MR. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 389 
limited amount from a limited mixture. So far as this goes, it is evidence against 
the usual hypothesis that in biological matters the chances of deviations on either 
side of the mean are equal, and the ‘‘contributory causes” independent and 
indefinitely great in number. Thus we appear in biological statistics to be dealing 
with a chance system corresponding, not to a binomial, but to a hypergeometrical 
series, such as that discussed in § 11. 
If it be remarked that Type IV. dismisses at once the problem of range from 
biological investigations, we must notice that, while this is theoretically correct so 
long as we are dealing with the continuous curve by which we replace the hyper- 
geometrical series, it is not true the moment we fall back from the curve on the point 
series (see p. 361). Ifthe 7 of that page (or the gn) be an integer, the series is limited 
in range. It seems very possible that discreteness, rather than continuity, is charac- 
teristic of the ultimate elements of variation ; in other words, if we replaced the curve 
by a discrete series of points, we should find a limited range. It is the analytical 
transition from this series to a closely fitting curve which replaces the limited by an 
unlimited range. Exactly the same transition occurs when we pass from the sym- 
metrical point binomial to the normal curve. Thus, while Type I. marks an absolutely 
limited range, the occurrence of Type IV. does not necessarily mean that the range 
is actually unlimited.* 
For the equation to the curve we have 
x= 11°69583 tan 0, 
y= 107°4706 cos 14427349 Cmecnies 
the origin being at a distance *803515 on the positive side of the centroid vertical. 
The normal curve as well as the curve of Type IV. are shown (Plate 11, fig. 8). The 
result in both cases is quite good for this type of statistics—z.e., the skulls came from 
eight different districts and include 100 female skulls. With the planimeter the areal 
deviation in both cases = 6°8 square centims., giving in either case an average per- 
centage error of 7:56. That the generalised curve does not in this case give a 
decidedly better result than the normal curve I attribute to the heterogeneity of the 
material. It clearly accounts better for the extreme dolichocephalic and brachy- 
cephalic skulls than the normal curve. The same 900 skulls have been fitted with a 
normal curve by Strepa,t but neither the constants of his normal distribution nor 
* I reserve for the present the fitting of hypergeometrical point series to statistical results. The 
discussion is related to curves of Type IV., as the fitting of point binomials to curves of Type HI. It 
will, I think, throw considerable light on the nature of chance in the field of biological variation, 
especially with regard to limitation of the material to be drawn upon, to which I referred above, and 
which, I believe, finds confirmation in skull statistics. 
+ “Ueber die Anwendung der Wahrscheinlichkeitsrechnung in der anthropologischen Statistik,” 
‘ Archiv fiir Anthropologie,’ Bd. 14. Braunschweig, 1882. 
