EVOLUTION 477 



tions on stature in man and on coat-colour in Basset hounds, 

 tells us that the fractions we are to take of H^, H.„ H.^, 



. . . H^Q, . . . etc., or the quantities 7^^, 7.,^, y.^-^, . . . 



710 1~ • • • ^^^ respectively J, J, ^, • • • toV?" • • • Now, 

 since each H is the mean of all the ancestry of a par- 

 ticular generation, we can state this law in the following 

 form : ^ — 



Each parent contributes on an average one-quarter 

 or (0.5)'", each grandparent one-sixteenth, or (0.5)*, and 

 so on, and that generally the occupier of each ancestral 

 place in the «th degree, whatever be the value of ?/, con- 

 tributes (o-S)"'" of the heritage. 



Now let us consider this law in a rather different form. 

 We will replace the mid-parental variabilities by the 

 variabilities of the race in the generations to which the 

 successive mid-parents belong ; let these be cr^, a.,, 0-3, 

 . . . o-jjj, . . . then by p. 476 the relation above may be 

 written : — 



/...y^ X J2 X ^H, + 7, X ( V2y3 X ~U, + y, x ( ^2)3 x^H,+ . . . 

 "^1 '^•z "3 



+ riox(v/2rx^H,o+ • • • 

 "^10 



It is clear that Mr. Galton has taken the coefficients 



7iX -s/2, 72X(x/2)', %X(^2f, . . . j^^X(y/2y'' .. . 



to be ^, -4, ^, ■ • • T0V4' • • • ^•^- ^ certain geometrical 

 series. More generally, let us take them to be any 

 geometrical series whatsoever, represented by 



ya, ya-, ya^, . . . ya^^ . . . 



or 



/i = y\a -H, + a2-H., + a3-H.,+ . . . +alO^H,„+ . . .). 

 'i o-l ' °-2 - ^s " o-io '" J 



This result I term the /azu of ancestral heredity. 



It makes no assumptions (i.) as to whether the type is 

 or is not altering, for H^, H^', H,, . . . H^^, . . . etc., may be 

 measured from different means for each generation, or (ii.) 

 as to whether the variability a^, a^, cr„ ... o-^^ ... of each 



1 Gallon, Royal Society Proceedings, vol. Ixi. p. 402. 



