215-217] Ejection in Filaments 219 



nature this would mean that equilibrium could be established only when 

 A had become so great, and therefore the density so great, that the ordinary 

 gas laws would be departed from. A filament of line density less than 2k 

 cannot rest in equilibrium at all under natural conditions ; it would scatter 

 into space. Thus for an isothermal filament the exact critical value of T is 

 seen to be 



2k 2C 2 



in which the gravitation -constant 7 has been restored. This exact value 

 differs by a factor | from the general approximate value obtained in 

 equation (540). 



217. Thus the lowest line density for which condensation can occur at 

 all will be comparable with that given by equation (541). Multiplying this 

 by expression (539) which gives the length of filament which goes to form a 

 single condensation in the nebular arms, we find that the minimum mass in 

 one such condensation must be comparable with 



(542)< 



For line densities much greater than this, the filament may become 

 transversely as well as longitudinally unstable, the mechanism again being 

 that already explained in 160. If the filament is of n times the critical 

 line density 2C 2 /3y, the linear dimensions of its cross-section will be of 

 the order of n^Cy~^p~^, which is 2n times the critical length (539) at 

 which wave-motion becomes unstable. Thus when a filament has a line 

 density much beyond the critical line density 2<7 3 /37, tne motion may be 

 supposed to be one in which nuclei of condensation form both laterally and 

 transversely, their average distance being comparable with that given by 

 formula (539), namely %Cy~^ p~^. The average mass surrounding each 

 nucleus will be p times the cube of this expression or 



.............................. (5*3), 



which is comparable with our former expression (542), although less by a 

 factor f . 



Inserting our previous conjectural values G = T6 x 10 5 and p 1'5 x 10~ 17 , 

 the numerical value of expression (542) is found to be 16 x 10 34 , or eight 

 times the mass of our sun, while the numerical value of expression (543) is 

 about three times the mass of our sun. 



Thus we are again led to the conclusion, reached rather more vaguely in 

 215, that the nuclear condensations in the arms of spiral nebulae are of 

 mass comparable with our sun. 



