194 
Skew Variation, a Rejoinder 
squares, equilateral triangles, etc. etc. In all these results I found the distribution 
asymmetrical, but the most probable value was not the geometrical mean, nor the 
distribution the Galton-McAlister curve. One of the striking defects of the curve 
was its high contact at both ends. The tlistributions clearly often corresponded 
to curves in which the contour cut the axis at a finite angle. Another point was 
that the skewness was in the opposite direction to that presupposed by the 
reasoning from which the curve is deduced. It was precisely this experience 
which showed me that putting x = ^\og{^/a) in the Gaussian curve is not a 
sufficient generalisation. 
Ranke and Greiner, to say notliing of Fechner himself, are remarkably vague 
as to the accurate determination of the position and constants of the Galton- 
McAlister curve. McAlister gives no clear description of how the curve is to be 
placed if neither the mode nor start of the range is known. I think it desirable 
therefore, having regard to the inferences I wish to draw, to give the fitting by 
my method of moments. I write the curve : 
^j^^e-ti}'''^ (xi). 
Differentiation shows us at once that the distance x,,„o of the mode from the 
origin is given by: 
= '^a'"' (xii). 
Integrating the expression iV/i,/ = 1 yxMx we find if iV = total frequency : 
JO 
y, = Ni{s/2^c) (xiii), 
and generally : 
fi,i — a^'e^ (xiv). 
Thus the distance from the origin to the mean, x^^c, i^ given by 
x,ac = aeh''' (xv). 
Now write e'"= A- and we have if be a moment coefficient about mean : 
yu,., = fij — fjbi '- = a-X (A. — 1) \ 
P--.! — M-3 - S/i/yu./ + 2fMi '" = aV\ {X* - 3X.- + 2A,) I . ..(xvi). 
yU,4 = yU-/ — 4/X;;'yUi' + 6yU-^Vl'' '^f^l' = l-'-'^^^ (A." — 4\-' + 6A, — 3)j 
Forming the usual constants of frequency we have : 
A = MaV/"/ = {X~1){X + 2Y (xvii), 
7] = /3.,-3 = {\-l) {X-' + 3\- + 6A. + 6) (xviii), 
where /^.2 = ya4/(U,2-; 
d = x,,,/, — x,no = a (V\ — \~^) (xix), 
Y = , (XX). 
^ ^X-1 ^ ^ 
