754 



KEPORT — 1898 



If, therefore, F^ be expanded in a series of sines, as indicated in (3), the sine 

 functions may by (4) be expressed in tesseral harmonics, and each coefficient of 



n + 'J 



Tn" in the final representation of V -will only depend on — j— ^ ; coefficients of the 



sine series if n is even, and on 



n-l 



coefficients if n be odd. Thus, the first 



coefficient, which is of the third degree, only requires one coefficient in the series 

 of sines, and will be independent of all the others. 



For the simplest case a- = ; equations (4) resolve themselves into the well- 

 Icnown ones : 



!sin. = P.-|p,-J^P,-. 

 ^sin2. = 3P,4P3-f^P,. 

 |sin3. = 5P.4Jp, 



(5) 



-sin4^ = 7P3-^-P- 



Hence, to expand a function into zonal harmonics, we may, if chief attention 

 is to be directed to the first few terms, express it in a series of the sines of multiples 

 of 6, and substitute the above values. P„ only occurs in the first equation, Pj 

 only in the second, P2 only in the first and third, and so on. 



We may show in a similar manner that if a is odd, we must use (2) (the 

 cosine form of the series) for F„. 



For the expansion of 



T''8in(9 = sin''+i ^ 



= a„+i cos (w + 1) 5 + ff„_i cos {>i — l)6+ ... 



leads to the conclusion that 



f+i 

 Tn" COS pddfx = when p>7i + l and when p + ?? is even. 



We find thus for o- = 1 



^cos^=|t,'+|^T/+ ... 



(6) 



TT 



— 00825= -3T,'+^T/+ — T-' + 

 ir ' 4 ^ 128 



??cos35= _5T/+15t/+ . . . 

 jr 16 



^ cos 46= -7X3'+ J^T/+ ... 

 277r =* 20 " 



where T/ will only occur in the first and third equations ; and generally a term 



of degree n will occur in - equations if w be even, and in -— — equations if ti be 



odd. 



In the case of the magnetic potential the quantities which are directly observed 

 are the forces which are connected with the potential V by the relations 



X = ' 



dV 



Ysin<9: 



dY 



dd " ' dX 



the radius of the sphere being, for simplicity's sake, taken as unity. 



(7) 



