MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 
83 
Suppose to be the standard deviation for the errors in a quantity x, and IL y 
the correlation coefficient for errors in two quantities x and y. 
Further let 
11 - . 
where subscripts and dashes may be attached to H to correspond to like distin¬ 
guishing marks attached to h. 
Since 
2/q - N 1 f*i _ ij;2 j / . v 
= dx .< V1 -). 
we have at once 3n 1 = NHjS/q, 
and tj h = ^/(NF^).(vii.). 
Similarly, 8 n 5 = — NH 3 SA 3 , whence : 
^ = W(NH.).(viii.). 
Further, we have = — S fll S„ J R„ 1 i, I /(N 2 n i H 3 ).(ix.) ; 
but, as is shewn in Part VII., § 4, 
^ 2 _ ? h ( N ~ ^ 2 _ %( N ~ 7 h) / \ 
— ]y > yj- .V x -/’ 
N N H — _ \ 
A,i l 4 3 r''# 1 »3 — V X W # 
Thus we find 
Probable error of /q = '674492/^ 
_ -67449 ± /yl -Vq) , jj x 
Probable error of 7 j 3 = ]q \/ .(xiii.). 
3 
Correlation in errors in h l and h s , or Rqq, is given by 
Si.SaA*, = ^ .( xiv -)- 
Let u = li z — /q, u = /q' — h{ be the ratio to the respective standard deviations 
of the ranges corresponding to the groups rq and n 2 '. Then 
2/ = V + V - 22 7ii 2,R v , 3 
_ 1 [?q(N - »i) , %(N — n s ) _ 2ypq1 
N>L HP ^ H 3 2 HVIJ ’ 
whence, if v be a proportional frequency = ?i/N, we readily find 
Probable error of « = ^ {j£i+gl ~ (| + ■ ■ • («■)• 
Probable error of «' = A; + ^ - (i + ^)*}‘. . . (xvi.). 
M 2 
