512 Mr. W. Sutherland on the Vii 



causes the spheres to behave as regards collisions as if they 

 were larger spheres devoid of force, the diameter-squared 

 (2a) 2 being enlarged in the proportion l-\-2mf(l/2a)/Y 2 : 1. 



Hence, in the theory of viscosity as worked out for force- 

 less molecules, we need only increase the square of the 

 molecular sphere-diameter in this proportion to take account 

 of molecular force. As the expression diminishes with in- 

 creasing V 2 , that is with increasing temperature, we see at 

 once why the apparent result of increasing temperature was 

 to make the molecules shrink : increase of temperature does 

 not make the real molecules shrink (at least to the extent 

 imagined) , but produces shrinkage of the imaginaiy enlarged 

 forceless spheres which could exhibit the same viscosity as 

 the real molecules. 



So far we have considered only a typical case of two 

 molecules : to obtain the effect of molecular force in the 

 average case we should have to calculate, in accordance with 

 Maxwell's law of the distribution of velocities amongst the 

 molecules, the number of pairs that have relative velocities 

 between V and V + dY and sum for all values of V. This pro- 

 cess can easily be carried out when necessary, but it will be 

 quite accurate enough for our purpose to assume that the pair 

 of molecules we have studied is an average pair, that is, a pair 

 which, has the square of the relative velocity equal to the 

 average value of the square of the relative velocities for all 

 the molecules ; this is proportional to the mean squared 

 velocity, and according to Maxwell's law of velocities is equal 

 to twice it, in the usual notation Y 2 = 2v 2 . 



Now if there are n spheres of radius a moving about in 

 unit volume with Maxwell's distribution of velocities of which 

 the average is v, then the average number of collisions per 

 second per sphere is 2 J mra 2 v when the spheres are forceless ; 

 when the spheres attract one another it becomes 



2W -„( 1+ wW). 



This number is fundamental in the kinetic theory of matter, 

 though more spoken of under another form, namely the mean 

 free path of a sphere ; accordingly we can state the highly 

 convenient result that all the investigations of the founders 

 and developers of the kinetic theory on the properties of 

 gases which depend on the mean free path or mean number 

 of collisions of forceless molecules can be applied to attracting 

 molecules by simply replacing a 2 by a 1 l-\-2mf (l/2a)/Y 2 . 

 The chief properties depending on number of collisions are 



