1884] 1"^ [Kirkwood. 



present dimensions. It is now proposed to find their original or maximum 

 values. 



In astronomy, as in other branches of physical science, many well- 

 known facts remain still unexplained. This is true not only in regard to 

 the fixed stars and the nebulae, but within the narrower limits of the solar 

 system. Recognizing the impossibility of accounting for present relations 

 without considering the causes which operated in the distant past, 

 astronomers have attempted to trace the process of formation from the 

 primal chaos down to the origin of the youngest planet. In the theory of 

 Laplace, the planets were formed Jrom nebulous rings successively aban- 

 doned in the plane of the solar equator. The present writer, while not 

 rejecting the nebular hypothesis itself, has indicated certain objections to 

 the special form in which it was proposed by its celebrated author.* These 

 difficulties, encountered in the theory of formation from rings, are avoided 

 by supposing each planet at its origin to have been separated from a very 

 limited arc of the equatorial protuberance. In either case, however, the 

 dimensions of the primitive planet would be necessarily restricted by the 

 law of gravitation. 



It is sufficiently obvious that an original planetary mass in a nebular 

 state could not have retained its continuity of form beyond a certain de- 

 terminable limit ; in other words, that it would have been changed into a 

 ring by the attraction of the central body. The main design of the follow- 

 ing paper, after finding in several cases the limits of equilibrium, is to 

 trace, if possible, certain unexplained facts to their origin in these primi- 

 tive relations between the various members of the solar system. 



Limits of Planetary Equilibrium. 

 If two nebulous bodies, M and vi, revolve about a common centre of 

 gravity, the disturbing force of M on the superficial stratum of m is the 

 diflFerence between the attraction of the former on the nearest point of the 

 surface of the latter and that on its centre of gravity. The same is true, 

 mutatis mutandis, in regard to the disturbing influence of m on M. If, 

 then, 



a = the distance between the centres of J!/ and m, and 

 X = the distance from the centre of the former to the limit of 

 equilibrium of the latter, we shall have 



Af 



— 7 = the attraction of Mon the centre of gravity of m, 



M 



—,2 = that on the nearest point of the surface, and 



M _ M 



the accelerating force of J/ on the portion of the surface of 

 m between the two centres ; but as these forces from M and 

 m are in equilibrium, the neutral point, or the limit of w, 

 may be found from the equation 



* Proceedings of the American Philosopliical Society, Vol. xviii, p. 321, and 

 Vol. xix, p. 1.5. 



FROC. AMER. PHILOS. 80C. XXIt. 118. N. PRINTED MARCH 5, 1885. 



