Prof. D. N. Mallik on Mutual Induction. 365 
whore 0= /between the perpendiculars to the two circles, 
$'" defining the plane containing these lines, 
/•'= Z between any radius of the second circle and the 
axis of the first circle §>' defining the plane 
containing these lines. 
.'. integrating for 6 between and '2tt) 
-e have 
¥ 
h 
a+l 
M = ±Tr-u±n { n + l) . ^-„- P,(cos 0) J P.OM/ij P„OW 
=^W(l-^)(l-^)2(|)"P„(co S «)^ . 0, 
which is a well-known result. 
Ex. 2. Mutual induction between two parallel rect- 
angular wires symmetrically situated and carrying unit 
currents. ■ 
Let x. y, Ji. and x\ y\ h ', be the coordinates of any two 
points in the rectangles of which the wires form the contours. 
Thpn M--CL j_ f dxdydx'dy' 
This is directly integrable, for 
aIld flog \<JW+rf + u}du 
= u log ( \/1r + "' + «) — V ^" + w 2 . . . . (2) 
But Ave may also proceed as follows : — 
W J 
a/* ' a/t'J J x /x'-~+y-+(h-h r f 
— a —6 v t7 \ y 
where « s £„ ^^ 
B d f(«+««-e-«»)(e+^-r-* 6 ] <2a'<2y' 
J 
if 2a, 26 are the sides of the first rectangle. 
Phil. Mag. S. 6. Vol. 15. No. 87. 3/areA 1908. 2 C 
