﻿Virial of a Mixture of Ions. 



5G9 



nearest — ion, is unspecified, may now be obtained by 

 summing the different values which (35) takes up for each 

 possible position p 2 of B 2 , i. e., for the values of p 2 from 2 to 

 jt?3 — 1 inclusive ; this gives 



.2V7 1 P2- 1 



e n (p n ) ...e 2 (pz) 2 e 2 (p 2 ) 2 *i(/>i). 



This process is evidently general. Carrying it out for all 

 the ions, we have for the relative probability of an arrange- 

 ment of m nearest ions containing n — ions, in which the 

 positions of the u — ions are all unspecified — we shall call 

 th-is/n(m) — 



m p„-l p 3 -l Pv-1 



f n {m)= X e n (p n ) 2 e n -i(p n -d 2 e 2 {p 2 ) 2 e{(p^ (36) 



1 O .1 



Vn—n 



JP„_1=»-1 



2V 



!H = 1 



The quantity fi(m) may be considered a function of the 

 natural number ???, the nature of the function depending on 

 the natural number n. (It is of course also a function of 

 the independent variable li.) It may be tabulated for dif- 

 ferent values of m and n, by employing on the numbers for 

 ei(p) of Table II. a process of successive additions similar to 







Table III. c 



tf n (m) [7i = -3]. 







m. 



/iW- 



/>(»). 



AM- 



/*(»)• 



Um\ 



A( m )- 



./». 



Mm). 



1 ... 



rooo 

















2 



176275 



1000 















3 ;;; 



2-3995 



276275 



•97100 













4 ... 



2-9473 



5-0449 



3-7338 



•90411 











5 ... 



3-4275 



7*6667 



87787 



4-5474 



•7962 









6 ... 



3-8515 



10-4995 



162597 



13-3261 



5-106 



•679 







7 ... 



4-2295 



13-4489 



26-1216 



29*586 



18-228 



5-378 



•5621 





8 ... 



4-5689 



16-4472 



38-182 



55*308 



47824 



22-950 



5-372 



•4527 



9 ... 



4-8751 



19-4455 



52191 



91-950 



103*132 



' 70-253 



26-981 



5-126 



10 ... 





22-4084 



67-871 



140-416 



194*079 



173-385 



95-339 



29-863 



11 ... 







84-939 



201-093 



330-349 



367*464 



267-28 



120-867 



12 ... 









273 926 



520-68 



695*064 



634-74 



382-47 



13 ... 











772-37 



1203-81 



1329*81 



101301 



14 ... 













1943-57 



2525*64 



2342-82 



15 ... 















4433-21 4868*46 



10 ... 















9277*46 



that adopted in the formation of the ordinary arithmetical 

 triangle. Table III. shows values of f n (in) for 7i = *3. The 

 construction of the table may be most simply explained by 

 observing that, as follows at once from (3b'), 



fn(m)=Jn(m—l) +/»-i(m — l)e»(m). 



