Mathematical Contributions to the Theory of Evolution. 153 



family is about 4*5 ; but if we confine ourselves to one sex, we must 

 exclude all sterile marriages and all not leading to two brothers, or 

 two sisters. We might then very well take n = 6, which gives 0*4 for 

 the fraternal correlation. Thus we might expect in the case of exclu- 

 sive inheritance that the fraternal correlation would lie between 0*4 

 and 1, according as to the degree of prepotency of one or other parent 

 in the individual marriage.* 



Thus our theory of exclusive inheritance is not, like that of blended 

 inheritance, incompatible with observed facts, i.e., high values of 

 parental correlation and values substantially less than unity of the 

 fraternal correlation. But for such cases we must deny the existence 

 of any regular and continuous law of reversion. We should have to 

 look upon reversion, if it occurred at all, as merely an irregular and 

 infrequent phenomenon. 



On the other hand, if we differentiate the taking after parents from 

 the reversion to ancestry as phenomena of a quite distinct nature, our 

 theory will enable us to surmount, for some cases at least, those 

 diflBculties in ancestral correlation, which arise when we take Mr. 

 Galton's Law in its original form to cover both blended and exclusive 

 inheritance. I illustrate this from data for the coat-colour of Basset 

 Hounds in the following section. 



(8) Application to basset Hounds. — Understanding that I was 

 desirous of testing my theory on a character which was definitely 

 exclusive, Mr. Galton, with his invariable kindness, at once placed at 

 my disposal his material on Basset Hounds. The reader will remember 

 from the statements in Mr. Galton's own memoir (' Roy. Soc. Proc.,* 

 vol. 61, p. 403) that these dwarf bloodhounds are either lemon and 

 white, or black, lemon, and white ; and here, as in Mr. Galton's work, 

 they will be classified as non-tricolour and tricolour, or by the symbols 

 N and T. In dealing with the offspring I was in many cases unable 

 to determine the sex of the dog, as that information is not in the stud 

 book,t and all individuals are not again recorded as sires or dams, 

 nor do they possess obviously male or female names. Thus in my 

 principal tables all the offspring of both sexes are clubbed together. 

 To measure the legitimacy of this, I have formed separate tables of 

 the two sexes in the case of sires and dams. Further, in dealing with 

 great grandparents there were so few of each of the eight individual 

 types alone, that I have formed merely one table, that of great grand- 

 parent and offspring, disregarding the line of descent. 



* Tlie mother and father may be eqtiipotent on the average, but in the individual 

 family one or other be markedly prepotent. It is to this prepotency of the 

 individual, regardless of sex, that the increase of fraternal correlation beyond 0*4 

 is very probably due. 



t Sir Everett Millais, ' The Basset Hounds Club Eules and Stud Book,' 1874— 

 1896. 



