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NA TURE 



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thus the whole swarm would probably settle down to the 

 condition of convective equilibrium throughout. 



It may be conjectured, then, that the best hypothesis in 

 the early stages of the swarm is the isothermal-adiabatic 

 arrangement, and later an adiabatic sphere. It has not 

 seemed worth while to discuss this latter hypothesis in 

 detail at present. 



The same investigation also gives the coefficient of 

 viscosity of the quasigas, and shows that it is so great 

 that the meteor-swarm must, if rotating, revolve nearly 

 without relative motion of its parts, other than the motion 

 of agitation. But as the viscosity diminishes when the 

 swarm contracts, this would probably not be true in the 

 later stages of the history, and the central portion would 

 probably rotate more rapidly than the outside. It forms, 

 however, no part of the scope of this paper to consider 

 the rotation of the system. 



The rate of loss of kinetic energy through imperfect 

 elasticity is next considered, and it appears that the rate, 

 estimated per unit time and volume, must vary directly 

 as the square of the qaasi-pressure, and inversely as the 

 mean velocity of agitation. Since the kinetic energy lost 

 is taken up in volatilizing solid matter, it follows that the 

 heat generated must follow the same law. The mean 

 temperature of the gases generated in any part of the 

 swarm depends on a great variety of circumstances, but 

 it seems probable that its variation would be according to 

 some law of the same kind. Thus, if the spectroscope 

 enables us to form an idea of the temperature in various 

 parts of a nebula, we shall at the same time obtain some 

 idea of the distribution of density. 



It has been assumed that the outer portion of the 

 swarm is in conveciive equilibrium, and therefore there is 

 a definite limit beyond which it cannot extend. Now a 

 medium can only be said to be in convective equilibrium 

 when it obeys the laws of gases, and the applicability of 

 those laws depends on the frequency of collisions. But 

 at the boundary of the adiabatic layer the velocity of 

 agitation vanishes, and collisions become infinitely rare. 

 These two propositions are mutually destructive of one 

 another, and it is impossible to push the conception of 

 convective equilibrium to its logical conclusion. There 

 must, in fact, be some degree of rarity of density and of 

 collisions at which the statistical treatment of the medium 

 breaks down. 



I have sought to obtain some representation Of the state 

 of things by supposing that collisions never occur beyond 

 a certain distance from the centre of the swarm 



Then from every point Of the surface of the sphere, 

 which limits the region of collisions, a fountain of 

 meteorites is shot out, in all azimuths and at all inclina- 

 tions to the vertical, and With velocities grouped about a 

 mean according to the law of error. These meteorites 

 ascend to various heights, without collision, and, in falling 

 back on to the limiting sphere, cannonade its surface, so 

 as to counterbalance the hydrostatic pressure at the 

 limiting Sphere. 



The distribution in space of the meteorites thus shot 

 out is investigated in the paper, and it is found that near 

 the limiting sphere the decrease in density is somewhat 

 more rapid than the decrease corresponding to convective 

 equilibrium. 



But at more remote distances the decrease is less 

 rapid, and the density ultimately tends to vary inversely 

 as the square of the distance from the centre. 



It is clear that according to this hypothesis the mass of 

 the system is infinite in a mathematical sense ; for the 

 existence of meteorites with nearly parabolic and hyper- 

 bolic orbits necessitates an infinite number, if the loss of 

 the system shall be made good by the supply.^ 



But if we consider the subject from a physical point of 



' It must also be borne in mind that the very high velocities which occur 

 occasional y in a medium with perfectly elastic molecules, must happen with 

 great rarty amongst -meteorites An imract of such violence that it ought to 

 generate a hyperbjlic velocity will probably merely cause fracture. 



view, this conclusion appears unobjectionable. The 

 ejection of molecules with exceptionally high velocities 

 from the surface of a liquid is called evaporation, and 

 the absorption of others is called condensation. The 

 general history of a swarm, as sketched at the beginning, 

 may be put in different words, for we may say thai at first 

 a swarm gains by condensation, that condensation and 

 evaporation balance, and finally that evaporation gains 

 the day. 



If the hypothesis of convective equilibrium be pushed 

 to its logical conclusion, we reach a definite limit to the 

 swarm, whereas if collisions be entirely annulled the density 

 goes on decreasing inversely as the square of the distance. 

 The truth must clearly lie between these two hypotheses. 

 It is thus certain that even the small amount of evapora- 

 tion shown by the formulre derived from the hypothesis 

 of no collision must be in excess of the truth ; and it 

 may be that there are enough waifs and strays in space 

 ejected from other systems to make good the loss. 

 Whether or not the compensation is perfect, a swarm of 

 meteorites would pursue its evolution without being 

 sensibly affected by a slow evaporation. 



Up to this point the meteorites have been considered 

 as of uniform size, but it will be well to examine the 

 more truthful hypothesis that they are of all sizes, grouped 

 about a mean according to a law of error. 



It appears, from the investigation in the paper, that the 

 larger stones move slower, the smaller ones faster, and 

 the law is that the mean kinetic energy is the same for all 

 sizes. It is proved that the mean path between collisions 

 is shorter in the proportion of 7 to 11, and the mean 

 frequency of collision greater in the proportion of 4 to 3, 

 than if the meteorites were of uniform mass equal to the 

 mean. Hence the numerical results found for meteorites- 

 of uniform size are applicable to non-uniform meteorites 

 of a mean mass about a third greater than the uniform 

 mass ; for example, the results for uniform meteorites of 

 3| tonnes apply to non-uniform ones of mean mass a 

 little over 4 tonnes. 



The means here spoken of refer to all sizes grouped 

 together, but there is a separate mean free path and mean 

 frequency appropriate to each size. These are investigated 

 in the paper, and their values illustrated in a figure. It 

 appears that collisions become infinitely frequent for the 

 infinitely small ones, because of their infinite velocity, 

 and again infinitely frequent for the infinitely large ones, 

 because of their infinite size. There is a minimum 

 frequency of collision for a certain size, a little less in 

 radius than the mean radius, and considerably less in 

 mass than the mean mass. 



For infinitely small meteorites the mean free path 

 reaches a finite limit, equal to about four times the grand 

 mean free path ; and for infinitely large ones, the mean 

 free path becomes infinitely short. It must be borne in 

 mind that there are infinitely few of the infinitely large 

 and infinitely small meteorites. Variety of size does not 

 then, so far, materially affect the results. 



But a difference arises when we come to consider the 

 different parts of the swarm. The larger meteorites, 

 moving with smaller velocities, form a quasi-gas of less 

 elasticity than do the smaller ones. Hence the larger 

 meteorites are more condensed towards the centre than 

 are the smaller ones, or the large ones have a tendency to- 

 fall down, whilst the small ones have a tendency to rise. 

 Accordingly, the various kinds are to some extent sorted 

 according to size. 



An investigation is made in the paper of the mean 

 mass of meteorites at various distances from the centre, 

 both inside and outside of the isothermal sphere, and a 

 figure illustrates the law of diminution of mean mass. 



It is also clear th.it the loss of the system through 

 evaporation must fall more heavily on the small meteorites 

 than on the large ones. 



After the foregoing summary, it will be well to briefly 



