293—¿ui] Physical Interpretation 03o 
critical figure (k = 2‘2) at which the pseudo-spheroid is just about to elongate 
into a pseudo-ellipsoid. This critical figure is that shewn in fig. 43 (p. 263), 
and its shape is about E 6T, but it is extraordinarily difficult to calculate and 
draw the figure with any accuracy. 
When k= oo, the corresponding figure is the Maclaurin spheroid of shape 
E 4 2, while when /c = l*2, the figure is the limiting Roche’s equipotential 
shewn in fig. 41 (p. 253) ; the shape of this is E 33. 
300. The S and SB branches, as well as the E branch, of the Y-shaped 
diagram, permit of interpretation as the configurations of rotating masses. 
Imagine that we gradually increase the rotation of a mass arranged 
according to the simple Roche’s model in which the dense central nucleus is 
so small that it may be treated as a point. Its configuration passes through 
a sequence of figures of increasing ellipticity until it reaches the lenticular 
figure of shape E 3*3 shewn in fig. 41 (p. 253) ; at this co 2 /27ryp = 0*3607 5. Further 
rotation does not increase the ellipticity of figure beyond this, but, as we have 
seen, causes an ejection of matter from the sharp edge which forms the equator 
of the lens. The matter so ejected describes orbits in the equatorial plane and 
as there are no forces to move it out of this plane, it forms a thin annular 
layer of matter in this plane. While this ejection of matter is in progress, the 
remaining central mass retains the shape of the critical lens-shaped figure, 
while c» 2 /27ryp retains the constant value 0*36075, the increase in tu 2 being met 
by a proportionate increase in /5, caused by light matter being removed from 
the atmosphere while the whole of the dense central mass remains. 
In its essentials, the series of configurations so obtained consists of a central 
lenticular figure of shape E 3'3, extended in its equatorial plane by an annular 
layer which revolves around it much as Saturn’s rings revolve around Saturn. 
Apart from details of structure, this gives the general characteristics of the 
nebular configurations which Hubble classifies as S, the normal spirals. 
The generalised Roche’s model gives a similar series of configurations, as 
has already been indicated in fig. 54. An additional series is however possible 
for this model, since the central mass may possess so much angular momentum 
that the Maclaurin spheroids give place to Jacobian ellipsoids. The figure then 
becomes a pseudo-ellipsoid with three unequal axes, and may, if its atmosphere 
is of sufficient extent, proceed to shed matter in two streams from the ends of 
its major axis. The general type of configuration obtained in this way is 
shewn in fig. 55 (p. 336), which represents a cross-section in the equatorial plane- 
These configurations reproduce the general features of the nebular configura 
tions which Hubble classifies as SB (barred spirals). In the limiting case in 
which there is no atmosphere at all, these configurations reduce to ordinary 
Jacobian ellipsoids. 
301. We have already noticed (§ 235) that the gap between the simple 
Roche’s model and the incompressible model can be bridged in other ways than