1902.] 



On Spherical Harmonic Analysis. 



101 



the longitude, the results of the investigation may be put into the 

 form 



r> = rt + l 



f n = 2 when n is even, tr odd. and p odd. 



p = o 



= «+i 



= 2 ,, n „ odd, o- „ „ 7; even. 



j» = i 



2> = 1 

 = + l 



= 2 hi si „ w „ odd, o- „ „ ^ even. 



i> =2 



In these equations the factors (/^/^ are the coefficients of the 

 Fourier series (see § 2), and the quantities 1%, n% 9 771%, s%_ are 

 numerical quantities, which (as well as their logarithms) are given in 

 tables at the end of the paper as far as n = 12, o- = 12, p = 12. By 

 means of these tables the numerical work is reduced to a minimum, and 

 the coefficients of the series may be obtained as far as terms of the 

 12th degree. 



8. The proposed method is specially adapted to deal with problems 

 like that of terrestrial magnetism, in which the function to be 

 obtained as a series of spherical harmonics is not given directly, 

 but by means of its differential coefficients. The force directed to 

 the geographical north may by Fourier's analysis be obtained as 

 a sum, the terms of which have the form cos <r<f> cos pO, and sin <rcf> 

 cos p0 when or is even, and the form cos <r<f> sin pO, sin <r<f> sin pO when o- 

 is odd. Integrating with respect to 0, the magnetic potential is obtained 

 in a form such that the transformation into the series of spherical 

 harmonics may be proceeded with. A separate expression of the 

 magnetic potential is derived from the force directed to the geographical 

 east. 



n ,, even, cr even, ,, p odd. 



