154 Prof. Ludwig Boltzmann on the 



ease, both in the case of spheres and of bodies which possess 

 no axis of rotation, did not then think of considering the allied 

 and almost similar case, namely that of the various perfectly 

 smooth and rigid solids of revolution which differ from the 

 spherical form. He would then have obtained exactly the 

 desired value 1*4 for the ratio of specific heats. 



The proof which Maxwell gave at that time for his law of 

 distribution of kinetic energy was afterwards shown by him- 

 self to be insufficient ; and in a second paper* he gave an 

 exact proof that the distribution of vis viva found previously 

 was a possible one, i. e. that when once set up among the 

 molecules of the gas it would not be altered by their impact. 



This proof as well as Maxwell's law itself is capable of 

 considerable extension, and he pointed out its very close con- 

 nexion with a far more general theorem which applies to any 

 system in which any forces operate f. In the attempt to 

 make this latter theorem still more general, however, Max- 

 well J committed an error in assuming that by choosing 

 suitable coordinates the expression for the vis viva could 

 always be made to contain only the squares of the momenta, 

 this assumption being, as Lord Kelvin has shown §, in general 

 incorrect. But the mistake may be easily corrected by a 

 slight modification of Maxwell's conclusions. To demon- 

 strate this let us turn to Maxwell's paper just quoted above, 

 and write with him (Sci. Pap. ii. p. 720) b i} b 2 , b 3 . . . b n for 

 generalized coordinates, and a l9 a 2 . . . a n as the corresponding 

 momenta. We must then stop at Maxwell's equation (42) 

 (/. c. p. 724), and so write for the vis viva 



T = i[ll]a 1 2 +[12]a 1 « 2 .... 



All Maxwell's conclusions as far as equation (29) inclusive 

 are perfectly accurate. In order to correct the further con- 

 clusions we might write the following deduction in place of 

 that of Maxwell, from that point onward. Let us suppose a 

 to be a linear function of the momentum a, and thus write 



k-n 



«A= 2 c hk^k- 

 k=\ 



Then w T e can always ciioose the coefficients c so that (1) their 



* Phil. Mag. [4] vol. xxxv. (1868) ; Sci. Pap. ii. p. 26. 



t Boltzmann, Wiener Sitzungsberichte, vols, lviii., Ixiii., lxvi., Ixxii , 

 lxxiv., lxxv. 



\ "On Boltzniann's Theorem on the Average Distribution of Energy," 

 Canib. Phil. Trans, xii. part 3 (1878) ; Sci. Pap. ii. p. 713. 



§ « Nature,' loth August, 1891. 



