[FIELDS] EXISTENCE OF THE POWER-SERIES 83 
Cr PC CS le 
CS CRC SET, 
9. = 
CS CR Ce SET. 
We shall examine the value c’, more closely. We have p < r. 
Putting s=p in (8.) we get 
n / 
10. F) (11) =>) aa at-? g¢it-Pef, (2). 
t=p ; 
= 
From (5.) we have for each element in this sum €, + (f — p) CS © — pe 
and furthermore for ¢ > 7 we have c, +(t—-p)c>c—pc For the 
coefficient of 2°0-?¢in F®) (u,) then we find 
at! 
t—? 
i Desi" gr (0), 
where we employ the notation g; (o) to designate o or f; (0) according 
as ¢,+tc¢ >c¢,or=c,. This expression is however the pth derivative 
with regard to a of the left-hand side of equation (7.) and must 
therefore have a value other than o since a is, by hypothesis, a p-fold 
root of this equation. The coefficient of 3607? in F®) (w;) is then 
different from o and we therefore have 
12: Cr = G5 = pc 
à Se Goi e. 
From this we derive ~ 5 ? =c. But from (9.) for any value of 
Coca 
s we have —— <c and consequently 
EGG. C0 
13: ° 2 <= 2 
Ss Pp 
Furthermore for s>p, we see that 
14. GE no < Co — Co 
5 Pp 
on remembering c' >c¢,. Combining this inequality with the pre- 
ceding inequality we find 
Co = 6 CRE. 
no: SD RS y en 
S D 
From this we see that the equation (4.) for v7, has for amplitude a 
number 7’ = p. This number 7’ namely will be the greatest value of s 
oP 
, 
3 (ap ae ck . 
for which Ser = c where 
f 4 LA , , , LA , , 
Eee te, Cote Co—e CA aG 
16. c= Max.{°° he 
1 2 s | 
: ; 2G 
That c’ > cis readily seen. Namely we have c’ = —°——" where- 
Nig I ol ME 
as from (9.) we have c = 
r (2 
