414 
MR. W. D. NIVEN ON CERTAIN DEFINITE INTEGRALS 
z = - — — aj sec i \i 
.dz — — aj sec i fj tan xpdip 
and the corresponding density is —2 cot 0 .— 
° 17 T aj cos y 
Hence the potential at all points inside the radius a is 
a sec 2 yjr 
J o v 7 / 2 +# 2 + (A ~ a i sec f) 
(d) 
This result may also be proved from the consideration that the expression for the 
magnetic force is continuous as we pass from one form of potential to the other. 
§ 27. The results (c) and (cl) give the mutual potential energy between two circular 
currents placed in any positions whatever, in which the smaller circle is either wholly 
outside or wholly inside the radius of the other. For the sake of simplicity we shall 
suppose that the axes of the two currents intersect. Let the radius of the larger be 
A and of the smaller a, and let u, v be the distances of their centres from the point of 
intersection, the axes being inclined at an angle cos -1 /r. Then, if unit current cir¬ 
culates in each, and if the smaller circle lies wholly inside the sphere of radius A 
containing the larger, the mutual potential energy is given by 
M=4«AfT 
.a J n 
j cos (f) sec 2, *y 
'0*0 \/(u + Aj sec -*y) + (v + aj cos <fif — 2 (u + Aj sec \p) (v + aj cos cj))/x 
If this series be expanded in harmonics 
B 0 P 0 +B1P1+ • • • +B„P W + . . . 
' 7r fj cos cf) sec 2 \]r(v + aj cos ) 
o'o (u + Aj sec -v | t ) B+1 
d(f)dxp 
we Jiave 
B„ = 4 «A 
-circlip 
The series in the form of zonal harmonics is given by Clerk Maxwell (‘ Electricity 
and Magnetism,’ vol. ii., p. 303). * 
On comparison with his series, since 
j j cos cp([x-\-jv cos 
J O 
7r dP n 
ir 
n +1 d/jb 
when we take account of the above value of B yi we deduce 
f’ 7 sec 2 yjrd\p 
J 0 (/u, + vj sec y\r) 
7r 
dP 
‘ M+1 n V dfj, 
true for integral values of n, from 1 upwards. 
