174 
Miscellanea 
Thus the hypergcometrical series, and not the point binomials (or their limits either the 
normal curve or Type III skew curve), form the general solution to the problem of random 
sampling. 
If we wish to con.sider the odds against any observed deviation from the most probable result 
for a class frequency, we must accordingly endeavour to determine the value of the first s terms 
of the above hypergeometrical series. But the labour of such an investigation is great and we 
are naturally thrown back, as Laplace was, on the discovery of an integral which will replace the 
finite diflerence series. 
I have shown* in an earlier paper what are the values of the moments of the hypergeometri- 
cal series. In the notation of the present memoir, we have 
t^2 = c'hipq (\ - (xiii), 
+ 3^,(.-2)(l-^-^[-^+^J)} (XV). 
The mean value is at a distance c{\+nq) from the left-hand zero start of the series, i.e. cnq 
from the value when the samite consists wholly of individuals with the character, and this is 
identical with the mean value calculated on the basis of the binomial {p + qY. 
If n and N are both large, but still commensurable, the above results reduce to the simpler 
forms : 
ti.2 = c^npq(\-~] (xvi), 
tiz = chipq{p-q){\--^(\-^ (xvii), 
It will thus be clear that when the sample is commensurable with the population from which 
it is drawn, the standard deviation of the class frequency must no longer be taken slnpq, but 
I'Jnpq^N —n)l{N —V), a result which even if we now use tables of the probability integral will 
gi\'e us a very different value for the probable error in the class frequency. But it is clear that 
we ought not to use such pi'obability integral tables, we ought to replace the sum of the first 
s terms of the hypergeometrical by an integral which gives the value of this sum with a degree of 
accuracy similar to that with which the probability integral in like case gives the symmetrical 
binomial. But such integrals representing the areas of certain curves fitting closely to the 
hypergeometrical series were provided by my memoir of 1895 1. 
It is there shown that if 
p lie between \ + ^ + ^) ' 
the sums of the series are closely given by the areas of the curve 
?/ = ^!>_e-^tan-ix/« ^xix)^ 
i+'ir" 
a 
* Phil. Mag. Feb. 1899, p. 239. 
t Phil. Trans, loc. cit. p. 361 and seq. 
