250 A Blometrical Study of Conjugation in Paramecium 
Obviously then the proper thing to do is to enter pach pair twice, once with A 
as the first variable and once with B as the first. This will result in making the 
table symmetrical* with the totals for the rows and columns equal. In each case 
in the present paper I have first formed correlation tables with A as the first 
variable, and deduced from each such table its correlation coefficient r. Then in 
those cases where we were dealing with the same character in both individuals of 
the pair the tables were made symmetrical and the coefficients of correlation again 
calculated. In the case of the symmetrical tables the coefficient was not calculated 
directly from the table but by a formula which is derived from a more general 
theorem given by Pearson -f" for determining the effect on the frequency constants 
of adding together diff'erent samples of material. He shows that if we let x and x 
be measures of two organs, and there be iV^ pairs of organs formed by i heterogeneous 
groups containing n-^, n^, n.^, ... etc., pairs with means mj, m/, m^, m^, niz, m^, ... 
etc., standard deviations a^, o-/, cr.,, cr./, 0-3, 0-3', ... etc., and correlations i\, r^, r^, ... 
etc., and M, M' be the means of the whole community, 2, 2' the standard deviations 
and R the correlation, then 
EIX'N = S (naa'r) + S [n (m - M)(m' - M')] (i) 
where S denotes summation with regard to all i groups. 
In the case of the symmetrical table clearly the following relations will hold. 
N=2n, 
M = M', 
% = X, 
i = 2. 
Equation (i) will then become 
iJSW = 2na<T'r + 2n (m - M)(m' - M), 
whence, dividing by 2n we get 
= a-a'r + (m - M){7n' - M). 
. ,^ m + m 
But since M = — ^ — 
we have RV = aa'r - i!!!^^ (ii). 
On p. 278 of Pearson's memoir above referred to the values of and 2'^ are 
given as follows : 
{na'-) 8[npnq{mp-mqY] 
^ - N ^ ' 
_ 8 (na-'^) S {tipUg (n i p - rnqf } 
* The reason for using such symmetrical tables was first pointed out by Pearson, Phil. Trans. 
Vol. 197 A, p. 293. 
t Phil. Trans. Vol. 192 A, p. 277. 
