328 GEORG DUNCKER [November 



tioii. This done, the results may be further dealt with. Firstly, all 

 the observed variants have to be arranged in series according to their 

 numerical value, and the frequency of each among the (71) investigated 

 individuals has to be determined. Thus we get the empirical series of 

 variation of the character in n individuals. Weldon [20], for example, 

 counted the number of the dorsal rostral teeth in 915 individuals of 

 Palaemonetes varians, whence he got 



A series of variation may be graphically represented by noting the 

 variants in the order of their numerical values as points of equal 

 distances on an abscissa, and by erecting ordinates from each of these 

 points which represent by their length the relative (percentage) 

 frequencies of the corresponding variants. Straight lines drawn 

 between the free ends of each two adjacent ordinates, together with 

 the abscissa, will give the outline of the polygon of variation of the 

 character. The average value of the character, that is, the arithmetical 

 mean of the variants, corresponds to a point on the abscissa (M) ; the 

 ordinate erected to the latter is the centroid vertical of the polygon 

 (y c ). The summit of the polygon of variation usually lies near the 

 centroid vertical ; its variant has been called the mode ; but the 

 mode is neither more " typical " nor more " normal " than any other 

 variant existing. 



The single ordinates of frequency are generally lower the more 

 distant they are from the centroid vertical. The polygon of variation 

 is broad and low when there is great variability in the character, 

 but high and narrow in the opposite case. The best and simplest 

 expression of the degree of variability of a given character is the 

 square root of the average square deviations of its variants from 

 the mean. This value, which may be called the index of variability 

 of the character (e), corresponds to a piece of the abscissa ; it is 

 expressed by the same unit as the variants of the character. The 

 above -cited series of variation of Palaemonetes has the index of 

 variability, "8627 rostral teeth. 



While the average values of a character may differ widely in 

 different form -units of the same species, the indices of variability 

 remain fairly constant l not only in the form-units of the same species, 



1 Examples : 



I. Number of fin rays in dorsal fin anal fin 



Me M e 



of Pleuroncctes flesus, Baltic 39 '46 1'483S 



„ North Sea 41 '56 1-7739 



,, Plymouth 61-7214 2'3895 43-6098 1-6026 



amcricanus (Bvmpvs [4]) 65"06 2-4467 48"62 1-8188 



,, Rhombus maximus (Petersen [14]) 62-98 2-2533 45-86 1-6792 



II. Number of rostral teeth dorsal ventral 



in Palaemonetes varians (Weldon [20]) 4-3137 0-8627 1-6948 0-4799 



vulgaris 8-2819 0-8145 2-9781 0*4477 



