356 MR. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
Now compare this property of the polygon with that of the curve: 
y = OOS 
We have by differentiation : 
slope of curve __ 2 abscissa 
ordinate ~ 2c 
Hence: this binomial polygon and the normal curve of frequency have a very close 
relation to each other, of a geometrical nature, which is quite independent of the 
magnitude of n. In short their slopes are given by an identical relation. By a 
proper choice of o and y, we can get the normal curve to fit closely the point- 
binomial, owing to this slope property, without any assumption as to the indefinitely 
great value of n. It is this geometrical property which is largely the justification for 
the manner in which statisticians apply, and apply with success, the normal curve to 
cases in which 7 is undoubtedly small. No stress seems hitherto to have been laid 
upon the fact that the normal curve of errors besides being the limit of a symmetrical 
point-binomial has also this intimate geometrical relationship with it.* 
(8.) Now let us deal with the skew point-binomial in precisely the same manner as 
we have dealt with the symmetrical binomial. Taking its form to be e(p + q)", we 
have, if x,= 7 X cand \ = q/p: 
VERS Oh 2G eae Nie OG ae Dy es ae) 
Yap tye  ¢(m—r+)aAfr+17 cAam+1)+7r0—A)) 

Let us write Ay = 4, — Yn Av =e. 
Ya = z (Yr aa oF Y,), Nee ee 45 (2, ap il aF Dyn 
Then X,..,/e=7-+ 3, and: 
* The following table shows the closeness of frequency within a given range as determined by the 






binomials :-— 
= See = Le aa 
Frequency per cent. 
Range oF Normal curve. 
deviation. 
| el =p 110. | qd ae 15) 20 
aa a Ee 
| 3 | 24 23 24 
5 | 37 3 38 
7 50 52 52 
11 | 71 73 73 
V5 | 87 87 87 
21 96 96 96 
33 100 100 100 



Here the distribution of 100 groups each of 100 events is seen to be practically the same whether we 
take n = 10 orn=om. 
