SYSTEMS OF MATING 51 



Under random assortment [c = (1/2), / = (1/2)], this reduces to 



X 2 - (X/2)[l + (a + b)/2] - 1/4[1 - (a + b)/2] = 0, 



which agrees with the exact recurrence formula given previously, 1442 



P = [1 - (N m + N f )/4N m N f ]P' + [(N m + N f )/8N m N f ]P". 



There is considerable simplification in the case of one male (a = 0) . The limiting 

 case of an indefinitely large number of females (b = 1) will be considered. 



X 4 - (X 3 /4)(5 - Ac) - (X 2 /4)[l - 2(1 - c)l] 



+ (X/16)(l - 2c) (7 - 10c + 8c 2 ) + (1/16)(1 - 2c)(l - I) = 0. 



The case in which all males are dd and all females Dd is given by interpreting a as 

 (N f — l)IN f , b as (JV m — l)/N m , c as recombination rate in oogenesis, and / = 2c(l — c ) 

 in this sense. The case of one dd male and indefinitely many Dd females is more 

 complicated than the converse. 



X 5 - X 4 (l - c){2 - c) - (X 3 /4)[c + 2(1 - c){\ - I) - 4(1 - c) 3 ] 



+ (X 2 /16)(l - 2c) [2 (3 - 21) + 4(1 - c)[c + 2(1 - c) 2 ] 

 + (X/16)(l - 2c) [2(1 - I) - 2(1 - c) 2 (3 - Ac) + 2cl] 



- 1/8(1 - 2/)(l - c) 2 = 0. 



Both of these reduce to the equation 



X 2 - 3/4X - 1/8 = 0, 

 X = (1/8) [3 + Vl7] = 0.8904, 



as shown previously, if c = (1/2). 



In the case of brother-sister mating (a = 0, b = 0), 



X 4 - X 3 (l -c) - (X 2 /4)[> + 2(1 - c){\ - I)] 



+ (X/8)(l - 2c) (3 - 21) + 0/8)0 - 2c) (1 - I) = 0. 



The equation obtained by substituting X = ju/2 is again an exact divisor of the 

 equation (quintic) derived by Bartlett and Haldane 56 and by Green and Doolittle 

 from the matrix of mating types. The exact recurrence equation for P in terms of 

 preceding P's is given as usual by replacing the highest power of X by P, the next 

 power by P', and so forth. The results are again in exact agreement. 



The limiting values of the ratio of heterozygosis in successive generations, X, are 

 shown for several values of c in table 1 2 in the cases of male Dd x female dd (or the 

 reciprocal), male Dd and indefinitely many females dd and the converse. They are 

 less efficient in reducing heterozygosis than if both parents are always Dd. The last 

 is the least effective. 



