TO THE THEORY OF EVOLUTION. 
25 
d 2 z 
(J 1 j/J 1 y 
cPcf) 
clXp d'j. ^ 
clz clef) 
djOUgf' d/Xjj 
Thus finally 
1 d 2 z 
Z clXpdXq 
dz 
dT 
d 2 (f> , d4> def) 
— | — ' 
(JjXpdjq dXp dXq 
d 2 z 
n 
dxpdx q 
(lxxii.). 
In other words, the operator djdr pci acting on 2 can always be replaced by the 
operator d 2 jdx p dx q . 
Let d/dp M denote the effect of applying the operator djdr pq to z, and putting r pq 
zero after all differentiations have been performed, then the effect of this operator will 
be the same as if we used cl 2 jdx p dx q on z, putting r pq zero before differentiation. 
Generally, let F be any series of operations like d/dr Pi , then we see that 
F 
d 
d 
d 
dr 
= F 
d 2 
pi 
d 2 
dr pqt dr P u q n 
d 2 
z 
N 
dxx/dxq dxpdx-q, 5 dx. p „dx ri „ 
(27 r)- n 
Now let F he the function which gives the operation of expanding z by Maclaurin’s 
theorem in powers of the correlation coefficients, fie., 
then 
z = e M’" afe 
F = e s *( r - , 5&)> 
N 
(27T)»« 
This is the generalised form of result (xiv.) reached above. 
N 
Now let 
Zn = 
■° (27r) 
. - hSiW) 
then z 0 is the ordinate of a frequency surface of the nth order, in which the distribution 
of the n variables is absolutely independent. We have accordingly the extremely 
interesting geometrical interpretation that the operator 
„S Ar„' d2 
applied to a surface of frequency for n independent variables converts it into a surface 
of frequency for n dependent variables, the correlation between the sth and fith 
variables being r ss ,.* 
* I should like to suggest to the pure mathematician the interest which a study of such operators would 
have, and in particular of the generalised form of projection in hyperspace indicated by them. 
VOL. CXCV.—A. E 
