VARIABLE STARS 201 
perhaps also by increased density, the requirement that shall have the 
same phase with T x is naturally fulfilled. 
The mode of operation is easily realised. At the time of greatest 
compression heat is being generated in a star at more than the average 
rate needed to replace loss; this strengthens the ensuing expansion. At 
the time of greatest expansion there is a net loss of heat which diminishes 
the opposition to the ensuing compression. 
The condition for an exact balance just maintaining the pulsation is* 
e “ ei + S = 0 (136-1). 
Thus for a mean region we have by (134-25) 
= — 2-2 [| x ] c cos nt (136-2), 
which may be compared with 
Pi = - 3-5 [| x ] c cos nt, T x = - 1-2 [&]<. cos nt. 
The rate of liberation of subatomic energy must increase nearly pro 
portionately to the square of the temperature or to the two-thirds power of the 
density in order to keep the pulsations going. 
There seems to be no possible cause for decay of a mechanical pulsation 
other than the leakage of heat. There are practically no viscous forces 
operating in a symmetrical pulsation. It would seem that if e increases 
faster than T 2 pulsation must occur. This condition of the star may be 
described as one of “ over-stability.” In the usual kinds of instability a slight 
displacement provokes forces tending away from equilibrium; in over 
stability it provokes restoring forces so strong as to overshoot the corre 
sponding position on the other side of equilibrium and set up an increasing 
oscillation. 
The limiting condition e x = 2T x is slightly modified according to the 
values chosen for ft and T, but no very substantial alteration is possible 
except in stars of small mass. I think we can safely say that if e increases as 
fast as p or as T s every star of mass greater than Sirius will be set pulsating. 
The application of this condition to non-pulsating stars is quite as 
interesting as its application to pulsating stars. It gives an upper limit 
to the rate of variation of e with p and T in ordinary non-Cepheid stars, 
and limits the kind of method by which subatomic energy may be liberated. 
For example, if the liberation depends on encounters of two agents 
(electrons and atomic nuclei) the number of such encounters per unit mass 
will be proportional to the density and presumably the simultaneous 
increase of temperature will also stimulate the liberation of energy. But 
we see that this increase with density and temperature is too rapid to be