Febbuabt 7, 1919] 



SCIENCE 



135 



(just as the radius of the earth exceeds that 

 of the orbit which the earth's center describes 

 about the center of gravity of the earth and 

 moon). If, as is very probable, their periods 

 of rotation and revolution are the same, the 

 actual motion of one of these stars would 

 closely resemble a rotation about an axis pass- 

 ing nearly, but not quite, through its center, 

 and there would be no " leading side " in the 

 sense assumed by the theory. 



Moreover, the rotation of so large a body 

 would give rise to an equatorial velocity so 

 large as to make all the lines in the spectriim 

 wide and diffuse; whereas they are actually 

 narrower than in most stars. 



It therefore appears improbable that these 

 bodies are reaUy spectroscopic iinaries, and 

 more likely that the line displacements arise in 

 some other manner than from orbital motion. 



A further argument against both these 

 theories is that, if the direction of the rota- 

 tion or orbital motion in any system should 

 be reversed, the resulting variation would show 

 a slow rise to maximum and a rapid fall — 

 which has never been observed. If the varia- 

 tion arose from orbital motion the same effect 

 would be produced by observing the star from 

 any point on our line of sight, but behind it. 

 That variations of this sort should be of equal 

 geometrical probability to the others, and yet 

 never be observed among a hundred cases, is 

 altogether beyond reason. 



Though these theories appear therefore to 

 be unsatisfactory, it must be frankly admitted 

 that the pulsation hypothesis is not yet in a 

 position to explain positively the form of 

 the light-curve or the apparent variations in 

 radial velocity. So far, all that can be said 

 for it is that it does not seem to fall foul of 

 any obviously fatal difficulties, and that it 

 appears likely to be fairly flexible. 



A detailed mathematical study of the pos- 

 sible modes of vibration of a compressible, 

 gravitating, radiating and possibly rotating 

 mass of gas may lead to the solution of the 

 problem. In spite of the evident difficulty of 

 such a disciision it is to be hoped that it will 

 soon be attempted — ^perhaps by the new and 



brilliant school of English mathematicians of 

 which Eddington and Jeans are the leaders. 



In the present state of our knowledge, the 

 following admittedly speculative considera- 

 tions may be of interest. 



From Shapley's careful work it appears 

 that, among the Ceplieids proper, the absolute 

 magnitude and color index are practically 

 linear fimctions of the logarithm of the 

 period. For a star of color index 0.75 (similar 

 to the sun's) the period is 7 days, and the 

 median absolute magnitude ■ — 2.3, corresponds 

 to a luminosity 700 times that of the sun. It 

 seems fairly safe to assume that the surface 

 brightness of such a star is equal to that of 

 the sun, so that its diameter may be estimated 

 as 26 times the sun's. 



Cepheids of longer periods are brighter and 

 redder, the absolute magnitude decreasing 

 (numerically) by 1".0, and the color index 

 increasing by 0™.4 if the period is doubled. 

 This increase of color index indicates almost 

 certainly a decrease of surface brightness. 

 From the existing data, it sems probable that 

 the change in surface brightness, expressed in 

 stellar magnitudes is fully three times that 

 in the color index. To obtain a total lumin- 

 osity one magnitude greater, with a siu^ace 

 brightness 1.2 magnitudes less, the diameter 

 must be increased 2.7 times. Hence it appears 

 probable that a Cepheid of 14 days period is 

 of about 70 times the sun's diameter, while 

 one of 40 days' period (about the longest 

 usually met with, of about twice the diameter 

 of the earth's orbit — very large, it is true, but 

 undoubtedly still of stellar and not of nebular 

 dimensions. 



A typical cluster variable 100 times as 

 bright as the sim, and of color index 0.15, 

 should be of about four times the sim's s\ir- 

 face brightness, and four and a half times its 

 diameter. 



If now we adopt the pulsation hypothesis, it 

 follows that, as in the case of all gravitational 

 oscillations, the period will be inversely pro- 

 portional to the square root of the density. 



Hence doubling the period corresponds to a 

 four-fold diminution o* the density. But we 



