356 
The Great Nebulae 
[CH. XIII 
Thus the physical conditions at the centre of a nebula must be precisely 
the same as those we have found to prevail at the centres of the white dwarfs, 
and, apart from their rotation and outer structure, the extra-galactic nebulae 
may be regarded as being merely white dwarfs of colossal mass. 
326. If the foregoing calculation had been presented in another order, we 
could have traced the changing physical conditions which succeed one another 
as the nebula shrinks, and would have found that the final “ white dwarf” state 
was reached when the nebula had shrunk to a radius of the order of magnitude 
of400 parsecs, this being the radius that corresponds to a “ white dwarf” tempe 
rature. On the hypothesis we are now considering, it is this that determines 
the dimensions of the great nebulae, and accounts for their being all of ap 
proximately equal size, as shewn in Table XXIX. 
The equilibrium of the nebula along its polar axis is determined by the 
usual hydrostatic equation 
into which the rotation does not enter at all. Thus when a nebula shrinks 
in the way just imagined, the final length of its polar axis ought to be 
approximately independent of its velocity of rotation. This latter quantity 
determines the ratio of the two axes, but the length of the minor axis must be 
determined almost entirely by the amount of matter in the mass. 
Thus we can imagine the original nebular medium to condense, under 
gravitational instability, into nebulae of approximately equal mass M, rotating 
with different angular velocities. The minor axes of these masses will be all 
equal, but their major axes will depend on the varying degrees of rotation. 
Or, more precisely, the minor axes will be ranged about a mean with the 
same dispersion no matter what their rotation, while the equatorial extension 
is determined by the rotation. 
This is exactly what Hubble’s measurements reveal in actual nebulae. 
327. If the rotation of the nebulae represents angular momentum arising 
from random currents in the primaeval nebular medium, then the angular 
momenta of the different nebulae about any one axis in space ought to shew 
a Gaussian distribution. The total angular momenta M ought accordingly to 
shew a Maxwellian distribution of the form 
M 2 e~* M dm (327T), 
the axes of rotation being oriented at random. If the nebulae are treated as 
being all of the same mass and dimensions, M is proportional to the angular 
velocity, and so by equation (303-3) to e^, the square root of the ellipticity, 
so that the law of distribution of ellipticities ought to be of the form 
e* e~ kt de (327-2).