SYSTEMS OF MATING 53 



random assortment (c = 1/2), r s0 , and r so become the same and so do r 00 , r 0o , and r 

 The characteristic equation reduces to: 



1/2) - X 1/2 



1/2 (1/4)4 - X (1/4)4 

 a -X 



= 



X 3 - X 2 [(2 + 4)/4] - (X/8)(2 - b + 2ab) + (\/8)ab = 0. 

 This agrees with the exact recurrence equation given previously for P as follows : 1426 



(N f +l) _ (N f -l){N m -l) _ 



8N f { } 8N m N f { J ' 



In the case of a single male (a — 0, r ss = 1), the last two rows and columns in the 

 P matrix drop out, and the characteristic equation becomes: 



X 4 - X 3 (l - c)[\ + (1 - c)b] - X 2 (l - c)[(l - /) - (1 - c) 2 b] 



+ X(l - 2c)[(\ - I) + (1 - c) 3 b] - (1 - 2c) 2 (\ - c) 2 b = 0. 



The limiting ratios of heterozygosis in successive generations are given for various 

 values of c in table 12. Fixation of sex-linked genes under forced heterozygosis of 

 females is a very slow process. 



With brother-sister mating (both a = 0, b — 0), the characteristic equation 

 reduces to the form given by Bartlett and Haldane. 56 



X 3 - X 2 (l - c) - X{\ - c){\ - I) + (1 - 2c) (1 -0=0. 



Fixation is again interfered with by forced heterozygosis of a gene of interest much 

 more than in the case of autosomal genes. 



Details of the various important systems of the backcross type discussed by Green 

 and Doolittle will not be elaborated. The gametic diagram can easily be constructed 

 and coefficients can be assigned to each path under the necessary assumption of a 

 hypothetical, symmetrical, total population in which gene frequencies remain constant. 

 The recurrence formulae for P present no difficulty. Under simple backcrossing 

 P = (1 — c)P'. Under the cross-intercross system, the result is the same except that 

 P' here refers to the uniting gametes of the preceding intercross, two generations back. 

 Under the cross-backcross-intercross system, P = (1 — c) 2 P' in a cycle of three genera- 

 tions from P' to P. In these cases the proportions of incrosses in the crosses to the iso- 

 genic lines are readily found by path analysis since they involve only two varying 

 gametes. The results all agree with those of Green and Doolittle. 



Dr. Pilgrim: There is no question concerning the logic of the breeding methods 

 discussed, but the mathematics is based on the assumption of randomness, and these 

 conclusions are only valid in the absence of selection. In the case of selection for the 

 homozygote, homozygosity may be achieved in much less than the predicted time; 

 and, in the case of high selective value for the heterozygotes, it is conceivable that 



