CONTKIBUTIONS TO THE THEORY OF EYOLUTIOK. 
223 
(6.) So far as the data at my command Avent, the dwarfs and giants appeared to 
deviate from the regression line in a remarkably symmetrical manner on opposite 
sides of it. In other words, the branches of the curve on opposite sides of the axis 
of y appeared to be centrally symmetrical or congruent. Thus the form of the curve 
was reduced to y — x^<f) (x^, y"). 
(c.) It follows from this that the asymptotes of the curve, besides x = 0, will be 
given by (j) (x~, y~) = 0. The problem then turns on what are the probable asymp¬ 
totes. Now if we examine the regression formula for an organ A on an organ B, it 
is of the form : 
where A,,; and B,,, are the mean organs, cr^ and o-j the standard deviations, and the 
coefficient of correlation. Now no amount of selection of either A or B, or any 
other organs, as to size only, would influence in the case of normal correlation 
but it would change the constant term A,„-B,,^. Hence, if we were to take the 
line of regression for an extreme population of d^varfs alone, or of giants alone, it 
would seem quite possible that r„j^(Talo-b might have remained constant, while the term 
-B„j changed. But these lines of regression Avould be the asymptotes of the 
recjuired curve. It was thus suggested to me that the asymptotes might be parallel 
to the line of regression of the normal population. On examining the points corre¬ 
sponding to giant and dwarf statures plotted to long bones, this hypothesis seemed 
to be highly probable. Accordingly the form of the curve finally selected to represent 
the extended curve of regression was 
y = ex' {Jr - y-), 
where the axis of x is the linear line of regression for normal stature, and the axis of 
y is the perpendicular to it through the mean normal stature of the French.* 
[d.) A diagram was now formed by plotting to half life-size centim. for 1 centim.) 
the points representing giants and dwarfs, and the lines of regression for the normal 
population were drawn. The y and x for the point for each giant for each bone were 
then read oft’, and these formed the data from Avhich the constants of the four curves 
of the above type were then determined. For this determination only giants over 
200 centims. were selected. The class of what mav be termed sub-giants, with 
statures from 180-200 centims., were put on oneside. Such individuals, termed giants, 
appear in both the Bonn and Munich anthropological catalogues, but the “Korperliinge” 
there given can hardly represent the living stature ; it is very probably only a skeleton 
* Some shifting of the origin would prohahly have improved my results, but the data were not 
sufficient to justify such extra labour. 
