CONTRIBUTIONS TO THE THEORY OP EVOLUTION. 
265 
I now proceed to the standard deviations for the three cases, and the additional 
case for parents. 
o-:; = S[(2/-M2)-z]/N; . (ix.). 
0-1= S{X.'r{2/-Mhr2]/N'. 
= S{x(^ - M2 + M2 - M'2)“2 ]/MiNi 
S{x(y - M,y2} + 2S{:c(.y - M,)^} (M. - + M, (M 2 - 
- M,N, 
Whence, after some reductions, we find : 
/O '>11 
fr 2 = 0-2-^ 1 + r 
(zl 
SI (x - Md ((y - M,.)^ - U- ^ (x - Md^) 2 
(x.). 
Now for a nearly straight line of regression : 
y _ M2 = r {x - Md + ^ 
where 77 is uncorrelated with x — Mj. It follows accordingly that S{(;r — Md^77z} and 
S{(a: — Mi)77“z} will both vanish, since 8(77) for an array and S (.t — Mj) for the 
whole correlation surface will be zero. Hence the summation term in (x.7 is either 
absolutely zero or extremely small. We have accordingly : 
/o 
^ 2 
. (xi.). 
Before we proceed to determine cr'd and a'l it seems simplest to find the coefficients 
of correlation rj r and r". We have ; 
r=S[(a;- M,) (y - M2) z}/(N,cr,cr2) 
To find r we have : 
r = S{Xa;z(x - M'd (y - M'2)}/(N'io-do-'2). 
Now 
7/ - M2 = (x - Md + y, 
(xii.). 
where 77 is sensibly un-correlated with x — M,. Hence : 
Ndo-'.o-V = sjxxz (x - M'd fr- (x - Md + M2 - Md + y ) j- 
L \ °'i J J 
Expanding, the summations with y vanish, and 
VOL. CXCII.-A. 2 M 
