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GRIFFITH CONRAD EVANS
May 11, 1887-December 8, 1973
BY CHARLES B. MORREY
GRIFFITH CONRAD EVANS was born in Boston, Massachu-
setts on May ~ I, 1887 and diecl on December 8, 1973.
He receiver! his A.B. degree in 1907, his M.A. in 1908, and
his Ph.D. in 191~0, all from Harvard University. After receiv-
ing his Ph.D., he studied from ~ 9 ~ 0 through ~ 9 ~ 2 at the
University of Rome on a Sheldon Traveling Fellowship from
Harvard. He began his teaching career in 1912 as assistant
professor of mathematics at the newly established Rice In-
stitute, now Rice University, in Houston, Texas. He became
professor there in 1916 and remained with the Institute until
1934. While he was at Rice, he was able to attract outstanding
mathematicians, such as Professor Mandelbrojt of the Uni-
versity of Paris, and young mathematicians, such as Tibor
Rado and Car! Menger, to Rice as visiting professors. Long
before Evans left Rice it was internationally known as a center
of mathematical research.
Evans was brought to the University of California at
Berkeley in 1934 as a result of a nationwide search; he ar-
rived with a mandate to build up the Department of Mathe-
matics in the same way that Gilbert Lewis had already built
the chemistry faculty. Evans struggled with himself to effect
the necessary changes with justice. His innate sense of fair-
ness, modesty, and tact, as well as his stature as a scientist,
127

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128
BIOGRAPHICAL MEMOIRS
brought eminent success. By the time he retired in 1954, he
had had the satisfaction of seeing the department evolve into
one of the country's major centers of mathematical activity.
His retirement did not diminish his interest in science nor
subtract from his pleasure at seeing others achieve goals he
cherished.
A few years before World War Il. Professor Evans and
others on the Berkeley campus recognized the importance of
the fields of probability and statistics, and Professor Jerzy
Neyman was brought to that campus by Evans in 1939 to
organize the Statistical Laboratory. A period of rapid growth
followed; by the close of World War IT the Laboratory had
transformed Berkeley into one of the three principal centers
of probability and statistics in the country. The size and im-
portance of the Laboratory continued to grow, and a separate
Department of Statistics was established in 1955.
Shortly after coming to Berkeley, Professor Evans inau-
gurated a seminar in mathematical economics, which he gra-
ciously held in his home once a week. This seminar became
internationally known, providing an inspirational educa-
tional activity and establishing a tradition of mathematical
economics on the Berkeley campus that continues to the pres-
ent. The seminar was attended by both students and faculty
and promoted a friendly atmosphere in the department.
FUNCTIONAL ANALYSIS
In the first decade of the century, while Evans was a
student, functional analysis was beginning to attract the inter-
est of the mathematical community. Classical analysis was
concerned with functions of real and complex variables,
while functional analysis was concerned with functionals, that
is, functions of"variables" that may themselves be ordinary
functions or other mathematical entities. For example, iff

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GRIFFITH CONRAD EVANS
denotes any ordinary function continuous for O c
may clefine a functional F by the equation
rl
F(f) = | f (x)dx.
Jo
129
xc 1,we
Evans began his career as a research scientist before he
received the Ph.D. degree. He published his first paper in
1909. During the ensuing ten years, he contributed a great
deal to the development of the general field of integral equa-
tions and more general functional equations. His principal
results concerned certain integro-ctifferential equations and
integral equations with singular kernels. His interest in this
field had been greatly stimulated by his contact with Profes-
sor Vito Volterra at the University of Rome. He received
early recognition for this important work in 1916 when he
was invited to cleliver the prestigious Colloquium Lectures
before the American Mathematical Society on the subject
"Functionals and their Applications" (see bibliography,
1918).
POTENTIAL THEORY IN TWO DIMENSIONS
In 1920 Professor Evans published the first of his famous
research papers on potential theory. He was among the first
to apply the new general notions of measure and integration
to the study of classical problems. In the course of this re-
search, he introduced many icleas and tools that have proven
to be of the utmost importance in other branches of mathe-
matics, such as the calculus of variations, partial differential
equations, and differential geometry; for example, he used
certain classes of functions that are now known as "Sobolev
spaces."

