Conducting Sphere on Magnetic Induction. 167 



The most general solution of these equations is given by 

 (c/. Recent Researches, art. 370), omitting the time factor, 



P = 2 { (n + 1)/ M _x(*,) ^f -«*V»+%+i(*r) ^ ^} 



+V.(^)(y s -*^ • ■ 



(2) 



Q and R satisfy similar equations. &)/, w n are arbitrary 

 solid spherical harmonics of degree n, and f n (kr) satisfies 



d% 2Q + 1) <#. 



r/r 



+ 



-f-£% = 0. 



(3) 



In the sphere r can vanish, and we must take that solution 

 of (3) which does not become infinite when r is zero, 



, ,, . / 1 d \ n sin kr 



''< f*W= {**!£) ST 



also 1 d.f n _ (4) 



kr d . kr ~ /n+1 

 *V/ w+1 +(2ra + l)/ M +/»-i=0 



Now the currents induced in the sphere are entirely due 

 to magnetic forces outside the sphere, therefore (Recent 

 Researches, art. 319) there can be no radial currents in the 

 sphere, and therefore the radial electromotive intensity must 

 vanish. 



Hence the origin being at the centre of the sphere, we 

 must have 



•-P+^Q+*R = 0; 

 r r r 



therefore from (2) 



^— } {/;-i(*'-) + *v/n + i(*»-) w=o, 



w(w + l)(2n + l) 



f n {kr).<o n '=0 



therefore ooJ — O. 



Hence, restoring the time- factor, 



p=2/ " (Ar) (4- 2 |) Mrf "' 



(5) 



N2 



