A SWARM OF METEORITES, AND ON THEORIES OF COSMOGONY. 
53 
, Biv 
-x> 
or 
Bto 
SWr 
= e 
— 3xx-l/io 
Now it is obvious that ; and. therefore, 
S 71 = t^-^xaV/'o 
But, by the dehuitioii of/’(a;), 
hence, 
Sitfj = Sx ; 
Bn = e Sx. 
(55) 
This is the law of frequency of mass x to x + Sx at radius ?•. 
Now, if m, mg be the mean masses at radii r and a respectively, 
m = 
e f {x) dx 
dx 
(56) 
and, if we put y = 0, we obtain wIq from the same formula. 
It is also clear that, if tv be the total density of the swarm at radius r, 
r 
IV = \xdn= \ X f {x) dx . (57) 
J Jq 
By the definition of y, and in consequence of the supposed spherical arrangement of 
matter, we have 
y = I q-j ( I 47r/ri(’r"cZr 
^ a')"' \ j 0 
If this value were substituted in (57), we should obtain a very complicated 
differential equation to determine w, the solution of which is hopelessly difficult. 
We may, however, assume without much error that the w in the integral expressing 
y is the density of meteorites, all of which are of the same size tn, and wliich are 
agitated with mean kinetic energy ^Jiq. If this density be written w, xve then clearly 
have 
X = ~ 
K 1 w 
log —. 
d m _ Wy 
The values of w and Wq may be extracted from Table III. of solutions in § 6. 