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130
B IOGRAPHICAL MEMOI RS
Introduction to Potential Theory.
The central iclea in poten-
tial theory is the notion of the potential of a distribution in R 3.
Given a distribution of mass g, we define its potential U by the
equation
~ I) U(`M) = J |MP|-ig(P)r1P
w
(W =R3,M = (x~y~z),P = (671~))
whenever this is clefinecI. In case g is Hoelcler continuous ~ for
all P and vanishes outside a compact set, then U is of class C 2
and its second derivatives are Hoelcler continuous.2 In this
case:
(2)
U (M)—Uxx (x,y,z) + US (x,y,z) + Uzz (x,y,z)
= - 4~g(M), M = (x,y,z).
A solution that satisfies (2) with AU (M) = 0 on some domain
is sail! to be "harmonic" on that clomain. Such a function has
derivatives of all orders.
The funciamental problem in potential theory is the
DirichIet problem. Roughly speaking, this consists in proving
the existence and uniqueness of the function U that satisfies
Laplaces equation on a given domain G. is continuous on G
(the closure of G), and takes on given continuous boundary
values on the boundary FIG of G.
Another problem, the Neumann problem, is to show the
existence (and uniqueness except for an arbitrary acIditive
constant) of a function V that satisfies Laplaces equation on
G. is continuously differentiable on G. and for which the
iA function g is Hoelder continuous on a set S if, and only if,
I¢(P) - ¢(Q) I C L |PQ|"
for some constants L and ,u, with 0 < ,u < 1 and all P and Q are both on S.
See Oliver Dimon Kellogg, Foundations of Potential Theory (New York: Dover,
1929), p. 38 or 152, for instance: "This could be called a 'classical result.'"

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GRIFFITH CONRAD EVANS
13
outer normal derivative dV/6n takes on given continuous
values on dG.
The function u (r, B), defined by3
(3) u (r, B) =
~ 2~ Jo
f(ff),
(l_r2) f(~)d¢, r < 1
l+r2 - 2 cos(¢—B)
if r = 1,
is the solution of the DirichIet problem in the case where G is
the unit circular disc in R.,. In case
(4) ~ g (by d ~ = 0 and ~ V ( 1,0) d ~ = 0.
~ BIG
~ At,
the solution of the Neumann problem with boundary values
g (~)-on dG proceeds as follows. Let
(5) v (r,H) =
J1 27r
log [1 +r2 - Or cos (¢'—B) ] go) d ¢.
AT 0
It is easy to see that rvr is harmonic on G. and
(6) rvr (r,8)
=
27r To 1 + r 2 - Or cos (+—H) g (I) d
1 J 27J
The first term on the right in (6) is the solution of the Dirich-
let problem with boundary values g (I). If (4) holds, the sec-
ond term is zero and V is one of the desired solutions.
Among Evans' first results were those concerning the
function
3 this is Poissonts Integral Formula.

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32
BIOGRAPHICAL MEMOIRS
(7) ~ (r, B) = 2—1 ( 1—r 2) [ 1 + r 2 - 2r cos (+—B) ] - 1 dF (¢ ),
where F(O is of boundecT variation ant] periodic. Evans
prover! the following:
· the function u (r, fit) is harmonic in G
ran
,] |u(r,0) |a,Cis bouncled for r ~ 1;
o
· u (r, B) = u 1 (r,0)—u 2 (r,8) each u ~ being harmonic and
non-negative on dG;
· if P = (1,0 is a point on dG,whereF(¢~) is continuous
anc]F'(¢~) exists and F'(O =f (O. then u (r,8) If (A as
(r,0) ~ (1,¢~) "in the wide sense"; i.e., (r,H) > (1,¢~)
remaining In any angle with vertex at (1,¢~).
· If F and F' are continuous, then (7) recluces to the
solution of the DirichIet problem with continuous
boundary valuesf(¢)).
.
, the unit disc in R2;
Conversely, if we assume that u = u ~—u 2 where each
u i—O and is harmonic on G. then u is given by (7).
Early Discussion of the Dirichlet Problem.
The first attempt
to solve the DirichIet problem was made by Green in 1828.4
His method] was to show the existence of a Green's function
of the form
G (Q,P ) = - + V ( Q. P ), r = (P,Q)
This function is the Green's function for the region R and the
pole P. In terms of this Green's function we have
U(P) = - 4 ji U(Q) ~ G(Q,P) dS,
s
4See Kellogg, Foundations of Potential Theory, p. 38.

