239-241] Summary 265 
If inequality (240*3) is not satisfied, d 2 A/dt 2 must be positive, so that the 
mass must continually increase its rate of expansion, or, if it is contracting, 
the contraction will be checked and ultimately replaced by an expansion. 
According to the ideas of Laplace and Roche the ring of matter which 
was thrown off from the sun, and ultimately formed the planets, was rotating 
at one time as a closed ring with approximately the same angular velocity 
as the main mass of the sun. If p s was the mean density of the sun, <w 2 would 
be given by 
a ) 2 = 0'36075 x 27r r yp 8 
whence, from inequality (2403) 
p > 0-36075 p s (240-4). 
This shews that unless the ring condensed at once so as to have a density 
of at least a third of the mean density of the main mass, it could not rotate 
steadily but would continually expand under the centrifugal forces arising 
from its own rotation. 
Summary. 
241. This and the preceding chapter have been occupied with an investi 
gation into the configurations assumed by masses rotating freely in space under 
their own gravitational forces. Before leaving the theoretical discussion, and 
turning our attention to the actual problems of astronomy, it may be profitable 
to summarise the main theoretical results which have been obtained. Some of 
these results have been quite general, but we have also investigated in detail 
the behaviour of certain simplified model masses. These models have been four 
in number : 
(A) The incompressible model, consisting of a mass of homogeneous 
incompressible matter of uniform density. 
(B) Roche’s model, consisting of a point nucleus of very great density, 
surrounded by an atmosphere of negligible density. 
( C ) The generalised Roche’s model, consisting of a homogeneous incom 
pressible mass of finite size and of finite density, surrounded by an atmosphere 
of negligible density. 
(D) The adiabatic model, consisting of a mass of gas in adiabatic 
equilibrium, so that the pressure and density are connected at every point by 
the relation p = Kp K , where K and k retain the same values throughout the 
mass. 
The two models A and B are limiting cases of the more general models C 
and D. If s denote the ratio of the volume of the atmosphere to that of the 
nucleus in the generalised Roche’s model G, then model '0 degenerates into 
model A when s = 0, and degenerates into model B when s — oo . Similarly 
the adiabatic model D degenerates into model A when 7 = oo and into model 
B when 7 = 1 ^ (cf. § 228). The relation between the four models is represented 
diagrammatically in fig. 44. 
J 
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