2H8 MR. G. UDXY YULE ON THE ASSOCIATION 



Now if we use ABC for son, father, grandfather, and the capitals to denote " ahove 

 average," Greek letters " below average," we want 



(ABC)/(BC) 

 (BC) = 2985 at once. From 11, p. 263, 



2 (ABC) = (AB) + (BC) - (aC) 

 = 2985 + 2985 2260 

 = 3710 



(ABC) = 1855, 



chance required = 1855/2985 = "6214. 



If only the father be known to be above average, chance of son being above average 

 = 2985/5000 = '5970. 



If, on the other hand, we ask what is the chance that the child will be above 

 average if both the father and mother are so, we have, assuming the correlation with 

 both parents to be the same, using B for father, C for mother, and assuming no 

 assortative mating : 



2 (ABC) = 2985 + 2500 - 2015 = 3470 

 (ABC) = (1735) 



chance = 1735/2500 = '6940. 



But if there be perfect assortative mating 



2 (ABC) = 2985 + 5000 2015 



= 5970 (ABC) = 2985 

 chance = 2985/5000 = '5970. 



Thus, if there be no assortative mating, a selection of father and mother is better 

 than a selection of parent and grandparent ; but not so if there be assortative mating 

 to any great extent. 



15. The relations that we have dealt with in the preceding pages have a general 

 bearing on the theory of certain multiple integrals. If we imagine, as we have 

 already done on several occasions, that the distribution of frequency is really con- 

 tinuous and the points of division between A and a. B and ft, &c., arbitrarily fixed, 

 then any ultimate frequency like (ABCD) (AySCS), for example, is equivalent to the 

 multiple integral expressing the total frequency contained within the four axes of 

 the frequency surface (or hyper-surface), taking each of these axes in either the 

 positive or negative direction. 



Now we know that there are 2" ultimate groups (or multiple integrals of the 

 above kind) to be formed from n variables, all of these groups being in the general 

 case independent. Suppose the function expressing the distribution of frequency to 

 contain m constants that remain in the expression, ^(a^a^a^ . . . ), for the multiple 

 integral. Then we have the equations 



