y. 
On the Secular Variations and Mutual Relations of the Orbits of the Asteroids. 
By SIMON NEWCOMB. 
( Communicated April 24, 1860.) 
Wuen we consider the number of members of which the group of asteroids is com- 
posed, and the narrow limits within which they are included, it is impossible to avoid 
the conclusion that their proximity is the result of a common element in the deter- 
mining reasons which fixed the respective positions of their several orbits, which com- 
mon element has some special relation to that portion of space in which the orbits of 
the asteroids are found which it has not to other portions of space. 
The object of the present paper is to examine those circumstances of the forms, 
positions, variations, and general relations of the asteroid orbits which may serve as a 
test, complete or imperfect, of any hypothesis which may be made respecting the cause 
from which they originated, or the reason why they are in a group by themselves. 
It may not, however, be out of place to begin with some general considerations respect- 
ing these hypotheses, and the nature of the methods by which they may be tested. 
Every a posteriori test is founded on the supposition that the hypothesis to be egen 
necessarily or probably implies that certain conditions must be fulfilled by the asteroids 
or their orbits. 'The tests consist in observing whether these conditions are fulfilled. 
The conditions may be divided into two general classes, — those which are rigorous and 
necessary, and those which are merely probable. The former class consists of those 
which follow immediately and necessarily from the hypothesis itself; the latter, of those 
which are deducible from it by the principle of random distribution. The nature of 
the latter conditions will be clearly seen from the examples which will be given of 
their deduction from hypotheses. : 
Two hypotheses worthy of consideration have been promulgated respecting. the 
