164 THE TIDAL PROBLEM. 



centrifugal component of rotation first comes into equality with the cen- 

 tripetal force of gravitation and constitutes a condition precedent to the 

 separation of any mass of gas as a body. 



In the depths of the gaseous spheroid the paths between encounters 

 may be assumed to be relatively short and hence straight, since gravitation 

 can not sensibly affect paths of brief duration. At higher levels, the free 

 paths grow progressively longer, and at length horizons may be reached at 

 which the attenuation permits free paths of such length and duration that 

 they may be appreciably curved by the gravitation of the spheroid. At 

 still greater heights the attenuation reaches such a degree that curved 

 paths come to dominate and, at a certain stage of rarity, a portion of the 

 molecules rebounding from encounters in outward directions, find no 

 molecules in their paths, and therefore hold on their courses until arrested 

 and turned back by gravitation, if its force be sufficient, or else they pass 

 on beyond the limit of the spheroid's control. Theoretically, under 

 the Boltzman-Maxwell law of molecular distribution, a certain small per- 

 centage of molecules should reach the parabolic velocity of the spheroid 

 and escape, but for the purposes of the present discussion this fraction need 

 not be considered independently of a larger class to be described presently, 

 with which it may be merged as having like influence on the moment of 

 momentum of the spheroid. 



Such of the outward-bounding molecules as are arrested by the sphe- 

 roid's gravitation obviously turn back toward the spheroid without a re- 

 versing encounter and thus describe elliptical loops. In this they differ 

 markedly from the molecules in the depths of the gaseous spheroid, whose 

 paths are sensibly straight and whose courses are terminated by encounters 

 at either end.^ In the elliptical courses the outward movement is ter- 

 minated by a gradual decline in the molecule's speed until its outward 

 progress is reduced to zero, when there follows a new movement inward 

 accelerated by gravitation. 



If the to-and-fro, collisional activity of molecules constitutes the essen- 

 tial characteristic of a gas, the outer border of the strictly gaseous part of 

 the spheroid should be placed at the transition zone where the molecules 

 cease to-and-fro passages between encounters and begin to describe ellip- 

 tical loops limited outward by gravitation, but this demarcation is rather 

 a matter of convenience than an essential in the consideration of the modes 

 of action. 



In the course of their outgoing and incoming movements, the molecules 

 pursuing elliptical paths are subject to collision with one another. The 

 phases of such encounters may vary indefinitely and the velocities of the 

 rebounding molecules may represent an indefinite variety of interchanges 

 of kinetic energies. Inspection shows that some of these molecules must 

 rebound toward the gaseous spheroid, that some must take distinctively 

 new elliptical paths, while a certain proportion will inevitably be thrown into 

 courses more or less tangential to the surface of the spheroid, and some of these 

 may have sufficient velocities to assume orbits about the spheroid, and thus form 



* This distinction has been drawn by G. Johnston Stoney, Astrophys. Jour., vol. XI, 

 1900, pp. 251 and 325, and elsewhere. 



