cases of such a departure in Pupa, Clausilia, and various Bulimi, where the enveloping cone of the spire is not a right cone but a cone whose sides are curved. It is plain that this condition may arise in two ways: either by a gradual change in the ratio of growth of the whorls, that is to say in the logarithmic spiral itself, or by a change in the velocity of {550} translation along the axis, that is to say in the helicoid which, in all turbinate shells, is superposed upon the spiral. Very careful measurements will be necessary to determine to which of these factors, or in what proportions to each, the phenomenon is due. But in many Ammonitoidea where the helicoid factor does not enter into the case, we have a clear illustration of gradual and marked changes in the spiral angle itself, that is to say of the ratio of growth corôÙreôÙsponôÙding to increase of vectorial angle. We have seen from some of Naumannãs and Grabauãs measurements that such a tendency to vary, such an acceleration or retardation, may be detected even in Ammonites which present nothing abnormal to the eye. But let us suppose that the spiral angle increases somewhat rapidly; we shall then get a spiral with gradually narrowing whorls, and this condition is charôÙacôÙterôÙisôÙtic
of Oekotraustes, a subgenus of Ammonites. If on the other hand, the angle öÝ gradually diminishes, and even falls away to zero, we shall have the spiral curve opening out, as it does in Scaphites, Ancyloceras and Lituites, until the spiral coil is replaced by a spiral curve so gentle as to seem all but straight. Lastly, there are a few cases, such as Bellerophon expansus and some Goniatites, where the outer spiral does not perceptibly change, but the whorls become more ãembracingã or the whole shell more involute. Here it is the angle