318 ^ Prof. Karl Pearson. 



shall, on the hypothesis of the normal law holding for frequency of 

 offspring, have 



where y is the deviation of some character based upon both jDarental 

 organs from the value which gives the maximum fertility, and s is its 

 standard deviation. 



Thus, c and a denoting constants, 



y = /{nil + xi, riio + q:-^ 



= Co + CiXi + CoXo + higher terms in Xi and X2 ; 



hence y- = ao + aiXi + a2Xo + a^xi- + 2(^X1X2 + a5o:2', 



if we neglect higher powers of .Ti and X2. This will, as a rule, be 

 justified if Xj and 0:2 are small as compared with m-^ and 1112. 

 We conclude that : 



n no expt. ^—^2 (^^<j + + + ^-s-'-'i" + 2«i'''i'''2 + «3f'2-)J • 



If we multiply this by z, we have the distribution of parents and off- 

 spring, allowing for a varying fertility. Let this be z\ then it will be 

 at once obvious that 



z' = const. X expt. [ - (quadratic expression in 'Xi^XoyCs)]. 



Hence if we integrate for Xi and xo, so as to get the distribution of the 

 offspring, we find it again given by a normal curve, i.e., a curve sym- 

 metrical about its mode. Tims a p'ogressive change, hut no differentiation, 

 can he produced by reproductive selection. 



This is the proof, of which I merely stated the result in my " Note 

 on Reproductive Selection" of 1895."^ The same view was again 

 expressed in my memoir on " Genetic (Reproductive) Selection " of 

 1898.t While reproductive selection is invaluable as an aid to 

 natural selection, alone it can only progressively modify not differen- 

 tiate a race. For such differentiation we should have to suppose some 

 much more elaborate relation between fertility and the complex of 

 parental organs than is indicated by a normal chance distribution. 



3. In order to show what would occur supposing fertility reached a 

 maximum with homogamous unions, I do not simply take y to be the 

 difference of the parental organs, for it is quite conceivable that the 

 organs may be sexual characters, and differ not only in magnitude but 

 even qualitatively. I accordingly suppose the fertility to be a maximum 



* 'Eoy. Soc. Proc.,' toI. 59, p. 303. 

 t ' Phil. Trans.,' A, vol. 192, p. 314. 



