The Equipartition Law in a System of Particles. 593 



are now the respective probabilities that given particles of 

 mass m a or mass m h will have momenta between yfr and 

 i/r + <f^r, the only change in the expressions being the sub- 

 stitution of the volumes V a and Vj in the place of the one 

 volume Y. Farther more, this distribution law will evidently 

 lead as before to the equality of the average values of 



m a v 2 , m h v 2 



and 



y/l-V 2 v/1-^ 



Since, however, the spaces containing the two kinds of 

 of particles are in thermal contact, their temperature is the 

 same. Hence we find that the equality of the average values 



of — ° is the necessary condition for equality of tempe- 

 rature. This evidently leads at once to the very important 

 corollary that for any given system of particles at a definite 

 temperature the distribution of momenta and hence the total 

 energy content is independent of the volume. We may now 

 proceed to the derivation of relations which will permit as 



to show that the important quantity — -° =1 is directly 



L v / 1 — v-J aVi 



proportional to the temperature as measured on the absolute 

 thermodynamic temperature scale. 



Pressure exerted by a System of Particles. 



We first need to obtain an expression for the pressure 

 exerted by a system of N particles enclosed in the volume V. 

 Consider an element of surface dS perpendicular to the X 

 axis, and let the pressure acting on it be p. The total force 

 which the element d$ exerts on the particles that impinge 

 will be pd&, and this will be equal to the rate of change of 

 the momenta in the X direction of these particles . 



Now by equation (11) the total number of particles ha vino- 

 momenta between ty x and ^\r x + dy\r x in the positive direction is 



* J+x Jo Jo 



But icd$ gives us a volume which contains the number of 

 particles having momenta between yfr x and tyx + dtyj. which 

 will reach dS per second. Hence the number of such 



* The system is considered dilute enough for the mutual attractions of 

 the particles to he negligible in their effect on the external pressure. 



Phil Man. S. G. Vol. 28. Xo. 166. Oct. 1914. 2 Q 



