THE AMERICAN NATURALIST 



[Vol. L 



peared, by examination of Sturtevant's extensive back- 

 crosses, especially of those involving three pairs of fac- 

 tors at once. As the results did not conform to the for- 

 mula, it was not published, but as Trow has since raised 

 this question publicly and the adherents of the reduplica- 

 tion hypothesis are still discussing it, it may not be out of 

 place to have given an analysis of it here, and to recall the 

 fact that it had already been tried and rejected. Besides, 

 as will appear below, a discussion of the relations which 

 would exist if crossings-over were independent of one 

 another is a necessary preliminary for a treatment of the 

 relations which do exist between linkage values. 



The results showed that double crossing-over does not, 

 as a rule, occur <as frequently as would be expected if, 

 as the above formulae assumed, it were purely a matter 

 of chance whether or not two cross-overs happen coinci- 

 dently. In a sense, then, the occurrence of one crossing- 

 over interferes with the coincident occurrence of another 

 crossing-over in the same pair of chromosomes, and I 

 have accordingly termed this phenomenon "interfer- 

 ence." The amount of interference is determined by 

 comparing the actual per cent, of double cross oxers with 

 the per cent, expected if crossings-over were independ- 

 ent, i. e., if they had a purely chance distribution with 

 reference to each other. Xow, the per cent, which would 

 occur on the latter expectation lias already been given by 

 formula 1 as per cent. AB X per cent. BC. If, then, the 

 observed per cent, of double cross-overs were divided by 

 per cent. AB X per cent. BC, we would obtain a fraction 

 showing what proportion of the coincidences which would 

 have happened on pure chance really took place. This 

 ratio of observed double cross-overs to the chance expec- 

 tation appears to me to furnish the most useful measure 



cent, and it mav be called the relative coincidence, or 

 simply -coincidence." If the -coincidence" is low, this 

 means that there has been much interference, for most of 

 the double cross-overs expected on chance were prevented 

 from appearing; conversely, if coincidence is high, the 



