82 Prof. L. Boltzmann on some Questions 



system, "when these .are not free to collide with one another. — 

 In fact, if (to take an extreme case) the particles of one system 

 were so small, in comparison with the average distance be- 

 tween any two contiguous ones, that they practically had 

 no mutual collisions, they would behave towards the par- 

 ticles of another system much as Le Sage supposed his ultra- 

 mundane corpuscles to behave towards particles of gross 

 matter &c." 



Against this reproach I endeavoured to defend both myself 

 and those who in their later investigations have followed a 

 similar line of thought (who are probably intended by Prof. 

 Tait's word " commentators ") in the paper " On the Assump- 

 tions necessary for the Theoretical Proof of Avogadro's Law/' * 

 where on page 629 I have proved f the following general 

 equation: — 



2 dE 



IT (It 



Jo Jo Jo** 



In obtaining this equation I have generally made precisely 

 the same assumptions as Professor Tait, except that I have 

 not made the least assumption with reference to the relative 

 magnitude of the diameters X and A of the molecules of the 

 two gases, nor as to the magnitude of the ratio Nj : N 8 . For 

 the stationary condition dE : dt must vanish ; whence it 

 follows, as I have repeatedly shown, that : — 



(1) Each of the gases assumes Maxwell's distribution of 

 velocities (or, as Prof. Tait says, the special condition) ; 



(2) The mean kinetic energy of a molecule is the same for 

 both gases. 



Simply to show why, in my former papers, I have not spe- 

 cially mentioned the cases where X . A or Nj . N 2 have very 

 small or very large values, I remark that these propositions 



* L. Boltzmann, Wien. Sitzb. Bd. xciv.'p. 613 (1886); Phil. Mag. [5] 

 vol. xxiii. p. 305 (1887). 



t The calculations here referred to, as -well as those upon the Heat- 

 equilibrium of Heavy Gases (cf. § 4 of the text), may he considerably 

 simplified hy the use of Lorentz's method, as fully explained in that para- 

 graph. I am indebted to Mr. Burbury for kindly calling my attention to 

 the possibility of further simplifications, especially by use of the extremely 

 useful proposition, " that for the relative velocity after impact with given 

 magnitude and direction of velocities before impact, but with optional 

 line of centres, every direction is equally probable, and I shall probably 

 return to all these points on some subsequent occasion. 



