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253] Internal Condition of Rotating Stars 
The investigations of the present chapter have provided a picture of the 
same star when in rotation. We have found that viscosity has so slight an 
effect in the core that the whole life of the star is inadequate for it to establish 
uniformity of angular velocity. The different layers of the core continue to 
rotate as though viscosity were non-existent, with angular velocities deter 
mined mainly by the past history of the star. On the other hand, viscosity and 
the braking effect of the transmission of radiation are extremely potent in the 
envelope, and we have seen that in a comparatively short time the angular 
velocity (o in the envelope will conform to a law which, to a sufficient approxi 
mation for our present purpose, may be expressed in the form co oc 1/r 3 . 
In Chapters vm and ix we studied the configurations assumed by masses 
rotating with angular velocities which were supposed uniform throughout each 
mass. For a fairly incompressible mass we found series of configurations which, 
under slow rotation, approximated to spheroids, and with higher velocities of 
rotation were pseudo-spheroids, pseudo-ellipsoids, or pear-shaped figures, 
according to the amount of angular momentum with which the system was 
endowed, the series finally ending, for high angular momentum, in two 
detached and separate masses revolving about one another. If, however, the 
compressibility approximated to that of a perfect gas, the configurations were 
pseudo-spheroidal throughout, the series of configurations coming to an abrupt 
end through the pseudo-spheroid forming a sharp edge along its equator, so 
that its form became that of a double convex lens. Beyond this no configurations 
existed in which the mass rotated with uniform velocity throughout; if the 
mass acquired further angular momentum beyond that corresponding to the 
lenticular figure, it shed matter from its equator, and this shed matter rotated 
at rates different from that of the main mass. 
We have seen that the angular velocity cannot be uniform in an actual 
star. If the star is spherical, or approximately spherical, in shape we have seen 
that the angular velocity in its outermost layers will fall off approximately as 
l/r 2 . When the star is not of spherical shape this law is no longer generally 
true, but the same physical principles from which we deduced it shew that in 
the equatorial plane of the star the angular velocity must fall off as 1/ra- 2 , where 
nr is the distance from the axis of rotation. When once this law is established, 
the lens-shaped figure with a sharp edge can never be reached. The lens-shaped 
figure is determined by the condition that centrifugal force exactly balances 
gravity on its equator. Now if co falls off as l/®- 2 , centrifugal force to 2 ® must fall 
off as 1 /® 3 , and so falls off more rapidly than gravity, which only falls off about as 
rapidly as l/® 2 . In an actual rotating star the law <w oc l/® 2 is reached with 
infinite rapidity in the outermost layers of all, so that the lenticular figure can 
never be reached, and the shedding of matter from the equator of a lenticular 
figure (which, incidentally, formed the basic conception of Laplace’s nebular 
hypothesis) cannot occur in an actual star. 
Thus if a star acquires an ever-increasing amount of rotation, its outer