426 THE AMERICAN NATURALIST [Vol. L 



this length to occur as for those 8 to 10 units long. Sim- 

 ilarly, those 12-14 units long may be three times as numer- 

 ous, for they may pass through f, g, or h, and so with each 

 increment of length, up to 20, there will be an equal addi- 

 tional amount of chance for a double cross-over of that 

 length (passing through the required sections, Cj — v and 

 s — r) to occur. Thus our curve of probability rises in 

 regular steps from 8 to 20; if we could have divided the 

 distance c x — v into an infinite number of parts, instead of 

 into 8, these steps would each be infinitely small, and so 

 we should have a straight line rising from 8 to 20. 



Beyond this point the rise in probability ceases; a 

 double cross-over between 22 and 24 units long has no 

 more chance of happening than one of 20-22 units. Refer- 

 ence to the figure will show that a double cross-over of 

 20-22 units passing- through any of the regions from c 

 through h will separate s from r and thus fulfill the re- 

 quirements, but a double cross-over 22-24 units long, 

 while it has the additional alternative of passing through 

 b, can not pass through h without its second point of 

 crossing-over falling to the right of section s — r. Sim- 

 ilarly, one 24-26 long may not pass through g or h, though 

 it may pass through any region from a to f ; double cross- 

 overs of all these lengths therefore have the same chance 

 of occurring, and our curve along the corresponding ordi- 

 nates would hence be a horizontal line. 



Double cross-overs longer than this would have less and 

 less chance of occurring; one 26-28 long could only pass 

 through regions a — e, one 28-30 only through a — d, and 

 so the curve falls again in a straight line to the zero level 

 at 36. 



The same rules can be shown to apply to all cases : the 

 curve starts at a place on the abscissa representing the 

 distance apart of the innermost factors involved (in the 

 above case this distance was v — s, =8); it rises in a 

 straight line for a distance equal to the length of the 

 smaller section involved (above, this was the distance 

 s — r > = 12, so that the line rose to point 8 -f 12, =20) ; 



