358-360] The Pulsation Theory 
387 
the agreement appears very much less good, as has already been indicated in 
Table XXXI. 
The mathematical theory of pulsating spheres has been discussed very 
fully by Eddington*. Apart, however, from all detailed mathematical treat 
ment, the pulsation theory of Cepheid variation appears to encounter a serious 
and probably fatal objection at the very outset, which makes all further 
mathematical treatment superfluous. 
As in § 98, the total flow of radiation from the star is 
4nrr 2 H = — 
ar*T*« dT 
p 2 dr 
(3601), 
where a is a constant, and all the other quantities are evaluated near, but not 
quite at, the surface of the star. As the star pulsates, all the quantities on the 
right will vary harmonically, and I have shewn f that the variations of T and p 
will be in phase with one another, so that the variations in 47 rr 2 H, the total 
emission of radiation, will also be in phase with T and with R, the radius of 
the star. Since this total emission is also proportional to R 2 T e i , the variation 
in T e must also be in phase with R. 
Now the spectral lines of a Cepheid shew displacements which indicate a 
rhythmical advance and recession. On the pulsation theory this must be 
caused by the changes in the value of R. But the displacement of the 
spectral lines shews that the changes in T e and in the emission of radiation 
are nothing like in phase with R ; in general they are almost exactly a quarter 
period out, being in phase not with R but with dR/dt. 
This objection, which I first pointed out in 1926 was also advanced in 
dependently by Reesinck§ and its validity was finally conceded by Eddington |j. 
The pulsation theory might possibly be saved, as Eddington has remarked, it 
the displacements of the spectral lines had been wrongly interpreted, but the 
following considerations suggest that the hope is a very slender one. 
Since the bolometric luminosity at any instant is proportional to R 2 T e 4 , 
we have, as in equation (359T), 
m vi8 = Am — 10 log T e — 5 log R + constant (360-2), 
where Am is the bolometric correction. This depends only on T e , so that if 
m vis were in phase with R, as the pulsation theory requires, both would be 
in phase with T e and so with the spectral type of the star. Shapley^i has 
examined the spectral changes of 20 Cepheids in detail and finds that they do 
not coincide with those of m v ¡ s . I have found** that the light curves of most 
* M.N. lxxix. (1918), p. 2, lxxix. (1919), p. 177 and The Internal Constitution of the Stars, 
Ch. vixi. 
f Ibid, lxxxvi. (1926), pp. 86 and 574. Ï l.c. p. 90. 
§ Onderzoekingen over Ô-Cephei en overhet Cepheidenprobleem. (Dissertation, Amsterdam, 
1926), and M.N. lxxxvii. (1927), p. 414. 
|| M.N. lxxxvii. (1927), p. 539. 
** M.N. lxxxv. (1925), p. 810. 
If Astrophys. Joum. xxiv. (1916), p. 273.