TO THE THEORY OF EVOLUTION. 
37 
Illustration II .—Our analysis opens a large field suggested by the following 
problem:— What is the chance that an exceptional man is born of an exceptional 
father ? 
Of course much depends on how we define “ exceptional,” and any numerical 
measure of it must be quite arbitrary. As an illustration, let us take a man who 
possesses a character only possessed by one man in twenty as exceptional. For 
example, only one man in twenty is more than 6 feet 1"2 inches in height, and such a 
stature may be considered “ exceptional.” In a class of twenty students we generally 
find one of “ exceptional ” ability, and so on. Accordingly we have classed fathers and 
sons who possess characters only possessed by one man in twenty as exceptional. We 
first determine h and k, so that the tail of the frequency curve cut off is fj of its 
whole area. This gives us h = k = 1'64485. 
Next we determine HK = ^-e - - (7i ' +7l ' 2) , and find log HK = 2 - 026,8228. 
L7T 
Then we calculate the coefficients of the various powers of r in (xix.). We find 
log \hk = -131,2225. 
log|(/F - 1) (/c 2 - 1) = 1-685,5683. 
log |- {hr - 3 )(¥ - 3) = 3-990,1176. 
log T 2 o (h* ~ Gh* + 3) - 6F + 3) = 1*464,4772. 
Iik - 
log 720 “ 10/i3 + 15 )(^ 4 “ 10 ^ + 15 ) = 2-925,6367. 
It remains to determine what value we shall give to r, the paternal correlation. It 
ranges from "3 to "5 for my own measurements as we turn from blended to exclusive 
inheritance. Taking these two extreme values we find 
—— -0046344 or -0096779. 
N 3 
But ^, = — — an d the second term is the chance of exceptional 
fathers with exceptional sons, when variation is independent, i. e. , when there is no 
heredity, = f 6 X -fo — '0025. 
Thus cZ/N = -007134 or ’012178 ; 
accordingly b/ N = "042866 or "037822. 
Hence we conclude that of the 5 per cent, of exceptional men "71 per cent, in the 
first case, and 1"22 per cent, in the second case, are born of exceptional fathers, and 
4"29 per cent, in the first case and 3'78 per cent, in the second case of non-exceptional 
fathers. In other words, out of 1000 men of mark we may expect 142 in the first case, 
