272 
PROFESSOR KARL PEARSON, MATHEMATICAL 
second set. Let n be the number of brothers in an array, and therefore -gn {n — 1) the 
number of pairs of brothers in the array. Let cr and cr' be the standard deviations 
of the two sets of brothers, and r the coefficient of correlation between brothers for 
the organ in question. Let S denote a summation with regai'd to all pairs of brothers 
in the community, and S with regard to all brothers in an array. Let N be the 
total number of brothers in the community. Then if we selected our pairs of brothers 
for tabulation at random {e.g., not by seniority or other character), we should find 
m — m and a' = cr. Further, by definition of correlation 
Nrcrcr' = S (a: — m) [x — m) — [x — M + M — m) [x' — M' + M' — m), 
where M and M' are the means of the two sets of brothers in any array and are 
clearly equal. 
Further, S (x — M) ~ S {x' — M') = 0, when summed for an array, and 
^ {x — M) {x' — M') = 0, for there is no correlation witliin the array when the 
deviations are measured from the mean of the array. Hence : 
or 
Nrcrcr' = S {^n [n — 1) (M — m) (M' — w')}, 
Nrcr" = S [^n {n — 1) M"} — 2??iS {^n [n — 1) M] + w'N ; 
S {^n (n — 1) M} = Nri. 
S - 1) M^l/N - 
'T* fy •••••. 
cr*' 
This can be written 
o / o 
r = a-Jcr- . 
but 
Thus, finally. 
(xxiii.). 
(xxiv.) 
where cr„ is the standard deviation of the arrays concentrated into their means and 
loaded wdth their sizes ; cr is the standard deviation of all brethren loaded wdth the 
number of times they are counted as brethren ; m is the mean of all the offspring 
loaded with the number of times they are counted as brethren. 
Let cTo be the standard deviation of offspring, and p the correlation between parent 
and offspring, then the standard deviation of an array of offspring, if correlation be 
sensibly linear,* will be cto \/(l — p^). We have, further. 
m = S (x) = SS (x - M + M) = S {^n {n - 1) M], 
Ncr^ = S (x - mf = SS (x - M + M - m)^- S [S (x - M)- + (n - 1) (M - mf]. 
%{x — M)“ = ^71 {n — 1) cTo (1 — p*)- 
* See, however, the tirst footnote p. 270. 
But 
