300 Mr. S. H. Burbury on some Problems 



position that Maxwell's distribution is stationary. The con- 

 verse (b) is founded on Boltzmann's proof (Sitzungsberichte, 

 Vienna, 1872), but is much simplified by using V and R for 

 variables. 



We will prove these theorems, (a) and (b), first on the 

 hypothesis that the molecules are to be treated as elastic 

 spheres. We will suppose two classes of molecules having 

 mass M and m respectively. It will be sufficient if we prove 

 our propositions for the encounters of M with m. 



Let F(xt/zj dxdydz, or F .dxdydz, be the number per 

 unit of volume of molecules of mass M whose velocities are 

 represented by lines from the origin to points within the ele- 

 ment of volume dx dy dz&txy z. Similarly f(x r y f z') dx 1 dy' dz', 

 or/, dx' dy' dz', is the corresponding number for the m's. If 

 xy z be denoted by P, and x' y' z' by p, we will write F P and 

 f p for F and/. 



Let C be the point £ 97 J. Consider all the pairs, M and m, 



which have O C for common velocity. About C as centre 

 describe two spherical shells, one with radii R . . .R + dR, the 



M 



other with radii r . . . r -\ dH. Let P be a point in the first 



m L 



shell. The common velocity being OC, and the velocity of M 

 being OP, the velocity of m is Op, where p is a determinate 

 point in the second shell, namely in PC produced so that 



7=Vr = — And so f„ can be expressed as a function of £ r\ £ 

 Lr m. 



and of xy z, the coordinates of P. 



The effect of an encounter between M and m under these 

 circumstances is, without altering V, to substitute for TCp 

 some other common diameter of the spherical shells as the 

 relative velocity. Let it be T'Cp'. Then all directions of 

 P'C//, given PCp, are equally probable. 



The number ol pairs M and m per unit of volume and time, 

 having OC for common velocity, for which the relative velocity 

 is FCp(dS) before, and VGp (dS') after, encounter, dS and 



