In a tubular spiral, whether plane or helicoid, the consecutive whorls may either be (1) isolated and remote from one another; or (2) they may precisely meet, so that the outer border of one and the inner border of the next just coincide; or (3) they may overlap, the vector plane of each outer whorl cutting that of its immediate predecessor or predecessors.
Looking, as we have done, upon the spiral shell as being essentially a cone rolled up, it is plain that, for a given spiral angle, intersection or non-intersection of the successive whorls will depend upon the apical angle of the original cone. For the wider the cone, the more rapidly will its inner border tend to encroach on the outer border of the preceding whorl.
But it is also plain that the greater be the apical angle of the cone, and the broader, consequently, the cone itself be, the greater difference will there be between the total lengths of its inner and outer border, under given conditions of flexure. And, since the inner and outer borders are describing precisely the same spiral about the pole, it is plain that we may consider the inner border as being retarded in growth as compared with the outer, and as being always identical with a smaller and earlier part of the latter.
If λ be the ratio of growth between the outer and the inner curve, then, the outer curve being represented by
r = a eθ cot α ,
the equation to the inner one will be
r′ = aλeθ cot α , or
r′ = a e(θ − β)cot α ,
and β may then be called the angle of retardation, to which the inner curve is subject by virtue of its slower rate of growth. {542}