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 molluses, 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 molluses 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 Y-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 Y-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 Y-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 bé 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- 
