PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 287 
Hence we have by symmetry the equations, 
x i = X i l ~ r i~)’ h = X ( l - 
*T X i + v i v s = X r i> ^ X 2 + v z v i = XW 
X s = X (! - r s), 
V 3 X 3 + VyVz = X^3- 
We easily deduce 
X S ( ? ’l ” r 8 r s) = ( X 2 X 3 “ ^l 2 ) (^Al + ^ 3 ) — (*b X 2 + w) A3 X 3 + v l v %) 
v i = X (n - ?Ws)> 
^]X> 
or, 
and similarly 
Finally, 
or, 
= X ( r 2 ~ *3^), *3 = X ( r 3 ~ ^ 2 ). 
x 2 (( r i - *2*3) (! - W*) + (’2 “ *Vs) ( r 3 - *V’ 2 )} = X r i> 
X (1 “ r i 2 - ’h 2 - ^3 2 + 2 r x r 2 r 3 ) = 1 . 
Thus all the constants are determined, and we have, 
p — n / X 
(27r) 3/ V l0 - 2 o-3 
-k { — 2 (! - n 2 ) + ^ (1 - r a 2 ) + 4o - ’•a 2 ) - 2 (n - r 2 r s ) AL-2 (r 2 -r 3 r,)-^- - 2(r 3 - , . , 
<■ <r^ <V VT ®2<*3 <r 3 <r, <Vi > aXOydZ. 
This agrees with Bravais’ result, except that lie writes for r x , r 2 , r 3 the values 
2 (yz)f{na 2 o- 3 ), etc., which we have shown to be the best values (see loc. cit., p. 267). 
Obviously we have the following general results. If X x be the standard deviation of 
a group of x-organs selected with regard to values h 2 and ho of y and. z, 
Sl v'fxfl 
Tl__ . /I - »V ~ ?’ 2 3 - Ls 2 + 2 Ws 
r^w)i ~ ^ V rwr 
and if /q be the deviation of the mean of the selected x-crgans from the x-mean of the 
whole population 
h — r A “Tl'j Tit I fAH JT '3 Tl /, 
1-7-d + 1 -V ^ 
Expressions of the form ^will be spoken of as coefficients of double correl; 
tion, and expressions of the form - — as coefficients of double regression." 
1 — rf (t 2 
* [The above values for Sj and h x are still true, as Mr. G. U. Yule points out to me, whatever he the 
law of frequency, provided the standard-deviations of all arrays be the same and h x be a linear function 
of ho and A,.] 
