135, 136] 
Stability of Liquid Stars 
149 
J 
IO 
By differentiation of this equation we obtain 
while the relation R 3 p = constant gives 
1 dp _ 3 
1 ie. - _ 3 
p dR R 
Equation (136T) now becomes 
(136-2). 
The total outward flow of radiation across a sphere of radius r in the 
interior of the star is 
in which ^ denotes differentiation along the radius of the star. From this 
we readily obtain 
When the star is in equilibrium, E—G, so that equations (136-2) and 
(136"4) give 
Our second stability criterion (D > 0) merely expressed that the term in 
square brackets on the right of this equation must be positive for stability, 
or, what is the same thing, that the right-hand member of the equation must 
be negative. But as we have seen that, quite apart from this special model, 
Since R increases as we pass to the right of the diagram shewn in fig.'9, 
the stable ranges will be those in which E increases more rapidly than G as 
we pass to the right. 
To a first approximation the first (thermodynamical) stability criterion 
required that G should be independent of changes in the density and 
temperature of the star. If we neglect the dependence of G on density and 
temperature, dG/dR is zero, and the condition for stability assumes the simple 
form that dE/dR must be positive—for a stellar configuration to be stable, 
the emission of radiation must decrease as the star contracts. This is the 
(136-3), 
d 
EdR T dR pdR + R 
1 dE _ 7 + n dT 2 dp 1 
(136-4). 
~ (E-G) = | (3a + £-n)+ ~~(7 + n-0) (136-5). 
the stability criterion depends only on the sign of ^ (E- G), it is clear that, 
in general, the second stability criterion takes the form that 
^ (E — G) must be positive for dynamical stability.