MATHEMATICS: W.L.HART 311 
integral form for the remainder. The ^ mean value ' case of the general 
theorem is 
Theorem 5. Suppose that the function f (^) and its partial derivatives 
df/dXi (i = 1, 2, . . . ) are Ci in R and that | ^/ (^) I converges 
uniformly for all ^ in R. Then for of R, 
Systems of differential equations of the form (1) were first considered 
by H. Von Koch^ who treated a case of an analytic type. A very simi- 
lar problem was considered later by F. R. Moulton.^ Infinite systems 
of a linear form were discussed by E. H. Moore^ as a special case of a 
more general investigation made in the sense of Moore's General 
Analysis.^ 
In the present paper, the problem (1) is treated by a generalized 
Picard Approximation method. The coordinates Xik (t) of the approxi- 
mations (t) to a solution of (1) are defined formally by the equations 
^0 (0 = (^u ^2, . . . ), 
Xik (0 = x'o + fi [ h-i {t), t] dt, k=l,2,,... 
The existence theorem obtained is 
Theorem 6. Suppose in (1) that the fi are defined and Ci in 
T:\t — tQ\^rQ', R :\xi — ai\-^ri (0 < ri < r; r finite), (5) 
and that there exist positive functions Aij (t, ^, ^0 defined and Ci for t in T 
and in R. Assume that 
I fi a, t) -/,- (?', t) I s (J, i, f) \xj-x,\. (6) 
Suppose, moreover, that for (z = 1, 2, . . . ) and for all admissible values 
of it, i, J') 
converges uniformly, and that the maxima Mi of the \fi /)| and the maxima 
Ki of the Vi satisfy Mi ^ n M, 2Ki ^ ri K, with K and M finite. Then 
the approximations (4) exist and converge to a function ^ (t) for \ t — to [ 
sufficiently small. Moreover, Xi (t) is continuous in t and ^ (t) is the unique 
continuous solution of (1). 
