Note 507 — Chapter XI · THE LOGARITHMIC SPIRAL

In the case of Turbo, and all other “turbinate” shells, we are dealing not with a plane logarithmic spiral, as in Nautilus, but with a “gauche” spiral, such that the radius vector no longer revolves in a plane perpendicular to the axis of the system, but is inclined to that axis at some constant angle (θ). The figure still preserves its continued similarity, and may with strict accuracy be called a logarithmic spiral in space. It is evident that its envelope will be a right circular cone; and indeed it is commonly spoken of as a logarithmic spiral wrapped upon a cone, its pole coinciding with the apex of the cone. It follows that the distances of successive whorls of the spiral measured on the same straight line passing through the apex of the cone, are in geometrical progression, and conversely just as in the former case. But the ratio between any two consecutive interspaces (i.e. R3 − R2 ⁄ R2 − R1) is now equal to ε2π sin θ cot α , θ being the semi-angle of the enveloping cone. (Cf. Moseley, Phil. Mag. XXI, p. 300, 1842.)