412 
MR. W. D. NIVEN ON CERTAIN DEFINITE INTEGRALS 
Hence writing herein cid, b9, multiplying by — and integrating, we find for the 
potential at (fg, h) the expression 
\ eO' 4 cos Kh~ c sin ^f^edddrp .(a) 
0 Jo 
The expansion of the exponential leads of course to the series (48) given above, and 
it is easy to find thereby the expressions for the action of an elliptic current on a 
magnet placed at any distance from a coil outside a certain boundary. In like manner, 
subject to a similar restriction, we might find, as in the case of two ellipsoids, the 
action of one elliptic current on another. Within the boundary referred to, the 
expressions we have obtained are no longer convergent, and in the case of the elliptic 
current the potential in the neighbourhood of the centre of the circuit must be found 
by an independent process. 
The case of the circle, however, admits of a simple and complete determination. 
Potential of Electric Currents in Circular Circuits. 
§ 26. Reverting to the integral (a) let us put c=0; then the potential at (f g, h) 
due to a circle of unit density is given by 
V=a 2 \ \ e ja6 cos *7h ^ddddxfj 
Jo Jo 
To integrate this in regard to 0 we observe first that 
fl of of 1 
0 ^d 6 =---+- % • 
J o u n~ vr 
Let us next consider the integral 
/"2jt rin e k cos \\i 
&k cos 't'clxh or COS lb - Tclxii 
Jo r Jo T COS ‘\]r 
Integrating by parts we find 
e k cos ^ tan \p 
~ n r^Jc sin 3 ylr , . 7 f 27 r sin 2 . 7 
+ cos +d\b — — ffp cos *dxb 
o Jo cos -vp Jo cos--»p 
'[Jr 
The first term becomes infinite when x p—g and when we ma y replace it by 
