TO THE THEORY OF EVOLUTION. 
7 
Here the left-hand side is known, and since h and k are known, we can find the 
coefficients of any number of powers of r so soon as the first two have been found, 
from (xii.) and (xiii.). 
Accordingly the correlation can be found if we have only made a grouping of our 
frequencies into the four divisions, a , b, c, and cl. 
If h and k be zero, we have from (xvii.) and (iv.) 
H = K = 
1 
\/27T 
a c — b -j- cl — -g-N. 
The right-hand side of (xix.) is now 
V —|— rn V ^ —|— 
O 
or equal to sin 1 r. 
Hence 
r — sin 27r 
{ad — be) 
N 2 
— COS 77 
(i + 5 
(xx.), 
which agrees with a result of Mr. Sheppard’s, ‘ Phil. Trans.,’ A, vol. 192, p. 141. We 
have accordingly reached a generalised form of his result for any class-index whatever. 
Clearly, also, r being known, we can at once calculate the frequency of pairs of organs 
with deviations as great as or greater than h and k. 
§ (2.) Other Series for the Determination of r. 
For many purposes the series (xix.) is sufficiently convergent to give r for given 
h and k with but few approximations, but we will now turn to other developments. 
We have by (vii.) 
[ U dr = <r^ 2+ ^ (u 0 r + ^ 
~b • • • + u n 
/y>n +1 
\n + 1 
Put x = h, y = k, and write for brevity 
€ = 
ad — be 
N 2 HK 
(xxi.), 
It follows at once from (xix.) that 
e = e i(A2 + i ' 2) f lJ dr 
J 0 
— e Uh*+hO 
vM - 
rh=ji<r+v-*-Wdr 
