r = 2 a b2 cos θ ⁄ ((b2 − a2) cos2 θ + a2).
Obviously, in the case when a = b, this gives us the circular system which we have already considered. For other values, or ratios, of a and b, and for all values of θ, we can easily construct a table, of which the following is a sample:
| θ | a ⁄ b = 1 ⁄ 3 | 1 ⁄ 2 | 2 ⁄ 3 | 1 ⁄ 1 | 3 ⁄ 2 | 2 ⁄ 1 | 3 ⁄ 1 |
|---|---|---|---|---|---|---|---|
| 0° | 1·0 | 1·0 | 1·0 | 1·0 | 1·0 | 1·0 | 1·0 |
| 10 | 1·01 | 1·01 | 1·002 | ·985 | ·948 | ·902 | ·793 |
| 20 | 1·05 | 1·03 | 1·005 | ·940 | ·820 | ·695 | ·485 |
| 30 | 1·115 | 1·065 | 1·005 | ·866 | ·666 | ·495 | ·289 |
| 40 | 1·21 | 1·11 | ·995 | ·766 | ·505 | ·342 | ·178 |
| 50 | 1·34 | 1·145 | ·952 | ·643 | ·372 | ·232 | ·113 |
| 60 | 1·50 | 1·142 | ·857 | ·500 | ·258 | ·152 | ·071 |
| 70 | 1·59 | 1·015 | ·670 | ·342 | ·163 | ·092 | ·042 |
| 80 | 1·235 | ·635 | ·375 | ·174 | ·078 | ·045 | ·020 |
| 90 | 0·0 | 0·0 | 0·0 | 0·0 | 0·0 | 0·0 | 0·0 |
Fig. 288.
The coaxial ellipses which we then draw, from the values given in the table, are such as are shewn in Fig. [288] for the ratio a ⁄ b = 3 ⁄ 1, and in Fig. [289] for the ratio a ⁄ b = 1 ⁄ 2 ; these are fair approximations to the actual outlines, and to the actual arrangement of the lines of growth, in such forms as Solecurtus or Cultellus, and in Tellina or Psammobia. It is not difficult to introduce a constant into our equation to meet the case of a shell which is somewhat unsymmetrical on either side of the median axis. It is a somewhat more troublesome matter, however, to bring these configurations into relation with a “law of growth,” as was so easily done in the case of the circular figure: in other words, to {565} formulate a law of acceleration according to which points starting from the origin O, and moving along radial lines, would all lie, at any future epoch, on an ellipse passing through O; and this calculation we need not enter into.
Fig. 289.
All that we are immediately concerned with is the simple fact that where a velocity, such as our rate of growth, varies with its direction,—varies that is to say as a function of the angular divergence from a certain axis,—then, in a certain simple case, we get lines of growth laid down as a system of coaxial circles, and, when the function is a more complex one, as a system of ellipses or of other more complicated coaxial figures, which figures may or may not be symmetrical on either side of the axis. Among our bivalve mollusca we shall find the lines of growth to be approximately circular in, for instance, Anomia; in Lima (e.g. L. subauriculata) we have a system of nearly symmetrical ellipses with the vertical axis about twice the transverse; in Solen pellucidus, we have again a system of lines of growth which are not far from being symmetrical ellipses, in which however the transverse is between three and four times as great as the vertical axis. In the great majority of cases, we have a similar phenomenon with the further complication of slight, but occasionally very considerable, lateral asymmetry.
In certain little Crustacea (of the genus Estheria) the carapace takes the form of a bivalve shell, closely simulating that of a {566} lamellibranchiate mollusc, and bearing lines of growth in all respects analogous to or even identical with those of the latter. The explanation is very curious and interesting. In ordinary Crustacea the carapace, like the rest of the chitinised and calcified integument, is shed off in successive moults, and is restored again as a whole. But in Estheria (and one or two other small crustacea) the moult is incomplete: the old carapace is retained, and the new, growing up underneath it, adheres to it like a lining, and projects beyond its edge: so that in course of time the margins of successive old carapaces appear as “lines of growth” upon the surface of the shell. In this mode of formation, then (but not in the usual one), we obtain a structure which “is partly old and partly new,” and whose successive increments are all similar, similarly situated, and enlarged in a continued progression. We have, in short, all the conditions appropriate and necessary for the development of a logarithmic spiral; and this logarithmic spiral (though it is one of small angle) gives its own character to the structure, and causes the little carapace to partake of the characteristic conformation of the molluscan shell.