PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 299 
Thus a deviation of the p th organ lying between v and v -f- dv from the mean of 
these organs will occur with a frequency varying as 
, jw + sow} 2 
clve . 
Now let the selected correlation surface centering round h z , h 3 , ... be given by 
2 = constant X e (Un ' Xl2 + + ■ • • + + •■••), 
Then the total frequency of the p th organ lying between v and v + dv = 
Constant X dt 
oo -CO -CO 
... , ... 
CO — J oo — 
To carry out the integrations, let us first transfer the expression in the exponential 
power to its “ centre,” writing v — £ = u, and x{, x 3 , ... as the coordinates of 
the centre. 
To find the !£ centre” we have the equations : 
(it ~ S (0 1 a: 1 , ))/cr^ = a n x^ + a-upl + ct ls x 3 + . . . , 
02 (w - S (0 1 ® 1 , ))/o- 3 = + Cl22 X 2 + “ 23 ^ 3 ' + ■ . ■ , 
03 ( u ~ s (0i«i / ))/o- 3 = a 3l Xy + C( 32 x 2 ' + C( 33 x 3 ' + . . . , 
hence 
Ax{ = (0jA 1]L + 0 oA 19v + 0 3 A 13 + . . .)(u — S (0 1 a , 1 / ))/cr 2 , 
AOC 2 ' = (0^2! + 02^22 + 03^-23 + • • -){ U ~ S (01^1 '))/c°> 
AX 3 = (0!A 31 + 02A 32 + 0 3 A 33 -j- . . .)(« — S (01^10)/ 0 " 3 , 
where A is the determinant of the ft’s, and the A’s are its minors, clearly ay = ay 
and Ay- = A y. Multiplying these equations by j.3 U 0 2 , 0 3 . . . respectively, and 
adding we find 
hence 
where 
cr 3 AS (0 1 as 1 / ) = {0 tA u + 02 2 A 23 + 0 3 2 A 33 + . . 
+ 2 A 12 0]0o + 2 A 13 0j0 3 + . 
= {S (0i S A n ) + 2 S (A 12 0 1 0 2 )}(w - 
S <AV) = ^ X 
..} (u - s (A*',)) 
- Sift*',)), 
X = s (AW n ) + 2 S (a b Aft). 
