A. H. Tkow 295 



therefore it would be an error to try and .shew the possible mathematical 



relations which may exist between them. The frequency of a double 



cross-over can perhaps be calculated, like .i double event, fi-om the 



fre(|uency of the corresponding single cross-overs, but it is exti-emeiy 



doubtful whether we are justified in assuming any mathematical 



relationship between a no cross-ovei' (whole distance) and a single 



cross-over (part distance). But the numbers of the gametes alone, 



even were such mathematically related to each other, would not suffice 



C O 

 to enable us to calculate the error of the ratio -- k— j^ due to the 



N. C. O. 



presence of double crossing over. We should require to know at least 

 the total number of the factors located in the chromosome as well. 



But there are greater difficulties involved in the hypothesis. We 

 have seen that there must be as many crossing over places as there are 

 known factors, less one. By hypothesis, these factors are not recognisable 

 unless they are separable in the process of crossing over. Hence we 

 can fairly assume that multiple crossing over must be frequent ; if the 

 F-i plant is heterozygotic for the twenty-six factors A to Z, there must 

 be twenty-five crossing over places, and the chromosome would some- 

 times appear as a closely wound spiral, thus : 



etc. 



In such a scheme as this, with the maximum of crossing over, there 

 are cross-overs and non-cross-overs in abundance, but for each type of 

 cross-over, there is no corresponding non-cross-over. Crossing over is 

 absolute, i.e. 100 / , for 



Ab^Ad Az 1 C. O. 

 AB~ AD'-- AZ~'0^K~CrO.' 



Every alternate factor, too, apjjears to be absolutely correlated ; thei-e 

 is no crossing over at all in Morgan's sense. 



It can be shewn that for each type of crossing over — simple, double, 

 triple, etc. — there is a different system of ratios ; and as a number of 

 these systems must be in operation in any one experiment, whether 

 recorded or not, it is practically impossible to find the true percentage 

 of cross-overs due to a single crossing over by deducting those C. O.'s 

 and N. C. O.'s which are due to multiple crossing over. All that we 

 can say positively is, that double crossing over as compared with single 



