224 The Law of Ancestial Iferediti/ 



the one bdow it. Jlen who live tu be part'Uts aiv a luore select groiip than their 

 offspring and this is still niore truo of gi-andpait-nts and liigluT ancestiy. Td be 

 a great-great-grandpaiunt lueans to have a healthy stock, and such nicn form a 

 selection only of the conimunity iu each geueration. Hence ueither for man 

 nor the thoroughbred horse ueed wo ä priori expect to find, for the correlation 

 coefficients as actiially deteiinined, the relation Ji + J„ + Jj+ ...= l ov e = l — p 

 satisfied. The actual statistics for man and the horse conclusively prove that it is 

 not satisfied. If we bad a popnlation iu which all niembers were randomly matcd 

 and were parents of equal fi-rtility ; further if every generation had the same mean 

 and variability, then we might have e=l — p, but tlien the correlations of aneestry 

 and oflspring would probably take very different values froni what we find theni 

 tp have in the cases of man and horse under existing eircumstances. 



(7) Sn])posiiig all niating to be raiidom * we can find e and p and all the iTs 

 by the method of my first paper on the " Law of Ancestral Heredityf." In the 

 notation of the present paper, if «/•" be the correlation coefficient of "the ?( + Ith 

 mid-parent " with the offspring, aud if J,j = e'p'''~', we find : 



r=p' + e' (3), 



— « + e'p'w '( 1 —p'r) = — e' (4); 



whence : 



, 1 +r- -2ar , 



p--p +1 = (5). 



"^ r — a 



Hence taking a = 'ö and r = § as close eiiongh to the observed values we have : 



p' = -22öl, e'=-441ä. 



Formiila (i) of p. 217 iising " niid-parents " now becomes: 



p, = -4415 ll h,„ + -2201 J h,, + (•22.-.1 f g /,,„ +...]. 



Here —,, S», —3 ... are the Standard deviations of the mid-parental gnmps, and 

 S, = (Ts/(V2)", if a-g be the Standard deviatiou of the sth generation of ance.stors. 

 Thus:' 



p, = -441 .0 X V2 -(-" //^, + -2251 V2 """ h„, + (-2201 s'2f -° /,,,, + . . .1 

 = -6244 1^" //„ +-:ll8t^"//„ +C3184)=°""/v,+ ...l 



(ff, " O-., ' ff:, ] 



Clearly e= '6244 is not cqual to 1 — p or "6816, or JIr Galton's hypothesis is 

 certaiuly not satisfied either for man or horse. But since we know in these cases 



• R. S. Proc. Vol. CG, p. l-l'.l. 

 t Ji. S. Proc. Vol. Ü2, p. 394. 



