284 Mr. J. H. Jeans on the Vibrations 



To obtain the number o£ collisions having values of g 

 greater than G, this has to be integrated through all possible 

 positive values of g, n, v, subject to 



u<g<v, g>G. 

 Integration with respect to g gives 



in which 



[r/]=0 when r<G, 



Q/] = v^ — Q^ when v > G, u < G, 



[(/•^] = v'^ — 71^ w^hen ic > G. 



On integrating (17) with respect to it, v we shall obtain 

 the number required. When Gr is very great the only inte- 

 grands which are o£ importance are those for which u^-\-v^ is 

 in the neighbourhood of its least possible value G", i. e. those 

 in the neighbourhood of u = 0, v = G. In (17) we may there- 

 fore replace [v~ — u^) by G", and [g^^ by r'' — G^and therefore 

 by 3G"(v — G). We may also suppose the integration to 

 extend from v=:G to v=yo , and from u = to w=co . The 

 integral is therefore 



of which the value when G is laro-e is found to be 



o" 





in which the last factor is obviously of preponderating 

 importance. 



For air under normal conditions we have 



approximately. The value of G must^ as v/e have seen, be 

 comparable with 2 X lO''. If we actually take this as the 

 lowest value for which the vibrations are appreciable, we get 

 A7?2G^ = 240,000 and 



^-IhraG-^ _ ^-120,000 



If w^e take half the foregoing value for G, 



^_|fe;HG2_. ^ — 30,000 



so that the dissipation of energy takes place e^^>^^^^ times more 

 slowly in the former case than in the latter. If G were 

 comparable with the m.ean velocity in the gas, the energy of 

 the gas w^ould probably be reduced to half its value in a 



