•J04 PROFESSOR TAIT ON THE 



It is exceedingly improbable (when we think of the mechanism involved) that any really 

 simple expression will give a fair agreement with an isothermal throughout the whole 

 range of volumes which can be experimentally treated. 



From the general results of Part III. of this paper we see that the term 



J2(mu 2 ) 



in the virial equation must, when molecular forces are taken into account, contain a, 

 term proportional to the number of particles which are at any (and therefore at every) 

 time within molecular range of one another. Hence if, when the volume is practically 

 infinite, we have for the mean-square speed of a particle 



(where n is the whole number of particles), we shall have, when the volume is not too 

 much reduced, no work having been done on the group from without, 



i2(mu 2 ) = E + 



v + y ' 



where C and 7 may be treated as constants, the first essentially positive if the 



molecular force be attractive, the second of uncertain sign. Even if the volume 



be very greatly reduced it is easy to see, from the following considerations, that 



a similar expression holds. The work done on a particle which joins a dense group 



is, on account of the short range of the forces, completed before it has entered much 



beyond the skin, and is proportional, ceteris paribus, to the skin-density. Hence 



the whole work done on the group by the molecular forces is (roughly) proportional 

 to 



Vp.p , 



the first factor expressing the number of the particles, the second the work done on 

 each. But, as we are dealing with a definite group of particles, the first factor is 

 constant, so that the whole work is directly as p , or inversely as (say) v + y, because 

 p <p. But the work represents the gain in kinetic energy over that in the free 

 state, so that this mode of reasoning leads us to the same result as the former for 

 the average kinetic energy of all the particles. 



In so far as R depends on the molecular attraction, the term 



JE(Rr) 



is evidently proportional, per unit volume of the group, to the square of the 

 density : — for the particles, in consequence of their rapid motions, may be treated 

 as occupying within an excessively short time every possible situation with regard 

 to one another. Thus, as regards any one, the mass of all the rest may be treated 



