based upon Electromagnetic Theory. 



573 



of the circular motion may now be written from (3) as 



follows : — 



F = 



{ — x -f-ajSi — a 3 S 3 cosa) — R/8 8 2 cos « 



*i * 8 (l--A 2 ) 

 (RA 2 )*(1-/V) t 



— AA [(C,C 2 cos a + S 1 S 2 )(— oj + axS, — a 2 S 2 cos a) 



- a 2 S r C 2 2 cos a 



+ Ci r 2 S 2 1 C 2 COS 2 a -f- OjOs^COS a— ^C^ COS a] i 



+ [(-y+aiO ] -a,C s ) + RAS 2 -i9iA[(OiC a co<« 



+ S 1 S 2 )(-y + a 1 C 1 -a 2 C 2 ) 

 + a 2 S]S 2 C 2 — a 2 S 2 2 C] cos a — S 2 Oi# + SiS 2 y] j 



-f (-«-a 2 S a sina) — R/S 3 C 2 sina — ^^[(Ci'OsCOSa 



+ S!S 2 )( — ^ — a 2 S 2 sina) 



— ajSiOi 8 sin a + a 2 S 2 C]C 2 sin a cos a 



■f OiCV^sin a — SjCgysina] # [ • 



. . . (20) 



The quantity RA 2 , which occurs in the coefficient of (20), 

 may be found in a similar manner to RA! above, giving 



RA 2 = R — q 2 . R/c = R — /3 2 [ — C 2 d' cos a + S 2 y — C 2 z sin a 



+ a 1 Si.G 2 cos.a— %S 2 Ci]. . (21) 



For brevity, write v for the quantities within the last 

 bracket of (2.1), giving 



RA 2 =K- j3 2 v, (22) 



whence (RA 2 ) ~ 3 = (R-£ 2 v) ~ 3 , .... (23) 



and expanding into series 



2T}-5„2 



(RA 2 )- 3 = R- 3 + 3ftR- 4 r+6/3 2 2 R 



+ 10/3 2 3 R- V+ 15/3 2 4 R- V. . . . (24) 



The powers of R may be obtained from (13) in terms of 

 u as follows : — 



R- 3 = s- 3 (l-3u/2 + 15w78- 35w 3 /16 ) . (25) 



R- 4 = 5- 4 (l~2w + 3ti 2 -4w 3 ) .... (26) 



R- 5 = *- 5 (l— 5m/2 + 35w 2 /8- 105it 3 /16 ) . (27) 



R- 8 = «- 8 fl-3u + 6u s -10u 8 ...-.) (28) 



