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.

358 PROF. LAMB, AN ELECTROMAGNETIC ILLUSTRATION OF THE THEORY
where the values of Bn and Cn are as given by (31), (32), and T2, U, have the forms given
in (43), (44), then since
ff{sin 0 cos os P s 0)}( dcs = n (n + 1). 2 1.....................(64)
the expression (62) reduces to
4k 2 (2n + 1) Bn + 2}................................. (65)
The proper standard of comparison here is the energy which is propagated per unit time
across unit area in the primary waves represented symbolically by (37). On the scale
of our formulæe this is c/87r. Hence, if I denote the ratio which the energy scattered per
unit time bears to the energy-flux in the primary waves, we have
I 2 (2n +1) B,j2 + |CJI 2}........................ (66).
For example, in the case to which the formuloe (33), (34), refer, the constants K and
/L for the sphere being not greatly different from unity, we have
2K —1 2/-1
9k= 2 K-3 2 k3a=.. (67),
B l — kaL, Co1.+ 2n=............................(67),
and thence
= 3 7ra2 + ca) 4...........................( 8)t.
6. We may proceed to examine more particularly the case where K is a large
number, whilst t/ is (for simplicity) put = 1. The types of free vibration which can exist
in the absence of extraneous disturbance are found by making T =O, Un=0 in (31)
and (32). In the first type we have
haf,' (ha) + (n + 1) %, (th) K af,' (ka) + (n + 1)fn (a) (),
(ha) f=K (k)a)
solutiowhere,s of this to beation for which is small. Onare speially concerned to fin have
solutions of this equation for which ka is small. On this hypothesis we have
ha-r4' (ha) + ( + 1) +n (ha) _
Irn,(ha) ~ -(n?d~ ha.*..........................(70),
%,, (ha)
nearly. This is satisfied approximately by ha =z, where z is a root of
n ()= 0............................................(71),
and more exactly by
ha=......................................(72).
* Ferrers, Spherical Harmonics, 1877, p. 86.
t This agrees with a result given by Lord Rayleigh, Phil. lMacg., April, 1899, p. 379.