view is basically similar to Huxley's l^vpothesis 

 (1932) of differential length growth and width 

 growth of moUuscan shells. Owen correctly 

 points out an error in Lison's interpretations that 

 the lines of equal potential activities involved in 

 the secreting of shell material at the edges of the 

 valves are parallel to each other. This is obvi- 

 ously not the case since all lines of growth of the 

 lamellibranch shell radiate from a common node 

 of minimum growth near the umbo. For this 

 reason the comparison of bivalves can be more 

 conveniently made by using radial coordinates as 

 has been shown by Yonge (1952a, 1952b). 



The mathematical properties of shell surfaces 

 are of interest to the biologist because they may 

 provide clues to understanding the quantitative 

 aspects of the processes of shell formation. It can 

 be a priori accepted that any organism grows in 

 an orderly fashion following a definite pattern. 

 The origin of this pattern and the nature of the 

 forces responsible for laying out structural ma- 

 terials in accordance with the predetermined 

 plan are not known. The pattern of shell 

 structure is determined by the activities at the 

 edge of the shell-forming organ, the mantle. At 

 the present state of our knowledge it is impossible 

 to associate various geometrical terms which 

 describe the shape of the shell with concrete 

 physiological processes and to visualize the 

 morphogenetic and biochemical mechanisms in- 

 volved in the formation of definite sculptural and 

 color patterns. The solution of this problem will 



be found by experimental and biochemical studies 

 wliich ma_y supply biological meanings to abstract 

 matliematical concepts and equations. Experi- 

 mental study of the morphogenesis of shells 

 oft'ers splendid opportunities for this type of 

 research. 



GROWTH RINGS AND GROWTH RADH 



Nearly 250 years ago Reaumur (1709) discovered 

 that shells grow by the accretion of material 

 secreted at their edges. Since that time this 

 important observation has been confirmed by 

 numerous subsequent investigations. The rings 

 on the outer surfaces of a bivalve shell, frequently 

 but incorrectly described as "concentric", rep- 

 resent the contours of the shell at dift'erent ages. 

 Rings are common to all bivalves but are partic- 

 ularly pronounced on the flattened shells of 

 scallops, clams, and fresh-water mussels. De- 

 pending on the shape of the shell, the rings are 

 either circular or oval with a common point of 

 origin at the extreme dorsal side near the umbo 

 (figs. 30 and 31). The diagrams clearly show 

 that the rate of growth along the edge of the 

 shell is not uniform. It is greater along the 

 radius, AD, which corresponds to the directive 

 axis of Lison, and gradually decreases on both 



Figure 29. — Normal axis and the two growth components 

 in the shell of scallop. LS — plane perpendicular to the 

 plane of the generating curve; X — turning point of the 

 concave side of the shell shown at right; M and O — aux- 

 iliary radii; P — transverse component; R — radial com- 

 ponent; UY — normal axis. From Owen (1953). 



Figure 30. — Diagram of a circular bivalve shell of the 

 kind represented in Perten, Anomia, and young C. 

 virginica. Radii extending from the umbo to the 

 periphery of the generating curve are proportional to 

 the rate of growth at the edge of a circular shell 

 Radius AD corresponds to the directive axis of Lison. 



26 



FISH AND WILDLIFE SERVICE 



