17, 18] The Statistical Method 17 



The volume of the cylinder is therefore Fcr 2 cos ddwdt, so that for any single 

 molecule of class A, the probability that .the centre of a molecule of class B 

 shall lie within this cylinder at the beginning of the interval dt is, in 

 accordance with 15, 



vf(u', v', w') du'dv'dw'Va- 2 cos0do)dt. 



This, then, is the probability for each molecule of class A that a collision 

 of class a shall occur during the interval dt. The number of molecules of 

 class A is vf(u, v, w) dudvdw per unit volume, so that the " expectation " of 

 the total number of collisions of class a which occur in time dt per unit 

 volume will be 



v*f(u, v, w)f(u, v', w')Va-*cos0dudvdwdu'dv'dw'da)dt (4). 



18. We now consider a second type of collision, class y8. This is to 

 be a type of collision through which a molecule enters into class A, and 

 is to be defined as a collision in which the three following conditions are 

 satisfied : 



(i) After the collision, one of the molecules is to be of class A. 

 (ii) After the collision the second molecule is to be of class B. 



(iii) The direction of the line of centres at impact is to satisfy the same 

 condition as for a collision of class a (p. 15). 



The velocities before the collision can be found without trouble. For the 

 relative velocity can be divided into two parts the one in the common 

 tangent plane through the point of contact of the spheres, and the other 

 along the line of centres. Of these, the former remains unaltered by the 

 collision, while the latter is reversed in direction, but remains unaltered in 

 magnitude. Now the normal relative velocity after impact must, in virtue of 

 the three conditions satisfied, be the same as for a collision of class a before 

 impact. It must, therefore, be V cos 6. We shall denote V cos by W, and 

 therefore have, by equation (3), 



W = - V(l\ + mfj,+ nv) 



= l(u u') + m(v v')+n(w w'). 



Let u, v, w and u', v', w' be the components of the velocities of two 

 molecules such that after a collision along a line of centres having direction 

 cosines I, m, n the velocities are u, v, w and u', v', w', then by what has just 

 been said we must have . 



u =u -IW = u -{I 2 (u - u')+ lm(v - v') + ln(w -w')} (5), t 



u'=u'+lW = u'-{l*(u'-u) + Im (v'-v) + In (w'-w)} (6). 



The number of collisions per unit volume, sach that before collision the 



components of the velocities lie between u and u + du, etc., and u' and 



j. 2 