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GRIFFITH CONRAD EVANS
133
where S is the boundary of R. This development is based,
however, on the existence and differentiability of G (Q,P),
which is obtained using physical considerations and so is not
logically suitable for a mathematical derivation.
In 1913 Lebesgue gave an example of the impossibility of
the solution of the DirichIet problem.5 The region R can be
obtained by revolving about the x-axis the area bounded by
the curves
y =e-"x, y = 0
and x = I.
This type of region is called a Lebesgue spine. It can be
shown that the region obtained by revolving about the x-axis
the area bouncled by the curves
y = An, y = 0, X = 1, n > 1
is a regular region; i.e., the DirichIet problem is always
solvable.
The Logarithmic Potential Function. A similar theory holds
for the two-dimensional situations. One considers the loga-
rithmic potential function in R 2, defined by
(~) U(M)=
JW [ g(MP\)] g`P'~P M=(xy) W=R2
whenever this is defined. If g is Hoelcler-continuous for alIP
ant! vanishes outside a compact set, then U is of class C 2, and
its second derivatives are Hoelder-continuous everywhere. In
this case,
(9) AU (M) = Uxx fx,y) + UP ~x,y) = - 2 g (M), M = (x,y).
A solution of (9) that satisfies AU (M) = 0 on some domain is
find., p. 285, 334.

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134
BIOGRAPHICAL MEMOIRS
said to be harmonic on that domains such a function has
derivatives of all orders.
The Dirichiet and Neumann Problems in Space. The solution
of the DirichIet problem in the unit sphere S is given bye
, .
(,lO)u(M) = ~
~ f (M)
~ (4~-i or [~l—r2) (MP)-3f (P) dS, O ~ r ~ I,
s
(r = 0M), M = (x,y,z)
, r = 1 (S = S1, Sr = dB

GRIFFITH CONRAD EVANS
unit ball, be given by the formula
( 13) u (M) = (41r) -1 Ji ( 1 r2) (MP)-3 dG (P)
for some distribution G (e) on S. is that
J~J ~u(M) HIS be bouncled for O Or c 1,
s
135
or that u = u ~ —u 2 where u ~ and u 2 are non-negative and
harmonic on B (0,1). If F(e) is a distribution on S. and if
iim (par pa) -I |F [B (P. p)] | =f (P), then u (M) Of (P ) as M >
p to
P in the wide sense (i.e., M remains in a cone with vertex
atP).
The Riesz Theorem. A function V is said to be "super-
harmonic" on a domain Q if, ant] only if, (i) it is lower semi-
continuous and ~ + ~ on Q. and (ii) V |M |—its mean value
over the surface of any sphere with center M that lies with its
interior in Q.
Professor Evans proved that any potential function of a
positive mass is superharmonic on any clomain on which it is
defined. Evans also gave the simplest proof of the following
theorem due to F. Riesz:
Suppose u is superharmonic on a domain Q. and D iS any
domain, the closure of which is compact and lies in Q. Then
U (M) = U (M) + V (M) M ~ D
where U is the potential of a positive mass on D and v is
harmonic on D .7
7 F. Riesz, "Sur des functions superharmoniques et leur rappaport a la theorie du
potential,"Acta Math, 48 (1926):32~43; 54 (1930):321-60.

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136
BIOGRAPHICAL MEMOIRS
Connection with Sobolev Spaces.
In addition, Evans proved
the following important theorems: Suppose U is super-
harmonic on a domain Q. Then U (x,y,z) is absolutely contin-
uous in each variable for almost all pairs of values of the other
two and retains this property under one-to-one changes in
variables of class C i.8
Finally Professor Evans proved the following theorems:
Suppose U is superharmonic on some domain ant! Up(M)
denotes the average of U over the surface OB (M,p); then
Up(M) is continuously differentiable over any domain Qpo
(which consists of all M such that B (M spot c Q) anct V UO >
VU in L., on any such domain. A necessary and sufficient
condition for a potential U offfe) to have a finite DirichIet
integral is that Jew U(M) Offer exist. In this case U must
belong to the Sobolev space H ~ on interior domains. Evans
proved many more similar theorems.
A Sequence of Potentials (A Sweeping Out ProcessJ. Evans
gave a simple proof that the limit of a non-decreasing
bounded sequence of potential functions of positive mass
each distributed on a fixed bounded closed set F is itself a
potential of positive mass F. The limit of a non-increasing
sequence of such functions, however, is not necessarily super-
harmonic (since the limit of a non-increasing sequence of
lower-semicontinuous functions is not necessarily lower-
. .
sem~cont~nuous
Nevertheless, Evans showed how to associate a particular
type of positive mass distribution with a particular type of
non-increasing sequence of potential functions on a
bounded, closed set F. To do this, Evans let U , U.,, ..., be
a non-increasing sequence of potentials of positive mass dis-
tributionsf~,f.,, ..., respectively on F. Let UO be the limit
X See bibliography entries of 1935 for the three-dimensional case and those in 1920
for the two-dimensional case.

