141, 142] 



Deviations from Boyle's Law 



123 



Now if B is in a position such as I. in fig. 5, in which its centre is at 

 a distance greater than <r from the boundary of the volume fl', the propor- 

 tion of configurations represented in the generalised space in which condition 

 (290) is satisfied or, what is the same thing, in which the centre of A lies 

 within a sphere of radius cr surrounding B is equal to the ratio of the 

 volume of this sphere, to the whole volume which is available for the centre 

 of A, is therefore equal to fTrcr 3 /!!'. If, however, B is in a position in 



III 



FIG. 5. 



which its centre is at a distance less than a from the boundary, including 

 being in a position such as II. in fig. 5, in which its centre is at a distance 

 less than \<r from the boundary, the proportion in question is less than that 

 just found, since it is only possible for A to lie in that portion of a sphere 

 of radius a- surrounding B which lies inside the volume H'. Ultimately, when 

 B is in position in., the proportion is only one- half of the above quantity, 

 namely, 7ro- 3 /n', for it is now only possible for A to lie inside a single 

 hemisphere of the sphere about B. On averaging over all positions of B it 

 is obviously legitimate to disregard the exceptional cases II. and in., and so 

 we arrive at the result that the proportion of cases in which condition (290) 

 is satisfied is fTrer 3 /!!'. Thus if condition (290) defined the only region to be 

 excluded, the whole integral H /Ar would have to be reduced by ^ira s l^l' N ~ l . 



There are, however, ^N(N \) such conditions, corresponding to all 

 possible pairs of molecules. The total reduction is therefore ^N (N 1) 

 times the foregoing amount. From this must be subtracted a quantity 

 representing regions in which two of the conditions of the type of (290) are 



