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THE AMERICAN NATURALIST [Vol. LIT 



Let 



x = exchange ratio of one pair, 

 v = exchange ratio of other pair 

 (so selected that x^y). 

 Then for the cross-over ratio (C) of the two pairs, the 

 formula is 



' * C = x + y-2y. 



An example or two will make the use of this formula 

 clear. Suppose that the exchange ratio of pair A-a is 

 %; of B-b it is %. Then 



C = % 4- 3 /t - 2( 2 / 5 . 3 / 7 ) = 17 / 3 5 = .486 

 Again, let 



x = .31, y = .34 

 C = .31 + .34 - 2 ( .31 X .34) = .439 



(It is customary to express the results as percentages; 

 thus the last example would give a cross-over ratio of 

 43.9 per cent. For our purposes it is more convenient to 

 leave them as decimals.) 



Now this formula has certain characteristics and limita- 

 tions that allow us to bring the theory on which it is based 

 to a test. The theory is that each pair has its character- 

 istic exchange ratio; if that be the case, this formula 

 holds. 



We shall set forth certain of the important relations 

 between cross-over ratio and exchange ratios, revealed 

 by this formula; then show how these provide a test for 

 the theory which the formula expresses. To aid in the 

 comprehension of these relations, we give a table showing 

 all cross-over ratios for two pairs of characters, resulting 

 from the combinations of exchange ratios varying by 

 tenths from (no exchange) to 1 (all exchange). The 

 table illustrates all the relations to be deduced from the 

 formula. 



