306 Prof. L. Boltzmann on the Assumptions necessary 



a generalization thereof. We must remember that analysis 

 oan deal only with systems more or less analogous to the 

 molecules of nature, but not with the molecules themselves. 



The behaviour of hot bodies is certainly influenced by heat- 

 radiation, probably also by movements of electricity &c. — 

 ■conditions which have not been taken into account by me, 

 nor in any other mechanical theory of heat. Absolute agree- 

 ment with facts cannot therefore be expected. It is therefore 

 only possible to consider (1) whether the propositions stated 

 really follow logically from the assumptions made, and 

 (2) whether the analogy between the properties of the 

 system considered and those of hot bodies undoubtedly exists. 



This analogy can only be made perfectly clear by reference 

 to all my papers, and in particular to those on the second law 

 of Thermodynamics. 



Although Mr. Burbury* has made well-founded objections 

 to the propositions of Prof. Tait, it appears to me desirable to 

 discuss still more rigorously the question, Which of Prof. 

 Tait's assumptions are really necessary to the proof ? In doing 

 this I will follow Prof. Tait's method, and at first will treat 

 only some special cases, in order not to become unintelligible 

 by too great a generalization. Although he does not expressly 

 say so, Prof. Tait yet implicitly assumes that two molecules 

 upon impact behave like elastic spheres. For under any other 

 law of mutual action Prof. Tait's equations on p. 346 would 

 only hold good in case the quantities which he denotes by u 

 and v were the components in the direction of the apsides ; and 

 the calculations on p. 347 would then not be applicable to 

 these equations. I will make the same assumption, and use 

 the notation of my Theory of Gaseous Diffusion, part i.f 



I have there treated the impact of two elastic balls as 

 generally as possible with reference to the Theory of Gases. 

 The figures and formulas are certainly a little copious, but 

 are generally applicable, and, as I believe, also clear, when 

 once their meaning has been comprehended. 



Let two elastic spheres (molecules) of masses m and M 

 impinge upon each other. Let v = D,v and V = HV be their 

 velocities before impact (see fig. 1), and v / = 0,v l and V' = OV ; 

 their velocities after impact, 8 the sum of radii of the spheres, 

 r = vY their relative velocity before impact, and suppose their 

 relative velocity after impact to have the direction r i =v'Y r . 

 Let OC be the line of centre of the spheres at the moment of 

 impact. 



* Phil. Mag. [5] vol. xxi. p. 481 (1886). 



t Sitzber. d. Wien. Akad. d. Wissensch. vol. Ixxxvi. p. 63 (June 1882). 



