EQUATION AND ITS TRANSFORMATIONS. 
771 
The new series, multiplied by the factor eh, thus becomes 
gx l+l 
=gQ- 
i +1 
1 — —r ax 
l+l 
(7+ l)(7 + 2) ah? 
V {i + l){i + %) 2! 
eh 
Denoting then by P' the finite part of P, the series being supposed to end at the 
term immediately preceding the first term which contains a zero factor in the 
numerator, viz. putting 
tv -[A i , 7(7 — 1) a?x 2 . N . i\ a)x l \ 
~ x l 1_ i“ + i(i^Ij it • • 1 +(_) m-i) . ■ ■ (i-i(i-i)} irr ’ 
we have found that 
P—P +</Q—U. 
Similarly, if R' denotes the finite part of Pi, the series ending at the term immediately 
preceding the first term which contains a zero factor in the numerator, we find that 
R=R / —(/S=U, 
and also, as before, 
Q=S=V. 
The proof in (1°) that P=U does not apply as it stands when p — i, but it can be 
extended so as to include this case by putting p=i-\-h, and making li indefinitely 
small. The equality of P and U for all values of p may however be proved without 
the use of limits by showing that the coefficient of x" in Ue - " is equal to the coefficient 
of x n in P. To prove this ; first suppose n to be even and = 2m, then the coefficient of 
a 2m x~ m in a^TJ e~ ax is equal to 
i _!_ I i ■ i J_ i 
(2m)! p—\ 2 2 (2m — 2)! (p —i)(p— f) 2f2! (2m—4)! 
+ (—y --- 1 — 
v (p—i)(p-¥) ■ ■ ■ (p—m + i) 2~ m .m ! 
__1_1_ \ (p-h)(P~i) ■ ■ ■ (p-m + j) 
2m(2m —1) . . . (m+1) {p — \)(P~ I) • • • (P~ m + \) 1 m! 
t Tp-#) • • • (p - m + \) . , 2m(2m-l) ... (m + 1) 1 1 
\ m h (m —1)! ■■■-!-{ ) ml 2 2m J‘ 
The last term 
_ / w ( 2m! ) . JL = /_w (m-D(m-l) . . ■ j 
V ) ( W !)2 2 2m V ; m! 
5 G 2 
