1036 PROFESSOR TAIT ON THE 



Now the average value of v/v is obviously 



Jvv]V sin /3d8 



JvviVq sin BdB 



where /3 is the angle between the directions of motion, so that 



vv 1 sin fid/3 = v dv . 

 Hence the average above is 



A 



- y^v ^h - l 



r- 



,*dv ~ V3 ~ J*E r l - 2 



vv 



Hence the mean of the free paths during a given period becomes 



1 J2s. 



J2n7rs 2 3 ' 



that is, it is shortened in the ratio 



~[ — -mrS 3 : 1 

 o 



or 



1 — 4 (sum of vols, of spheres in unit vol.) : 1 = 1 — ^ : 1 say. 



Hence the number of collisions per second, already calculated, is too small in the 

 same ratio. 



Thus the value of 2(R) in § 30 must be increased in the ratio 1 : 1 — y , and the virial 

 equation there given becomes 



If this were true in the limit, the ultimate volume would be double of that before calcu- 

 lated, i.e. 8 times the whole volume of the particles. 



§ 62. Another mode of obtaining the result of § 61 is to consider the particles as 

 mere points, and to find the average interval which elapses between their being at a 

 distance s from one another and their reaching the positions where their mutual distance 

 is least. The space passed over by each during that time will have to be subtracted 

 from the length of the mean free path calculated as in § 11 when the particles were 

 regarded as mere circular discs. 



The average interval just mentioned is obviously 



, / 8 cos 6 . sin cos QdQ „ 



it pi 3u 



/ sin 6 cos Odd 



