156 Prof. Ludwig Boltzmann on the 



belonging to the momentoid u n lies between the values k n and 

 k n + dk m we again arrive exactly at Maxwell's expression (51) 

 (/. c. p. 725). 



The expressions (52) and (53) (I. c. p. 726) of Maxwell's 

 paper thus represent the mean and maximum values of the 

 vis viva belonging to any momentoid. Instead of the law of 

 Maxwell that the mean vis viva has the same value for every 

 coordinate, we now obtain the law that the mean value of the 

 vis viva belonging to all momentoids is the same. 



Since the number of momentoids is always the same as that 

 of the degrees of freedom, the law given by Maxwell at the 

 commencement of his paper (J. c. p. 716) still remains true, 

 viz. that the mean kinetic energies of two given parts of the 

 system are in the ratio of their respective degrees of freedom. 

 The kinetic energy T k of any portion whatever may therefore 

 contain the products of different pj/s, the terms p& being the 

 momenta of the general coordinates of that portion. But T k 

 must not contain the product of a term p k and another mo- 

 mentum which is not included among the terms p k . As a 

 special case, the law will apply without any modification to 

 so-called polyatomic molecules of a gas, whose condition is 

 expressible by generalized coordinates. 



As 2 ^j- is equal to 2 a ? ^r hi all cases, my proof of the 



second law* will still be correct, provided that by q h we un- 

 derstand, not the momenta belonging to the coordinates p fl) 

 but the momentoids. 



On the Special Cases suggested by Lord Kelvin as 



Test-cases. 



§ 2. Motion of a Material Point in a Plane. 



I believe that with these modifications Maxwell's proof of 

 the laws enunciated in the previous paragraphs is a satisfactory 

 one ; but in addition I have already given another proof from 

 a quite different point of view f. I believe therefore that its 

 truth as a law of analytical mechanics can hardly be called 

 into question %. As 1 myself arrived at my theorem only 



* Borchard-Kronecker's Journal, vol. c. p. 201 (1885). 



t " On the Equilibrium of Heat between Polyatomic Gas-molecules. 

 Part I. Motion of the Atoms iu the Molecules/' Wien. Sitz.-JSer. vol. iii. 

 9th March, 1871. " Some General Theorems on Equilibrium of Heat," 

 ibid. 13th April, 1871. In the latter paper I first made use of generalized 

 coordinates. 



X It is an entirely different question, whether such systems present a 

 sufficiently close analogy with hot bodies. This question cannot be 

 discussed here; cf., however, Beibl. v. p. 403 (1881). 







