ACTIVITIES IN APPLIED MATHEMATICS

1946-1947

NATIONAL APPLIED MATHEMATICS LABORATORIES

OF THE

NATIONAL BUREAU OF STANDARDS
PREFACE

This is a summary of the mathematical activities carried on during the fiscal year 1947 in a unit of the Office of the Director of the National Bureau of Standards . This unit of the Office

of the Director was known within the Bureau by the code designation DA-2, and included as a component the well-known Mathematical

Tables Project in New York City. The entire unit has since been

incorporated into a new Division of the Bureau known as the

National Applied Mathematics Laboratories.

The projects described in this report are representative of

the present work of the National Applied Mathematics Laboratories.

It is the mission of the Laboratories to perform research and

provide services in various quantitative branches of mathematics,

placing special emphasis on the development and exploitation of

high-speed numerical analysis and modem statistical methodology.

The Laboratories maintain an expert computing service of large

capacity, and provide consulting services in classical applied

mathematics and in mathematical statistics. These services are available primarily to other federal agencies, but under certain

circumstances it is possible to perform work for industrial laboratories and universities.

Inquiries concerning the availability of the services of the

National Applied Mathematics Laboratories, or concerning the details of any of the projects described in this report, should be

addressed to the National Applied Mathematics Laboratories, 415

South Building, National Bureau of Standards, Washington 25, D.C.

J . H. Curtiss, Chief

National Applied Mathematics

Laboratories

Approved:

E. U. Condon, Director

National Bureau of Standards

October 15, 1947
U. S. DEPARTMENT OF COMMERCE

W. Averell Harriman, Secretary

NATIONAL BUREAU OF STANDARDS

E. U. Condon, Director

ACTIVITIES IN APPLIED MATHEMATICS

1946-1947

CONTENTS

Page No.

1. Administrative and Planning Activities 1

2. Research and development Projects Completed

during the Fiscal Year 1947 7

3. Research and Development Projects Under Way

at the End of the Fiscal Year 1947 9

A. Projects on which substantial progress

was made during the fiscal year .... 9

B. New Projects which were still in the

planning stage at the end of the

fiscal year 18

4. Consultation -and Advisory Services 20

A. Statistical engineering 20

B. Computing machinery 22

5. Participation in the Activities of Outside

Technical Groups ..... ..... 23

A. Committee activities 23

B. Papers presented at meetings 24

National Applied Mathematics Laboratories of the National Bureau of Standards
\
1. ADMINISTRATIVE AND PLANNING ACTIVITIES

The Office of the Assistant to phe Director, DA-2, consisted

during the fiscal year 1947 of the organization set forth in

Table I. It will be noted from the table that certain units were activated during the fiscal year* The growth in the Mathematical

Tables Project was the result of taking over the New York Project

of the Navy Department Hydrographic Office, which throughout the

war had been supervised by personnel of the Mathematical Tables

Project on loan.

Table I . Organ:Lzation of DA-2

Number of Persons

Unit 1 July 1946 30 June 1947

1. Administrative

(Washington) 3 5

2. Mathematical Tables Project

(150 Nassau St., NYC) 48 60

3. Statistical Engineering

Group (Washington) 0 6

4. Machine Development Group

(Washington) 0 4

The administrative head of these mathematical activities was

Dr. J. H. Curtiss, Assistant to the Director. The Director of the

Mathematical Tables Project was Dr. Arnold N. Lowan, and his two

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ACTIVITIES IN APPLIED MATHEMATICS

chief assistants were Dr. Gertrude Blanch and Mr. Milton Abramowitz.

The chief of the statistical group was Dr. Churchill Eisenhart,

who reported for duty on 1 October 1946. The head of the Machine

Development Group was Dr. E. W. Cannon, who reported for duty on

1 July 1946. His chief assistant was Mr. Albert Calm, who joined

the Bureau staff on 13 January 1947.

The efforts of the administrative unit were focused during

the fiscal year on the task of making plans for a new division of

the Bureau, which was to contain the above organization, and

which was to be so designed that it could serve as a central

applied mathematics laboratory for the entire Federal scientific

program.

The idea of a central computation laboratory with an auxiliary mission of developing high speed computing machines had

arisen in the Planning Division of the Office of Research and

Invention (now the Office of Naval Research) of the Navy Department in 1945. In the early spring of 1946, the Chief of Naval

Research had enlisted the aid of the Bureau of Standards in

studying the problem of establishing such a facility.

