506 



THE LOGARITHMIC SPIRAL 



[CH. 



Mg. 246. 



In such a spiral, radial growth and growth in the direction of 

 the curve bear a constant ratio to one another. For, if we consider 

 a consecutive radius s^ector, OP' , whose increment 

 as compared with OP is dr, while ds is the small 

 arc PP' , then 



drjds = cos a = constant. 



In the concrete case of the shell, the distribution 

 of forces will be, originally, a little more compli- 

 cated than this, though by resolving the forces in 

 question, the system may be reduced to this 

 simple form. And furthermore, the actual distri- 

 bution of forces will not always be identical; 

 for example, there is a distinct difference between the cases (as 

 in the snail) where a columellar muscle exerts a definite traction 

 in the direction of the pole, and those (such as Nautilus) where 

 there is no columellar muscle, and where some other force must 

 be discovered, or postulated, to account for the flexure. In the 



most frequent case, we have, as 

 in Fig. 247, three forces to deal 

 with, acting at a point, p : 

 L, acting in the direction of 

 the tangent to the curve, and 

 representing the force of longi- 

 tudinal growth ; T, perpen- 

 dicular to L, and representing 

 the organisna's tendency to grow 

 in breadth ; and P, the traction 

 exercised, in the direction of the 

 pole, by the columellar muscle. 

 Let us resolve L and T into 

 components along P (namely 

 A' , B'), and perpendicular to P (namely A, B) ; we have now only 

 two forces to consider, viz. P — A' — B' , and A — B. And these 

 two latter we can again resolve, if we please, so as to deal only 

 with forces in the direction of P and T. Now, the ratio of these 

 forces remaining constant, the locus of the point 'p is an equiangular 

 spiral. 



