302 



Prof. K. Pearson. Contributions to the [Mar. 5, 



wishing at present to publish my complete work on this subject, I 

 should like to put on record the following conclusions already 



reached : — 



(2) Let any organ in individuals of one sex be selected, and let y 

 be the fertility of an individual, whose organ differs x from the 

 mean organ of all mated individuals. Let M,» be the mean organ for 

 all mates,* M p be the mean organ for all parents, i.e., a mate reckoned 

 once for each, offspring. Let M be the mean of the offspring 

 for the same or any other organ, taking one or any other number 

 equally from each mated individual, let M x be the mean of ali off- 

 spring. Let (T m , (Tp, <r , c x be the corresponding standard deviations, 

 reckoned from the formula : a % = (sum of squares of deviations) -f- 

 (number of individuals), and without regard to any special law of 

 variation, such as Laplace's law of errors. 



Let r be the coefficient of correlation between parent and offspring, 

 each parent being given only one or, at any rate, an equal number of 

 offspring, i.e., r is the coefficient of pure heredity for the organs in 

 question, supposing fertility to be uniform, or at any rate to have no 

 correlation with the organ or characteristic under i avestigation. Let p 

 be the correlation between fertility and the given organ in the parent, 

 and let v equal the coefficient of variation of fertility in the parent, 

 i.e., if y m be the mean fertility: v = of/y m , where of is the standard 

 deviation of parental fertilities. Let ij = y — y»i be the deviation from, 

 mean fertility of the parent with organ x. The values of r and p are 

 to be calculated from the formulas — 



Sum of (deviation of offspring x deviation of parent) 



Number of pairs of offspring and parent Xff x a m 



Sum of (deviation of mate X deviation of mate's fertility) 



Number of mated pairs x <r m x <r/ 



w^here, in r , each parent is to be taken only once, or at any rate the 

 same number of times. 



Thus r and p are absolutely independent of any special distribu- 

 tion of variation. 



Then the following results hold if n be the number of mated 

 pairs : — 



M p == M m +pVff m (i). 



C\ = 0* m (1-yoV) (ii). 



ny m 



restraint on fertility. As those couples who fall into this component leave no 

 offspring, they cannot give rise to a new species. 



* If there be preferential mating, Mm will not be the mean organ for all indi- 

 viduals. I have adopted the mate mean in order to free the investigations from the 

 influence of this portion of sexual selection. 



