30 GENETIC STOCKS AND BREEDING METHODS 



for m = 0, 1 , 2, 3, The probability of heterozygosity is the same as the probability 



of backcrosses among the crosses, i.e., h m = r m . This probability takes the successive 

 values 



0, 3/8, 3/32, 3/128, 3/512,..., 



for m = 0, 1, 2, It may be seen at once that when m > 0, 



h m+1 = (l/4)*m, 



when <7 = 1 and c — 1/2. 



The characteristic root of the determinant formed from the q and r rows and 

 columns of the cycle matrix (table 9) is 



Xt = (1 - c)\ 



Pm + l 

 ?m + l 



r m + l 



Table 9 

 Cycle matrix for the cross-backcross-intercross system 



A. 



(i -cY - x (i/2)(i - cy 



2c{l - c) 2 {2 - c) c(\ - c) 2 (2 - c) - X 



X = (1 - c) 2 , 



= 



This is the factor of proportionality between successive values of q and r, 



<7m + l = ^-<7m> 

 r m + 1 = Ar m- 



It also gives the relationship between successive values of A, the probability for hetero- 

 zygosity, since h m = r m . When c = 1/2 



h m + 1 = (1 -c)*h m = (1/4)A B> m>0 



confirming the result of the preceding paragraph. 



Again, the cycle matrix may be obtained more easily by multiplying the generation 

 matrices in reverse order to that in which they appear. 



The cross generation matrix is 



Gi = /l 1/2 



(I l/2\ 

 \0 1 1/2J* 



