30 
PROFESSOR G. H. DARWIN ON THE MECHANICAL CONDITIONS OF 
9. On the Criterion for the A^jAicahility of Hydrodynamics to a Swarm of Meteorites. 
The question at issue is to determine within what limits the quasi-gas formed by a 
swarm of colliding meteorites may be treated as a plenum, subject to the laws of 
hydi’odynamics. The doctrines of the nebular hypothesis depend on the stability of 
a rotating mass of fluid, and that stability depends on the frequencies of its gravita¬ 
tional oscillations. Now the works of Poincare and others seem to show that 
instability, at least in a homogeneous fluid, first arises from one of the graver modes 
of oscillation, and the period of the gravest mode does not difler much from the 
period of a satellite grazing the surface of the mass of fluid. Then, in order that 
hydrodynamical treatment should be applicable for the discussion of such questions of 
stability, the mean free time between collisions must be small compared with the 
period of such a^ satellite. Another way of stating this is that the mean free path of 
a meteorite shall be but little curved, and that the velocity of a meteorite shall be but 
little changed by gravity in the interval between two collisions. This must be fulfilled 
not only at the limits of the swarm, but at every point of it. The condition above 
stated will be satisfied if the space through which a meteorite falls from rest, at any 
part of the swarm, in the mean interval between collisions is small compared with the 
mean free path. If this criterion is fulfilled, then, in most respects which we are 
likely to discuss, the swarm will behave like a gas, and we must at present confine 
the consideration of the matter to this general criterion. 
It would be laborious to determine exactly the space fallen through from rest, 
because gravity varies as the meteorite falls, but a sufficiently close approximation 
may be found by taking gravity constant throughout the fall and equal to its value at 
the point from which the meteorite starts. 
We have already denoted by g tlie value of gravity at any part of the swarm, and 
have tabulated it in Table III. in terms of G or yuMja^. 
Now the mean interval is T = Lfr^SlSir). Hence, if D be the distance fallen in 
this time. 
D 
11 
But 
L = 
l( 
WnUp 
m 
\ 
M] w 
Therefore, 
I) 
_ 1 W 
m 
I” Stt 
L " 
2 a M 
' ^ 
18/3^’ 
[v\ 
OTT 
y 
and = 
1 
[gm 
l Mq m 
2a M 
r- 
(37) 
The factor F-^ has been tabulated above in Table III., and it increases from the 
centre to the outside. 
This criterion may be regarded from another point of view, for, if the meteorite be 
