OF ARTS AND SCIENCES. 417 



Foui' liuiiclred and eiglity-iirst meeting* 



April 24, 1860. — Special Meeting. 



The President in the chair. 



The Corresponding Secretary read letters relative to the 

 exchanges of the Academy. 



Mr. S. Newcomb read a paper on the secular variations and 

 mutual relations of the orbits of the Asteroids, of which the 

 following is an abstract : — 



The tests to which it is possible to put any theory respecting the 

 common origin of the asteroids may be divided into two classes. The 

 one is per se demonstrative, and consists in determining by actual trial 

 whether the orbits of the asteroids could ever have fulfilled the condi- 

 tions which may be required by the hypothesis. It may, however, be 

 subject to practical difficulties in its applications, which impair its de- 

 monstrative vigor. 



Another method is furnished by the method of probabilities, and 

 might, perhaps, if the asteroids were sufficiently numerous, approach 

 very nearly to certainty in its results. It is founded on the suppo- 

 sition, that the hypothesis examined will imply a high probability of 

 some general relationship among the orbits of the asteroids, if the latter 

 are sufficiently numerous; and consisting in observing whether any 

 such relationship is observable between the elements of the asteroids. 



Take, for example, the hypothesis of Olbers. It seems highly prob- 

 able that, if this hypothesis be true, the fragments of the exploded frag- 

 ment would have been thrown promiscuously in every direction ; and 

 that the velocities of those thrown in one direction would not differ ma- 

 terially from that of those thrown in another. From this hypothesis 

 we easily deduce the following conclusions : — 



1. The exploded planet, supposing it to have moved in an orbit with 

 a small eccentricity, was at a distance from the sun about equal to the 

 average distance of the asteroid, or 2.6. 



2. The asteroids, of which the mean distance is nearly equal to 2.6, 

 ouo-ht on the whole to have much smaller eccentricities than those at a 

 greater or less distance. 



3. The eccentricities ought on the whole to be more than twice as 

 great as the inclinations ; and the average quantity by which the mean 



VOL. IV. 53 



