74 On Errors of Random Sampling 
been to show that, under certain conditions, the probabilities of different values of 
r can be represented by the ordinates of a "normal" curve*. The nature of the 
assumptions involved will be placed in the clearest light by the following 
considerations. 
Substituting 0, 1, 2, ... m for r successively in (1), reducing to B and then to 
r functions and finally evaluating each term, we have for the chances of 0, 1, 2, . . . m 
successes in a sample of m after a first sample n = p + q : 
C f 1 + - p + 1 + m ( m ~ 1 )(j ) + 1 )(P + 2 ) I (2) 
J "\ l\q + m 2l(q + m)(q + m-l) "') y h 
where = T(q + m + l)T(n + 2 ) 
0 r (q + 1 ) r (n + m + 2) 
We may notice that, if p and q are both very large as compared with m, 
(2) reduces to 
ffY" ( X + m + + etc. 
q 1 ! 2 lq 
-) fl + -1 = (» + q) m , where jo = - and « = - (2) Us. 
nj \ qj r 1 r n n 
The conditions for the approximation of this binomial to the " normal " curve 
have already been noted. 
More directly, the approach of (2) to a " normal " form can be examined by 
treating the series in brackets, which is a hypergeometric series having as 
parameters 
a = — m, ft =p + 1, y = — (q + m), 8 = 1, 
by the method of moments and then noting the conditions under which the 
momental constants /3 1 and fi 2 approximate to the values 0 and 3 respectively. 
This method was adopted by Pearson who had, several years before the date of 
the publication last cited, obtained the moment coefficients of a hypergeometric 
series f. 
The results are that : 
fl + 2 (w " 1) Y 
Pl m(p + l)(q + l) m — 1 W ' 
A = 3 1 
1 + 
n + 3 
m-1 /, 8 
n -f 4 V m — 2 n + o 
w — 1 
1+ ,7T3 
(» + 2) 2 n + 4 V 7i 4- o , . ( 
m(j)4 l)(g+ 1) , vi -I 
n + 3 
* See, for instance, Czuber's Wahrscheinlichkeitsreehnung (1903 Edition), pp. 151 etc. 
t Karl Pearson, "On Certain Properties of the Hypergeometrical Series, and on the fitting of such 
Series to Observation Polygons in the Theory of Chance," Philosophical Magazine, 1899, p. 236. 
