156 
Liquid Stars 
[CH. V 
hand side of the equation has no very great effect on the equation as a whole, 
the graph of the left-hand side being, in its general nature at least, similar to 
what it is when this term is absent. 
On the other hand, the term in 3X/3T on the right-hand side of the 
equation has a profound effect, with the result that the equation always has 
a root even for stars of the smallest mass. 
For by the definition of A, (equation (138*1)), 
p — \T X x a constant, 
so that, by logarithmic differentiation, 
_»A + Td\\ 
t v 3x dr) 
l dp 
pdf~TV ' 3X3 T) 
The effect of introducing the term in 3X/3 T is accordingly to replace X by 
XT d P „»a i 1 d P 
Sp dT ’ an< ^ * s P ro P or fio na l to ~ . 
As the deviations from the gas-laws become greater, dp/dT falls steadily 
below the value it would have if the gas-laws were obeyed, and would finally 
vanish in a state in which the density increases no further, heating now 
being accompanied by expansion as in a solid or liquid. But the two sides of 
equation (143T) will have become equal at some point before this state is 
reached, so that the deviations from the gas-laws again diminish until (1 — x) p 
reaches a minimum value. 
Thus we see that for every star, whatever its mass, and for every ring of 
electrons, (1 — x) p must in time pass through a minimum. But the more the 
mass of the star falls short of the critical limit calculated in § 140, the greater 
the deviations from the gas-laws must be before (1 — x) p reaches a turning 
point. The occurrence of the turning point in an actual star is of course 
dependent on deviations of this magnitude being attained; it may well be 
that such deviations are not attained in a given star through the next ring 
of electrons being ionised first. This reservation cannot apply to if-ring 
ionisation since there is no further ring of electrons to be ionised and 
calculation shews that nuclear disintegration is non-existent, or at any rate 
unimportant, even at the central temperatures of the white dwarfs. Thus the 
if-ring ionisation must reach a turning point in every star, and pass through 
maxima and minima in succession. But as the masses of actual stars are all 
far below the critical value (X= 0 - 09) calculated in § 140, it appears that there 
must always be very great departures from the gas-laws before the turning 
point for AT-ring ionisation occurs. 
144. We proceed to consider the stability of the various types of con 
figuration which have been under discussion. 
If for the moment we neglect all dependence of the rate of generation of