IN PHYSICAL ASTRONOMY'. 
605 
iL,= - -L( (26 i — 30 i 2 + 8 i 3 ) R { + (9 - 2 7i+ 12 i 2 ) a — 
48 l da 
+ (6 i — 6) 
a 2 d 2 R, a 3 d 3 J t 2 1 
d a 2 d 
1 M 
a 3 / 
which agrees with the expression given by Burckhardt for (7J/ (0) ), Mernoires 
de rinstitut, 1808, Second Semestre, p. 39. 
Similarly 
_ a- d- i?,,, (2 i — I) ad R iq , (4i 2 — 5 i) 
— 4 d a 2 2 da 4 
_ a/ 2 d 2 i2, , (2 i — 1) a, d , (4 i- — 5 i) R t 
lj 4 da, 2 2 da, 4 
If i = 2, 
2 -^51 — 
2R 
19 
R, = - 
a-d-R l0 x 3 ad R ]0 ^ 3 p 
4 da 2 + 2 da + 2 19 
a, 2 d 2 -R, 3 a, d i? , 3 „ 
4 d a, 2 2 d a, 2 1 
t,, s _ 3a s 3.5 a 4 _ 3.3.5 .7 as 
a, 4 a, 3 2 . 4 . 6 a/ 2.4.4.6.8a, 7 
— &c. 
In the Lunar Theory, the higher terms may be neglected; and taking 
3 eft 
*. = - Sr- ^ is evident that -^19 and R bl are each equal to zero. This 
4 a, 
theorem, however, cannot be extended to the other terms, and therefore in the 
Planetary Theory the coefficient corresponding to the argument 2 t — 2 x-\-2 z 
or 2 — 2 to-,, in the development of R, (which term is important as regards the 
secular inequalities,) does not vanish. 
If the coefficients of the wth argument in the expressions for and a 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 
2d7 2 
-ii-t-ii+2/dR + r— = 0 
raj dr 
and 
^ = A_l /A5 dI 
d t r 2 r\/ d A 
d 2 r 3 $— 
T 
d t 2 
+ 2 J* dR + 
rd R 
dr 
4 i 
MDCCCXXXII. 
