368 NOTES ON THE ABELIAN INTEGRALS.— [58 
and taking the sum of all the equations of this form, the last term disappears on 
account of the factor x x —x in the denominator, and the result is 
N ^ _ iN ^ ^ fa i i (n V(fa) 1' _ iy fa 
a-x dt 2 ** a - x dx {F'x) 2 + 5 r (a - x ) F'xj 2 (a - x) 2 {F'x) 2 * 
This being premised, assume 
which, by differentiation, gives 
y = f(Fa), 
and thence 
dy _ x y 1 dx 
dt 2 y ~‘ a — x dt 
= -^2 
VP) 
(a — x) F’x ’ 
d 2 y _ 1 dy y 1 fa y 1 /d#\ 2 1 N 1 d 2 x 
di 2 2 dt a — x dt 2 ^ {a — x) 2 \dt) 2 ^ (a — x) dt 2 ’ 
that is 
fay 
dt 2 
i v [n _VO)_V _ i v y fa i v y 1 d?x 
v (« - X) F'x) {a - x) 2 (F'xf a-x dt 2 ‘ 
Substituting the preceding value of 
2 
1 
a — x 
d 2 x 
dt 2 * 
we have 
that is 
d 2 ^ 
dt 2 
+ 2M^ 
/# 1 d 
(a - x) 2 (,FV) 2 a-x dx {.Fxf ; 
> +2^1 T ^ r ^Ll o. 
(a — xf {F'x) 2 (a — x) da; {F'x) 2 \ 
Now the fractional part of is equal to 
fa 
+ 
1 d fx 
(a — x) 2 {F'x) 2 a — x dx {F'x) 2 ' 
Also if L be the coefficient of x 2n in fa, the integral part is simply equal to L, 
(since {Fa) 2 is a function of the order 2n, in which the coefficient of a 2n is unity). 
Hence the coefficient of y in the last equation is simply 
-L; 
or we have