SYSTEMS OF MATING 1 1 





Since 



V n = A»V , 



P n = A-*V, = A^A^Vo. 



Since V = /1\ , p n = Mi" + b s2*2 n + ■■■ + U" 



where b si is the element in row s, column i of A 1 . 

 For all inbreeding systems, 



X s _ j n = for n of any appreciable size, 



p n ~ 1 + Xj"*,!. 



From equation (9), it follows that the number of generations n required to exceed the 

 frequency a of incrosses can be approximated by setting 



X! n £ sl + 1 = a. 

 Then 



n log Xj = log 6 sl - log (1 - a), 

 and 



log6 sl - log(l - a) 



logXj 



(10) 



Using the nearest integral value of n, we can calculate the true p n from equation 

 (6). The value of n for which equation (9) is satisfied should be within one or two 

 generations of the approximation derived from (10) in all cases. 



In summary, then, the frequencies of the various mating types can be obtained in 

 any generation, if the frequencies in the preceding generation are known, by using 

 equation (2). Equation (3) allows the frequencies of mating types to be derived in 

 some advanced generation n from the frequencies in the initial generation. This, 

 however, involves raising the G matrix to the power n, which may be a difficult task. 

 This difficulty may be avoided by deriving a diagonal matrix from the G matrix; it is 

 easy to raise a diagonal matrix to any power. Doing so requires us to obtain the 

 roots of the original G matrix and to derive a new matrix, A, which is used to trans- 



