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THE AMERICAN NATURALIST [Vol. L 



to the sum of the number of crossings-over between A 

 and B, and between B and C, minus all those crossings- 

 over contained in the cases where coincidence occurred, 

 and in which A and C, therefore, failed to separate,— i. e., 

 minus twice the number of cases of double crossing-over. 

 Hence, if formula 1 is correct, then it must also be true 

 that per cent. AC = per cent. AB +per cent. BC — 2 

 (per cent. AB X per cent. BC) (formula 2). This for- 

 mula was originally expressed not only in the above 

 terms, where the "per cent, of separations" (i. e., ratio 

 of separations to the total number of cases) is used as the 

 index of separation frequency, but also in terms of the 

 so-called "gametic ratio" — the ratio of cases of non- 

 separation to those of separation — for this was the way 

 of indicating degree of linkage then used by all investi- 

 gators of the subject. The latter index gives much more 

 complicated formulas, however, and so it was pointed out 

 at the same time that per cent, of separations would 

 afford a much more useful measure of linkage. 



Later, Trow also worked out and published the same 

 formula (no. 2)— in terms of the "gametic ratio"— and it 

 is generally known as "Trow's special hypothesis" (17). 

 But on the reduplication hypothesis held by Trow, and 

 by the other English geneticists who do not accept the 

 chromosome explanation, the formula would be supposed 

 to result, not from the fact that crossing-over between A 

 and B was independent of that between B and C, but 

 from the fact that "reduplications" AB and BC were 

 independent, not being disturbed by any "primary redu- 

 plication" AC. Adherents of the reduplication hypothe- 

 sis have been much concerned as to whether or not their 

 results confirmed the assumptions made in Trow's for- 

 mula, and have in one or two instances calculated that 

 they did. Let us examine for a moment the requisites 

 for proving such a conclusion. As above shown, the 

 whole matter turns on the frequency of coincidence of 

 separations AB and BC (i. e., on the frequency of 

 "double crossing-over") and the question can be settled 

 by determining directly the amount of this coincidence. 



