330 
PROFESSOR C. RIVEN ON THE INDUCTION OF ELECTRIC 
sphere may be readily deduced by the principles which have been already explained. 
I may observe that U„ is to be expressed in the system of tessera! harmonics 
S(A cos m<£+B w sin mcf)) P,/ 
0 
The rest of the process needs no remark. 
Spherical shell of finite thickness. 
§ 9. Let the external and internal radii be b, a ; all the conditions of the problem 
will be satisfied by the same general assumptions as to the forms of the electro¬ 
magnetic momentum of the currents, the current function and of the potentials of the 
external magnetism. The characteristic equation remains 
er 
47T 
V 
and, as in the former cases, since there is nowhere flow along the radii to the common 
centre of the surfaces and no free electricity, the currents are entirely confined to the 
interior of the shell, and we shall have as before for boundary conditions, that the 
vector potential of the currents in the shell and its differential coefficients sustain no 
sudden change at crossing the surfaces. Outside the shell and within its inner 
surface, the vector potential is given by expressions of the same form as in the 
substance, viz.: 
(1 p dV 
F=0, G=-, H=^j, 
sin 6d<p a 6 
but in the space free from currents 
V C P = 0. 
At the boundaries all the conditions are satisfied by having 
dV 
P and 
dr 
continuous when r=a and when r=b; we must therefore take 
o o* | 
within sphere of radius a, P = SCr"Ude“ A V 
in substance of shell, P = S.(AS„-f- BT„)U, t e~ A2 'U [* . 
I 
without sphere of radius b, P~ SDr " 1 . U„e A ^ 
(58) 
