CHAPTER XI THE LOGARITHMIC SPIRAL

of the spiral shell. They are independent of the shape and size of the embryo, and depend only (as we shall see better presently) on the direction and relative rate of growth of the double contour of the shell.

A geometric diagram showing the relationship between radial distance $r$, angle $\theta$, and the angle $\phi$ on a curve.

Now that we have dealt, in a very general way, with some of the more obvious properties of the logarithmic spiral, let us consider certain of them a little more particularly, keeping in {532} view as our chief object the inôÙvesôÙtiôÙgaôÙtion (on elementary lines) of the possible manner and range of variation of the molluscan shell.

There is yet another equation to the logarithmic spiral, very commonly employed, and without the help of which we shall find that we cannot get far. It is as follows:

This follows directly from the fact that the angle öÝ (the angle between the radius vector and the tangent to the curve) is constant.

For, then,

therefore

and, integrating,

As we

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