544 
PROFESSOR H. LAMB ON ELECTRICAL 
4 
11. In the whole of the preceding investigations it has been assumed that the 
quantity of § 2 may without sensible error be put =0. I proceed to sketch the 
method to be pursued when we do not make this assumption, confining myself for 
simplicity to the case of p=l everywhere. The fundamental equations to he satisfied 
are:—for the spherical conductor (18) and (19); for the surrounding dielectric (21) 
and (22). 
In the solution of the First Type the values of F, G, H and of a, b, c inside the 
sphere are then given by (27) and (34), respectively. Outside the sphere we shall now 
have 
?=Ujr)(4 r 4^, 0V)( 2/ |- Z | y )x_._ 1 . . . (105), 
where X„, X_„_ x are solid harmonics of the degrees indicated. The values of G and 
H may be written down from symmetry. We thence find 
f dX ,*2 r 2»+3 ( 1 -) 
a = - {( ,l + W-M + J x X.’- 2 - 1 } + terms in X.,.,. (106), 
with symmetrical formulae for b, c. The “ terms in X _ w _ 1 ” are to be derived from the 
preceding line by writing — n — 1 for n throughout. 
The continuity of F, G, H at the surface of the sphere requires 
M m )xn— V'*(iR)x,»+i/»_„_ 1 (/R,)x_»_ 1 .(107), 
when r=R. The continuity of a, b, c requires in addition 
^-i(/dd)x. = ^-iO'R) X «+ ‘ ‘ ' 
In the solutions of the Second Type the forms of F, G, H, a, b, c inside the sphere 
are given as before by (40) and (43) ; whilst in the dielectric we shall now have 
4> — f I- > /( - k c l- , -«_i 
and 
(109), 
F =-if 
d 
dx 
oVl+i (jr)pad 
2 «— 1 
2 n+ 1 . 2 n + 3 r “ +lw ' ’dx " 
+ terms in . (HO), 
with symmetrical formulae for G, H. The symbols <E>», <E>_„_ 1} D. tl , 0._ n _ y stand for solid 
harmonics of the algebraical degrees indicated by the suffixes. The foregoing expres¬ 
sions make 
. i. 
