IN PHYSICAL ASTRONOMY. 
603 
r d R a d R 
d R 
= —iR* 
dr da d A 
Multiplying; by means of Table II. Phil. Trans. 1831, p. 238, we find 
a d R 0 
R — 
ad/?, 
d a 
2 d a 
a d R q 
3 a d R 
Jo 
2 d a 
2 d a 
a d R 3 
i R 
3adft, 
5 i /?, 
2 d a 
C ilj 
4 d a 
4 
a d R 4 
-f“ i -R 4 — 
Sadi?! 
5 iR, 
2 d a 
4 da 
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 
, j _ a d Koo 3 a d R a 17 a d i? 3 71 a d i? 0 
38 — 2 da 4 d a 16da 24 d a 
I find on reference to the development in question 
■^38 — 
24 a 3 
Ran — 
16 a 3 
R s — 
a- 
8a y 3 
R,= 
2 a 3 
whence 
Ro = 
a- 
4 ~a} 
d R i0 _ a°- 
da 8 a 3 
a d R s _ a 2 
da 4 a 3 
a d ftp _ a g 
da a 3 
ad R 0 _ a- 
d a 2 a) 
which values satisfy the equation above, for 
4 _ 1 3 _ 17 71 
24 2-8 4-4 16 + 24 -2 
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. 
