b= 



Illustration of the Isothermal Formula. 305 



+ 7rlPB? ^ (L-$)Twf(v)¥(v,w) sin f cos f dv olio df, . (8) 

 and this takes by (7) the form 



f i )V=NE-J r Nc(2E-E c ) + ^§§imio 2 .f{v)F{v,iu)dvdw, (9) 



where 



ft)=f7rNR 3 (10) 



The double integral in (9) is seen to represent (apart from 

 the factor 1STg>/2V) the mean value of the relative kinetic 

 energy for all ideal combinations of two points we are able 

 to imagine throughout the gaseous mass ; but it is clear that 

 this mean is essentially different from the mean relative 

 kinetic energy for interacting molecules, or molecular systems. 

 It follows that we have to write (2E— E e ) for the value of 

 the double integral. For suppose we are computing all pos- 

 sible imaginary pairs of molecules throughout the gas ; the 

 frequency of systems in which true physical connexion occurs 

 among our ideal pairs cannot depend upon the velocity 

 of the centre of inertia of the pairs, hence E e added to the 

 mean value of \mw 2 must be equal to 2E. We have, then, 



| P V=NE + iN(2E-Ej(|-c). . . . (11) 



Now write for simplicity, 



and 



e (?- c ) = v' (13) 



so that 

 2weNR 2 $ , j 1 2R ^ S ^ - T \wf(v)¥(v,w) sin yfr cos f dv dw df; (14) 

 equation (11) takes the form 



JpV=BE(l+y), (15) 



similar to those obtained by Van der Waals, Lorentz, and 

 other authors. It would seem, however, to deserve here 

 special attention, that b, according to (14), may be either 

 positive, or zero, or negative, as the law of force varies ; and 

 still more suggestive is the further possibility of b changing 

 its sign, for a given law of force, at some definite temperature 

 or temperatures. 



