264 PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
(4.) Special Case of Tiro Correlated Organs. 
(a.) Theory. —Let x and y be the deviations of a pair of organs (or measurable 
characteristics) from their respective means. Let oq and cr 2 be the standard devia¬ 
tions of x and y, treated as independent variations. Let N be the total number of 
pairs and z X BxBy the frequency of a pair falling between x and x 8x, y and 
y + %y> then, by Bravais’ form, 
2 _ Q X g-Li-c 2 +2/i.r^+f 1. 2 >X) 
where g v g. 2 , and h are constants. 
Integrate 2 for all values of y from — a to + a, and we must have the normal 
curve of ^-variation, hence 
ATI = ( ,h 0 “ h*l9i9i)- 
- ,0 i 
Similarly integrating 2 for all values of x, we have 
vr 3 = 9-2 (! - W/g.gf 
ZjU o 
Now integrate 2 for all values of x and y to obtain the total frequency, and we 
have 
N = CiTjs/ggg. z — hr. 
If we now write r for — hj f g x g 2 , we can throw 2 into the form 
N 1 _. c _ *- a _ y- 7 . 
2 - ~ / , C 2 l<ri 2 (l—r 2 ) <r I a - 2 (1 -r 2 ) + ov*(1 -r-)I 
27 rcqoq v L — r- 
(h.) On the best Value of the Correlation Coefficient .— This is the well-known 
Galtonian form of the frequency for two correlated variables, and r is the Galton 
function or coefficient of correlation. The question now arises as to what is 
practically the best method of determining r. I do not feel satisfied that the 
method used by Mr. Galton and Professor Weldon will give the best results. The 
problem is similar to that of determining a for a variation-curve, it may be found from 
the mean error or the median, but, as we know, the error of mean square gives the 
theoretically best results. 
Let the n pairs of organs be x x , y x , x. : , y. 2 , ,r 3 , y 3 , &c 
observed series for a given value of r varies as 
then the chance of the 
