588 Mr. S. H. Burbury on the 



If the two spheres composing a " system " collide ivith 

 each other, the " system " remains on the same path before, 

 daring, and after that collision, only the phase changing. 

 But if either sphere collides ivith a third sphere, the " system " 

 changes its energy, and therefore its path. In this case no 

 pair of spheres or " system " passes on the same path through 

 all phases consistent with conservation of energy. Rayleigh's 

 argument is therefore inapplicable to systemsof this description. 



9. It is to be noted also that, even if we regard the system 

 only while it continues on the same path, during an interval 

 which includes a collision between m and M, the reasoning- 

 still fails because E is not the only constant before and after 

 collision. The square of the relative velocity R of m and M 

 is also constant, namely 



R2 = ( M _ "[J) 2 + (v - V) ' 2 + (w - W) *. 



Therefore / is a function, not of E only, but of E and R 2 . 



And for the same reason I think it must fail as applied to 

 the translation velocities of any group of spheres, because E 

 will always have a companion constant representing conserva- 

 tion of momentum. 



It is useful here to compare Rayleigh's equation /=/' 

 with that which Boltzmann obtains in the corresponding case, 

 namely F/=F'/'. In Boltzmann's notation F, F' relate to 

 M spheres before and after collision, and f, f similarly relate 

 to m spheres. Rayleigh's equation admits of solution in the 

 form 



f= Ae- A ( T + KR2 >, 



where A and K are constants. Boltzmann's equation admits 

 of no solution for our present purpose except 



F/=Ae-* T . 



Boltzmann, if his fundamental assumptions are true, 

 proves the law mi* 2 =MU 2 , while Rayleigh's method if 

 applied to any finite group of spheres as a " system" fails to 

 prove it. 



lU. In order rightly to apply Rayleigh's argument, we must 

 treat as one '" system " all the elastic spheres (if our mole- 

 cules are such) in the field. Or it must be a material system, 

 which, however its parts may act on each other, is, and for 

 ever remains, subject to no external influences. And it 

 passes in cycle through all phases which can be reached from 

 its initial phase with E constant. I think we have no cage 

 for such a bird. 



Nevertheless Rayleigh's reasoning, or Maxwell's pp. 553— 

 554 7 must be accepted as proving that throughout the path 



