XI] OF VARIOUS CEPHALOPODS 553 



determining the constant angle of the .spiral would not in these 

 cases be accurate enough to enable us to measure the length of 

 the coil: we should have to devise a new method, based on the 

 measurement of radii or diameters over a large number of whorls. 

 The geometrical form of the shell involves many other beautiful 

 properties, of great interest to the mathematician, but which it 

 is not possible to reduce to such simple expressions as we have 

 been content to use. For instance, we may obtain an equation 

 which shall express completely the surface of any shell, in terms 

 of polar or of rectangular coordinates (as has been done by Moseley 

 and by Blake), or in Hamiltonian vector notation. It is likewise 

 possible (though of little interest to the naturalist) to determine 

 the area of a conchoidal surface, or the volume of a conchoidal 

 solid, and to find the centre of gravity of either surface or solid*. 

 And Blake has further shewn, with considerable elaboration, how 

 we may deal with the symmetrical distortion, due to pressure, 

 which fossil shells are often found to have undergone, and how 

 we may reconstitute by calculation their original undistorted 

 form, — a problem which, were the available methods only a little 

 easier, ^yould be very helpful to the palaeontologist; for, as 

 Blake himself has shewn, it is easy to mistake a symmetrically 

 distorted specimen of (for instance) an Ammonite, for a new and 

 distinct species of the same genus. But it is evident that to deal 

 fully with the mathematical problems contained in, or suggested 

 by, the spiral shell, would require a whole treatise, rather than 

 a single chapter of this elementary book. Let us then, leaving 

 mathematics aside, attempt to summarise, and perhaps to extend, 

 what has been said about the general possibilities of form in this 



class of organisms. 

 ■» 



The Univalve Shell: a summanj. 

 The surface of any shell, whether discoid or turbinate, may be 

 imagined to be generated by the revolution about a fixed axis of 

 a closed curve, which, remaining always geometrically similar to 

 itself, increases continually its dimensions : and, since the rate of 

 grow^th of the generating curve and its velocity of rotation follow 

 the same law, the curve traced in space by corresponding points 

 * See Moseley, op. cit. pp. 361 seq. 



