PROCEEDINGS 
OF THE 
NATIONAL ACADEMY OF SCIENCES 
DIFFERENTIAL EQUATIONS AND IMPLICIT FUNCTIONS IN 
INFINITELY MANY VARIABLES 
By William L. Hart 
The paper of which this note gives a brief abstract, has three main 
divisions. In the first section certain fundamental theorems are de- 
veloped concerning a type of real-valued functions of infinitely many 
real variables. In the second section there is considered the problem of 
infinite systems of ordinary differential equations. 
^=f.(xi,X2, . . .;t), Xi(tQ)=ai, i = 1,2, . . (1) 
at 
in which the fi are of the type treated in the initial theorems. In the 
third section of the paper the fundamental problem of impHcit function 
theory in this field is discussed for a system of equations 
fi {xi,X2, . . .;yi,y2, . . .) = 0, i = l,2, . . ., 
fi (ai, (Z2, . . .;bi,b2,. . .) = 0, 
where the Xj are independent variables and the yj are to be determined. 
The results of all three sections of the paper include as special cases 
the corresponding theorems on functions of a finite number of variables. 
The region of real points ^ = {xi, X2, . . . ) in the space of infinitely 
many dimensions in which the functions considered are supposed defined, 
is of the type specified by 
R:\ Xi — ail^n, i = 1, 2, . . . , 0 < r ^r, r finite. 
In the first part of the investigation theorems are derived for func- 
tions completely continuous according to the 
309 
Volume 2 
JUNE 15. 1916 
Number 6 
DEPARTMENT OF MATHEMATICS. UNIVERSITY OF CHICAGO 
Received by the Academy, April 15, 1916 
