MOTIONS IN A SPHERICAL CONDUCTOR. 
545 
a=-?Uir)[yj-zj y 
terms in 
with symmetrical formulae for b and c. 
The continuity of </> at the surface requires 
(no. 
.( 112 ), 
when r=R. The continuity of F, G, H requires 
—^+(^+ ^■) x l J n-l{^>) Q, » 
= —^+( n + 1 ) x l J H-i{jR)C l , l + (n+ 1 ) 2n-1.2n+l^- H ^^ n - u ~ 1 ’ ' ( 113 )’ 
and 
7»2-P2 <J) ,;2T>2 
- V+i. 2»+3 ,,, * +i( * b) “* = —ir~ V+i.2 B +3 ,;, - +1 5 , K)Q.-"’/'-.- 2 (iR)n—i ■ (m)- 
Adding (113) and (114), and taking account of (112), we find 
{&fh//„(£It) -f- (n -j-1 )i//„ (Alt)} (o n 
— {j R) + (^+1 )*/lOR)} R«+ {jR'p'-n-\{) R) — nip-n-iUR )} (115),'“' 
where some reductions have been effected by means of (29) and (30). 
The continuity of a, b, c requires 
PR%(A:R)6)„=/R,%(yR,)0^+/It 3 ^„_ 1 (yR)fl_ w _ 1 . . . . (116). 
12. Let us now apply the foregoing results to the case of free motion. A certain 
relation must then hold between the surface values of X ;i and X_ /i _ 1 , and also between 
those of VL a and 0_„_ 1} viz.: a relation expressing that the disturbance at infinity in 
the dielectric is finite. It may be shown that F, G, H are determinate wlren the 
values of acF-f-yG-f-sH and of xa+yb+zc are known at every point of space. Now 
in the first type we have rrF+yG+cdI=0, and 
xa+yb+zc=—n.n+l.{ilj H (jr)X >l +xlj_ n _ 1 (jr)X_, l _ 1 } . . . (117). 
For large values of r we have 
'l'n{jr) = (—) n .3.5 ... 2n+l. 
sin (jr + n^ 
U r ) 
?£+l 
(118). 
Equations (109) and (115) express that the tangential components of current just outside and just 
inside the sphere are in the ratio of j~ to /c 2 . This may also be easily deduced from the fundamental 
equations (18) and (21). 
