26 The Law of Distribution of Velocities [CH. n 



Let P be any one of these points, so that OP represents the velocity of 

 the corresponding molecule. Let OR represent the velocity u , v , w of the 

 vessel, then RP will represent the velocity of the molecule relatively to the 

 vessel. After collision with the element da-, the normal component of this 

 velocity will be reversed, while the tangential component will persist un- 

 altered. Hence if TRS is a plane through R parallel to the element da, the 

 relative velocity after impact is RP' where RP' is the image of P in the 

 plane TRS. 



The small parallelepiped in which Pmust lie if the corresponding molecule 

 is to belong to class A will have as its image in the plane TRS a second 

 parallelepiped which is obviously of the same volume as the former. Let us 

 denote the two parallelepipeds by a, 0, and when the velocity of a molecule is 

 such that the line representing it has its end within (3, let us say that the 

 molecule is of class C. Then we have seen that a molecule of class A is 

 changed by collision into a molecule of class C, and from symmetry it is 

 obvious that the converse is true. 



The number of molecules of class A which collide with the element da- in 

 time dt is equal to the number of molecules of class A which lie within a 

 certain cylinder at the beginning of the interval dt. Similarly the number of 

 molecules of class C which collide with da- in the same time dt is equal to the 

 number of molecules of class C which lie within a second cylinder at the same 

 instant. 



The former cylinder is of base da- and of a height equal to the distance 

 described by a molecule of class A in time dt measured normal to the element 

 da: The second cylinder is of the same base da, but of height equal to the 

 distance described by a molecule of class C in time dt measured normal to da. 

 Since the normal velocity is the same for a molecule of class A as for one of 

 class C, the heights of these two cylinders are the same, and since their bases 

 are the same, their volumes will be the same. 

 <\, 



The density of molecules of class A in the first cylinder is, in accordance 



with equation (33) 



and since in fig. 2 the coordinates of P are u, v, w while those of R are 

 M , V Q , w , we have 



= (u - w ) 2 + (v - v ) 2 + (w- w ) 2 , 



and the density of molecules of class A in the first cylinder is 



vAe- hmRp2 x volume of a ........................ (35). 



C 

 Similarly the density of molecules of class 15 in the second cylinder is 



vAe -hmRp'* x volume of /3 ........................ (36). 



