180 
Skew Variation, a Rejoinder 
we may look at it in this way: If S.r,, be the variation in the neighbourhood of a^,, 
then hxf, is not independent of x^, hut correlated with it. We may have a perfectly 
continuous jiopulation from dwarfs to giants, but it does not follow that the actual 
tendency to vary of dwarl's and of giants is identical. All proofs that I have seen 
of the normal cnrve fail in this respect. They assnme that the character x is due 
to a number of increments, which are due to an indefinitely large number of 
independent cause groups. They assume that Srf'o is not correlated with the already 
accrued value x^. All processes like those of Edgeworth, Galton, Kapteyn and 
Fechner are really devices for getting over this gradual change in the tendency 
to vary from point to point of the range. It appears to me best to directly 
acknowledge and face this difficulty by selecting a fitting function for F {X). 
If we drop now the distinction between A' and x as unnecessary we reach as our 
frequency equation : 
1 dij X 
y dx~ a- J'F {■'■)' 
or if we use Maclaurin's theorem for F{x): 
1 di/ — X . . ... 
— ^ = (vii). 
y dx cTg- (1 + aiX + a^a? + cizU? + . . . ) 
Now I have sliown* how to determine the successive constants a^;, o-„-ai, 
o-„-rt.2, etc. Further all these constants but o- are zero, when the distribution is 
normal, and the series will be found to converge rapidly, when the distribution is 
from the largest term of the binomial, then X,.= c„ V(7i + 1) x function of i' = f„ ^(it f say, and 
conversely 7' = a function of A',./ Jc,, J[n + 1)| . Divide both sides of the above equation by i-,, , which may 
be written on the left \X, and we obtain : 
A'AA- I («+ 1) ,v/{r) 0 {r)j{rl(n + \) - hV 
Put a^^ = hJn + \i•^^ and F {r) for the expression /(;•) 0 (/-)/] /-/(yt + 1) - i} which does not become 
infinite with 2r=;f + l, because A',, and therefore f(r) vanishes for this value of r, and accordingly /(?•) 
contains 2r ~ (h + 1) as a factor. We then have : 
Ay _ _ X,, 
Now make n infinite and <"„ vanishingly small, then we have if (r„= 2 + c,, be still finite 
dY _ X 
a result in agreement with the above investigatiou. In other words this, and not the Ciaussian curve, 
is the generalised frequency curve we reach if we directly abrogate the third Gaussian principle, that 
contributory increments of the variate are independent. Of course the first two Gaussian principles 
simultaneously disappear. This view of the matter occurred to me many years ago, when considering 
Hagen and Crofton's proofs of the Gaussian law. It was expressed in my memoir of 181)4 by the 
statement that we require curves produced by conditions in which the contributory cause groups are not 
independent, i.e. in which an increment 5x to the variate x deijends upon the value of x, or is cor- 
related with it. My method of reaching such curves, however, was a direct appeal to discrete series in 
which such a condition was fulfilled. 
* "Mathematical Contributions to the Theory of Evolution, XIV." p. 0. Dulau and Co., London. 
