incidence. The plane of closure of the valves 

 originates at the umbo and passes between the 

 edges of the two opposing valves when they are 

 closed and touching each other. The angle of 

 incidence, as defined by Lison, is the angle between 

 the plane of closure and the directive plane. In 

 round and symmetrical shells of scallops, pearl 

 oysters, and other bivalves the directive plane is 

 perpendicular to the plane of closure and the 

 angle of incidence is 90° (fig. 26). In the shells 

 of CanUiim orbita, the directive plane forms an 

 acute angle of 81 ° and is much smaller in elongated 

 shells such as Fimbria fimbriata and Trapezium 

 ohlomjnm. The comparison between the shells 

 can easily be made by recording the contours at 

 the free margins of the valves and determining 

 the angle of incidence. 



To determine the shape of logarithmic spiral of 

 the valve the shell may be sawed along the direc- 

 tive plane (fig. 27) and the section oriented with 

 the umbo O at lower left. If Si and S2 are respec- 

 tive lengths of the two radii the value of para- 

 meter p can be computed by using the fundamental 

 equation of logarithmic spiral, 



„ log. Si— log. S2 



(logarithms in this equation are natural, to base e) . 



In resume, Lison attempted to prove that the 

 form of the shell in which the generating curve is 

 confined to one plane is determined by three 

 conditions: (1) the angle of the dii'ective spiral, 

 (2) the angle of incidence, and (3) the outline of 

 the generating curve. 



Further attention to the problem of the shape 

 and formation of the bivalve shell was given by 

 Owen (1953). In general he accepted Lison's 

 conclusions and stated that "the form of the valves 

 should be considered with reference to: (a) the 

 outline of the generative curve, (b) the spiral angle 

 of the normal axis, and (c) the form (i.e., plani- 

 spiral or turbinate-spiral) of the normal axis." 

 The normal axis is considered by Owen with 

 reference to: (1) the umbo, (2) the margin of the 

 mantle edge, and (3) the point at which the great- 

 est transverse diameter of the shell intersects the 

 surface of the valves. Thus, it can be seen from 

 this statement that Owen's "normal axis" does 

 not coincide with Lison's directive plane except 

 in bilaterally symmetrical valves (fig. 28). Ac- 

 cording to Owen's view, the direction of growth 

 at any region of the valve margin is the result of 

 the combined effect of three different components: 



(a) a radial component radiating from the umbo 

 and acting in the plane of the generating curve, 



(b) a transverse component acting at right angles 

 to the plane of the generating curve, and (c) a 

 tangential component acting in the plane of the 

 generating curve and tangentially to it. The 

 turbinate-spiral form of some bivalve shells is due 

 to the presence of the tangential component which 

 in plani-spiral shells may be absent or inconspicu- 

 ous. Likewise, the transverse component may be 

 greatly reduced or even absent in the valve. 

 Thus, from this point of view the great variety of 

 shell forms may be explained as an interaction of 

 the three components (fig. 29). Owen's point of 



Figure 27. — Valve of a shell sawed along the directive a.xis 

 describes a plane logarithmic spiral. According to 

 Lison (1942). OM — radius vector; T — tangent; 

 — umbo; V — angle between the two radii. 



poslenor 



Figure 28. — Comparison of directive plane of Lison with 

 normal axis (Owen). A — shell not affected bj' tangen- 

 tial component; B — shell affected by tangential com- 

 ponent. 



MORPHOLOGY AND STRUCTURE OF SHELL 



25 



