802 THE EQUIANGULAR SPIRAL [ch. 



The corresponding values of A are as follows : 



Ratio (\) of rates of growth of outer and inner 



border, such as to produce a spiral with interspaces 



between the wliorls, the breadth of which 



intei-spaces is a mean proportional between the 



Constant angle (a) breadths of the whorls themselves 



90° 1-00 (imaginary) 



89 0-95 



88 0-89 



87 0-85 



86 0-81 



85 0-76 



80 0-57 



75 0-43 



70 0-32 



65 0-23 



60 018 



55 013 



50 0090 



45 0-063 



40 0042 



35 0026 



30 0016 . 



As regards the angle of retardation, y, in the formula 

 / = Ae^^o*«, or / = e(^-^)cota^ 

 and in the case 



r' = e(2^-y)cota^ ^j. _ log A - (277 - y) cot a, 



it is evident that when y = 27t, 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 

 y wfll tend towards infinity — tend that is to say towards a condition 

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

 condition is not infrequently approached in nature. I take it that 

 Cyfraea is such a case. But the nearly parallel-sided cone of 

 Dentalium, or the widely separated whorls of Lituites, are cases 

 where A nearly approaches unity in the one case, and is still large 

 in the other, y being correspondingly small ; while we can easily find 

 cases where y is very large, and A is a small fraction, for instance 

 in Haliotis, in Calyptraea, or in Gryphaea. 



For the purposes of the morphologist, thoji, 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 



