CHAPTER VIII 



THE EVOLUTION OF GASEOUS MASSES 



GENERAL THEORY 



188. We may begin with a consideration of the general motion of a 

 cloud of nebulous astronomical matter. This may be supposed constituted 

 either of gaseous molecules or of dust particles ; for convenience we shall 

 speak of the separate particles as molecules. 



The equations of motion of a single molecule are 



d*x v 

 m d? = 6 ' 



whence we obtain, by direct algebraic transformation, 



This is the equation used by Clausius to establish his celebrated theorem 

 of the Virial. Its importance in theoretical astronomy has been pointed out 

 by Poincare* and Eddingtonf. 



Summing the three equations such as (501), we find 



On further summing this equation for all the molecules, or other particles 

 of the mass under consideration, we obtain 



zZ) .................. (502), 



where / is the moment of inertia about the origin, defined by 



/ = 2ra (# 2 + f 4- z z ), 



and T is the kinetic energy of translation of the molecules of the gas. The 

 last term 2 (xX + yY + zZ) is the Virial of Clausius ; call it V. 



To evaluate the virial, we fix our attention on two particles of masses 

 m lt m 2 at the points x lt y lt z^ and x z , y z , z^ respectively. Let the force exerted 

 by the second on the first have components H, H, Z, so that the force exerted 



* Leqom sur les Hypotheses Cosmogoniques, p. 94. 

 t Monthly Notices R.A.S. 76 (1916), p. 525. 



