FOUNDATIONS OF THE KINETIC THEORY OF GASES. 265 



as diffused uniformly through the space they occupy. In volume v, therefore, the 

 amount is as vp 2 . But, in the present case, the quantity vp is constant, so that, 

 ao-ain, the approximate value of the term is directly as p, or inversely as v. But, 

 once more, we must allow for the bounding film (though not necessarily to the same 

 exact amount as before), so we may write this part of the term as 



V + a 



But there is another part (negative) which depends on resilience. This is (§ 30) pro- 

 portional to the average kinetic energy, and to the number of particles and the number 

 of collisions per particle per second. The two last of these factors are practically the 

 same as those employed for the molecular attraction. Hence the whole of the virial term 

 may be written as 



A-e(E + C/(i;+y)) 

 v + a 



Thus if we write again A and C for 



A-\ and C+- 



a — y a~y 



respectively, the complete equation takes the form 



C A-eE 



pv = E + 



V + y v + a ' 



which is certainly characterised by remarkable simplicity. 



69. We must now consider how far it is probable that the quantities in the above 

 expression (other than p and v) can be regarded as constant. E, of course, can be altered 

 only by direct communication of energy ; but the case of the others is different. 

 Generally, it may be stated that there must be a particular volume (depending 

 primarily upon the diameters of the particles) at and immediately below which the 

 mean free path undergoes an almost sudden diminution, and therefore we should ex- 

 pect to find corresponding changes in the constants. In particular, it must be noted 

 that some of them depend directly on the length of the free path, and that somewhat 

 abrupt changes in their values must occur as soon as the particles are so close to one 

 another that the mean free path becomes nearly equal to their average distance from 

 their nearest neighbours. For then the number of impacts per second suffers a sudden 

 and large increase. Thus, in consequence of the finite size of the particles, we may be 

 perfectly prepared to find a species of discontinuity in any simple approximate form of 

 the virial equation. From this point of view it would appear that there is not (strictly) 

 a " critical volume " of an assemblage of hard spheres, but rather a sort of short range 

 of volume throughout which this comparatively sudden change takes place. Thus the 

 critical Isothermal may be regarded as having (like those of lower temperature) a finite 



VOL. XXXVI. PART II. (NO. 10). 2 S 



