260 THE AMERICAN NATURALIST [Vol.LII 



high or very low), they can not together give cross-over 

 ratios of the more intermediate values. For example (as 

 our table shows), if two factor pairs each give, with any 

 other, cross-over ratios below .10, they can not give 

 together a cross-over ratio lying anywhere between .18 

 and .84. If the two pairs each yield any cross-over ratios 

 lying below .20, they can not give together a cross-over 

 ratio lying between .32 and .68. These and many similar 

 relations, illustrated in the table, are inherent in the 

 theory we are considering, but are completely opposed to 

 what is found in nature. 



IV 



These facts completely refute any theory which holds 

 that the observed constant cross-over ratios between pairs 

 of factors are the result of constant exchange ratios be- 

 tween the two members of a given pair— exchange ratios 

 that are characteristically diverse for the different pairs 

 (such theories as that outlined by Goldschmidt, 1917). 

 The refutation is independent of the question of the 

 nature of the forces involved ; whatever the forces, if they 

 give constant average exchange ratios for each pair, the 

 results are bound to be inconsistent with the observed 

 cross-over ratios. No theory will hold that does not 

 provide for diverse relations between the different factors 

 in the same chromosome, such that some tend to cling 

 together more frequently than others. 



Possibly some elements of the theory that diverse ex- 

 change ratios are characteristic for different pairs might 

 be retained, if there be added provision for modification 

 of the exchange ratio in a given pair, depending on 

 whether or not exchange occurs in some other pair. It 

 might be held, for example, that A and a are more likely 

 to exchange if in the same cell B and b have exchanged ; 

 or the reverse. This would give a theory of mixed type, 

 which added to the forces regulating the exchange be- 

 tween two members of a pair, other forces causing two 

 given pairs to tend to do the same thing, or the opposite 



