A SWARM OF METEORITES, AND ON THEORIES OF COSMOGONY. 
9 
Then the equation for the hydrostatic equilibrium of the gas is 
— ^ + 4 TT /r f ivr^ dr = 0. 
10 dr J 0 
( 12 ) 
It is obvious that-^ is equal to the whole mass enclosed inside radius v, and 
fjbw dr 
this relation will hold however the equation be transformed, provided we do not 
multiply the equation by any factor. 
In consequence of the relation between p and iv this may be written 
Jc 
d , 47ryu. p c 7 
0 . 
If Pj be the mean density of the mass we have 
9k 
4: TT p = 
Pl«l 
/3-papp^ 
Hence, we may write the equation (12) in the form 
r” d 9 f'" VO d 
ka^ 
Now, let 
r- d . 9 p VO d -j~\ 
— — 
r 
and the equation becomes 
w 
Pi 
10 3 H/f ^^Vi if 
€^1 , 
- 
= 0. 
By dilferentiation we obtain the equation 
tMi 4_ fl'' _ A 
dxp o:p 
(13; 
(14) 
It is obvious from (13) that ^ M-^ dyjdx-^ is the mass enclosed inside radius a/cc^, 
and therefore (Bp dypdx-^ is equal to unity when x ■= 1. 
A general analytical solution of (14) does not seem to be attainable, and recourse 
must be had to numerical processes. Although this is an equation of the second 
degree, and its general solution must involve two arbitrary constants, wm shall see (as 
pointed out by M. Bitter) that the general solution, as applicable to our problem, 
may be deduced from one single numerical solution. M. Bitter proceeds by a 
graphical method, which he has worked with surprising accuracy. I shall therefore 
adopt an analytical and numerical method, which, although laborious, is susceptible of 
greater accuracy. 
MDCCCLXXXIX.-A. C 
