certain Bessel Integrals. 333 
those in use at present ; the cases examined are those involving- 
two coaxial coils treated simply as cylindrical current-sheets. 
The Bessel integrals used here occur frequently in other 
physical problems ; it has been thought better to give the 
investigation of these separately in the first section, where 
the series are obtained and examined independently of the 
applications made later. 
§ 1. Some Integrals involving Bessel Functions, 
The integrals which we have to evaluate are of the type 
-i 
1=1 e J 1 (p)J 1 {\p)p dp, 
and we require their expression in series suitable for large 
and for small values of p. 
When p is zero we have the following known integrals : — 
J ' 2 K?=i ; • (2) 
S l (pa)3 1 (jih)^t = \l, h>a (3) 
fi lb 
^0)^0^)% = ~ f </(a?-<?)Q?-<*)dc, b>a 
r 1 J Q 
9, 
= 3^{(& 2 +« 2 )E-(&'-« 2 )F}, 
where E, F are complete elliptic functions of modulus a/b. 
Also, putting k for a/b, we have in the last case 
(ww £ — (i - n *- m hi ~ 2-4 i6 - ) 
_ fl g, {1.3.5....(2r-3)}'(2r-l) „ 
.■=0 2 2r+1 r!(r+l)! 
Phil. Mag. S. 6. Vol. 15. No. 87. March 1908. i A 
