SYSTEMS OF MATING 21 



Table 6 

 Matrices for the backcross system 



To obtain the frequencies of the various mating types in generation n, P l5 the 

 vector of frequencies in generation 1, is first obtained from equation (2). Pj will be 

 of the form 



Pi = (A o tx). 



We then form the vector P* = (p x /j, transform to V*, obtain V* and convert 

 to P*. These vectors are the same as the vectors operated with earlier, in equations (2) 

 through (6) ; the * notation is introduced merely as a reminder that we are operating 

 with only two of the original four mating types. 

 Note that 



the matrix A being raised to the power (n — 1 ) , since one generation has been accounted 

 for already by calculating P x . P* gives only the frequencies p n of incrosses and t n of 

 backcrosses of the second type; the frequencies of the other two types of matings possible 

 in G are, as has been demonstrated, zero in G n . The frequencies p n and t n calculated 

 by this method agree perfectly with those calculated by repetitive application of formula 



(2). 



The number of generations required to obtain a given percentage of incrosses can 

 also be calculated for the backcross system. When c = 1/2, 5 generations are required 

 to obtain a frequency of incrosses greater than 95 per cent and 8 to obtain a frequency 

 greater than 99 per cent. When c = 1/10, 29 generations are required to obtain 95 

 per cent incrosses, and 44 to obtain 99 per cent. It should be noted that if the gene of 

 interest is recessive, test generations will be required, thus increasing the time required 

 to obtain a given percentage of incross matings. 



