SELECTION ON THE VARIABILITY AND CORRELATION OF ORGANS. 
_ Clearly, when x! 
. x'„ are constants, the distribution of x\ is of the form 
constant X expt. 
x\ + 
.Pm " 
3 
Or, re-introducing the o-^, cto, . . . a,,, we have a distribution about the point given by 
Xi = 
R ] o I P l 3 
- I T> “ T" + 
R 
11 
E 
11 
cr.. 
I Epj x,i ^ 
“b T* ~ I 
± 1 ,, rr.. / 
11 
with standard deviation 
o-\ = cxi v/R/Il 
11 
(xxiv.). 
(b) Given n variables, what are the mean values rii^ + x^, ni^ -f- Xo, the standard 
deviations a-'\, cr'b, and the correlation of two of them, when we give definite 
values arg + Ag , aij. + /i4. . . . . -f- to the remaining (n — 2) variables ? 
In this case Ave have from (i.) 
2 = expt. — 
+ 2 (cjgAg -b -b . . . -p 
“1“ ^ “1" ^34^0 ~b • • • “b (^'hyb,>) G 
+ terms not involving x^ and x’.,} . 
(XXV.). 
Writing for the coefiicient of .r^, and for that of x.,, we have for the centre 
./ij — 
Xi = 
x-^ — 
C'i-1^32 ^2^12) 
^1iG2 ^ 12” 
bl^23 — b2^ J ’ 
/IfiyJLo — En^jRi., 
— cr 
i-"l 
P P 2 
12 
X.T 
a'.,. 
( lC^Cj2 -f- C'j^lVo) 
o 
b2~ 
^11^2 
X.-) — 
_ / %^ii 7 
'll, n 'G / ’ 
Ra/jlPn 12 
— ^2-1 
RjiR-.o — Ifio” pJ 
liy transferring to the minors of R and the cr’s. Or 
/ / / 1 , \ 
P ij> 'h 
X^ = 
cr 
l-l 
R' 
X.7 = 
— cr.^S., 
P sp Ih 
R " <r , 
(xxvi.). 
Here R" is the detei'ininant formed liy striking out the first two rows and 
columns of R ; p'\p is the minor obtained by striking out the second row and column 
from R, and then the first roAv and column ; pb;> the minor obtained by striking 
out the first row and column, and then the second row and column. But a 
comparison Avith (xxiv.) shoAvs us that these values for and au are precisely Avhat 
Ave should have obtained for the regression equations of the 1st and 2nd variables 
respectively alone on the other n — 2 variables. Thus the existence and the 
correlations of an haAm no effect on the value of nor those, of X|.on. the value of x^. 
VOL. C€.—A, 
C 
