Analysis of Heterogeneity in Some Simplest Chaeacters 197 



spot may be entirely absent or present as a simple round area and that 

 it may retain its round form and increase greatly in size. As in the spot 

 c it also changes now in one direction, now in another, or in two or more 

 at the same time, but these directions are never chance or haphazard, but 

 always are along certain invariable lines. Sta- 

 tistically, a lot of 500 h spots shows a typical 

 curve of distribution, plus and minus (fig. 19). 

 In L. multitceniata spot h shows the same con- 

 dition present or absent in its entirety, present 

 in the simplest condition as a simple round spot, 

 often of considerable size as a round spot; at 

 other times pushing out in different directions, ^o 

 but never in chance or irregular directions. The 

 statistical statement from a lot of 500 left spots 

 offers nothing of interest; it is simply the plot- 

 ting of a lot of areas having upper and lower 

 limits — a mode, a mean, and coefficients not 

 worth calculating. 



It would be easy to go on endlessly accumu- 

 lating data of this kind, and I could fill many 

 pages with the mere statement of the facts 

 gathered concerning spots occupying the posi- 

 tion 16c in chrysomelidae, and also in other 

 families of Coleoptera. Naturally, not all show the same conditions, but all 

 give the same results in principle, namely, that the common variations of these 

 simple indivisible color-characters present two superficially distinct variation 

 phenomena — diversity with respect to area in units of some measure, and in the 

 arrangement and relation of the character, both external and internal. 



It is obvious from the data given that the common differences of simplest char- 

 acters are not all in line, plus and minus, but that there are changes first in one 

 direction, then in another, or in one or more directions at the same time. If the 

 biometric definition of fluctuations be true, then all changes found that are not 

 plus and minus in terms of quantity, but are multifarious in direction and rela- 



Fig. 19. — Statistical treatment 

 of a uniform lot of 500 spot b 

 areas In L. utidecimlitieata from 

 Tierra Blanca. 



Fig. 20. — Spots having the same area, but not the 

 same shape or direction of variation ; statistically 

 unlike things are often confused on the basis of mea- 

 surement and treated as lilie. 



M % 4. f 



tion, are mutations, and only those measurements of the area of pigmented sur- 

 face are fluctuations and are of necessity in line, because forced there by the 

 method of description. Area, moreover, is not an accurate method of describing 

 the condition of these spots, and this I have shown in figure 20, where there are 

 shown spots all of the same measured area, but their shape is quite diverse and 

 might be productive of quite diverse results. Statistical methods force these 

 spots into the same class and count them in statistical analysis as all alike. They 

 are decidedly unlike. 



