350 
MATHEMATICS: F. R. MOULTON 
SOLUTION OF AN INFINITE SYSTEM OF DIFFERENTIAL 
EQUATIONS OF THE ANALYTIC TYPE 
By F. R. Moulton 
DEPARTMENT OF ASTRONOMY. UNIVERSITY OF CHICAGO 
Presented to the Academy, April 20, 1915 
Consider the infinite system of differential equations* 
^ =/, (/; X,, X,, - ■ .) =a, +/f> + . . . (»• = 1, 2, ■ . • ), (1) 
where a» is a constant and fi^i^ is the totality of terms of /» which are 
homogeneous in t, Xi, X2, ... of degree 7. That is, //^^ is a linear func- 
tion of the infinitely many variables /, Xi, x^^ . . . ; fx^^ is a quadratic 
function of the same variables; and so on. From the analogy with 
analytic functions of a finite number of variables, /»• will be said to be 
of the analytic type. 
It is assumed that the following hypotheses are satisfied: 
{Hi). Xi = 0 (i = 1, 2, . . .) at t = 0. 
(^"2). Finite real positive constants Cq, Ci, c^, . . . ; ^o, ^1, ^2, . . A and 
a exist such that 
s = c^t -\- CiXi + C2X2 + . . . (2) 
converges if 
\t\ ^ro, \xi\ Sri {i = 1, 2, ...) (3) 
and such that Ansi dominates Ji^i^ and ^ Ana. 
Since the series (2) converges if the relations (3) are satisfied, a finite 
constant M exists such that 
^ = ''T.^''t'A''T~A •••<!• (4) 
M M M 
Consequently if if |/| ^ro and If ^ f,- are satisfied, then 
\fi\^ArAa + S + S' + - ■ .} = ^r,|a + ^|- (5) 
* The problem of an infinite system of differential equations was first treated by E. H. 
Moore in his paper read at the Fourth International Congress of Mathematicians, held at 
Rome in 1908. His treatment was in the sense of General Analysis (cf. Introduction to a 
Form of General Analysis, New Haven Mathematical Colloquium, 1906), in which a general 
variable p is used in place of the index i of this paper having the range of positive integers, 
and his functions were not limited to those of the analytic type. The solution was made to 
depend upon the solution of an integral equation, in general non-linear, of the Volterra 
type. Simplifications and extensions of this theory were presented before the National 
Academy of Sciences, at Chicago, December 9, 1914. 
