and growth in the direction of the curve bear a constant ratio to one another. For, if we consider a consecutive radius vector, OPÿ£¢ãý, whose increment as compared with OP is dr, while ds is the small arc PPÿ£¢ãý, then
In the concrete case of the shell, the distribution of forces will be, originally, a little more complicated than this, though by resolving the forces in question, the system may be reduced to this simple form. And furthermore, the actual distribution 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, pã₤:ã₤L, acting
in the direction of the tangent to the curve, and representing the force of longitudinal growth; T, perpendicular to L, and representing the organismãs 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 Aÿ£¢ãý, Bÿ£¢ãý), and perpendicular to P (namely A, B); we have now only two forces to consider, viz. Pã₤ãã₤Aÿ£¢ãýã₤ãã₤Bÿ£¢ãý, and Aã₤ãã₤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. {507}
Furthermore we see how any slight change