XI] OF BIVALVE SHELLS 561 



"discoid" shell ; and furthermore we have numerous intermediate 

 stages, on either side, shewing right and left-handed spirals of 

 varying degrees of acuteness*. In this case, the angle 6 may be 

 said to vary, within the limits of a genus, from somewhere about 

 35° to somewhere about 125°. 



The angle of retardation (jS) is very small in Dentalium and 

 Patella ; it is very large in Haliotis. It becomes infinite in 

 Argonauta and in Cypraea. Connected w^ith the angle of retarda- 

 tion are the various possibilities of contact or separation, in various 

 degrees, between adjacent whorls in the discoid, and between 

 both adjacent and opposite whorls in the turbinated shell. But 

 with, these phenomena we have already dealt sufl&ciently. 



Of Bivalve Shells. 



Hitherto we have dealt only with univalve shells, and it is in 

 these that all the mathematical problems connected with the 

 spiral, or helico-spiral, are best illustrated. But the case of the 

 bivalve shell, of Lamelhbranchs or of Brachiopods, presents no 

 essential difference, save only that we have here to do with two 

 conjugate spirals, whose two axes have a definite relation to one 

 another, and some freedom of rotatory movement relatively to 

 one another. 



The generating curve is particularly well seen in the bivalve, 

 where it simply constitutes what we call "the outline of the shell." 

 It is for the most part a plane curve, but not always ; for there 

 are forms, such as Hippopus, Tridacna and many Cockles, or 

 Rhynchonella and Spirifer among the Brachiopods, in which the 

 edges of the two valves interlock, and others, such as Pholas, 

 Mya, etc., where in part they fail to meet. In such cases as these 

 the generating curves are conjugate, having a similar relation, but 

 of opposite sign, to a median plane of reference. A great variety 

 of form is exhibited by these generating curves among the bivalves. 

 In a good many cases the curve is approximately circular, as in 

 Anomia, Cyclas, Artemis, Isocardia ; it is nearly semi-circular in 

 Argiope. It is approximately elliptical in Orthis and in Anodon ; 

 it may be called semi-elliptical in Spirifer. It is a nearly rectilinear 



* See figures in Arnold Lang's Comparative Anatomy (English translation), n, 

 p. 161, 1902. 



T. G. 36 



