396 VIII. NEWTONIAN POTENTIAL FUNCTION. 



and any derivative of a harmonic is itself a harmonic, so that 



<~>a )/? cy 



JL y 



I ^ /? ^ v ' n J 



is a harmonic of degree -(**+ ^ Since tO VQ = C 

 sponds the harmonic F_i = -> we have 



If ^ he any constant direction whose direction cosines are 



cds (\ x) = \ , cos (^ y) = % , cos (^ *) = !, 



^+*$if*l? 



and J- ^ is a harmonic of degree - 2, and to it corresponds the 



harmonic, 



93) ^ ; 



which is of the first degree. Since ^ 2 + mf + V = l, the harmonic 

 contains ftw arbitrary constants, and multiplying by a third, 4, we 

 have the general harmonic of degree 1, in the form 



94) "i 



If in like manner ^, h s , . . . h n , denote vectors with direction 

 cosines Z 2 , m 2 , W 2 , . . l nf m n> n n- 



d 



is a spherical harmonic of degree - (n + 1) and to it corresponds 



95) ^ = ^ +1 ii-i(7> 



a harmonic of degree n, and since every h introduces two arbitrary 

 constants, multiplying by another, A, gives us 2n + 1, and we have 

 the general harmonic of degree n in the form, 



The directions ^, fc 2 , . . .'* are called the ewes of the harmonic. 

 To illustrate the method of deriving the harmonics we shall find the 

 first two. 



