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THE AMERICAN NATURALIST [Vol. LII 



and B-b; the latter of these ratios depends on the former. 

 We shall call the former the exchange ratio; the latter is 

 commonly known as the cross-over ratio, which we will 

 designate by the letter C. 



The exchange ratio signifies the ratio of the number of 

 exchanges between A and a to the total number of germ 

 cells: 



Exchanges 

 Exchange Ratio = ^otal Number 



The cross-over ratio (C) signifies, of course (following 

 Morgan and the general usage), the ratio of the number 

 of cross-overs to the total number of germ cells or 



Total Number 



Goldschmidt (1917, page 90) has given a formula for 

 the cross-over ratio resulting from any two exchange 

 ratios, and has computed the resulting cross-over ratios 

 from certain assumed exchange ratios. We shall give the 

 formula a simpler expression than Goldschmidt has done ; 

 one that will enable us to determine its properties and 

 limits of performance. 



In cross-over ratios we deal with two pairs of char- 

 acters, which we may designate A-a and B-b. Let x 

 signify the exchange ratio for one of the pairs ; and let y 

 signify the exchange ratio for the other pair. Thus, if 

 A and a interchange in one third of all cases, this pair's 

 exchange ratio x will be one third (or .33y 3 ) ; while if B 

 and b interchange in two fifths of all cases, its ratio, y, 

 will be two fifths (or .4). For convenience we will always 

 choose x and y in such a way that if there is any differ- 

 ence, x will designate the smaller ratio. That is, x will 

 always be equal to or less than y. 



Now, suppose that originally the first chromosome of 

 the pairs bears the two factors A and B, the second a and 

 b (as in I, above). After crossing over in the proportion 



