Gaseous Stars 
90 
[ch. Ill 
the value of n already obtained in equation (83T); for a star of large mass 
(X c small) it gives the same value for n as equation (841). 
In the special case in which ¡x and cFG have uniform values throughout 
the star, so that j = l = 0, this reduces to 
- 8+ r£& < 85 ' 3 >- 
Thus as X c varies from oo to 0—i.e. as we pass from stars of very small 
mass to stars of very great mass, n varies continuously between n = 3'25 and 
n—'l, the two limiting values already obtained. 
We can obtain somewhat higher accuracy as follows. The contributions 
4:7rpr z dr made to the total mass by small equal steps dr in r, increase from 0 
at the centre (r = 0 ) up to a maximum, and afterwards decrease until they 
again become zero at the surface. A study of Emden’s solutions suggests that 
the maximum occurs near to a value r of r such that half the total mass of 
the star is contained within the sphere of radius r. Thus we may say that 
the main part of the star’s mass is concentrated about the value r' of r. 
A further study of Emden’s solutions shews that the temperature at a 
distance r from the centre ranges from 0-59 T c for stars of very small mass 
up to 0'61 T c for stars of very great mass. We shall obtain sufficient accuracy 
for our present purpose by supposing this temperature to be uniformly equal 
to 0'60 T c . To the accuracy of this approximation, half of the star’s mass is at 
a temperature greater than 0'60:!r c , whilst half is at a lower temperature, the 
temperatures of different elements of the star being ranged fairly closely 
around the value 0'60 T c . If X' is the value assumed by X at a distance r' 
from the centre, equation (82*7) gives the relation 
V (V + 1) = (0-60)* " 1 X c (X c +1) (85-4). 
When fi and cFG have uniform values throughout the star, l = 0, and this 
equation becomes 
V (V +1) = 0-775\ c (X c + 1) (85-5), 
which reduces further in the special cases of X large and X small, giving 
for a star of very small mass, X' = 0'880X C , 
„ „ „ large „ X' = 0-775 X c . 
Clearly on replacing X c by X' in formulae (85'2) and (85‘3) we shall obtain 
a better approximation to the arrangement of stars of moderate mass. 
When n= 5, the standard differential equation (65’5) can be solved in 
finite terms, the solution being Schuster’s solution 
C 
-i 
P = P<