XI] OF VARIOUS UNIVALVES 555 



form is seldom open to easy mathematical expression, save when 

 it is an actual circle or ellipse ; but an exception to this rule may- 

 be found in certain Ammonites, forming the group "Cordati.'" 

 where (as Blake points out) the curve is very nearly represented 

 by a cardioid, whose equation is r = a (1 + cos 6). 



The generating curve may grow slowly or quickly ; its growth- 

 factor is very slow in Dentalium or Turritella, very rapid in Nerita, 

 or Pileopsis, or Haliotis or the Limpet. It may contain the axis 

 in its plane, as in Nautilus ; it may be parallel to the axis, as in 

 the majority of Gastropods ; or it may be inclined to the axis, as 

 it is in a very marked degree in Hahotis. In fact, in Hahotis 

 the generating curve is so obhque to the axis of the shell that 

 the latter appears to grow by additions to one margin only (cf. 

 Fig. 258), as in the case of the opercula of Turbo and Nerita 

 referred to on p. 522 ; and this is what Moseley supposed it to do. 



A 



Fig. 284. A, Lamellaria perspicua; B, Sigaretus lialiotoides. 

 (After Woodward.) 



The general appearance of the entire shell is determined (apart 

 from the form of its generating curve) by the magnitude of three 

 angles ; and these in turn are determined, as has been sufficiently 

 explained, by the ratios of certain velocities of growth. These 

 angles are (1) the constant angle of the logarithmic spiral (a); 

 (2) in turbinate shells, the enveloping angle of the cone, or (taking 

 half that angle) the angle {d) which a tangent to the whorls makes 

 with the axis of the shell; and (3) an angle called the "angle of 

 retardation" (j8), which expresses the retardation in growth of 



male than in the female. But this difference is not to be detected in the young; 

 in other words, the form of the generating curve perceptibly alters with advancing 

 age. Somewhat similar differences in the shells of Ammonites were long ago 

 suspected, by D'Orbigny, to be due to sexual differences. (Cf. Willey, Natural 

 Science, vi, p. 411, 1895; Zoological Results, p. 742, 1902.) 



