XI] 



OF BIVALVE SHELLS 



825 



margins of the shell, or in other words the successive positions 

 assumed during growth by the growing generating curve; and we 

 have a good illustration, accordingly, of how it is characteristic of 

 the generating curve that it should constantly increase, while never 

 altering its geometric similarity. 



Underlying these lines of growth, which are so characteristic 

 of a moUuscan shell (and of not a few other organic formations), 

 there is, then, a law of growth which we may attempt to enquire 

 into and which may be illustrated in various ways. The simplest 

 cases are those in which we can study the lines of growth on a more 

 or less flattened shell, such as the one valve of an oyster, a Pecten 

 or a Tellina, or some such bivalve mollusc. Here around an origin, 

 the so-called "'umbo" of the shell, we have a series of curves, some- 

 times nearly circular, sometimes elliptical, often asymmetrical; and 

 such curves are obviously not "concentric," though we are often apt 

 to call them so, but have a common centre of similitude. This 

 arrangement may be illustrated by various analogies. We might 

 for instance compare it to a series of waves, radiating outwards 

 from a point, through a medium which offered a resistance increasing, 

 with the angle of divergence, according to some simple law. We 

 may find another and perhaps a simpler illustration as follows: 



In a simple and beautiful theorem, Galileo shewed that, if we 

 imagine a number of inclined planes, or gutters, sloping downwards 

 (in a vertical plane) at various angles 

 from a common starting-point, and if 

 we imagine a number of balls rolling 

 each down its own gutter under the 

 influence of gravity (and without 

 hindrance from friction), then, at any 

 given instant, the locus of all these 

 moving bodies is a circle passing 

 through the point of origin. For the 

 acceleration along any one of the 

 sloping paths, for instance AB (Fig. 

 400), is such that 



AB = ig cos d.t^ 



Therefore 



^ig.AB/AC.t^. 

 t^=2lg.AC. 



