TO THE THEORY OF EVOLUTION. 
21 
/E., 7 t\ -p. Og cot % 
— tan - - = cot D-wAr 
\ N 2 / oq sin D 
Now eliminate oq/oq between (lix.) and (lix. bis ); then 
E[ 7T^ , J n /Eg 7r\ 
(tan y + cot D ) ( tan ( k 2 j + Cot 
1 
sin 2 D 
Whence we deduce 
and, therefore, 
cot D = cot 
Ej + Eg 7T 
r = cos D = cos 
N 2 ’ 
Ej + Eg 7T 
N 2 
(lix. bls ). 
(lx.). 
Substituting for D in (lix.) we find further 
7T 
vN 2 )/ 
cr i . / E, 7T \ / 
— cot X COS (—'-)/ cos 
Ej 7T 
N 2 
(lxi.). 
Thus Equations (lx.) and (lxi.) give the coefficient of correlation and the relative 
variability of the two characters. The latter is, I believe, quite new, the former novel 
in form. 
If we call m L the frequency in the angle y (AOjc of the figure above), then it is easy 
to see that E x = 2 (n 2 — n x ) = N — 4rq, and similarly E 2 = N — 4 m v Thus 
(E x + E 2 )/N = 2(N — 2 {n 1 + m x ))/N. But n l -fi m x is the frequency in the first 
quadrant. This Mr. Sheppard terms P, while that in the second he terms lb We 
have thus (E x -p E 2 )/N = 2R/(Pt + P), or 
r 
= cos 
R 
E + P 
7 T 
(lxii.), 
i.e., Mr. Sheppard’s fundamental result # (‘ Phil. Trans.,’ A, vol. 192, p. 141). 
We can, of course, get Mr. Sheppard’s result directly if we put y = 0, when we 
have at once E x = 2(R — P), E 2 = N = 2(R + P), and the result follows. 
Equation (lxi.) may also be written in the form 
^ = cot y sin ^2 tt) j sin 2 tt) .(lxiii.). 
If we put y = 0, then m 1 becomes zero, and the right-hand side of (lxiii.) is 
indeterminate. If we proceed, however, to the limit by evaluating the frequency in 
an indefinitely thin wedge of angle y, we reach merely the identity oq/oq = oq/oq. 
Hence there is no result corresponding to (lxi.) to be obtained by taking 
Mr. Sheppard’s case of y = 0. 
The following are the values of the probable errors of the quantities involved :— 
* In the actual classification of data (lx.) and (lxii.) suggest quite different processes. We can apply (lx.) 
where (lxii.) is difficult or impossible, e.g., correlation in shading of birds’ eggs from the same clutch. 