The study soon revealed that something more fundamental than

merely a large computation laboratory was needed, and that the

development of high-speed tools for computation carried with it

an obligation to perform research in the theory and disciplines

needed for the most effective use of these tools. The investigation was brought to a conclusion in the fiscal year 1947 with

the publication of a document entitled "The National Applied

Mathematics Laboratories - A Prospectus". The functions of the

Laboratories, as set forth in the Prospectus, would be as follows:

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ACTIVITIES IN APPLIED MATHEMATICS

(a) to conduct basic research in various fields of mathematics heavily drawn upon by the physical and engineering sciences, placing special emphasis on the development and exploitation of high-speed numerical analysis

in these fields;

(b) to provide a consulting service in special problems in

such fields of mathematics;

(c) to develop and construct tools (such as tables and automatic high-speed computing machines) for work in such

fields of mathematics;

(d) to conduct theoretical and in-service training in such

fields of mathematics, and in particular, in the disciplines needed for the effective use of automatic highspeed computing machines;

(e) tc provide an extensive general-purpose computing service

specializing in digital, methods of calculation;

(f) to prepare reports, monographs, manuals, indices, and

other types of publications, setting forth the results

of the work of the Laboratories and providing expository

treatments of technical material lying within the scope

of the Laboratories.

As a corollary to these activities, the Laboratories would

provide facilities for competent scholars to investigate and

develop in their own special fields of research the mathematical

techniques studied by the Laboratories. The major items of

equipment would be three automatic electronic digital computing

machines. One of these would be the machine being constructed by

the Bureau for the Office of Naval Research (see Section 3 of

this Report) .

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ACTIVITIES IN APPLIED MATHEMATICS

Table 2. Organization of the Proposed National Applied Mathematics Laboratories

Section

Man years

Fiscal 1948 Fiscal 1949 and

thereafter

0. Administration

(Washington, D.C.) 6.00 6oQO

1. Institute for Numerical

Analysis

(California) 25.00* 28.50*

2. Computation Laboratory

(New York City and

Washington, D.C.) 61.00 47.00

3. Statistical Engineering

Laboratory

(Washington, D.C.) 6.32 8.32

4. Machine Development Laboratory

(Mathematics Group)

(Washington, D.C.) 4.00 4.00

Totals 102.32 93.82

^It is expected that there will be some part-time employees, so the

number of persons on duty in Section I will be somewhat in excess

of these figures.

An outline of the organizational plan proposed in the "Prospectus" is given in Table 2. A part of the plan, not shown in

the table, consisted in forming an "Applied Mathematics Executive

Council" composed of representatives of interested Government

agencies and private organizations. The members were to be appointed by the Director of the Bureau. The functions of the

Council would be to give advice and guidance at the policy level

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ACTIVITIES IN APPLIED MATHEMATICS

and to approve the budget.

It was decided after a prolonged investigation of sites for

Section 1 to place this unit on the campus of the University of

California at Los Angeles. The controlling factors were a favor^-

able offer from UCLA for permanent quarters, and the interest of

the Army Air Forces in having a computing center available in

Southern California.

The "Prospectus" was approved by the appropriate authorities

in the Commerce Department and Navy Department. Representatives

of a number of other agencies and private institutions also indicated general concurrence. The "National Applied Mathematics

Laboratories” were accordingly established along the lines of the

"Prospectus" as Division 11 of the National Bureau of Standards

on 1 July 1947.

Other important activities in the field of mathematics not

mentioned specifically in the later sections of this report were

the following:

T?he Establishing of the Applied Mathematics Colloquium Series

This is a series of lectures on applied mathematics intended

primarily for Bureau personnel, and given by outstanding applied

mathematicians connected with other institutions. The fiscal

year 1947 was devoted to applied mathematical statistics. The

lectures and attendance were as follows:

Lecture Attendance

"The Combination of Data from Tests Conducted at

Different Laboratories" by Prof. P.C. Mahalanobis,

F.R.S. 100

"Statistical Quality Control" by Prof. Holbrook

Working 65

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ACTIVITIES IN APPLIED MATHEMATICS

Lecture ( continued) Attendance

"Some Problems in the Analysis of Sets of Consecutive Measurements" by Dr. Merrill M. Flood

