394 VIII. NEWTONIAN POTENTIAL FUNCTION. 



Examples. Differentiating the arbitrary homogeneous function, 

 and determining the coefficients, we find for n = 0, 1, 2 7 3, the 

 following independent harmonics: 



n = a constant, 

 n = l x, y, e, 



n = 2 x 2 y 2 , f , xy, yz, zx, 



n = 3 3x*y-y 3 , 3x 2 2-z 3 , 3y 2 x-x*, 3y*0-0*, 3z 2 x-x 3 , 3* 2 y-y 8 , xyz. 

 If we insert spherical coordinates r, ft, <p y 



x = r sin # cos tp, 



y = r sin # sin <jp, 



3 = r cos -9* 

 the harmonic V n becomes 



V n = rY n (fr,<p) 



where Y n is a homogeneous function of the trigonometric functions 

 cos &, sin # cos <p, and sin # sin 9. Y n being the value of V n on the 

 surface of a sphere of unit radius, is called a surface harmonic. 

 The equation Y n = represents a cone of order n, whose inter- 

 section with the sphere gives a geometrical representation of the 

 harmonic V n . 



If u and v be any two continuous functions of x } y, z, 



d*(uv) d*v , ndu dv , d*u 

 s . ; = u ^- 9 -f 2 o Q h v 05- 

 dx* dx* ' dx dx ^ ox* 



86) 



Put u = r m , and since 



2~ 



.m 2 



- = mr 7 " ~'x 

 x 



we get 



87) z/(r m ) = 3mr m ~ 2 + m (m 2) r m ~* (x 2 + y 2 -\- z 2 ) 



If V n is a harmonic of degree n, 



88) z/(r m F) = ^ 



by virtue of equations 84) and 85). 