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GRIFFITH CONRAD EVANS
145
holds. Carrying out the differentiation with respect to t in (3),
we get for Euler's equation
(4)
where
(5) ~T, p, means
means —, etc.
~p
~P'P' d P2 +~P,P dP ='7Tp,
62~
(6 ')2' ~P'P means
82~
dp'dp
~T(p,p') =p(ap +b +hp')—A(ap +b +hp')2—B(ap +
b + hp ')—C and the derivatives in (5) are the indicated partial
.
' as independent variables.
derivatives regarding p and p
Carrying out the differentiations in (5), we get
(6) ~p,p, = - 2Ah2, ~p,p = h (1 - 2aA),
rrp = a (2p—B) +hp '(1 - 2aA) +b (1 - 2aA)—3a2Ap.
Setting ~Ip/dt = p ' in (4) and using (5) and (6), Euler's equa-
tion becomes
—2Ah 2p,, + h (1 - 2aA)p' = a (2p —B) + hp'(1 - 2aA)
+ b ( 1 - 2aA ) - 2a 2Ap
—2Ah 2p', = 2a ( 1 —aA )p + b ( 1 - 2aA ) —aB
2a (1—aA) b (1 - 2aA)—aB
= > p " = - p +
= M p —N
which is reduced to the form
2Ah2 -, i.e.,p"
dt2 f(P)

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46
BIOGRAPHICAL MEMOIRS
This is solvable by standard methods in differential equa-
tions.
This is one of the simplest cases. More sophisticated theo-
ries involving such things as taxes, tariffs, rent, rates of
change, transfer of credit, the theory of interest, utility, the-
ories of production, and problems in economic dynamics
were worked out by Evans.
Evans' scientific career resulted in over seventy substantial
published articles, four books, and several ciassifiecl reports.
It should be added, since it is such a rare occurrence among
mathematicians, that he continued his productive work for
many years after his retirement. He gave a number of invites!
actresses in Italy and elsewhere cluring that period.
Professor Evans was elected to the National Academy of
Sciences in 1933 anti became a member of the American
Academy of Arts and Sciences, the American Philosophical
Society, the American Mathematical Society (vice president,
192~26; president, 1938-401; the Mathematical Association
of America (vice president, 1934), and the American Associa-
tion for the Advancement of Science. He was a fellow of the
Econometric Society.
Evans was invited to give addresses in connection with the
Harvard Tercentenary and the Princeton Bicentennial Cele-
bration. He was also asked to give the Roosevelt Lecture at
Harvard in ~ 949 and was Faculty Research Lecturer in Berk-
eley in 1950 and was awar(lecl an honorary degree by the
University in 1956. The Griffith C. Evans Hall on the Berke-
ley campus was cledicatect in 1971.
During World War I, Evans served as a captain in the
Signal Corps of the U.S. Army. During World War Il. he was
a member of the Executive Board of the Applied Mathemat-
ics Pane! anct was part-time technical consultant, Ordnance,
with the War Department. He received the Distinguishecl
Assistance Award from the War Department in 1946 and
received a Presidential Certificate of Merit in 1948.

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GRIFFITH CONRAD EVANS
147
The charming hospitality of the Evanses is remembered
with pleasure by those fortunate enough to have been guests
at their home. And Evans' own keen, dry sense of humor was
much appreciated by his many friends and associates.
Professor Evans married Isabel Mary John in 1917. They
had three children, Griffith C. Evans, Jr., George William
Evans, and Robert John Evans and many grancichildren.

