376 ELECTROSTATICS AND MAGNETISM. [FT. II. CH. IX. 



cylinder, so that the potential F and its derivatives are finite 

 and continuous at the surfaces of the cylinder. 



Let it be developed at the outer surface in the infinite series 

 of harmonics 



Then at points for which p < a it is given by the series, 



(2) V a =T a + ^T 1 + P lT 1 + ...... 



Ct (I- 



The potential F* is represented by three different develop- 

 ments in the three different regions, (1) p>a, (2) a> p>b, and 



(3) p < b. We will distinguish these by an affix. Since F$ 

 vanishes at infinity, we have outside the cylinder 



(3) VM = ZA n p-"T a . 







In the substance of the cylinder we must take 



(4) V^= 



o 

 while in the cavity, since Vi is finite at the center, 



(5) V^ 



o 



Since Vi is continuous, at the surface p = awe have*Fi (1) = F^ (2) , 

 and as this must be identically true for all values of <j> we must 

 have for every term the coefficients of T n equal. 



(6) A n ar n = B n a n + C n a~ n . 



In like manner, at the surface p b, we have for every term, 



(7) D n b n = B n b n + C n b~ n . 



Beside the conditions of continuity, we have at each surface of 

 the cylinder 



for the whole potential F= F + Vi. The potential of the ex- 

 ternal field being continuous, with its derivatives as well, we have 



(9) + = 0> 



