17tf Mr. S. H. Burbury. [Nov. 10, 



zero and (d^jdt)^ d^fdt being positive, will undergo collision 

 within the time St after that instant. Therefore d^/dt or R measures 

 the frequency of collision. 



8. From the linear equations connecting Vi ---- v n with 

 Si ____ S H -i and R, we can find, say, v l as a linear function of 

 Si '. . . . S_i R, and v\ as the same function of Sj . . . . S M _i and R. 

 Therefore v? v\ 2 = 4R2/*S, and (v? v'fiR = 4R 2 2/iS, where the 

 fts are functions of the coordinates and constants. Now let 

 Si .... S n _i go through all values consistent with 



and let 0(Si .... S n -\)dSi .... dS n -i be the number in unit volume 

 of systems for which they lie between 



Si and Si -MS, &c., 



given E and R and the coordinates within certain limits. 

 Then 



JJJ. . . . Ov-*; 7) R0 (Si-S^ 



= 4R 2 JJJ .... (Si-SH-0 2/tS dS x . . . . dS,,-,. 



Now, in the Maxwell-Boltzmann distribution, 0(Si . . . . S_i) is a 

 function of the kinetic energy only, and, therefore, constant through- 

 out this integration. Therefore 



JJJ .... ( vf-v'V R dS t . . . . dS tt _i = 4R 2 JJ . . . . 2 /iS dSn . . . . dS n -i 

 % =0, 



because for every set of values of Si . . . . S_! there is included in the 

 integration another set with reversed signs. 



Now JJJ ---- Ox 2 v'^jE, dSi ---- ^S n _! expresses the mean value of 

 t?i* tf'j* for all collisions, given E and R, and since it is zero, the 

 distribution of velocities is not altered by collision, or the Maxwell- 

 Boltzmann distribution, given existing, is not affected by collisions. 



9. Certain examples are given showing the values of S!....S w _i 

 and R in given cases, viz. : I. Elastic spheres of masses M and TO. 

 II. Elastic spheres colliding with spheroids. 



10. Professor Burnside's problem of a set of equal and similar 

 spheres, each of which, instead of being homogeneous, has its centre 

 of inertia at a small distance c from the centre of figure. Discus- 

 sion of his result, which does not agree with the Maxwell-Boltzmann 

 doctrine, owing, as I believe, to an oversight. 



11. A general proof is now given of the permanence of the distri- 

 bution, viz., if there be a set of systems called system M, each having 



