96, 97] NEWTONIAN POTENTIAL FUNCTION. 187 



If ^! be any constant direction whose direction cosines are 



) /i \ 



and 37- ( - ) is a harmonic of degree 2, and to it corresponds the 



Ofli \T / 



harmonic, 



<" *- 



which is of the first degree. Since Zj a + mf + n^ = 1, the harmonic 

 contains two arbitrary constants, and multiplying by a third, A, 

 we have the general harmonic of degree 1, in the form 



If in like manner h 2) h 3 ...... h ny denote vectors with direction 



cosines 1 2) ra 2 , n 2 ...... Intent n n> 



_a_ _a_ _a /i\ 



a/ij dh 2 " "dh n \r) 

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



( 12 \ V -r"* 1 ^ 



dh.dh," ~dh n (r)> 



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

 constants, multiplying by another, A, gives us 2n 4- 1, and we 

 have the general harmonic of degree n in the form, 



Jr . a a a /i\ 



(I3) ...... 



The directions A 1? /i 2 , ...... h n are called the axes of the har- 



monic. To illustrate the method of deriving the harmonics we 

 shall find the first two. 



/ lx 

 = A ^(~^ 



89 



/ 7 a a a\ /, a a a\ /i 



f tj^- -l-.wii ^- + *h 5- U 2 5- + ^2 5- +^25- - 



\ 3* ay fa) \ ox oy dz/ \r 



