Now in the three cases (a, b, c) represented in Fig. 278, it is
plain that rÿ£¢ãý
and
The case in which ö£eÿ£¢2üã₤cotã₤öÝ =ã₤1, or ãlogã₤ö£ =ã₤2üã₤cotã₤öÝã₤logã₤öç, is the case represented in Fig. 278, b: that is to say, the particular case, for each value of öÝ, where the consecutive whorls just touch, without interspace or overlap. For such cases, then, we may tabulate the values of ö£, as follows:
| Constant angle öÝ of spiral | Ratio (ö£) of rate of growth of inner border of tube, as compared with that of the outer border |
|---|---|
| 89ô¯ | ôñ896ã |
| 88ã | ôñ803ã |
| 87ã | ôñ720ã |
| 86ã | ôñ645ã |
| 85ã | ôñ577ã |
| 80ã | ôñ330ã |
| 75ã | ôñ234ã |
| 70ã | ôñ1016 |
| 65ã | ôñ0534 |
We see, accordingly, that in plane spirals whose constant angle lies, say, between 65ô¯ and 70ô¯, we can only obtain contact between consecutive whorls if the rate of growth of the inner border of the tube be a small fraction,ãa tenth or a twentiethãof that of the outer border. In spirals whose constant angle is 80ô¯, contact is attained when the respective