XI] 



IN ITS DYNAMICAL ASPECT 



505 



For, in the logarithmic spii'al, when a tends to 90°, the expression r = a^^°^"' 

 tends to r = a (1 4- ^ cot «); while the equation to the Archimedean spiral is 

 r = bd. The nummulite must always have a central core, or initial cell, 

 around which the coil is not only wrapped, but out of which it springs ; and 

 this initial chamber corresponds to our a' in the expression r = a' + a6 cot a. 

 The outer whorls resemble those of an Archimedean spiral, because of the 

 other term a6 cot a in the same expression. It follows from this that in all 

 such cases the whorls must be of excessively small breadth. 



There are many other specific properties of the logarithmic 

 spiral, so interrelated to one another that we may choose pretty 

 well any one of them as the basis of our definition, and deduce the 

 others from it either by analytical methods or by the methods of 

 elementary geometry. For instance, the equation r — a^ may be 

 written in the form log r = 6 log a, or ^ = log r/log a, or (since a is 

 a constant), 6 = k log r. Which is as much as to say that the 

 vector angles about the pole are proportional to the logarithms 

 of the successive radii ; from which circumstance the name of the 

 "logarithmic spiral" is derived. 



Let us next regard our logarithmic spiral from the dynamical 

 point of view, as when we consider the forces concerned in the 

 growth of a material, concrete spiral. 

 In a growing structure, let the forces of 

 growth exerted at any point P be a 

 force F acting along the fine joining P 

 to a pole and a force T acting in a 

 direction perpendicular to OP; and let 

 the magnitude of these forces be in the 

 same constant ratio at all points. It 

 follows that the resultant of the forces 

 F and T (as PQ) makes a constant 

 angle with the radius vector. But the 

 constancy of the angle between tangent 

 and radius vector at any point is a 

 fundamental property of the logarithmic spiral, and may be 

 shewn to follow from our definition of the curve : it gives to the 

 curve its alternative name of equiangular spiral. Hence in a 

 structure growing under the above conditions the form of the 

 boundary will be a logarithmic spiral. 



