﻿BRITISH FOSSIL CEPHALOPODA. 23 



and those in which it is small as longicones ; whilst among coiled forms many 

 attempts have been made to separate those that are involute, or have a rapid rate of 

 increase, from those that are more uncovered. 



(c.) The curve of motion. — This is usually assumed to be an equiangular spiral, in 

 which the straight line may be included as an extreme case. On this assumption, I 

 have indicated, in the paper above referred to, how the spiral of any particular shell 

 can be ascertained from fragments. In practice, however, only a rough approxima- 

 tion can be arrived at among the Palaeozoic Nautilidse ; for, partly owing to irregu- 

 larities of contour, partly to the unknown amount of imbedding in the rock, and 

 partly to a natural variation in specimens of the same species, no measures on il]- 

 preserved shells are sufficiently reliable for calculation. To account for the devia- 

 tions from the true equiangular spiral in recent shells, Professor Naumann 1 supposed 

 the curve of motion to be one, allied only to the equiangular spiral, which he called 

 the " conchospiral." This may be a perfectly true supposition, but the curve is less 

 manageable, and the equiangular spiral is quite a close enough approximation when 

 we are dealing with fossil shells which cannot be observed with minute accuracy. In 

 the case of coiled shells in which more than half a whorl is preserved, the ratio of 

 two diameters of the whorls, that is, the breadth of the last whorl divided by that of 

 the whorl at the opposite end of the same diameter, gives us the element of the curve 

 as compared with another of the same kind, whether it be an equiangular or a 

 concho-spiral. This, therefore, is one of the essential elements of the form of the 

 species. When the shell is open, so that it has no diameter, an analogous method 

 is inconvenient, and the curvature is sufficiently defined by the ratio of the mean 

 radius of curvature to the mean breadth of the whorl. The curvature of course, 

 by the nature of the curve, decreases with the growth ; but, as in the case sup- 

 posed, only a small portion of a whorl exists ; the circle which has the same curva- 

 ture as the middle portion seldom deviates much from the general outline. But earlier 

 parts of the same shell having a greater curvature than the later, the absolute radius 

 is not sufficient, but its ratio to the corresponding breadth must be given, which is 

 the same for all parts of the same shell. 



Besides the minor variations that this element shows, there are, in the case of the 

 Lituites and others, those sudden cessations of curvature which produce the long, 

 straight body chamber. Also in Trochoceras, the spiral is not in a plane, but 

 forms a helico-spiral. In this case we may state, as a fourth element of the shape, 

 the elevation in the height of the apex above the median plane of the base, divided 

 by the diameter of the base. In the Silurian species, however, this is seldom 

 necessary, as it requires a careful examination to ascertain that there is any want of 

 symmetry at all. 



On the direction of the curvature. — Seeing that it is known that the curvature of 



* ' Die Cyclocentrische Conchospiral.' 



