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111. Rakerian Lecture, 1917. — The Configurations of Rotating 
Compressible Masses. 
By J. H. Jeans, M.A., F.R.S. 
Received May 14,—Lecture delivered May 17, 1917. 
1. On the supposition that astronomical matter may be treated as incompressible and 
homogeneous, a single star rotating freely in stable equilibrium can be spheroidal or 
ellipsoidal, but of no other shape, while Darwin has shown that both components of 
a binary star must be very approximately of the ellipsoidal shape. 
Both for the interpretation of astronomical observations and for the more general 
purposes of cosmogony, it becomes of importance to examine how the sequence of 
figures assumed by an ideal homogeneous mass will be modified by the compressibility 
and non-homogeneity of actual astronomical matter. 
The general mathematical problem of determining the configurations of stable 
equilibrium of the most general compressible mass is one of great complexity, but 
some important simplifications can be introduced by the use of general considerations. 
These are discussed in §§ 2-5 of the present paper. In § 10 we abandon the general 
problem and turn to a detailed study of the configurations possible when the 
compressibility is such that pressure and density are connected by the law 
p = Kp y — cons., 
where k and y are constants, this of course including the important case of a gas in 
convective equilibrium. The results obtained are summarised, and their astronomical 
bearings discussed, in §§ 45-58. The paper is arranged so that these last sections 
contain the main results of the paper in a form which is free from mathematical 
technicalities ; it is hoped that they will prove intelligible to readers who have 
omitted the more mathematical sections. 
2. For a mass of matter of the most general kind, rotating with angular velocity 
w about the axis of z, the equations of relative equilibrium are three of the type 
.(i 
.( 2 ) 
[Published April 2, 1919. 
dp dQ , 
' "aV &c -’ 
dx 
where 
VOL. CCXVIII.-A 563. 
Q = V + ^oo 2 (x 2 + y~). 
Y 
