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.

360 PROF. LAMB, AN ELECTROMAGNETIC ILLUSTRATION OF THE THEORY
on the right hand of (82) never exceeds unity; but it becomes equal to unity, and the
intensity of the scattered waves is therefore a maximum, when
G (ha)= 0.................................................(83),,ha,,n' (ha) + (n + 1) n (a) (la) K 1 a,,' (ka) + 1) l r, (a)(84).
~o~~r ~ - k -............(84).
When K is large, the lower roots of this, considered as an equation in ha, are easily
seen to be real and to be very approximately equal to the real parts of the roots of (71).
When the period of the incident waves is such that (83) is satisfied exactly, we have
Tn =- iT,............................................(85).
If the incident waves be plane, the dissipation-ratio (68) takes the form
2 (2n + 1) r 2 + 1.................................86)
k2 2r(
If we compare this with (68), we find that in the case n = 1 the effect of synchronism
is to increase the dissipation in the ratio
9 (ka)-6.
The wave-length of maximum scattering is of course very sharply defined. If we
put
ha = (1 + e) z.............................................(87),
where z is a root of (84), and e is a small fraction, I find
gfla(ha) (a.:... (2n- }
g (Itct ) =- -,^ e ). G (/ta)= (ACa2n+î _ (ha). 2e1.........(88),
3~. 3.. (21- (ka)~approximately, whence......................... (89).
1+.... iK.
(ka)2n+l
For example, in the case n = 1 the dissipation sinks to one-half of the maximum when
the wave-length deviates from the critical value by the fraction (ka)3/K of itself.
The second type can be treated in a similar manner. Writing (32), with /t = 1, in the
form
U - g (hha)
Un G (ha)-ig (ha)......................
the equation G (ha)= O which determines the wave-lengths of maximum dissipation may
be written
kn-i(ha) _ _ (ha)
~fr (ha) tqL, (ka) *****.*(9..