212 The Evolution of Rotating Nebulae [CH. ix 



proof, has shewn that when a ring is formed, the mean density p A of the ring 

 must always be greater than o> 2 /27r. By a simple extension of Poincare's 

 method, a much more general result may be obtained. 



Consider the equilibrium of any mass whatever which is rotating approxi- 

 mately as a rigid body with angular velocity co. Let the motion be referred 

 to axes rotating with uniform velocity co, and let u, v, w be the velocity relative 

 to these axes at any point of the mass. Let us assume that u, v, w are small 

 compared with cor, the velocity of rotation. 



Then the equations of motion are of the form 



du dv i dp 



-17 =-5- + (D'X- ^~ . 



at ox p ox 



Differentiating with respect to x, y, z and adding, we obtain, on using 

 Poisson's relation V 2 F= - 4-7T/3, 



d (du dv dw\_ ra A 3p\ , !/! %A , 9 i ld P 



+ + ~ P ~ " + + 8* (p 



Let us multiply by dxdydz, and integrate throughout the whole volume 

 of any detached mass ; let us further transform the first and last integrals by 

 Green's Theorem. We obtain 



j l!(lu + mv + nw) dS = j [J(2a> 2 - 4mp) dxdydz + fj - |? dS, 



where d/dv denotes differentiation with respect to the inward normal. 



The integral on the left measures the rate of expansion of the volume A 

 of the mass under consideration, so that the equation may be put in the 

 form 



Now p vanishes at the boundary of the mass and must, if disintegration 

 is not to occur, be positive at all points inside. It follows that dp/dv must be 

 positive at every point of the boundary, and hence that the final term in 

 equation (537) must be positive. Thus if p < o> 2 /27r, the whole right-hand 

 member of the above equation is positive and d 2 A/dt 2 must therefore be positive. 

 If the mass is relatively at rest, it starts expanding ; if it is already expanding, 

 it expands still more rapidly ; if it is in process of contraction, the contraction 

 is checked. A condition for a steady state is that d*A/dt 2 shall vanish, and 

 this clearly requires that p shall be greater than &r/27r, which is Poincare's 

 result. 



211. Returning now to the particular problem in hand, co will be sup- 

 posed to be the angular velocity of the Laplacian ring. In the earliest stages 

 of the motion this must be very approximately equal to that of the main 



