38 
PEOFESSOR A. E. H. LOVE ON THE INTEGRATION OF THE 
fundamental equations are included among those obtained in § 1 0, by proper choice of 
the function (/)q. 
Taking the centre of the spheres as origin, let denote any spherical solid 
harmonic, and write 
. /v\ / \ /I 3\"siiit 
-Y’ 
T /y\ / \ , /I ^ \" COS f 
"^(0 - (-)' -Y~ ’ 
^'^u(0 + = X’i(0 ’ 
(/ 
A and B being constants. Then, in the modes of oscillation under discussion, we 
have 
(u, V, ?v) 
= cos K Gt. (kc) . 
a 
dy ’ " dx 
] 
cos K ct 
with similar formulae for g and h ; the value of k and the ratio A : B are to he found 
from the boundary conditions, which are that tlie electric force is normal to the 
bounding surfaces. 
The vector (/', g, h) has a radial component of amount 
— n {n + I) cos K Gt r~^x>‘ . (SO). 
If then we form a new vector from (/’, g, A), by resolving this radial component in 
the directions of the axes, and subtracting its resolved parts from f, g, A, the new 
vector will have the same tangential components at any sphere as { f,g,h) has ; the x- 
component of the new vector is 
w 
+ 1 
= 0 , 
i'll (0 = - 
{2n + l)'pn{0 = '/'ii-i (0 + C"’/'»+i(0> 
and these equations are also satisfied by and by y, provided the constants A and B are supposed not to 
change with n. Reductions made by using these equations will be introduced without remark. 
