62 
PROFESSOR G. H. DARWIR ON THE MECHANICAL CONDITIONS OF 
are small. Since this “theoretical value of the kinetic energy” is zero at the centre, 
there must have been diffusion of energy from without inwards, and considerations 
of the same kind show that when a planet consolidates there mmst he a cooling of 
the middle strata both outwards and inwards. 
We must now consider the nature of the criterion which determines whether the 
hydrodynamical treatment of a swarm of meteorites is permissible. 
The hydrodynamical treatment of an ideal plenum of gas leads to the same result 
as the kinetic theory with regard to any phenomenon involving purely a mass, when 
that mass is a large multiple of the mass of a molecule ; to any phenomenon involving 
purely a length, when the cube of that length contains a large number of molecules; 
and to any phenomenon involving purely a time, when that time is a large multiple 
of the mean interval between collisions. Again, any velocity to be justly deduced 
from hydrodynamical principles must be expressible as the edge of a cube containing 
many molecules j^assed over in a time containing many collisions of a single molecule ; 
and a similar statement must hold of any other function of mass, length, and time. 
Beyond these limits, we must go l)ack to the kinetic theory itself, and in using it 
care must be taken that enough molecules ai‘e considered at once to impart statistical 
constancy to their properties. 
There are limits, the]i, to the hydrodynamical treatment of gases, and the like must 
hold of the parallel treatment of meteorites. 
The principal question involved in the nebular hypothesis seems to be the stability 
of a rotating mass of gas ; hut, unfortunately, this has remained up to now an 
untouched field of mathematical research. We can only judge of probable results 
from the investigations which have been nmde concerning the stability of a rotating 
mass of liquid. Now, it appears that the instability of a rotating mass of liquid first 
enters through the graver modes of gravitational oscillation. In the case of a 
rotating spheroid of revolution the gravest mode of oscillation is an elliptic deforma¬ 
tion, and its period does not differ much from that of a satellite which revolves round 
the spheroid so as to graze its surface. Hence, assuming for the moment that a 
kinetic theory of liquids had been formulated, we should not be justified in applying 
the hydrodynamical method to this discussion of stability unless the periodic time of 
such a satellite were a large multiple of the analogue of the mean free time of a 
molecule of liquid.^ 
Carrying, then, this conclusion on to the kinetic theory of meteorites, it seems 
probable that liydrodynamical treatment must be inapplicable for the discussion of 
such a theory as the meteoric-nebular hypothesis, unless a similar relation holds good. 
These considerations, although of a very general character, will afford a criterion of 
the applicability of hydrodynamics to the discussion of the mechanical conditions of a 
swarm of meteorites in the kind of problem suggested by the nebular hypothesis. 
* If the molecules of liqiiid describe orbits about one another, the analogue would probably be the 
mean periodic time of one molecule about another. 
