594 On tlie Laio of the Partition of Energy. 



established the law in the form mu i = 7n'u' 2 . But these 

 relations cannot be true generally, because the coefficients b 

 are functions of the diameters of- the molecules if elastic 

 spheres, or of their radius of action if centres of force. The 

 law is therefore not generally true in the form mu 2 =- m'u 12 . 



The following method suggests itself: — 



Since Q is constant we have 



_, /dQ, du dQ, dv dQ div\ _ _ 



\ du dt ' dv dt dw dt ' 

 Also by conservation of energv 



du dv . dw\ 



2 



I au av . aw\ „ 



mu — - + mv ~r + «w — r / — u. 

 \ dt dt dt' 



If we can deduce — ccmu. &c, this together with A proves 



du ° L 



the law of equal partition. I think, however, the deduction 

 is unsound, for the reasons given above, and at p. 102 of my 

 " Treatise on the Kinetic Theory of Gases/' 



II. By introducing the coefficients b in the index Q we 

 have given to -any two molecules very near each other a 

 common velocity on average, and so diminished the energy 

 of the motion of any molecule m x relative to its neighbours. 

 If we prove that the amount by which the energy of this 

 relative motion is diminished is J(wj|:i + i , j»7i-+ u 'i£i)> this 

 expression would denote the energy of the stream. And now 

 our equations (B) would express the law of equal partition 

 in the only sense in which we could expect it to be true 

 (see art. 2) — that is, as exclusive of the energy of the stream. 



But in fact ^( w ifi -f v 1 t] 1 + w-£ x ) does not express the loss 

 of energy of the motion of m x relative to its neighbours, and 

 therefore does not express the energy of the stream. To see 

 this it is sufficient to consider the case in which all the 

 molecules have the same mass. Then the energy of relative 

 motion in question is for 7n s 



im 1 B 1 2 = im 1 ( Ul -u f ) 2 + {v,-v') 2 + {w l -w') 2 }, 



where all molecules except m l are included in the summation 

 for u r , v' , w' . — That is, 



iwilV = \ m x % { (u 2 + u' 2 ) + O 2 + v' 2 ) + (w 2 + w 12 ) } 

 — m{Z (uu' -f- vv' + vow') . 

 If .every b is zero the last term is zero. The loss of the 

 energy of relative motion due to the b coefficients is therefore 



mjX (uu' + vv' -f ww'), 

 which is no longer zero when the &'s are not zero. 



