( 495 ) 



Finally the question might be raised, when we want to consider 

 the separate velocities, whether it is possible to arrive at the 

 equally possible combinations under discussion on another suppo- 

 sition a priori about the chances of every value for the velocity than 

 the one indicated by Boltzmann and J BANS. The supposition must of 

 course be such, that the chance is independent of the direction of 

 the velocity, so that the chance of a velocity c, at which the point 

 of velocity falls into a certain element of volume dzd^dZ, may be repre- 

 sented by f(r)dlJ> t dl. When we moreover assume that the probabilities 

 for .the different molecules are independent of each other, the probability 

 of a certain combination of velocities is proportional tof(t\)f(i\) . ■•/(<-'„), 

 and this must remain the same when the kinetic energy L, or 

 because the molecules are assumed to be equal, Se 1 remains the 

 same. For every change of c/ : and c/ into c'k and c'i, so that 

 c'jfc + c% l == c' 8 fc -j- c ' % h nmst f( c k)-f(ci)=f(c'k).f{c'i). This is an 

 equation which frequently occurs in the theory of gases, from which 

 follows f(c) = ae bc *. As a special case follows from this: /(c) = a, 

 i. e. the assumption of Boltzmann and Jeans, that the probability a 

 priori would be equal for every value of the velocity. 



§ 2. In the second place I wish to make some remarks in con- 

 nection with the proof that Boltzmann gives in his "Gastheorv*, 

 that for an "ungeordnetes" gas with simple suppositions on the nature 

 of the molecules in the stationary state Maxwell's distribution of 

 velocities is found. Dr. C. H. Wind shows in his above-mentioned 

 paper that in this Boltzmann makes a mistake in the calculation of 

 the number of collisions of opposite kind. Boltzmann, namely, assumes, 

 that when molecules whose points of velocity lie in an element 

 of volume da, collide with others whose points of velocity lie in 

 du> lf so that after the collision the former points lie in <ho' and the 

 latter in ito/, now the elements of volume day and dio', dto l and «2co x ' 

 w T ould be equal, so that now dto' do>\ = dw dio 1 . He further assumes 

 that when molecules collide whose points of velocity lie in dta' 

 and t/to/. they will be found in dm and dio l after the collision. 

 These last collisions he calls collisions of opposite kind. Wind now 

 shows that this assumption is untrue ; dm is not =r dto' , do> l not 

 = d€t>\, nor even dtodto^ = doj'doj^', except when the masses of the 

 two colliding molecules are equal J ). 



Further the points of velocity of colliding molecules which lay 



in du>' and da>/, do not always get to doj and <ho x after collision, 



*) I point out here that even then it is not universally true, but only when the 

 elements of volume du and du\ have the shape of rectangular prisms or cylindres 

 whose side or axis ha? the direction of the normal of collision. 



33* 



