ON THE KINETIC THEORY OE GASES. 205 



tween a pair of spheres, and the summation includes all 



pairs. If the spheres be material particles without sensible 



dimensions S^R7' vanishes, there being no finite forces. If 



they be spheres of diameter "c,"and subject to collision, 22R.7' 



is proportional to zZvnr. For consider two equal spheres 



A and B, each of mass m. Let p be their relative velocity. 



About the centre of A describe a sphere with radius q, i.e., 



twice the radius of A or B. Consider all radii of that 



sphere which make with p angles between and 6 + dO. 



Let v be the volume in which the spheres, 11 in number, 



are supposed to move. Then the chance that the centre of 



B shall at a given instant be within the element of volume 



277-t; 2 cosO sin ft dO pdt 

 27rc 2 cos sin dO pat is 



and if this is the case, A and B will collide within the time 

 dt after the given instant, and the relative velocity in line 

 of centres, namely, p cosO, will be reversed. Suppose that 



reversal to be affected by the large finite force 



acting on each sphere during the time 2c//, then R, in the 



. nip cosO 

 expression Rr, is , ; also r = q, and the number of 



n 2 

 pairs of spheres in volume v is — ; so 



r 



cos 2 sin dO 



7l 2 



3^2 



2^2 Rr = — VI 2 irq J p 



2V r 



n 2 



^mtirg'p' 



n z - 



= —ml-rcqHr, 

 v 



because P 3 = 2u 2 , and since 2mu 2 = nmu 2 the virial equation 

 becomes 



Ipv = 2Zwir - 1 + — - ) = Zw;r \ 1 + - if v = f tt€ j 3 or 



four times the volume of one of the n spheres. The 

 effect is that if the gas were compressed at constant 

 temperature, the pressure would increase rather more 

 rapidly than it should do were the law pv = C/ accurate. 

 In fact the deviations from the laws of Boyle and Charles 

 that require explanation are in the opposite direction. 



