Manchester Memoirs, Vol. Ixiv. (1920) No. 2 7 



which have attained any given length, and have been utilised for 

 the construction of Figs. 3 and 4. In Fig. 3 the mean width, 

 and in Fig. 4 the mean depth, is plotted against the length. The 

 same figures show also the range in width and in depth which was 

 found amongst shells of each length — and again, the actual values 

 may be found by reference to Tables I and II. 



It should be observed that this method is only applicable in 

 the case of species which show no tendency towards division into 

 two or more groups : any such tendency is at once brought out 

 by the solid distribution figures, and when evidence of a differen- 

 tiation of this kind is found the ontogeny of the species cannot 

 be directly studied by the method outlined above. 



The remarkable smoothness of each of the ontogeny curves 

 provides a very striking justification for the method by which 

 they have been obtained. The discrepancy between the actual 

 mean width or depth values and the corresponding points on the 

 mean curves upon which they appear to lie is throughout very small. 

 The two curves may now be considered somewhat more closely. 



At the outset it may be remarked that the curves confirm 



the statement of Day that the ratios „ r . , , and ^ , decrease 



3 Width Depth 



throughout the life of the individual. But they show very 

 definitely that the relationship which exists between Width and 

 Length is by no means similar to that which is found between 

 Depth and Length — a feature that the purely qualitative method 

 used by Day could not be expected to bring out. 



Fig. 3 shows that the width and length are related through- 

 out life by a linear function, which may be expressed : — 



where /=length in mm., z£/=width in mm. 



Two points of interest arise in this connection. Obviously, 

 the line corresponding with this function does not pass through 

 the origin : the width is directly proportional to a quantity which 

 is roughly one millimetre less than the length as measured, i.e., 

 from the brachial umbo to the fold. Consideration of the form of 

 the shell affords a fairly simple explanation of this. The brachial 

 umbo projects to a slight extent behind the hinge-line; the ex- 

 tremities of the width axis lie in the plane of the hinge-line and 

 the anterior margin, not in that of the brachial umbo and the 

 anterior margin. Thus it appears highly probable that the width 

 is directly proportional to the length from the hinge-line to the 

 anterior margin. This suggestion is supported by the fact that 

 1 mm. is quite a fair average value for the difference between 

 the lengths as measured from the hinge-line and from the brachial 

 umbo respectively. No such correction for the length is neces- 

 sary in the case of the relationship between depth and length. 



The average values of the ratio ^ T e "f - are less than those 



Width 



