The Adiabatic Model 
263 
238, 239] 
tc — 22 will begin to lose matter equatorially at precisely the moment at 
which the pseudo-spheroidal form becomes unstable and gives place to the 
pseudo-ellipsoidal form. 
I have calculated the coefficients which occur in the equation of the 
boundary for the critical case of re — 2*2, and find the shape of boundary to 
be that of the outermost curve in fig. 43, but unfortunately it is not possible 
to draw the figure with much accuracy in the neighbourhood of the sharp 
edge. The interior curves are equipotentials and so are also surfaces of 
constant density and temperature. 
239. For the special value re = 2’2, equation (2381) becomes 
= 0-18712 + 0-06827 ( p -°^) + 0-03022 (>—Y + (239-1). 
27ryp \ Po ) V Po / 
The general series of which the first three terms are here written down 
is probably convergent up to the value (p 0 — <r)/p 0 = 1, but it is not easy to 
determine the value to which it converges. At a guess the value of tf/Ziryp 
appears to converge to about 0"31. 
In discussing the incompressible mass, the Maclaurin spheroid was found 
to become unstable when the rotation was given by 
= 018712. 
Z.TT'yp 
In Roche’s model, which, in a sense, represents the extreme limit of com 
pressibility, the rotation at which the mass began to break up was given by 
= 0-36075. 
2t TT'yp 
In the present model, we have modified the physical conditions until the 
two processes occur simultaneously for the same value of &> 2 /27ry/5; it is then 
natural that this value of <« 2 /27r7p should be intermediate between the values 
0-18712 and 0-36075. 
Equation (2391) and the more general equation (238*1) shew that the 
first effect of compressibility is to increase the value of w^/'Iir^p at which the 
pseudo-spheroidal form first becomes unstable. Or, to put it in another way,