145-147] General Theory 145 



As in 141, the values of p and dp/dn are determined at the boundary. 

 It follows from equation (398) that the arrangement of density is determined 

 throughout the layer of matter of type a. At the surface of transition to the 

 next layer, p and dil/dn are continuous, so that p and dp/dn are determined 

 at the boundary of layer 6, and so also through this layer. Hi fhis way the 

 configuration can be built up layer by layer, the configuration being uniquely 

 determined when the order of the layers is determined. 



The order of the layers is determined by conditions of stability. We shall 

 return to a discussion of this matter in the next chapter. For the present 

 we may notice that any arrangement will be unstable if energy can be gained 

 by an interchange of any two layers, the instability shewing itself by the 

 creation of convection currents which result in the actual interchange of the 

 layers in question. Thus the only arrangement of layers which can be stable 

 is that for which the potential energy is a minimum. For this arrangement 

 the results already obtained for a homogeneous compressible mass remain 

 true ; in particular the configuration is uniquely determined, and is stable as 

 regards internal vibrations, when the values of V T and &> and the shape of the 

 boundary are given. 



The Two Mechanisms of breaking up 



147. There are two conditions that must be satisfied by a configuration 

 of equilibrium ; the equations of equilibrium must be satisfied, and also p 

 must be positive everywhere. Now the linear series so far discussed have 

 been series of configurations such that the conditions of equilibrium have 

 been satisfied everywhere, but we have not introduced the condition that 

 p must be positive everywhere. Any region on these series in which p is 

 anywhere negative will represent configurations in which the equations of 

 equilibrium are satisfied but which are physically impossible through negative 

 pressures being demanded. Hence if at any point on a linear series the 

 pressure becomes negative, the series may be supposed to be abruptly termi- 

 nated at that point : the configurations beyond are of no physical interest. 



It is easily seen that p cannot change sign at a point in the interior of 

 the mass, for p can only change sign by first vanishing, and the points at 

 which p vanishes determine the boundary. But close to the boundary p can 

 change sign by passing through a zero value ; when this happens dp/dn 

 vanishes at the point of the boundary in question. Thus the normal force 

 9H/dtt vanishes, which means that the gravitational attraction of the mass is 

 just neutralised, and ultimately outbalanced, at this point by centrifugal and 

 tidal forces. When this happens a stream of matter will be thrown off from 

 the point in question. 



j. c. 10 



