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THE AMEBIC AN NATURALIST [Vol. LII 



change ratio is less than V 2 , the other greater than Y 2 . 

 That is : If C> V 2 , then x < V 2 , J > V2. This also was 

 proved under 3. 



6. Conversely to 5, when one exchange ratio is less than 

 V 2 , the other greater than V 2 , the cross-over ratio is 

 greater than %. That is : If x < V 2 , y > %, then C > %. 

 This also was proved nnder 3. 



All these relations are evident in the table. 



7. When the cross-over ratio is less than V 2 , the two 

 exchange ratios are either both equal to or less than the 

 cross-over ratio ; or both equal to or more than the com- 

 plement of the cross-over ratio. They can not have any 

 value lying between the cross-over ratio and its comple- 

 ment. That is : When C < V 2 , either x and y each ^ C or 

 x and y each m 1 — C. 



This is an extremely important principle, on which the 

 final test of the theory depends. It is proved as follows : 



In 3 we saw that if the cross-over ratio is less than y 2 , 

 either x and y are both less than V 2 ; or both of them are 

 greater than V 2 . 



{a) Let us take first the case where x and y are each 

 less than V 2 . In this case, in the formula C = x -f y — 

 2xy, the quantity 2xy is smaller than x, and smaller than 

 y. For since x is less than y 2 , 2x is less than 1, whence 

 it follows that 2xy is less than y ; and the same reasoning- 

 shows that 2xy is likewise smaller than x. Hence the 

 formula for C subtracts from the sum of x and y a quan- 

 tity smaller than y ; it therefore leaves a quantity larger 

 than x; and the same reasoning shows that it leaves a 

 quantity larger than y. Only in the limiting case that 

 x = does y = C. 



(b) Take next the other possible case, in which x and 

 y are both greater than V 2 . In this case 1 — x and 1 — y 

 are both less than V 2 . Thence it follows (by the reason- 

 ing just employed) that 



(l-x) + (l-y)-2(l-x)(l-y) 

 is greater than 1 — x and greater than 1 — y. But, as 

 was seen in (1), 



