In this case, let us call OA =ã₤R, OC =ã₤Rÿ£¢1 and OB =ã₤r. We then have
And
whence, equating,
The corôÙreôÙsponôÙding values of ö£ are as follows:
| Constant angle (öÝ) | Ratio (ö£) 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 |
|---|---|
| 90ô¯ | 1ôñ00ã (imaginary) |
| 89ã | ãôñ95ã |
| 88ã | ãôñ89ã |
| 87ã | ãôñ85ã |
| 86ã | ãôñ81ã |
| 85ã | ãôñ76ã |
| 80ã | ãôñ57ã |
| 75ã | ãôñ43ã |
| 70ã | ãôñ32ã |
| 65ã | ãôñ23ã |
| 60ã | ãôñ18ã |
| 55ã | ãôñ13ã |
| 50ã | ãôñ090 |
| 45ã | ãôñ063 |
| 40ã | ãôñ042 |
| 35ã | ãôñ026 |
| 30ã | ãôñ016 |
As regards the angle of retardation, öý, in the formula
and in the case
it is evident that when öý =ã₤2ü, that will mean that ö£ =ã₤1. In other words, the outer and inner borders of the tube are identical, and the tube is constituted by one continuous line.
When ö£ 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 öý will tend towards infinityãtend 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