." The straight line manual is

divided into ten chapters whose titles are: Chapter I - General introduction

II - X - Without error - Classical ideas

III - Several values of y for each x IV - Bivariate normal population

V - Both variates in error VI - Several lines - The analysis of variance

VII - Orthogonal polynomials - Curvature

VIII - Transformations

IX - Rejection of data

X - Cumulative data

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.

10
Status of Projects 11 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. For a description of the continuing activity

on this task, see the Oct-Dec issue, p. 12.

Publications:

(1) Chaines de Markof dans les ensembles abstraits et applications aux processus avec regions absorbantes et au probleme des boucles. (Formerly,

"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. Annales de l'Institut Henri Poincare

Jit, 1^5-199 (1955).

(2) A note on the normal distribution. S. Geisser. To appear in the Annals

of Mathematical Statistics.

(3) A comparison between the mean successive differences and the root mean

square successive difference. S. Geisser. Withdrawn from publication.

(b) Inequalities for probabilities associated with the multivariate normal

distribution. I. R. Savage. Withdrawn from publication.

STUDIES IN THE MATHEMATICS OF EXPERIMENT DESIGN

Task 1103-10-1107/53-1

Origin: NBS Authorized 10/15/52

Manager: W. S. Connor

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

Status: CONTINUED. R. C. Burton has written a note on "Some

existence theorems for dualized designs."

Publications:

(1) Partially replicated Latin squares. W. J. Youden end J. S. Hunter

(American Cyanamid Co.) Biometrics 1

1

, 399-^05 (Dec. 1955).

(2) Contributions on partially balanced incomplete block designs with

two associate classes. W. H. Clatworthy. To appear as NBS Applied

Mathematics Series ^7.

(3) Exact tests of significance for combining inter- and intra-block

information. M. Zelen. Submitted to a technical journal.

(*+) On the analysis of covariance. M. Zelen. Submitted to a technical

journal.

(5) Fractional factorial experiment designs for factors at two levels.

To appear as NBS Applied Mathematics Series *+8. (6) Design of experiments in research and development. W. J. Youden.

To appear in Proceedings of the 1955 Conference on the Design of

Experiments in Army Research, Development, and Testing.

(7) Some examples of the use of high speed computers in statistics.

J. M. Cameron. To appear in Proceedings of the 1955 Conference on the

Design of Experiments in Army Research, Development, and Testing.

(8) The principle of randomization in the design of experiments. C. Eisenhart.

To appear in Proceedings of the 1955 Conference on the Design of

Experiments in Army Research, Development, and Testing.
12 Status of Projects

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. J. Lieblein consulted with E. L. Brancato and

W. F. Gauster of the Naval Research Laboratory on the application of

extreme-value theory to dat8 on breakdown voltage of transformer oil.

Dr. Lieblein also corresponded, among others, with N. T. Uzgoren

(University of Istanbul, Turkey) on theoretical aspects of the asymptotic

development of the distribution of extreme values, and with E. Gassner

(Darmstadt, Germany) on the fit of a formula used in fatigue-life testing

of ball bearings.

The final report (see J8n-Mar 1955 issue, p. 33) to Subcommittee

7 (Load Ratings) of the American Standards Association Committee B-3 (Ball

and Roller Bearings), on "Statistical investigations of the fatigue life of

deep groove ball bearings" has been developed into an article for submittal

to a technical journal.

Publication:

(1)

Application of extreme value theory to cobbles and boulders in gravel

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

Submitted to a technical journal.

STUDY OF NON- PARAMETRIC STATISTICAL TECHNIQUES

Task 1103-10-1107/56-170

Origin: NBS Authorized 12/15/55

Manager: Joan R. Rosenblatt

Full task description: Oct-Dee 1955 issue, p. I

1* Status: CONTINUED. A paper is being prepared on the power of some rank order two-sample tests.

A talk on the efficiency of statistical tests was given at a Statistical Engineering Laboratory Seminar on February 17.

Work was continued on the distribution of Burrow’s statistic.

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—two-sample case.

I. R. Savage. To appear in the Annals of Mathematical Statistics.

(3) Simultaneous tests of randomness and other hypotheses. I. R. Savage.

To appear in the Journal of the American Statistical Association,

under the title "On the independence of tests of randomness and other

hypotheses."
Status of Projects 13

MEASUREMENT OF RELIABILITY

Task 1 1 03- 1

0

- 1 1 30/ 56-1 82

Origin: NBS Authorized: 3/23/56

Manager: M. Zelen

Objective: To conduct research on the mathematical and statistical

aspects of the definition, measurement, and specification of the reliability

of components and assemblies, with particular attention to the evaluation

of the reliability of electronic components and systems.

Background: The improvement of the reliability of electronic

equipment, civilian as well as military, is one of the major problems of

science and technology today. It is highly desirable that electronic

computers and communications equipment operate with satisfactory reliability

over relatively long periods of continuous or intermittent use. On the

other hand, electronic systems of guided missiles, for example, are required

to withstand a long "shelf-life" yet perform with high reliability during

their very brief period of use. While it is widely recognized that reliability

is primarily a probability problem, and that the measurement of reliability

is largely a statistical problem, only a beginning has been made in applying

recent developments in statistical inference, experiment design, and statistical

analysis, to the definition and measurement of the reliability of complex

assemblie s

.

Status: NEW. Examples of the literature on measurement and prediction

of reliability are being collected and reviewed.

Work is being done on methods for obtaining confidence limits

for the product of binomial parameters.
*+. MATHEMATICAL PHYSICS

RESEARCH IN MATHEMATICAL PHYSICS AND RELATED FIELDS

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

Origin: NBS Authorized 9/1/5*+

Manager: R. F. Dressier

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

Status: CONTINUED. New methods have been developed by P. Henrici

to speed up the convergence of the Quotient-Difference Algorithm and to

apply it to the numerical determination of the residues of a meromorphic

function. Also some applications were made of I. N, Vekva's theory of

elliptic differential equations with analytic coefficients. Several

representations for the complex Riemann function of the equation

Au + ~yfy

+ = 0,

have been obtained.

As a continuation of previous work on addition theorems P. Henrici

has obtained several new addition theorems for Lommel functions. These

are of importance in the solution of the inhomogeneous wave equation.

Publications:

(1) Comparison of theories and experiments for the hydraulic dam-break wave.

R. F. Dressier. Publication No. 38 de 1 'Association Internationale

d 'Hydrologie (Assemblee generale de Rome, 195*+? tome III, p. 319~328).

Extrait

.

(2) Entropy changes in rarefaction waves. R. F. Dressier. Submitted to a technical journal.

(3) A discussion of the paper, "Design of corrugated diaphragms," by

J. A. Haringx. R. F. Dressier. To appear in the Transactions of the

American Society of Mechanical Engineers.

(*+) On generating functions for the Jacobi polynomials. P. Henrici. Pac.

J. Math. 5, ,923-931 (1955).

(5) On the representation of a certain integral involving Bessel functions

of hypergeometric series. P. Henrici. Submitted to a technical journal.

(6) On the domain of regularity of generalized axially symmetric potentials.

P. Henrici. To appear in the Proceedings of the American Mathematical

Society.,

(7) The quotient-difference algorithm. P. Henrici. In manuscript.

RESEARCH IN ELECTROMAGNETIC THEORY

Task 11 0*+-1 0-51 60/ 5*+-*+7 Origin and Sponsor: Diamond Ordnance Fuze Laboratory, Authorized 6/29/5*+

Department of the Army Revised 9/29/5*+

Manager: F. Oberhe ttinger

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

Status: INACTIVE.
Status of Projects 15

RESEARCH IN MATHEMATICAL ELASTICITY

Task 1 104-10-51 60/55-85

Origin: NBS Authorized 12/27/54

Sponsor: Office of Scientific Research, ARDC, USAF

Manager: R. F. Dressier

Full task description: Oct-Dec 1954 issue, p. 30

Status: CONTINUED. The second phase of the corrugated diaphragm

project is being continued, —to analyze a sequence of diaphragms with a fixed shape for a range of different thicknesses in order to determine the

relative importance of bending and stretching effects. The problem

originally solved by R. F. Dressier and W. Cahill was for a thickness

h = .007 inches. Now the case 3h has been completely solved -also and

exhibits the following interesting property: For average thicknesses,

all stresses causing , stretching vary with the same frequency as the corrugation

profile shape, whereas all bending stresses vary with a double frequency*

However, for the thick diaphragm the present results indicate that the

double frequency behavior almost disappears for all bending stresses. The

next case, now being analyzed, corresponds to h/3. For this problem all

four homogeneous solutions were computed, but the general solution is not

yet completed.

