266 



Prof. K. Pearson. On a Criterion which may [Mar. 4, 



or 



Pfc 



Similarly, 



Pmc 



Here we have supposed that although the fathers of ''father's 

 offspring " and of " mother's offspring " will, when weighted with their 

 offspring, be unequal in number (i.e. % and n 2 ), yet their variabilities 

 are the same, and similarly for the mothers. This is equivalent to 

 supposing no correlation between the dominant effect of a parent and 

 his or her deviation from type. Otherwise we cannot put ay in the Si 

 sum the same as ay in the So sum. 



We are now able to write down the general regression equation of 

 bi-parental inheritance, i.e., 



? m - Pfc ~ Pfm Pmc <r c , Pm c ~ Pfm Pfc °" c 



Zp - fib JO + 2 ■ !Ji 



f - Pmf °~f f - p-fm Vm 



where z p is the probable value of the character in offspring of parents 

 of characters x and y. Hence, if we remember that Tf. m = p fm , we have 

 on substituting from (ii) and (iii) : — 



bp = m + % + * S» Ej/) x+( n S^- R 1M + * Zh E 2 ,„ ) y... (iv), 



\n o-f n. cry J \n cr m , n a- m J 



where Eiy, E im are the bi-parental co-efficients {ry-r m f ri m )/(l - r m 2 f) 

 and (r lm - r m fr^)/(l - r m 2 f), of the "father's offspring" and Eoy, 

 E 2m similar quantities for the " mother's offspring." 



Now, fixing our attention for a moment on " father's offspring," 

 we should expect parents of characters x and y to produce an array of 

 father's offspring with a mean : 



m = mi + Ei/ x + E lm °^ y (v), 



ay o- m 



and with a standard deviation s 1 given by 



s i 2 = °"ci 2 (1 - r v 2 - r lm 2 - r mf 2 + 2r lf r lm r rnf )/(l - r m 2 /) . . . (vi). 



Similarly the arrays of " mother's offspring" for parents of the same 

 characters would have a mean : 



/x 2 = m 2 + K 2 y x + R 2 m— y (vii), 



ay a m 

 and a standard deviation s 2 given by 



s 2 2 = o- C2 2 (1 - r 2 / 2 - r 2m 2 - r m / 2 + 2r 2 fr 2 mrmfW ~ r mf 2 ) - • • (vm). 



n cr c 



n <r c 



(ii). 



+ 



n cr c 



(iii; 



