Curves formed by Cephalopods and other Mollusks. 247 



the axis. The genus Trochoceras indeed lies between them 

 in this respect. 



7. Similar methods to the above may be adapted to other 

 generating curves besides the ellipse. For the circle, of course, 

 we have only to put k = 1. In many cases no simply expressed 

 equation can represent the shape ; but there are a few that do 

 admit of this. Thus, in the Ammonites of the group Cordati 

 the curve is nearly represented by the cardioid whose equation 

 is 



/3 = a(l+ cos </>). 



Here the retardation of the pole from the apex being 



2a=e 9cota (e pcota — l) = 2ae ecot0C say, and taii7 = 0, 

 the equation to such an Ammonite will be 

 {(r — e ecota )(r — (l + a)e ecota ) + z 2 } 2 = * 2 e 2ecot0l {(r--e 6cota y + z 2 }, 



8. It is to be noted that, if a plane section through the 

 middle point of growth have an elliptical or any other curved 

 outline, the trace on any other plane through the same point 

 will not be of the same character ; and though the assumption 

 (say) of an elliptical section in a plane perpendicular to the 

 direction of growth of the middle point may be as near an 

 approximation to nature as the one chosen, and the corre- 

 sponding equation may be worked out, yet the results are 

 quite unmanageable, and have no particular relation to the 

 laws of growth already enunciated. The curve assumed in 

 the above investigation is that exposed by a longitudinal sec- 

 tion of the shell through its axis. It has therefore no neces- 

 sary connexion with the form of the aperture, which may be 

 in another plane, or, indeed, have a curve of double curvature 

 for its outline. 



This difference is particularly to be noticed in conical shells, 

 such as the Orthoceras, which correspond to the value a = 0. 

 In these we cannot take a plane through the pole (i. e. the 

 apex) as the surface of growth. Nor can we consider one 

 side more retarded than the other. Fundamental differences 

 are thus revealed between the laws of growth of such shells 

 and those which are curved. We may, however, connect them 

 both theoretically and by natural links. To turn an equian- 

 gular spiral about its pole through a given angle is to bring 

 an earlier point in the curve into any particular radius ; so 

 that the apical angle of a conical shell corresponds to the 

 angle of retardation in a discoid one : only in the latter case 

 the inner curve stops on the extreme radius of the outer, while 

 in the former it is continued to be of the same length. In 

 some Cyrtocerata and in the opercula of Grasteropods we may 