"Random Processes" by Prof. Harald Cramer

104

97

"A Statistical Technique for Analytical Data"

by Dr. William J. Youden 75

The Initiation of The Applied Mathematics Series

This new series of publications of the Bureau is intended to serve as a vehicle for manuals, expository treatises, and tables

produced by the National Applied Mathematics Laboratories. The

first number of the series, which should appear in the fall of

1947, is a mathematical table particularly useful in nuclear

physics in the construction of certain types of chain-reacting

piles. The title is

"Tables of the Bessel Functions Y0 (x) ,Y 1 (x) ,K0 (x) ,K X (x) , 0^ x^ 1"

Courses given in the Graduate School of the National Bureau of

Standards

"Elements of Statistical Inference" by Churchill Eisenhart;

two terms; attendance: 74 students, in 1st term; 59 students

in 2nd term.

"Correlation and Regression" by Dr. J.H. Curtiss; one term;

attendance: IS students.

"Vector Analysis" by Dr. E.W. Cannon; one term; attendance: 2S

students.

"Modem Operational Methods in Engineering" by Dr. E.Wc Cannon; one term; attendance: 8 students.

"The Differential Equations of Mathematical Physics" by Dr. E.W.

Cannon; two terms; attendance: 8 students, first term; 9 students,

2nd term.

In addition, a seminar on numerical analysis and related

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ACTIVITIES IN APPLIED MATHEMATICS

aspects of function theory was conducted at the Mathematical

Tables Project.

(1) Tables of Spherical Scattering -Functions for Complex Arguments, for Naval Research Laboratory.

(2) Computations related to the hydraulic analogy of shock wave

intersections, for the Bureau of Ordnance, Navy Department.

(3) Estimation of parameters of distribution of maximum wind velocities at various stations, for the Weather Bureau,,

(4) Various sets of Loran Tables for the Hydrographic Office,

Navy Department.

(5) Table of Spherical Bessel Functions, Vol. II.

(6) Tables of the function

(7) Tables of circumferences and areas of circles to six decimal

places, for the Bureau of Ordnance, Navy Department.

(8) Tables of the altitude and azimuth for selected groups of six

stars, for the Hydrographic Office, Navy Department.

(9) Checking of tables in forthcoming OSRD book entitled "Selected

Techniques of Statistical Analysis".

(10)

Tables of

for various values of wQ and a. (The project included research

2. RESEARCH AND DEVELOPMENT PROJECTS

COMPLETED DURING THE FISCAL YEAR 1947

1

Z

(11)

Solutions of the equation of the "Human Centrifuge"

W

2

+ W

4 = (wQ + Ott) 4

7
ACTIVITIES IN APPLIED MATHEMATICS

leading to the reduction of the problem to the solution of

w

2 + w

4 = t4

for which an asymptotic expansion was obtained.) for Special

Devices Center, Office of Naval Research, Navy Department.

112) Tables of the confluent hypergeometric function. (The Project included research leading to new approximations to the

function.) For the Statistical Research Group, Columbia

University (OSRD contractors) .

(13) Tables of the Bessel Functions Jv

(x) for fractional values

of v

.

(14) Tables of operating characteristics of certain double sampling

plans, for the War Department.

(15) Tables of conversion angles for converting a rhumb line

course into a great circle course, for the Hydrographic

Office of the Navy Department

(16) Table of intensity functions for complex indices of re- fraction 0

(17) Formulas for the percentage points of the distribution of

the arithmetic mean in random samples from certain symmetrical universes.

(IS) "Acceptance Sampling by Variables, with Special Reference to

the Case in which Quality is Measured by Average or Dispersion" by J. H. Curtiss. To be published in the Journal

of Research and in a Supplement to the Journal of the American Statistical Association.

(19) "The Assumptions Underlying the Analysis of Variance" by

Churchill Eisenhart. Published in Biometrics, March 1947.

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ACTIVITIES IN APPLIED MATHEMATICS

3. RESEARCH AND DEVELOPMENT PROJECTS UNDER WAY AT

THE END OF THE FISCAL YEAR 1947

A* Projects on Which Substantial Progress Was Made During the

Fiscal Year„

(1) Mathieu Functions I„

An 8-place table of the first fifteen odd and sixteen even characteristic values b of the Mathieu differential equation

y* + (b - s cos 2

t) y = 0

for s ranging from 0 to lOfr at various intervals, was published for limited circulation in AMP report 165. 1R in September, 1945. The manuscript of the Fourier coefficients

for periodic solutions of period n and 2n has been completed.

