82 PROFESSOR K. PEARSON OX A GENERALISED THEORY OF ALTERNATIVE 



mothers of q 1 allogenic units in a population with (n 1) couplets in their 

 constitution 



_ _!-! v ,, v r Q-2(-l)-/>-7-l 



X C-i,j>, o A O,,_ 1,2-1,0 



Hence 



,-1=1-1 



S { X G.-1,!,,,-.! X C R ,_!_,+!,.} = X C M _ lij))0 X C H _i, ,_,,.. 



Now 



4 x 3 2-p-j xf= -i" X f.,,,, . X (,,, 3 2 " 

 Thus 



fCn, r ,o X 0,,,,., 



or, if'/' denote the above series, we have 



v _ // (n - />) , 

 ^ ii. 



This lends us to 



4 f/ (' 1>) 



- 



0,0 



This is a most remarkable result, for it shows that the regression surface is not a 

 plane but a hyperboloid. Let us measure all the quantities in deviations from the 

 mean of the general population, i.e., put m /lt/ m' fr/ -f i", 1> = p' + i, 5 = </ + i"-' 

 We find 



This foi-mula reconciles at once the Mendelian and C4altonian positions. When the 

 number of couplets is large, parents having a number of allogenic couplets comparable 

 with n are vanishingly small in number. The standard deviation a- of the population 



is \/'.\nj 4- (see p. 57). 

 Hence we may write 



But // can only once per thousand cases be as big as 3cr, and accordingly when n is 

 large, both terms of the product will be small. In this case the surface becomes 

 practically planar, and we have the Galtonian result of 1885,* that the offspring from 

 the Galtonian "midparent" are one-third nearer the general mean of the population. 

 On the other hand, when n is small we see that for midparents not differing too 

 widely from the population mean, Galtonian regression of the value ^ holds, but that 



* ' Natural Inheritance,' p. 97. 



