that ratio will determine the character (that is to say, the constant angle) of the spiral; and the spirals so produced will include, as special or limiting cases, the circle and the straight line.
We may dispense with the hypothesis of bending moments, if we simply presuppose that the increments of growth take place at a constant angle to the growing surface (as AB), but more rapidly at A (which we shall call the ãouter edgeã) than at B, and that this difference of velocity maintains a constant ratio. Let us also assume that the whole structure is rigid, the new accretions solidifying as soon as they are laid on. For example, {501} let Fig. 242 represent in section the early growth of a Nautilus-shell, and let the part ARB represent the earliest stage of all, which in Nautilus is nearly semicircular. We have to find a law governing the growth of the shell, such that each edge shall develop into an equiangular spiral; and this law, accordingly, must be the same for each edge, namely that at each instant the direction of growth makes a constant angle with a line drawn from a fixed point (called the pole of the spiral) to the point at which growth is taking place. This growth, we now find, may be considered as effected by the continuous addition of similar quadrilaterals. Thus, in Fig. 241, AEDB is a quadrilateral with AE, DB parallel, and with the angle EAB of a certain definite
magnitude, =ã₤ö°. Let AB and ED meet, when produced, in C ; and call the angle ACE (or xCy ) =ã₤öý. Make the angle yCz =ã₤angle xCy , =ã₤öý. Draw EG , so that the angle yEG =ã₤ö°, meeting Cz in G ; and draw DF parallel to EG . It is then easy to show that AEDB and EGFD are