d 1] 
1902 - 3 .] On the Series y = l+ F ([a][/3][y]) • pj, + , etc. 443 
In these series p[m 1 — 1] is the lowest exponent and occurs only 
in a single term ; then, since is arbitrary, denote it by A (not 
zero) and choose m l so that the coefficient of viz. 
r i f b -+■ TYi-t — 1 { p. 
W{ n f =0 
which is 
p m i - 1 y pb+m l -n_\ m 
p — 1 X 1 , p b + m i — 1 . 
K= CO 
therefore 
. pb+m i - n+K - 1 _ 1 . pn+1 _ J 
-tb+m.+K-l 
1 • p l - 1 . 
. . . p n+K - 1 
. . . p K - 1 
pn(a -n) — 0 
m Y — 0 
m 1 = n — b 
m x — n — b - 1 
m 1 —n — b - k - 1 make zero factors in 
the numerator, and are possible values of m 1 , when n is not a 
positive integer. If, however, n be integral, 
K] 
b + m 1 - 1 ) 
n | 
reduces to 
pm 1 _ 1 pb+m l - 1 _ | . pb+m 1 - 2 _ ] 
p —Ip — 1 * p 1 — 1 . . 
and the possible values of m 1 are 
0 
1 -b 
•2 - b 
. p b +m l -n _ J 
. . . p n - 1 
n - b 
If we use the value m 1 — 0 , the relation (6) shows that the series 
V is 
y 
■ • (O 
and the differential equation of which this is a solution is 
