IN PHYSICAL ASTRONOMY. 
33 
R = - (r + -*rl! / I + 3 cos (2 X - 2X, + 2 S\) \ 
4 r/ L J 
_ _ { 1 + 3 cos (2 X — 2 X,) } / 2 r $ r + Jr 2 ! + — sin (2 X — 2 X,) 5 r 5 X 
4r, 3 *- J r i 3 
+ Al!(cos(2X-2X / ) (Jx) 2 
Z r t 
Srss + r 3 
Neglecting the terms multiplied by & -A and hx, 
_ _ 3 r 2 + 3 cos (2 X — 2X,)} j 2 _ 
3 r 2 
R=-3r' 1 -J -'^T sm(2X-2X ; ) (rJ-L)iX, 
+ 4 Acos(2X-2X ; ) (Jx) 2 
- r i 
d R and r (Vp) ma Y be obtained from R as before. 
d R 3 r 2 • /0 , 0 , 
-=-- sin (2 X — 2 X,) 
dX 2 r, 3 V " 
dR = 3 {2r3r + *r 2 } sin ^\-2X l ) + 6 cos ( 2 \ , 2 ^ r $r$\-^sin (2X-2X,) (Jx) 2 
r / J r t r t 
Neglecting as before the terms multiplied by § A and l X, 
= A-4 Sin (2X-2X,) ( r ay-- 6 ^ cos . ^- 2 ^ (tJ-L^X 
-^4 sin (2 X — 2 X ; ) (Sx) 2 
Retaining the terms depending on the cube of the disturbing force, 
d 2 r 3 S -4 3d 2 r 4 
d 2 r- ____ 
adl 2 dr 2 
H)‘ 
2 d 2 r 5 
2 d r- 
OlV 
A_il_ J £ +2 yi R + r (^) = o 
Fortunately this series does not appear to contain the quantity {^d«| 
MDCCCXXXII. 
