148 On the Theories of Multiple and Partial Contingency 
a population of uniform sex and let there be three characteristics : (i) frequency 
of age groups, (ii) frequency of occupational categories and (iii) frequency of survival 
and of death, the latter classified by various special disease classes — the whole, 
say, representing the returns for one year of a large area or country. We then 
require to determine, whether the like contingency solid for a sub-district, or for 
another population entirely, may be considered as significantly different from the 
above general population, i.e. we require to find the probability of its being a 
random sample of this general population. Now we may do this in the most 
universal manner, by assuming that not only survivals and deaths, but that age 
groups and occupations all have fretiuencies, which are random samples of the above 
general population. But this is not very often what we require; we admit that 
the age distribution is differentiated, we admit that the occupational frequencies 
are peculiar to the locality, and we ask whether, notwithstanding these differences, 
the death distribution is to be considered as a random sample. 
In other words, we do not only fix the size of our sample M ; we fix all one 
face — that of age groups and occupational categories of our contingency solid — 
and ask what is the distribution of samples of M taken from this solid, subject 
to the linear conditions that the totals of age-occupational categories are constant. 
For example, if A be age and B occupation, we make constant for all values 
of u and v, but 
where %,j6^,Cj i''^ the frequency of the uvs cell, c denoting the category C of type of 
death and survival. 
Now clearly in making this investigation we shall be studying the mean square 
contingency and the resulting probability of a ^partial sample — a sample of survival 
and death-type in a population of constant age groups and occupational classes. 
Again, we might treat a population as a sample with only constant age groups or 
only constant occupational frequencies and again investigate its probability as 
a sample with regard to deaths and occupations or with regard to deaths and age 
groups respectively. These would be partial contingencies of the order a or j8, 
while the previous partial contingency was of the order a -f j8, a being the number 
of categories in A (i.e. age groups) and ^ being the number in B (i.e. mortality 
and survival classes). 
(4) Now the value of given above, and of the frequency surface (xi), was 
discussed by me in the year 1900* from the general normal frequency surface by 
evaluation of determinants. The demonstration depends on two hypotheses: 
(i) The approach of the binomial M {p + qY to the normal curve 
V^nV npq 
We know that this is true if n be considerable and neither p nor q very small, 
* Phil. Mag. Vol. l. p. 157, 1900. I have recently given a more elementary proof with the probable 
error of P : see Phil. Mag. April, 1916. 
