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



By differentiating any harmonic F any number s of times, multiplying by r 2a ~ l and 

 differentiating again sl times, we obtain more harmonics of degree zero. 



n= -1. Any harmonic of degree zero divided by r or differentiated with respect to 

 x, y or z, e.g. 



n= - 2. By differentiating harmonics of degree - 1 with respect to x, y or z we obtain 

 harmonics of degree - 2, e.g. 



x y z z , ,y z . r + z z 

 -^> ^ ^, -stan" 1 ^-, -rjlog - , -;. 

 r 3 ' r 3 ' r 3 ' r 3 x* r 3 s r-z' r 2 



n = l. Multiplying harmonics of degree - 2 by r 3 , we obtain harmonics of degree 1, e.g. 

 X) y, 0, z tan" 1 ?, z log - 2r. 



00 T Z 



Rational Integral Harmonics. 



239. An important class of harmonic consists of rational integral algebraic 

 functions of x, y, z. In the most general homogeneous function of x, y, z of 

 degree n there are J (n + 1) (n -f 2) coefficients. If we operate with V 2 we 

 are left with a homogeneous function of x, y, z of degree n 2, and therefore 

 possessing ^n(n 1) coefficients. For the original function to be a spherical 

 harmonic, these ^n (n 1) coefficients must all vanish, so that we must 

 have ^n(n l) relations between the original J(n-fl)(n + 2) coefficients. 

 Thus the number of coefficients which may be regarded as independent in 

 the original function, subject to the condition of its being a harmonic, is 



or 271+1. This, then, is the number of independent rational harmonics of 

 degree n. 



For instance, when n = 1 the most general harmonic is 



Ax + By + Cz, 



possessing three independent arbitrary constants, and so representing three 

 independent harmonics which may conveniently be taken to be x, y and z. 



When n = 2, the most general harmonic is 



aa? + % 2 + cz 2 + dyz + ezx +fay, 



where a, b, c are subject to a + b-i- c = 0. The five independent harmonics 

 may conveniently be taken to be 



yz, zx, xy, x 2 - y 2 , a? z*. 



When n = 0, 2n -f 1 = 1. Thus there is only one harmonic of degree zero, 

 and this may be taken to be F= 1. 



Corresponding to a rational integral harmonic V n of positive degree n, 



y 

 there is the harmonic -^^ of degree (n + 1). These harmonics of degree 