On the vibrating triangular wing project, I. Rhodes and U. Hochstrasser

satisfactorily rectified the difficulty in the code for obtaining higher

eigenvalues and eigenvectors for the basic 78x78 asymmetric matrices.

On small test matrices the second eigenvalues were obtained with high

accuracy; this code is to be applied to the large matrices describing the

wing problem.

J. L. Synge worked on a geometrical approach to problems of

forced vibrations in elasticity, the solution appearing as the intersection

of two orthogonal linear subspaces in a suitable function space. This

approach led to methods of approximate solution. A paper on this work is

to be presented at the Eighth Symposium on Applied Mathematics, Chicago,

April 12-13.

J. L. Synge has applied tensor methods to the study of the

general theory of linear elastic waves in completely anisotropic media,

developing a theory of Rayleigh surface waves in such media. A report on this investigation was prepared.

J. L. Synge and W. Cahill engaged on an approximate numerical

solution on SEAC of the elastic torsion problem for the hollow square.

The solution uses the Prager-Synge hypercircle method, which gives upper

and lower bounds for the torsional rigidity. Final calculations on the SEAC

for this problem are in progress.

Publications:

(1) Investigations of the properties of corrugated diaphragms. W. A. Wildhack

(NBS OBI), R. F. Dressier, and E. C. Lloyd (NBS Mechanics Division).

To appear in the Transactions of the American Society of Mechanical

Engineers. (Now available as ASME Preprint No. 5 5—A— 1 81 .

)

(2) Stationary principles for forced vibrations in elasticity and electromagnetism. J. L. Synge. Submitted to a technical journal.

(3) Elastic waves in anisotropic media. J. L. Synge. In manuscript.
16 Status of Projects

FOURIER TRANSFORMS OF PROBABILITY DISTRIBUTION FUNCTIONS

Task 11 04-10-51 60/56-1 5*+ Origin: NBS Authorized 9/30/55

Sponsor: Office of Naval Research

Managers: F. Oberhettinger , J. Lieblein

Full task description: July-Sept 1955 issue, p. 20

Status: CONTINUED. Further examples have been added to the

collection of Fourier transforms of distribution functions. This completes

the present compilation, which contains some previously underived results.

The whole material is in systematic form ready for book publication when

the accompanying text is written.

RESEARCH IN FLUID DYNAMICS OF TWO-PHASE FLOWS

Task 1 1 0^-1 0-51 60/56-1 55

Origin: Office of Naval Research Authorized 9/30/55

Sponsor: " " Manager: R. F. Dressier

Full task description: July-Sept 1955 issue, p. 21

Status: CONTINUED. Relations between the states on both sides of an interface separating two phases of a moving fluid have been determined

for the cases in which the phase transition is brought about by heat

removal and/or cold liquid addition at the interface. If the specific

volume of the liquid state is neglected in comparison with the specific

volume of the vapor state and the enthalpy of the liquid state is given

as a linear function of the temperature alone, then an explicit solution

has been found for the liquid state variables in terms of the other parameters.

For more accurate results, a rapidly converging iterative scheme has been

developed which utilizes directly thermodynamic tables of the vapor and

liquid. Some calculations have been made for the specific case of a steamwater interface. For a stationary interface and fixed thermodynamic state

of the vapor, upper and lower bounds for heat removal or mass addition

rates have been found as a function of the incoming steam velocity. Conversely,

for given heat removal or mass addition, bounds have been obtained for the

steam velocity. A paper on this first aspect of the investigation has been

prepared

.

Publication:,

(1) Condensation interfaces in two-phase flow. P. Chiarulli and R. F. Dressier.

In manuscript.

RESEARCH IN MATHEMATICAL GEOPHYSICS

Task 1 104-10-51 60/56-1 56

Origin: The American University Authorized 9/30/55

Sponsor: Office of Naval Research

Manager: R. Stoneley

Full task description: July-Sept 1955 issue, p. 21

Status: CONTINUED. The investigation of the variation of amplitude

with depth for Rayleigh waves in a continent with a double surface layer

has been completed by R. Stoneley and U. Hochstras ser . The results show

that a nodal point for the amplitude may exist in either of the two surface
Status of Projects 17

layers depending upon the relative sizes of the various parameters. This

work was presented at the meeting of the Seismological Society of America,

Reno, Nevada, March 30-31.

A paper dealing with the hydrodynamical aspects of turbidity

currents in oceans was also completed by R. Stoneley.

The problem of the propagation of Rayleigh waves across an ocean floor in which there are two rocky layers has been programmed by U. Hochstrasser

for computation by the SEAC.

Four lectures by R. Stoneley presented at an NBS seminar on the

general theory of elastic surface waves have been written up as notes by

P. Henrici. This will also be incorporated in a Monograph on elastic waves which is now in preparation.

Publications:

(1) The transmission of Rayleigh waves across an ocean floor with two surface

layers. Part I. R. Stoneley. To appear in the Bulletin of the

Seismological Society of America.

(2) On the propagation of turbidity currents. R. Stoneley. To appear in

the Vening Meinesz Jubilee Volume.

(3) Lectures on elastic surface waves. R. Stoneley. In manuscript.

RESEARCH IN RADIATION THEORY

Task 110^-10-5160/56-175

Origin: NBS Authorized 12/28/55

Sponsor: Office of Naval Research

Manager: F. Oberhe ttinger

Full task description: Oct-Dec 1955 issue, p. 18

Status: CONTINUED. Investigations concerning the asymptotic

behavior of certain integrals describing the far zone field of electromagnetic

waves diffracted by a wedge are continuing. The case of an incident plane wave field (see J. Math. Phys. 1956) is being extended to take care

of an incident cylindrical and spherical wave field. This involves reduction

of integrals of the form

1(6, a)

1 2ct sin ( 7Tb /a,

)

H(x) dx

COSh(7Tx/ct) - COS (TTb/cc)

with

H(x)

H^^tk^^+y^ 2

+2^cy=>' cosh x)^]

or

exp(-ik[/3 2+ /^ 2+(z-z * )^+2 p

p

x cosh x]^)

5 To

* 5

1 P +p +(z—z') +2pp' cosh xj 2

into an asymptotic series for large P-
5. MATHEMATICAL AND COMPUTATIONAL SERVICES

11 02-4-0-51 2'6/50-1 3 RAY TRACING

Origin and Sponsor : NBS, Section 2.2

Manager : E. Haynsworth, P, Walsh

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

Status : Continued. It was found necessary to compute the inverse of a given matrix (n^28) which was stored in fixed binary point form. A

code was written in two parts: (1) To read in the matrix and convert it

to floating binary point form. A previously written code for computation

of the inverse in floating binary point form was then used. (2) To

convert the inverse of the matrix from floating binary point form, and to

transfer the matrix to wire by rows. The two sections of the code are being checked out.

11 02-^0-51 26/51 -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. Additional analyses were performed to determine the

structures of the spurite and dimethoxybenzophenone crystals.

1 102-4-0-51 26/52-4-4- CALCULATIONS FOR d SPACINGS

Origin and Sponsor : NBS, Division 9

Full task description : Oct-Dec 1951 issue, p. 4-7 Manager : R. Zucker

Status : Continued. Calculations for d-spacings were performed for 3

tetragonal, 9 hexagonal and 20 orthorhombic crystals. Also determination

of unit cell constants was carried out by least squares fitting to

measured d-spacings for 7 tetragonal, 6 hexagonal, and 23 orthorhombic1 crystals

.

11 02-40- 5 126/53-2/ 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. Thermodynamic functions were re-computed for zinc (Z T)

,

making use of revised input constants. Under another code thermodynamic

functions were computed for trans-dif luor odichloroethylene.

1102-^-0-5126/53-51 RADIATION DIFFUSION

Origin : NBS, Section 4-.

8

Sponsor : Atomic Energy Commission

Manager : J. Doggett (4-. 8)

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

Status : Continued. The data obtained from the Mott formula code for the

elastic scattering of electrons and positrons by nuclei have been checked

by interpolation schemes and reruns. A manuscript on the results has been

18
Status of Projects 19

submitted to a technical journal.

The basic codes for the generation of collision density patterns

of gamma radiation by Monte Carlo methods have been revised. Preliminary

runs indicate an increase by a factor of two in the operational efficiency

of the code.

