THE SHELLS OF MOLLUSCS 155 



the other hand, will produce elongated cones, as in many of 

 the early paleozoic Cephalopods. 



(2) Constant differential ratio as regards growth in the median 

 plane. If the length-width growth-ratio remains constant, 

 but the absolute magnitudes of both components are greatest 

 at one margin, least at the opposite margin, and are inter- 

 mediately graded around the two sides of the mantle, our 

 cone will be distorted, and, so long as the growth-ratios con- 

 cerned remain constant, will grow into a true logarithmic 

 spiral in a single plane. Examples are provided by Nautilus, 

 the great majority of Ammonites, and Dentalium. 



Our first growth-ratio will still decide the form of the cone, 

 but now that the cone is distorted this will be measured by 

 the ratio of the shell-diameter at any place to the length of 

 the shell measured from its origin along the curve of the spiral. 



The first and second ratio together will decide the tightness 

 with which the spiral is coiled. There are six main possibilities : 



(1) As limiting factor, with median growth-ratio = i-o, or 

 in other words, no growth-gradient from one end of the median 

 plane to the other, a cone results. 



(2) When the growth-ratio is low, only slightly above unity, 

 the curve is very slight. In such cases, the mathematical 

 properties of the logarithmic spiral being what they are, a 

 many-whorled structure will never be produced, as the radius 

 of even the second whorl would be of relatively immense 

 extent, and no organism could do more than produce a portion 

 of the first whorl. Such forms are realized in the rhinoceros 

 horn or, among Molluscs, in the shell of Dentalium. 



(3) With increasing values of the ratio, the radius of suc- 

 cessive whorls rapidly decreases. The next possibility is 

 therefore a shell with more than one whorl, but with no contact 

 between each whorl and the next. This condition is rare, but 

 is realized, e.g. in certain Ammonites. 



(4) As the ratio increases further, a specific value will even- 

 tually be reached which allows the outer margin of each 

 whorl to be precisely in contact with the inner margin of the 

 whorl following. This condition is not infrequently realized. 

 The precise value of the median growth-ratio needed to produce 

 this result will of course vary with the value of the previous 

 (length-width) growth-ratio. With a narrow elongated cone, a 

 much higher median growth-ratio will be required just to effect 

 contact between whorls than with a broader, less elongated 

 cone (see Fig. 74). 



