Equilibrium of Vis Viva. 163 



The integration of this last equation over all possible values 

 of the variables involved in it shows us at once that the 

 quantity S log % suffers no alteration by internal motion of 

 the doublets, and the same will be true, of course, for any 

 central motion. All that now remains is to determine the 

 effect of the collisions. 



Let us now introduce instead of it, V, \V, g, h, k, the absolute 

 velocities u, v, te- 9 u ,r , v" , w' f . Since 



(m + m n )g = mu + m n u n , U = u n — u, 



it follows that 



dg ch\ = du du" '. 



If we write for the number of doublets in unit volume, 

 whose variables x, y, z, u", v", w n , u, v, w, lie between the limits 

 x and x + dx . . . w and w + dw, the value F (#, y , z, u" v", to 1 ', 

 u, v, w) dx . . . dw, where 



' / „ mil + m"u" ^ 



^V m + m J 1 



then 



S iog%=JF log F dx . . . dw=t log F, 



in which the summation again extends to all the doublets 

 contained in the unit of volume. Now denote by 8{Z log F 

 the increase in 2 log F produced during time St by the impact 

 of doublets on each other, by 8 2 S log/ the increase in 2 log/ 

 during the same time, produced by the impact of single atoms 

 on each other, and by S 12 (S log F + 2 log/) the corresponding 

 increase of the quantity in brackets, produced by the impact 

 of doublets and single molecules. 



To calculate S 12 (2logF + 21og/) we must sort out, from 

 all the impacts between a shell and a single atom in unit 

 volume during time St, those collisions for which the velocity- 

 components of the shell at the moment of the collision (and 

 likewise before) lie between the limits u and 11 + du, v and 

 v + dv, iv and iv + dw, the velocity-components of the nucleus 

 being between u n and u" + du", v" and v" + dv' ! , w" and iv" + dw u , 

 the coordinates of the nucleus relatively to the shell being 

 between x and x + dx, y and y + dy, z and z + dz ; further, the 

 velocity-components of the common centre of gravity of the 

 shell and the single atom are to lie between p and jt> + d^>, </ 

 and q + dq, r and r + dr, and the direction of the line of centres 

 of the impinging atoms at the moment of impact are to lie 

 within an infinitely narrow cone placed in a definite direction 

 in space and having an infinitely small angle dX. The velocity- 

 components of the single atom at the instant when the impact 