1102-40-5126/54-13 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. Analysis of contract award problems continued to be

processed on SEAC as they were submitted by the sponsor. During this

quarter, the awards for approximately 15 contracts were determined.

3711-60-0009/5^17 DEPOLYMERIZATION

Origin : NBS, Section 7.6

Manager : U. Hochstrasser

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

Status : Continued. Some additional cases were run with new values of the

parameters <r and 6 as needed in further investigations under way by

the originator.

371 1 -60-0009/55—22 ENERGY DISTRIBUTIONS ON OPTICAL IMAGE

Origin : NBS, Section 2.2

Manager : L. S. Joel

Full task description : Jan-Mar 1 9 54 issue, p. 43

Status: Continued. The new codeues verified as correct for W = arctan Cv/x)

W = 9x(x^-ry2) ^

3711-60-0009/55-30 SPECTRUM ANALYSIS

Origin : NBS, Division 4

Manager : S. Prusch

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

Status : Continued. Searching for pairs of lines for approximately 14 common differences was carried out for Ruthenium I.

3711-60-0009/54-38 COMPRESSIBILITY FACTORS OF DRY AIR • Origin : NBS, Section 3.2

Manager : M. Paulsen

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

Status : Inactive. For status to date, see Apr-June 1955 issue, p. 12.

1102-40-5126/55-39 MOLECULAR VIBRATIONS

Origin and Sponsor : NBS, Section 3.2

Manager : K. Goldberg

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

Status : Inactive.

1102-40-5126/54-43 CHARACTERISTICS OF CONDUCTING RESISTORS

Origin and Sponsor : NBS, Section 1.6

Manager : B. G. Urban

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

Status : Inactive. For status to date, see Apr-June 1955 issue, p. 12.
20 Status of Projects

3711-60-0009/55-53 ELECTRONIC FUNCTIONS

Origin : NBS, Section *+.5 Manager : W. G. Hall

Full task description : Apr-June 1955 issue, p. 12

Status : Continued. The tabular material was typed, and the final readying

of the manuscript for publication was almost completed.

1102-^0-5126/55-58 ELECTROMAGNETIC RADIATION FROM LIGHTNING

Origin and Sponsor : NBS, Section 82.1

Manager : B. G. Urban

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

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

1102-^0-5126/55-61 ELASTIC CROSS SECTION FOR NEUTRON SCATTERING

Origin and Sponsor : Naval Research Laboratory

Manager : I. Stegun

Full task description : Oct-Dec 1955- issue, p. 18

Status : Inactive. For status to date, see Apr-June 1955 issue, p. 13.

3711-60-0009/55-66 RECONSTITUTION OF MONOCHROMATIC LIGHT INTENSITIES

Origin : NBS, Division 30

Manager : H. Bremer

Full task description : July-Sept 1955- issue, p. 18

Status : Inactive. For status to date, see Oct-Dec 1955- issue, p. 19.

3711-60-0009/55-68 CRYSTAL STRUCTURE CALCULATIONS

Origin : NBS, Division 9

Manager : R. Prosen

Full task description : Jan-Mar 1955 issue, p. 18

Status : Continued. A general program applicable to all crystallographic

space groups is being written that will calculate least squares refinements

for atomic positions as well as individual anisotropic temperature factor

parameters for each atom. Code-checking of the general structure factor

routine is progressing satisfactorily. The code for the general Fourier

routine to calculate three-dimensional electron density maps has been

punched and is ready for checking;

1102-^0-51 26/55-7*+ LIQUID-VAPOR TRANSITION, II

Origin and Sponsor : Naval Medical Research Institute

Manager : S. Prusch

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

Status : Inactive.

3711-60-0009/55-75 PARAMETER OF THE DISPERSION EQUATION FOR OPTICAL GLASS

Origin : NBS, Section 2.2

Manager : R. Zucker

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

Status : Continued. For a given set of observed wavelength A, and index of

refraction n, the constant of dispersion k^ was determined by the method of

least squares, using the original code. A new code was written and checked

to set up a table of the computed values of n , 5 k. 2 c n

c = 1 + tl -§ o

i=1 X

2

- A?

The table of n was prepared for A = . 5*+( .02)2( . 2)20( . 1 )

IfO for given values

of Aj and k

i#
Status of Projects 21

1102-^0-5126/55-81 COMBINING TESTS FOR SIGNIFICANCE

Origin ; NBS, Section 11.3

Manager ; L. S. Joel

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

Status ; Continued. The bulk of the results for the table have been

transmitted to the sponsor. It remains to compute and check scattered

value s

.

3711-60-0009/55-82 THERMOMETER CALIBRATIONS

Origin : NBS, Section 3.1

Manager ; S. Frusch

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

Status ; Continued. Calibration tables were run on approximately 49

thermometers under'test.

1 102-40-51 26/55-88 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 : In a c t ive

.

1102-^0-5126/55-103 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. New data from the sponsor was processed on UNIVAC using

codes already set up.

11 02-^0-51 26/55-1 05 AERONOMIC TIDAL WINDS OF THERMAL OilGIN

Origin and Sponsor : NBS, Section 82.1

Managers ; B. G. Urban, H. H. Howe (82.1)

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

Status ; Inactive. For status to date, see Jan-Mar 1955 issue, p. 26.

1102-^0-5126/55-113 REACTOR DESIGN

Origin : Westinghouse Atomic Power Division

Sponsor : Atomic Energy Commission

Manager ; U. Hochstrasser

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

Status ; Continued. Some changes in the code were made in order to be able

to iterate on the equations for high and low energy level flux

separately. A change in the computation of the eigenvalues was prepared.

Several additional runs with the original code were made in order to

complete the study of the best choice of an over-relaxation factor.

1102-40-5126/55-117 ATTENUATION OF PRESSURE PULSES OF FINITE AMPLITUDE

Origin : NBS, Section 3.2

Sponsor ; Bureau of Aeronautics, U. S. Navy

Manager ; M. Psulsen

Full task description : Apr-June 1955 issue, p. 18

Status ; Inactive. For status to date, see July-Sept 1955 issue, p. 30.
22 Status of Projects

1 102-40-51 26/55-1 1 8 THERMOMETER CALIBRATION, II

Origin and Sponsor : NBS, Section 3.1

Manager : B. G. Urban

Full task description : Apr-June 1955 issue, p. 18

Status : Continued. An additional code to compute certain constants and

differences formerly computed by hand was completed and checked out.

Tables for eight thermometers were computed and transmitted to sponsor

A supplementary code to edit finished results is being written

1102-40-5126/55-121 ELECTRON PENETRATION

Origin : NBS, Section 4,8

Sponsor: Atomic Energy Commission

Manager : S. Peavy

Full task description : Apr-June 1955 issue, p. 19

Status : Continued. Preparation of the code continued.

1102-40-5126/55-122 SOLUTION OF NORMAL EQUATIONS

Origin : Advisory Committee on Weather Control

Sponsor : IT. S. Weather Bureau

Manager : M. Newman

Full task description : Apr-June 1955 issue, p. 20

Status : Inactive.

1102-40-5126/55-123 TEMPERATURE DISTRIBUTION

Origin : NBS, Section 6.4

Sponsor : NBS, Section 6.4

Manager : W. G. Hall

Full task description : Apr-June 1955 issue, p. 19

Status : Inactive. For status to date, see Oct-Dec 1955 issue, p. 24.

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

Origin : NBS, Section 3.2

Sponsor : Bureau of Aeronautics, U. S. Navy

Manager : J. Cooper

Full task description : Apr-June 1955 issue, p. 20

Status : Reactivated. The project was reopened to compute values of

the integrals R^ and R

2 for new parameters. In addition, the code was modified so that the times taken to fill containers could be computed

directly. This involved computing the following quantities for each

value of a, instead of R^ and R

2

.

t.| - 8-73 R-j + t c

where

and

a R 0 + t

cL C 14.022 [ (1-x

c

) 1/7

r ..128(1 + , OOga ) _ 8 L 1+. 528a

- 1 ]

0.005]

Calculations were made for 12 values of a ranging from 0.1130

to 0.905-0 and lor ji = . 533( .05) .033, .963, .983.”
Status of Projects 23

1102-^0-5126/55-126 AERODYNAMIC HEATING

Origin and Sponsor : Diamond Ordnance Fuze Laboratory, Dept* of the Army

Managers : B. Walter, R. Capuano

Full task description; July-Sept 1955 issue, p. 31

Status : Continued. Tw

was calculated with the following modifications:

B h At increased by 2°/o and T AUJ increased 8°F for each of the four

0 e aw missile trajectories. The corresponding variations of Tw

and of the Tw

intersections were calculated and compared with the modifications

described in the Oct-Dec 1955 issue, p. 2b,

A table of W as a function of L, M, and AS was calculated for

each of the four trajectories from the equation,

M _ W

AW = -i—1- , LAS

+ &

where M = 19.62 cos [ —

^—]»

L = (CD

1 +r

D2

) 9

,

1 +f2 )1 * l86,10_3 ’

AS = VAt,

and tables of 9 C^ were given. Based on the computed values of W,

the original velocity was then checked against the velocity obtained

by the relation,

AV =_

V

1 AW

W, [1 -

1bAW

W„ 5 ( w7

)2 " hffi3 + •••]>

where W^ = V^.

