146 Compressible and Non- Homogeneous Masses [CH. vn 



Clearly it is of the utmost importance to cosmogony to know under what 

 conditions streams of matter will be thrown off in this way. We have found 

 two ways in which a mass can break up the one by fission, and the other by 

 the ejection of streams of matter from a point or points on its boundary. In 

 an actual astronomical mass, which will happen first ? 



This question is one which can only be solved by detailed analysis in 

 special cases. Now there is an infinite variety of arrangements of compressible 

 matter possible, while the solution of even a single case is a problem of con- 

 siderable difficulty and complexity. It therefore behoves us to choose the 

 special cases which we attempt to solve with skill and care, so as to economise 

 labour as much as possible. 



148. Compressibility of matter is of course associated with variations of 

 density in the compressible mass, and the greater the compressibility of the 

 matter, the greater these variations of density will be. In Chapters III VI 

 we have solved the problem in the special case of a mass having no com- 

 pressibility and so having no variations of density. 



This problem formed in a sense a limiting case of the problem of the 

 motion of a compressible mass. At the other end of the general problem 

 there will be another limiting case in which the compressibility is so great 

 that infinite variations of density may be expected. Mathematically this 

 limiting case may be specified by the condition that the density is infinite or 

 zero at different places. Physically, as we shall now see, this limiting case is 

 not so artificial as its mathematical specification might lead us to suppose. 



149. For a mass of gas at rest in isothermal equilibrium, the density at 

 great distances from the centre falls off' as 1/r 2 . The general law of density 

 has been obtained by Darwin* and others t. But without detailed analysis 

 it is clear that, at a sufficient distance from the centre, the law of density 

 must become | 



P = p (a*/r*), 



so that, when viewed from a very great distance, the density may be regarded 

 as infinite at the centre and zero everywhere else. The total mass is how- 

 ever infinite, so that a finite mass of gas in isothermal equilibrium will be 

 of zero density everywhere. 



Similarly for a mass of gas in adiabatic equilibrium with the ratio of the 

 specific heats <y equal to 1J, the law of density is 



* " On the Mechanical Conditions of a Swarm of Meteorites and on Theories of Cosmogony." 

 Phil. Trans. 180 A (1889), p. 1, and Coll. Works, iv. p. 362. 

 f For detailed references see Darwin's paper. 

 J L. c. p. 377. 

 A. Schuster, Brit. Ass. Report, 1883, p. 428. 



