206, 207] Comparison with Observation 209 



The value of T O depends only on the value of o> 2 at the boundary of the 

 figure, being determined by the relation M <w 2 VT^, where w is the angular 

 velocity at the sharp edge ^ = w . Clearly a lessening of o> 2 as we pass 

 inwards from the sharp edge -or = tn- will result in a lessening of the value of 

 the integral, and so in an increase in the value of r corresponding to any 

 given value of -nr, leading to the result stated. 



207. In most of the nebulae shewn in Plate III, as indeed in almost all 

 known nebulae, the evolution has proceeded somewhat beyond the formation 

 of the sharp edge; matter has, on our interpretation, already been ejected 

 from this edge. Thus before attempting a fuller interpretation of observed 

 nebulae, we ought to return to the theoretical problem, and attempt to trace 

 out the motion which is to be expected after the formation of the sharp edge. 



Let us consider first what may be expected to happen to the main 

 mass. The ejected matter, as soon as it is of sufficient amount, will exert 

 gravitational forces on the remainder of the mass, but in the earliest stages 

 of the motion, so long as the total mass of ejected matter is still small, the 

 gravitational field set up by this ejected matter may be neglected, and the 

 main mass may be supposed to be acted on solely by its own gravitation. 

 As the mass slowly shrinks, the radius of the critical circle on which centrifugal 

 force just balances gravity will also slowly shrink. Matter will be gradually 

 thrust across this circle much in the same way in which water would gradually 

 drip over the edge of a slowly shrinking cup*. 



During this early stage, it is impossible for the sharp edge ever to dis- 

 appear. For if at any instant it did so, the main mass would at once become 

 a^ain a mass rotating freely in space under its own gravitation ; the slightest 

 amount of further shrinkage would produce an increase of rotation which 

 would again result in the formation of a new sharp edge, and the ejection of 

 more matter. Thus the motion is one in which the main mass shrinks, 

 keeping always a sharp edge. Just enough matter must be ejected through 

 this edge for the main mass to remain always of the form of the critical figure 

 of equilibrium, the condition for this being that equation (534) shall always 

 remain satisfied. In the motion before .the sharp edge was formed, the 

 angular momentum of the mass remained constant, so that &> 2 /27r/5 increased 



* Poincare" (Lemons, p. 25) appears to follow Roche in believing that fairly violent oscillations 

 might be set up in the main mass, so that the rate of ejection of matter would periodically over- 

 shoot the amount necessary for equilibrium. Thus after an eruption there ought to be a period 

 of quiescence until the angular velocity has again overtaken the ejection of matter ; after this 

 another period of eruption, another period of quiescence and so on. 



Although every opinion expressed by Poincare must be considered with the greatest respect, it 

 is extremely difficult to find any dynamical justification for these supposed violent oscillations. 

 The oscillations of the main mass appear to be thoroughly stable (this has been rigorously proved 

 for Eoche's model in 150), and it is- hard to find any agency capable of forcing oscillations of any 

 but infinitesimal amplitude. 



j. c. 14 



