ON OUR KNOWLEDGE OF THERMODyNAMICS. 83 



by collisions and by the free motion of the molecules. This angular velocity 

 might, therefore, follow any law of distribution wliatever, and that law 

 would be permanent. Neither does the conclusion hold good if the 

 centres of mass and of figure coincide. 



The arguments of Boltzmann might have been rather more conclusive 

 if they had shown at lohat stage of the process these exceptional cases had 

 to be excluded. Burnside did worse than this, for he said : ' Hence the 

 three equations ' (now known to be wrong) ' are a solution ; and, there- 

 fore, must be the solution of the problem of the special state.' 



The cases of perfectly rough spheres or circular discs having both their 

 normal and tangential coefficients of restitution unity furnish interesting 

 examples for solution. We may imagine the spheres and discs covered 

 over with perfectly elastic fine teeth, or minute projections by whose 

 action the tangential components of the relative velocity ai-e reversed at 

 impact, and it is no longer necessary to suppose the molecules to have a 

 ' bias ' in order to have a transference of energy between the translational 

 and rotational kinds. 



The Functional Equation for Colliding Bodies in General. 



38. The earliest investigation of the law of distribution for molecules 

 other than point-atoms (or smooth hard spheres, whose centres of mass 

 and figure coincide) seems to be that of Boltzmann in 1871.' 



At the present time the simplest and best treatment of the general 

 problem for colliding bodies with any number of degrees of freedomis that 

 given in Burbury's paper ' On the Collisions of Elastic Bodies.' ^ The 

 methods there used are perfectly general, and include Watson's proofs for 

 lop-sided circles and spheres as particular cases without the attendant 

 complications in the formuliie, which arise from writing down in full the 

 special forms of the various expressions assumed in those investigations. 



39. The assumptions involved in proving the Boltzmann-Maxwell Law 

 for colliding bodies seem to me to resolve themselves into the following : — 



(i) That the law is not meaningless. The expression (26) or 



Ne-'"^-^'(7j;, . . . dp„dq, . . . dq,, . . (29) 



must represent a definite number of molecules. 



Hence in a volume element so small that x may be considered constant 

 over it, there must be a very large number of molecules moving about 

 with all possible momenta, and out of these a large number must have 

 their remaining co-ordinates and momenta distributed within the corre- 

 sponding small multiple differential. The law will obviously not hold 

 at points where x, the potential of the field, becomes infinite or discon- 

 tinuous. Moreover, the collisions must be sufficiently frequent to admit 

 of a similar law being applicable to the colliding molecules. 



(ii) That any molecule has a chance of colliding with any other mole- 

 cule. Hence the frequency of distribution of the molecules must depend 

 on their actual st^e, and not on their past history or future prospects of 

 colliding with any particular set of other molecules. As Burbury has just 



' ' Ueber das Wiirmepleichgewicht zwi.schen mehratoniigcn Gasmolekulen,' Sitzber. 

 dvr k. Wiener Alcad., Ixiii. (ii.), p. .397. 

 ''■ Phil. Trans. R.S., 1 892, pp. 407-422. 



G 2 



