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THE AMERICAN NATURALIST [Vol. LIV 



as an illustration, to show what kind of discrepancies are 

 meant. Discrepancies of exactly the same character and 

 direction appear when the diagrams obtained by this 

 method from experiments involving three points close 

 together are compared with those from other exper- 

 iments having three points far apart; that is, the former 

 figures are repeatedly found to be nearer in form to a 

 straight line than the latter; in such three-point exper- 

 iments, moreover, highly extensive counts have been 

 made, involving altogether (in the published experiments 

 on the first chromosome), approximately a hundred 

 thousand flies, and thousands . of double cross overs. 



5. The relation which has just been described, whereby 

 the larger frequencies of separation are relatively 

 smaller than could be directly represented in a curve 

 constructed on the basis of the small frequencies, is due, 

 according to the phraseology of the Drosophila workers, 

 to the fact that the relative frequency of double crossing 

 over ("coincidence") is so much larger for large fre- 

 quencies than for small ones. Castle realizes to a cer- 

 tain extent the difficulty which this circumstance entails 

 for his models, and he endeavors to meet it by means of 

 the "subsidiary hypothesis" that the breaks in his 

 models are more frequent in certain directions than in 

 others. This assumption would, in some measure, ex- 

 plain away in a formal manner certain of the discrep- 

 ancies (although cases of "triple crossing over" still 

 remain an insurmountable obstacle), but the adopting 

 of any such hypothesis really amounts to cutting away 

 the ground from under the main theory of "propor- 

 tionate representation," for the hypothesis involves an 

 abandonment of the claim that the model represents each 

 frequency by a proportionate distance between the nodes. 

 For it is evident that if, in a given region, breaks in one 

 direction are more frequent than in another, then points 

 in this region which are an equal distance apart will be 

 separated with different frequency according to the di- 



