16 
ON CERTAIN DEFINITE INTEGRALS. 
[3 
we find 
and we hence obtain 
da 
22 
Ç + h 2 
2 
a 2 
(î+a s )’ 
Hence the function 
vx£= 
f* 00 
d<f> 
J 0 
(£ + A 2 ) 2 
l 
(£ + $) V{(£ + ^ 2 + $) •••} 
(observing that differentiation with respect to £ is the same as differentiation with 
respect to </>) becomes integrable, and taking the integral between the proper limits, its 
value is 
2 X«£ ^ f+h 2 + 4x °' ^ 
a- 
(f + A*)* 
where 
We have immediately 
or 
whence 
x ’ ( fVKI+A’Hf+V)-)' 
*** s (Fr*) +4 *'*— 4 f ; 
pao 
I dcf). 
J 0 
(f+ *) v((f+A!+ ^ •••) fv «F + ui(f+V) ••■} + 
6 2 
(£ + V) ! 
Hence restoring the value of V, and of the first side of the equation, 
/f.— 
+ 
{(a-x) 2 + (b - y) 2 ... 
iin+9 
hh, ... ir 
in 
{ 
d 2 d 2 V- 1 
+ 
2 2 5 _î . 1.2...g.r(-|n + g') \da 2 db 2 " 
with the condition 
PVKf + h‘) (f+V) -1 + w+w + 
a- 
b 2 
~r 
+ 
... = 1; 
Ç + h 2 £ + hf 
from which equation the differential coefficients of £, which enter into the preceding 
result, are to be determined.