Mathematical Contributions J,o the Theory of Evolution. 163 



tion, which is more stringent for the males than the females, appears 

 to be, theoretically at least, a mistake when we remember that the 

 potency of the female is thrice that of the male in coat-colour. 



Within the groups of grandolFspring and siblings the differences are 

 hardly significant enough for special conclusions to be drawn. 



if)) Turning next to the parental groups, we see that (i) the sires and 

 dams, neither of v/hich can be considered to form a more modern 

 group than the other, have yet remarkably different values of x, that 

 for the sire being about seven times as great as that for the dam. The 

 sires are thus far more stringently selected than the dams, and a great 

 deal of this difference must undoubtedly be due to the lesser variability 

 of the sires. Here again the breeders, if they are selecting at all for 

 coat-colour, would appear, at least theoretically, to be in error, 

 (ii) Grandsires, ^ and ? , appear to be less variable than the sires, 

 and granddams, and $ , less variable than the dams. This may 

 be due to the original paucity of the breed, or be an instance of the 

 general rule to which I have elsewhere referred, i.e., that parents are a 

 selection out of the general population, and so less variable than their 

 offspring. 



(c) But this rule meets with a remarkable exception in the case of 

 $ parentage; both granddams and dams are more variable than their off- 

 spring, and very significantly so. An examination of dam and female 

 offspring shows that the ? offspring have a value of x double as great 

 as that of their dams. With few original dams, it is difficult to under- 

 stand how they could be more variable than their offspring. Consider- 

 ing the great prepotency of the dam, it is difticult to attribute this 

 increase of x entirely to the action of the less variable sire ; one is 

 more or less forced to believe that there is a process of stringent 

 selection of the offspring which are entered on the record going on, 

 and that thus a group of dams possibly fairly variable, and with a 

 not very marked tendency to melanism, is represented in the next 

 generation by offspring of a more stringently selected character ; the 

 stringent selection of sires may have contributed, but can hardly be 

 the sole source of this change. 



A further conclusion is worth noting : Parents, whether male or 

 female, when they have male are apparently more variable than when 

 ■they have female offspring. 



(10) General Results. 



(a) The laws hitherto propounded for blended inheritance do not 

 appear to cover the cases of exclusive inheritance, e.g., such cases as 

 eye-colour in man, coat-colour in horses or hounds, &c. 



(h) The law of ancestral heredity must be distinguished from a law 

 of reversion. Neither seem to fit the facts if we adopt the amounts of 

 heritage, |, -^V, &c., from parent, grandparent, great grandparent, 

 &c,, originally taken as a first approximation by Mr. Galton. 



