388 



Prof. Karl Pearson. 



determined. Then there will be 2 S_1 male and 2 s ~ l female sth. 

 parents, and the "mid-parent" will be defined to be an individual 

 having a deviation from the mean of the sth. parental generation 



_ 



This is a somewhat more general definition than Mr. Galton's. 



Since 8(h) = 0, S(h') = 0, when the summation extends over all 

 individuals of the sth generation who are parents, it follows that the 

 mean deviation of all possible sth mid-parents is zero.* 



Next let us find the standard deviation 2 5 of the sth mid-parents. 

 If their number for the population be N, then 



NxZ s * = -^ s S\ l h s + 2 h 5 + 3 h s + .... +m (Ji s + Ji s + Ji' JI + ....)}*. 



A 



Now, if there be assortative mating, S( q h s . q h' s ) will equal Na s o r s e St 

 where e s is the correlation coefficient for assortative mating in the 

 sth generation and (t s , a' s the standard deviations of male and female 

 mates in that generation. Further, S( q h s . q h s ) and S( q ti s . qh's) will 

 not be absolutely zero, because assortative mating would mean a class 

 mating into a like class leading to a correlation between relations 

 in law ; but these sums would be of the order e s 2 , and since, at any 

 rate for man, e s appears to be very small, we may neglect them to a 

 first approximation. "I" Hence 



NX 2/ = i {2*- 1 Syj+^r 1 SG^) 2 + 2m2^S(^ . q h' s )}, 



A 



or 2/ = ^— (<r. 2 + m 2 a s 2j t2ma s (T' s e s ) (i). 



A 



Now let us take m, which is at our choice, equal to <x s /o\j, then 



2, = |^(lW * (ii). 



In the next place let us find the coefficient of correlation between 

 two mid-parents, say those of the nth and sth generations. We 

 have 



N X ^, s ^npns = ~ S[{ i^ + 2 ^, + 3^a+ +m (ih' s + z ti s +3li s + .. .)} 



A' A 



X {ih u + 2 h n + 3 h n + +m( 1 h' n + 2 h' 3l -\-Ji'n+ )}]. 



* [Reproductive selection, which would weight particular parents, is neglected, 

 or, if not, the h's must be measured from the weighted means of all 5th parents. 



f Here, as later, I exclude the effects of in-and-in breeding ; this case requires, 

 special treatment. I hope shortly to pviblish fuller data for sexual selection in 

 man, based upon a wider system of measurements than are dealt with in my 

 memoir of 1895. 



