TEE STATISTICAL STUDY OF EVOLUTION. 455 



ancestral intermediate condition; i. e., they correspond to the right- 

 and left-hand polygons of Fig. 7. Consequently we may expect them 

 to he skew in opposite directions and so we find them to he. For ex- 

 ample, Bateson has measured the horns on the heads of 343 rhinoceros 

 beetles; the frequency curve is shown in Fig. 8. The left-hand 

 polygon has a skewness of -)- 0.48; the right-hand polygon of — 0.03. 

 One might infer that the right-hand form, the long-horned beetles, 

 had diverged less from the ancestral condition than the short-horned 

 beetles. Again, my pupil, Mr. Garber, has obtained a bimodal distribu- 

 tion polygon in the length of the chinch bug's wing (Fig. 9). The 



Fig. 10. Fkequkncy Poly- 

 gon OF THE Number of 

 Petals in Buttercups. 



65 

 60 

 55 

 50 

 45 

 10 

 55 

 50 

 25 

 20 

 15 

 10 

 5 



ta <VS 50 55 60 65 70 15 60 .•%vr. 



Fig. 11. Polygon of frequency of 



LENGTHS IN MILLIMETERS, OF PECTEN SHELLS 

 G.iTHERED AT RANDOM FROM A FISHERMAN'S 

 SHELL HEAP. 



short- winged form has a skewness of -j- .44 ; the long w^inged form of 

 — .43. In this case also the ancestral form lies between the present 

 modes. It is obvious that we may get cases in which two modes, repre- 

 senting conditions in different places, have moved, to different extents, 

 in the same direction. Thus the index (breadth -^- length) of the shell 

 of Littorina, a marine snail, as measured by Bumpus, has at Newport 

 a mode of 90; at Casco Bay of 93. The skewness is positive in both 

 •places and greater (-(- .24) at the more southern point than at Casco 

 Bay (+.13). This result indicates that the Littorina came from a 

 more northern home, for which we have confirmatory historical evi- 

 dence, and that these ancestral races were rounder, having an index 



