166 THE TIDAL PROBLEM. 



cules whose projections have forward components are regarded as fulfilhng 

 these conditions, nearly half the molecules given outward projections will 

 be included. 



Encounters and rebounds in gases under familiar terrestrial conditions 

 range into the billions per second, but encounters are much less frequent 

 in rare gases. Excluding outward flights that do not properly belong to the 

 true gaseous state, the maximum period between encounters is less than 

 the rotation-period of the spheroid, and the average period is much less 

 than that. Since interchanges are thus frequent, all molecules are liable to 

 receive a forward impulse within a brief period. 



Molecules which simply receive a forward and outward projection 

 greater than that requisite for free circular revolution about the spheroid 

 do not, however, enter upon free orbits, directly, in most cases, because the 

 paths on which they enter, the orbital in type, normally lead back to their 

 starting-points, and in nearly all cases they cut the spheroid before they 

 return to these points and thus complete a free orbit. If this were univer- 

 sally and inevitably true, the way to free orbits along this Hne of evolution 

 would be effectually barred. There are three lines of escape from this 

 result, the first and second of which are probably unimportant; the third 

 is probably effective. 



1. The first is the case in which molecules receive impulses from molec- 

 ular interaction in Unes tangent to the points of impact, and hence take 

 elliptical paths about the spheroid which return tangentially to the points 

 of impact as their peri-spheroidal cUmax. Their liability to encounter the 

 spheroid is thus limited to these tangential touches which, in the rare con- 

 dition of the gas at these vanishing points of gaseous organization, will 

 not necessarily involve capture. Such molecules will not, however, be free 

 from collision with the molecules pursuing elliptical loops above the gaseous 

 spheroid. 



2. By hypothesis, the spheroid is shrinking, and if the rate of shrinkage 

 is appreciable during the free flight of molecules whose paths only slightly 

 cut the surface of the spheroid such shrinkage may leave these paths free, 

 so far as the gaseous spheroid is concerned. 



3. The two cases just named are perhaps more serviceable in deJEining 

 conditions where gradations rather than sharp limits prevail than as 

 sources of free orbital paths. The most important case is built upon the 

 action of the ultra-gaseous molecules outside the gaseous spheroid, as Hm- 

 ited above. These molecules start from the outer part of the spheroid— 

 strictly, from all depths from which there is an open path outward in the 

 line of their projection— and pursue elliptical courses with return to the 

 spheroid, except in the cases just noted. We have seen that, in the rep- 

 resentative case of the solar system, the rotational velocity is so great, 

 relative to the average molecular velocity, that most of the molecules will 

 pursue forward courses, even when directed backward, and hence will be 

 moving in harmonious directions. Considered as independent molecules, 

 they constitute a corona of particles rising in curves, predominantly at low 

 angles, and descending at similar angles to the spheroid. They are liable 

 to collide in these courses and a certain percentage of coUisions is inevi- 



