688 



SCIENCE. 



[N. S. Vol. VII. No. 177. 



4 to 6 are bimaximal curves. All sucli 

 curves indicate that the material is not 

 homogeneous ; that there is a tendency to 

 break up into two races or species with dif- 

 ferent modes and different indices of varia- 

 tion. 



The relationship of the two groups indi- 

 cated in these curves is not equally close in 

 all. Thus in curve Fig. 2 the two races 

 are hardly separable. In Fig. 6 they ap- 

 pear as distinct, almost completely segre- 

 gated species. These cases differ both in 

 the degree of isolation and the degree of 

 divergence of the constituent races. "We 

 need quantitative expressions for these two 

 qualities. 



The degree of isolation may be measured 

 by the depth of the depression between the 

 maxima. By depth of depression we mean 

 the distance of the deepest part of the de- 

 pression below the level of the lower maxi- 

 mum. This depth may be expressed in 

 per cents, of the length of the shorter mode. 

 It is clear that in Figures 2 and 3 there is 

 no real depression, in Figure 4 one is just 

 appearing ; Figures 5 and 6 represent cases 

 of successive increase in the depth of the 

 depression reaching 82% in Figure 6. 

 This ratio of the depression to the length 

 of the shorter mode may be called the 

 Index of Isolation. 



The degree of divergence between two 

 groups may be measured by the distance 

 between their modes. This distance must, 

 however, be expressed in a unit independ- 

 ent of the particular units employed in 

 measuring the characters of the species. 

 The unit must be some quality of the curve. 

 The variability of the curve is expressed, 

 as we have seen, by the half-range. * We 

 may use as our unit the average of the two 

 half-ranges of the broader curve when 

 they are both approximately known, other- 

 wise to its outer half-range. The diver- 

 gence between the races will then be ex- 



* Or thrice the standard deviation. 



pressed as the ratio of the distance between 

 the modes to the half-range,* or thrice the 

 standard deviation, of the broader curve. 

 This may be called the Index of Divergence. 



These two indices, however, are not inde- 

 pendent but are curiouslj^ bound together. 

 Thus if two equal, symmetrical curves with 

 the same variation overlap so that the inner 

 end of each curve just touches the mode of 

 the other — in other words, when the Index 

 or Divergence, is 100, the Index of Isola- 

 tion will be found to be about 56. If the 

 curves are of very different area or form, 

 the Index of Isolation may be, with the 

 Index of Divergence still at 100, diminished, 

 but where large numbers are used it will 

 rarely, in practice (provided the curves 

 are symmetrical) be less than 50. 



The question arises- whether it would not 

 be necessary to draw curves for many char- 

 acters. Practically it will not be necessary, 

 for confluent species are usually separated 

 chiefly by one most distinctive character. 

 This character may be termed the chief dif- 

 Jerentlal. It may be used alone to measure 

 the isolation and divergence of the groups, 

 to test their specific value. 



Again, how are the individuals which are 

 measured for the differential to be selected ? 

 They must be taken methodically at ran- 

 dom. This sounds paradoxical, perhaps, 

 but it is not. One takes methodically at 

 random when one lays a yard stick on a 

 grass plot and plucks those blades which 

 lie nearest the inch divisions, or gathers 

 field mice from traps set in a straight line 

 at distances of one mile apart, or at the 

 angles of hundred-mile rectangles, and so 

 on. The individuals measured are rigidly 

 taken on some other basis than their own 

 characters. It will not, of course, always 

 be possible to have individuals gathered so 

 rigidly at random. This is only the ideal 

 which can rarelj' be realized. 



In plotting results the actual and not the 

 percentage frequency of each class should 



