26 GENETIC STOCKS AND BREEDING METHODS 



This result shows that one cycle of the cross-intercross system is genetically equivalent 

 to one generation of the backcross system. 



The cycle matrix for the cross-intercross system can be obtained by directly reason- 

 ing what proportion of each type of mating a given type will yield in the subsequent 

 cycle, two generations later. It is possible, however, to avoid the confusion attendant 

 on the attempt to reason through two generations. A generation matrix can be con- 

 structed for each generation in the cycle. By multiplying these matrices together in 

 order opposite to that in which they occur, the cycle matrix is obtained. Thus, for a 

 cycle of k generations, 



C = GfcGfc.!- . .G l5 



where C is the cycle matrix, and G; is the generation matrix for generation i in the 

 cycle. 



In the case of the cross-intercross system, the matrix for cross generations is 



while that for intercross generations is 



G 2 = 



and 



C = GaGi. 



From the matrices given in tables 7 and 8, P m can be calculated for any m, given 

 P . Also the number of cycles required to obtain any given percentage of incross 

 matings can be calculated. For both calculations, we must keep in mind that one cycle 

 is the equivalent of two generations. 



Table 8 

 The A, A, and A -1 matrices for the cross-intercross system 



