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THE AMERICAN NATURALIST [Vol.LII 



(in which x and y arc proper fractions). 



1. Two exchange ratios, x and y, give the same cross- 

 over ratio (C) as do their complements, 1 — x and 1 — y. 



For 



x + y - 2xy = (1 - x) + (1 - j) - 2(1 - x) (1 - y), 

 as will be seen by performing the operations indicated in 

 the second member of the equation. But this second 

 member is the value of C for exchange ratios 1 — x and 



For example, the two exchange ratios .2 and .3 give the 

 same cross-over ratio as do the two exchange ratios .8 

 and .7 ; for both cases C = .38. This relation is seen in 

 the symmetrical constitution of the table; the cross-over 

 ratio resulting from .1 and .2 is the same as that from 

 .9 and .8; the cross-over ratio resulting from exchange 

 ratios .4 and .7 is the same as that resulting from .6 and 

 .3, etc. The rule holds equally for values not found in 

 the table; thus the cross-over ratio resulting from .011 



