56 
PROFESSOR G. H. DARWIN ON THE MECHANICAL CONDITIONS OF 
Thus, if m be the mean mass at radius r, 
(61) 
where a is the smaller of f and M. 
If we put X = 6 in (61), we obtain ??Iq, the mean mass at radius a. 
It is clear also that, if lo be the total density of the swarm at radius r, 
(62) 
By definition of y, and in consequence of the supposed spherical arrangement of 
matter, we have 
^ f’ fo 
If this value were substituted in (62), we might obtain a complicated difierential 
equation for w. It is clear, however, that an adequate approximation maybe obtained 
by assuming that the lo in the integral expressing y is the density of meteorites, all 
of which are of the same size m, arranged in a layer in convective equilibrium, and 
with kinetic energy of agitation at the limit r =- a equal to g Iiq. 
If this density be written w, and if v^ be the mean square of velocity of 
agitation at radius r, we have, by (60), and in consequence of the relationship 
(w/Wo)* = (v/vo)^ 
and 
" K 
Let 
1 
/3 
• ' G Tt' 
for brevity ; then, adopting the law of frequency f{x) = x (M — x), as before, we 
have for the mean mass at radius r 
(03) 
where a is equal to the smaller of M and B. 
j’“x2(M-rr)(^l -^Jdx 
I cc (M — x) ^ dx 
