Statistical Studies on the Variation of Stagbeetlks. 



$7 



The curve given from the data above illustrated (Fig. S.), affords us an 

 excellent example of the unimodal type which is nearly symmetrical on both 



sides of the mode, 33min ; the mean is 

 32.7mm aiul the median 33mm. Thus the 

 theoretical mode coincides almost com- 

 pletely with that of the curve, so that it 

 stands ver}- near the normal curve. 



By the calculation of the value of 



J ~ - '^ the standard deviation was 



obtained as 2.14. 



Comparing this cur\-c with that of 

 the females of Cladognat/ms, we see that 

 here the apical angle is very acute instead 

 of being an obtuse angle as in the case 

 of Cladognatlms. The conclusion is, there- 

 fore, justified that the \'ariation in the 

 females of Liccanits fluctuates normally 



Curve showing frequency 

 of various grades of body lengtli 

 of females in Liicaims. Figures 

 on axis represent units and num- 

 bers as before. 



between the two extremities, 25mm and 39mm. 



Bateson (1894) was the first who made out in his study of stagbeetles 

 of Xylotnipes, and earwigs, the existence of two apices in the frequency of 

 variation. Bateson's case is based upon the length of the horns of 342 male 

 beetles and the length of the forceps of 5 S3 male earwigs. In the present 

 investigation, we took the body length as the chief factor, because the speci- 

 mens are abundant enough to ascertain the curve which Bateson made out 

 otherwise. Our curve is, like his, diviorphic in Cladognatlms at least. 



In the dimorphic curve which de Vries (189S) gave for Chrysantliamwt 

 iegetuin, the author ascribes its two apices to tlie influence of the mixed fac- 

 tors of two different species, assuming the first maximum to be caused by the 

 plants with 13 radial petals and the second by those of 21. In the course 

 of his experiments with culture selections, from plants with dimorphic curve, 



