548 PROFESSOR H. LAMB ON ELECTRICAL 
xa 0 + yh Q +zc 0 =tfa(jr) ®„.(129), 
*P 0 +3/Qo+ zR o=.(130), 
where ®„, are solid harmonics of positive degree n. Now %a-\-ijb-\-zc and 
ccP-j-yQ+zR must differ from the above by terms representing a disturbance 
propagated wholly outwards. But 
oca+yb+zc= —n.n+l.{ fa ( jr ) X w + fa»-i( jr) X_„_ 1 }, 
xP+yQ+zR=—\.n.n+l.{fa(jr)n„+fa n _ 1 (jr)a_ n _ l }. 
The condition that 
should represent a disturbance travelling outwards may be shown to be 
3.5 . . . 2n+l( , X re +~ , d *\ 9 rX_„_ 1 =0 . 
\ n.n + 1 ] 1.3 . . . zn — 1 
(131), 
where in the harmonics X,„ &c., r is supposed put = R. Similarly we have, on the 
same understanding, 
3.5 .. . 2»+l[ j)+i ■ 
(132). 
The equations (107), (108), and (131) determine X„, X_„_ 1 in terms of ®„ ; whilst 
(115), (116), and (132) determine oi n , O*, 0_»_ 1 in terms of Z n . Thus the complete 
solution of our problem is effected. 
Introducing the consideration that is small, we find, in the solutions of the 
second type, 
1 Z„ 
ft«= — 
approximately, and thence 
Jc 2 fa(Jcr)(o n 
n.n + 1 X 
2n+l p 
\*.n + IX n ' 
(133), 
by (115) and (116). If cr,, denote the w th harmonic constituent of the surface dis¬ 
tribution of electricity, we.deduce 
1 cl<T n 1 2n +1 
<T n = - — = 
X dt 477-?;-R n 
z. 
(134). 
