Fig. 269.
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 investigation (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:
r = εθ cot α .
This follows directly from the fact that the angle α (the angle between the radius vector and the tangent to the curve) is constant.
For, then,
tan α (= tan ϕ) = r dθ ⁄ dr,
therefore
dr ⁄ r = dθ cot α,