Then L was increased by 2°/o, which is equivalent to increasing

the ambient density by 2°/o, and the change in V was determined in

accordance with the above equations.

The values of Tty as a function of time were calculated for the

modified density and velocity conditions in accordance with the procedure

previously outlined (Oct-Dec 1955 issue) for two of the ranges. The

intersections of the two strip temperatures were then compared with those

obtained for the unmodified conditions.

1 102-^0-5126/55-127 VIBRATIONS OF CIRCULAR DISC

Origin and Sponsor : Evans Signal Laboratory

Manager : U. Hochstrasser

Full task description : July-Sept 1955 issue, p. 32

Status : Continued. A set of solutions for the equations described in

the July-Sept 1955 issue has been obtained. The code has been changed

in order to eliminate the singularities on the lefthand side of the

mentioned equation.

3711-60-0009/56-131 RAY TRACING

Origin : NBS, Section 2.2

Manager : E. Marden

Full task description : July-Sept 1955 issue, p. 33

Status : Continued. Code checking is in progress.
24 Status of Projects

1 1 02-40-51 26/56-1 33 COMPLEX EIGENVALUES

Origin : Moore School of Engineering

Sponsor : Office of Naval Research

Manager : I. Stegun

Full task description : July-Sept 1955 issue, p. 33*

Status : Continued. The largest eigenvalues for two matrices of order ten were obtained, together with the successive vectors.

1 1 02-40-51 26/56- 1 34 TRANSMISSION-DELAY TIMES

Origin : NBS, Section 82.1

Sponsor : NBS, Section 82.1

Managers : B. G. Urban, H. H. Howe (82.1)

Full task description : July-Sept 1955 issue, p. 3^

Status : Continued. The bulk of the tables have been computed.

1102-40-5126/56-136 CALCULATION OF WAVE FUNCTIONS BY HARTREE METHOD

Origin : Naval Research Laboratory

Sponsor : " " " Manager ; S. Peavy

Full task description : July-Sept 1955 issue, p. 34

Status : Continued. The new code was checked out, and the results were transmitted to the sponsor. The simplification of the equations reduced

the SEAC time considerably. This made it possible to carry out the

computation for certain cases to many iterations.

1102-40-5126/56-137 STABILITY OF SUPPORTED PLATES

Origin and Sponsor : National Advisory Committee on Aeronautics

Manager : R. Zucker

Full task description : July-Sept 1955 issue, p. 35

Status : Continued. Computations were continued for r=3 and r=2 for various

combinations of k

c

, k

g

, and The calculations are now extended for

negative values of k and k . As answers are obtained, they are submitted

to the sponsor.

c s

1102-40-5126/56 139 STUDY OF INTERNUCLEAR POTENTIAL FOR H.

Origin and Sponsor : NBS, Section 3.2 J

Manager : E. Haynsworth

Full task description : July-Sept 1955 issue, p. 36

Status : Inactive. For status to date, see Oct-Dec 1955 issue, p. 26.

1102-40-51 26/56-140 MULTIPLE CORRELATION ROUTINES

Origin and Sponsor : Agricultural Economics Division, Department of

Agriculture

Managers : H. Bremer, M. Paulsen

Full task description : Oct-Dec 1955 issue, p. 26

Status : Continued. A program has been set up for the sponsor to do the

the computations on SEAC for least squares regression problems and

the evaluation of correlation coefficients. This program consists of

three basic routines.

For each regression problem the variables may or may not undergo

any of the following transformations, —logarithm of original variables,

the first difference of the variable or its logarithm. Using these

transformed variables the moment matrix is then computed. This part of

the program is designated as Routine I.

Routine II consists of existing codes for matrix inversion and
Status of Projects 25

is used to compute the inverse of the moment matrix.

Routine III takes the inverse matrix and computes the

a) regression coefficient

b) standard errors of the partial regression coefficient

c) partial correlation coefficient

d) multiple correlation coefficient

e) standard errors of estimate and the constant for

the regression equation.

So far nine problems have been submitted by the sponsor and were processed by the above program. Each problem took approximately 20 to

40 minutes depending upon the number of variables and number of observations

It is hoped that this running time will be reduced by about one third when

the routines are better integrated. Also considerable time will be saved

if a matrix conversion code is written so that the entire problem can be

done in the internal memory,

It is planned that some additional routines will be written to

aid the sponsor in fitting systems of equations by the limited information

approach.

1 102-40-51 26/56-143 CHARGED PARTICLE TRAJECTORIES IN THE MAGNETIC FIELD

OF THE EARTH

Origin and Sponsor : Naval Research Laboratory

Manager ; J. H. Wegstein

Full task description : Oct-Dec 1955 issue, p. 27

Status : Continued. The trajectories for 22 more sets of initial conditions

were computed with the NORC and were sent to the sponsor.

1102-4-0-5126/56-147 GROUND WAVE ATTENUATION

Origin and Sponsor : NBS , Section 82.1

Managers : B. G. Urban, H. H, Howe (82.1)

Full task description : Oct-Dec 1955 issue, p. 29

Status : Inactive. For status to date, see Oct-Dec 1955 issue, p. 30.

1 1 02-1 0-51 26/56-1 51 MULTIPLE SCATTERING IN CROSS SECTION MEASUREMENTS

Origin and Sponsor : Naval Research Laboratory

Manager : I. Rhodes

Full task description : Oct-Dec 1955 issue, p. 30

Status : Continued. Of the 16 sets of parameters, assigned computations on four have been completed. Computing time takes from 25 to 40 minutes,

depending on the number of trials required.

1102-40-5126/56-1 52 COST ACCOUNTING OPERATIONS FOR HIGH SPEED COMPUTERS, II

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

Manager : G. Hawkins

Full task description : July-Sept 1955 ifsue, p. 38

Status : Continued. The code for editing and presenting the output was

finished. Because time on the UNIVAC in the Washington area is

practically impossible to obtain, a total of only three hours of code

checking was available during the quarter. Debugging of the codes was

thus delayed.

11 02-40-51 26/56-t 57 DAMAGE ASSESSMENT

Origin and Sponsor : Office of the Assistant Secretary of Defense (Supply

and Logistics)

Managers : A. J. Hoffman, H. Bremer

Full task description : Oct-Dec 1955 issue, p. 31

Status : Continued. Meetings have been held with the sponsors in order to
26 Status of Projects

reformulate the model so that new and more extensive programs could be

be written that will more nearly represent the problem. Some of the

input programs are being written at present. There is also an interim

code to be written which will adapt routines of the past to conform with

a better representation of the input data.

1102-^0-5126/56-158 FEDERAL RESERVE DATA

Origin and Sponsor : Federal Researve Board

Manager : E. Marden

Full task description : Oc^-Dec 1958 issue, p. 31

Status : Continued. Five codes were written, together with some needed

auxiliary service routines that were unavailable at the UNIVAC installation

where these tabulations were performed.

It became increasingly difficult to obtain sufficient UNIVAC time

to accomplish the desired objectives, and even time on auxiliary

machines was in short supply.

The Air Force card-to-tape machine was used to convert the

information on the approximately 200,000 punched cards to 55 UNIVAC

magnetic tapes, which were then processed (and to some extent compressed)

to form new data tapes for use in the actual tabulations. UNIVAC f s at the Census Bureau, the Army Map Service, and the Du Pont Company (in

Newark, Delaware) were used for this purpose. Tabulations, making use

of the processed data, are under way on the Du Pont UNIVAC.

