A SWARM OF METEORITES, AND ON THEORIES OF COSMOGONY. 
37 
distribution of density which is deduced on the hypothesis that all the meteorites are 
rising. Again, since every element of the sphere shoots out a similar fountain, and since 
collisions are precluded by hypothesis, we need only consider the velocity along the 
radius vector. As far as concerns the distiibution of density, it is the same as if each 
element shot up a vertical fountain; but, of course, in determining the vertical 
velocity, we must pay attention to the inclination to the vertical at which the 
meteorite was shot out. 
The mass of the matter inside the sphere, whose attraction affords t^ e principal 
part of the force under which the meteorites move, is siy M, and, for the sake of 
simplicity of notation, we shall take ‘Zi^Mja as being unit square of velocity. 
Now, let j> (r) be the potential at the point whose radius is r, and suppose that a 
meteorite is shot out from a point on the sphere with a velocity u, and at an inclina¬ 
tion e to the vertical; then, if r, 9 be the radius vector and longitude of the meteorite 
at the time t, the equations of conservation of moment of momentum, and of energy 
are — 
.d9 
r- — = ua sin e, 
) ('’ d!) “ = u~- (a). 
If we write f{r) — cf) (a) — (r), and eliminate cW/dt, we get 
cJt 
= r (ir —/ (?7 ) — siiv e}7 
/*• 
Now, we are to regard drjdt as the vertical velocity in a fountain squirting up from 
a point on the sphere. Then, since = it follows that at the foot of the 
fountain drjdt is equal to ahi cos e. If, therefore, S be the density at the height r, 
and §0 at the foot, the equation of continuity is 
Therefore, 
Sq n~u cos e. 
S a~u cos e 
Sq r [d — /{r) ) — siid e}^ 
But now let us suppose that tire meteorites are not only shot out at inclination e, 
but at all possible inclinations from 0° and 9Oh It is then clear that this expression 
must be multiplied by sin ede, and integrated. Hence, if S now denotes the integral 
density, 
J 0 r [d {u~ — f{r) ) — u-cd siid e}‘^ 
