586 THE HON. J. W. STEUTT ON THE VALUES OF A DEFINITE INTEGEAL 
From the first term we have simply — L As for the second, 
1 +a ,2 +,r 
G'+y)} 
:(l+* ! ) *{l - § ]+ (+ ) + + if Tf (+ ) 
1.3.5 / 1 \ 3 | 
2 3 (3 (1 +x i ) s \y~'y) * * * * "j 
The term in 
•V+f+f+- • • •) independent of y is 1. 
The term in (V+J) 0-+2/+/+* • •) is 1 + 3, and generally, if n be odd , the part 
independent of y in 
(y+fo-y)-' + (H-i)"+2*. 
Thus the term in j (1— y)~' independent of y 
=( 1 +* 3 ) 
-Hi 
1 ' i-2. 
1.3.5 a? 3 1 
.+a? 2 2'^ — 2 3 (3_ (l+a? 2 ) 3 2 
t — + an even function of x 
\ 
J 
r-. 1 2a? 1.3 (2a?) 2 1.3.5 (2a?) 3 ^ 
(l+a? 2 ) -^| 1 — 2 ‘ 1 +^ 2 + 2 2 i 2_(l + a? 2 ) 2 ~ 2 3 |+ (1+^1 
+ + an even function of x J 
l 
(1 +a? 2 ) ~? 
2 
[jl + r+T 2 } + even function of x~^ 
1 1 
"2 1 +a? 
+ an even function of x. 
Finally, therefore, 
A,= - 
1 +a? f 
\/x 
|«0 + ®2-^ 2 + ®4# 4 + ...j, 
where a m a 2 . . . are unknown coefficients, of which we will only determine a 0 by a reference 
to the expression from which A, was obtained, which shows that 
so that 
— -|+« 0 — 
Aj — 2 *</ oc-\- ct 2 x^ cl 2 x~ + d^x^ -\-ttyx % -\~ .... 
Multiplying by Vy and replacing e and e\ 
A */ y—^e-^-a^d + o. 2 e z e l2j r a i e 4 e' 3 + a 4 e s e u + a 6 e 6 e' s + a 6 e 7 e' 6 -\- .... 
