﻿Virial oj a Mixture of Ions. 563 



The coefficient of: u n /n\ in this equation is 

 / T(m) y- 1 r(m-^) 



When m is large the first factor of this becomes practically 



ra-,1 < ?i - 1 



identical with m 3 , and the second factor with m~~r, as 

 may readily be seen by using the exponential expansion of 

 T(m). Each coefficient reduces, therefore, practically to 

 unity, and the expression in the brackets of: (24) becomes 

 identical with the expansion of e' a . Hence, 



Y(m) = k m e-\ . . . (25) 



We have by (22) 



^ /^f ,. . . (2G) 



/4?rU 



XT) , 



a p here stands for the charge on the pth nearest ion, A w , 

 multiplied by the total charge on the ions which are inside 

 A/s sphere (see equations (IS) and (19)). This is valid 

 only for the provisional units temporarilv adopted in which 

 w and 2N/V were put equal to unity. We shall now revert 

 to the ordinary units, and also understand by a p a purely 

 numerical factor, expressing the excess of the number of the 

 ions of like sign to A p over that of the ions of unlike sign, 

 which are nearer to A than A, ; , A being included in counting 

 this excess. , a p will be called for shortness the " excess of 

 like ions inside the joth," it is a fixed quantity for a definite 

 order of succession of the signs of the ions. 

 Under these conditions (26) becomes 



/4,r 2N\V » T(p-X) 

 "-\T'.y). w^-TiJj- 



(that q 2 jw and 2N/V come in as they do is evident from the 



dimensions). — J, * is a purely numerical function ofp, 



which may be easily tabulated. We shall call it hence- 

 forward Up. Values of u p are shown in Table 1. 



Write h for l-.y- >^rr) 1 7t is a function of 2N/V the 



concentration of the ions, and it is the variable in terms of 

 which the virial will be ultimately expressed. 

 With this notation (25) becomes 



VI 



— h S ctplip 



F(m)=k w e p=i (27) 



2 P 2 



