certain Bessel Integrals. 343 
for purposes o£ calculation the series can be written in the 
following way : 
M/S7i 3 a 2 n l n 2 = h 1 --b 2 \](u-v)- ~d(u*-v*) 
+ ^ (c 2 + ^)(« 5 -» 5 ) - i «(3« 2 + «! 2 )(« 7 - « 7 ) + ■ • • ■], • (23) 
where 
c = ab ; d=a 2 + b 2 ; u= . r ; v= 7,7 * 
/i 3 — % li 2 -\-li\ 
We shall test this formula by the following numerical 
example which has been used to compare other series : 
7j 2 = 15 cm. ; hi = 2'5 cm. ; 6 = 5 cm. ; a = 4 cm. ; 
nj = 10, and n 2 = 40 turns per cm. 
Then we substitute in (28) the following values : 
c = 20 ; d = <±l ; w= — ; 27 = 
•> 9 
5 ; U= 35- 
Calculating the terms shown, we find to the order indicated 
M = -0012000 henry. 
From the form of the series we see that this is larger than 
the true value ; and in fact, by taking an extra term we find 
to the same order 
M= -0011999 henry. 
Rosa and Cohen * have calculated the same example by 
three different series using a similar number of terms and 
give the results : — 
M. Series. 
•001199896 Roiti. 
•00119990 Searle and Airey. 
•00119989 Russell. 
§ 5. Sltort Coil outside a Long Coil. 
With the same notation, suppose li 2 is small and li x large. 
It has been thought that in this case the formula for M is 
* Loc. cit. 
