26 
PROFESSOR G. H. DARWIN ON THE MECHANICAL CONDITIONS OF 
§ 7. On the Kinetic Energy of Agitation and its Distribution in an Isothermcd- 
Adiabatic Sphei'e of Gas. 
We shall now consider what would be the distribution of kinetic energy in the 
nebula if each meteorite (or molecule) were to fall from infinity to the neighbourhood 
where we find it, and were to retain that energy afterwards. This will give the dis¬ 
tribution of energy in a swarm of the supposed arrangement of density, if the rate of 
diffusion of kinetic energy were to be infinitely slow, and if there were no loss of energy 
through imperfect elasticity. 
The square of the velocity of a satellite in a circular orbit is one half of the square 
of the velocity acquired by the fall from infinity to the distance of the satellite from 
the centre. If the concentration has proceeded as far as radius r, and if a meteorite 
falls from infinity to distance r, then, if U be its velocity, and v the velocity in a 
circular orbit at distance r. 
1 jn 
o 
fxM dy 
X 
dtXj 
1 . 
3 
vfx 
cG 
dx’ 
in the isothermal sphere, 
= -f . a; ^ = -f vfx y , in the adiabatic layer. 
CC CtCG 
dx' 
In these formulae, by the definitions of y and 2 :, 
y — lorn ( ^ ) in the first, and 2 = in fb© second. 
From these formulae a” was computed in Table III. The value of or ^U~ gives 
what may be called the theoretical value of the kinetic energy, because it gives us a 
measure of the amount of redistribution of energy by diftusion and loss of energy 
by imperfect elasticity, which must take place before the whole system can assume the 
form of an isothermal adiabatic sphere. 
We will now go on to consider the total potential energy lost in condensation. 
We have seen that the potential energy lost by the fall of a single meteorite is 
\vqX dyjdx in the isothermal part, and §V(fx dzjdx in the outer part. 
Now, in the isothermal part a spherical element of mass is 
dx, 
and the energy lost by its fall is 
dx dx~ 
