OCCURRING IN SPHERICAL HARMONIC ANALYSIS. 
415 
If the smaller circle is wholly outside the sphere of radius A, the value of M given 
above is inapplicable, and we ought to take a double application of the formula (c). 
§ 28. In the important case of two coaxal circular currents, if b be the distance 
between them, the mutual potential energy is given by 
M=4« A 
j cos 0 sec 3 0 
J 0 Jo& +i(A sec -yjr—a cos 0) 
dtpclxp 
(e) 
provided the sphere of radius A encloses the smaller circle. 
Performing the integration in regard to 0 we find 
M 
= 47TC< | 
17 cos <f>(a cos 0 + bj) 
0 \/A 3 — (a cos 0 + bj) 
=40 
With the view of simplifying this let us put a cos 0+5/= A cos ( x-\-yj ), so that 
a cos 0=A cos x cos yj and — bj=A sin x sin yj. Thus x and y are not independent, 
and by the substitution proposed it can readily be shown that M takes the form 
8 77 A* | 
cos^ x sm x 
\/ A 3 cos 4 x— (<x 3 + A 2 + b 2 ) cos 3 x + w 
<lx 
the limits of integration being determined as follows: 
When 0=^", and when 0=0, A 3 cos 4 x — (a 3 + A 3 +6 3 ) cos 3 cc+a 3 =0. 
The roots of the last equation are 
jy 2 — cos 3 6 X 
_ A 3 + a 2 + b 2 + v/( A 3 + « 3 + b 2 ) 2 - 4a 3 A 3 
2A 3 
_ o a _ A 3 + a 2 + b 2 - (A 3 + ft 3 + b 2 ) -4a 3 A 3 
rf= cos 3 do 
2 A 3 
Since p is greater than unity we must take q as the value of cos 6 when 0=0. 
Hence the value of M is 
8nA ' 
cos j x sm x 
cos -1 cos 2 .)!? 3 - cos 2 .r) 
dx 
= 8 77Ay 3 1 2 , C0S f-- ==dd 
] V p 2 — <± COS" v 
(f) 
(J 
where F and E are complete elliptic integrals whose modulus is 
