14 
ON CERTAIN DEFINITE INTEGRALS. 
[3 
Let the variable x, on the second side of the equation, be replaced by cf>, where 
f +h ‘ . 
f + A ! + <i>’ 
we have without difficulty 
dV _ , hli J ... 7f jn a f 00 d(f) 
da = ~ {n ~ Z) r (inr~ J 0 (T+FT^TV^ ’ 
where & = (£ + h 2 + <f>) (g + h 2 + (f>)... 
and similarly 
dV_ i) ,hh / ... 7r~ n b f x dcf) 
Ob (n_ ' T(ih)' Jo(T+W+W7®’ 
&c 
From these values it is easy to verify the equation 
y _ (tt — 2) hh / ... 7r* n f 00 / a 2 b 2 \ dcf) 
2T(JtX) Jo \ £ + h 2 + <f> t; + h 2 + <f> ) 
dV 
For this evidently verifies the above values of 
vanishes; and we have 
> &c. if only the term ^ di; 
dV_ (n — 2) hh / ... 7r in f” d / a 2 \ 1 
dç ~ 2r (in) Jo 0 ‘ di V f+¥Tt 7$ ; 
or, observing that 
d a a 2 \ 1 d /- a 2 \ 1 
and taking the integral from 0 to oo , 
dV (n — 2)hh / ... ir in / a 2 b 2 \ 1 
2r(in) V |+/i 2 f + V''7V{(f+>)(^ + /q 2 )...} ’ 
in virtue of the equation which determines 
No constant has been added to the value of V, since the two sides of the- 
equation vanish as they should do for a, b... infinite, for which values £ is also infinite 
and the quantity 
1 - 
Ç+h 2 +<f> 
1 
7T®Y 
which is always less than 
1 
V(^) 
, vanishes. 
Hence, restoring the values of V and <ï>, 
//-< * times) 
dx dy ... 
(n — 2) hh / ... nt 
2F (in) 
in /.00 
{(a-x) 2 + (6- y) 2 1 
b 2 
d<t> 
f+/t ! +0 f + v+0 "V V{(? + ^ + <M(f+ V + 4>)-l