310 Prof. L. Boltzmann on the Assumptions necessary 



the magnitudes of the velocities v and V and their angle T 

 are given. This quotient is 



mM r Mr' 2 1 



m 



This agrees exactly with the first expression for // — 1\, 

 which Stefan* finds in his treatise on the dynamical theory 

 of the diffusion of gases towards the end of the second part, 

 and which, after correction of some printer's errors, is as 

 follows : — 



for, in Stefan's formula, 



v 2 Y 2 



^i = w 2"? 4 = Mp a: 1 a: 2 +y 1 y 2 + z 1 z 2 = vYt=vrg + v , 



and 



Y 2 =v 2 + r 2 + 2vrg. 



But the first conclusions of Stefan's would apply only 

 where the molecules act upon each other with a force in- 

 versely proportional to the fifth power of the distance. In 

 this case the factor r disappears from the expression dZ, and 

 on further integration in respect of dT the terms 



^i<% +W2 + 2 1*2 

 disappear. 



With elastic spheres, on the other hand, the impacts are so 

 much the more probable the greater the relative velocity r ; 

 therefore, for each of the products x x x 2) y^y^i z \ z -ii positive 

 values are more probable than negative values, and we cannot 



therefore suppose r 



c x 1 ^ 2 +y 1 y 2 + z 1 Z 2 = 



on the average. In fact, if we wish also to integrate express- 

 sion (4) with reference to T, we must observe that 



r 2 = v 2 + V 2 — 2vVt and rg = Yt—v. 



Integration gives, therefore, 



the upper limit is Y + v, the lower Y—v or v — Y according 

 as Y > v or the reverse. In the first case the integral has 

 the form 



f v d Y* + cY* + |tf^ (k 5 + 2v 3 V 2 + vY*) . . (5) 



— A7, 5 . 



* Sitzbcr. d. Wien. Akad. d. Wissensch. vol. lxv. April 1872. 