This manuscript will be combined in one volume with the table

of characteristic values, and certain joining factors*. An

introduction to this volume was in process of preparation at

the end of the fiscal year 1947.

(2) Mathieu Functions II

The periodic solutions

cc Se r (s,t) = Z De2 n+p cos (2n + p) t n= 0

OC So (s,t) = Z Do2 sin(2n - p) t

n= 1 corresponding to the characteristic values under (1) are

being tabulated for p = 0 or 1; r = 0(1)15; t = 0°(lo )90°,

to six decimal places over the same range of the parameter

s as in the tables under (1) . This work is being carried

out on IBM machines and was approximately 35% completed at

the end of the fiscal year.

^

Tables of Bessel Functions Jv

( x)

;

+ v = 1/3, 2/3, 1/i, 3/ 4,

This is intended as a companion volume to the one under

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ACTIVITIES IN APPLIED MATHEMATICS

COMPLETED PROJECTS (13), containing values of Jv

(x), and will

be of similar scope. Differencing of the tables of I

v

(x) was 25 per cent completed at the end of the fiscal year.

^

Tables of Intensity Functions

Computation for certain given complex indices of refraction

have been completed. A final manuscript of the results was being prepared at the end of the fiscal year. A detailed

description of the material for both real and complex indices

of refraction with the definition of the tabulated functions

is contained in the July, 1946 issue of Mathematical Tables

and Other Aids to Computation.

(5) Tables of E^lz), (z = x 4- iy)

E1 (z) = ( e~ u/u)du

Table I. E1(z) + loge z, x = [0(.01)l]; y

= [0(.02)lj 6D]

Table II. E^z), x= [0(.02)4]; y

= [0 (.02) 3 (.05) 10; 6D]

Table III. e

xE1 (z), x= [0(.02)4]j y = [0 (.02) 3 (.05) 10; 6D]

Table I is completed. Tables II and III were about 75 per

cent completed at the end of the fiscal year.

(6) Table of Jacobi Elliptic Functions

Work on tables of sn(u,k), cn(u,k), dn(u,k) for k

2 = 0(»01)l,

,n/2

u = pK, p = 0(.0l) 1, K = J

dcp

was

K

2 sin2

*?

about 50 percent completed at the end of the fiscal year.

(7) Tables of Bessel Functions Y0(z) and Yj_(z) . The tables give the real and imaginary parts of the functions

Y 0

(z) andY1

(z) for z = pe

iC

^, p = [0(.01)10]; <p - [0°(5°)9Qu ]o

- 10 -
ACTIVITIES IN APPLIED MATHEMATICS

Punch cards for all entries have been prepared. Differencing

of the table was being carried out on the tabulator during

June, 1947.

(8) Table of Gamma Functions for Complex Arguments

Computations of logr(z) , z = x + iy for x = [9(,l)10];

y

= [0(.1)10] were about 35 per cent completed at the end of

the fiscal year. They will serve as pivotal values to obtain

logT(z) for x = [0 (. l) 10] ; y

= [0(.l)10].

(9) Tables of Characteristic Values for Spheroidal Wave Functions

Systematic work was in progress on the determination of

the characteristic values of the wave equation for the case

of the prolate spheroid during the latter part of the fiscal

year. The differential equation under consideration is':

In June, 1947, work was proceeding on the characteristic

values for in = 1, for orders 1 = 0, 1,2,3. For m = 1, 1 = 0,

at various intervals in c

2

. For 1 = 1,2,3, the values have

been computed up to c

2 = 60. Preparatory work had been

carried out on the characteristic values for in = 0.

(10)

Tables of the Coulomb Wave Functions

Systematic work is in progress on the tabulation of the

regular solution FL

( n ,P) of the differential equation

which arises in the quantum mechanical treatment of the

problem of two particles moving in a Coulomb field of force.

