508 
PROFESSOR CLERK MAXWELL ON THE ELECTROMAGNETIC FIELD. 
We obtain M by considering the following conditions:— 
1st. M must fulfil the differential equation 
dm dm l <m 
■ da 2 + dtf +0 da ~ U ’ 
This equation being true for any magnetic field symmetrical with respect to the common 
axis of the circles, cannot of itself lead to the determination of M as a function of a , 
and b. We therefore make use of other conditions. 
2ndly. The value of M must remain the same when a and a x are exchanged. 
3rdly. The first two terms of M must be the same as those given above. 
M may thus be expanded in the following series : — 
8 r ( ~2 a ' 16 d z 32 a 3 j 
— irah 4 1 1 b 2 -3{a-a*) _ 1 (6b°~~ (a-a^ia-a,) , » \ 
\ 2 a ' 16 a 2 48 a 3 j 
(113) We may apply this result to find the coefficient of self-induction (L) of a circular 
coil of wire whose section is small compared with the radius of the circle. 
Let the section of the coil be a rectangle, the breadth in the plane of the circle being 
c, and the depth perpendicular to the plane of the circle being b. 
Let the mean radius of the coil be a, and the number of windings n ; then we find, 
by integrating, ^ 
L= F?J JjJ M (^ %'y')dx dy dx' dy\ 
where M [xy cdy') means the value of M for the two windings whose coordinates are xy 
and x'y ] respectively ; and the integration is performed first with respect to x and y over 
the rectangular section, and then with respect to x' and y' over the same space. 
L = Ann 2 a jlog e — + ^ — ^ cot 2 0 — ^ cos 2 0 — ^ cot 2 0 log cos 0 — ^ tan 2 0 log sin 6 
+- 
logy(2 sin 2 ^+l) + 3-45+27-475 cos 2 0-3-2(|-^ 
sin 3 0 lcos 4 0, 
log cos 6 
cosfl 1 5 sm 2 S & 
13 sin 4 9 . .) . D 
+T^ lo g sm 7+ &e - 
Here a= mean radius of the coil. 
„ t— diagonal of the rectangular section =\Z^ 2 +c 2 . 
„ 0= angle between r and the plane of the circle. 
„ n= number of windings. 
The logarithms are Napierian, and the angles are in circular measure. 
In the experiments made by the Committee of the British Association for deter- 
mining a standard of Electrical Resistance, a double coil was used, consisting of two 
nearly equal coils of rectangular section, placed parallel to each other, with a small 
interval between them. 
