and
λe2π cot α
1.
The case in which λe2π cot α = 1, or −log λ = 2π cot α log ε, is the case represented in Fig. [278], b: that is to say, the particular case, for each value of α, where the consecutive whorls just touch, without interspace or overlap. For such cases, then, we may tabulate the values of λ, as follows:
| Constant angle α of spiral | Ratio (λ) of rate of growth of inner border of tube, as compared with that of the outer border |
|---|---|
| 89° | ·896 |
| 88 | ·803 |
| 87 | ·720 |
| 86 | ·645 |
| 85 | ·577 |
| 80 | ·330 |
| 75 | ·234 |
| 70 | ·1016 |
| 65 | ·0534 |
{544}
We see, accordingly, that in plane spirals whose constant angle lies, say, between 65° and 70°, we can only obtain contact between consecutive whorls if the rate of growth of the inner border of the tube be a small fraction,—a tenth or a twentieth—of that of the outer border. In spirals whose constant angle is 80°, contact is attained when the respective rates of growth are, approximately, as 3 to 1; while in spirals of constant angle from about 85° to 89°, contact is attained when the rates of growth are in the ratio of from about 3 ⁄ 5 to 9 ⁄ 10.
Fig. 280.