along one diameter on both sides of the pole. The ratio between alternate measurements is therefore the same ratio as Moseley adopted, namely the ratio of breadth between contiguous whorls along a radius vector. I have then added to these observed values the corôÙreôÙsponôÙding calculated values of the angle öÝ, as obtained from our usual formula.
There is considerable irregularity in the ratios derived from these measurements, but it will be seen that this irregularity only implies a variation of the angle of the spiral between about 85ô¯ and 87ô¯; and the values fluctuate pretty regularly about the mean, which is 86ô¯ã₤15ÿ£¢ãý. Considering the difficulty of measuring the whorls, especially towards the centre, and in particular the difficulty of determining with precise accuracy the position of the pole, it is clear that in such a case as this we are scarcely justified in asserting that the law of the logarithmic spiral is departed from.
In some cases, however, it is undoubtedly departed from. Here for instance is another table from Grabau, shewing the corôÙreôÙsponôÙding ratios in an Ammonite of the group of Arcestes tornatus. In this case we see a distinct tendency of the ratios to
| Breadth of whorls (180ô¯ apart) mm. | Ratio of breadth of successive whorls (360ô¯ apart) | The spiral angle (öÝ) as calculated |
|---|---|---|
| 0ôñ25 | ã | ããô ãô |
| 0ôñ30 | 1ôñ400 | 86ô¯ã₤56ÿ£¢ãý |
| 0ôñ35 | 1ôñ667 | 85ãô 21ã |
| 0ôñ50 | 2ôñ000 | 83ãô 42ã |
| 0ôñ70 | 2ôñ000 | 83ãô 42ã |
| 1ôñ00 | 2ôñ000 | 83ãô 42ã |
| 1ôñ40 | 2ôñ100 | 83ãô 16ã |
| 2ôñ10 | 2ôñ179 | 82ãô 56ã |
| 3ôñ05 | 2ôñ238 | 82ãô 42ã |
| 4ôñ70 | 2ôñ492 | 81ãô 44ã |
| 7ôñ60 | 2ôñ574 | 81ãô 27ã |
| 12ôñ10 | 2ôñ546 | 81ãô 33ã |
| 19ôñ35 | ã | ããô ãô |
| Mean | 83ô¯ã₤22ÿ£¢ãý |
increase as we pass from the centre of the coil outwards, and consequently for the values of the angle öÝ to diminish. The case is precisely comparable to that of a cone with slightly curving sides: in which, that is to say, there is a slight acceleration of growth in a transverse as compared with the longitudinal direction.
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