148 
PROFESSOR H LAMB OX ELLIPSOIDAL CURREXT-SHEETS. 
we find 
A = 4 W 5 ,»-i)Q,'({ 0 ).cn 
B = M4 2 - 1) P,/(«.CI. '' 00 > 
Owing to the symmetry about the axis, there will be no difference of electric 
potential, and the electromotive force of induction will be everywhere in the direction 
of the current. The magnetic induction across any element of the surface will be 
<m 
dVy 
d-Sf, ds„ 
w dQ i , 
~ rff A *- 
l) 
(56) 
Substituting from (53), and integrating over the portion of the surface bounded by 
a parallel of latitude and including the positive pole, we find for the total induction 
through the parallel 
P n (p) djl . C. 
If the system of currents defined by (51) remain always similar to itself, the electro¬ 
motive force round a parallel is equal to — d/dt of this. The electromotive force at a 
point is derived by dividing by 2ttJc (£ 0 2 — 1) (1 — p~). Since 
we obtain in this way 
477" 
[ P„ (fi) djx = 
J ix 
1 — [d d¥ n (yU.) 
n (n + 1) dp 
(57) 
Equating this to 
/ d<f> 
p dd: 
where d(f>/ds^ has the value (52), and p = p/eur, we find that all the conditions of the 
problem are satisfied, provided 
where 
dC , C 
*+ 7 = 0 ’ 
i) : p,;(£ 0 ) q; ( 4 ). 
(58) 
* We may prove in a similar manner that in the case of an ellipsoidal shell with unequal axes 
magnetised everywhere in the direction of the normal so that the “ strength ” 0 is proportional to a 
Lamp’s function, the potential on either side is everywhere proportional to 0. 
