4Q2 Dr. F. Galton. The average Contribution of each 



"broods and in different places and years. No statistical results of 

 any consistence or value could be obtained from them. Latterly, 

 while engaged in planning another extensive experiment with small, 

 fast-breeding mammals, I became acquainted with the existence of a 

 long series of records, preserved by Sir Everett Millais, of the colours 

 during many successive generations of a large pedigree stock of 

 Basset hounds, that he originated some twenty years ago, having 

 purchased ninety- three of them on the Continent, for the purpose. 

 These records afford the foundation upon which this memoir rests. 



The law to be verified may seem at first sight too artificial to be 

 true, but a closer examination shows that prejudice arising from the 

 cursory impression is unfounded. This subject will be alluded to 

 again, in the meantime the law shall be stated. It is that the 

 two parents contribute between them on the average one-half, or 

 (0*5) of the total heritage of the offspring; the four grand- 

 parents, one-quarter, or (0'5) 2 ; the eight great-grandparents, one- 

 eighth, or (0'5) 3 , and so on. Thus the sum of the ancestral contribu- 

 tions is expressed by the series {(0'5) + (0'5) 2 + (0"5) 3 , &c.J, -which, 

 being equal to 1, accounts for the whole heritage. 



The same statement may be put into a different form, in which a 

 parent, grandparent, &c, is spoken of without reference to sex, by 

 saying that each parent contributes on an average one-quarter, or 

 (0'5) 2 , each grandparent one-sixteenth, or (0*5) 4 , and so on, and that 

 generally the occupier of each ancestral place in the nth degree, 

 whatever be the value of n, contributes (0'5) 2 * of the heritage. 



In interbred stock there are always fewer, and usually far fewer, 

 different individuals among the ancestry than ancestral places for 

 them to fill. A pedigree stock descended from a single couple, m 

 generations back, will have 2™ ancestral places of the mth order, but 

 only two individuals to fill them ; therefore if m = 10 there are 1024 

 such places ; if m = 20 there are more than a million. Whenever 

 the same individual occupies many places he will be separately 

 rated for each of them. 



The neglect of individual prepotencies is justified in a law that 

 avowedly relates to average results ; they must of course be taken 

 into account when applying the general law to individual cases. No 

 difficulty arises in dealing with characters that are limited by sex, 

 when their equivalents in the opposite sex are known, for instance in 

 the statures of men and women. 



The law may be applied either to total values or to deviations, as 

 will be gathered from the following equation. Let M be the mean 

 value from which all deviations are reckoned, and let D l5 D 2 , &c, be 

 the means of all the deviations, including their signs, of the ancestors 

 in the 1st, 2nd, &c, degrees respectively ; then 



£(M + D0 J -«M + D a )+&c. = M+(iDi + iDa + &c<) 



