Mathematical Contributions to the Theory of Evolution. 393 



gives us sufficient equations to find the coefficients of total correlation 

 between the offspring and the successive mid-parents. Equations (v) 

 will then enable us to find the coefficients of correlation between an 

 individual and any individual ancestor. But these in their turn will 

 suffice to determine all inheritance whether direct or collateral (see 

 below). In short if Mr. Galton's law can be firmly established, it is 

 a complete solution, at any rate to a first approximation, of the ivhole 

 problem of heredity. It throws back the question of inheritance upon 

 two constants, which can be once and for all determined ; herein lies 

 its fundamental importance. I must confess that this element of 

 simplicity was at first my chief difficulty in accepting the law as laid 

 down in the paper on Basset hounds, and I even yet have a certain 

 hesitation, owing to an apparent difference in collateral heredity in 

 different social classes, and also to the apparent numerical value of 

 the inheritance of fertility in man. 



(5) I shall next obtain a solution of equations (ix). Taking the 

 minors of the n + 1 constituents of the first row, namely, R 00 , R i, 

 R 02 , .... Hon, and multiplying them in succession by the constituents 

 of the 2nd, 3rd .... n + 1th row of R, we have by the ordinary theory 

 of determinants the system : 



/J 1 R + Roi+/>lR()2 + / J 2R()3+ • • • • +/>«_lRoM = 0. 

 ^RoO + ^Al +Ro2 + />lRo3+ • • • • + pll-lR>Q)l — o. 



^Rqo + /> ? _iRoi + Pq-'iR>02 + Pq-zR>03 + .... +/>«_gRo« = 0. 



/^R o+/>tt_lRol + / , «_2R'02 + / ) «-3R'03+ ...» +RoJ2 = 0. 



Divide each of these equations by R 00 and let us use instead of (ix) 

 the somewhat more general system which will allow us to consider 

 one or two limiting cases, and rather more generally than Mr. 

 Galton has done "to tax the bequests of each generation," as he 

 expresses it :* 



|==-iA f^=-7/?, Ss=-7A *>... (x), 



-K'OO -K'oo -ttoo 



where 7 and (3 are two constants ; we then find : 



* * Natural Inheritance,' p. 135. 



