222 Methods for the Solution of Special Problems [OH. vm 



then, by equation (171), 



2n 4- 1 f fl = 

 a n = 5 I aP n (cos 0) d (cos 0) 



^ ./ = TT 



1 /"0=0 



=- o-o ^ (cos (9) d (cos 0) 

 J e=a 



(cos a) - P n+l (cos a)} 



by equation (165), except when n = 0. For this case we have 



Thus 



= J<7 (1 cosa)+ 2 jlJU^cosa) ft +1 (cosa)j-ft(cos^) . 



It is of interest to notice that when 6 = a, the value of a- given by 

 this series is O- = ^CT Q , as it ought to be (cf. expression (172)). 



The potential at an external point may now be written down in the 

 form 



......... (176), 



and that at an internal point is 



......... (177). 



On differentiating with respect to a, we obtain the potential of a ring 

 of line density o- acfa. At a point at which r > a, we differentiate ex- 

 pression (176), and obtain 



Fm'n a. n=<*> //^\ n +i ~~\ 



V = 27raer da - + 2 ^ (cos a) sin a ( - ) P n (cos (9) , 



L ^ n=l v / 



or, putting ao- c?a = r and simplifying, 



P n (cos0) ............ (178). 



Obviously the potential at a point at which r < a, can be obtained on 



i f a \ n+l v, ! r \ n 

 replacing by . 



260. These last results can be obtained more directly by considering 

 that at any point on the axis = the potential is 



v _ 27rar sin a 



Vr 2 + a 2 2ar cos a ' 

 or, if r < a, 



T 27rcw sin a = /r\ % 



F= -- - - 2 ^(cosa) - , 



