304 Combination oj Linkage Values 



Therefore 



X — I z , since x and y vanish together, and py <1. 



J Q 1— pt 



Hence if p were constant we should have 



— 1 1 — e~^^ 

 «=---iog«(i-i>yXor2/ = — -— (1). 



Since however p varies between and 2, the values of x 'must lie 



— 1 1 — e~^^ 



between ^.and -Q-loge(l — 2y), those of y between x and — - — ; the 



equation 



3/ = ^ (2) 



being nearly accurate for small values of x and y, the equation 



X _ g-2a; _ 1 



y= —2 — >orx=^\oge(l-2y) (3) 



for large values of x and y, as is obvious, since for large values of x, 

 y approaches the value '5 asymptotically. The equation (2) corresponds 

 to Morgan's summation formula m + n, the equation (3) to Trow's 

 formula m-{-n — 2mn. 



The equation (3) may be deduced more directly as follows for a 

 perfectly flexible chromosome : 



Let a length x of the chromosome be considered as divided into 

 a very large number iV of small equal portions. Then the chance of a 



X 



cross-over in each of these is approximately ^. Hence the chance 

 (5f a cross-over in t of these segments and no more is 



When N becomes infinite the limiting value of this expression, i.e. 

 the probability of exactly t and no more cross-overs in a length x, is 



"^--tT W- 



Hence the value of y for a given value of x is the sum of the 

 probabilities of all odd numbers of cross-overs. 



.-. 3/ = Ci-Hc3-f C5-i-C7-[- 



^ fx x^ x^ x' \ 



= ^ (r! + 3-! + 5-!+7-!+ ) 



= e~^ sinh x 



= —^ (3). 



