152 On the Theories of Multiple and Partial Contingency 
be to that of the class Cq in the total population in the ratio of the partial sample 
m to the total sample M. Without this condition it is not possible to replace 
Mps by injps'- Assuming these conditions to be satisfied, then samples of the 
size m in classes C-i , C^, ■•■ C j picked out of very large samples of M will reproduce 
the same distribution of frequencies in those classes as samples of m picked out of 
an indefinitely large population with the same relative frequencies in those classes. 
But in the case of samples of M, the deviations have their correlations zero 
for the classes C^, Cg, ... C„, or they will be approximately distributed by the 
product of their independent probabilities. The standard deviation being V nipj 
and the mean mp^ > we see that the frequency distribution would really follow 
a Poisson's binomial limits but as shown by L. Whitaker* this binomial limit is 
approximately Gaussian with fairly low values of ; see the Diagrams for 
mp/ = 10 and = 30 in the plate of her memoir. We may accordingly therefore 
take the distribution of the frequency in the sth class or cell to be given by 
1 (wig - iTig)^ 
z, = e 2 (xxvii), 
and the general distribution to be 
W'2tt' y/m^f-^'p2 ... p{ 
where 
— m„ 
.(xxviii). 
Here, if the size of the sample only be fixed, we shall have 
S (tHs) = m = 7nS {ps) = S {Ms), 
or S {nis — nis) = 0. 
If we take X, = 
we have: = + X^^ + ... + X^^ (xxix) 
subject to the condition: 
Vnij^Xi + Vm^X^ + ... + VmiXi = 0 (xxx). 
It is clear that equal to a constant gives a sphere in l-io\d space, and that 
(xxx) is a plane passing through its centre, and therefore cutting the sphere in l-io\d 
space in a sphere of the same radius in {I — l)-fold space. Hence if we desire to 
find the volume of the frequency surface (xxviii) which lies outside a value of 
X = Xo subject to the condition (xxx), all we have to do is to transfer to polar 
* Biometrika, Vol. x. p. 36. If m^ = mp^' be the mean, Pj^ = llin^=^^-3, so that ^i = -03, j32 = 3-03 
already for = 33. 
