OF ROTATING COMPRESSIBLE MASSES. 
201 
and probably dividing into two detached masses after passing through a series of 
pear-shaped configurations. In the present investigation it has been found that, 
except for an alternative possibility to be discussed later, a shrinking compressible 
mass will experience a very similar sequence of changes. It will pass through 
a sequence of figures which from their similarity to the spheroids and ellipsoids of 
incompressible masses, may be described as pseudo-spheroids and pseudo-ellipsoids. 
The pseudo-spheroids become unstable when a certain degree of flatness is reached 
and give place to a series of pseudo-ellipsoids, these in turn become unstable when a 
certain degree of elongation is reached and give place to a series of pear-shaped 
figures which probably end by fission into detached masses. 
46. It may simplify the presentation of the detailed results obtained, if we confine 
our attention at first to the innermost strata, this being supposed to mean strata so 
near the centre (which is taken as origin) that terms in x i , x 2 y 2 , &c., may be neglected 
in comparison with terms in x 2 , y 2 , z 2 . It has been found that these strata will be 
first truly spheroidal, then truly ellipsoidal, then pear-shaped, each change of figure 
corresponding, as in the incompressible mass, to a passage through a point of 
bifurcation of the system as a whole. 
The problem has been studied in detail for a pressure-density law of the form 
p — — cons.,.(175) 
and special attention has been paid to the exact position of the configuration at which 
pseudo-spheroidal configurations give place to pseudo-ellipsoidal ones. This configu¬ 
ration we may refer to as the ellipsoidal point of bifurcation ; its special importance 
will appear later. The corresponding configuration for an incompressible mass is the 
Maclaurin-Jacobian point of bifurcation ; at this point the semi-axes a 0 , c 0 and the 
angular velocity w 0 are given by 
a 0 = 1T972, c 0 = 0-69766, 
so that the equation of the boundary is 
x 2 + y 2 z^_ 
a 2 c 2 
For a compressible mass, the equations of the innermost strata of constant density 
at the ellipsoidal point of bifurcation have been found to be 
2 2 
^-^[1 — 0-0163076 —(0*00384 + 0-01850 (y —2)) e 2 —...] 
+ ^[l+0'05634e + (0-01598 + 0-06565 (y —2)) e 2 +...] = &=£, . (176) 
c 0 p 0 — <T 
where p 0 is the density at the centre, cr that at the surface, and e stands for (p 0 — cr)/p 0 . 
For the important case of a gaseous mass, e is of course equal to unity. 
Wo 2 /2t r = 0-187 12 /q, 
