322 NATURAL SCIENCE. May, 



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 1 be the mean of all offspring. Let 

 cr,„, <t p , <r , (r x be the corresponding standard deviations, reckoned from 

 the formula : a- 2 — (sum of squares of deviations) -s- (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 investiga- 

 tion. 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., \iy m be the mean fertility : v = a-/!y m , where ay is 

 the standard deviation of parental fertilities. Let y' — y — y m be the 

 deviation from mean fertility of the parent with organ x. The values 

 of r and p are to be calculated from the formulae — 



Sum of (deviation of offspring x deviation of parent) 

 Number of pairs of offspring and parent x a x a- m 

 _ Sum of (deviation of mate x deviation of mate's fertility) 

 Number of mated pairs x cr m x ay 

 where, 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, = M m + P vo- m fi). 



°* P = "Mi-p^-f S (*^) (ii). 



ny,n 



Mj == M + r pv<r .• (iii). 



°-i 2 = <r * (i - r « + v? ?>-!) (iv). 



V cr m V 



The first three equations are true whatever be the distribution of 

 variation in mates, parents, offspring, and fertility ; the fourth 

 equation assumes the standard-deviation of a fraternity or an array of 

 offspring to be o- 2 (i — r 2 ). This result would flow for normal corre- 

 lation between organs in parent and offspring, a type of correlation 

 which holds closely for inheritance in the case of man. It would also 

 flow from any law of variation which gave a constant coefficient of 

 regression and a constant standard deviation for the array. What, 

 however, is the important point is this, that no assumption has been 

 made with regard to the nature of the fertility correlation. This is 

 essential, as in the case of man this correlation is certainly, like the 

 distribution of variation in fertility, markedly skew and not normal in 

 character. Our equations accordingly amply cover facts, which they 



