298 PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
It is clear that we have here to deal with a skew curve of the kind discussed in my 
second memoir, and the intensity-age distribution must be a skew correlation surface 
to give rise to such a curve. The full treatment, accordingly, of morbid inheritance 
requires a discussion of skew correlation, I hope to be able to return to it again 
when dealing with the general theory of disease distributions. Meanwhile, the 
considerations of this section are based on an approximate theory, which, however, 
can hardly fail to give the main outlines of the subject, if a more accurate develop¬ 
ment might be requisite when actual statistics were forthcoming to be dealt with. 
(10.) Natural Selection and Panmixia. 
(a.) Fundamental Theorem in Selection. —The general theory of correlation shows 
us that taking p + 1 correlated organs, if we select p of them of definite dimensions, 
the remaining organ will follow a normal law of distribution, of which the standard- 
deviation and mean can be determined. Now, in the problem of natural selection, 
we do not select absolutely definite dimensions, and the p organs selected may be 
specially correlated together in selection, in a manner totally different from their 
“natural” correlation or correlation of birth. We, therefore, require a generalised 
investigation of the following kind : Given p + 1 normally correlated organs, p out 
of these organs are selected in the following manner : each organ is selected normally 
round a given mean, and the p selected organs, pair and pair, are correlated in any 
arbitrary manner. What will be the nature of the distribution of the remaining 
(p -f- 1 ) th organ ? 
Geometrically in ^-dimensional space we have a correlation surface of the p th 
order among the p organs, and out of this, with any origin we please, we cut an 
arbitrary correlation surface of the p th order—of course, of smaller dimensions—the 
problem is to find the distribution of the (p + l) th organ related to this arbitrary 
surface cut out of what we may term the natural surface. 
If the p organs are organs of ancestry—as many as we please—and the (p + l) (h 
organ that of a descendant, we have here the general problem of natural selection 
modified by inheritance. 
We will distinguish the two correlation surfaces as the unselected and the selected. 
Let / 3 t , / 3 . 2 , / 3 3 , . . . be the regression coefficients of the (p + l) th organ on the p organs 
for unselected correlation, then for values of the p organs /q, h. 2 , h s , . . . from their 
respective means, the (p + l) ttl organ will have a distribution centering round 
/Vh + AA + A; A + • • • , and a standard deviation cr given by the general theory of 
correlation (i.e., the S.D. of the array). Similarly, for values /q + aq, h. z -fi aq, 
hf-\- aq . . . of the p organs, the (p -f l) th will have a distribution with standard- 
deviation cr and centre 
fii (hi + aq) + / 3 2 (/q + x i) + A3 (A + ^3) "h • • ■ — £ + S (/^aq), say. 
