the Kinetic Theory of Gases. 347 



that if we write (1) in the form 



Pu 2 -uv-Qv^uv = 0, (2) 



the terms on the left are equal multiples of the average energy 

 of a P and of a Q respectively. Thus Maxwell's Theorem is 

 rigorously true, though in a most unexpected manner. There 

 must surely be some extremely simple and direct mode of 

 showing that u 2 — uv is independent of the mean-square speed 

 of the system of Qs. Meanwhile I give the heads of the 

 investigation by which I arrived at the result just stated. 



5. It is shown that, if a particle move constantly with velo- 

 city v among a system of other particles which are in the 

 " special " state, the fraction of the whole of its encounters 

 which take place with particles whose velocity is from i\ to 

 i'i + dvi, and whose directions of motion are inclined to its 

 direction at angles from ft to ft + dft, is proportional to 



6-^ 2 v ^v dvism /3 dft ; 

 where 



v = v/ v 2 + Vi — 2vv 1 cos ft 



is the relative velocity, and 3 / 2g is the mean-square speed of 

 the spheres of the system. 



The line of centres at impact depends for its positions upon 

 the condition that the line of relative motion of the centre 

 of one of the impinging particles may pass, with equal proba- 

 bility, perpendicularly through all equal areas of a diame- 

 tral plane of the sphere, whose radius is the sum of the radii, 

 and which is concentric with the other particle. 



The number of particles of the first system which have 

 speeds between v and v + dv is proportional to 



e-P» 2 v 2 dv, 



where the mean-square speed of the system is 3 / 2p. 

 Taking these things into account, it is found that 



^_ p + 2q 

 2p{p + q) 

 and 



so that 



-2 1 



u^ — uv= -> 



p 



which gives the result stated above. 