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48
BIOGRAPHICAL MEMOIRS
BIBLIOGRAPHY
1909
The integral equation of the second kind, of Volterra, with singular
kernel. Bull. Am. Math. Soc., 2d ser., 41:130-36.
1910
Note on Kirchoffts law. Proc. Am. Acad. Arts Sci., 46:97-106.
Volterra's integral equation of the second kind, with discontinuous
kernel. Trans. Am. Math. Soc., 11: 393-413.
1911
Volterra's integral equation of the second kind, with discontinuous
kernel, Second paper. Trans. Am. Math. Soc., 12:429-72.
Sopra l'equazione integrale di Volterra di seconda specie con un
limite del'integrale infinito. Rend. R. Accad. Lincei C1. Sci. Fis.
Mat. Nat., ser. 5, 20:409-15.
L'equazione integrale di Volterra di seconda specie con un limite
del'integrale infinito. Rend. R. Accad. Lincei C1. Sci. Fis. Mat.
Nat., ser. 5, 20:656-62.
L'equazione integrale di Volterra di seconda specie con un limite
del'integrale infinito. Rend. R. Accad. Lincei C1. Sci. Fis. Mat.
Nat., ser. 5, 20:7-11.
Sul calcolo del nucleo dell'equazione risolvente per una data equa-
zione integrale. Rend. R. Accad. Lincei C1. Sci. Fis. Mat. Nat.,
ser. 5, 20:453-60.
Sopra ['algebra delle funzioni permutabili. R. Accad. Lincei C1. Sci.
Fis. Mat. Nat., ser. 5, 8:695-710.
Applicazione dell'algebra delle funzioni permutabili al calcolo delle
funzioni associate. Rend. R. Accad. Lincei C1. Sci. Fis. Mat. Nat.,
vol. XX, ser. 5, 20:688-94.
1912
Sull'equazione integro-differenziale di tipo parabolico. Rend. R.
Accad. Lincei C1. Sci. Fis., Mat. Nat., ser. 5, 21:2~31.
L'algebra delle funzioni permutabili e non permutabili. Rend. Circ.
Mat. Palermo, 34: 1-28.

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GRIFFITH CONRAD EVANS
1913
149
Some general types of functional equations. In: Fifth International
Congress of Mathematicians, Cambridge, vol. 1, pp. 389-96. Cam-
bridge: Cambridge University Press.
Sul calcolo delta funzione di Green per le equazioni differenziali e
intergro-differenziali di tipo parabolico. Rend. R. Accad. Lincei
C1. Sci. Fis. Mat. Nat., ser. 5, 22:855-60.
1914
The Cauchy problem for integro-differential equations. Trans.
Am. Math. Soc., 40:215-26.
On the reduction of integro-differential equations. Trans. Am.
Math. Soc., 40:477-96.
1915
Note on the derivative and the variation of a function depending o
all the values of another function. Bull. Am. Math. Soc.,2d ser.,
21:387-97.
The non-homogeneous differential equation of parabolic type.
Am. J. Math., 37:431-38.
Henri Poincare. (A lecture delivered at the inauguration of the Rice
Institute by Senator Vito Volterra. Translated from the
French.) Rice Inst. Pam., 1 (21: 1 33~2.
Review of Volterra's "Le~cons sur les fonctions de lignes." Science,
41(1050):246-48.
1916
Application of an equation in variable differences to integral equa-
tions. Bull. Am. Math. Soc., 2d ser., 22:493-503.
1917
I. Aggregates of zero measure. II. Monogenic uniform non-analytic
functions. (Lectures delivered at the inauguration of the Rice
Institute by Emile Borel. Translated from the French.) Rice
Inst. Pam., 4( 1 ): 1-52.
I. The generalization of analytic functions. II. On the theory of
waves and Green's method. (Lectures delivered at the inaugura-
tion of the Rice Institute by Senator Vito Volterra. Translated
from the Italian.) Rice Inst. Pam., 4~11:53-117.

