142 Prof. Tait on the Foundations of 



their action as is the tendency to recovery, a local " special " 

 state is maintained in every region of the space occupied by a 

 gas or gaseous mixture. This may be, and in the cases now 

 to be treated is, accompanied by a common translatory motion 

 of the particles (or, of each separate class of particles) in the 

 region — a motion which at each instant may vary conti- 

 nuously from region to region, and may in any region vary 

 continuously with time. 



A troublesome part of the investigation is the dealing with 

 a number of complicated integrals which occur in it, and 

 which (so far as I know) can be treated only by quadratures. 

 All are of the form 



I 



vv r 



where v is that fraction of the whole number of particles of 

 one kind per cubic unit whose speeds (relatively to those of 

 the same kind, in the same region, as a whole) lie between v 

 and v + dv ; and 1/e is the mean free path of a particle whose 

 speed is v. Throughout the paper regard has been had to the 

 fact that e must be treated as a function of v. So long as the 

 particles are of the same kind, or at least of equal mass if 

 of different diameters, such integrals are easy to evaluate ; 

 but it is very different when the masses differ in two mixed 

 gases. In what follows, the merely numerical factor of the 

 expression above will be denoted by C r , so that the value of 

 the expression is, when the masses and diameters are equal, 

 Gr/nTTsVf! 2 , and the introduction of different diameters merely 

 introduces another factor. Here 3/2h is the mean square speed, 

 n the number of particles per cubic unit, and s their common 

 diameter. 



When the masses are unequal there will, in general, be 

 different mean free paths for particles of two different kinds, 

 and the integrals cannot be simplified in the above way. In 

 this case the integrals will be expressed as j r , 2 @r« 



(1) In the first part of the paper I showed that the Virial 

 equation is, for equal hard spheres exerting no molecular 

 action other than the impacts, 



nW/2=%p(Y-2n7rs d /3), 

 where n is the number of particles, P the mass of one, s its 

 diameter, v 2 the mean-square speed, p the pressure, and V the 

 volume. The quantity subtracted from the volume is four 

 times the sum of the volumes of the spheres ; and I pointed 

 out that this expression exactly agrees in form with Amagat's 

 experimental results for hydrogen, which were conducted 



