SUBJECT TO FORCES OF ANY KIND. 353 



If the right-hand side of the equations (7) and (8) are constant, the left- 

 hand sides will also be constant ; and since l u m,, j are independent of l a m u n, 

 we must have 



F^AMW and F, = AM,V? (9), 



where A is a quantity independent of the components of velocity, or 



(10), 



(n). 



This result as to the distribution of the velocities of the molecules at a 

 given place is independent of the action of finite forces on the molecules during 

 their encounter, for such forces do not affect the velocities during the infinitely 

 short time of the encounter. 



We may therefore write equation (1) 



dN=Ce u "*+++*>dd n dldxdydz (12), 



where C is a function of x, y, z which may be different for different kinds 

 of molecules, while A is the same for every kind of molecule, though it may, 

 for aught we know as yet, vary from one place to another. 



Let us now suppose that the kind of molecules under consideration are 

 acted on by a force whose potential is \j>. The variations of x, y, z arising from 

 the motion of the molecules during a tune St are 



&c = &, By = T)8t, Sz = t,St (13), 



and those of 17, in the same time due to the action of the force, are 



**$*:- -g* <"> 



If we make c = logC (15), 



The variation of this quantity due to the variations 8x lt Sy lt 8z lt S$ u Si/,, S l( is 



(.. dc dc dc\ s AHflt d$ d$ r d\fi\ - 

 dx V dy dz) \ dx ^ dy dzj 



.(17). 



VOL. ii. 45 



