﻿22 BRITISH FOSSIL CEPHALOPODA. 



becoming elliptic, and vice versa, or the subquadrate becoming more rounded. 

 These peculiarities, combined with others, give rise to such genera as Phragmoceras. 



(6.) The law of variation. — The section usually increases in size at a uniform 

 rate from near the apex to the aperture. In the case of straight or slightly curved 

 shells this rate may be expressed either by the vertical angle of the cone, or by the 

 difference of two corresponding diameters of the section divided by the distance 

 between them. 1 Thus, if two given diameters at a distance of 2 inches are 13 

 lines and 10 lines respectively, the rate of increase is 3 in 24 or 1 in 8. If 

 the section be circular, any diameter will give the same rate of increase ; but if 

 it be elliptic, we must specify which diameter is taken, since it is obvious that 

 the longer diameter will increase more rapidly than the smaller on the same shell. 

 When the shape of section is known, either will give the true form of the shell. 

 When there is any curvature, the increase of the transverse diameter is the most 

 convenient to measure, the distance between the sections being taken along the 

 curved line. When the shell is a coiled one, the ratio of the breadth of the last 

 whorl to the whole diameter, which involves but does not depend wholly upon the 

 rate of increase, may be conveniently taken to represent it. 2 It is worth noticing 

 that the ratio of the breadth of a whorl at one end of a diameter to that at the 

 opposite end of the same diameter does not depend upon the rate of increase at all, 

 but only on the curvature of the spiral. 



The rate of increase thus measured can only have reference to the general form, 

 for it is seldom entirely constant. The tip of the shell is always, so to speak, 

 rounded off, so that the increase is rapid at first, and only settles down later on to its 

 typical amount. On the other hand, the greater part of all the changes taking 

 place in the body-chambers of the adults is due to the variations of this element, 

 such as the contraction towards the aperture in the Phragmocerata. Indeed, 

 scarcely any shell retains the same rate of increase throughout, but it generally 

 diminishes, or more rarely increases, at last. In specimens agreeing in all other 

 respects we also find that some have a greater and some a less rate of increase — and 

 a certain range of variation must be allowed, especially as some of the difference 

 may be due to compression. Nevertheless, this element forms one of the most 

 important criteria of species, for in some of the smooth Orthocerata it becomes 

 almost the only thing left to judge by. The general amount of the rate of increase 

 has also been made use of to group the species of various genera. Thus, among the 

 uncoiled forms, Barrande has distinguished those in which it is great as brevicones, 



1 In the descriptions given by Barrande the ratio of the two diameters is given, which is of no 

 value, since on the same cone the ratio of two diameters at a given distance varies with their 

 position ; and the same ratio may be found on different cones if taken at suitable positions. 



2 For the mathematics of these curves see my paper in the Phil. Mag. 1878, "On the Measure- 

 ment of the Curves formed by Cephalopods and other Molluscs." 