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1918
Harvard college and university. Intesa Intellet., 1:1-11.
Functionals and their Applications Selected Topics, Including Integral
Equations. Amer. Math. Soc. Colloquium Lectures, vol. 5, The
Cambridge Colloquium. Providence, R.I.: American Mathemat-
ical Society. 136 pp.
1919
Corrections and note to the Cambridge colloquium of September,
1916. Bull. Am. Math. Soc., 2d ser., 25:461~3.
Sopra un'equazione integro differenziale di tipo Bocher. Rend. R.
Accad. Lincei C1. Sci. Fis. Mat. Nat., vol. XXVIII, ser. 5,
33:262-65.
1920
Fundamental points of potential theory. Rice Inst. Pam., 7:
252-329.
1921
Problems of potential theory. Proc. Natl. Acad. Sci. USA, 7:89-98.
The physical universe of Dante. Rice Inst. Pam., 8:91-1 17.
1922
A simple theory of competition. Am. Math. Monogr., 29:371-80.
1923
A Bohr-Langmuir transformation. Proc. Natl. Acad. Sci. USA,
9:230-36.
Sur l'integrale de Poisson generalisee (three notes). C. R. Seances
Acad. Sci., 176: 1042-44; 176: 1368-70; 177:241-42.
1924
The dynamics of monopoly. Am. Math. Monogr., 31:77-83.
1925
I1 potenziale semplice ed it problema di Neumann. Rend. R. Accad.
Naz. Lincei C1. Sci. Fis. Mat. Nat., ser. 6,2:312-15.
Note on a class of harmonic functions. Bull. Am. Math. Soc.,
31: 14-16.

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151
Economics and the calculus of variations. Proc. Natl. Acad. Sci.
USA, 11:90-95.
The mathematical theory of economics. Am. Math. Monogr., 32:
10~10.
On the approximation of functions of a real variable and on quasi-
analytic functions. (Lectures delivered at the Rice Institute by
Charles de la Vallee Poussin. Translated from the French by
G. C. Evans.) Rice Inst. Pam., 12~2~:105-72.
Enriques on algebraic geometry. Bull. Am. Math. Soc., 31 :449-52.
1927
With H. E. Bray. A class of functions harmonic within the sphere.
Am. I Math., 49: 153-80.
Review of "Linear integral equations" by W. V. Lovitt. Am. Math.
Mon., 34:142-50.
The Logarithmic Potential. Am. Math. Soc. Colloquium Publications,
vol. 6. Providence, R.I.: American Mathematical Society.
150 pp.
1928
Note on a theorem of Bocher. Am. I. Math., 50:123-26.
The Dirichlet problem for the general finitely connected region.
In: Proceedings of the International Congress of Mathematicians,
Toronto, vol. 1, pp. 549-53. Toronto: University of Toronto
Press.
Generalized Neumann problems for the sphere. Am. I. Math.,
50: 127-38.
The position of the high school teacher of mathematics. Math.
Teacher, 21 :357-62.
1929
Discontinuous boundary value problems of the first kind for Pois-
son's equation. Am. J. Math., 51: 1-18.
With E. R. C. Miles. Potentials of general masses in single and
double layers. The relative boundary value problems. Proc.
Natl. Acad. Sci. USA, 15: 102-8.
Cournot on mathematical economics. Bull. Am. Math. Soc.,
35:269-71.

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152
BIOGRAPHICAL MEMOIRS
1930
With W. G. Smiley, Jr. The first variation of a functional. Bull. Am.
Math. Soc., 36:427-33.
The mixed problem for Laplace's equation in the plane discontin-
uous boundary values. Proc. Natl. Acad. Sci. USA, 16:620-25.
Mathematical Introduction to Economics. New York: McGraw-Hill.
177 pp.
Stabilite et Dynamique de la Production dans l'Economie Politique. Me-
morial fasicule no. 61. Paris: Gauthiev-Villars.
1931
With E. R. C. Miles. Potentials of general masses in single and
double layers. The relative boundary value problems. Am. I.
Math., 53 :493-516.
Zur Dimensionsaxiomatik. Ergebnisse eines mathematischen kollo-
quiums. Kolloquium, 36: 1-3.
A simple theory of economic crises. Am. Stat. J., 26:61-68.
Kellogg on potential. Bull. Am. Math. Soc., 37:141~4.
1932
An elliptic system corresponding to Poisson's equation. Acta Litt.
Sci. Regiae Univ. Hung. Francisco-}osephinae, Sect. Sci. Math.,
6~11:27-33.
Complements of potential theory. Am. T. Math., 54:213-34.
Note on the gradient of the Green's function. Bull. Am. Math. Soc.,
38:879-86.
The role of hypothesis in economic theory. Science, 75:321-24.
1933
Complements of potential theory. Part II. Am. }. Math., 60:29~9.
Application of Poincare's sweeping-out process. Proc. Natl. Acad.
Sci. USA, 19:457-61.
1934
Maximum production studied in a simplified economic system.
Econometrica, 2: 37-50.
1935
Correction and addition to "Complements of potential theory."
Am. }. Math., 62: 623-26.

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