THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 25 
Multiplying the series [4 6 + 6. cos 6 + b®. cos 20 + 6”. cos 36 + ete. ] 
by [5 B® + B® cos (e+: Q) + B®. cos 2(e + Q) + ete.], 
noting that 6 = Q@— ¢, and arranging the terms with respect to cos7#, sin 7, 
we find 
@'\ — i KO) AO 1 D)  {O 
(“) = 130, 6 +B, 459.6 
A 
4 fb. © + (B© +b) cD + (6 +4 B®) c] cos 6 
a |f + (6 — 6%) s® + (6 —}®) s@] sind 
+ [62.6 + (6M + B®) c& + (6 + 6) c®] cos 26 (31) 
tall + (6 — B®) s+ (6 — 6%) 8] sin 26 
+ [6%. 6 + (6 + B%) c + (6 + B®) e®] cos 36 
FE $24) 8-4 (HY —B) 9°7] sin 30 
a= ete. ete. 
Now let 
i; cos K; — £9, 6 aL (G52 = Hem) ree) + (Gee) + Be) ce) ) (32) 
k,sin K; + OBEY) 5 4 GH) 5 J 
and we find 
(“) = k; [cos K;. cos 20 + sin K;. sin. 76] 
= k,cos (#0 — K,) = k;. cos («@ — ve’ — Ki). (33) 
Subtracting and adding the angle 2g, this becomes 
(5) = k,cos|¢(Q—g)—K + (ig—t’) | 
— k,cos [*(Q@—9) —K.| cos.¢(g—e’)—k;. sin | i(Q-9) — K;| sin.7(g —e’) (34) 
If we put 
(©) 9 a: 
A, . = 7 i, C08 [(O.=9) = all | 
5 : (35) 
Ae esi (0) eel ] 
A. P. S.=— VOL. XIx. D. 
