Miscellanea 
241 
The individual ancestral correlations are found from 
wv 
■ ' =-6(^)p-«. 
Hence the individual parental correlation is "3 and the geometrical ratio 
It is thus clear that Galton's Rule is not absolutely consistent with the Mendelian somatic 
correlations J, J x ^, etc., although it is not very far removed from them. 
The variability of an array of brothers from same ancestry is found from (xi) to be 
•8944<r 0 , 
or, the reduction in variability by continued selection will not exceed 11 °/„. 
The correlation of brothers 
= i? = ( j 3-,)/ N /,*-2,/3+l 
= -4472. 
Application II. Suppose the individual ancestral correlations follow the simple Mendelian 
i i 
3) s 
Then 
/•p_ 8 =(V2)P-«| x ^ 
i Y"« 
3VV27 ' 
2 i 
Hence a = -, /3= = "7071, 
"3' M "V2" 
^ = •3100, yJ? = -3971. 
whence we deduce 
Our series is therefore 
z 0 - 5=o= -561 7 (^'i - Ji) + -2463 (x 2 - x 2 ) + -1080 {x 3 - x 3 ) + -0474 (# 4 - x t ) + etc., 
this giving slightly more influence to the parents and less to the ancestry than Galton's Rule. 
The correlation of brothers is - 4896, giving 
2 0 = o- 0 x -8720, 
or a reduction of variability of about 13°/ 0 . 
Not much stress, however, must be laid on these results for a single Mendelian unit character. 
They are the furthest from true linear regression of the mid-parent and theoretically only two 
grades of the character occur. 
Application III. Suppose the correlations are given by the Mendelian gametic values 
h h 8. etc - 
•We have r p _ 9 =(V2)*-«x = J . 
Hence a=l, $ = ^2 | • 
It follows that ); = 0, yr) = ft = ^ • 
Hence the prediction formula reduces to 
.Vq — ,r 0 = Xp — X p , 
or the parents suffice to determine the offspring. This is in complete accordance with the 
Mendelian idea of the gametic parental constitution determining the offspring. 
Biometrika vm 31 
