XI] OF SHELLS GENERALLY 803 



or overlap of the successive whorls, are simply the outward ex- 

 pression of a varying ratio in the rate of growth of the outer as 

 compared with the inner border of the "tubular shell. 



The foregoing problem of contact, or intersection, of successive 

 whorls is a very simple one in the case of the discoid shell but 

 a more complex one 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 360°, and the shape of 

 the whorls need not be considered. 



As the angle of retardation diminishes from 360°, the whorls stand 

 further and further apart' in an open coil; as it increases beyond 360°, 

 they overlap more and more ; 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 Httle more obscure in Nautilus pompilius. 



In the turbinate shell the problem of contact is twofold, for v.\, 

 have to deal with the possibihties of contact on the same side of 

 the axis (which is what we have dealt with in the discoid) and also 

 with the new possibiHty of contact or mtersection on the opposite 

 side; it is this latter case which will .determine the presence or 

 absence of an open umbilicus. 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 

 eUipse, then contact will be found to depend on the angle which 

 the major axis of this elhpse makes with the axis of the shell. The 

 question becomes a compHcated 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 fine upon the shell, 

 whjch is called the "suture" by conchologists. It is 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 



