204 
Skew Variation, a Rejoinder 
variate arc correlated with the value of the variate already reached*. Galton and 
Fechner made the increment proportional to the variate. But in our ignorance of 
the actual nature of variation in organisms, we have no reason at all for making 
such a narrow assumption. We can to please our critics put the matter as I have 
already indicated in the Gaussian form. We simply assume that if the causes of 
variation in the immediate neighbourhood of the character x„ remained the same 
for the whole range they would give a normal curve, hence we should have a 
relation of the form : 
Idy _ X 
y dx cr'^ " 
They do not, however, remain the same ; the tendency to vary at x is a function 
of X, in other words a = cr^'^ F {x), where F{x) is an arbitrary function. We have 
then 
Idy _ X 
y dx a^^F (x) ' 
This is a result as general as Edgeworth's and more so than Kapteyn's or 
Fechner's. We now take the simplest possible functional series for F{x), i.e. 
= tto + ctiX + a-iX- + — 
The coefficients a^, cii, a.,... can be found at once in terms of the moments*!*, and 
my special curves result if we stop at a.,. Against going to higher powers are the 
objections I have raised in my memoir on skew correlation!, namely (i) that the 
higher powers involve moments of the 5th and higher orders and their probable 
errors are very large, (ii) that it has not yet been shown that going to does not 
suffice to describe all the types of frequency which occur in conmion practice. 
The above is tlie simplest and most general form into which I would put my 
theory of asynnnetrical frequency for those who feel compelled to approach all 
frequency from the Gaussian standpoint. 
D. Specific Criticisms of Ranke and Greiner on my Theory. 
I think these may be summed up as follows : 
(a) That all distributions of variates are continuous, and that accordingly 
no curves, however closely they may approximate to finite discontinuous series 
like the binomial and the hypergeonietrical series, can be applicable to variation in 
nature. 
* Suppose we draw cards from a pack, and wish to consider the chance of s being of one suit, we 
may do so by drawing one card at a time, observing it and returning it, and then drawing again. 
Here there is not correlation between the successive contributions to r. Or we may draw the r cards, 
without replacing the individual ; here the successive contributions are correlated with the previous 
contributions, and the third Gaussian principle is upset. 
t " Mathematical Contributions to the Theory of Evolution, XIV. On the General Theory of Skew 
Correlation and Non-Linear Regression," Drapers' Company Research Memoirs. Biometric Series, ii. 
(Dulau and Co. 1904) p. 6. 
X Ibid. p. 7. 
