532 



THE LOGARITHMIC SPIRAL 



[CH. 



view as our chief object the investigation (on elementary hnes) 

 of the possible manner and range of variation of the molluscan 

 shell. 



There is yet another equation to the logarithmic spiral, 

 very commonly employed, and without the 

 help of which we shall find that we cannot 

 dr get far. It is as follows : 



"^^ ^ _ ^ecota 



This follows directly from the fact that 

 the angle a (the angle between the radius 

 vector and the tangent to the curve) is 

 constant. 



For, then, 



Fig. 269. 

 and, integrating, 



tan a (= tan ^) = rddjdr, 

 therefore drlr = dd cot a, 



log r 



or 



r = € 



6 cot a, 



Scot a. 



As we have seen throughout our prehminary discussion, the 

 two most important constants (or chief "specific characters," as 

 the naturalist would say) in any given logarithmic spiral, are 

 (1) the magnitude of the angle of the spiral, or "constant angle," 

 a, and (2) the rate of increase of the radius vector for any given 

 angle of revolution, 6. Of this latter, the simplest case is when 

 6 = 277, or 360° ; that is to say when we compare the breadths, 

 along the same radius vector, of two successive whorls. As our 

 two magnitudes, that of the constant angle, and that of the ratio 

 of the radii or breadths of whorl, are related to one another, we 

 may determine either of them by actual measurement and proceed 

 to calculate the other. 



In any complete spiral, such as that of Nautilus, it is (as we 

 have seen) easy to measure any two radii (r), or the breadths in 



