No. 615] THEORIES OF CROSSING OVER 



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chromosomes, are now following the same one. The 

 cross-overs in II, above, are those indicated by the -f 

 sign ; there are five of these. The ratio of the number of 

 these new combinations (5) to the total number of germ 

 cells (12) is the cross-over ratio; in this case the cross- 

 over ratio 'is % 2 , or .417. 



Examination of this case will illustrate an important 

 fact. A cross-over is produced only when one of the two 

 pairs exchanges while the other does not. In the last pair 

 to the right, in the example given above, the members of 

 both pairs exchange places, but this does not give a cross- 

 over—since A and B are still together, as they were 

 before the double exchange. 



Now if the number of exchanges for each pair of cells 

 is different from that in the example given above, the 

 resulting cross-over ratio will be different. By suppos- 

 ing each pair of factors, xY-a, B-b, C-c, D-d, etc., to have 

 its own characteristically diverse frequence of inter- 

 change of its members, all sorts of cross-over ratios could 

 be obtained, varying from to 1; that is, from no cross- 

 overs to all cross-overs. The question in which we are 

 interested is, could the observed cross-over ratios in such 

 an organism as Drosophila be accounted for in this way \ 



It is to be noted that the problem as we take it up is 

 independent of the nature of the forces that hold A and a 

 (and the other factors) in their places, and that permit 

 them to exchange in a certain proportion of cases. These 

 forces may be utterly heterogeneous in the different 

 cases; they may turn out to be of any kind whatever, so 

 far as this examination goes. We ask merely whether, 

 if the forces, whatever they are, give a constant average 

 proportion of interchanges characteristic for each pair, 

 they can yield the cross-over ratios actually observed. 



II 



It is evident that on this theory there are two kinds of 

 ratios to be dealt with: the ratio of the number of inter- 

 changes of A and a (characteristic for each pair), and the 

 ratio of the number of cross-overs, between two pairs A-a 



