269 
1907-8.] The Problem of a Spherical Gaseous Nebula. 
we have the following expression for g, in any spherical mass, m, having 
throughout equal densities, p, at equal distances, r, from the centre : — 
9 E/e 2 5'6.e 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 
o dt 3 k-1 \ r -j 2 3 k- Ira 
dr 5*6. e kS o 5’6.e kS 47 r 
(17). 
Differentiating (17) with reference to r, we find 
-,dt 
dr 
3 k - 1 
5*6. e kS 
r z p 
§ 23. By (6), and (7), of §§ 4, 5, we find 
sty 
p “Va) 
where 
k- 1 
Eliminating p from (18) by (19), we find 
r*—~\ 
dr } 
rH K 
/t-2 
where 
2 _ 5-6.e (k+ 1)A* 
3S K 
§ 24. By putting 
we reduce (21) to the very simple form 
r _ cr 
X 
■ (18). 
■ (19), 
• ( 20 ). 
• ( 21 ), 
• ( 22 ). 
• (23), 
dx 2 
■ ( 24 ); 