1102-^0-5126/56-162 STRESSES IN A WALL RESTING ON A FOOTING

Origin and Sponsor : NBS, Section 10.1

Manager: I. Stegun

Objective : To evaluate on SEAC coefficients in a stress function 0 whicn

is used in obtaining the stresses and strains due to shrinkage in a concrete wall resting on a footing.

Background : The interior of a wall can be considered to be an elastic

slab which obeys the equation of elasticity. In the equation

^ + 2

^

2 -

2. + ^ = °> ax ax ay ay

(i)

0 is the stress function which gives the stresses in the wall by the

following relationships:

a 20

<TX = ~ stress i- n x-direction,

ay

0- = 4j = stress in y-directi.n,

ax ^ _ _ a^0 _ xy” axay

shear stress.

No single expression satisfying equation (1) above and the boundary

conditions of the problem could be found. Therefore it has been decided

to obtain a solution in the form of a series of expressions, each of

which will identically satisfy equation (1) and the sum of which can be

made to satisfy the boundary conditions to any desired degree of accuracy.

The equation for the stress function used is

0 =

n~r1 > 3 ? • •

• o—x1,2,...

®mn sin mn

m7rx sin MX.

b
Status of Projects 27

n-1

n^+mt + (-1) 2 pr Wt m n m

(n 88d) "

Od2

f

+ n

2

£)

2

nVmt

n+( ~ 1)2 ns

m

8

">n (m2 b

+ n

2 £,2

(n even) a b'

The coefficients k , i . t , p. and s are to be determined by the m m n rn boundary conditions. This necessitates the solution of *+0 simultaneous

equations for a wall twice as high as it is long, and of 60 simultaneous

equations for a square wall. (For the computation of the stress function,

see task 1 1 02-40-51 26/55-88, Jan-Mar 1955 issue, p. 22.)

Status : New. The solutions have been obtained for the rectangular wall

(40 simultaneous equations) and the square wall (60 simultaneous equations).

The matrix inversion codes were employed restricting the number of

vectors to two—the vector of the unknowns and a check vector.

1102-40-51 26/56-1 63 ANGULAR DISTRIBUTIONS AND POLARIZATION EFFECTS IN

NUCLEAR SCATTERING

Origin and Sponsor : Naval Research Laboratory

Manager : I, Stegun

Full task description : Oct-Dec 1955 issue, p. 32

Status : Continued. The code was completed, and various sample problems

are being run to check approximation formulae.

1102-40-5126/56-165 INTEGRALS FOR SCATTERING FUNCTIONS

Origin and Sponsor : Naval Research Laboratory

Manager : R. Zucker

Full task description : Oct-Dec 1955 issue, p. 32

Status : Continued. For

^ » 1, the value of

•“LIMIT = ^|- Ip • p

2 [ln(>+e2

5 )-1]

was verified for several sets of p and t . The code was checked out, and

a sample case was run for a given set of parameters.

1102-40-5126/56-166 SCF-LCAO SOLUTION OF SOME HYDRIDES

Origin and Sponsor : NBS, Section 5.9

Manager : E. Haynsworth

Objective : To find the eigenvalues, €• , and eigenvectors, C- of the

secular equation

(Ht-G)C = esc

where H, G, and S are nxn symmetric matrices, n ^ 6. H end S are constant,

but the elements of G are quadratic functions of the eigenvectors C.

.

Background : The eigenvectors are the coefficients of the atomic orbital in

the LCAO approximation to a molecular orbital, and the secular equation

results from minimization of the expression for the ground state energy

for the molecules considered with respect to the coefficients, (See

C,C. J. Rootham, Rev. Modern Phys. 2^, 69 (195D.) The prnblem waa
28 Status of Projects

requested by Morris Krauss (5.9). . , w . ..

Status: New, The main code has been written but has not been checked^

iTs~~a* whole [ Most of the subroutines involved, such as matrix multiplication

and addition, calculation of determinants, orthonormalization of vectors,

have been checked.

1102-^0-5126/56-169 CRYSTAL FIELD EFFECTS FOR ATOMS

Origin and Sponsor : NBS , Division 3

Manager : I. St e gun . Ob iec t ive : To compute eigenvalues and eigenvectors of matrices as functions

of parameters describing various crystal fields.

Background: The calculations arise in the study of the effect of the

^ combinations of crystal field and spin-orbit coupling on the optical

and magnetic properties of solids. The problem was transmitted by

C. M. Herzfeld (3.0). . - ,, Status: New. For heavy atoms the matrix for the combined crystal fieia

effect and spin-orbit coupling is a 33x33 matrix. This matrix factors

directly into three 8x8 matrices and one 9x9. The eigenvalues and

eigenvectors were obtained for ten 8x8 matrices and five 9x9 matrices.

For nitrogen atoms the eigenvalues and eigenvectors were

obtained for fifteen 5x5 matrices.

1 1 02-50-51 26/56-1 71 COLLISION INTEGRALS USED IN TRANSPORT THEORY

Origin and Sponsor : NBS, Section 3.2

Manager : J. Cooper

Full task description : Oct-Dec 1955 issue, p. 33

Status : Continued. A short code to compute the value of

X = 7T- 2b

L

00

m

_dr/r!

Vi-o*/ - m

for various values of g and b was written to determine the behavior

of x using the Morse potential:

0(r) = W*(1 r) (e^(1 r) -l).

To find r (the largest root of the denominator) the equation r ^.=r -f(r ), m n+ in n where f(r) is the expression under the radical sign in the above equation,

was solved by iteration. In order to improve convergence it was found

convenient to replace f(r) by X (, r )_ and iterate using the form

f

1 (r

)

n+1 = r n " f(r

n

) f'(r„)

Convergence using this form in conjunction with an iteration

accelerator technique was rapid, and the process converged for all cases

tried except when /3 in the Morse potential was large.

The integration was performed using l6-point Gaussian quadrature

on the interval 0 to 1/r m

, transforming the integral using p

- 1/r.

However, due to the singularity at the upper limit the accuracy was not

sufficient. The code has now been modified to compute the integral using

the transformation 1 = sin P which removes the singularity except for

r r m certain values of the parameters b, g and y3.
Status of Projects 29

1102-40- 5126/56-172 NUMERICAL EVALUATION OF SPECIAL INTEGRAL EXPRESSIONS

Origin and Sponsor : Diamond Ordnance Fuze Laboratory, Department of the Army

Managers : P. Henrici, L. Joel

Objective : To evaluate some principal value integrals of the form

P I °° */

2

2

I

o (xy) x dx

Jo x -»

p where f (x ) is an elementary differentiable function for CKx<oo

.

Background : Integrals of this type occur in the theory of elastic vibrations.

In a report by P. M. McQuinlan, "Elastic theory of soil dynamics," p. 3

(Symposium on Dynamic Testing of Soils, ASTM Special Technical Publication

No. 1 56 , 1953 ) » simple cases of transmission of tremors or vibrational

disturbances through surface soils are presented. The current study is an attempt to complete the analysis of one unresolved aspect in this

report as expressed in the above integral. The integral gives the soil

vibration amplitude at a distance from the source as 8 function of soil

parameters and experimental conditions. The results of this evaluation

of Quinlan's integral will be compared with other experimental data

obtained at the NBS in an attempt to check the validity of the analysis.

The problem was requested by E. N. Skomal.

Status : New. In order to avoid the singularity in the numerical integration,

one introduces the function

g(x 2

) =

x -a

The integral can then be written

g(x2

)T0 (xy)x dx + f (a

2

) ~~2 2 J

0 (xy)x dx

X f’

where P

{ } denotes the principal part of the integral. Here the first

integral has a regular integrand, and the second can be evaluated

explicitly in terms of the Neumann functions.

1102-40- 5126/56-173 COMPUTATIONS FOR JOB SATISFACTION INVENTORY

Origin and Sponsor : Public Health Service

Manager : S. Prusch

Objective : To determine and compare distributions of data in a job

satisfaction inventory, and to compute the^2 distribution for each of

21 5 items in the inventory.

Background : The computations arise in connection with the study of a job

satisfaction inventory for five groupings of personnel in three hospitals.

An attempt is being made to check the relative validity of the 215 items

with a view to reducing the number to 100 . To aid in this it is necessary

(1) to determine how the distribution of individuals with favorable and

unfavorable responses to each item compares with the distribution of

individuals with high and low scores, and (2) to determine how the

distribution of individuals with favorable and unfavorable responses

to an item compares with the distribution of individuals who are satisfied

or dissatisfied as determined by special criteria. This was specifically

requested by B. Buchbinder of the Public Health Service.

