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.’ The 
first three data of our example (Palaemonetes) are :— 
MSA 3157, e€="8627, SA — Waar 
the curve itself is a curve of type iv. (Pearson [12]) of the form 
y = Yo(COS Des 70 
where y,, m, and 7 are constants, @=/ (#) the variable. The error 
between the empirical and the theoretical series of variation amounts 
to only 3% of the total number of individuals, viz. :-— 
Variants 5) 3 4 5 6 7_ rostral teeth). 
Emp.trequency . 0 2 18 4123 #372 349 50 TT) individuals) 
Theor. frequency (y) ‘1 1:7 18°3 122:2 374°6 345°9 51:7 -5J ( ; 
From the curve of variation the probable range of variation of the 
1 The magnitude of the error is, ceteris paribus, inversely proportional to the square root 
of the number of investigated individuals. 
