1904.] serve to Test various Theories of Inheritance. 267 



Now, if v 1 and v 2 be the numbers of children in the array of x y 

 parents belonging to either group, and v = v 1 + v 2 , we shall have for 

 the standard deviation of the total offspring of x y parents : 



"V = V 1 { Sl 2 + - Z p f\ + V 2 {Sj + (/Xo - Z p f] (ix), 



where z p is given by (iv) and is the mean of the whole array. 



If the relative influences of mother and father depended upon their 

 characters, we could go no further with (ix) until this had been 

 determined. If, however, we suppose this influence on the average to 

 be not sensibly dependent on the characters x and y, we may write 

 m/n = vi/v and n 2 /n = v 2 jv. Substitute from (iv), (v) and (vii), we 

 find after some reductions : 



_ wS + n&l + r _ + Aj ^ _ K <) x 



+h lm ^i-B >2m a ^)A 2 (x). 



\ °"m 0"m/ J 



To verify this equation I transferred to the mean, summed for every 

 possible array, i.e., for all values of x and y, and divided by the total 

 number of arrays. The left-hand side should be o- c 2 , the right-hand 

 side became, after some considerable reductions, identical with the 

 right-hand side of (i), as it should be. 



We have in equation (x) accordingly, the variability of an array of 

 all offspring on the hypothesis that the children may be divided into 

 two groups, differently related to the two parents. We see at once 

 that the variability of an array will depend on the actual values of 

 the parental characters, unless : 



and RlmCc./cm = K 2m o- C2 /a- TO . 



But this is asserting that the bi-parental regression co-efficients for 

 the two groups are the same, or, as we may put it, that there is no 

 distinction between the parental influences in the two groups. This 

 is the case usually assumed under the "Law of Ancestral Heredity" with 

 its constant variability within the limits of random sampling for the 

 arrays. In every other case the arrays alter in variability with the 

 magnitude of the parental character. 



Let us look at the matter from another light and sum. S 2 ^ for 

 every value of y only, or we can obtain the same result by direct 

 investigation, namely, we find if be the standard deviation of an 

 array of offspring due to fathers of character x, 



v 2 _ ^i^, 2 (! - r if) + % 20- C2 2 (1 - r 2f 2 ) ■ 



Zj x — . -f 



n 



^\ mi - m , + (r v ^-r v <) x y (xi). 



n- I \ (Tf o-f/ J 



