Prof. Clausius on the Conduction ofReat by Gases. 4.23 



The cause of their occurrence depends, in the case before us, 

 upon the fact that when two molecules, coming from different 

 sides, strike each other within the stratum, the molecule which 

 comes from the warmer side has in general a greater velocity 

 than the one which comes from the cooler side. The magnitude 

 of this difference is determined by the distances from the stratum 

 in question of the points at which the said molecules commenced 

 their motions ; and since the distances through which the mole- 

 cules move between each two impacts are in general very small, 

 this difference must also be very small, so that we can regard the 

 mean value of this difference as a magnitude of the same order 

 with the mean excursions (Weglange) of the molecules. We 

 must now try to determine what influence this difference, exist- 

 ing before the impacts, exerts upon the motions after the impacts. 



§ 4. The behaviour of two impinging molecules is not in every 

 respect the same as that of two elastic spheres ; but we can never- 

 theless in many respects obtain a useful insight into the beha- 

 viour of molecules by starting from the consideration of elastic 

 spheres. The mutual action of two impinging elastic spheres is 

 very comprehensively treated by Maxwell in the memoir already 

 mentioned. I will here only quote a few principles, which may, 

 however, be considered as sufficiently well known without my 

 doing so. 



When two elastic spheres move with equal velocity in opposite 

 directions, and with their centres in the same straight line, so 

 that they strike each other centrally, they rebound from each other 

 in such a manner that each sphere moves back with the same 

 velocity in the direction of the point from which it came. But 

 if the spheres move, before the impact, still in opposite direc- 

 tions, but with their centres in two parallel straight lines instead 

 of in the same straight line, and so that the spheres consequently 

 impinge excentricaliy, they rebound again with equal velocities, 

 their centres again move in opposite directions in two parallel 

 straight lines ; but the direction of these straight lines is not 

 the same as that of the straight lines in which the centres moved 

 before the impact. The new direction depends upon the position 

 on the two surfaces of the point of contact ; and since the spheres 

 may strike each other on an infinite number of different points 

 of their surfaces, the rebound may also take place in an infinite 

 number of different directions ; and it can be easily shown that 

 each jjossible direction in space is equally likely for the motions 

 of the spheres after the impact. 



Let it now be assumed, as a general case, that the two equal 

 spheres move before the impact with any velocities whatever and 

 in any directions whatever. We will decompose the motion of 

 each sphere into two components. We will take as the first com-. 



