482 The Foundations of the Kinetic Theory of Gases. 



actual velocity of the P sphere before collision is represented 

 by pG and that of Q by qG. Then I propose to prove 



(1) If Maxwell's distribution prevail, for any given direc- 

 tion of pOq, 00 is on average of all the collisions in question 

 as likely to have any direction in space as any other. 



(2) Therefore on average of all these collisions the differ- 

 ence between the kinetic energies of the spheres before 

 collision is 



iP{Op 2 + OC 2 }-iQ{0^ + OC 2 }. 



(3) After collision Y f remains unaltered in magnitude and 

 direction. V remains unaltered in magnitude, but by the law 

 of impacts between elastic spheres is as likely to have any 

 direction in space as any other. Therefore after collision the 

 difference of energies on the average of all the collisions in 

 question is 



!P{0^ + OC 2 }-iQ{0,f + OC 2 }, 



or the same as before collision. Therefore the difference of 

 energies is on average unaffected by collisions. It follows that 

 Maxwell's distribution, if existing, is not affected by collisions. 

 It remains to give a proof of (1). About as centre 

 describe two concentric spheres, of radii 00 and OG + dY'. 

 Between them at C form an element of volume 



OCW. sin (£d(Ma, 

 where </> is the angle OOp, and « the angle between the plane 

 of COp and a fixed plane through pOq. Then, according to 

 Maxwell's law (which we assume to prevail), the chance that 

 the velocity of P shall be represented by a line drawn from p 

 to some point or other within that element of volume is 

 proportional to 



e-**.cp*oc*dY'am<l>d<l>da. 



And the chance that the velocity of Q shall be represented by 

 a line drawn from q to some point or other within that element 

 of volume is proportional to 



e-*Q- c *X)0W' sin <£#<&*. 



And therefore the chance that the velocities of P and Q shall 

 be represented by lines drawn from p and q both to the same 

 point within the element of volume is proportional to 



e-ACP.c^+Q.c^QCW sin <j> dcj> da. 



And this measures the chance that the angle COp shall lie 

 between <fi and (f> + d<\>. 



Now since =^ 7 P.Cp 2 + Q.0<? 2 is independent of 0. 



