M 
344 On certain Bessel Integrals. 
different from before and more complicated. But we have 
J,(\a) J,(\?0^ 1 d A e ~ X ' x ~ x ' ' da)1 
cO J -hi J-h> 
= 8^ab ni n 2 [|*i-^(^^-^^)ji(A)Ji(f /*)^]. (29) 
Comparing tins with (27) we see that h x and 7i 2 are merely 
interchanged ; so that there is a similar series to (28) for this 
case also. 
§6. Self-induction of a Cylindrical Coil. 
As a final example we consider the self-induction of a 
single-layer coil ; then if we have 
2h = length of coil ; a = radius ; 
N = 2n/i = total number of turns of wire; 
we can easily deduce from the integrals in § 4 an expression 
for the self-induction of a coil in the form 
L = 4ti V f Q J - _ * sinh (U) I J?(\a)d\ 
" Jo L^ *•" J 
=2 ^ir[ 1 -5Fi + ]Ej u * ' °^A m 
For the integral in (30J we can now use one of the 
series (8) or (18). 
If 7i>a, we have from (8) the series 
+*©'-£# + }• • • • < 3 » 
where the general term is given by 
2s I) (2s + 2)\ /a\ 2s+ ' 2 
2)! \(s + l)\< i 2 2± s +Ah) ' 
s\ (5 + 2)! K* + l) 
The first four terms of this series have been obtained by a 
different method by Russell *. We see that the series in 
general is simple and rapidly convergent for coils whose 
length is greater than their width. 
* Russell, Philosophical Magazine, vol. xiii. p. 445 (1907). 