Calculations are now being carried out for the regular solution

(1-x2

) w" - 2(m + 1) xw' + (bc

2x

2

) w = 0

the characteristic values have been computed up to c

2 = 600

2T] L (L + 1)

P

) y = o

FL = cL

pL+1(p

L’ FL = CL

pL<PL* for L = 0, n = 0(1)10

11 -
ACTIVITIES IN APPLIED MATHEMATICS

p = Q(.l)10. For the definition of <P L

’" and (P L see the article

"Coulomb Wave Functions in Repulsive Fields"- Physical Review,

vol « 49, Jan. 1936, by Yost, Wheeler and Breit. Tables of

the normalizing factor

C Q = exp(nh/2) |F(1 + in) I and o

q = argr(l + in)

have been prepared. The values were computed up to the point

where the expressions

/2nn exp(-3nr|/2) and (n/4) + n(logn-l) -l/(12n)

yield approximations to C Q and Qoi respectively, accurate to

8 significant figures. Values of P? Q and for n = 0(.1)10,

p - 0 (.05) 1.5 to 7 significant figures were being computed in

June, 1947. Values of P<P 0 , <P* f = F0 '/F0 for n = 3(1)10,

p = 0(.1)10 have been computed. Work was started for the same range of P and n on the tabulation of

df J_ d

a f _1_ dff . }_ d*f

dn # 2! dn 2 ’ 3i dna’ 4! dn 4

These derivatives will be used for interpolation in f(n,p)

as a function of n.

(11) Table of AntiloQarithms

A table of 10 x to 10 places of ciecimals for x from 0 to 1

at intervals of 10~ 5 was being prepared from a photographed

copy of J. Dodson's Antiloiarithmic Canon, London 1742* Hie

values from 0 to .9 have been transferred to punch cards and

differenced.

(12; Table of Sines and Cosines to Hundredths of a Deiree

Tables of Sines and Cosines to 15 decimal places with

second central differences. The proofreading and differencing of final manuscript was started in June, 1947.

12 -
ACTIVITIES IN APPLIED MATHEMATICS

(13) Tables of the "i" Integral

The integral defined by

g f

x t/v_ wdw

x

2 Jn 1 + w

2 [cos nw - — sin gw] 2

was being computed for various values of the parameters x and g. The case g = .i was being carried out by numerical

integration- The task was laborious due to the large

number of sharp peaks in the integrand. For g = .001 about

80% of the work was completed by the end of the fiscal year

by a process of obtaining analytic expansions for the

integrand in the neighborhood of the peaks. The remainder

of the integral will be evaluated by numerical integration.

(14) The Arithmetic Mean and Median of a Set of M Measurements

as Estimators of Parameters of Location, with Particular

Reference to the Problem of Anomalous Observationso

A statistical research project being performed by the

Statistical Engineering Laboratory. This project is aimed

at the studying of the sampling distributions of these

statistical estimators, through the evaluation of the

percentiles of their distributions, which represents a departure from the more usual method of studying such variables, by attempting to determine their moments. At the end

of the fiscal year, the work was 50 per cent completed.

Estimated time to complete the project 1 year.

(15) The 2x2 Table, from the Viewpoint of Design of Experiments,

Estimation, and Tests of Significance . Apaperby Dr. Churchill Eisenhart on the tests-o f- significance aspects of contingency tables, to be presented at the

September 1947 meeting of the Institute of Mathematical

Statistics. The design-of-experiments and estimation aspects were being covered by collaboration outside of the Bureau.

13 -
ACTIVITIES IN APPLIED MATHEMATICS

(16) Revision of Statistical Phases of Federal Specification

ZZ-R-601 ("Rubber Goods: General Specifications")

Tills project was undertaken in cooperation with the organic

and Fibrous Materials Division of the bureau. Progress during

fiscal year 1947 consisted of the following: A list of definitions of sampling and inspection terms, and three sets of

sampling inspection plans for different consumer requirements, were prepared for inclusion in the proposed revised

specification. A procedure was developed which, while not

leading to outright rejection of apparently anomalous observations, greatly reduces their influence on the final results

upon which decisions are based. The statistical parameters

of the proposed procedure were aeterndned from test data.

Future work involves extensive editorial work on the specification and preparation of instructions pertaining to the

selection and use of the various sets of sampling inspection

plans. The project was about 80 per cent completed at the

end of the fiscal year, and it is estimated to take about 3

months to complete it.

