Then, if ö¡ increase by 2ü, while r increases to rÿ£¢1ã₤,
which leads, by subtraction to
Now, as öÝ tends to 0, k (i.e. cotã₤öÝ) tends to ã, and therefore, as kã₤ãã₤ã, log(rÿ£¢1ã₤ãã₤r)ã₤ãã₤ã and also rÿ£¢1ã₤ãã₤rã₤ãã₤ã.
Therefore if one whorl exists, the radius vector of the other is infinite; in other words, there is nowhere, even in the near neighbourhood of the pole, a complete revolution of the spire. Our spiral shells of small constant angle, such as Dentalium, may accordingly be considered to represent sufficiently well the true commencement