SYSTEMS OF MATING 49 



Bartlett and Haldane 56 and Green and Doolittle arrived at a quartic equation in 

 terms of ju = 2X from the matrix of mating types in this case. On substituting X = fi/2, 

 the above cubic is found to be an exact divisor, so that again the conclusions on rate 

 of decrease of heterozygosis are in exact agreement. 



To hold together a line considerably larger than under brother-sister mating, 

 attention is directed to the limiting case of one male (a = 0) and exclusive mating with 

 half sisters (6 = 1) and also equal crossing over in both sexes. The exact recurrence 

 equation for relative heterozygosis is 



P = (3/2) (1 - l)P' - (1/16)[5(1 - I) 2 - 13P]P" - (3/16)(l - I) (I - 2l)P'" 



With random assortment, 



c = 1/2, I = 1/2: 



P = (3/4) P' + (1/8)/"' as given previously. 1443 The limiting rate with any value of 

 c is again given by the largest root of the characteristic equation 



X 3 - (3X 2 /2)(1 - /) + (X/16)[5(l - I) 2 - 13/ 2 ] + (3/16)(l - /)(1 - 21) = 0. 

 The values of X for several values of c are given in table 1 2 in comparison with those 



Table 12 

 The limiting ratio of heterozygosis in successive generations {PIP') UNDER various 



SYSTEMS OF MATING AND AMOUNTS OF RECOMBINATION 



t Matings of type dd<$ x Z)rf$ give the same results as those above for the reciprocal type. 

 Equal crossing over in both sexes is assumed. 



under brother-sister mating (matings Dd x Dd) and various other cases. Whereas 

 mating one male with many half-sisters is only slightly more than half as effective as 

 full brother-sister mating for loci with random assortment, it approaches equal effective- 

 ness for loci closely linked with the gene of interest. 



Green and Doolittle (and also Bartlett and Haldane) consider the case of brother- 

 sister matings of the type Dd x dd at a locus of interest. This can also be dealt with 

 by path analysis. First the more general case of a population of N m males, N f females 

 will be considered. All males are assumed to be Dd and all females dd, so that only 

 recombination in spermatogenesis (here c) is involved. Letting S and s in subscripts 

 represent spermatozoa carrying D and d respectively, there are six kinds of correlations 

 to consider (r So , r so , r ss , r Ss , r ss , r 00 ). There is no difficulty constructing gametic 

 diagrams in each case from which the formula for the correlations can be written from 



