Miscellanea 
173 
A similar difficulty arises whatever values wc take for p and q (lietween 0 and 1) if m itself be 
small, i.e. if we are dealing with random samples of small size. 
To surmount this difficulty we are compelled to return to the original binomial M {p + q)'\ 
Now the calculation of any number of the terms of this binomial is very troublesome, 
es[)ecially when n is large, but np small. Accordingly we need an integral which will stand as 
closely to the sum of the first s terms of this binomial for an^ values of n and p, as the normal 
l)robability integral does to the same sum when 7i is lai'ge and p moderate. This expression is 
directly and efi'ectively provided by the curve* 
y=y„e-v-'-(^l+0" (viii), 
where m = A:(— -) — 1 (ix), 
\npq nj 
and yo = -3//rt . «i"' + ie-"7l" (»« + !) (xi). 
c of course will usually be taken unity and the origin is the mode or maximum fi'equency. The 
areas of this curve give as completely as the probability integral does the odds against any 
observed deviation from the modal value. 
It will be obvious that to find the odds against any given deviation we require the ratio of an 
incomiDlete to a complete r-function. Numerical tables to assist the calculation of incomplete 
P- and incomplete /3-fvmctions are nearly finished and will be shortly published. 
Thus within these limits a solution is reached for the problem of the probable error of random 
sampling when n, p, q are anything whatever. 
(2) The whole of the preceding investigation is, however, subject to a limitation which 
often escapes notice. We have supposed the chance of any individual arising with the character 
of a given class to be^, and that this chance remains constant throughout the collection of the 
sample. This statement of the problem is however incorrect, when the size of the sample is in 
any manner commensurable with the total population from which it is drawn. Such cases are 
by no means uncommon in the treatment of vital statistics for the case of man. Further in the 
consideration of determinant theories of inheritance, when the character of the individual 
depends on the random sampling of a finite number of determinants, the size of the sample not 
being small as compared with the number of selectable determinants, we are again excluded from 
using either the jDrobability integral or the incomplete r-function for the determination of the 
distribution. 
For example, if a cell-division leads to the exclusion of n' determinants out of N=n + n' 
available determinants, where n and n' are commensurable, it is not possible to approach the 
matter as we have done above ; for in the cases treated n' is supposed large as compared with n. 
We accordingly reach the following more general problem : 
A population consists of N individuals, Np of which possess a given character and A^q do not, 
what will be the distribution of frequency in this character for M random samples of magnitude 
n which is commensurable with N 1 
The solution is of course the hypergeometrical series 
^ pN{pN-\)... {pN-n + ^) f qN n{n-l) <l^iq^'^) 
N{N-\)...{N-n + \) \ pN-n + \^ 1.2 {pN-n-{-\){pN -n + ^) 
n{n-\){n-2) qN(qN-l){qN-2) ] 
1.2.3 {pN-n + l){pN-n + 2){pN-n + 3y---j 
* Skew-Curve of Type III: see Phil. Trans. Vol. 186 A, p. 373. 
