XI] OF SHELLS GENERALLY 547 



also with the new possibihty of contact or intersection on the 

 02)posite 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 Blake's paper already referred to. 

 When one whorl overlaps another, so that the generating 

 curve cuts its predecessor (at a distance of 27r) on the same radius 

 vector, the locus of intersection will follow a spiral line upon the 

 shell, which is called the " suture " by conchologists. It is evidently 

 one of that ensemble of spiral lines in space of which, as we have 

 seen, the whole shell may be conceived to be constituted; and we 

 might call it a "contact-spiral," or "spiral of intersection." In 

 discoid shells, such as an Ammonite or a Planorbis, or in Nautilus 

 umbilicatus, there are obviously two such contact-spirals, one on 

 each side of the shell, that is to say one on each side of a plane 

 perpendicular to the axis. In turbinate shells such a condition 

 is also possible, but is somewhat rare. We have it for instance, 

 in Solarium 'persjpectivum, where the one contact-spiral is visible 

 on the exterior of the cone, and the other lies internally, 

 winding round the open cone of the umbilicus*; but this second 

 contact-spiral is usually imaginary, or concealed within the 

 whorls of the turbinated shell. Again, in Haliotis, one of the 

 contact-spirals is non-existent, because of the extreme obliquity 

 of the plane of the generating curve. In Scalaria fretiusa and 

 in Spirula there is no contact-spiral, because the growth of the 

 generating curve has been too slow, in comparison with the vector 

 rotation of its plane. In Argonauta and in Cypraea, there is no 

 contact-spiral, because the growth of the generating curve has 

 been too quick. Nor, of course, is there any contact-spiral in 

 Patella or in Dentalium, because the angle a is too small ever to 

 give us a complete revolution of the spire. 



* A beautiful construction: stupendum Naturae artificium, Linnaeus. 



35—2 



