CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 
261 
and offspring, and according to the Law of Ancestral Heredity the same for all 
organs. Then multiplying (ii.) and (iv.) together and summing we have : 
na-yCr.j'R = S {AyAy') = t Ax'i)) -f S (cpcfaS {Ax^^Ax^ + Aa-jAich)), 
where R is the coefficient of correlation between the characters y and y in parent and 
offspring. ■ Now ; 
S(AXiAa'j) 7i(T^^(r^,r 
S(Aa;iA.r2 ffi Ax.Ax,) — 
where and are what I have elsewhere termed coefficients of cross heredity. 
Now if the race be stable or sensibly stable for two generations we shall have for all 
organs cr^- = cr^. Hence : 
S(AaiAa:'i) = ncrl^ X r 
S (AaiAa2 ff Ax^^Ax j) RG'^^o'y.^ {i ff- i^ ^^'Px,x.p 
for it is shown in my memoir on the Law of Ancestral Heredity'"' that on a probable 
hypothesis : 
i {rx,x’, + X 
Th us we find on substitution : 
cryCTy R = r {t {d\a':,) + 2S ' 
But (iii.) and (iv.) show us that cr^ = <Ty, if there be no sensible changes in a 
generation. Hence; 
^y^y' = + 2S 
and 
R = 
Thus the character which is a function of physical organs is inherited at the same 
rate as those organs themselves. 
As we may not unreasonably consider fertility and fecundity to be functions of 
physically measurable oigans, even if we cannot specify which organs, we may, 
d pi'iori, expect fertility and fecundity to be inherited characters. 
(2.) P'roposition II.—To determine the numeilcal values of the changes in mean 
variation and correlation if fertility he mheriled. 
Let us first define two terms which will be frequently used in the sequel. 
(«.) The fertility of an individual shall be defined as the total number of actual 
offspring. 
* ‘Roy. Soc. Proc.,’ vol. 62, p. 411. The hypothesis yet awaits an experimental verification. The 
need to use it prevents Proposition 1. being self-evident. 
