IN PHYSICAL ASTRONOMY. 
603 
r d R _ a d R 
dr da dA — 
Multiplying by means of Table II. Phil. Trans. 1831, p. 238, we find 
R, 
a d R n 
d a 
R 3 =- 
adR, 
2 d a 
— i R l 
R,= - 
adR, 
2 d a 
+ i-R, 
a d Rj _ 3 a d R 0 
2d a 2 d a 
2 Rj — 
ad R s . j, 3 ad R { 
2 d a 1 J 4 d a 
5 i R x 
4 
2 R io — 
a d R 4 . „ 3adR x . 5iR, 
2d a 4 4da 4 
These equations may be formed at once from the Table by inspection, taking 
care to write R with the sign + in the term multiplied by i when the index is 
found in the upper line in the Table, as in the case of the argument (10) ; and 
with the sign — when in the lower, as in the case of the argument (9). The 
term multiplied by always takes its sign from the factor arising from 
. In what precedes, i is any positive whole number. 
By means of the Tables, any term in R depending on the eccentricities may 
be found at pleasure, and the development given in the Phil. Trans. 1831, 
p. 263, may be verified with great facility ; thus 
^ r _ _ a d R„ 0 _ 3 a d jR 8 _ 1 7 a d R^ _ 71 ad R a 
3b 2da 4da ]6da 24 da 
I find on reference to the development in question 
R 3S — 
24 a, 3 
R >o — 
16 a, 3 
R« = 
8a ( 3 
R„ = 
whence 
2 a, 3 
, d R m _ a 2 
da 8 a ( 3 
adR s a 2 
da 4a ( 3 
ad R„ 
da 
ad R 0 
d a 
which values satisfy the equation above, for 
4 _ 1 3 _ 17 71 
24 2-8 4-4 16 + 24 -2 
Rn 
a- 
4 a~ 3 
a- 
2a^ 
By successive substitutions in the expressions which have been given, it is 
* This is only a method of notation as regards the coefficients, which will be easily understood. 