(17) Sampling Plan for Testing Clinical Thermometers

Sampling procedures currently used' by Thermometry Section

of the Bureau for testing Veterans Administration clinical

thermometers consist essentially of 100 per cent inspection,

however, the specification referenced in the current Veterans

Administration contract calls for the termination of inspection and rejection of a delivery when the fraction defective

among the items tested exceeds a specified limit. The first

task performed under this project was to determine the operating characteristic curve of the inspection procedure of

the referenced specification. Then a sequential inspection

procedure permitting only a rejection of a delivery prior to

complete inspection and having approximately the same operating cnaracteris.tic curve was developed. In addition a sequential inspection procedure permitting acceptance or

- 14 -
ACTIVITIES IN APPLIED MATHEMATICS

rejection prior to complete inspection and having the same

operating characteristic curve was also developed. Special

studies of the reproducibility of the tests were undertaken.

It is estimated that the work was' about 50 per cent completed

at the end of the fiscal year, and that time needed for com- pletion will be about six months.

(18) The Theory and Practice of Hitting a Straight Line

This is a manual on certain statistical methods. Work on the project was initiated by Dr. J. H. Curtiss during the

fiscal year in connection with the course in Correlation and

Regression which he was giving in the NBS Graduate School.

A mimeographed draft of the Introduction and first chapter,

dealing with the case in which one variable is fixed without

error, was prepared and distributed to students in the course.

(19) Automatic Digital Computing Machine for the Bureau of the

Census,

This project has the objective of developing and constructing a large-scale, automatically-sequenced digital

computing machine suitable for the preparation of Census

reports and the performance of calculations arising in

connection with Census work. In accordance with an agreement worked out in consultation with the Bureau of the

Census and with the Office of Naval Research, during the

fiscal year 1947, the project was lira! ted to the preparation

of design specifications for this machine, together with the

performance of such component research as may be necessary

to insure sufficiency of these design specifications. On

October 24, 1946, a contract was entered into with the

Electronic Control Company, Dr. J. W. Mauchly, Jr., and

Mr. J. Presper Eckert, Jr., Proprietors, for the performance

of this work. The Electronic Control Company was furnished

by the Bureau with detailed performance requirements for the

machine, and began work immediately upon recruitment of

- 15 -
ACTIVITIES IN APPLIED MATHEMATICS

personnel, procurement of equipment, and applied research on materials. Although the contract was scheduled to run only

six months, it was deemed in the interest of the- Bureau to

extend the contract so that further research on components

could be performed by the contractors. At the end of the

fiscal year, work on the contract was about' 90 per cent

completed and the preparktion of a detailed report was in

progress.

The most important items of component- research performed

by the Electronic Control Company during the course of this

contract were as follows:

(a) Studies of the optimum frequency of operation of

mercury delay tanks.

(b) Tests on magnetic tape materials and reading and; recording heads.

(c) Development of a servo device for intermittent

tape feed.

(d) Studies of wireless typewriters to determine the

maximum speed of reliable operation. .

(e) Development of a deflection type electrostatic

memory tube.

In connection with this project, a study of the programming of Census problems on proposed computing machines was inaugurated at the Bureau. Mrs. Ida Rhodes of the Mathematical Tables Project was transferred temporarily from New

York City and was placed in charge of this project. The work

was concerned largely with the study of the techniques of

sorting and collating on a large scale. In connection with

this activity a weekly meeting was held at the Bureau on programming, attended by representatives of the Bureau of

the Census, the Office of Naval Research, the Social Security

Administration, and members of the Bureau staff. In addition

to the Census type of problem, basic specific mathematical

16 -
ACTIVITIES IN APPLIED MATHEMATICS

processes were investigated, including the solution of

various types of differential equations.

It is estimated that the entire Census machine project

was about 50 per cent complete at the end of the fiscal year,

and that it will take about 16 months to complete the project.

(20) Automatic Digital Computing Machine for the Office of Naval

Research.

This project has the obj ective of developing and constructing a large-scale automatically sequenced digital computing

machine suitable for general mathematical calculations. As

in the case of the Census machine, it was agreed that the

fiscal year 1947 should be devoted to the design phase of

the project. A letter containing performance specifications

for the machine and the form of contract desired by the

National Bureau of Standards was sent to all firms known to

be competent in the field of electronics and to have personnel

known to be qualified to design electronic digital computing

machines. The letter solicited an expression of interest in

contracting with the National Bureau of Standards for the

design of the Navy’s electronic digital machine. Either

because of the contract form or the nature of the work,

most of the manufacturers declined to participate in the

program and the field soon narrowed down to the Raytheon

Manufacturing Company of Waltham, Massachusetts, and the

Hughes Aircraft Company of Culver City, California^

An investigation by Bureau personnel resulted in the

conclusion that the Raytheon Manufacturing Company was better

qualified to perform the work, and accordingly a contract

effective 27 January 1947 was entered into with the Raytheon

Company for design specifications together with the necessary

component experimentation. Within the Raytheon Company the

contract was under the supervision of Dr. N. E. Edlefsen,

with Dr. C. V. L. Smith having responsibility for the mathematical phases of the work. Dr. J. Barnes of Tufts College

