48 GENETIC STOCKS AND BREEDING METHODS 



It is convenient to let l s = 2c s (l - c s ), l = 2c (\ - c ), and l m = c s {\ - c ) 

 + (1 - c s )c and a = (N m - l)/JV m , b = (N f - l)jN f . The equations for r N and r, 

 can be written from the appropriate averages. A further condensation of symbolism is 

 desirable. 



u = (l - L) 



V = l m 



w= (1/4)[(1 -/ s )(l -«) + (! -/.)(1 -*)] 



* = (l/4)[/,(l - a) + / (1 - A)] 



y = (1/4)[2(1 - / m ) + (1 - /.)« + (1 - O*] 



z = (l/4)[2/ m + / s a + / A] 



F = ur' N + vr\ 



r N = wF' +yr' N + zr\ + x 

 r, = xF' + zr' N + yr\ 4- w. 



Since the sums of the coefficients are equal to 1 in all cases, the constant terms disappear 

 on substituting r — 1 — P in each case. This leads to the P matrix, the characteristic 

 equation of which follows: 



= 



X 3 — X 2 (2jy) + X[z/ 2 — z 2 — uw — vx] + [uwy + vxy — uxz — vwz] = 0. 



The largest root gives the ratio of heterozygosis in successive generations. 



In the case of brother-sister mating (N m = N f =\,a = b = 0), this reduces to 

 the following which could have been arrived at much more simply if these assumptions 

 had been made in the first place : 



X 3 - X 2 (l - l m ) - (X/4)[l - (/ s + / )(1 - 2/ m )] + (1/8) (2 - l s - l ){\ - 2/ m ) = 0. 



It is interesting to note that, if there is random assortment in one sex (l m = 1/2), the 

 equation reduces to that for brother-sister mating with random assortment in both 

 sexes. 



If there is no crossing over in one sex as in Drosophila (c s = 0, l s = 0, l m = c ), 

 then: 



X 3 - X 2 (l - O - (X/4)[l - l {\ - 2c )] + (1/8) (2 - 0(1 - 2c ) = 0. 



On substituting X = fif2, this is an exact divisor of the quartic equation arrived at by 

 Bartlett and Haldane. 56 The conclusions on the rate of decrease of heterozygosis 

 are thus in exact agreement. 



If there is equal crossing over in both sexes (c s = c = c, l s = l = l m = 2c(l — c), 

 then: 



X 3 - X 2 (l - 1) - (X/4)[l - 2/(1 - 2/)] + (1/4) (1 - /)(1 - 21) = 0. 



