178 Skew Variation, a Rejoinder 
Now if we assume that the actually observed character is not x but X, and that 
X is some function of .r, we shall not in plotting the frequencies to A' obtain a 
normal curve, but we ought if 1" be the ordinate of this curve to have 
YdX = yd.r or ^"=2/^- 
Taking logarithmic differentials 
1 dY _\dy dx d^x j dx 
YdX 'ydxdX^dX'IdX 
_ 1 dx d'-x I'dx 
~~a:^^''~''''dJ^dX'^ dX- 
Assume x — m = f{X) and we find : 
1 dY X 
■(iv). 
YdX~ a^'F{X) 
where F (X) = Xf (Z)/{/(A) (/' {X)f - " (Z)}. 
The form has been so chosen that the origin is the mode, i.e. dY/dX vanishes 
with A'. The proposal to thus generally transform the Gaussian curve is due in 
a quite different form to Edgeworth*. Kapteyn following Edgeworth and without 
any acknowledgment takes : 
/{X) = ^{X + ^)<i 
where ^, k and q are constants to be determined. 
He therefore puts : 
F(X)= X{X^-k) 
' <T^{q-l)-mq^{X + Kyi-q^^{X + Kyi- 
Tliis is a somewhat complex expression. The resulting frequenc}' curve is 
i = 1 1, (A + k)'^ ^ e ^'^<' (v), 
and has been suggested by Kapteyn as a general form of the skew frequency curve. 
We shall consider it later. 
Galton and McAlister as early as 1879 took 
f{X) = h\oix~- ni, 
where h and a are constants. Ranke and Greiner, without apparently knowing 
the history of research in this field, take the same value and attribute to Fechner 
the well-known Galton-McAlister curve of the geometric mean which results. 
M^e find 
F{X) = ± ^, 
o-„'' -I- ¥ log — 
* He lias developed it iu a long series of papers published in the R. Statistical Society's Journal, 
