390 
Variable Stars 
[ch. XV 
95 per cent, of its total mass would have 61 times the mean density of the 
star, while if it had the configuration of the limiting Jacobian ellipsoid, its 
semi-major axis would be 047 times the mean radius of the star. Never 
theless, the general conception is not free from difficulties; for instance, if a 
Cepheid has a period of a month, equation (360T) shews that the mean density 
of its core ought only to be 0 ' 00002 , and it is not at present easy to imagine 
how matter of this low density can shew so little central condensation of 
mass that a Jacobian ellipsoid can be a possible figure of equilibrium. The 
difficulty is only one aspect of a wider one which affects the theory of liquid 
giant stars in general. 
362 . Since fission commences through a pseudo-ellipsoidal configuration, 
it can only begin in stars whose central cores are in a liquid or a semi-liquid 
state. Thus, on the fission theory, Cepheid variation can only occur in stars 
whose centres are near to the liquid state, and this would restrict it to stars 
lying on the left-hand edges of the various bands of stability in the tempera 
ture-luminosity diagram (cf. fig. 13, p. 161). 
If the mean spectral types and absolute magnitudes of normal Cepheids 
are mapped out on such a diagram* the majority are at once seen to cluster 
along the extreme left-hand edge of the Z-ring area of stability, which is 
precisely the type of position which the fission theory requires for them. 
Many of them seem actually to have overstepped the edge so that, if fission 
is in progress, the final product will be a binary star on the main sequence. 
Stars of the /3-Cephei type, which are generally regarded as Cepheid 
variablesf, occupy a corresponding position on the main sequence. The more 
normal cluster-variables conform less well to the anticipations of theory, some 
lying on the main sequence, some near the left-hand edge of the Z-ring 
branch, and some sprawling over the space between. Bailey J and Shapley§ 
have found that there are three distinct groups of cluster-variables, differenti 
ated by periods, form and range of variation. 
363 . As the process of fission progresses, the pseudo-ellipsoidal form must 
become unstable. The core of the star will now undergo the oscillations 
resulting from secular instability. At this stage the star will have two distinct 
periodicities, those of its rotation and of its secularly unstable pulsations. Otto 
Struve has suggested || that the rapid changes in the velocity and light curves 
of stars such as /3 Cephei, 7 Ursae Minoris, 12 Lacertae and others of the 
/3 Canis Majoris type, may be explained in terms of the superposition of 
periodic rotations and pulsations. This would obviously fit in exactly with the 
fission theory, these stars being interpreted as rotating masses executing the 
secularly unstable pulsations which immediately precede fission. 
* See, for instance, Bruggencate, Die Naturwissenschaften, xl. (1926), p. 910. 
t Henroteau, M.N. lxxxvi. (1926), p. 256, and B. H. Baker, Ast. Soc. Pac. xxxvm. (1926), p. 86. 
t Harvard Annals, xxxvm. (1902), p. 132. § Harvard Circular, cccxv. (1927). 
i| M.N. lxxxv. (1925), p. 75.