206 Prof. G. M. Minckin on the Magnetic 



In the small term we can put k r2 = -7-=, -^— = — cos 6, 



and i/r = # ? so that, neglecting quantities of the order — 3, 



= 2tt-2{^E^— +£+^(K-E)sin0 

 I It cos-^ 2 2a 



+ ~[3(2K-3E) sin (9 cos + (|-6>)(2K-E cos 2(9)1 j. (1: 

 The relation between ^jr and # is given by the equation 

 ¥ sin i/r= a/ t: = tan-> where X= z. PVQ, 



.*. sini/r= ^ tan-? 



.-. sin^=sin^|l+ |- cos 6>4~ (5 cos 2 <9-l) 1, . (20) 



as far as the second order of small quantities ; and the series 

 for sin -1 x in terms of x, or rather for x in terms of sin -1 x, 

 gives, to the same order, 



>=0+^sin0+^sin0cos0, . . . (21) 



a~~" ' 2a 2 



^sin 2 0- 



2a ba" 



"R 5"R2 



■. cos^=cos6>-^sin 2 0-^cos0sin 2 0. . (22) 



Also 



^ = - cos + 5 s i n 2 61 + ^L C os (9 sin 2 0; (23) 



It _ ££ _ c 



:a* 



so that (19) becomes expressed entirely in terms of 6. Thus, 

 -p— = — ( cos i/r + — cos sin 2 ), therefore 



-S-. T = "~l l + 7T-^sin 2 ^ 



It COSiJr \ ba 2 / 



since may be put for i/r in the term of the second order ; 

 and thus we have now the equation 



0=2tt-2( (l-E)^ + E0+|^Ksin0 



+ ^[(2K-E)(|-0) + (6K-E)sin0c OS 0]}, . (U) 

 which is, however, not yet in its simplest form. 



