SELECTIOX OX THE VARIABILITY AXD CORRELATIOX OF ORGANS. 
13 
Now if we differentiate the quadric to find its “ centre” we have n equations in 
. . . Xq . . . x,i, but the solutions of these, if . . . x^ were eliminated, are known 
to be the ‘‘centre” /q, /c, . . . hq. Hence we require only n — q equations involving 
^q+\ . . . £C„ and we can, put the /fs for the remaining values x^ . . . Xq. Let us take 
the differentials of the quadric with regard to Xq+^ . . . x,^, then the resulting 
equations involve none of the c’s, but only the c’s. They reproduce in fact (xxxiv.). 
But the values of Xq^^ . . . x„ found from (xxxiv.) are, we have seen, identical 
with the values of (xxxii.). Thus we have 
X 
E (^ + 
^g+\ 
E (q + lk + l,7y + l 
Ii (q + 2)j 
E (q + 'i'lt+^.g+i 
/q + 
/q -h 
{q + 1 ) 3 .?+! 
^'y+i 
E (q + Ik+i.i+i 
Co 
Ep2 + 2)2.,^3 
'^y+2 
E (2 + 
Co 
/q + . . . + 
ho + . . . + 
h(g+lh .?+! y?+i 7 I 
h (g F ^)?,g +2 J I 
h (^dl.« 7 h ( 5 ^) 3 , )J V,; J . 
cr.. 
J tl (7?-)^, ;i O'// 
(xxxvi.). 
These give the most general form of a theorem proved for a particular case in 
‘ Phil. Trans.,’ A, voL 187 , p. 300 , c (ii.). If systems of q organs he selected ivitJi any 
arbitrary variations and correlations out of complexes of n organs, then the mean sizes 
of the remaining n — q organs have jjreciscJy the same values as if the selection of all 
the. systeras of (j organs had been to one size and not varied cd)ont mean values. The 
arhilrary variations of the selected systems about these mean values, as well as the 
arbitrary correlations, have no influence on the mean changes of the n — q organs. 
Iveturning to equation (xxxv.) Ave know that if the determinant 
A = 
^1,1’ 
*^T 27 
^‘l :3 
• • ^X.n 
^2,0 
fo,2> 
■ • fo,,/ 
*^?2’ 
^?.3 
Oqq, 
• • Cfq^ ,i 
(xxxvii ) 
%+l,2) 
^l? + l ,3 
• • ^q+\,n 
^ n. 1 > 
G/,2) 
.3 
^n.q+X 
• • ^nn 
be formed, its 
constituents and 
not the 
linear 
terms in 
the 
exponential of (xxxv.) 
determine all the standard deviations and correlations. Let be the variation 
after selection of the organ ; then if u be one of the selected organs = s,„ if u 
be for one of the unselected organs has still to be found. Let x„v be the correlation 
