IN THE MOTIONS OF THE EARTH AND VENUS. 
107 
t- ^ { (13 + (L3 } cos { 13 (»' t + s') - 8 (» < + ,) } 
The terms added to N ' t + E' constitute the inequality in the epoch. 
52. The values of the elements for 17*50 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 n t increases by 2 nr ; its expression 
is therefore — . Consequently if we multiply the annual variations by we 
fjJ t 
shall have the variations in a unit of time : and if we multiply them by we 
shall have the variations in the time t. With regard to the quantities [*', &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 
B 1 
of Venus is X the mass assumed by Delambre, or 401£11 of the sun’s mass. 
1 | uj 
Laplace supposed it 3^ 37 of the sun’s mass : the comparison of these gives 
Laplace’s /// = — ,045. In using Laplace’s expressions, therefore, I shall sup- 
pose (Jj' = — ,045, and p, p", (jJ", &c. = 0 . For convenience, the centesimal * 
division will be retained. 
53. Thus we have 
650198000 . , 
71 ~ 099993009 X U 
e [ = 0,01681395 — 0,0000000729 X n't 
e = 0,00688405 — 0,0000001005 f X ri t 
f = 0,02960597 + 0,0000000172 X n't 
nr'— 109s, 5790 + 0,0000091017 X n't 
* 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 