Status ; New. The problem was coded, and code-checking is in progress.
30 Status of Projects

1102-40-5126/56-176 MODEL GOVERNMENT PAYROLL ON HIGH SPEED COMPUTERS

Origin and Sponsor : NBS, Section 12.5

Manager : G. Hawkins

Objective : To investigate the feasibility of translating the Government

payroll computations to electronic computer operations. This investigation

is to be effected by studying in particular the conversion of the operation

of the NBS payroll system (1) to a large general-purpose computer, and

(2) to a drum computer, and then comparing the efficiency of these two

systems with each other and with an IBM mechanized system.

Background : The NBS payroll system was recently mechanized with IBM

equipment. Biweekly time cards for each employee are submitted to the

Payroll Section. From these, time cards and paychecks are computed and

processed; and various division and bureau reports are derived and used

for time distribution analyses and general accounting procedures. The

present investigation aims to determine whether further time-saving and

efficiency is possible if the material were to be processed on a digital

electronic computer.

Status : New. Coding of the problem for UNIVAC is in progress.
6 . STATISTICAL ENGINEERING SERVICES

COLLABORATION ON STATISTICAL ASPECTS OF NJBS 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 ore representative examples:

(1) A 3x3 factorial experiment in 9 blocks of 3 units each was

analyzed for J. Wachtman, Section 9.1, to evaluate the

characteristics of the modulus of rupture of sapphire single

crystals with respect to loading rates and temperatures.

(2) Statistical analyses were made on SEAC for H. C. Allen,

Section *+.2; W. G. Eicke, Section 1.8; G. W. Cleek, Section

9.2; J. Hilsenrath, Section 3.2; A. Friedman, Section 3.2;

M. D. Burkhard, Section 6.1; and R. S. Jessup, Section 3.2,

In the last two instances special codes were written for SEAC.

(3) Members of the staff consulted with personnel of the Fire

Protection Section (10.2) on the scheduling of tests of fire

protecting finishes for building materials.

(^) W. J. Youden has been working with W. B. Mann, Section ^.9?

on experiment designs for the intercomparison of radium

standards from several national laboratories.

(5) An analysis of the data on lengths of meter bars was made

for Section 2.*+ to determine the improvement in precision

due to improved methods of marking lines on the newer bars.

(6) M, Zelen served as statistical consultant to the members of

Section 12.5 who are studying the sorting operations of the

Post Office.

Publications:

(1) National physical standards and design of experiments. W. J. Youden.

To appear in Revue de l'lnstitut International de Statistique.

(2) More scientists. W. J. Youden. Submitted to a technical journal,

(3) A simplified calibration of platinum versus platinum-rhodium thermocouple

B. E. Squires (3.1) and M. Zelen. In manuscript.

STATISTICAL SERVICES FOR COMMITTEE ON SHIP STEEL, NRC

Task 11 03-^0-51 05/52-1

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: CONTINUED. A multiple regression study of the effect of

chemical composition on ship steel was nearly completed.

31
32. Status of Projects

RESEARCH IN APPLICATIONS OF MATHEMATICAL STATISTICS TO

PROBLEMS OF THE CHEMICAL CORPS

Task 1103-4

- 0- 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. A report giving a collection of formulas for

computing probabilities from common statistics was submitted.

Work on tabulating a function related to problems of diffusion

in the atmosphere was begun.

STATISTICAL ANALYSIS AND DESIGN OF EXPERIMENTS

FOR THE U. S. GEOLOGICAL SURVEY

Task 11 03-^0-51 4-0/54-1

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

Department of Interior

Managers: C. Eisenhart, W. J. Youden

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

Status: CONTINUED. Some quick and easy methods for estimating the

direction of gradient and the properties of such estimators are being

worked out.

Data supplied by Dorothy Carroll of the Beltsville Laboratories

concerning counts of different kinds of minerals in samples taken from

tributaries of the South River in Virginia were analyzed for the most

abundant mineral, the "opaques," with the aim of deciding how many samples

should be taken from each location in order to get the most information for

a given cost.

Consultations were held with J. Hack on application of extreme

values and other statistical methods (including order statistics) to

pebble deposits.

C. Eisenhart visited the U. S. Geological Survey Center, Menlo

Park, California, March 19-24-, at their request (i) to acquaint the

geologists at Menlo Park with the uses and usefulness of statistical methods

in laboratory and field studies, and with the necessity of giving adequate

consideration to statistical aspects of such studies in the planning stage;

(ii) to discover in what particular phases of the work going on at Menlo

Park statistical methods might be most advantageously applied; and (iii),

to make some tentative plans for provision of statistical assistance to

the geologists at Menlo Park in the future. While there he gave five

lectures to the staff of the center as follows: (1) Evolution of statistical

method as a research tool; ( 2 ) Some canons of sound experimentation;

(3) Principles of sampling: (4-) Variance components and optimum allocation

of sampling effort; and (5) Experiment designs. On the final day he

presided over an informal meeting at which (1) representatives from the

Department of Statistics, Stanford University, and the Statistical Laboratory

University of California, Berkeley, outlined the- statistical courses of

potential value to geologists available at these two neighboring educational

institutions, and ( 2 ) geologists from the center outlined some of their

problems to the statisticians and sought their advice on them.
Status of Projects 33

MANUAL ON EXPERIMENTAL STATISTICS

FOR ORDNANCE ENGINEERS

Task 1 1 03-l+0-5l ^6/55-93

Origin and Sponsor: Office of Ordnance Research Authorized 12/29/5^

Manager: C. Eisenhart

Full task description: Oct-Dec 1951+ issue, p. 28

Status: CONTINUED. Drafts of the following sections have now been

completed: "Estimating average performance from a sample," "Estimating

the variability of performance from a sample," "Statistical tests concerning

averages and dispersions." A draft of the section "Description, prediction,

and correlation" is nearly completed, and some work has been done on the

sections "Simple control charts for means and ranges," "Estimating true

proportions or percentages," and "Statistical tests concerning proportions."
APPLICATION

of

NATIONAL BUREAU OF STANDARDS AUTOMATIC COMPUTER (SEAC)

The record of SEAC operations for tasks of the Applied Mathematics

Division for the period January 1 through March 31 is as follows:

Hours Used:

Task No. Title

Code

Checkina

Productive

Ooeration

NBS:

1104/55-55 Research in numerical analysis 6

5116/56-148 Nerve fiber reaction 20

1 1 1 0A7-2 Tables of Coulomb wave functions 9 25

1110/52-57 Sieverts integral 2

1110/53-51 Radiation diffusion III 17 42

1110/53-52 L-Shell conversion coefficients 24

1110/55-65 Automatic coding 2

5126/52-44 Calculations for d-spacings 14

5126/53-29 Dynamic behavior of aircraft structure 5 19

0009/54-17 Depolymerization 1 16

5126/54-30 Spectrometer analysis 1 1 0009/55-53 Electronic functions 10

0009/55-75 Parameter of the dispersion equation

1 5126/55-81 CoJn8£n?R§ 2

:*'£isti?*? Ef significance 55

5126/55-87 Crystal structure 10

5126/55-97 High temperature properties of air 3? 22

5126/55-118 Thermometer calibrations 14 4

5126/55-121 Electron penetration 2

5126/55-124 Evaluation of integrals related to

gas flow 6 4

5126/55-127 Vibrations of circular disc 10 19

5126/56-166 SCF-LCAO solution of some hydrides 13

0009/56-131 Ray tracing 69 00 \r\ 5126/56-134 Transmission delay times 1 5126/56-139 Study of internuclear potential for H^ 10 13

5126/56-150 Mathematical expression 3

5126/56-169 Crystal field effects for atoms 20

5126/56-171 Collision integrals used in transport 6 4

5126/56-179

theory

Normal propagation constant 27 51

5126/56-178 Cytac 3 2

0002/52-1 Statistical aspects of administrative 6 36

Misc

.

operations

Training 1 n Least squares 2

Other:

5126/53-45 Application of game theory 4 5

5126/54-13 Award of procurement contracts for

linear programming 12

34
35

Task No. Title

Hours

Code

Checkinq

Used:

Productive

Operation

Other:

51 26/55-1 04 "Fuse" problem 1 58

5126/55-113 Reactor design 3 78

5126/55-119 Field rocket problem 10

5126/56-1 30 Aircraft responses 6 2

5126/56-133 Complex eigenvalues 1 3

5126/56-136 Calculation of wave functions 6 32

5126/56-137 Stability of supported plates 22

5126/56-138 Crystal counter efficiency 3

5126/56-1^0 Multiple correlation routines 15 5 5126/56-149 Logical matrix multiplication 4 3

5126/56-151 Cross section measurements 1 4i

5126/56-163 Nuclear scattering 9 5 5126/56-165 Integrals for scattering functions 13 1 5126/56-167 Multiple regression 1 5126/56-172 Special integral expressions 1 5126/56-173 Job satisfaction inventory 9

5126/56-181 Coulomb - AEC 27

5160/55-85 Mathematical elasticity 3 106

5160/56-156 Mathematical geophysics 13 7
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.

