( 119 ) 



elements have been chosen quite arbitrarily, we easily see that the 

 quantity ^ ni +n a must be the same tor the greater majority of the 

 possible ways of choosing the elements in the spaces up to the 

 (»!+«, — l) th , and that we may therefore write 



X K, »,) = //„,+„, x (»i. «. — 1) (6) 



^ w ,+„ s being now a quite definite quantity, which it remains to 



determine. 



It is very easy to find a tirst approximation to its value. For this 



purpose we have on\y to neglect the fact that the above mentioned 



spheres in the (w, -f- n t ) th space intersect, Doing so, we iind 



Tr 4 4 



g„ l+ » 3 = V - n, - jt <j'— (n—1) - n e t * . . . (7) 



From (6) and (7) we deduce by successive reductions 



"2 



(8) 



XK»* s ) = X(n x )JJ[ f V — n^Jto* — (i> a — l)—na t 



where we have affixed to the sign of the product the highest value 

 that we have to give to the number denoted by the corresponding' 

 Greek letter. A similar notation will be used in later formulae. 



It is easily seen that, with the degree of exactitude to which we 

 have now confined ourselves, the value of / (nj is given by 



xK) = II^-^** 1 ') (9) 



In order to push our approximation further, we have to deter- 

 mine g Vl -\-> h more accurately. We must take into account that the 

 spheres mentioned above intersect, and that we have therefore sub- 

 tracted too much from the total volume. 



Now three cases are to be distinguished. 



1. Intersection between the spheres of radius a described around 

 the points corresponding to the centres of the molecules of the tirst 

 kind. The distance x of the centres cannot be less than <j, and must 

 be less than 2 <r. 



2. Intersection between the spheres of radius J, described around 

 the points corresponding to the molecules of the second kind. The 

 distance of the centres must lie between a t and 2 o t . 



3. Intersection between the spheres of radius a and j, described 

 around the points corresponding to the centres of molecules of the 

 first and second kinds respectively. The distance x of the centres 

 must lie between o and o t -\~ o. 



I shall determine the parts corresponding to these three kinds of 



