THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 51 
2 
For (") we have 
Gy = 1+ de?— 2¢ cos e + de cos 2e 
a 
But 
d [(r* : de 9 . 
"\ = % 5 = 22 sin 
a (3) 2e sin e (1—e cos « ) ay sin ¢, 
and 
“ (0) Oa - (1) 8), (2) o>. 
am eel wh oh sing +3[ Jy + J, | sin 29+ 4 | Jn + J;, | sin 3g + ete. 
Multiplying by 2e.dg we have for the integral of i (“) 
” 26 (0) 2) Qe (a) (3) j Qe (2) (4) 
= 0 J, th cos g——| Joy. + J, | COS A J; +3, |cos3g— ete. 
where c= 1 + 3e’. 
By means of (22) this becomes 
r\2 (1) (2) (3) 
(“) =1-+ $e— 4J, cos g —4J2 cos 2g — 4; cos 3g — ete. 
a 
In case of (om we have 
3é . cos «= $e (1 + cos 2e), 4¢ cos *e = ¢' (3 cos ¢ + cos Be), 
be’. cos *e = 3e' (3 + 4c0s 2c + cos 4e), 6e?. cose = 38° (10 cos e + 5 cos 3e + cos 5e), 
Té° cos °e = x5e° (10 + 15 cos 2e + 6 cos 4e + etc.) 
and hence 
(“)°H1 +438 + A + We + ote. 
+ [2e + 3e’ + $8 + ete.] cos « 
+ [3e + 20¢! + 195¢° + ete.] cos 2e 
+ [é + 2% + ete.] cos 3e 
+ [Be + 42e'+ ete.] cos 4e 
