Memoirs presented to the Cambridge philosophical society on the occasion of the jubilee of Sir George Gabriel Stokes, bart., Hon. LL. D., Hon. SC. D., Lucasian professor.

352 PROF. LAMB, AN ELECTROMAGNETIC ILLUSTRATION OF THE THEORY
where U, is a surface-harmonic. From these we deduce
Tr iafj /~ 7n) \ d d
= i2/î (h, (hr,)( d -C rd U1),. C.(2).
(ix dz
c X=- (hr) X y-d lir |
T-r Z~icrLL, / 7 x d d '\ i,x X + zz = O................................................... (23);
also yy - z,3 = {h ) + (Y + 1) i( (h1, )} y d z r &c., &c..........(24),
/-sY=-5 r2+ (hr)-,d,-,l- 2 U ), &c....., &c........(25).
It is known that the most general solution of our equations, consistent with
finiteness at the origin, can be built up from the preceding types, by giving n the values
1, 2, 3,....
2. Let us now suppose that a sphere of radius a, having the origin as centre, whose
electric and magnetic coefficients are K and,/, is surrounded by an unlimited medium
(the aether) for which K= 1 and u = 1. The disturbance in this medium rnay be regarded
as made up of two parts. We have, first, the extraneous disturbance due to sources
at a distance; this is supposed to be given. Secondly, we have the waves scattered outwards by the sphere.
The general expression for the extraneous disturbance is conditioned by the fact that
if the medium were uninterrupted the electric and magnetic forces at the origin would
be finite. It is therefore made up of solutions of the type already given, provided we put
K= 1, /= 1, and replace h by k, where
k = -/c.............................................(26 ).
As usual, 27r/k is the wave-length of plane waves of the period 2vr/-.
In the corresponding expressions for the divergent waves, we must further replace
(n(hr) by fn (kr), where fn is the function defined by (5). This is necessary in order that
the formulle may represent waves propagated outwuards, the complete exponential factor
being then ek (ct-r)
It is necessary to have soine notation to distinguish the surface-harmonics used to
represent different parts of the disturbance. Those harmonics which occur in the expression
for the imposed extraneous disturbance will be denoted by T_?, Un, simply; those relating
to the scattered waves by T,\, Un,; and those relating to the inside of the sphere by.T") 0''