218 The Evolution of Rotating Nebulae [OH. ix 



universe the " island universe " in which we live while the smaller nebulae 

 are of masses equal to thousands of that of our sun. The masses of the indi- 

 vidual condensations in the arms appear to be probably about comparable 

 with that of our sun a conclusion which we shall again arrive at, in a more 

 precise form, by a different path. 



216. So far the ejected matter has been supposed to form a definite fila- 

 ment. Now it is clear that a jet of gas ejected into a vacuum will merely 

 scatter into space under its own expansive forces, except when the mass is 

 so large that its own gravitational coherence is sufficient to outbalance the 

 expansive forces produced by molecular velocity. We must examine under 

 what conditions a jet will condense into a filament. 



Consider first the simpler problem of the conditions under which a fila- 

 ment, in existence, can continue in existence without scattering into space. 

 Let T be the line density, or mass per unit length of a uniform long filament. 

 The potential of this filament at a point near its surface will be of the order 

 of magnitude of jr, so that a molecule moving with velocity C will escape 

 altogether if 



JC'XyT, 



Thus if C is the mean-square molecular velocity near the surface of the fila- 

 ment, the filament will scatter into space unless (approximately) 



r>a 2 /2 7 ................................. (540). 



If the filament is assumed to be in isothermal equilibrium a more precise 

 result can be obtained. Let p = kp, so that & = J0 2 , then the potential 

 must satisfy the differential equation 



of which the appropriate solution for a long filament is 



J^_ Ar* 

 P ~ 2-Trr 2 (l+Ar c )*' 



where c and A are constants of integration. The mass of gas per unit length 

 is readily found to be r = kc. 



The density is finite at the origin only if c = 2 ; in all other cases there is 

 found to be a nucleus at the origin of line density k (1 -|c), and the mass of 

 this nucleus together with that of the surrounding gas will give a total mass 

 per unit length equal to k(l + c). 



Interpreted physically, this means that a filament of line density 2k can 

 rest in equilibrium with finite density at the centre and no tendency to 

 scatter into space. A filament of line density greater than 2k can rest in 

 equilibrium with no tendency to scatter into space, but mathematically there 

 will be zero density at the centre and a line charge repelling the gas ; in 



