52 
PROFESSOR G. H. DARWIR OX THE MECHAXICAL CONDITIONS OF 
result, that tlie meau mass in the centre of the swarm becomes infinite. The 
existence of very large meteorites in sufficient numbers to give statistical constancy 
in a volume which is not a considerable fraction of the volume of the whole swarm is 
physically improbable. We shall, therefore, treat the case best by absolutely 
excluding very large masses. When such masses occur, they must not be treated 
statistically ; this is a question which I hope to consider in a future paper. Had I 
foreseen this conclusion when the investigations of the last two sections were carried 
out, a different law of frequency of mass would have been assumed. But the results 
of those sections are amply sufficient to indicate the conclusions which would have 
been reached with another law of frequency, and, therefore, it does not seem worth 
while to recompute the results by means of a fresh series of laborious quadratures. 
Any law of frequency would suffice for our purpose which excludes masses greater 
than a certain limit and rises to a maximum for a certain mean mass. For the 
present, I do not specify that law precisely, but merely assume that at some radius, 
which may conveniently be taken as that of the isothermal sphere, where r = a, the 
number of meteorites whose masses lie between x and x -f- S* is f {x) Sx ; it is also 
assumed that x may range from M to zero. 
The meteorites whose masses range from x to x hx may be deemed to constitute 
a gas. Suppose that at radius r the number of its molecules per unit volume is hn, 
its density hiv, its pressure and let the same symbols, with sufiix 0, denote the 
same things at radius a. Since all the partial gases are in the permanent state, they 
all have the same mean kinetic energy of agitation, equal to \h, suppose. Throughout 
the isothermal sphere, this li is constajit, and equal, say, to but varies with the 
radius in the adiabatic layer over it. It follows, therefore, that the mean square of 
the velocity of the particular partial gas x to x Sx is equal to h/x, and the relation 
between Sj? and Siu is 
Let — X ije file excess of the gravitation potential of the whole swarm at radius r 
above its value at radius a. 
Then, since each partial gas behaves as though it existed by itself, the equation of 
hydrostatic equilibrium of the partial gas x to x + Sx is 
1 d 8p dx _ 
Sio dr dr 
The investigation must now divide into tAvo, according as whether we are con- 
widering the isothermal sphere or the adiabatic layer. 
The Isotheruial SiTiere. 
Here we have h a constant and equal to Uq, and Sp varies as Stv, so that 
* This M is not to be confused with M, the mass of the isothermal sphere. 
