will be a resultant force BG , acting in a direction intermediate between Bb and BD , and also a resultant, DH , acting at D in an opposite direction; and accordingly, after a small increment of growth, the growing end of the cylinder will come to lie, not in the direction BD , but in the direction GH . The problem is therefore analogous to that of a beam to which we apply a bending moment; and it is plain that the unequal force of growth is equivalent to a ã couple ã which will impart to our structure a curved form. For, if we regard the part ABDC as practically rigid, and the part BBÿ£¢ãýDÿ£¢ãýD as pliable, this couple {500} will tend to turn strips such as Bÿ£¢ãýDÿ£¢ãý about an axis perpendicular to the plane of the diagram, and passing through an intermediate point Fÿ£¢ãý . It is plain, also, since all the forces under consideration are intrinsic to the system , that this tendency will be continuous, and that as growth proceeds the curving body will assume either a circular or a spiral form. But the tension which we have here assumed to exist in the direction BD will obviously disappear if we suppose a sufficiently rapid rate of growth in that direction. For if we may regard the mouth of our tubular shell as perfectly extensible in its own plane, so that it exerts no traction whatsoever on the sides, then it will be drawn out into more and more elongated ellipses, forming the more and more oblique orifices of a straight tube. In other words, in such a structure as we have presupposed, the existence or
maintenance of a constant ratio between the rates of extension or growth in the vertical and transverse directions will lead, in general, to the development of a logarithmic spiral; the magnitude of