346 MR. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
boundary of the first rectangle, on the line of common bases, and let y, be the height 
of the 7™ rectangle, or 
T= eos GSP aE) ey 
Yr = ie a ) p r+1 q ui 

c 
while | 
yy, = ap"/c. 

<C x C3 CH CS CH CC® CC® CP 
N 
Let us find the values of 
where s is any integer, for values of s from 0 to 4. 
It is easy to see that 
E l @\ if @ 
Set 10\st — # = (g 5- — / nu 
SAS UA) |) Eee = (7 -) ---9(p +49)", 
where the operation d/dq is repeated s times. 
The operations indicated can easily be performed by putting g = e” when 

du 
Sy nrr\S 2 auc? d a 2 u\n 
Sine x (rey} =“ (4) fer(p + e9"}, 
and the successive values can be found by Lerenirz’s theorem. After differentiation 
we may putp+gqorp+e"=1. There results: 
YrC) ae 
> (KO x TON | ms xg: al SEN ops 
J) = ac + 8ng + n(n — 1) 3 
= (yc X (re)?) = ae {1 + 7ng + 6n (nv — 1) + n(n — 1) (n — 2) 3 
c)*) = act {1 + long + 25n (n — 1) @ + 102 (n — 1) (n — 2) 
+ n(n — 1) (mn — 2) (m — 8) g. 
Let NG be the vertical through the centroid of the system of rectangles, then 
clearly 
ON = = (y,¢ X re)/a =c {1 + ngt. 
