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.

350 PROF. LAMB, AN ELECTROMAGNETIC ILLUSTRATION OF THE THEORY
the general expression for the dissipation of energy by secondary waves, which is
obtained in ~ 5, is found to take a very simple form, and may have other applications.
Some notations which are of constant use in the sequel may be set down for
reference. We write
/ d\ sin 1 __ 4 )
sn) ( )1l.3... (2n+1) 2 32 ). (2(2)n+ 3) 2(2) + )a
d' cos 1.3..2(2n-1) ) 2 )
n,(~) = _^ ~6 =. 1 - 2(1 -2n) 2.4(1-2n2)(3- 2n) 2n (-.()
These may be taken as the two standard solutions of the differential equation
dI2F 2 (n + 1) dF
+ 2( + = = o.................................... ();
the solution rn(,) being that which is finite for =0. In the representation of waves
divergent from the origin we require the combination
/ d )~ne-i~
f ( ) =,- - = )- (T).......................... ()
The functions * (n), 1n (), fn(a) all satisfy formulae of reduction of the types
% I (=) —1%+~1 ( ~)....................................... (6),
n ' ( ) + (2n + ) k () = n- (r).............................. (7),
from which (4) can be verified.
We have also the formula
% ' (0) % - ())- n () 4 ' (>) = ( 2 +2..............................
1. The equations to be satisfied in a medium whose electric and magnetic permeabilities are K and /L may be written, as in Prof. Love's paper,
dy _d/ K da dy K d_), ca
c dy dz' c dz dx' c dx dy..dZ dY p dX dZ /. dY dX
- - a= -,- - -7-, — / /3=. =.............. (1 ),
c dy dz' c dz dx' c / dx dy
where (X, Y, Z) is the electric force, (a, 3, y) the magnetic force, and c denotes the
wave-velocity in the aether. Assuming a time-factor eit, we find
(V72+h)X=O, (72+ h2) =0, (V2 + h2) Z==O.....................(11),
dX dY dZ
with d.~ + d- + d - = ~dx dy dz,
where h2 = Kir2/c2...(13).
* See Hydrodynamics, ~~ 267, 305. t See Lord Rayleigh' Sound, ~ 327.