Participation in NBS Scientific Staff Meetings

STONELEY, R. (Cambridge University) The thickness of the continents.

February 10.

SYNGE, J. L. (Institute for Advanced Studies* Dublin) The concept of

time. March 9.

Applied Mathematics Division Seminar

ARONSZAJN, A. (University of Kansas) Survey of the work in progress on partial differential equations, in particular eigenvalue

problems at the University of Kansas. March 9.

Seminar: Special Topics in Numerical Analysis

(Offered by the Department of Mathematics of The American University in

collaboration with the Graduate School of the National Bureau of Standards)

TODD , J. (1) The theorems of Markoff. January 10. (2) The theorem of

Zolotareff. January 17.

Algebra Seminar

(Offered by the Department of Mathematics of The American University in

collaboration with the Graduatre School of the National Bureau of Standards)

GORN, S. (Moore School of Electrical Engineering, University of Pennsylvania)

Computations in cubic field. February 13.

NEWMAN, M. A theorem of Mordell concerning positive quadratic forms, I,

II. February 20, 29.

TODD , J. Number theory in quaternions, March 7.

TAUSSKY-TCDD, 0. Mordell’ s theory of positive binary quadratic .form.

March 1*+. MARSHALL, L.C. (Department of Defense) General theory of binary quadratic

form I, II. March 21 , 28.

36
Lectures end Symposia 37

Statistical Theory Seminar

(Offered jointly by the National Institutes of Health and the National

Bureau of Standards.)

CORNFIELD, J. (NIH) . Models in the analysis of variance. January 17.

ROSENBLATT, J. R. (NBS) The efficiency of statistical tests. February 17.

Applied Statistics Seminar

(Offered jointly by the National Institutes of Health and the National

Bureau of Standards.)

MEIER, P. (Johns Hopkins University) Some statistical aspects of safety

testing of vaccines. March 16.

Mathematical Physics Section Seminar

STONELEY, R. (Cambridge University) Theory of elastic surface waves. Three lectures: February 9, 16, 23.

SYNGE, J. L. (Institute for Advanced Studies, Dublin) Elastic waves

in anisotropic media. March 8 .

Papers and Invited Talks

Presented by Members of the Staff

at Meetings of Outside Organizations

ABRAMOWITZ, M. The evaluation of special functions on high speed computers.

Presented at the Moore School of Electrical Engineering,

University of Pennsylvania, Philadelphia, February 8.

CHIARULLI, P. Applied mathematics and the atomic bomb. Presented at a meeting of Sigma Pi Sigma, Howard University, Washington, D.C.,

January 18.

EISENHART, C. (1) Evolution of statistical method as a research tool.

(2) Some cannons of sound experimentation. (3) Principles of

sampling. (*+) Variance components and optimum allocation of

sampling effort. (5) Experiment designs. A series of lectures

presented to the staff of the U. S. Geological Survey Center,

Menlo Park, California, March 19-23.

HOCHSTRASSER, U. Methods and tools in applied mathematics. (1) Presented

at a meeting of Sigma Pi Sigma, Howard University, Washington,

D. C., February 17, and (2) Presented to the Research and

Development Group, Naval Powder Factory, Indian Head, Md.,

February 29.

HOFFMAN, A. J. (1) Cross sections of two families of sets. Presented at

a meeting of the Logistics Research Project, George Washington

University, Washington, D. C. , February 2, (2) Integral

solutions of linear programming problems. Presented to the

Industrial Engineering Department of Johns Hopkins University,
38 Lectures and Symposia

Baltimore, Md., February 15. (3) Representatives of sets.

Presented at a Combinatorial Problems Seminar, Princeton

University, Princeton, N. J., March 19.

STONELEY, R. (Guest Worker, from Cambridge University).

(1) The interior of the earth. Presented at the National

Academy of Sciences, Washington, D. C., January 10. (2) The

propagation of elastic waves over the surface of a cubic

crystal. Presented at the Exploration and Production Research

Laboratory, Shell Development Co., Houston, Tex., January 17.

(3) The thickness of continents. Presented at the Agricultural

and Mechanical College of Texas, College Station, Texas,

January 19. (4) The variation of amplitude with depth in

Rayleigh waves. Presented before the Seismological Society of

America, Reno, Nevada, March 30.

SYNGE, J. L. (Guest Worker, from Institute for Advanced Studies, Dublin).

(1) Classical dynamics and de Broglie waves. Presented at

New York University, February 6; at Princeton University,

February 7> at Brown University, February 9; at the Ballistic

Research Laboratory, Aberdeen Proving Ground, Md., February 21;

at the Carnegie Institute of Technology, Pittsburgh, Pa.,

March 1; at the Institute for Fluid Dynamics and Applied

Mathematics, University of Maryland, March 28. (2) Geometrical

optics in moving media. Presented at Harvard University,

Cambridge, Mass., March 6. (3) Classical dynamics and optics:

de Broglie waves and photons. Presented to the Department of

Mathematics, University of California at Los Angeles, March

13, I 1

*-. (4) Elastic waves in anisotropic media. Presented

at the Massachusetts Institute of Technology, Cambridge, Mass.,

March 7. Also presented at the Geophysical Institute, University

of California at Los Angeles, March 15-1 8.

TAUSSKY-TGDD , 0. (1) Pencils of matrices. Presented at the Institute for

Fluid Dynamics, University of Maryland, College Park, Md.,

February 17. (2) Characteristic roots of sums and products

of finite matrices. Presented at the Naval Ordnance Laboratory,

White Oak, Md., January 18.

YOUDEN, W. J. (1) Design of experiments in the physical sciences. Presented

at the Nuclear Power Division of the Bureau of Ships* Washington,

D. C. , January 3. (2) Interpretation of chemical data. Presented

before the local chapter of the American Chemical Society,

Morgantown, W. Va., January 9. (3) Control of experimental error

by statistical design. Presented before the local chapters of

the American Chemical Society in Charleston, W. Va.; Cincinnati,

Ohio; and Lexington, Ky., January 10-12.

(49 Making good measurements. Presented before the

Indianapolis Chapter of the American Statistical Association,

Indianapolis, Ind., January 13. (5) Experimental designs for

the physical sciences. Presented before a joint meeting of the

Minnesota ASQC and the American Institute of Chemical Engineers,

St. Paul, Minn., January 17> and also before a group of scientists

and engineers at the Westinghouse Electric Corp., Bettis Plant,

Pittsburgh, Pa., March 2.

(6) Improvement in the technique of learning. Presented

before the professional staff of the Baltimore County Board of

Education, Towson, Md., January 30. (7) Dice, data, and deduction

Presented before the Physics Club and Chemical and Mathematics

Societies of Trinity College, Washington, D. C. , February 1,

and also before the Mathematics Club, Woodrow Wilson High School,

Washington, D. C., February 28.
Lectures and Symposia 39

(8) Statistics for novices. Presented at the Sixth Annual

Convention Textile Division, ASQC, Charlottesville, Va.,

February 13.

(9) Design of experiments. Presented before the

Youngstown Section of ASQC, Youngstown, Ohio, March 1. (10) Design

of experiments in research and development. Presented before

the Washington Chapter, American Society for Metals, March 12.

(11) Cubes, colors, and computations. Presented before the

Roosevelt High School Science Club, Washington, D. C. , March 19.
Publication Activities

1 . PUBLICATIONS THAT APPEARED DURING THE QUARTER

1.3 Technical Papers

(1) Analytic study of war games. H. Antosiewicz. Naval Research

Logistics Quart. 2, 1 81—21 5 (Sept. 1955).

(2) Chaines de Markof dans les ensembles abstraits et applications 8ux

processus avec regions absorbantes et au probleme des boucles.

(Formerly, 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. Ann. Inst.

H. Poincare jj+, 1*+5- 199 (1955).

(3) Abscissas and weights for Gaussian quadratures of high order.

