16 The Law of Distribution of Velocities [OH. n 



The number of molecules of class A is vf(u, v, w) dudvdw per unit volume, 

 and each of these is capable of taking part in a collision of class a. Let a- be 

 the diameter of a molecule, and imagine a sphere of radius a- drawn round 

 each molecule and concentric with it. As the molecule moves, the sphere is 

 to move so as to remain concentric, but is not to rotate with the molecule. 

 If a collision of class a occurs, the centre of the second molecule that of 

 class B must lie on this sphere at the moment of impact, and further, since 

 condition (iii) is to be satisfied, must lie within a small element of surface of 

 area er 2 do>. In figure 1, the sphere of radius a is drawn thick. The other 

 spheres represent the two molecules just before and at the instant of collision. 



.ft,V 



FIG. 1. 



Supposing that a collision of class a takes place, we see that before collision 

 the second molecule must have been moving relatively to the first with a 

 velocity of which the components, except for infinitesimally small quantities, 

 were u u, v v, w' w, let us say a velocity V in a direction \, p,, v. 

 Hence at an infinitesimally small time St before collision, the centre of the 

 second molecule must have been upon a small element of area a^dw obtained 

 by moving the element cfidw from an initial position upon the surface of the 

 sphere through a distance VSt in a direction \, /*, v. If, therefore, 

 a collision is to take place within an interval dt the centre of the second 

 molecule must, at the beginning of this interval, have been inside the 

 cylinder which is described by moving the original element through a 

 distance Vdt in this same direction. 



The volume of this cylinder is equal to its base multiplied by its height. 

 The former is a^daa, the latter is Vdt cos 6, where is the angle between the 

 axis of the cylinder and a perpendicular to the base. The direction cosines 

 of the axis are X, /JL, v, those of the perpendicular to the base are of 



course I, m, n, so that 



cos 6 = (I \ + wyti + nv) (3). 



