No. 615] THEORIES OF CROSSING OVER 



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(l-x) + (l-y)_2(l-x)(l-y) = x + y--2xy s C 

 So that in this case C> 1 — x and C> 1 — y. 



Thus the fraction C is nearer to 1 than the fraction 

 1 — x. If therefore we subtract the fraction C from 1, 

 it will leave a smaller number than if we subtract the 

 smaller fraction 1 — x from 1. That is : 1 — C < x. 



And in the same way it can be shown that 1 — C < v. 

 Only in the limiting case that y = l does x = 1 — C. 



The general principle in this section can be expressed 

 as follows: 



When the cross-over ratio is less than y 2 , the two ex- 

 change ratios, x and y, either both differ from by less 

 than the cross-over ratio, or both differ from 1 by less 

 than the cross-over ratio. 



This relation is well seen in the table. For example, 

 for the cross-over ratio .38 the two exchange ratios are 

 either .3 and .2 (both less than .38), or they are .8 and .7 

 (both greater than .62, the complement of .38). 



8. Conversely to 7 : 



If both exchange ratios, x and y, are less than y 2 , both 

 are equal to or less than the cross-over ratio. 



If both exchange ratios, x and y, are greater than V->. 

 both are equal to or greater than the complement (1 — C) 

 of the cross-over ratio. 



9. When the cross-over ratio C is above V2, one of the 

 exchange ratios (x) is equal to or less than the comple- 

 ment of the cross-over ratio (1 — C), while the other (y) 

 is equal to or more than the cross-over ratio (C). 



Or otherwise expressed : 



When the cross-over ratio is above V2, one of the ex- 

 change ratios (x) differs from 1 by an amount equal to 

 or more than the cross-over ratio, while the other (y) 

 differs from by an amount equal to or more than the 

 cross-over ratio. That is, when C > V 2 , 1 — x m C, y < C, 

 or x 1 - C, y C. 



This can be proved by methods similar to those em- 

 ployed in 7. 



10. Conversely to 9 : 



