205, 206] Comparison with Observation 207 



and it seems permissible to identify these actual figures, at least conjecturally 

 and tentatively, with the theoretical figures shewn in tig. 41. On this 

 suggested interpretation then, the nebulae shewn on Plate III are masses of 

 gas, or possibly clouds of dust, in rotation. Rotation has actually been 



Fig. 41. 



observed spectroscopically in some nebulae, as for instance the last nebula 

 shewn on Plate III, namely N.G.C. 4594 and the Andromeda Nebula M. 31, 

 while it would be difficult to imagine any cause other than rotation which 

 could account for the flattened symmetrical shape of the remainder. 



One point of difference perhaps appears between the theoretical and the 

 actual curves. The photographs shew curves which are somewhat less blunt 

 near the equatorial edge than the theoretical curves ; in some of the photo- 

 graphic curves the boundary appears to become convex to its equatorial 

 section at points near this edge. 



The theoretical curves have been obtained on the supposition that the 

 angular velocity has everywhere the same value ; the mass has been assumed 

 to rotate as a rigid body. If shrinkage were an infinitely slow process, or if 

 the action of viscosity were infinitely rapid, a rotating and shrinking mass 

 would rotate at every instant like a rigid body, but in nature viscosity acts 

 so slowly in a mass of gas that we have to contemplate the possibility of 

 uniform rotation never becoming established*. 



To examine the effect of non-uniform rotation, we return to the funda- 

 mental equations (386) to (388) of Chapter VII. Assuming the pressure to 

 be a function of the density, these may be expressed in the form 



*MW 



dx J p dx 



da;J p 



and these will be the equations of relative equilibrium even when &) 2 varies 

 from point to point in the mass. 



Differentiating the x, y equations with respect to y, x respectively and 

 subtracting, and treating the two other pairs similarly, we find 



9ft) 2 _ 9co 2 9&) 2 _ ~ 

 'dy ()x uz 



so that ft) 2 must be a function of x*+y\ Thus the surfaces of constant 



* Cf. Poincare, Lemons sur les Hypotheses Cosmogoniques, p. 28. 



