PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 279 
any Jaw of frequency whatever which causes S (£ 77 ) = 0,—for example, if it be equally 
likely that 77 occurs with an equal negative or positive value of £,—will show that 
x and y are independent variations. Hence, if we define r = S (^/(woqoq) as the 
coefficient of correlation, we see that it has a significance extending much further 
than the normal law of error. Just as oq, oq are radii of gyration (and independent 
of any special law of error), so S (^ 77 ) is a product moment, and its vanishing marks 
the absence of correlation, or directions of independent variation. 
We see, then, that the coefficient of correlation may be found from 
r 
s+o j? q 0 /> o o 
- —/voy —/yg’s 
%fxfy a ‘l <r 'i 
3 
or by calculating standard deviations. 
The question naturally arises as to what is the best value of f(x, y ). This will 
often be already answered by the data themselves. A common case is that in which 
the variations in x and y are given, and the variation in their ratio or the index x/y 
is calculated. In this particular instance f x — M /m 1 and f y =— M /m. 2 . Hence 
o , o xro 
V + TV — v - 
r = —vT" 
We thus throw back the determination of correlation on ascertaining three 
coefficients of variation. 
This formula, while less general than the one previously given, in that we have 
neglected squares of small quantities, is more general in that we have not limited 
ourselves to any special law of frequency. 
(iii.) Example .—The formula may be illustrated by the following statistics taken 
from a not yet published paper on variation in man. r = coefficient of correlation 
between length and breadth. 
Adult Mall Crania. 
Professor Flinders Petrie’s newly discovered race.'" 
Length of skull . . . m 1 = 185*2777, oq = 5*7783, v l = 3T187 
Breadth of skull. . . m 2 = 135'0194, oq = 4*4076, v 2 = 3'4183 
Cephalic index, B/L . M = 72*9379, % = 2*8848, V = 3*9551 
r = *2705. 
* Professor Flinders Petrie kindly replied to my request for 100 skulls of a homogeneous race, 3,000 
to 4,000 years old, by bringing back to England the finest anthropological collection—skeletons as well 
as crania—known to me. The collection was packed and brought to England at the charge of Mr. 
A. B. Pearson-G-ee. Mr. Herbert Thompson has made a series of measurements on 301 skulls, $ and 
?, details of which will be published later, and the above constants are calculated from his measure¬ 
ments. The date of the new race is about 3000 b.c. 
