82-85] 
Configurations of Equilibrium 
89 
Equation (82 - 6) now shews that p ac T' n , where 
3-5-3 j-l , j 
n = 
+ 
.(84-1). 
(i -j)(i-j + 0 ' i -j 
When p, and cFG are assumed uniform throughout the star, j = l = 0 so 
that n = 7'0. Our analysis shews that in this case the star would arrange 
itself like a sphere of gas in adiabatic equilibrium with n = 7. The radius of 
such a sphere is, however, infinite, so that the star must find some means of 
evading the constitution which our analysis would try to impose on it. 
Stars of Moderate Mass. 
85. The case of a star of intermediate mass, in which X can neither be 
treated as very small nor very great, presents more serious difficulties. We can 
obtain the relation between p and T' by eliminating X between equation 
(82‘6) and (82‘8), but the result will not in general be of the desired form 
p oc T' n (851), 
which corresponds to a sphere of gas in adiabatic equilibrium and so, thanks 
to Emden’s tables, is suited to easy numerical computation. 
The p, T' curve which results from the elimination of X between the two 
equations just mentioned can always be made to agree with equation (851) 
at the centre of the star by a suitable choice of the constant multiplying T’ n 
in this equation. We can further make the tangents to the two curves coin 
cide at the centre of the star by a suitable choice of n. Moreover, the two 
curves agree in any case at the surface of the star (p = 0 , T' = 0). 
Thus by a suitable choice of n we can make the curves agree at three 
points, two of which are close to the centre of the star, round which the main 
part of the mass is congregated. In the special cases of X small and X great, 
the curves not only agree at these three points, but coincide throughout their 
lengths. 
For stars of intermediate mass we shall determine n in the way just 
described, and then assume the p, T' relation to be that given by equation 
(851). Analysis of definite cases suggests that this procedure gives quite a 
good approximation for stars of small and medium mass. 
The value of n will of course be the value of 
r dp 
p dT ,} 
obtained by differentiating equations (82 6 ) and f82‘8) and eliminating dX. 
After simple algebra its value is found to be 
21 
4 + X c (1 — j) 
.(85-2). 
3 + j 1 
* ~ r- j + 1 - j Ll + 4K (1 -j) + - 2 jJ 
In passing we may notice that for a star of small mass ( X c large) this gives