352 
MATHEMATICS: F. R. MOULTON 
Therefore if (10) converge for \t\ ^ p, then (6) also converge for at 
least the same values of 
It follows from (8) that 
1 ^1 _ 1 ^ = • 
fi dt r% dt dt 
The initial values of ^i, ^2, ... are zero; hence on taking ^(0)= 0, it 
follows that 
^, = (i = 1,2,- • •)• (12) 
Therefore each of equations (8) reduces to 
(13) 
dt I \-cot-Q) 
where 
C = CiTi + C2f2 + • • • , (14) 
which is a finite constant by (^"2). 
It follows from the ordinary theory for a finite number of differential 
equations that (13) has an analytic solution which converges if \t\ is 
sufficiently small. Therefore equations (10) and (6) converge for at 
least the same values of /. 
In general the Kmitations placed on / in order that the solution of 
(13) shall be known to converge are so restrictive that the corresponding 
Xi do not attain the boundary of the region for which the right members 
of (1) converge. The question arises whether the solution can be con- 
tinued beyond its original domain. 
Suppose equations (6) converge for t = to and let the corresponding 
value of Xi be Xi'^^K Suppose 
Co\to\ + CiW^l + C2W«^I + . . . = So<Si<l. 
Then let 
Xi = xi^^^ + yi, t = to + T. (15) 
The differential equations (1) become in the nev/ variables 
^^ = h + g^'> + g?>+- ■ ■ (i = l, 2, ■••), (16) 
where gi^J^ is the totahty of terms in the iih equation which are homo- 
geneous in T, yi, y2, . . of degree j. 
