20 PROFESSOR G. H. DARWIR OR THE MECHANICAL CONDITIONS OF 
wl\ p = Tims, for example, at the centre, p is 32-14, and when 
r = -4789 a it is 5-7417. 
The proportion of the mass M which is included in radius ajx is ^ dijjdx 
= dyja dx^ — dyjdx^. Hence, the masses may be computed. 
A.t any part of the isothermal sphere gravity g is to be found from 
or, expressing g in terms of G gravity at the surface, we have, since G = gMla\ 
£ 
G 
(27) 
The angular velocity of a body moving in a circular orbit at any part of the nebula, 
and its linear velocity v are also easily to be found. 
5. On an Atmosphere in Convective Equilihriiiin. 
I shall now suppose that a sphere of gas of mass M at minimum temperature is 
bounded by an atmosphere in convective equilibrium, with continuity of temperature 
and density at the sphere of discontinuity of radius a. Let Vq be the mean square 
of velocity of agitation in the isothermal sphere, and that at any other radius r. 
Then throughout the isothermal sphere — Vq, but in the layer outside gradually 
decreases to zero. 
Let Wy be the density and p)Q fhe pressure at radius a, and lo, p the same things at 
radius r. 
Then, if the ratio of the two specific heats be that deduced from the simj)le kinetic 
theory of gases, without any allowance for intra-molecular vibrations, we have that 
ratio equal to -f. 
Hence, 
and 
V 
also 
1 0 _ _ 1 0 
vj V 
Wr 
dp 
w 
A 
3 
—1 w “ d'W 
wy 
d 
Now, the equation for the hydrostatic equilibrium of the laj'er is 
+ /rdifi- 477/X 
w dr ' ' 
lor" dr = 0. 
(28) 
