290 
PROFESSOR K. PEARSON AND OTHERS ON 
fraternal correlation we have, since one set of brethren will have the same standard 
deviation as tlie other, if they be taken of the same sex. 
S (tj X z.) = n X a- X cr X H, 
where n is the total number of pairs of brethren. 
Hence 
It X or" X It = t {a/{x),x\)) + t {a/x,jS + -r'/r'y)) 
+ - (/SypS {y'py"p)) + - {y'py'q + y"py',i)) 
with the same notation as before and the same assumption as to no selective process 
l)etween the female and male reproductive cells. 
Now x'p, x"p ai'e undifferentiated like organs put forth l)y an individual. If ppp be 
the correlation of such : 
^iyxjpt j) = no-;Ppp, 
Similarly ^{y'py'p) = na'p'p'pp. 
In the next place we should have all pairs of brotliers : 
S (.r p.C Ij'j S (^X pX ^p(X qPprj , 
^ {y py ~ ^ (y pV i) ” i^p p,^ . 
AVe thus deduce 
cr- X It = S {^-ip-o-p'ppp) + 2 [iSp-o-'p-p'pp) 
H” {^p^^(^p(^^Ppri)~^ ■-‘^{/^p/^q^ p^ pj) ■•••■- (in.). 
Now if the degree of resemblance between undifferentiated like organs vary largely 
from organ to organ, we could proceed no further with this investigation. AA^e should 
liave shown that there was a relation, namely (iii.) between the I\ and tlie p’s, but 
as the characters in the sjDermatozoa and ova are unknow-n and unmeasurable, we 
could not j^roceed further. On the other hand, if the correlation between undif¬ 
ferentiated like organs is approximately constant, then we may replace correlation 
coefficients like ppp and p'p^ by their mean, value p, say. 
AVe have next to consider such correlations as Pp^ or p'p^j. These are the tAvo 
correlations betAveen tAvo different characters in two spermatozoa or o\'a from the 
same individual. Such correlations must Amnish (i.) if there be no organic correlation 
lietAveen characters of the same speiinatozoon or oAuiin, (ii.) if there be no correlation 
lietAveen the reproductive cells put foi'th by the same indi\udual. In other AA'ords, ppj 
must contain p and as factors, and since if these tAvo correlations are perfect pp,j 
must be perfect, it is a reasonable hypothesis to assume 
Ppq — p X Vpq . 
This, it Avill l)e S;een, is identical in form Avith the result I haA-e supposed to hold 
