428 THE AMEBIC AN NATURALIST [Vol. L 



shown by the dotted line. To make comparison with the 

 other curve legitimate, it had to be constructed by the 

 same method,— namely, by making a composite of indi- 

 vidual curves, each of which represented the probabilities 

 for a certain type of double cross-over— only, instead of 

 using the observed numbers of double cross-overs of the 

 different types, in constructing it, it was necessary to use 

 the numbers of double cross-overs of the different types 

 that would have been observed if there had been no inter- 

 ference. (This curve hence represents the results of a 

 chance distribution both among and within the various 

 regions.) In the case of each type of double cross-over, 

 the way to find the per cent, of individuals showing it that 

 would be produced if there were no interference, is to 

 multiply the total per cent, of crossing-over in the first 

 region by the per cent, in the second region, as explained 

 in section 4a. (Thus, the per cent, of double cross-overs 

 passing between A and B and between C and D equals per 

 cent, of cross-overs between A and B times per cent, of 

 cross-overs between C and D.) This per cent., then multi- 

 plied by the total number of individuals counted, gives the 

 number of such double cross-overs theoretically to be ex- 

 pected in the absence of interference. When such calcu- 

 lations for each different possible kind of double cross- 

 over have been made, and the individual curve for each 

 then made, the latter may be combined to form a com- 

 posite curve like the curve shown by the dotted line. 



The end desired is of course to compare the dotted and 

 the heavy-lined curves and see what proportion of the dou- 

 ble cross-overs various distances apart, that were expected 

 on pure chance, actually occurred. Therefore a new curve 

 (Fig. 13) may be made, representing this relative coinci- 

 dence, i. e., the per cent, which each frequency on the ob- 

 served curve formed of each frequency on the expected 

 curve (see sect. IVa). This curve consequently shows the 

 rise or fall of the index with which we are already famil- 

 iar, and which we have called simply "coincidence." 



Owing to the fact that not very large figures have so far 



