582 THE HON. J. W. STRUTT ON THE VALUES OE A DEFINITE INTEGRAL 
coefficient of e 5 
e ' 5 . 3.5 1 (l+e' 2 )i 7 1 (1 + e' 2 )^ , J_ (1 +e ,2 )^-l . 
~ll"^ 2 2 |2 ■ 3 ~e' 2 2*7 e' 4 ‘ H e'°' ’ 
coefficient of e 1 
e' 7 3.5.7 1 (l+d 2 )f l 7-9 1 (1 +e ,2 )£ 
15 2 3 (3 * 3 ^2 "t _ 2 2 ^2 ’ 7 e ' 4 
11 1 (l+e' 2 )¥ . J_(l+e*)¥-l . 
2 ‘ 11 e ' 6 '15 e' 8 * 
and so on. From these series the coefficients of e n e' n ' for moderate values of n and n' may 
be calculated with facility. 
It is desirable to know the limit of the integral i Q„Q n ,dp when n becomes very large, 
Jo 
n! remaining finite. A distinction is necessary according as it is the even or the odd 
suffix which is supposed to increase without limit. 
4m + 1 
J-2m + l 
m 
The whole coefficient of e 1 ' 1 is a sum of terms of the form — ■ , where ?n is zero, 
or any positive integer, each term multiplied by a numerical factor, which may be re- 
+ 1 
(1 +d 2 ) 2 
garded as a function of n and m. The general term in the expansion of v — 
powers of e 1 is 
jar-*.-, (4m + 1 ) (4m— 1) 3 .1.1.3... (2r-4m-3) 
6 ’ 
irrespective of sign. 
If we put 2r—2m — l=2n r — 1, it becomes 
fj2nt 
(4m + l)(4m— 1 ) ...3. 1 .1 .3 ... (2n! — 2m — 3) 
<L4.6...(2» , + 2 m) 
The coefficient of e <ln e , ‘ 2n '~ 1 is thus a series of terms of the form 
(4m + 1) 1 -1 (2n' — 2m—3) 
2.4.6... (2n' + 2m) ’ 
each term multiplied by a factor depending on n and m but independent of n'. The 
question is which term has the predominance when n' increases without limit l It appears 
that it is the one corresponding to m= 0 ; for the ratio of this to the general term is 
1.1.3.... (2n' — 3) 2.4 (2n' + 2m) 
(4m + 1) 1.1.... (2ii! — 2m — 3) 2.4...2n' 
{2rJ — 2m — 1) (2n'—3) X (2n! + 2) (2 n' -f 2m) 
(4 m+ l)(4m— l)...l 
a fraction which increases without limit with n'. 
