312 
MATHEMATICS: W. L. HART 
The linear system of Moore is not related to the system (1) of the 
preceding theorem but the results of Moulton, when restricted to reals, 
are a special case of the conclusion of Theorem 6. In addition to the 
results of Moulton, however, it follows that the unique analytic solution 
he obtained is, for real values of t, the only continuous solution of the 
system he treated. 
Under certain hypotheses in addition to those of Theorem 6, it is 
proved that the solution ^ (/) has continuation properties which reduce, 
for the finite case, to the ordinary theorem on the existence of a solu- 
tion extending to the boundary of the given region of definition of the 
system. 
It is also shown, by the aid of Theorem 5, that the formal hypothesis 
(6) can be replaced by an assumption concerning the existence and the 
values of the partial derivatives (^fifdxj (i, j = 1, 2, . . . ). 
Infinite systems of equations of the form (2) have been considered 
by H. Von Koch'' and R. d'Adhemar."^ Von Koch treated a system of an 
analytic type defined in the field of complex numbers and established 
the existence of an analytic solution. His work, however, is valid 
only if the sum of the numbers Vi of the region similar to R in which his 
system is defined converges in a very special manner. R. d'Adhemar 
treated a special type which arose in a problem he considered in integral 
equation theory. The results of Von Koch, when restricted to reals, are 
a special case of the theorem stated below. 
In the solution of the system (2), considered in the present paper, 
infinite sets of linear equations enter in a fundamental fashion. Such 
hypotheses are imposed that these linear systems come under the 
theory of infinite systems of linear equations with normal determinants. 
The method of solution of (2) for the yy is related to that used by Gour- 
sat^ in his solution of the finite case by a method of successive approxi- 
mations. An analogue for (2) of the fundamental theorem on implicit 
functions in the finite case, is obtained in the following form: 
Theorem 7. Suppose that, in (2), the functions fi and ^fi/dyj {i, j = 
1, 2, . . .) are defined and Cifor all points in 
S: I Xi - a, I ^ T: \ y, - bi\ ^ i = 1, 2, . . . , 0 < f, £ r, (7) 
and that the maxima Mi of the \ fi 77) | {r] = yi, 3/2, . . . ) satisfy 
Mi ^ri M (M finite). Assume that for all t?) in (7) 
(8) 
converges uniformly. Suppose that the normal infinite determinant 
