126 ON THE ORBITS OF THE ASTEROIDS. 
These equations may be so simplified as to materially facilitate the computation of 
the required quantities. The following equations are given by Laplace in § 49. 
p 
= a) 
Ia G—D Q4) MC? tee tts 
5.3 (i—s)a 
—3b — A 
ECE (0) . 
= Dé D (1—a ya r4 4 
By a comparison of these equations with the values of (0.1) and [0.1] given above, 
the latter reduce to 
s 
WE) 
al = "2^ e s [05] = 222 a oP. (3) 
With these equations we can obtain the numerical values of (0,5) and [0,2] with 
great ease, provided that we have tables which give the values of AO for different 
values of a, such as those published by Runkle in the Smithsonian Contributions to 
Knowledge. 
To integrate the equations (1) and (2), we shall suppose all the accented quantities, 
V, U, K, &c., p', &c., d, &c., given in terms of the time. This is admissible, because we 
neglect the action of the asteroids on the larger planets, and also on each other; we 
may, therefore, use the expressions for the elements of the larger planets in functions 
of the time, which are obtained in neglecting the action of the asteroids. These ex- 
-pressions are of the following form: — | 
M = N, sin (gt + 8) +, sin (gt + 8) + N, sin (get + &) + &e. 
| = N', cos (gt + 8) + IN, cos (811 + 8) + No cos (gt + &) + ec. 
h" = Ny sin (gt + 8) + N sin (gt + fi) + &e. > (4) 
i" = N",sin (gt + 8) + IN", sin (git + p) + &c. 
« &c. &c. 
p' = M sin y + M, sin (kt + 7) + M^ sin (Lat + ya) + &c. 
ql = M cos y + Mi cos (kit + 71) + Mi cos (k, t + 73) + &c. 
p" = M sin y + M", sin (t + n) + &c. (5) 
&c. &c. gon "n 
In the second members of these equations, all the symbols, except t, represent Kees 
constants. If we substitute the expressions (4) in (1), and put for brevity 
E, = [0,1] IN, + [02] N" + [0,3] IV", + &e. 
B= [0.1] Ni + [02] IP, E [038] 1, + ae. E 
&c. &c. | eat 6) 
b = (04) + (02) + &c. 
