186 
VARIABLE STARS 
and ST e /T e are nearly the same. The theoretical relation between them 
has not been worked out and there is no reason to expect exact equality. 
We might expect that the “eccentricity,” i.e. the deviation from simple 
harmonic oscillation, would increase with the relative amplitude; but there 
is no evidence of this in the Table. If there is such a tendency it is probably 
masked by the more prominent relation of eccentricity to period noticed 
by Hertzsprung (p. 180, footnote). 
The amplitude ST e of the temperature oscillation should also corre 
spond to the observed change of spectral type. The mean value ± -066 T e 
corresponds to a relative range -934 to 1-066 which represents an increase 
of about four-tenths of a type. The range given in our Table is generally 
greater than this. But a great deal depends on the characteristics chosen 
in assigning the type. For the types given in the Table the intensities of 
the hydrogen lines were used as the principal criterion. Adams and Joy* 
have shown that the hydrogen lines behave anomalously in the Cepheids, 
and the more general spectral features indicate a smaller change of type. 
For the mean of nine stars they found a range of six-tenths of a type from 
the hydrogen lines, and one-tenth from the general features of the spectrum. 
Adiabatic Oscillations of a Star. 
127. We shall now investigate the theory of the pulsation of a gaseous 
star. The exact theory of the changes of temperature and density, taking 
into account the flow of heat, involves differential equations of the fourth 
order which at present seem unmanageable. But the problem is simplified 
by noticing that owing to the high opacity of stellar material the oscil 
lations through the greater part of the interior are approximately adiabatic. 
We therefore start by considering adiabatic oscillations of a sphere of gas; 
we can afterwards calculate the flow of heat which would result, and 
determine whereabouts in the star it becomes so great as to render the 
adiabatic approximation invalid. 
Let P, p, T be the pressure, density, and temperature at a point 
distant | from the centre and let g be the value of gravity there. We fix 
attention on a particular piece of matter so that £ oscillates with the 
pulsation. Let f 0 , P 0 , p 0 , etc. denote the undisturbed values, and let 
f - fo = 8f = foil, P-P 0 = SP = P 0 P ± (127-1), 
and similarly for all the other variables. If the period of pulsation is 
27 r/n, , P x , etc., will contain a factor cos nt. We consider small oscillations 
and neglect the square of the amplitude. 
For adiabatic changes the pressure and density of a particular piece 
of matter are connected by 
Pozpy, 
* Proc. Nat. Acad. Sci. 4, p. 131.