66 
PROFESSOR G. H. DARWIN ON THE MECHANICAL CONDITIONS OF 
than tlie 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, then, 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 
hyperbolic 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 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 stated in § 2, may then be put in 
different words, for we may say that 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 betwmen these two hyjootheses. It is thus certain that even the very small 
amount of evaporation shown by the formulae derived from the hypothesis of no 
collisions must be in excess of the truth; and it may be that there are enough wmifs 
and strays in space, ejected from other systems, to make up for 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 is 
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 § 14, 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 greater in the proportion of 4 to 3, than if the 
meteorites were of uniform mass, equal to their mean. Hence, the previous numerical 
results for uniform size are applicable to non-uniform meteorites of 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 are a separate 
mean free path and a mean frecjuency appropriate to each size. These are investigated 
[* It must also be borne in mind that the very high velocities, which occur occasionally in a medium 
with perfectly elastic molecules, must happen with great I’arity amongst meteorites. An impact of such 
violence that it ought to generate a hyperbolic velocity will probably merely cause fracture.—Added 
Nov. 28, 1888.] 
