150 On the Theories of Multiple and Partial Contingency 
The distribution of frequency of the different classes in these samples of M will 
be given by the terms of the multinomial 
(xvi), 
the general term being 
u,\^^,u'^\...u^ ^V'"i^i'"iV'- ... ^^.-CV'CV^ ... (xvii), 
where 0^, C^, ... Ci are only logical symbols to denote that this general term is 
the frequency of the group, where the class occurs u, times in the sample. 
Clearly Uq + + ... + Ui = M. 
But (xvi) may be put into the form of a binomial, it is 
m=M I 
= ni\{M'-m) \ ^PoGor-'" iPiC, + p,C, + ... + |),C,)™...(xviii). 
Let (pi + P2+ ... + Pi) = ^, then the with term of the above series may be 
read as 
M I 
mliM-m)\ (PoCo)'''--'^-{p,'C, + p,'C\+ ...+p{Cr (xix), 
where p-^^ + p^ + ... + p{ = 1, 
and ^?'l = -^ = B=...=I^, = A=l-^>o (XX). 
P^ P2 Pa Pi 
Now it is clear that the factor 
{p,'c, + p,'c, + ... + p/c,r 
gives the frequency distribution of samples of m drawn from a population of 
indefinitely large size of which the proportions of the classes C^, C^, ... Gi are 
Pi > Pi > •••Pi which no class Cq occurs. But by (xx) these proportions are 
the same as in the original population which contains Cq. 
Hence if we take samples of M from an indefinitely large population with 
classes Cq, C^, C,^, ...Ci, those that contain m of the classes C^, C^, ...Gi will be 
distributed in the same proportions as if we had extracted m from an indefinitely 
large population consisting only of those I classes in the same proportions. 
Now thus far the nature of the class Cq is at our choice. In the original popula- 
tion N it appears with the total frequency Wq- Let + n' = N , and suppose 
is indefinitely greater than n' , then p^ will be indefinitely greater than 
P\> P2> ••• P>n- It follows that pe if s be not zero is very small compared to 
unity, because 
Po-^ Pl + P2 + ••■ + i?„ = 1. 
Hence in such a system from (xiii)'^^^, if s be not zero, 
a, = VMp, (xxi), 
and from (xv) to the same degree of approximation r^t = 0. That is to say, if 
Pq be large in taking samples of size M from an indefinitely large population, there 
will be no correlation in deviations in the frequency of the classes C^, ... Cj. 
