in the turbinate. For in the discoid shell contact will evidently take place when the retardation of the inner as compared with the outer whorl is just 360ô¯, and the shape of the whorls need not be considered.
As the angle of retardation diminishes from 360ô¯, the whorls will stand further and further apart in an open coil; as it increases beyond 360ô¯, they will more and more overlap; and when the angle of retardation is infinite, that is to say when the true inner edge of the whorl does not grow at all, then the shell is said to be completely involute. Of this latter condition we have a striking example in Argonauta, and one a little more obscure in Nautilus pompilius.
In the turbinate shell, the problem of contact is twofold, for we have to deal with the possibilities of contact on the same side of the axis (which is what we have dealt with in the discoid) and {547} also with the new possibility of contact or intersection on the opposite side; it is this latter case which will determine the presence or absence of an umbilicus, and whether, if present, it will be an open conical space or a twisted cone. It is further obvious that, in the case of the turbinate, the question of contact or no contact will depend on the shape of the generating curve; and if we take the simple case where this generating curve may be considered as an ellipse, then contact will be found to depend on the angle which the major axis of this ellipse makes with the axis of the shell. The question becomes a complicated one, and the student will find it treated in Blakeãs paper already referred to.
When one whorl overlaps another, so that the generating curve cuts its predecessor (at a distance of 2ü) on the same