1 42 PROFESSOR H. LAMB ON ELLIPSOIDAL CURRENT-SHEETS. 
The corresponding values of F, G, H, at the film are 
F = DNz 
G = - CNz 
IT = CM y - DLx_ 
If we further assume 
+ ft = (Aai + By)z,. 
and substitute in the equations 
/ / dF <?F d . . . n 
p “ = “ * - * ~ *+ * >■ &c - &c - ; 
(31) 
(32) 
then, equating coefficients, we obtain the following four equations to determine 
A, B, C, D 
p/ec 2 . 1) = — ip N. D — A ''l 
-p/ec 2 . C= ip N.C-B 1 
— p/ea 3 . D = Ip L . D — pi — A 
p/e/r . C = - ip M.C - (pi - B 
Hence 
V 
r 
(33) 
{f(^ + ?) + i P< M + N)}c=-V.I. 
j j if + ^) + *p (r + N) j d = pi. 
If t v To, denote the persistencies of free currents of the types 6 = C.r, 6 = D y, 
respectively, these equations may be written 
- ip ^ _ I 
1 + ip T a M + N i 
, r 
D _ F** . 1 
1 -\-ip To L + N J 
The components of magnetic force within the shell, due to the currents alone, are 
found to be 
a=(M+N)C, /3 = ( L + N)D, y — 0, 
so that we have for the total magnetic field inside 
a T~ a = 
I 
1 + ip 
ft + ft — 
il 
1 + ip t 2 
y + y = 0. 
(35) 
These diminish indefinitely as p increases. 
