The corresponding values of λ are as follows:
| Constant angle (α) | Ratio (λ) of rates of growth of outer and inner border, such as to produce a spiral with interspaces between the whorls, the breadth of which interspaces is a mean proportional between the breadths of the whorls themselves |
|---|---|
| 90° | 1·00 (imaginary) |
| 89 | ·95 |
| 88 | ·89 |
| 87 | ·85 |
| 86 | ·81 |
| 85 | ·76 |
| 80 | ·57 |
| 75 | ·43 |
| 70 | ·32 |
| 65 | ·23 |
| 60 | ·18 |
| 55 | ·13 |
| 50 | ·090 |
| 45 | ·063 |
| 40 | ·042 |
| 35 | ·026 |
| 30 | ·016 |
As regards the angle of retardation, β, in the formula
r′ = λeθ cot α , or r′ = e(θ − β)cot α ,
and in the case
r′ = e(2π − β)cot α , or −log λ = (2π − β)cot α,
{546}
it is evident that when β = 2π, that will mean that λ = 1. In other words, the outer and inner borders of the tube are identical, and the tube is constituted by one continuous line.
When λ is a very small fraction, that is to say when the rates of growth of the two borders of the tube are very diverse, then β will tend towards infinity—tend that is to say towards a condition in which the inner border of the tube never grows at all. This condition is not infrequently approached in nature. The nearly parallel-sided cone of Dentalium, or the widely separated whorls of Lituites, are evidently cases where λ nearly approaches unity in the one case, and is still large in the other, β being correspondingly small; while we can easily find cases where β is very large, and λ is a small fraction, for instance in Haliotis, or in Gryphaea.
For the purposes of the morphologist, then, the main result of this last general investigation is to shew that all the various types of “open” and “closed” spirals, all the various degrees of separation or overlap of the successive whorls, are simply the outward expression of a varying ratio in the rate of growth of the outer as compared with the inner border of the tubular shell.