SELECTION ON THE VARIABILITY AND CORRELATION OF OROANS. 
31 
^3 — ^13 
^ 1 . ~ /^li 
o■^ 
H" ^33 ‘^^^3 
h -f 'h 
(Tc) 
1 
I 
I' • 
J 
. (Ixvii.), 
the ^6“ are the partial regression coeilicieiits, and the whole solution can be expressed 
in terms of them. Thus : 
X 3 
-3 
= O- 3 M 1 - ^I3^‘l3 — A3^’33 + As" ( ^ ) + A 3 " ( — ) + -dl3^13A3 ~5 \‘ (Ixviii.) 
"^ ^±^31, — O's^r 1 '^31 /^ 13 '^’ir A3'^’31' a ^ 13 /^U ( ~ ) A A3/^3J' 
+ f »13 (A 3 A 4 + A 3 ^u) 
SiSj 
(Ixix.). 
— '^‘ 1^3 1 ^13 ^ P 13 A 3 
(Ixx.). 
Thus the whole series of results can be easily calculated, if the regression coefficients 
are first calculated. 
I may make some remarks upon these results. A formula equivalent to (Ixviii.) 
was first given by me in my memoir on “Heredity, Panmixia, and Regression” 
(‘Phil. Trans.’ A, vol. 187, i^. 303), and used for certain problems of inheritance, and 
conclusions drawn from (Ixix.) or (Ixx.) have been cited or indicated in other 
memoirs. 
Some interesting results follow at once. If the selection be very stringent, s^/cr 
and sJ(T = 0 sensibly, then all correlation between a selected and non-selected organ 
is destroyed. But 
‘■34 
_^'34 Da") ~t Da (^3^24 ~t _ 
\/(t '>‘\i ~ ^' 13 ” “t 2 ^’i2^’i 3^’23) Da" ^’ 24 “ ^’ 14 “ “t 
(Ixxi.). 
This is what I have termed a partial correlation coefficient—/.e., the correlation 
between C and D when fixed values are given to A and B. So far as I am aware, 
such coefficients were first directly used l)y Mr. G. U. Yule in certain economic 
problems.^ They are of very considerable interest, but for natural or artificial 
selection ai'e not quite so important as the generalised form (Ixix.), for we generally 
select about a mean value, and not absolutely at it. 
It will be noticed that the coefficient of correlation of two non-selected organs 
differs from the corresponding partial correlation coefficient by terms of the square 
order in s/a, hut the coefficient of correlation of a selected and non-selected organ 
* ‘Roy. Soc. Proo.,’ vol. 60, })p. 185, 488; ‘ Ecotiouiic Jouriicil,’ Decemlier, 1895, and December, 1896. 
