Figure 25. — Cross section of two valves of Cardium. 

 The similarity with the conchoid in figure 24 is super- 

 ficial. 



each other at the ends. The two branches of 

 conchoid (C and D) join together only in infinity 



(fig. 24). 



Lison pointed out that the shape of the shell 

 may be considered as a whole series (ensemble) 

 of arches, the curvatures of which are described 

 by logarithmic spirals of the same parameter which 

 have a common origin at the umbo. The latter 

 is their common pole. The arches terminate at 

 the edge of the valve. The contour of the valve 

 edge, frequently called the "generating curve", 



is usually confined to one plane parallel to the 

 plane of opening and closing of the shell. 



Among many spirals that can be drawn on the 

 surface of a shell only one is completely confined 

 to a single plane. This spiral was called by Lison 

 the "directive spiral"; its plane is the "directive 

 plane" of the shell. A& other spirals which can 

 be easily noticed on the shell surface as ridges, 

 furrows, or as pigmented bands deviate to the 

 right or left depending on which side of the 

 directive plane they are located (fig. 26). 



By mathematical analysis of the curved surfaces 

 of various bivalve species Lison arrived at the 

 general equation ^ of a valve. He observed that 

 by itself such an equation may not be helpful to 

 biologists unless it can be used for comparing the 

 shape of the individuals of the same species or in 

 maldng comparison between the different species. 

 Lison stated that in practice it is not necessary to 

 make the involved mathematical computations. 

 It is sufficient to compare certain "natural" 

 characteristics of shells, namely, the du-ective 

 plane described above, the plane of closure of 

 valves (or connnissure plane), and the angle of 



3 General equation of a valve as given by Lison (1939) is as follows: 

 CT=(ropx; w=w„+«; z = ZoeP'* in whicli p is a constant and ao, wo, and z„ 

 are the functions which express on cylindrical coordinates the form of tlie 

 free edge of the valve when the directive plane is located within the xy and 

 the origin of the coordinates is at the umbo. (Translation by Paul S. 

 Galtsoff.) 



Figure 26. — Directive plane of scallop shell, Peclen, viewed from hinge end 2a, and from the broad side 2b. Tin. 



arrows indicate the directive plane. (After Lison, 1939.) 



24 



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