17 -
ACTIVITIES IN APPLIED MATHEMATICS

and Dr. W. R. Fuller of MIT were working as consultants on

the contract.

The contractor immediately began work along two general

lines: (1) Mathematical study of requirements that computing

machines must fulfill and (2) Component research. In (1)

the emphasis was upon (a) the relative frequency of the

basic arithmetical operations (b) the number of references

to and depositions in the memory (c) the number of transfers,

both within the internal memory and between external and

internal memory, (d) the frequency of reading in data and of

reading out final results, (e) the need for using scale

factors. In component research, the company performed

advanced development work on electrostatic tubes and on

mercury delay registers.

At the end of the fiscal year, work on the Raytheon design

contract was about SO per cent completed. It was expected

that the design specifications will be available for study

by the Bureau electronic engineers and mathematicians in

the early part of August. These design specifications will

be built about the mercury delay line as an internal memory

device. However, some auxiliary work of this contractor in

the deflection type electrostatic tube memory device was so successful that an extension of the contract to develop this

device was being considered at the»end of the fiscal year.

At the end of the fiscal year the work on the entire

Navy machine project was about 40 per cent completed. Estimated time to complete the project, about 16 months.

Bo New Projects Which Were Still in the Planning Stage at the End

of the Fiscal Year

(1) Tables of Value in the Development and Application of Mathematical Statistics

Tables to facilitate the construction of random sampleso

Detailed specifications were drawn up by Messrs. Abramowitz,

- 18 -
ACTIVITIES IN APPLIED MATHEMATICS

Eisenhart, Laderman and Lowan of the Bureau in cooperation

with Professor J. W. Tukey of Princeton University at a con- ference held in New York City. The tables so formulated are not tables of random numbers but are special arrangements of

probability deviates or percentage points of the populations

concerned.

(2) Tables for the Occasional Computer

The size, scope, and objectives of such a set of tables

were discussed in general terms in the conference mentioned

in (1) . No final decisions were reached. The general idea

of a handbook for computers was considered at various other

times during the year. Dr. Wigner of the Oak Ridge Laboratories of the Atomic Energy Commission submitted, at the invitation of Dr. Lowan, a detailed outline for such a volume.

Members of the staff of the Mathematical Tables Project have

drawn up an alternative proposal somewhat different in scope

from Dr. Wigner* s outline. The general idea is to produce an enlarged and revised version of the well-known J ahnke-Emde

handbook

e

(3) Table of Surfaces of Constant Pressure

To find surfaces of constant barometric pressure of the

form

<P(x,y) = Ea 4 j x*yJ 0 g i + j < 3

by principle of least squares to fit observations of pressure, al ti tude, and wind conditions, taken at various weather

stations. Surfaces are to be found such that the following

expressions are minimized:

1) 10 E (tp - <p s)

2

+ S (u 2

s Bx By

2) -2. E (<P - T s

) 2 + E(v-± |2) 8 + E(u + I|£) 8

io s A Bx a By

In the above <Pg , u, v, and X are observed values and the

summations extend over given stations.

- 19 -
ACTIVITIES IN APPLIED MATHEMATICS

(4) Standard Sampling-Inspection Procedures t f mine; .

At the invitation of the War Department, Bureau personnel

participated in a two-day conference on the proposed standard

sampling-acceptance inspection procedure developed by a Task

Group of the War Department Inspection Advisory Council. The

preparation of standard sampling-inspection procedures and

tables for Federal Specifications was discussed subsequently

by the Director of the National Bureau of Standards, Dr. Mina

Rees, Head, Mathematics Branch, Office of Naval Research, and

Professor W. Allen Wallis, University of Chicago, formerly

Director, Statistical Research Group, Columbia University.

It was decided to proceed with the preparation of such tables

at the Bureau, and steps were taken to recruit additional

personnel in connection with the program.

