The Vortex Ring Theory of Gases, 



33 



linked together, then for a given value of a the velocity of the 

 molecule will be least when the links are so far apart that they do not 

 greatly affect each other's velocity ; in this case v will equal — 



-7— log — , 

 lira e 



the velocity of the molecule will be greatest when the n rings are 

 close together ; in this case v will equal — 



nm n 8 a 

 log: — 



2wa 



e 



So that if we integrate first with respect to v we must do so between 

 these limits. 



Since, however — 



T = Aa°v + + IMv 3 , 



we cannot perform the integration except between the limits zero and 

 infinity for both a and v ; if, however, n be large, or the molecule 

 complicated, the results got by integration between the limits — 



m n 8a j mn , 8a 

 log — and- — log — 

 lira e Lira e 



for v and zero and infinity for a will not differ much from those got by 

 integrating between zero and infinity for both a and v. 



The second term in the expression for the kinetic energy is very 

 small compared with the first, so that it may be neglected without 

 causing sensible error. We shall find it convenient to take as new 

 variables the two remaining terms in the expression for the kinetic 

 energy ; we shall call these new variables a. and respectively, where 

 a. denotes the energy in the fluid surrounding the ring, ft the energy 

 due to the translation al velocity of the ring, so that — 



Aa 2 v=a. 



and therefore 



dadv = — - — d xd3, 

 MAav 2 



so that 



3p-2 3/1-16 



where C is a new constant. 



Thus the number of molecules which have the energy in the fluid 

 surrounding them between « and a.-\-h%, and also the energy due to 

 the translational velocity of the ring between ft and ft + Sft is — 



VOL. xxxix. D 



