168 Prof. Ludwig Boltzmann on the 



impact. If we put the velocity of a single atom 



equal to c l9 then yf is clearly a function of c u but F may be 

 expressed as a function of the following six variables: — (1) the 

 two velocities c and a" of shell and nucleus, (2) their distance 

 apart p (measured from centre to centre), (3) the angles a and 

 a" which the directions of c and c" make with the line p, the 

 latter being supposed drawn from shell to nucleus, and (4) the 

 angle ft between the plane containing p and c and that con- 

 taining p and c". 



Now imagine a collision to take place, and denote the 

 values of these variables at the moment immediately before 

 the impact by letters without an affix, their values just after 

 the impact being distinguished by the affix. We can so 

 choose the direction of the line of centres at the moment of 

 impact and the direction of c i? that c, a, and ft take any re- 

 quired values c, a', and ft', provided these latter furnish a real 

 value Ci of the variable c ± after impact. This value will be 

 determined by the equation of vis viva 



nJci 2 -f md 2 = m'ci 2 + m<?. 



The values of the variables c", p, and a" will not be altered 

 by the impact. The first of the equations (10) can therefore 

 be written in the form 



F(e", «" ; p, c, «, /3) ./, (*0 = F(c", «", P , J, o>, 0) . 



*/7(v* 4 5 (e -^). 



This equation must be satisfied by all possible values of the 

 variables c", a.", p, c, «, ft, d , a!, ft', and c x ; from which it 

 follows easily that 



/xfo) = Ax e-K^i 2 , F = Ae~ Kmc2 . 



in which A x and h are simple constants, while A may contain 

 the variables c", p, and ex!'. 



It is therefore evident that the mean kinetic energies of a 

 shell and a single atom are equal, nnd that Maxwell's law of 

 distribution of velocities between shells and single atoms is 

 satisfied, without assuming the existence of impacts of shells on 

 each other, or of single atoms on each other. These assumptions 

 do not alter the distribution of vis viva in the slightest degree, 

 however, because the values of f 1 and F found above satisfy 

 identically the other relations demanded by the equations (10). 

 On the other hand, the condition that the nuclei should never 

 come into collisions of any kind does not prevent the law from 



