A SWARM OF METEORITES, AND ON THEORIES OF COSMOGONY. 
39 
therefore, 
u {u^ — f{r))' du = 
and the limits of x are oo to 0. 
Hence, the fii’st term is 
x~ dx, 
e ^ dx 
Again, for the second term put 
./OO 0 
1 - fd/rV 
= x~ 
and similarly introduce it into the second term, and we have 
- 1 - \^^xH-^dx. 
From these expressions we may omit the constant factors ; and, if iv be the density 
at height r, whilst Wq is the density at the sphere, 
r-y 
In this formula unit square of velocity is •2p,ili/a ; but we have elsewhere written 
v" = ; hence, if the special unit of velocity be given up, we may write yS' in 
place of 2v^, and the result becomes 
w 
— = e-3/0-)/2i;2 _ ] 
Wa 
10 
10 
0 
g-3/(r)/^2 _ 
Q-‘6f(rWHl-aW)^ 
(43) 
It is interesting to observe the connection between this law of density and that 
which would have held if the gaseous law (due to collisions) had obtained. In that 
case, since ^(f) [r) is the potential, we should have had 
1 dp 
10 dr 
2 
^ 0‘) = 0- 
Now, p = ^v^w, and, therefore, 
log V) - ^'^2 </> (r) = const. 
3 
= log«’0“ ^2^(4 
Tlius, 
or 
w 
Q-Z/{r):-2c^ _ 
