IN PHYSICAL ASTRONOMY. 
33 
R = - ( r + ( 1 + 3 cos (2 X - 2X, + 2 SX) \ 
4 rp L J 
_ _ { 1 + 3 cos (2 X — 2 A,) } f 2 r S r + S r- \ + — sin (2 X — 2 X ( ) S r 5 X 
4 r 3 L J L 3 
+ A4(cos(2X-2X ( ) (Jx) 2 
Z T t 
Sr == -r 2 $-|- + rs^J-iy 
Neglecting the terms multiplied by ci -|-and lx, 
_ _ 3 r i (l + 3 cos (2 X — - X,) } s 2 _ 3_r^ S J 
R=- “'TT sm (2 X 2 X ; ) (rS-^X, 
+ |^cos(2X-2X ; ) (Sx) 2 
d R and r (j-r) may be obtained from R as before. 
~ = All sin (2 X — 2 X,) 
dX 2 V " 
d^ _ 3{2r5r + 5r 2 } . , 0 O N 6 cos (2 X — 2 X.) } ,, 3 r 2 . ~ . . 
dT“ — - sm (2X-2X ( ) + lirSrSX- — sm (2X-2X,) (*X) 2 
‘i ‘ t T i 
Neglecting as before the terms multiplied by ^ and S X, 
(2 X — 2 \) 6l!£££l 2X-2XP 
9 r°- 
T r 7 
sin 
3 r 2 
sin (2 X — 2 X ; ) (£x) 2 
Retaining the terms depending on the cube of the disturbing force, 
d 2 7- 3 $4r 3d 2 r 4 /j. — ^ 2d 2 r 5 ^S— V 
-*■ -V ’>■ ?)=0 
2 d f* d 
2 d i 2 
dX' 
d £ 
_ h J , 1 r&R , 1 / />dR ,.l 8 l.l / RdR , f l 4 
i ~?s{ xy dT di + 2S=iy di 1 '* 1 } -^rsiy av d ‘/ 
1.1.3 f ^RaA 6 , o 
- 2 ^ 6^17 dA' d< / +&C - 
Fortunately this series does not appear to contain the quantity j 
MDCCCXXXII. 
