232 Methods for the Solution of Special Problems [OH. vin 



Since T^_! is a rational integral algebraic function of //, of degree n 1, it 

 can be expanded in the form 



T^_i 

 so that 



= S s {n (n + 1) - (n - s) (n - s + 1)} ^_ s . 



Comparing with (187), we find that a s = when s is odd, and is equal to 



2(2n-2s+l) 



5(271-5+1) 



when s is even. 



Thus 



- - -i 3(n _ 1 -3 5 (n _ 2) 



and 



271. When we are dealing with complete spheres it is impossible for 

 the solution Q n to occur. If the space is limited in such a way that the 

 infinities of the Q n harmonic are excluded, it may be necessary to take 

 into account both the P n and Q n harmonics. An instance of such a case 

 occurs in considering the potential at points outside a conductor of which 

 the shape is that of a complete cone. 



Tesseral Harmonics. 



272. The equation satisfied by the general surface harmonic S n is. 



As a solution, let us examine 



where is a function of only, and <1> is a function of <j> only. On 

 substituting this value in the equation, and dividing by 3>/sin 2 0, we obtain 



sin0 i 



e 8 



We must therefore have 



sin0 8 



|,(sin0~) + Mrc + l)sin 2 0=-K. 



OU \. 00 J 



