294 Prohable Errors of Calculated Linkage Values 



When T is the most probable value this probability is a maximum, 

 and hence the expression in brackets is a minimum. The condition for 

 this is 



(l-«^a+2(2-f^a)^a + (2+g(l-W 



Hence, putting ^2 + ^3 = ^i + ^4, 



^-^« + ^- 2VUg ^^)' 



while P=^/T. 



As our value of ta we may take t^ in the case of repulsion, ^ (^1 + ^4) 

 in the case of coupling. If greater accuracy is desired the value of T 

 thus obtained should be substituted for ta in equation (6), and a more 

 accurate value thus obtained. This proceeding is however rarely worth 

 while. 



The same value for Tis reached more directly by Bridges' method (5), 

 where T is calculated from the coefficient of association 



( AB) x(a h)-(Ab)x(aB ) _ 4>T-1 , . 

 (AB)x (ah) + {Ah) X {aB) ~ 21^+1 ' 



as may readily be seen on substituting the value ^ (2 + fj) for {AB), and 



so on. It is doubtful however if this method is any shorter than that 

 given above, unless a four figure table of values of P in terms of the 

 coefficient of association has been calculated in advance. 



To take a concrete example, Punnett(4) working with the coupled 

 factors B and L in sweet peas obtained the F^ zygotic series 



48315i:, 3905^, 3936X, 13386^. 



Here n = 6952, 



4xl338 _ 

 ^'~ 6952 ~ ^^^^' 

 ^^ = i(^^ + ^,) = -77475. 



^ 3 X -77475 X -779 6 + 2'774 75 x -7699 _ ^^.^ 

 •'• ^ == 2 + 4 X -77475 -'^^^^' 



If we substitute "7743 for ta in equation (6) we obtain no change 

 in the first four decimal places. 



