572 



THE LOGARITHMIC SPIRAL 



[CH. 



circular or nearly so, becomes flattened or compressed dorso- 

 ventrally ; and the angle, or rather edge, where dorsal and ventral 

 walls meet, becomes more and more drawn out into a ridge or 

 keel. Along the free margin, both of the dorsal and the ventral 

 portion of the shell, growth proceeds with a regularly varying 

 velocity, so that these margins, or lips, of the shell become regularly 

 curved or markedly sinuous. At the same time, growth in a 

 transverse direction proceeds with an acceleration which manifests 

 itself in a curvature of the sides, replacing the straight borders of 

 the original cone. In other words, the cross-section of the cone, 

 or what we have been calhng the generating curve, increases its 

 dimensions more rapidly than its distance from the pole. 



Fig. 298. Cleodora cuspidata. 



In the above figures, for instance in that of Cleodora cuspidata, 

 the markings of the shell which represent the successive edges of 

 the lip at former stages of growth, furnish us at once with a 

 "graph" of the varying velocities of growth as measured, radially, 

 from the apex. We can reveal more clearly the nature of these 

 variations in the following way which is simply tantamount to 

 converting our radial into rectangular coordinates. Neglecting 

 curvature (if any) of the sides and treating the shell (for simplicity's 

 sake) as a right cone, we lay ofl equal angles from the apex 0, 

 along the radii Oa, Oh, etc. If we then plot, as vertical equi- 

 distant ordinates, the magnitudes Oa, Oh ... OY , and again on to 

 Oa' , we obtain a diagram such as the following (Fig. 299) ; by 



