ON THE ORBITS OF THE ASTEROIDS. 125 
The subject may be arranged under the following heads : — 
§ 1. Computation of the rigorous expressions in terms of the time of the elements of 
the asteroids. 
§ 2. Of the possibility that the orbits of all the asteroids once intersected in a com- 
mon point. 
§ 3. Have the elements of the asteroid orbits ever been materially affected by a re- 
sisting medium? - 
§ 4. Of the relations among the mean distances, eccentricities, and inclinations of 
the orbits of the asteroids; and between their masses and the velocities wath which 
they must have been projected, if Olbers’s hypothesis be true. 
§ 9. Of certain observed relations among the asteroids which are the necessary or 
probable result of known causes, and therefore throw no light on the origin of the 
asteroids. 
$ 1. Computation of the Rigorous Expressions in Terms of the Time of the Elements of certain of the Asteroid 
Orbits, or of the Secular Variations of those Elements. 
To obtain the required expressions, we shall start from the expressions given by 
Laplace in the Mécanique Céleste, Liv. II. §§ 55 & 59. 
GE $(0,1) ch (0.2) ^ (0,3) T &c.i 1 —[0,1] P e [0,2] D — ée, (1) 
T = — 0:1) + (03) + (0,8) + &e4 A 01] + [02] W + &c. 
Y — —4(01) + (0.2) + &e§ g + (0,1) g + (02) e" + dec. | " 
7. 4(0,1) + (02) + do] p — (0,1) g — (02)  — &c. 
where 
À — e sin z, i= e cos sy 
p=isin Q, q = i cos 8. 
The unaccented symbols relate to the disturbed planet, which, in our present investi- 
gation, is supposed to be an asteroid ; the accented quantities relate to the disturbing 
planets. The expressions given by Laplace for the quantities (0,1) and [0,1] are 
8 m' n at 0) 
(04) — — 3 4= rg 
3am' n Í(14-d) 99 Hrd. 
[0,1] = — 2 (1— d , 
a representing the ratio of the mean distances of the disturbing planet and the as- 
teroid, and the other symbols being used in their usual signification. 
^ 
D 
