140 Mr. J. C. Maxwell on the Dynamical Theory of Gases. 



move with uniform velocity in straight lines, and 6 the angle 

 which determines the plane in which V and b lie. From V and 

 b 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 mole- 

 cule 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 disturb- 

 ing 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 vo- 

 lume be N 1; the mass of each being M,. The velocities of these 

 molecules will in general be different both in magnitude and di- 

 rection. Let us select those molecules the components of whose 

 velocities lie between 



f 1 and f l + <?(?!, Vi and Vi + d Vv ?i and & + «%, 

 and let the number of these molecules be d~N } . The velocities 

 of these molecules will be very nearly equal and parallel. 



On account of the mutual actions of the molecules, the num- 

 ber of molecules which at a given instant have velocities within 

 given limits will be definite, so that 



an^Mv&dWvM (2) 



We shall consider the form of this function afterwards. 



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

 volume be N 2 , and let ^N 2 of these have velocities between f 2 

 and f 2 + df 2 , r} 2 and drj 2 + y^ ? 2 and f 2 + d£ 2 , where 



^N 2 =/ 2 (f 2 i72? 2 )^2^2^&- 



The velocity of any of the d'N l molecules of the first system 

 relative to the d~N 2 molecules of the second system is V, and each 

 molecule M 2 will in the time St describe a relative path V67 

 among the molecules of the second system. Conceive a space 

 bounded by the following surfaces. Let two cylindrical surfaces 

 have the common axis YSt and radii b and b-\-db. Let two 

 planes be drawn through the extremities of the line YBt perpen- 

 dicular to it. Finally, let two planes be drawn through YSt 

 making angles <£ and (j)-\-d<p with a plane through V parallel to 