(5) Computing Machines for the Army Air Forces

During the last quarter of the fiscal year, the Bureau

made commitments to extend the scope of the work on the Nhvy

and Census machines to include such copies of these machines

as may be required by the Army Air Forces. Progress during

the fiscal year consisted largely of conferences with Air

Forces personnel regarding the feasibility of such a program.

Arrangements were made for the transfer of funds to carry on

the work.

4. CONSULTATION AND ADVISORY SERVICES

(A) Statistical Engineering

Dr. Churchill Eisenhart and his staff spent about 85% of

their time during the past year in statistical advisory work.

In addition. Dr. J. H. Curtiss gave some time to this type of

work. The clients of this service were for the most part personnel of the National Bureau of Standards. The more important jobs

20 -
ACTIVITIES IN APPLIED MATHEMATICS

were as follows:

(1) Preparation and Analysis of Industrial Samplini Plans

The usual acceptance sampling plans for hardwood plywood

were analyzed, and the results were placed in an appendix to a Commercial Standard for this material. Hie operating characteristic of a proposed sampling plan for microscopic analysis of

laundry starch, to he incorporated in a Federal Specification,

was determined. A new method of sampling and testing tuning forks

for the Veterans Administration was devised. Sampling plans for

various types of thermometers were set up. The sampling plan in

a proposed Commercial Standard for perforated clay pipe was ana- lyzed and an appendix to the Standard prepared. Operating characteristics of sampling plans in a proposed Army Specification

for the VT fuze were worked out.

Note: Two major projects in this category which began as con- sulting jobs are Nos. (16) and (17) in Section 3, part A, of this

Report.

(2) Analysis of Experimental and Test Data

(i) Studies of the Validity of Stated Inferences

A study was made of the accuracy and precision of the standard method of determining the wool content of blankets, for reference in a pending FTC lawsuit. Advice was given on a number of

occasions to various scientists in the Central Radio Propagation

Laboratory relative to their applications of statistical methods.

About fifteen (15) research papers submitted to the Research and

Testing Editorial Committee of the Bureau were reviewed from a statistical viewpoint, and minor corrections or revisions were

suggested to the authors. An extended study was made of a paper

- 21 -
ACTIVITIES IN APPLIED MATHEMATICS

on the effect of magnification on the precision of telescope

pointing; it was determined that a significant serial correlation

ignored by the author existed.

(ii) Analyses of raw data

True ranges of preignition ratings of different types of

automatic spark plugs were estimated from sample ranges, and an appropriate graphical representation was devised. A regression

analysis was performed on experimental data obtained in studies

of the physical properties of leather. A regression analysis of

data from 863 samples of manila rope was performed, using the

computing facilities of the Mathematical Tables Project, and a paper was written around the results in collaboration with the

person who obtained the raw data.

(3) Advice on probability theory

Assistance was given- to various personnel of die Bureau from

time to time on minor points in the theory of errors and tests of

significance. In response to a request from the Naval Research

Laboratory, a problem involving the distribution of a quotient of

Poisson variables was solved.

(B) Computing Machinery

Dr. E. W. Cannon and his assistant, Mr. Albert S» Calm, spent

about 75% of their time during the year in advising representatives of other Government agencies concerning the applicability

of automatic digital computing machinery to their activities.

The work on the Census machine was carried on in continuous con- sultation with that Bureau. Many consultations were also held

with the Office of Naval Research in connection with the machine

under construction for the Navy.

22 -
ACTIVITIES IN APPLIED MATHEMATICS

Dr. Eisenhart participated as a formal discussant in the

Symposium on Statistical Methods in Experimental and Industrial

Chemistry, held in conjunction with the April 1947 meeting of the

American Chemical Society, and again as a discussion leader in

the May and June meetings of the Washington Quality Control Society.

Presented by title at meetings of the American Mathematical

Society;

"The Approximation of Numbers as Sums of Reciprocals”,

by H. E. Salzer

"Table of Coefficients for Interpolating in Functions

of Two Variables", by H. E. Salzer

"Alternative Formulas for Direct Interpolation of a Complex Function Tabulated Along Equidistant

Circular Arcs", by H. E. Salzer

"An Alternative Definition of Reciprocal Differences", by

H. E. Salzer

"Tables for Facilitating the Use of Chebyshev’s Quadrature

Formula", by H. E. Salzer

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