THE SHELLS OF MOLLUSCS 161 



shells of Helix. When it is higher, the shell becomes more 

 pointed, as in Turritella. Here again, D'Arcy Thompson 

 gives detailed mathematical treatment, although for some 

 reason he has not reduced the degree of ' shear ' (distortion 

 of the primary spiral in a plane at right-angles to its own) 

 to terms of differences of growth-rate, as he has done for the 

 degree of coiling (distortion of the primary cone into a spiral). 1 



Typically, of course, the shape of any structure produced 

 by accretionary growth must remain constant so long as the 

 various growth-ratios concerned in its production remain 

 constant. But as a matter of fact, the various ratios often 

 alter with age or size, in some cases suddenly, in other cases 

 progressively. Thus certain Ammonites have their oldest 

 portions uncoiled, while the earlier-formed part of the shell 

 is of typical form — an example of sudden alteration. In 

 others, the ratio of the diameters of successive whorls does 

 not remain constant as in the true logarithmic spiral, but in- 

 creases progressively. This gradual change, due to a progres- 

 sive change in the shape of a fundamental growth-gradient, 

 is not infrequent, and may possibly prove to be associated 

 with senescence. Among Gastropods, the tapering of the 

 oldest part of the shell in Pupa, Clausilia, etc., in place of the 

 continued expansion to be expected if the growth-ratios remain 

 constant, is another case, and there are numerous other 

 examples. 



(4) Finally, there may be excess or defect of growth-ratio 

 at special points, not in connexion with the main growth- 

 gradient. This, of course, has its analogy in multiplicative 

 growth, e.g. in the development of markedly heterogenic 

 appendages which break the main growth-gradient of the 

 body, as in the right chelae of male hermit-crabs. 



The most obvious examples of such growth are found in 

 lamellibranch shells. The ' ears ' of the shells of scallops and 

 other species of Pecten are an excellent case. The general 

 growth-gradient is of usual Molluscan type, with high point 

 directly opposite the hinge, and a uniform and symmetrical 

 double gradient extending thence round both sides of the 

 shell. Just before reaching the hinge, however, the growth- 

 ratio, after sinking very low, increases rapidly and then 

 abruptly descends to zero, thus generating the ' ears ' of the 

 shell. 



1 Interesting numerical details concerning various structures of 

 logarithmic-spiral construction may be found in Petersen, 1921. 

 11 



