EQUATIONS OF PROPAGATION OF ELECTRIC WAVES. 
39 
or, what comes to the same thing, the new vector is 
cos K ct 3(w,j 3(u„ Sco,i' 
(2n + 
l)r«3r ' -h . 0 ^y H- (^ 0 ^. 0 ^, 3Jv2«+V_ 
and the condition that ( f, g, h) should be radial, at the conducting surfaces r = Vq 
and r = 1 \, is that 
X« Ml = 0 • 
. . (81) 
at r = 7’u and r = i\. These two equations determine the ratio A : B and the value 
of K. 
A"hen the conducting surfaces are very near together, we have approximately 
cr 
072 xAkv)] = 0 , 
or, in vii’tue of the differential equation, satisfied by y„, 
= n (n + 1),.(82), 
a result otherwise obtained Ijy Larmor."^ The ratio A : B is determined by the 
e(piati()n 
k~ {r'‘+‘7,(Acr)} + b£ ['r«+'Tq(/cr)] = 0, 
3r 
( 83 ). 
Wil 
ich holds for r = r... 
32. We consider, in particular, modes of oscillation, for which the axis z is an axis 
of symmetry. We take 
zjr = p , 0 ),, — r’‘V„ (p),.(84), 
where B„(p) is the zonal surface harmonic (Legendre’s coefficient) of order ?/. We 
find 
dz dy 
(-)” , 1 r 3“+9'-i 3"r-7 
= m’' F a?v + 2'='5c}- 
Now it is easy to showf that 
( — )“ 3«7’“® 
n ! G 2 " 
3 + 1) P« + P 
dl\-\ 
df. [’ • • • 
. (85), 
* ‘ London Math. Soc. Proc.,’ vol. 26. 
t One way of doing this is as follows :— 
( _ 3'!r-3 
-^— = the coefficient of //.” in the expansion of {x^ + y- + (■^ ~ ^d"} * > 
+ 7 + u - = 
also 
