=] 
oO 
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 
Finding the expression for (ZZ) first by the method of Hansen, we let 
A= = .k.cos (I—K), hi =". COS p. Cos 9’. k,.cos (II — K) 
1=*,.cosp.k.sin (I—K), U=*,. cos 9’. k,. sin (11 — K)), 
and have, if we make use of the eccentric anomaly, 
a 
(17) = h.cos oe cos f’ —ch(“)*.cosf’ 1. sin €. ( i)? COSN i 
r 
u/ 
+ 1.cos aa) pins ats op) )\.. Sle +h’. sine GC) _ sin 
cos ¢! 7 COS ¢ 7 cos ¢! 
Putting 
\ 2 
(2) cos f’ = y’,.cos gy’ + y’,. cos 29’ + y’;.cos 3g’ + ete. 
a’\2 sin f’ ; : Fi , : , pees , 
(“) oa a= Une RING) ae 0’5. sin 2g’ + 4,’. sin 3g’ + ete. 
- ¢ 
we find 
(1) = $ (hy’, —'8,) cos (— g'—e) + 3(ly’, —U0',) sin (—g' —e) 
—ehy’, cos(— gs) + el’, sin(— g's) 
+ ally’ +884) cos (gy — 2) + 3(/'1 +15) sin(  g’—e) 
; (1) 
+ 2(hy', —h'8’,) cos (— 2g’—«) + 2(ly’.— U8’,) sin (—29' — «) 
—4.ehy’,cos(— 2g )+ 4.eld’, sin (—29’_ ) 
+ A(hy’s + h's'2) cos (  2g'—e) + Aly’. + U8’,)sin( 29’ —e) 
+ ete. + ete., 
where 
(0) (2) (0) (2) 
— y 
04 = Jy, nN 9 DP i IN? === CAN 
(1) (3) (1) (3) 
= 3] Sa » |» 2 = 3 Jn eae: » | 
ete. ete 
A. P. S.— VOL. XIX. H. 
