208 The Evolution of Rotating Nebulae [OH. ix 



angular velocity must be circular cylinders about the axis of rotation*. This 

 being the case, the three equations of equilibrium (535) have the common 

 integral 



and the boundary of the mass must be one of the surfaces 



(536). 



This equation can be put in the alternative form 



F+ <^ 2 (a? + i/ 2 ) = cons. 



where o> 2 is used to denote the mean value of &> 2 at all points of the equatorial 

 plane inside a circle of radius (a? + yrf. 



A nebula shrinking homologously in the way described in the last chapter 

 would increase its angular velocity at the same rate throughout, so that 

 uniformity of angular velocity, if once established, would not be disturbed by 

 homologous shrinkage. But if a nebula has shrunk from an approximately 

 isothermal condition to one in which there is a rapid temperature gradient 

 from surface to centre, then the outer parts will have fallen in much more 

 than the inner parts. In the absence of any viscosity at all the conservation 

 of angular momentum would require that the parts furthest from the axis of 

 rotation should have a greater angular velocity than the parts nearer the 

 centre. In such a case w 2 would increase with # 2 + y 2 , and when viscosity 

 acts, but without sufficient power to produce absolutely uniform rotation, 

 this increase of o> 2 with oP + y* will still persist to some extent. 



Thus in a natural nebula, or other rotating mass of gas, we should expect 

 a* 2 to increase as we pass from the centre outwards. This is the type of 

 motion observed in the sun, while the measurements of Pease f on the rota- 

 tion of the Andromeda nebula suggest a similar increase of angular velocity 

 with distance from the centre. 



It will be readily seen that the effect of such a variation in o> 2 is to change 

 the critical equipotential from the theoretical curves shewn in fig. 41 in the 

 direction towards the photographic curves exhibited in Plate III. The 

 general principle is sufficiently illustrated by a consideration of Roche's 

 model ( 152). Let ^ stand for x*+ y z and let T O be the value of r at the 

 sharp edge of the critical equipotential. The equation of this equipotential is 

 seen from equation (536) to be 



* Poincare, I.e. p. 32. 



f Nat. Acad. Sciences, 4 (1918), p. 21. It would perhaps be straining the evidence to regard 

 the variation of w with distance as being definitely established, the more so as Pease himself 

 does not interpret his measurements in this way. 



