in the Kinetic Theory of Gases. 301 



dS r being small solid angles, is 



— — Fpf p d$7rs 2 p, that is F P ^ -— irs 2 p, 



where 5 is the sum of the radii of M and m. 



The number of pairs for which it is P'Cp'^S') before, and 

 PCp(dS) after, encounter is 



J Fp/^S' v*p, that is F P ,/ p ,^^ Wp. 



If, then, F-pf p = ~F-p>f p r y the number per unit of volume and 

 time of pairs for which the relative velocity turns, as the result 

 of encounter, from PCp to P'Cp' is equal to the number for 

 which it turns from ~P'Cp' to PC/?. And if this is true, 

 given C, for every two directions of PC/?, and for all positions 

 of C, it follows that the distribution of velocities is not affected 

 by encounters between M and m. 



Now F-pfp represents the chance that, given V = OC, the 

 relative velocity shall have direction PCp. And we see that 

 if, given OC, this chance be the same for all directions through 

 C, the distribution of velocities is not affected by encounters. 

 So (a) is proved. 



If for some two directions of the diameter Fp^i zfzF?f p we 

 proceed as follows, adapting Boltzmann's proof. 



Let 



R = ^dxdydz'F(xyz)(hgF(xyz)-l) 



+ ^ dx dy dzf(xy z)(\ogf(xyz) - 1), 

 the limits being in each case + <x> , or, as we will write it, 



H = #j^A,<fe{F(logF-l)+/(log/-l)}; 

 then 



§ = M^{flo g F + flo g/ }. 



Now F is supposed to vary only by encounters between the 

 M's and the rns. Therefore 



dF P 



where T P is the number per unit of time of encounters with 

 the m molecules which the M molecules with velocity OP 

 undergo, and T' P is the number per unit of volume and time 

 of encounters between M and m from which M issues with 

 velocity OP. Now P being given, and the velocity of M 

 being OP^ C or f w £ may have any position whatever, and 

 Phil Mag. S. 5. Yol. 30. No. 185. Oct. 1890. Y 



