IN PHYSICAL ASTRONOMY. 
605 
R. n = ~ ~l (26 i — 30 i 2 + 8 i 3 ) R, + (9-2 7i+ 12i 2 )a — 
4o L da 
+ (6 i-6) !££/! +?!£«.} 
a a 2 da 3 J 
which agrees with the expression given by Burckhardt for (J/ (0) ), Memoires 
de VInstitut, 1808, Second Semestre, p. 39. 
Similarly 
2 R 
51 — 
a 2 d 2 i2 lf) 
4d a 2 
(2 i — I) a d R 1q l 
2 da + 
(4 i~ — 5 i) R u , 
4 
0 p _ a/d 2 i2, t (2i — I) , (4 £- — 5i)R 
ZKl ‘- Td^~ 5 -~ + -- 
d a, 
If i = 2, 
2 = a d ' R '" + A + A 72 
51 4 da 2 2 da 2 19 
OJ? — «; 2 d 2 i?, , 3 a ( dfl, x 3 p 
-^ A~~~ T* "o- TZ + 7T n l 
4 d a/ 
R ' a, 
2 da. 
3.5 a 4 
4 a/ 
3.3.5.7 a ! _ &c> 
2.4.6a/ 2.4.4.6.8a/ 
In the Lunar Theory, the higher terms may be neglected; and taking 
/i] — — ~-, it is evident that /t 19 and i? 51 are each equal to zero. This 
theorem, however, cannot be extended to the other terms, and therefore in the 
Planetary Theory the coefficient corresponding to the argument 2 t — 2<r + 2 z 
or 2 to- — 2 rar /5 in the development of i?, (which term is important as regards the 
secular inequalities,) does not vanish. 
If the coefficients of the wth argument in the expressions for ^ and \ be 
called r n and X n , the Table which has been used for the preceding multiplica¬ 
tions may also be used (when the square of the disturbing force is neglected,) 
for the integration of the equations 
d 2 . r 2 _ jo. ju, 
2d t' 2 r a 
+ 2 \JdR + r^- 
= 0 
and 
= A_ A r**& t 
d t r 2 rj dA 
d 2 r s S — 
__ji -^$i + 2 y t dR+ 
rdR 
d r 
MDCCCXXXII. 
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