CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 
267 
exactly the result reached by a longer process in (vli.). Similarly (viii.) may be 
deduced from (i.). Applying this to find r" we have from (xiii.): 
r"cr''n/(T"i — ra-zjfy'u and therefore = ncr.^/cri 
(xiv.), 
a result which again extends the constancy of the regression coefficient under the 
action of reproductive selection. 
Next from (xi.) : 
cr 2 = CT 2 ^1+ r — i I 
= CTo 
/ 
' . or 
.."•2 
or using (xiii.) and rearranging : 
//o '> \ -t i '> 
cr 2 = cr-I i 1 + 
/ ^''2 
^ 1 
Again by interchanges in (ii.) : 
"2 '2 -1 1 \ I ' 
M\N'i 
S{(.r - M'dV} 
1 . 
(xv.). 
. (xvi.). 
Here z' stands for Xxz, and we should obtain a fourth moment of tlie original 
system of unweighted parents by substitution. But it is practically impossible to 
obtain a correlation table for such a system. Thus it is better to allow the sum¬ 
mation term to stand as it is, where it represents the third moment of a system 
of parents, weighted for fertility owing to the nature of the record, but not weighted 
with all their recorded offspring, (xvi.) is then a relation between the standard- 
deviations of parents weighted solely by forming a record and weighted both by this 
and by their offspring. 
Equations (i.) to (xvi.) contain the chief theoretical relations of our subject,* and 
I shall consider some points with regard to them in the following section. 
( 3 .) (a.) If we wish to ascertain whether fertility is inherited, we have to discover 
whether r is or is not zero. Now by (xiv.) r vanishes with both r' and r”, and accord¬ 
ingly either of these will suffice to answer the problem. Still better, we may ascertain 
the coefficient of regression, and then whether our statistics weight for progeny or 
not we shall obtain the same value. If there be no secular change taking place in the 
population, due to something else than reproductive selection, we should expect, 
provided the Law of Ancestral Heredity holds for fertility, that the regression will 
be near '3 for parent and offspring.! 
* Two of these formul®, (v.) and (xi.), were given, but in a less precisely defined manner, in my 
“ Note on Reproductive Selection” of 189G, ‘Roy. Soc. Proc.,’ vol. .59, p. GOG. 
t See “ Law of i\ncestral Heredity,” ‘ Roy. Soc. Proc.,’ vol. G2, p. 397. 
2 M 2 
