THE DYNAMICAL THEORY OF GASES. 37 



There will be similar expressions for the components of the final velocity 

 of M l in the other coordinate directions. 



If we know the initial positions and velocities of M l and M t we can 

 determine V, the velocity of M r relative to Jf 2 ; b the shortest distance between 

 M! and M t if they had continued to move with uniform velocity in straight 

 lines ; and <f> the angle which determines the plane in which V and b lie. 

 From V and 6 we can determine 6, if we know the law of force, so that the 

 problem is solved in the case of two molecules. 



When we pass from this case to that of two systems of moving molecules, 

 we shall suppose that the time during which a molecule is beyond the action 

 of other molecules is so great compared with the time during which it is 

 deflected by that action, that we may neglect both the time and the distance 

 described by the molecules during the encounter, as compared with the time 

 and the distance described while the molecules are free from disturbing force. 

 We may also neglect those cases in which three or more molecules are within 

 each other's spheres of action at the same instant. 



On the Mutual Action of Two Systems of Moving Molecules. 



Let the number of molecules of the first kind in unit of volume be N lt 



the mass of each being M^ The velocities of these molecules will in general 



be different both in magnitude and direction. Let us select those molecules 

 the components of whose velocities lie between 



and & + d, 17, and 17, + drj l} , and , + d lt 



and let the number of these molecules be dN^. The velocities of these molecules 

 will be very nearly equal and parallel. 



On account of the mutual actions of the molecules, the number of mole- 

 cules which at a given instant have velocities within given limits will be 

 definite, so that 



dN t *fi(fatt*t&& (a). 



We shall consider the form of this function afterwards. 



Let the number of molecules of the second kind in unit of volume be 

 N t , and let dN 3 of these have velocities between , and & + d 3 , 77, and 

 , and , + cZ.,, where 



