VARIABLE STARS 
203 
Here the dissipation — T x dQ is negative for the inner part of the star 
up to about i 0 — 2-5 and positive beyond, and it is touch and go whether 
the integrated dissipation is positive or negative. I think that, if allowance 
were made for the increase of e towards the centre, an actual integration 
for this example would show the dissipation to be negative, so that the 
pulsations would increase up to the natural limit. 
This method of accounting for Cepheid variation seems hopeful because 
it at the same time accounts for its rarity. It definitely requires a deviation, 
but not a very extravagant deviation, from the state of things found in 
the more typical stars. 
A point of some importance has been urged by S. Shinjo and by J. H. 
Jeans in criticism of the pulsation-theory of Cepheids. We have considered 
only the fundamental vibration of the gas sphere; but harmonics, with 
one or more nodes in the interior are also possible. If these are excited 
the light-curves and velocity-curves of the Cepheids will contain a number 
of incommensurable periods superposed. No such harmonics have been 
observed. To meet this criticism we ought to show that for reasonable 
values of the constants the fundamental pulsation is maintained and the 
harmonic pulsations are dissipated. The lengthy calculations necessary to 
investigate this have not been tackled*. 138 
138. Up to a certain point the suggestion made in the last section can 
be supported. The necessary reversal of phase of the opacity-variation is 
brought about by reduction of the exponent in the absorption law from 
| to below 3 combined with a low value of y ; low y results either from 
low T or high (1 — j8). 
A high value of 1 — ¡3 corresponds to large mass. Cepheid variation 
is not found in stars of low mass and the typical Cepheids of types F and 
G are super-giants. This is in favour of the theory. 
But in addition we must find some cause for the reduced exponent in 
the absorption law. Even for high mass it must be brought down to a 
value not much above Lowering of F is comparatively unimportant, 
though as a matter of fact any cause tending to reduce the exponent of 
the absorption law abnormally would be likely at the same time to reduce F. 
Consider a star say of mass 10. There is only one stage of its evolution 
at which it can be a Cepheid owing to the empirical luminosity-period 
relation (§123). Since the luminosity depends mainly on the mass, and 
* [Dr J. Woltjer has shown me some interesting calculations relating to the first 
overtone, which correspond to those in Table 26 for the fundamental. It appears 
that the amplitude remains relatively small between the centre and the first node 
and becomes very large between the first and second nodes. W hether we follow the 
theory of § 136 or § 137 the motor part of the star is near the centre and the dis 
sipating part towards the outside. It seems therefore that the dissipation is relatively 
greater for the overtones, so that their absence is accounted for.]