394 Lord Rayleigh : Problems in 



The solution is 



B=B /„O) (76) 



where B is independent of z and 



/M _i , n(n + l) (n-l)n(n + l)(n + 2) 

 JnK^J— L "i 27^ + 2.4.,s 2 " ■■■'» ^ ' 



as may be verified by substitution. Since n is supposed 

 integral, the series (77) terminates. For example, if n=l, 

 it reduces to the first two terms. 



The solution appropriate to the exterior is thus 



rv = B 8 n e^e-^^( i P)fn(^p ir )- • - • (78) 

 For the interior we have 

 rv^AoSneW* {e-r »/(ip)f H (ihph r ) —e r */(ip)f n ( — iip*r)}, (79) 



which may also be expressed by a BesseFs function of order 

 n + J. _ 



Jn like manner we may treat the problem in two dimensions, 

 where everything may be expressed by the polar coordinates 

 r, 6. It suffices to consider the terms in cos n6, where n is 

 an integer. The differential equation analogous to (69) is 

 now 



d 2 v 1 dv n 2 . /OAN 



d? + rd-r-? V = t P V ' • • • • < 8 °) 



which, if we take r^/(tp) =z, as before, may be written 



i^_ ("-*X»+i) (_■,,) =sip> . . (81) 



and is of the same form as (69) when in the latter n — \ is 

 written for n. 



As appears at once from (80), the solution for the interior 

 of the cylinder may be expressed 



v = A cos n6 e** J ' n (i s/2 p lf2 r), . . . (82) 



Jn being as usual the BessePs function of the nth order. 

 For the exterior we have from (81) 



rh = B cos nO && e~ r *Kw)f n _i (iipi r), . . (83) 



wh 



>re 



f , ( e) -1 , ^- V (4n»-l)(4n»-ffl) 



n-i w - 1 + T -y F + - 1>2>(ar) , 



(4»»_ P)(4n*-;5 2 )(4» 2 -5 2 ) 



