147-iso] General Theory 147 



Again, when the mass of gas is viewed from a sufficient distance, the value 

 of p becomes infinite at the centre and zero everywhere else. The same is 

 true for any value of 7 from 1 to 1. The mass is infinite when 7 < 1-J, but 

 becomes finite when 7 = 1J. 



This same model, in which the density is infinite or very great over a 

 point or small concentrated area but zero everywhere else, has been largely 

 utilised by Roche * in his researches on cosmogony. For convenience we may 

 refer to it as " Roche's model." Roche interpreted it physically as referring 

 to a small and intensely dense solid nucleus surrounded by an atmosphere of 

 negligible density. In Roche's model, the whole of the mass is supposed 

 concentrated at the centre ; in this respect it differs from a mass of gas in 

 isothermal equilibrium, although giving a faithful representation of an 

 adiabatic mass for which 7 = 1^. 



ROCHE'S MODEL 



150. We have seen that Roche's model and the incompressible model 

 form the two limiting cases of the general compressible, mass. The latter has 

 already been studied in detail ; it is natural to begin our investigation of the 

 compressible problem with a discussion of the former. 



Roche's model has one great advantage over the incompressible model. 

 For in studying the configurations and motion of an incompressible mass, one 

 of the main difficulties was found to lie in the determination of the gravita- 

 tional potential. Now in Roche's model no such difficulty occurs ; the mass 

 is supposed collected at one or more points and the gravitational potential 

 reduces to M/r, or to a sum of such terms in cases where there is more than 

 one nucleus. Thus, when there is only one nucleus involved, the quantity 

 which has been denoted by H assumes the simple form 



= + F T + io> 2 (tf 2 + 2/ 2 ) ..................... (400). 



For given values of V T and o>, the surfaces 11 = cons, will be a system of 

 equipotentials of the usual type ; since 11 is uniquely determined as a function 

 of x, y and z t two different equipotentials can never intersect. Of the system 

 of equipotentials only one is suitable for the boundary of the gravitating mass, 

 this being picked out by the condition that the volume enclosed by it shall 

 be just adequate to contain the whole amount of the compressible matter. 



* " Essai sur la Constitution et 1'Origine du Systems solaire" (1873). Acad. de Montpellier, 

 Section des Sciences, vm. p. 235. See also Poincare, Lemons sur les Hypotheses Cosmogoniques, 

 Chap. in. 



102 



