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



Interchanging r and a in equation (152) we find that, if r > a 



~*+P^+ (153). 



Va 2 -2arcos<9 + r 2 r 



We have already seen that the functions J?, 7J, ... are surface harmonics, 

 each term of the equations (152) and (153) separately satisfying Laplace's 

 equation. The equation satisfied by the general surface harmonic 8 n of 

 degree n, namely equation (136), is 



Sn 



In the present case P n is independent of <, so that the differential equation 

 satisfied by P n is 



or, if we write p for cos 8, 



This equation is known as Legendre's equation. 

 247. By actual expansion of expression (151) 



a \ a a/ .>\ a 



so that on picking out the coefficient of r n , we obtain 



Thus F^ is an even or odd function of p, according as n is even or odd. It 

 will readily be verified that expression (155) is a solution in series of 

 equation (154). 



Let us take axes Ox, Oy, Oz, the axis Ox to coincide with the line 6 = 0, 

 then ^r r cos 6 = x. Then it appears that P n r n is a rational integral function 

 of x, y, and z of degree n, and, being a solution of Laplace's equation, it must 

 be a rational integral harmonic of degree n. We have seen that there can 

 only be one harmonic of this type which is also symmetrical about an axis ; 

 this, then, must be P n r n . 



248. If we write 



(a 2 

 we have, by Maclaurin's Theorem, 



