﻿Vir'ial of a Mixture of Ions. 



507 



expression takes up in succession as p is given the values 

 from 21 — 1 to p z — 1 inclusive, and multiply all the equations 

 together. On the one side all the intermediate P's except 

 the first and the last cancel out, and on the other side the 

 exponents of e all add togeth< 



P(^! ...pi...p») 



ter, giving 

 1 



= e 



6 p=2i-l P 



Up 



(31) 



l\ Pl ...2i-l... 1 j n ) 



If pi is less than 2i — 1, the transformation is effected by- 

 giving p all values from pi to 2i—2, and we get an expression 

 which may also he represented by (31) if we note that a 

 general property of 2 considered as the sign of a summation 

 of a function of the natural numbers n between limits which 

 are both included in the summation, is, 

 n n — 1 w, — 1 



2 =- 2 



(32) 



(31) represents a function of pi and i, which can be calcu- 

 lated without difficulty when the value of the independent 

 variable h is given. We shall represent this function by 

 ?i(pi), oi* more generally, considering it as a function of the 

 natural numbers p, the nature of the function depending on 

 the natural number i, as €i(p), where 



P-I 



p=2i-l 



*i{p) 



9 



p-2i+2 



Up 



(33) 



Values of ef p) for different values of p and i are given 

 in Table II. for an assumed value of /i = 0*3. The meaning 



TableII. of^( ? >)[/i=-3]. 



p. 



*i(p)- 



e 2 (p). 



e s (p). 



e i(p)- 



e 5 (p). 



e*(p)- 



<h(p)- 



e*(p)> 



1 ... 



1-000 

















2 



•76275 



1-000 















3 ... 



•63675 



1-000 



0-97100 













4 ... 



•54781 



•95108 



rooo 



0-93110 











5 ... 



•48019 



•88957 



1000 



•97578 



0-88062 









6 ... 



•42395 



•82648 



•97578 



1-000 



•94783 



0-85276 







7 ... 



•37810 



•76578 



•93926 



1-000 



•98468 



•92035 



0-82790 





8 ... 



•33938 



•70889 



•89676 



•98468 



1-000 



•96398 



•89433 



0-80532 



9 ... 



•30618 



•65624 



•85177 



•95967 



1-000 



•98910 



•94156 



•86995 : 



10 ... 





•60777 



•80636 



•92863 



•98910 



1000 



•97303 



•91895 



11 ... 







•76169 



•89401 



•97049 



1-000 



•99169 



■95453 



12 ... 









•85747 



•94649 



•99169 



1-000 



•97873 



13 ... 











•91882 



•97708 



1-000 



•99336 



14 ... 













•95776 



•99336 



1000 



15 ... 















•98147 



1-000 



16 ... 

















•99453 



