406 MR. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
earlier will be found at a much higher percentage. ‘The whole frequency curve is 
sliding across from right to left. Now it is of interest to notice that in this, as 
in other cases where the trend of the variation is known @ priori, the skew curve is 
shifted away from the normal curve in the direction in which variation is taking place 
with lapse of time. It is not safe at present to extend this to all biological instances, 
but the result suggests, for example, that there is a secular progression towards brachy- 
cephaly in Bavarian skulls (fig. 8), towards reduced antero-lateral margin in crabs 
(fig. 4), towards increased height in St. Louis school-girls (fig. 7), and towards long- 
sightedness in Marlborough School boys.* I believe most suggestive and important 
results might be obtained for the theory of evolution, if we only had the series of 
skew curves for a biological case of progressive variation in the same manner as we 
have for pauper percentages. 
(35.) Example XV. The theoretical resolution of heterogeneous material into 
two components, each having skew variation, is not so hard a problem as might at 
first appear, and I propose to deal at length with the subject later. If there be more 
than two components, the equations become unmanageable. In this case however, if 
the components have rather divergent means, a tentative process will often lead to 
practically useful results. To illustrate this I propose to conclude this paper by an 
example of a mortality curve resolved into its chief components. By a mortality 
curve I understand one in which frequency of death (for 1,000, 10,000, or 100,000 
born in the same year) is plotted up to age. I have worked out the resolution for 
English males, and for French of both sexes. The generally close accordance of the 
results for both cases has given me confidence in their approximate accuracy. The 
method adopted was the following: An attempt was made to fit a generalised 
frequency curve to the old age portion of the whole mortality curve, the constants of 
this curve being determined from the data for four or five selected ages by the method 
of least squares; the frequency curve so determined was subtracted from the total 
curve, and a frequency curve fitted by the same method to the tail of the remainder. 
This second component was again subtracted and the process repeated, until the 
remainder left could itself be expressed by a single frequency curve. The com- 
ponents thus obtained were added together, and a tentative process adopted of 
slightly modifying their constants and position, so that the total areas of the com- 
ponents and of the whole mortality curve coincided. It was soon obvious that no 
very great change either in the constants or position was permissible, if the sum 
of the components was to give the known resultant curve, hence I feel very confident 
that whatever be the combination of causes which result in the mortality curve, that 
curve is very approximately to be considered as the compound of five types of 
mortality centering about five different ages. The allied character of the results 
obtained for both French and English statistics confirms this view. 
* Dr. Rozerts’ statistics, which I have reduced to skew curves, but have not reproduced in this 
memoir. 
