PRODUCED BY ALTERNATING ELECTRIC CURRENTS. 
307 
but the same plan will not avail when applied to find the new couple (by directly 
differentiating the couple due to the current I cos pt), owing to the interaction of the 
currents set up by the two elementary currents of the electromagnet. 
We shall have to use our previous result, that if the external field on the surface of 
a shell has momentum 
% [('M m cos nd + 'N„ sin nd) sin pt + ('Q„ cos nd + 'R„ sin nd) cos pt], 
the couple is by (vii.), 
irapard 
On reference it will be seen that due to the current I cos pt, we had 
v , r aa' iiira'rja' a'b' T . 
Ml = - —jyT- ^ I + cubes. 
, aa' 4 z 7 ra'pa' a'c T 
N =4----1 . 
^ d 3 D\ p'* 
_ ‘ 2 ab T ( aa' 87 f~a'~p 2 a’V T 
^ 1 + ^ 1 • 1 • 
V p 2 ac t- aa' 87rV 3 p 3 a'c r 
1 ~ + y 1 _ d 3 " T>\ ‘ ’ • 
Now, with the electromagnet instead of the single current (remembering 
p cos y —b, p sin y = c), — I will become 
+ A 
or 
A 
sm« 2b . , . 
— —— H-; + O sin a 4- c cos a) 
P" p" 
sin a 2 cos 7 . -' 
—r + — r~ sm a + y 
Also — I will become 
P 2 
[cos a — 2 sin y sin a -|- y] 
y c 
and for — I, — I, we have merely to add dashes to p and y. 
If we omit terms of the eighth degree in a v a, the value of the couple will be 
2 r 2 