P. Davis and P. Rabinowitz. J. Res. NBS ^6, 35_37 (Jan. 1956),

RP2645.

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

P. Davis and P. Rabinowitz. J. Wash. Acad. Sci. j+6, 12-17 (1956).

(5) Comparison of theories and experiments for the hydraulic dam-break

wave. R. F. Dressier. Publication No. 38 de 1 'Association

Internationale d'Hydrologie (Assemblee generale de Rome, 195*+?

tome III, p. 319-328). Extrait.

(6) Heat flow in a fluid with eddying flow. W. H. Durfee (now at

Mt. Holyoke College). Condensed version appears in J. Aero. Sci.

23, 188-189 (Feb. 1956).

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

Pac. J. Math. 5, Supp. 2, 911-913 (1955).

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

J. Assoc. Comp. Mach. 3? 6-9 (Jan. 1956).

(9) Automatic computations with power series. P. Henrici. J. Assoc.

Comp. Mach. 10-15 (Jan. 1956).

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

Pac. J. Math. 923-931 (1955).

(11) Commutators of A and A*. T. Kato (University of Tokyo) and

0. Taussky. J. Wash. Acad. Sci. .|+o, 38-1+0( 1 956)

.

(12) Pairs of matrices with property L, II. T. S. Motzkin (University

of California at Los Angeles) and 0. Taussky. Trans. Amer. Math.

Soc. 80, 387-^01 (Nov. 1955).

(13) An. alternative proof of a theorem on unimodular groups. M. Newman.

Proc. Amer. Math. Soc. 6, 998-1000 (1955).

^0
Publication Activities 4l

(14) The normalizer of certain modular subgroups. Canadian J. Math. 8,

29-31 (1956).

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

fields. M. Newman and 0. Taussky. Comm. Pure Appl. Math. (NYU)

2) 85-91 (1956).

(16) On asymptotic series for functions occurring in the theory of

diffraction of waves by wedges. F. Oberhettinger . J. Math. Phys.

i!+, 245-255 (Jan. 1956).

(17) Determinanten mit ueberwiegender Hauptdiagonale und die absolute

Konvergenz von linearen Iterationsproze ssen. A. M. Ostrowski,

Comment. Math. Helv. 32) 175-210 (1956).

(18) Contributions for inverse osculatory interpolation. H. E. Salzer,

(Department of the Army). J. Res. NBS £6 , 51-54 (Jan. 1956),

RP2647.

(19) A note on group matrices. 0. Taussky. Proc. Amer. Math. Soc. 6,

984-986 (1955).

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

(American Cyanimid Co.). Biometrics 1

1

« 399-405 (Dec. 1955).

2. MANUSCRIPTS IN THE PROCESS OF PUBLICATION MARCH 31) 1956

2.1 Mathematical Tables

(1)

A table of the coefficients of the powers n ('£') . M. Newman. To

appear in the Proceedings of the Royal Netherlands Academy of

Science s

.

2.2 Manuals, Bibliographies, and Indices

(1) Analyzing straight line data. F. S. Acton. To appear as a book

in the Applied Statistics Series of the Wiley Publications in

Statistics

.

(2) Fractional factorial experimental designs for factors at two

levels. W. S. Connor, M. Zelen, L. Deming, and W. H. Clatworthy.

To appear as NBS Applied Mathematics Series 48.

2.3 Technical Papers

(1) On the differential equation x + k(f(x) + g(x)x)x = ke(t).

H. A. Antosiewicz. Submitted to a technical journal.

(2) Stable systems of differential equations with integrable forcing

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

Mathematical Society.

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

Submitted to a technical journal.

(4) Dilworth's theorem on partially ordered sets. G. B. Dantzig (RAND)

and A. J. Hoffman. To appear in "Papers on linear inequalities

and related systems," Annals of Mathematics Study 38.

(5) Uniqueness theory for asymptotic expansions in general regions.

P. Davis. To appear in the Pacific Journal of Mathematics.
b2 Publication Activities

(6) Complete sequences and approximations in normed linear spaces.

P. Davis and K. Fan. Submitted to a technical journal.

(7) Some Monte Carlo experiments in computing multiple integrals.

P. Davis and P. Rabinowitz. To appear in Mathematical Tables and

Other Aids to Computation.

(8) Entropy changes in rarefaction waves. R. F. Dressier. Submitted

to a technical journal.

(9) On systems of linear inequalities. K. Fan. To appear in"Papers

on linear inequalities and related systems," Annals of Mathematics

Study 38, ed. by H. W. Kuhn and A. W. Tucker.

(10) Systems of inequalities involving convex functions. Ky Fan.

Submitted to a technical journal.

(11) A note on the normal distribution. S. Geisser. To appear in Annals

of Mathematical Statistics.

(12) A matrix with real characteristic roots. K. Goldberg. To appear

in the Journal of Research (NBS).

(13) The formal power series for log e x

e^. K. Goldberg. To appear in

the Duke Journal of Mathematics.

(I 1*) On the domain of regularity of generalized axially symmetric

potentials. P. Henrici. To appear in the Proceedings of the

American Mathematical Society.

(15) On the representation of a certain integral involving Bessel functions

by hypergeometric series. Submitted to a technical journal.

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

(Bryn Mawr College). To appear in "Papers on linear inequalities

and related systems," Annals of Mathematics Study 38.

(17) Systems of distinct representatives and linear programming.

A. J. Hoffman and H. W. Kuhn (Bryn Mawr College). To appear in

the American Mathematical Monthly.

(18) The number of absolute points of a correlation. A. J. Hoffman,

M. Newman, E. G. Straus, 0. Taussky. To appear in the Pacific

Journal of Mathematics.

(19) On the Hilbert matrix. T. Kato. Submitted to a technical journal.

(20) Applications of extreme value theory to cobbles and boulders in

gravel pits. W. C. Krumbein (Northwestern University) and

J. Lieblein. Submitted to a technical journal.

(21) Solvability and consistency for linear equations and inequalities.

H. W. Kuhn (Bryn Mawr College). To appear in the American Mathematical

Monthly.

(22) An inclusion theorem for modular groups. M. Newman. Submitted to a technical journal.

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

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

Society.
Publication Activities 5-3

( 25-) On the existence of identities for the coefficients of certain

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

Mathematical Society.

(25) The diophantine equation axm-byn=c. M. Newman. Submitted to a technical journal.

( 26 ) On the Lerch zeta function. F. Oberhettinger . To appear in the

Pacific Journal of Mathematics.

(27) On Gauss' speeding up device in the theory of single step iteration.

A. M. Ostrowski. Submitted to a technical journal.

(28) On the convergence of Gauss' alternating procedure in the method

of the least squares, I. A. M. Ostrowski. Submitted to a technical journal.

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

To appear in the Annals of Mathematical Statistics.

( 30 ) On the independence of tests of randomness and other hypotheses.

I. R. Savage. To appear in the Journal of the American Statistical

Association.

(31) The transmission of Rayleigh waves across an ocebn floor with two

surface layers. R. Stoneley. Submitted to a technical journal.

( 32 ) Algebraic equations satisfied by roots of natural numbers.

E. G. Straus (University of California at Los Angeles) and 0. Taussky.

To appear in the Pacific Journal of Mathematics.

(33) 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.

(35-) Generation end testing of pseudo-random numbers. 0. Taussky and

J. Todd. To appear in the Proceedings of a Symposium on Monte

Carlo methods, held at Gainesville, Fla., 1955%

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

solution of partial differential equations. To appear in the

Proceedings of a Symposium on Partial Differential Equations held

at Berkeley, California, 1955.

(36) National physical standards and design of experiment. W. J. Youden.

To appear in Revue de L'Institut International de Statistique

(The Hague).

(37) Exact tests of significance for combining intra- and inter-block

information. M. Zelen. Submitted to a technical journal.

(38) On the analysis of covariance. M. Zelen. Submitted to a technical

journal.

2.5 Miscellaneous items

(1) Development of automatic computers. F. L. Alt. To appear in the

Proceedings of the American University Institute on Electronics in

Management, held in Washington, D. C. , November 1955.
l+*+ Publication Activities

(2) Contributions on partially balanced incomplete block designs with

two associate classes. W. H. Clatworthy. To appear as NBS

Applied Mathematics Series *+7.

(3) More scientists. W. J. Youden. Submitted to a technical journal,
THE NATIONAL BUREAU OF STANDARDS

Functions and Actirities

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 ($1.25) and its Supplement ($0.75), 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.