SELECTION ON THE VARIABILITY AND CORRELATION OF ORGANS. 
5 
This determines the central co-ordinate for any variable Xg for a given value of 
a-j and x^. 
Now let us transfer the quadric to aq, x^ . . . a„ as origin. It may be written 
Q = S^-1' (ciyXi + Coya., + 
= (a^ -b a^ 
+ 
+ C^iX-i + C^fjX^ - 4 - 
+ Cg^a 3 + 
ICx^Xi + c-i\X2 + Csiaj + 
"1" 1 /I 
, /Cioaj 4* Co^a., + Cgoa, + 
+ a., ( 
\ + Csia 3 4- 
Making use of the n linear equations (ii.) and (iii.): 
-j- ('nqX,^ 
-J- C.,iqXn 
4 “ Ch). 
+ ^n\^n ' 
— C)iiX ji 
4~ 
4~ CjjnX ,j 
Q — ^1=3 (^7 ~b ^ (?) 3 “b r ”b • • • “b «) 
4“ a^ (a 4" 4 + • • • + Cupc',) 
4" (/3 4“ Cgoa'g 4" cqoa ^ -b • • • + Cnpx.',^ 
= X'q {c^qX\ 4- C^,qX\ 4" ... 4" Cn,fc',) 
“b o.Xy “b /3a.i. 
(vii.). 
For arranging vertical columns in rows, the remaining terms are 
+ S2 = 3 (w)) 
+ (^hr'^1 + 4- iVq)) 
4“ ^'u 4“ “b ^^^ = 3 
each line of whicli vanishes Ijy the equations (ii.) for the centre. 
Accordingly ; 
Q = Q'’ “b aaq 4“ ^x.^, 
where Q' is a quadratic function of a'g, x\, . . . a'„, not involving a^ and a, at all. 
Hence : z = Zq 
Now integrate 2 with respect to all the variables a'g, x\, . . . a'„ from — co to 4“ 00 , 
keejjing a^ and a^ constant. 
Then, although the origin is a function of aq and ag. 
cannot involve a^ and ao but only Cg,^, .. , c„^, &c. ; let the result be Then : 
— + Px.,) 
