290 



THE AMERICAN NATURALIST 



[Vol. L 



frequent it tended to be. In the former case coincidence 

 would rise to a certain level, as distance between the points 

 of crossing-over considered increased, and would after 

 that remain constant ; in the latter case it would rise pro- 

 gressively, and might or might not reach or pass the 100 

 per cent, level. 



On the other hand, if crossing-over is due to a breakage 

 of tightly twisted threads, not so many different kinds of 

 variation of coincidence, with increase in distance, would 

 be theoretically possible, but a condition something like 

 the one last mentioned must always obtain. For, on 

 scheme 3, the interference of a breakage with the tightness 

 of twisting and consequent chance for another breakage 

 must decrease progressively at greater and greater dis- 

 tances from that breakage; coincidence would thus rise 

 until finally it reached the 100 per cent, level expected on 

 chance. It would never rise much 2 beyond this, as one 

 break could never make another more likely to occur; 

 neither could coincidence fall once more, with a still 

 greater distance (as it could on the loop scheme, after a 

 "modal distance" had been reached). If, therefore, it 

 should be found that, for certain (modal) distances be- 

 tween two points of crossing-over, coincidence ran well 

 above 100 per cent., or that, beyond certain distances, co- 

 incidence fell again, there would be good evidence that 

 crossing-over did not occur at a stage of tight twisting. 

 If, on the contrary, it were found that crossing-over coin- 

 cidence rose progressively with distance, until it reached 

 the 100 per cent, mark, but neither went much 2 beyond 



2 Even on scheme III, coincidence could finally rise slightly above 100 

 per cent., for although one break (I) could not help another (K) to occur, 

 no matter how far away the latter (K) might be, still it might, by pre- 

 venting the occurrence of other breaks (J), in between these two, give more 

 chance for the occurrence of the break farther off (K), since in this way 

 the interference of breaks J with K (which is stronger than the interference 

 of the more distant I with K) is removed. Thus break K might occur more 



rise above 100 per cent. However, it would be easy to distinguish between 

 the slight rise in coincidence above 100 per cent., due to this cause, and the 



separated by a distance about equal to the modal length. For, in the first case, 

 considering only gametes in which no crossing-over at all tooh place in between 



