4 NILS ODHNER, NORTHERN AND ARCTIC INVERTEBRATES. V. PROSOBRANCHIA. 1 DIOTOCARDIA. 



and within definite limits. As a rule in constructing such variation curves it is 

 presupposed, that all the specimens investigated are at the same stage of development, 

 so that all changes referable to growth are eliminated. Thus for molluscs, where 

 growth continues constantly throughout life, such a method would obviously be im- 

 practicable. In such cases it is impossible to determine an average value of length 

 or indeed of any such other characters, as are influenced by growth, for the absolute 

 minimum and maximum of length etc. do not indicate the extent of variation, but 

 rather the extreme values of the growth. 



Briefly, growth is a new factor to be considered in such investigations, which 

 stands in an unknown ratio to variation; for young forms may vary in different 

 ways from older ones and the growth is not uniform for all the organs and dimen- 

 sions during the whole of life, and thus the proportions may be subject to change, 

 as age increases. For this reason, also, it is impossible to find any ratios (e. g. length 

 to breadth) that occur so constantly both in young and full-grown specimens, that 

 one is entitled to speak of a median value for them during the whole of life. 



For molluscs and other forms with a permanent growth another method of ex- 

 pressing variation must be employed, and in respect to them I have proceeded in 

 the following manner: — 



In a coordinate system the horizontal or X-axis may be taken to represent the 

 absolute length, or sometimes height, of the specimens, expressed in mm, while the 

 vertical or F-axis may be taken not as indicating the number of specimens (as 

 usual), but the absolute amount of some other measurement such as the breadth or 

 height of the aperture; two or more such quantities may be examined and re- 

 presented on the F-axis. A large number of specimens in all the varying stages of 

 development must be measured and the numbers expressing the respective characters 

 must be marked in the coordinate system. They fall within zones of a certain 

 breadth, which thus give us a conception of the extent of variation during all the 

 stages of development. By connecting all the extreme values, curves are obtained, 

 which mark the limits of the variation. Between these lies an average value, attained 

 by the majority of the specimens. 



The extent and the median value of variation are available as a characterisa- 

 tion of the entire material. By taking more characters into account on the F-axis 

 the material will be further defined in those several respects. The relative positions 

 of the variation zones form another basis for establishing the distinction existing in 

 different classes of material. 



It is evident that the greater the number of specimens, the more exact will 

 the result be. Some of the values arrived at in this work for the variation limits 

 might very probably be subject to some slight adjustment, were the measurement 

 of a larger number of shells carried out ; they must be regarded as only approximate 

 in character and in some cases should only with reserve be used for comparisons. 



In order to illustrate the above account some examples are given below in a 

 graphic form. 



Figs. 1 — 3 represent the variation of Margarita groenlandica in different dis- 



