Fraternal and Parental Correlation Coefficients 
For the S.D. of y we find by introducing s in (1) 
""^ n<i-\l'+{q-l)rY 
(c) The hitandard Deviation of &-. 
The S.D. of the &■ of our sample is found from o-^'^'^ = o-* - {&-)-, where the latter 
term is already known. From (2) we find for the calculation of cr^ 
n^a^ = (nq -1)2 (yr) - 22 {m,) -2S{y,yd (4). 
and from this 
nya^ = {nq - If (2 {y^')T + 4 (2 (y^y,))' + 4 {S iy,y,)y + 
- 4 (nq -1)2 (y,') 2 {y,y^ - 4 (nq -1)2 (2/1^) 5f (^/.^Z^) + 82 (y,y,) S (t/.t/,) . . .(5). 
For the calculation of the mean values contained in this equation, the six pro- 
ducts of product sums must be examined. We find 
(2 (y,')r- = 2 (y/) + 22 (y^y^) + 2S (y.^yi) 
(2 (y,y.;)f = 2 (yr2//) + 22 (y^^y,y,) + 62 (yMl/^) + (yMyd' 
s s 
- (i/f ) S (yiyd = 2 (y.'y;) + 2 (y^^y./y,) + S (y,'y,y,) 
...(6). 
When the multiplication of products containing the factor S(yiy.^ is carried out, 
it is clear that we need not consider such sums of products where the product con- 
tains a factor which is uncorrelated with all the other factors of the product, 
because the mean values of such product sums are 0. In the products 2 [y^) 8 (y-^y^ 
and 2 (y^y-^ S(y^y.2) all the sums of products are of this kind, the factors being distri- 
buted either in two classes of which one contains 3 and the other 1 factor or in 
three classes with respectively 2, 1 and 1 in each. 
We therefore find 
iViy^^f = S iy'y^') + 28 (y.Hj^y,) + 4S (y.WM) + «i\ 
l(y,^)S(y,y.^ =a, ' ^ ^^^^ 
^{yiy2)S(y,y.^^a.j, 
where the mean values of the a's for the population are 0. 
Let us denote the product moment corresponding to yi^y-i^yz^yi'^ by finmpq if 
all factors belong to the same class and in the opposite case let us insert ' ' or ' s ' 
as denoting different or same class. 
* In the sums ,S' all factors of a product are supposed to belong to different classes except those which 
are denoted by an 's' inserted between them, as belonging to the same class. 
