68 
Proceedings of Royal Society of Edinburgh. [sess. 
that the main effect is merely to alter the value of the disposable 
quantity A, below. But, if if/ (r) he the molecular force (attractive) 
at distance r, there is also a positive term proportional to 
where s is the diameter of a particle, fx the range of the molecular 
forces, e and g functions of r. The factor e/(V - g) represents the 
density, taken as (on the time average) uniform throughout the 
shell r,r + dr round any one particle. er 2 dr/(V-g) is thus pro- 
portional to the average mass distant r to r + dr from any particle; 
and the remaining factor, r, is in the virial expression. 
With these preliminaries it is obvious that we may now write the 
virial equation approximately as follows : — 
volume is indefinitely great — i.e ., when the number of particles at 
any time within molecular range of one another is an infinitely 
small fraction of the whole. 
quasi-constants a and y can be treated as equal only in the very 
special case in which the molecular force is supposed to act 
(impulsively) at one definite distance. If it increase rapidly as 
particles approach one another, y must apparently he less than a, 
and may be negative. Both depend upon temperature. But, since 
1 3 depends directly on the distance at impact, while y and a depend 
indirectly on the range of the molecular force, it is not possible to 
rank them all in order of magnitude. 
This equation is, at first sight, quite different t in form from that 
of Van der Waals, even as modified by Clausius, and I think it 
undoubtedly represents more accurately than do these the true 
result of the application of the Yirial. It introduces five constants, 
of which three at least are entirely independent, so that (when 
expressed as a cubic in Y) it can necessarily (in an infinite number 
of ways) be made to have three equal roots, corresponding to the 
critical point. 
But we have a difficulty of a new and serious kind which requires 
suppose, 
where E is proportional to the whole kinetic energy when the 
Erom the mode in which they are obtained, it is clear that the 
