20* MAXWELL'S LAW 391 



molecules, arid u } , v^ w } , u. 2 , v%, w^ . . . the components of their 

 respective velocities. For a mixture of two gases we have there- 

 fore six equations to take into account ; for a mixture of three 

 gases, nine equations ; &c. 



In addition to these equations, which specially hold for each 

 species separately, there is still a condition relative to the whole 

 number of particles which has to be fulfilled, namely, that the 

 kinetic energy of the whole system shall possess a given value. If 

 we denote by N lt N z , . . . the numbers of particles of each kind, and 

 by N their total number, so that 



N = N, + N 2 + . . . , 



and if we further write E lt E 2 ,. . .for the mean values of the 

 energy of a particle of each kind, and E for the mean value for 

 them all, so that 



NE = N } E l + N 2 E 2 + . . . , 

 then 



NE = ^.m^u^ + V + Wi 2 ) + iS.w 2 (w 2 2 -I- v 2 a + w z *) + . . . 



where, as before, N and E are given magnitudes. 



If now with respect to this enlarged number of equations of 

 condition we seek the function F which determines the distribu- 

 tion of the different values of the components u, v, w by the same 

 methods of the Calculus of Variations as before in 12*, there is 

 in the calculation only a difference in the number of the variations 

 that remain arbitrary and of those which are determined by them. 

 The form of the equations therefore remains the same ; the number 

 of constants only is increased. For each kind of molecules we 

 obtain differential equations of the form 



= 1 f + SM,- ft) 



as before, wherein a, ft, y may have different values for different 

 kinds; the constant k however has the same value for all the 

 kinds of molecules, since it is the elimination-coefficient by which 

 the last equation of condition is brought in which takes into 

 account all the particles alike. 



