228 Methods for the Solution of Special Problems [CH. vin 



We have to find a potential which shall have, say, unit value over r = a, 

 and shall vanish over r = b + e^. Assume 



V=~ 



when B and D are small, then we must have 



These equations must be true all over the spheres, so that the coefficients 

 of P and the terms which do not involve 7J must vanish separately. Thus 



+ a-l0, ^ + Da = 0; 



T+O-o. -++^-* 



From the first two equations 



b -a 



A= ^' 



and this being the coefficient of - in the potential, is the capacity of the 



condenser. Thus to a first approximation, the capacity of the condenser 

 remains unaltered, but since B and D do not vanish, the surface distribution 

 is altered. 



FURTHER ANALYTICAL THEORY OF HARMONICS. 

 General Theory of Zonal Harmonics. 



268. The general equation satisfied by a surface harmonic of order n y 

 which is symmetrical about an axis, has already been seen to be 



() ............... (182). 



One solution is known to be P^, so that we can find the other by 

 a known method. Assume S n P n u as a solution, where u is a function 

 of p. The equation becomes 



and, since P n is itself a solution, 



Multiplying this by u and subtracting from (183), we are left with 



