498 Lord Rayleigh on the Pressure of 



supposing n=4z. It might seem that in a rare gas, whenever 

 the virial depends sensibly upon what occurs during the 

 encounters of simple pairs of molecules, there must be pro- 

 portionality to v" 1 , so that (14) would apply. If, as Maxwell 

 supposed, w = 5, 



2p<j>( P ) <x t- 1 . Ti, (15) 



in agreement with a result obtained by Boltzmann for this 

 case. If we retain n=5, but leave the relation to v open, we 

 get from (12) 



tpcj>(p) ac v-'.T 1 -** (16) 



If we now discard the supposition that the dependence upon 

 v follows the law of v~ 8 , we may interpret (]6) to mean that 

 considered as a function of v and T, the virial is limited to 

 the form 



2p^(p)=T.F(«Tt), .... (17) 



F denoting an arbitrary function of the single variable vT+. 



And more generally, whatever n may be, we find from (12) 

 that the virial is limited to the form 



2 P <pO>)=T.F U-5- (18) 



\ f^l 



A further generalization may be made by discarding 

 altogether the supposition that $(/?) is represented by any 

 power of p. In this case it is convenient to write 



^(p) = -f,'f(pja), (19) 



where a is a linear quantity. Here / itself may be supposed 

 to be of no dimensions, while p! has the dimensions of a 

 force. The virial is a function of pi ', a. m, Y, v ; and since 

 its dimensions are those of energy, i. e. of mV 2 or T, we may 

 write 



2 /? *(p) = T.F(At / ;a,m,V,f;), 



where F is of no dimensions. It is easy to see that //, ?n, 

 and V 2 . can occur only in the combination p//m~V 2 or ////T. 

 To make this of no dimensions, we introduce the factor a. 

 Thus F becomes a function of a, v, and pfa/T, in which 

 again v can occur only in the form a? /v. Accordingly 



Sp+"W=T.F(J,^), .... (20) 

 F being in general an arbitrary function of two variables. 





