M) PROFESSOR TAIT ON THE 



seemed as if the greater masses would have on the average less energy than 

 the smaller. These are two of the pitfalls to which I have alluded. Another 

 will be met with presently.] 



20. But these first impressions are entirely dissipated when we proceed to 

 calculate the average values. For it is found that if we write (1) in the form 



Pu 2 — uv — Qv 2 — uv = 0, 

 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, in default of anything more simple, 

 I give the investigation by which I arrived at the result just stated. 



21. Suppose a particle to move, with constant speed v, among a system of 

 other particles in the " special " state ; the fraction of the whole of its 

 encounters which takes place with particles, whose speed is from i\ to v x + dv x 

 and whose directions of motion are inclined to its own at angles from 

 /3 to fi + dfi, is (§ 8) proportional to 



or as we may write it for brevity 



v^v sin 6 d8 . 



This is easily seen by remarking that, by § 8, while the particle advances 

 through a space hx, it virtually passes through a layer of particles (such as those 

 specified) of thickness vfix/v. Here (§ 3) 3j2k is the mean-square speed of the 

 • (articles of the system. 



Let the impinging particle belong to another group, also in the special 

 state. Then the number of particles of that group which have speeds between 

 v and v + dv is proportional to 



s- hv Vdv= v> 

 as we will, for the present, write it. 



Now let V, Y v V , in the figure, be the projections of v, v v v on the 



unit sphere whose centre is ; C that of the 

 line of centres at impact. Then VOVj = /3. 

 Let V OV = a, VqOV! = a v V OC = y, and 

 VV C = (f>. The limits of y are and n/2 ; 

 those of (J> are and 2tt. Also the chance that 

 C lies within the spherical surface- element 

 sin ydydfy, is proportional to the area of the 

 projection of that element on a plane perpen- 

 dicular to the direction of v , i.e., it is proportional to 



cos y sin y dyd(p . 



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