76 PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
exponential equations, the solution of which seems more beyond the wit of man than 
that of a numerical equation even of the ninth order. 
(3.) In any given example the conditions will be sufficient to reduce the suitable 
roots of this equation very largely, possibly to two or even one. These limiting 
conditions will be considered later. A suitable root of this equation leads to a 
quadratic for the areas of the two component normal curves. This quadratic is funda¬ 
mental, and appears to be highly suggestive for the problem of evolution. We have 
two cases : 
(i.) Both its roots are 'positive. 
In this case the given frequency-curve is the sum of two normal curves. The 
units of the frequency-curve may be considered as composed of definite proportions of 
two species, each of which is stable about its mean. The process of differentiation 
here appears complete. 
(ii.) One root is positive and the other 'negative. 
The given frequency-curve is now the differe'nce of two probability-curves. The 
probability-curve, with positive area, may possibly now be looked upon as the bh’th- 
population (unselectively diminished by death). The negative probability-curve is a 
selective diminution of units about a certain mean ; that mean may, perhaps, be the 
average of the less “ fit.” 
It is possible that in some numerical cases solutions of both the types (i.) and (ii.) 
will be found to exist, but I imagine that in most cases of a well-marked and charac¬ 
teristic asymmetrical frequency-curve, either only one type of solution will exist, or, 
if two types do exist, then one will give a much better agreement with the actual 
shape of the curve than the other. That the two types of solutions should exist side 
by side occasionally is, perhaps, to be expected. In such cases we have examples of 
groups, which are, perhaps, in process of differentiation into separate species by the 
elimination of members round a selected mean. 
(iii.) From the nature of the problem, the case of both roots negative does not 
occur. 
We now pass to the solution of the problem : 
Given an asymmetrical frequency-curve to break it up, if possible, into two com- 
qoonent probability-curves, or into two normcd curves. 
[ ( 4 .) Preliminary Definitions and Problems. 
(i.) Given any curve ABC, and the line y'y' , if we take the sum of the products of 
every element of area by the uth power of the distance of the element from the line 
y'y’ , we form the 7rth moment of the area about the line yy. 
Clearly, if y be the length of a strip parallel to y'y' and x its distance from y'y, then the 
7 ?th moment — \x^‘y dx, the integration extending all over ABC, or from A to C in our 
case, where the curve is always bounded by a straight line, AC, perpendicular to y'y. 
