46 
PROFESSOR K. PEARSOX OX THE TXFLUEXCE OF XATURAL 
_ _ _ _ 
make y some fimction of\r, say y —f{x) or y + y' =f[x) 4- f («) + ^ f (•'■) + •. • 
by Taylor’s theorem. But y = f (.r) and x is small, so that our attempted relation 
will he of the form : 
y' — wx'. 
Hei'e m is the slope of a line to which we endeavour to confine the selected organs. 
But we shall not he quite able to exactly hit this relation off; when x = e, w-e shall 
find that y' — me fi- y, where p is a small unavoidable error in selection of y' uncor¬ 
related with e. Thus, if and .9o be the selected variabilities, we shall have : 
Therefore : 
o 
6 a 
is(P)=- S(,«e+,)=, V 
0 
Further : 
Pic = 
S{(?/ie + ?/)e} 
rnsy 
C Q 
ms. 
and therefore : - 8 {y~) = (1 — piz)- 
Or, Pjo is at once obtained from the slope of the line m, by wdiich we endeavour to 
fix the relationship of the organs A and B. Or, again, we may look upon 
s. 2 s/l — Pi-}" as a quantity measuring the mean divergence of the B organ from that 
absolute fulfilment of the relationship between A and B which we are striving to 
attain. Thus p ^.2 is a quantity which naturally arises in every attempt, whether 
artificial or natural, to select organs having a definite relationship to each other. 
IMuch the same considerations arise when we select three or more oro-ans. In each 
case the selected coefficients of correlation are constants which enable us to express 
(i.) to a first approximation the form of relationship we are aiming at, and (ii.) the 
average degree of divergence from absolute fulfilment of this relationship. 
Thus, without regard to any particular distrihution of frequency, the i’s and the 
p’s are the approjiriate constants to express approximately the nature of any form of 
natural or artificial selection. 
(11.) O/} the ProhahiUty of Survival and the Surface of Survival Rates. 
In the course of the present jDaper I have assumed that when measurements are 
made on any population for a complex of n organs, the frequency surface may be 
taken as approximately normal. If this liolds for the population before and after 
selection, and measurements made on many groups at difierent periods of life seem 
to indicate that it is approximately true, it follows that we can determine the form 
of the probability of survival as a function of the means, variations, and correlations 
of the selected and unselected populations. 
