300 Combination of Linkage Values 



lowing proportions if it were of composition ABG.ahc, and similarly 

 for other compositions : 



Actually the last class has been shown to be in defect in many 

 cases. This has been thought to be due to the loops formed by the 

 chromosomes during synapsis having a modal length (3). If this were 

 so, we should expect to find an excessive number of double cross-overs 

 when the distance between the loci of A and G was equal to twice the 

 modal distance between points of crossing over. This phenomenon 

 has however not been recorded. The shortage of double cross-overs 

 can equally well be explained by the mere rigidity of the chromo- 

 somes, which makes sharp bending difficult. In the sex chromosome 

 of Drosophila the ratio ^ of observed to calculated numbers of double 

 cross-overs is '58 : 1 for eosin (white), vermilion, and sable (4) (where 

 m + n = -406), and -21:1 for vermilion, sable, and bar (3) (where 

 m + n = -239). 



If the calculated number mn of double cross-overs occurred, the 

 cross-over value for A and C would be equal to the total number of 

 single cross-overs, i.e. to m (1 — n) -]- (1 — m) n, or m 4- n - 2mn. 



If double cross-overs were impossible, but the full numbers of single 

 cross-overs occurred, as would happen if the chromosomes were straight 

 rigid rods, the cross-over value for A and G would obviously he m-\-n 

 (Morgan and Bridges' formula). 



Finally if double cross-overs were impossible, and in every case 

 where one should have occurred according to the calculation above, 

 a single cross-over took its place, the cross-over value for A and G 

 would be m -I- n — 2mn + mn, or m +7i — mn. This case might be 

 approximately realised if the chromosome could not form loops shorter 

 than some definite length. 



Hence the cross-ov«r values for A and G should be approximately 

 mi-n when m and n are small, m + n— %nn when their sum is large, 

 and m-\-n — mn for intermediate values. 



Table I contains the observed values (5) for all triads of factors in 

 the sex chromosome of Drosophila for which each of the cross-over 

 values exceeds '1 (10 7o)- The first column gives the three factors 

 con6erned in each case ; the second and third columns give the cross- 



^ Called by Muller the "coincidence." 



