96-98] Maxwell's Treatment of Equipartition 87 



dimensions. Then the necessary and sufficient conditions for a steady state 

 are 



in the latter of which the change in p l includes that caused by the formation 

 and dissolution of double molecules. 



98. To determine the relation between p 1 and p 2 , we make the assump- 

 tion of 15, namely, that the gas is in a state of molecular chaos. Having 

 made this 'assumption we proceed to calculate the number of encounters of a 

 given kind which occur in an interval dt. If 1, 2 %m ar e the internal 

 coordinates of a molecule, the number of molecules per unit volume for which 

 i> & m> u , v , w lie within a range 



d%\d%2 ' d 2n dudvdw ........................ (207), 



will be Pidgid^ . . . dg m dudvdw. 



Hence, as in expression (4), the number of collisions in time dt for which 

 the coordinates of the first molecule lie within a range (207), and those of 

 the second within a similar range in which the variables are accented, while 

 the line joining their centres meets a unit sphere in a given element of 

 surface dw, will be 

 p l p 1 ' Va z cos d d^d^ 2 ... dijvidgidffz ... d^'^dudvdwdu' dv'dw'dwdt ....(208). 



This number of collisions must however be equal to the number of double 

 molecules which cross a certain element of the surface S in the 4>n + 9 dimen- 

 sional space in time dt, and this number will be 



.................................... (209), 



at 



where dS is the element of the surface 8 representing collisions of the type 



\ 



in question, and =- is the velocity in this space at the element of surface dS 



measured inward along the normal. The equation of the surface 8 being 

 equation (204), we may clearly suppose the normal to be the shortest distance 



from dS to 



x-x' = Q, y-y' = Q, z-z'=Q, 



and therefore write 



e 2 = (x - a/) 2 + (y- y') 2 + ( z ~ -sO 2 - 



, 9e x x' 3 . f. 



Thus 57 = ~ 07 ( x -#)+ 



dt e dt ^ 



= X (u - u'} + . . . 



= Fcostf . .................... (210), 



