184 THEORY OF NEWTONIAN FORCES. [PT. I. CH. IV. 



vv vv vv_ 

 8* 2 + 8^ + S?"!"- 



ar ar 8F 



(2) fl?-5- + y^-+s-5-=wF. 



3# " 9y 9.3 



The general homogeneous function of degree n 



+ a 2/ n 



contains 1 + 2 + 3 +rc + l = (+l)(n + 2)/2 terms. The sum 



of its second derivatives is a homogeneous function of degree n 2 

 and accordingly contains (n 1) n/2 terms. If the function is to 

 vanish identically, these (n 1) n/2 coefficients must all vanish, so 

 that there are (n l)w/2 relations among the (n + l)(n + 2)/2 

 coefficients of a harmonic of the nth degree, leaving 2^ + 1 

 arbitrary coefficients. The general harmonic of degree n can 

 accordingly be expressed as a linear function of 2^ + 1 inde- 

 pendent harmonics. 



EXAMPLES. Differentiating the arbitrary homogeneous function, 

 and determining the coefficients, we find for ?i = 0, 1, 2, 3, the 

 following independent harmonics : 



n = constant 



n = l x, y, z 



n=2 # 2 - 2/ 2 , 2/ 2 -2 2 , x y> y z > zx 



n = 3 3^y-2/ 3 , 3ate- z s , %fx - a?, %fz - z\ 



3z*x a?, 3z*y y 3 , xyz. 



If we insert spherical coordinates r, 0, <j>, 

 x=r sin 6 cos 0, 

 y = r sin 6 sin </>, 

 = r cos 

 the harmonic V n becomes 



