Referring to the definitions of the s's we see that Su¢y is
'
a function of the s's. The series (8) as it ncw is can be written
5 3 up " and must be changed to : S ugpx® in erder to reduce it tc
th 1

a power series whose coefficients are functions of the s's. Some
conditions which permit the interchange of order of sumpation are:

1. Let {uihx” canverge in Ry and logll + v¢(x)] converge
unifermly in R, <R,. (The Weierstrass thecrem'® requires this condi-
tien, and states that then 3 §1qhx” converges in R, and is equal to
Hhy T

2. Let %u;hxh copverge in R, and %Ivi(x)l cenverge uniformly
in Ry. (If Slu (x)lcenverges uniformly, then so does Zlog(l + y (x)] .
The conclus;on follows from 1 above.)?*

3. Let %Iuihxhlt Uy converge, and let {U{ converge, (This
is Cauchy's theorem.'®)

4. Let Sugp s , Sugs® , and 2;u¢hx” , all converge,
1 n

5. Let 5 lugpa"| converge.®
ih

If the elewents cof the array (1) are chosen so that any one
of these five conditions is satisfied, then we cap sum the series
(6) by cclumns. Because of the conditicns already impesed on page 11
on the elements cf the array (1), conditicns 1, 2, and 4 are
alreauy partly satisfied, for §u4hxh converges and equals lcg[l +vy].
1f we proceed under one of these three conditions the least restric~
tive additional conditions would be needed; but 2 is stronger than
1, since if we have 2 satisfied then 1 is also satisfied, but not

conversely. For definiteness we shall precceed under condition 1,

 

2% Harkness and Morley, sec, 81. Knopr, sec. 249, r. 480.
g Pierpont,II, seo, 209, 124. Swail, seo, 258, 257,

M Bromwich, sec, §2, Swail, seo, 232, 224,

3 Bromwjoh, eec, 20. Smail, sec. 213,

a5 Bromwich, sec, 32. Swpail, sec, 230.