536 THE LOGARITHMIC SPIRAL [ch. 



curvature had already greatly diminished. That is to say we might suppose 

 that, however small the angle a, and however rapidly the whorls accordingly 

 increased, there would nevertheless be a manifest spu'al convolution in the 

 immediate neighbourhood of the pole, as the starting point of the curve. 

 But it may be shewn that this is not so. 



For, taking the formula r = ae^*'"'*^", 



this, for any given spiral, is equivalent to ae . 



Therefore log {r/a) = kd, 



or, Ik = . --—. . 



log (r/a) 



Then, if 6 increase by 27r, while r increases to r^, 



1_ 6 + 27r 

 k~ log {rja)' 



which leads, by subtraction to 



1 X^ . log (rjr) = 277. 



Now, as a tends to 0, k (i.e. cot a) tends to oo , and therefore, as k ^ go , 



log (rjr) >■ GO and also r-^/r > oo . 



Therefore if one whorl exists, the radius vector of the other is infinite; 

 in other words, there is nowhere, even in the near neighbourhood of the 

 pole, a complete revolution of the spire. Our spiral shells of small constant 

 angle, such as Dentalium, may accordingly be considered to represent suf- 

 ficiently well the true commencement of then- respective spirals. 



Let us return to the problem of how to ascertain, by direct 

 measurement, the spiral angle of any particular shell. The 

 method already employed is only applicable to complete spirals, 

 that is to say to those in which the angle of the spiral is large, 

 and furthermore it is inapplicable to portions, or broken fragments, 

 of a shell. In the case of the broken fragment, it is plain that the 

 determination of the angle is not merely of theoretic interest, 

 but may be of great practical use to the conchologist as being the 

 one and only way by which he may restore the outline of the 

 missing portions. We have a considerable choice of methods, 

 which have been summarised by, and are partly due to, a very 

 careful student of the Cephalopoda, the late Rev. J. F. Blake*. 



* On the Measurement of the Curves formed by Cephalopods and other Mollusks 

 Phil. Mag. (5), vi, pp. 241-263, 1878. 



