PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 301 
/+ Vi a n + l 2 a \2 = Vi a n + V-z a n + a is, x s + a H x ^ + • • • 
9 + Vi a i2 + r )2 a 22 = Vi a 21 + V2 Cl 22 + a 23 X S + a U X l + • • • 
0 = ^x^o] + + a S 3 X S + a ?,i X -L H~ • • • 
0 = 7j x a u -f- ^ 4 < 2«42 + a i3 x z H~ a 44> x 4i + • • • 
where f and g are written for a x gx ! 3 -f a 14 x' 4 + . . . and a 23 af 3 -f a 24 af 4 + . . . respec¬ 
tively. Solving, we find 
Hence 
An (/ + ^Ai + 1 ?#i2) + A 13 (g + + h2 a 22) — V\ A, 
A-21 (j + Vian + 172^12) + A 33 (p + -f 172^22) = ^2 A - 
9 = 
Vl ^-%2 V 2^~12 ^ 
An A g2 — Ajg^ 
V2^-n ~ ^l^i a ^ 
AnA 22 — A 13 - 
7 7l a ll r l 2 a 12 > 
1 ?i a l2 ” V 2 a 22 ' 
But the exponential expression with its origin changed is given by 
z — constant X c ( - auri>2 + + a&vi* + fm + n) 
X e~- + o t rf + • • • + 2^.% +.. .) 
Integrating between the limits ± 00 for all the variables cc 3 , aq, x 5 
have the correlation surface for g v g 2> or substituting for f and g 
we 
shall 
z = constant X e “ 1 - (AA-MVa) 
hj! , V „ A i2 ■) 
i A n A 22 A 11 Aoo ) 
Comparing this with the formula on p. 264 , we see that if p 13 be the correlation 
coefficient of aq, x . 2 and s L , s. z their standard deviations 
Pi2 2 = A 12 ~/A 11 A 2 3 sg = A u /A s 2 — A 22 /A or P\2 s i s 2 = A 12 /A . . (e). 
Thus we conclude that the standard deviation for the organ resulting from the 
selection is given by 
V = cr + figsg -f fi. 2 s 2 + . . . + 2fi l f3. : p n s 1 s 2 + • • • 
Here cr, / 3 lf fi 2 . . . refer to the natural or unselected correlation surface, and 
s l5 s 2 , ... p l2 .. . to the selection correlation surface. 
(b.) Edgeworth’s Theorem .—We may stay for a moment over the results (e) above 
in order to deduce Professor Edgeworth’s Theorem,* which we shall shortly require 
to use. By the theory of minors (Salmon’s ‘ Higher Algebra,’ 1866 , p. 24 ) we have 
* Briefly stated with some leather disturbing printer’s errors in the ‘ Phil. Mag.,’ vol. 34, p. 201, 1892. 
