1396.] 



Mathematical Theory of Evolution. 



303 



M, = Mo+TopvttQ (iii). 



oi = °o 2 (l — r 2 -Hv — J (iv). 



\ ff » / 



The first three equations are true whatever be the distribution of 

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

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

 offspring to be tf 2 (l- — 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 certainly in the case of man this correlation is 

 like the distribution of variation in fertility, markedly skew and not 

 normal in character. Our equations accordingly amply cover facts, 

 which they could not cover had they been solely based on the usual 

 ur normal theory of correlation. 



(3) By simply forming the means for any organ (or characteristic) 

 for mates and for parents, we can ascertain from Equation (i), if 

 there is or is not any sensible correlation between that organ (or cha- 

 racteristic) and fertility. Equation (ii) enables us to verify the value 

 found for since a p and a m are easily calculated when we know the 

 distribution of fertility. If the correlation were normal S(x 2 y) 

 would be zero, and this term it may reasonably be expected will never 

 be very large. When p has been found from Equation (i), then 

 Equations (iii) and (iv) give us Mj — M and a x - <x , or the measures 

 of reproductive selection in its action on the mean and variation of 

 successive generations. 



(4) I have applied these results to the only cage— that af man— in 

 which statistics are at present available. 



I find for upwards of 4,00(J families, principally of Anglo-Saxon 

 race, v = 0'692, and for 1,842 families of Banish race, v = G'652, 

 This, considering difference of race, is a very satisfactory agreement. 

 In the next place there appears to be a significant difference 0'278" 

 between the mean height of mothers of daughters and the mean 

 height of wives. Thus we have pva m = 0*278", and since a m =z 

 2 303", it follows that pv = 121. Now, the coefficient of variation 

 for fertility in daughters is not quite the same, but still very nearly 

 the same as that for fertility in general. We therefore find that 

 p = 0'175 to 0*186, according as we use the first or second value of 

 v given above. We therefore conclude that there is a sensible 

 correlation {circa 0*18) between fertility and height in the mothers 

 of daughters. 



