ON CERTAIN DEFINITE INTEGRALS. 
[From the Cambridge Mathematical Journal, vol. ill. (1841), pp. 138—144.] 
In the first place, we shall consider the integral 
V =[[... (n times) dy."‘ 
li '{(a-xf + (b-yf... 
the integration extending to all real values of the variables, subject to the condition 
rn 2 7/2 
—V ——b < or = 1 
and the constants a, b, &c. satisfying the condition 
a 2 6 2 
We have 
— (n — 2) j)... (tl times) 
2hh, ... 7r^ n a f 1 
(a — x)dxdy ... 
{(a - xf + (b -yf ... 
¿c n 1 dx 
tin ’ 
(n 2) V(£ + h*). r (in)J 0 v [{£+h 2 + (h;- - h 2 ) tf 2 } {£ + k 2 + (V- /i 2 )a?)...] ’ 
£ being determined by the equation 
+ 
6 2 
... = 1, 
Z + /u %+h? 
by a formula [see p. 12] in a paper, [2], “ On the Properties of a Certain Symbolical 
Expression,” in the preceding No. of this Journal: £ having been substituted for the rj 
of the formula.