CONTRIBUTION'S TO THE THEORY OF EVOLUTION. 
275 
circumstances, completely screen, in a certain number of cases, both the potential 
fertility and the real fecundity of man. Precisely similar circumstances, which will be 
considered more at length later, hinder our obtaining in horses a true measure of 
fecundity for all cases. We are thus really dealing with a mixture of correlated and 
apparently uncorrelated material. In what manner does the influence of this 
mixture effect our results ? 
Let a group N consist of Ux + ^2 + ^3 + pairs of individuals. Of these, in the 
case of pairs, both individuals have the true value of the character under investi¬ 
gation recorded ; in the case of lu pairs, neither have the true value recorded; in the 
case of ^3 pairs, it is the first individual of the pair which has a true recorded value, 
and the second an apparent or fictitious value; lastly, in ^4 cases, let the fictitious 
value be in the first and the real value in the second individual of the pair. Then 
there will be no correlation between individuals in the groups lu, n^, Hi. Let r be 
the correlation in the group Hi and R that observed in the whole group of 
N = ni -f- iio + 5^3 + ^ 4 * Let x be the measure of a character in the first, x in the 
second individual. Let M and M' be the means of the total groups of the two 
individuals and their standard deviations. In group let the corresponding 
quantities be mi, m'l, o-j, cr'i, and a similar notation hold for the other sub-groups. 
Then mi = and o-j = 0 - 3 ; m 2 . = mi and a -2 = 0 - 4 ; while m'j = m '4 and cr', = cr' 4 ; 
771 2 771 3 , cr 2 •— O' 3 . 
We have at once : 
,, 7!,7H.J + Uomo -f 7J37??3 + (Ui + n^) Vli + (u., -f lUo 
M = -^^^- =: -^^ , 
vii + 7?Zo -f 773 -f- 71 j til -f n., + n-i ■+ Ui 
while 
_ (til + nj) m'l + {n, + n^) m' , 
til + '>h + % + Wj 
Further : 
(tii + 712 + 72.3 -f 71 . 4 ) XX'R = S (a? — M) [x — M'), 
by the usual properties of product moments 
= tiiCTiir' it ' -h Til {till — M) {m'l — M') + no {mo — M) {m '2 — M') 
+ ??3 (7713 — M) {m '2 — M') -h Hi {nii — M) {m'i — M') 
= TliCTiCrV’ + 7ip77.i77l'i fi- 77277l2777h + ^ + nitnitn i 
— M {uitn I + 7727n'2 + 77377^3 fi- tiirn'i) - M' (77l777i -f- tlomo -f- 7737773 -p 7747774 ) 
-f MM' (77i + 772 + 773 - 1 - Ui) 
— 1 ^' “1“ 1 + n2m2m'o + + tiiniirn'i — MM' ( 77 , + 772 + 773 + 724 ). 
Substituting the values of M and M' and using the relations between the 777 ’s, we 
find after some reductions : 
2 N 2 
