TO THE EVALUATION OF DEFINITE INTEGRALS. 
271 
41] 
But writing u 2 , v 2 for A, y, in the formula (5) (u and v being supposed positive), the 
integral in this formula is 
\ZirT(i + ±) 1 
T (i + 1) v (u + v) 2i ’ 
hence, after a slight reduction, 
V = 
S 
(~) K r (i + A + 1) A> 
or finally 
wr (i +1) T (i + 1) T (A + 1) {{u + v) 2 } A ’ 
V = 
r (i + 1) v {('u + v) 2 + A) 1 
.(16), 
a remarkable formula, the discovery of which is due to Mr Thomson. It only remains 
to prove the formula for W. Out of the variety of ways in which this may be accom 
plished, the following is a tolerably simple one. In the first place, by a linear trans 
formation corresponding to that between two sets of rectangular axes, we have 
W = 
dx dy ... 
{(x — \JA) 2 + y 2 ... + n 2 Y ’ 
or expanding in powers of A, and putting for shortness R = x 2 + y 2 ... + u 2 , the general 
term of W is 
(—V/4 A. r (j + A T <x) 02(7 f r 2<r i A cr r ], r J„. 
' ’ r*T(X-<r + l)r(a7 + l) J axay... 
the limits being as before x 2 + y 2 + ... = To effect the integrations, write \/g *Jx, 
&c. for x, y ... so that the equation of the limits becomes x + y+...= 1. 
Also restricting the integral to positive values, we must multiply it by 2 2i+1 : the 
integral thus becomes 
equivalent to 
i.e. to 
g<r+i+ij æ cr iy ^ ... + y ...) + u 2 } 1 K <r dxdy ... 
t +i+h j 1 W + u 2 )-^-* de ; 
F (i + cr + I) 
Hence, after a slight reduction, the general term of W is 
F(^ + A + q-) f w+o--i/ & + u 2\-i- 
n { } * r(a- + l)r(X-cr + l)r(i + a-+i)J 0 ^ + 
d£, 
where a may be considered as extending from 0 to A inclusively, and then A from 
0 to oo. But by a formula easily proved 
¿YV u 2 )-1 = ^ r (A +1) T (i + A + ^) x 
du) 
S(-)° 
Ti 
T (i + \ + a) 
F (<r + 1) T (A — cr + 1) F (i + cr + ^-) 
^(l + w 2 )- 1 '-