698 Lord Kelvin : The Problem of 
of the sun (or any other spherical star) for all the time before 
the central density had come to be as large as *1. 
§ 22. To pass from the case of convective equilibrium in 
a gaseous atmosphere so thin that the force of gravity is 
practically constant throughout its thickness, to the problem 
of convective equilibrium through any depth, considerable in 
comparison with the radius, or through the whole depth 
down to the centre, provided the fluid is gaseous so far, we 
have only to use (13) and (14), with the proper value of g, 
varying according to distance from the centre. Remembering 
that we are taking g in terms of terrestrial gravity, and that 
the mean density of the earth is 5*b\ in terms of the standard 
density of water, which we are taking as our unit density, 
we have the following expression for g, in any spherical 
mass, m, having throughout equal densities, p, at equal 
distances, r, from the centre : — 
3 & rr2 P 
9 ~~R/e 2 ~ 0'6.e r 2 
m/r 2 
(16), 
where E denotes the earth's mass, and e the earth's radius. 
This expression we find by taking g as the force of gravity 
due to matter within the sphere of radius r, according to 
Newton's gravitational theorem, which tells us that a 
spherical shell of matter having equal density throughout 
each concentric spherical surface exerts no attraction on a 
point within it. Using this in (13) of § 13, with dz= —dr ; 
multiplying both members by r 2 , and introducing m to denote 
the mass of matter within the spherical surface of radius r, 
we find 
At 3 jfe— 
— r z — — — 
dr b'6.e kS 
if' 9 3 k-lm , . 
"J drT ^ = ¥67elS^r ' (17) * 
Differentiating (17) with reference to r, we find 
d f «dt~] 3 Jc — 1 9 x., ox 
§ 23. By (6), and (7), of §§ i, 5, we find 
where 
"=j=i ( 2 °)- 
