■* s ï: J 
148 
Liquid Stars 
[ch. V 
ages of a star the intersections ST may coalesce in two adjacent positions 
of equilibrium as at B or C. When such a situation arises, the star can move 
from one of these configurations to the adjacent one without any forces of 
restitution coming into play, so that B and G represent configurations of 
neutral equilibrium. 
If G denote the rate of generation of energy of the whole star, and E its 
total rate of emission in any configuration, the tangents of the slopes of the 
lines of generation and emission in fig. 9 are dG/dR and dE/dR. In a con 
figuration of neutral equilibrium these slopes are the same, so that 
A transition from dynamical instability to stability or vice versa ac 
cordingly occurs whenever 
passes through a zero value, and it only remains to determine which sign of 
this quantity corresponds to stable and which to unstable configuration. We 
can easily do this by considering the special model already discussed in which 
the pressure and density are connected by the law p oc p l+s T. 
136. With reference to the special model considered in the last chapter, 
in which the pressure is determined by the law p oc p 1+s , the stability criterion 
which we now have under discussion requires that D in equation (108‘2) 
shall be positive. When D is negative the star becomes unstable through 
the time-factor for the corresponding expansion or contraction of the star 
assuming the form e 0t with 6 positive. Changes from stability to instability 
or vice versa occur in configurations for which D vanishes ; when D = 0, 6 = 0 
and the star is in neutral equilibrium. We have just seen that configurations 
of neutral equilibrium are precisely those at which two intersections of the 
emission and generation lines such as S, T coalesce in our diagram. In other 
words, they are the configurations such as B and C in which the tangents to 
the lines of generation and of emission coincide. 
When the rate of generation of energy G is assumed to be propor 
tional to p a T ? , we have, at each point of the star, 
1 dG _a dp B dT . 
GdR~ P dR + TdR 1 ° 
where d/dR denotes differentiation with respect to the different values of 
R in the various configurations which are possible for a star of given mass. 
These configurations have been seen to fall into homologous series, along 
each of which 
1 _ 3iX 
R x + 4 T = constant.