334 
The Great Nebulae 
[ch. XIII 
as to how far the observed figures of the regular nebulae can be explained as 
the figures assumed by masses rotating under their own gravitation. The 
theoretical material necessary for such an inquiry has already been assembled 
in Chapters vm and ix. 
In Chapter ix we examined the configurations assumed under rotation by 
a mass of matter of a type which we described as the extended Roche’s model, 
this consisting of a central nucleus of homogeneous incompressible matter 
surrounded by a light atmosphere of negligible density. The possible con 
figurations for a given mass rotating with a given uniform angular velocity are 
of the general type of those shewn in fig. 54. Here the shaded part represents 
the central mass, which assumes the shape of the Maclaurin spheroid appro 
priate to the given rotation, while the outer curves are the external closed 
equipotentials of this mass in rotation. 
The rotating mass may have any one of 
these external equipotentials as its boun 
dary, and selects that particular one whose 
volume is just adequate to contain its own 
atmosphere. If even the outermost of the 
closed equipotentials is inadequate to con 
tain the whole atmosphere, the mass fills 
the outermost lenticular shaped equipotential, and the remainder spills over 
into the equatorial plane. 
We notice at once that this series of equipotentials have very much the 
shape of the “ elliptic ” nebulae which occupy the lower half of the Y-diagram. 
A mass rotating with given angular velocity assumes these various forms 
according to the extent of the atmosphere which surrounds it. We add a bit 
more atmosphere to an E 3 figure and it becomes E 4 ; subtract a bit and it 
becomes E 2. But the sequence of figures possible for a given velocity of 
rotation is limited at both ends ; it is limited at one end by the bare Maclaurin 
spheroid and at the other by the last closed (sharp-edged) equipotential of the 
Maclaurin spheroid. 
The limits in both directions can be extended by varying the angular 
velocity of rotation ; an increase of speed changes an A 1 3 mass into E 4, while 
a decrease of speed changes an E‘S mass into E 2. With zero rotation every 
mass, no matter how great or how small its atmosphere, becomes E 0, so that 
this fixes the limit in one direction. The limit in the other direction is that of 
the sharp-edged equipotential surrounding the critical Maclaurin spheroid 
which is just about to elongate itself into a Jacobian ellipsoid. This is not, 
however, of shape E 7 but of shape E 5 3. 
The same general sequence of configurations is exhibited by almost any 
model in which the mass is well concentrated towards the centre, as for 
instance the adiabatic model discussed in § 235. The limits here are again 
E0 at one end, and at the other end the sharp-edged equipotential of the