330 GEORG DUNCKER [november 



the series of variation of numerical characters, and have found that 

 the actual magnitude of the frequencies of variants corresponds to the 

 law of probability of combinations, which law Pearson [12] has recently 

 expressed by his general probability curve (curve of variation). This is, 

 as far as I know, the first substantiated mathematical law of biological 

 processes. So in investigating a series of variations we have next to 

 find the probability curve determining the shape of its polygon of 

 variation. But this demands a consideration of the already somewhat 

 compendious mathematical literature of the subject, which I cannot 

 now discuss. Pearson's methods I have recently described in a manner 

 especially suited to the needs of biologists [7]. 



Curves of variation are symmetrical if the two groups of causes 

 of variation are equal in number ; asymmetrical, if the latter are un- 

 equal ; in the single form-unit they always show one summit. In 

 symmetrical curves the maximal ordinate and the centroid vertical are 

 identical ; in asymmetrical curves their distance apart is greater the 

 more asymmetrical the curve. The ratio between this distance and 

 the index of variability gives an abstract number, the index of 

 asymmetry of the curve (A), which is, corresponding to the position 

 of the centroid vertical to the maximal ordinate, either positive or 

 negative. Positive asymmetry of a curve means that there are more 

 negative than positive causes of variation, while negative asymmetry 

 implies the contrary. 



The question as to the variation of a numerical character within a 

 form-unit is therefore to be answered by giving the average value, the 

 indices of variability and of asymmetry, and the formula of the curve 

 of variation of this character. These four data given, the series of 

 variation can always be reconstructed with only a small error, which 

 decreases as the number of investigated individuals is increased. 1 The 

 first three data of our example (Palaemonetes) are : — 



M = 4-3137, e = -8627, = A-\L735; 

 the curve itself is a curve of type iv. (Pearson [12]) of the form 



y = 7/ o (cos0fV e 



where y , m, and r are constants, =f (x) the variable. The error 

 between the empirical and the theoretical series of variation amounts 

 to only '3°/ of the total number of individuals, viz. : — 



•I/O 



Variants .012 3 4 567 rostral teeth). 



Emp. frequency . 2 18 123 372 349 50 1 1 , individuals) 

 Theor. frequency (y) '\ 1-7 18-3 122-2 374-6 345-9 51-7 -5) <, inamauais > 



Prom the curve of variation the probable range of variation of the 



1 The magnitude of the error is, ceteris 2Mribus, inversely proportional to the square root 

 of the number of investigated individuals. 



