64 PROFESSOR G. H. DARWIN ON THE MECHANICAL CONDITIONS OF 
of a planet at distance 16, revolving about a central body with a mass equal to 
46 per cent, of that of the Sun, viz., 5^ kilometres per second; and in the adiabatic 
layer it diminishes down to zero at distance 44-g-. This velocity is independent of the 
size of the meteorites. 
The mean free path between collisions ranges from 42,000 kilometres at the centre, 
to 1,300,000 kilometres at radius 16, and to infinity at radius 44-|-. The mean 
interval between collisions ranges from a tenth of a day at the centre, to three days 
at radius 16, and to infinity at radius 44^. The criterion DjL ranges from g-o-^-oo 
the distance of the asteroids, to 3 -(Tq-^ at radius 16, and to infinity at radius 44^. 
The criterion AjC somewhat smaller than DjL. All these quantities are ten times 
as great for meteorites of 3-|- kilogrammes, and a hundred times as great for meteorites 
of 3-g- tonnes. 
From a consideration of the table it appears that, with meteorites of 3| kilogrammes, 
the collisions are sufficiently frequent, even beyond the orbit of Neptune, to allow the 
kinetic theory to he applicable in the sense explained. But, if the meteorites weigh 
3|- tonnes, the criteria cease to be very small about distance 24 ; and, if thej^ weigh 
3125 tonnes, they cease to be very small at about the orbit of Jupiter. 
It may be concluded, then, that, as far as frequency of collision is concerned, the 
hydrodynamical treatment of a swarm of meteorites is justifiable. 
Although these numerical results are necessarily affected by the conjectural values 
of the mass and density of the meteorites, yet it wms impossible to arrive at any 
conclusion whatever as to the validity of the theoiy without numerical values, and 
such a discussion as the above was therefore necessary. If the particular values used 
are not such as to commend themselves to the judgment of the reader, it is easy to 
substitute others in the formulm, and so submit the theory to another test. 
I now pass on to consider some results of this view of a swarm of meteorites, and 
to consider the justifiability of the assumption of an isothermal-adiabatic arrangement 
of density. 
With regard to the uniformity of distribution of kinetic energy in the isothermal 
sphere, it is important to ask whether or not sufficient time can have elapsed in the 
history of the system to allow of the equalisation by diffusion. 
In § 11 the rate of diffusion of the kinetic energy of agitation is considered, and it 
is shown that, in the case of our numerical example, primitive inequalities of distribu¬ 
tion would, in a few thousand years, be sensibly equalised over a distance some ten 
times as great as our distance from the Sun. This result, then, goes to show that we 
are justified in assuming an isothermal sphere as the centre of the swarm. As, how¬ 
ever, the swarm contracts, the rate of diffusion duninishes as the inverse power of 
its linear dimensions, whilst the rate of generation of inequalities of distribution of 
kinetic energy, through the imperfect elasticity of the meteorites, increases. Hence, 
in a late stage of the swarm inequalities of kinetic energy would be set up ; thus, 
there ^vould be a tendency to the production of convective currents, and the whole 
