MR. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 357 
n(n + 1)— (+a) (A — 3) 
c 
NO ae 
> aay, [X; : 
Bex tes ae 4 =a) (A 4) 

or, if X42, = X%.,-—e(4+¢(n4+ ))), 
— ean XG 
Tay - Oss 
paint e+ (p— OX ra 

feu | ne y Xia 
—— eae 
at Xp, 
angi eee 
(p—e D4 
The curve which has the same law of slope as this skew binomial is : 
ify= 
I= YO (1 + x/aye7%, 
(9.) This curve accordingly stands in the same relationship to the skew binomial 
as the normal curve to the symmetrical binomial.* There are several points, however, 
to be considered with regard to it. In the first place it is usually assumed that n is 
indefinitely great and c indefinitely small, and then it is supposed that we may 
neglect (p — q) cX’..; as compared with pg (n + 1)c*, and so we deduce the normal 
error curve whether p be equal to q or not. But I contend that this is unjustifiable 
except for very small values of X’.,:, When the deviation X’ is considerable and 
¢ vanishingly small, X’ will be an indefinitely great multiple of ¢; c must be in fact 
the unit in which X’ is measured and unless p = q, the ordinary normal curve is only 
an approximation, even if 7 be large, near the maximum frequency. In the next 
place, when we speak of 7 being large, are we quite clear as to what we mean in the case 
of physical or biological frequency curves? We speak of a multiplicity of small 
“causes” determining the actual dimensions of an organ, or the size of a physical 
error, or the height of the barometer. But it is less clear why this multiplicity 
should be identified with the infinite greatness of n. If we take Dr. VENN’s 
frequency curve for barometric height, we see that the closest point-binomial is by no 
means consistent with either p= q, or with n being indefinitely great, Further, 
many statistical results in games of chance are given with great exactness by the 
normal curve, although we are then able to show that n is quite moderate. 
Now, it is true that the biological and physical statistics to which we are referring, 
give essentially continuous curves, but it does not seem to follow of necessity that 7 
must be infinite ; while their frequent skewness sufficiently indicates that the neglect 
* Note again the deviation of the constant pq (n + 1) c? from its usually adopted value pgnc?. 
