IN THE MOTIONS OF THE EARTH AND VENUS. 
107 
+ 
39 
n 12 a' 
— Q — q t 
(1 3 n' — 8 n) 2 ' (13 n 1 
+ 
2 p 
cos {13 (ri t + s') — 8 (n t + s)} 
The terms added to N 't + E' constitute the inequality in the epoch. 
52. The values of the elements for 1750 and their annual variations are 
given by Laplace in the Mecanique Celeste, 2 me Partie, Livre 6, N os 22 and 
26. To give them the form necessary for our purpose, we must from the varia¬ 
tion in a Julian year deduce the variation for a unit of time. Now a Julian 
year is (nearly) the time in which the angle ri t increases by 2 t ; its expression 
is therefore Consequently if we multiply the annual variations by we 
• e • • • • • 7l! t 
shall have the variations in a unit of time : and if we multiply them by (T ^, we 
shall have the variations in the time t. With regard to the quantities [ri, &c. 
introduced by Laplace for the purpose of altering his assumed masses if neces¬ 
sary, it may be observed that the only planet which materially affects the 
changes of the elements, and whose mass is known with certainty to require a 
change, is Venus herself. The investigations of Burckhardt and Bessel lead 
to the same conclusion as my own (Phil. Trans. 1828), namely, that the mass 
8 1 
of Venus is -y- X the mass assumed by Delambre, or -j of the sun’s mass. 
1 | uj 
Laplace supposed it of the sun’s mass: the comparison of these gives 
Laplace’s [ri = — ,045. In using Laplace’s expressions, therefore, I shall sup¬ 
pose [ri = — ,045, and [ri ', [ri", &c. = 0. For convenience, the centesimal * 
division will be retained. 
53. Thus we have 
6501Q8000 
~ 39oyy3ooy 
X ri 
e' = 0,01681395 — 0,0000000729 X n't 
e = 0,00688405 — 0,0000001005 f X ri t 
f = 0,02960597 + 0,0000000172 X rit 
™'= 109^,5790 + 0,0000091017 X rit 
* Borda’s tables, published by Delambre, have been used in these computations, 
f The variations of the elements of Venus do not agree with those of Lindenau’s tables. 
p 2 
