certain Bessel Integrals. 337 
Substituting these values of y 1 and y 2 in (13) we find that 
the terms in log e and inverse powers of e cancel ; then 
making e indefinitely small the terms involving positive 
powers of € vanish and we obtain as far as terms in p 4 
-i 
erPvJ-?{v)dn 
I 
HI 1 + i>'- mf) '».-•! -'- h?+ m/\ ■ <I6) 
Further, if we integrate (11) with respect to p and put 
the constant of integration equal to \ on account of (1), we 
obtain 
= ~ % {{p + iqp 3 - mip) l0 *4 -\~p + 2M8^ 5 } - l(17) 
Integrating again with respect to p and taking account of 
(2) we find 
8 
, 4 7T 1 , , 1 1 fi ~l 
+ 8~r-P _ 4^ + 256^ + 3072 ?/ 
it 1 ,"/ 8 1\ 
7T 13 2 
ti^l+D-^K-g)}-^ 
In the same way, if r p 2 + (1— X) 2 is small compared with X, 
and if X > 1, we obtain from (11) the more general 
expansion 
X 
p 2 +(l-\)« 
L 1 + 16 X 10241 X J J l088 VpT 
l y« + (l-X)« 31 f p« + (l-\n » , 
1 16 X + 20481 X J • • <- 19 ) 
Finally, integrating (19) with respect to p and taking 
account of (3) we can obtain a similar series for the integral 
fV^J^J^)^ (20) 
Jo r 
