546 THE LOGARITHMIC SPIRAL [ch. 



it is evident that when j3 = 2tt, that will mean that A = 1. In 

 other words, the outer and inner borders of the tube are identical, 

 and the tube is constituted by one continuous hne. 



When A is a very small fraction, that is to say when the rates 

 of growth of the two borders of the tube are very diverse, then 

 j8 will tend towards infinity — tend that is to say towards a con- 

 dition in which the inner border of the tube never grows at all. 

 This condition is not infrequently approached in nature. The 

 nearly parallel- sided cone of DentaUum, or the widely separated 

 whorls of Lituites, are evidently cases where A nearly approaches 

 unity in the one case, and is still large in the other, jS being 

 correspondingly small; while we can easily find cases where j8 is 

 very large, and A is a small fraction, for instance in Hahotis, or 

 in Gryphaea. 



For the purposes of the morphologist, then, the main result 

 of this last general investigation is to shew that all the various 

 types of "open" and "closed" spirals, all the various degrees of 

 separation or overlap of the successive whorls, are simply the 

 outward expression 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 the suc- 

 cessive 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 

 will stand further and further apart in an open coil ; as it increases 

 beyond 360°, 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 



