( 496 ) 



so thiil another definition is necessary for collisions of opposite 

 kind, viz. such for which the points of velocity get in du> and dat^ 

 after the collision. Wind proves further that the number of collisions 

 of opposite kind is all the same represented by the expression which 

 Boltzmann had found for it. 



It is then easy to change (what Wind does not do) the proof 

 given by Boltzmann in § 5 of his "Gastheory", that Maxwell's 

 distribution of velocities is the only one possible, in such a way that 

 it is perfectly correct. But the error in question makes itself felt all 

 through Boltzmann's book. Already with the proof of the i/-theorem 

 given in more analytical form in a footnote to § 5 we have some 

 difficulty in getting rid of this error. 



We meet the same thing when the molecules are treated as centres 

 of force, and when they are treated as compound molecules. At the 

 appearance of the second volume of his work, Boltzmann had taken 

 notice of Wind's views, but the inaccurate definition for collisions 

 of opposite kind has been retained '). 



In connection with this error, made by Boltzmann in a geometrical 

 treatment of the phenomena of collision, is another error of more 

 analytical nature, so that also Jeans, who treats the matter more 

 analytically, gives a derivation which in my opinion is not altogether 

 correct. Though preferring the geometrical method, Boltzmann repeat- 

 edly refers to the other 2 ). The method would then consist in this, 

 that the components of the velocities after the collision êVS'IWiS'i 

 are expressed by /i^^ï^C,) and then by means of Jacobi's func- 

 tional determinant d§'dri'd^d9 1 dri' l d^ l is expressed in d^dtid^d'^di^d^ . 

 We find then that here this determinant is = 1 and so 



d^drld^d^\drl x d 1 S l = dgdrid^d^d^d^ or du>'do>\ = doidto l . 

 The number of collisions of opposite kind =f'F\du)'do}' 1 o*c/ cos dd)dt 

 according to Boltzmann, and so also = /'F'^dtodto^ij cos 9-dXdt. In 

 this the mistake is made, however, that d^d^d^d^ l d'rl l d^\ the 

 volume in the space of 6 dimensions that would correspond with the 

 volume d^drid^d^d^d^ before the collision, is thought as bounded 

 by planes such as $,' = c, which is not the case. Jeans too equates' 

 the products of the differentials, in which according to him, <$;'... d$\ 

 being arbitrary, the d§ . . . c?S must be chosen in such a way, 

 that the values of §' . . . 5\ calculated by the aid of the functions 

 §'=ƒ(§...£,) etc. fall within the limits fixed by dg etc *\ This, 

 however, is impossible. 



!) Gf. § 78, 2nd paragraph. 



~) Gf. among others volume I, p. 25 and 27. 



3 ) Gf. 'The dynamical Theory of Gases" p. 18. 



