FOUNDATIONS OF THE KINETIC THEORY OF GASES. 87 



(a) It is the rate at which momentum passes across a plane unit area ; or 

 the whole momentum which so passes per second. [It is to be noted that a 

 loss of negative momentum by the matter at either side of the plane is to be 

 treated as a gain of positive.] 



In this, and the other investigations which follow, we deal with planes sup- 

 posed perpendicular to the axis of x ; or with a thin layer bounded by two such 

 planes. 



The average number of particles at every instant per square unit of a layer, 

 whose thickness is Bx, is n&x. Of these the fraction 



7T 



have speeds from v to v + dv. And of these the fraction 



sin /3 d/3/2 



are moving in directions inclined from /3 to /3 + d/3 to the axis of x. Each of 

 them, therefore, remains in the layer for a time 



Sx/v cos & 

 and carries with it momentum 



Vv cos ft 



parallel to x. Now from /3 = to fi = ~x we have positive momentum passing 



towards x positive. From /3 = -„- to /3 = tt we have an equal amount of 



negative momentum leaving x positive. Hence the whole momentum which 

 passes per second through a plane unit perpendicular to x is 



cos 2 $ sin fidfi = gPwjj 2 " 

 o ~s 



where the bar indicates mean value. That is 



2 

 Pressure =p — o (Kinetic Energy in Unit Volume). 



(b) Or we might proceed as follows, taking account of the position of each 

 particle when it was last in collision. 



Consider the particles whose speeds are from v to v + dv, and which are con- 

 tained in a layer of thickness Sx, at a distance x from the plane of yz. Each 

 has (§ 10) on the average ev collisions per second. Thus, by the perfect re- 

 versibility of the motions, from each unit area of the layer there start, per 



second, 



nvevSx 



such particles, which have just had a collision. These move in directions 

 uniformly distributed in space ; so that 



sin j3 dfi/2 



