OF ROTATING COMPRESSIBLE MASSES. 
207 
of about one-quarter of that of the substance in its solid state, and for the permanent 
gases this may be taken to be about a quarter of the density of water. 
Thus a mass of gas will lose matter equatorially until it has shrunk to a density 
of about a quarter of that of water, after which it will elongate and divide up. 
Assuming the relation w 2 = 0'36 x 27 rp, the mean density p and the period of rotation 
in days (P), will be connected, so long as the mass is losing matter equatorially, by 
the relation 
p = 0-035 -P P 2 , 
the density p = i corresponding to a period of about 9 hours and p = \ to a period 
of 6^ hours. When this stage is reached the process of elongation followed by 
fission begins. Assuming that when fission is complete we have two stars of 
approximately equal mass and mean densities p — \ revolving round one another 
almost in contact, the period of this system would be about a day. 
55. The critical density which we have conjectured to be about one quarter is 
perhaps not far from that of the average B-type star.* Thus, subject to the 
assumptions on which we have been working, fission ought to begin at about B-type. 
This is in very close agreement with the results obtained by Campbell in his 
“ Second Catalogue of Spectroscopic Binary Stars.”! 
It is not, however, in agreement with the results obtained by Shapley J in his 
“ Study of the Orbits of Eclipsing Binaries.” Adopting Bussell’s view of stellar 
evolution, our result would show that giant stars (except, possibly, of A-type) 
should be pseudo-spheroids; only B and dwarf stars could form binaries. Shapley 
discusses 93 systems ; 88 are of B or dwarf type, but only 21 have densities greater 
than 0'316, and only 57 have densities greater than O’l. For one star, W Crucis, 
of which the orbit has been determined by Bussell, § the density of the brighter 
component appears to be of the order of 0'000002. We are led to inquire under 
what physical conditions, different from those we have assumed, it can be possible 
for fission to occur while the density is still far below the value to which our 
analysis has led. 
56. Consider the simplest problem of a spherical mass at rest, the equations of 
equilibrium being 
dp _ av 
dr P dr 
where G is the gravitation constant. If p 0 is the pressure at the centre and B the 
radius, we obtain, on integrating from the boundary to the centre, 
p 0 = 47tG | — 2 J pr 2 dr dr. 
* Russell estimates the density of a giant A-type star at 0 -1 (‘ Nature,’ 113, p. 282). 
f ‘Lick Observatory Bulletin,’ 1910. 
\ ‘Princeton Observatory Contributions, No. 3 ’ (1915). 
§ ‘ Astrophysical Journal,’ 36, p. 146. 
2 E 2 
