786 
MR. J. W. L. GLAISHER OH RICCATTS 
Transform the integral by assuming x=~; this equation then becomes 
f x~ n ~ l e~ x '-*dx= £r(£n)a^M+0(7i)N. 
■* o 
whence, changing the sign of n, 
f a?- 1 e-*'-*dx=$r(-±n) a*N +<£( - n) M; 
J o 
and it follows therefore that (f>{n)=^T( — ^n). 
Thus for all values of n (except, of course, n= an even integer) 
x a -\- x ^dx=\Y{\n)W^\Y(-ln)o.^ .(7) 
o 
1 
2 
r {in 
n ~ 1 (-Xx)4 -' 3 ) (2*Y_ (n-l)(n-3)(n-5) 
m-P (n-l)(n-2) 21 (n-l)(n-2)(n-3) 
(2a) 3 
+ &c. 
+ ir(-i n y 
n+1, , (n+l)(n + 3) (2a) 3 _ (?t+!)(?;. +3)(^+5) (2a) 3 j , 
n+l^ ^(n + l)(n+2) 2 , ( to+1) ( w+2 )(» + 3) 3! + J 6 
i r (^){i+^T 
t/o \ , (tt-l)(rc-3)(2«)» 
(h- 1)(^-3)(»-5) (2a) 3 
(w-l)(»-2)(w—3) 3! C ' 
+£ r (-^) a ’ 
. ^ + 1 / -, \ ■ (^+l)(?^+3) (2a) 3 . Q& +!)(?& +3)(u +5) (2a) 0 . 
^n + V ) ' t (n + l)(n + 2) 2! " l “(w + l)(?H-2)(w + 3) 3! 
the series extending to infinity in every case. 
The method by which the fundamental formula (7) has been obtained is open to 
some objections. These will be noticed, and a complete proof of (7) given, in art. 28. 
21 . We have 
F(—W) 2 T(l-\n) 
T{^n) n rQn) 
2 
nsin^nrr (r(^?i)} 2 ’ 
7T 
since r(m)r(l— m)= -; 
v ' ' ' sm mir 
and, if n is an uneven integer, this 
9 _ i 
— (_\K«+i) _ - ± - = (_W«+i)--- —a, 
> n {V^bH---Un-2)r 1 ' 1 3 .3 3 .5 3 ... n~ 7l 
r/ l being as defined in art. 16, when a is put =2. 
