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