CONTRIBUTIONS TO THE THEORY OF EVOLE^TION. 
273 
Thus : 
or ; 
and r in ay be written : 
Here cr^j can be found from the arrays, and o-q and p will in many cases have been 
previously ascertained. 
(B.) To find the correlation between “uncles” and “ nepheivs” {“aunts” and 
“ nieces ”) from the means of the corresponding arrays. 
Let Tij be the number of uncles in an array, m be the number of nephews in the 
associated array, so that n^no is the number of pairs of uncles and nephews provided 
by the associated arrays. Let N = S (n-in.,) be the total number of pairs of uncles 
and nejihews in the community under consideration. Let x be the measure of the 
organ or character in the uncle, x in the nephew. Let M and M' be the means of 
two associated arrays of uncles and nephews respectively. Let m and m be the 
means of all uncles weighted with their nephews and all nephews weighted with 
their uncles respectively, and let (t, a be the corresponding standard deviations 
under the same circumstances ; r the correlation of uncle and nephew. Then : 
Nr'cro-' = S (x — m) {x — m!) = SS (rr — M + M — m) {x — M' + M' — ni). 
Now S (x — M) = S (x' — M') = 0, and within the arrays there is no association 
of individual uncles with individual nephews, i.e., ^ {x — M) {x — M') = 0. Thus ; 
Nct^ = Ncrs (1 - p^) + NcrL 
O'- = O-^ 1 — p-) + cr“ 
r = 
o’d (1 — P') + o-r, 
(xxv.), 
(xxvi.). 
Nr'orcr' = S {npi2 (M — m) (M' ~ m')] = S {n{tifKM.') — Nwm', 
since 
Thus : 
m ■= S(wiW2M)/N, ni = S (n,n2M.')/N. 
S — mm' 
aa 
(xxvii.). 
If cr„ and a a be the standard deviations of the means of the arrays of uncles and 
nephews and B the correlation of these means, the numerator is clearly Ilcr„cr'o. 
Thus : 
’ _ "p a 
era- 
(xxviii.). 
Here the numerator as a whole or in parts is easily found from the means of the 
VOL. cxcii. —A. 2 N 
