500 THE LOGAEITHMIC SPIRAL [ch. 



will tend to turn strips such as B'U about an axis perpendicular 

 to the plane of the diagram, and passing through an intermediate 

 point F' . 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 

 elhpses, 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 that ratio will determine the character (that is 

 to say, the constant angle) of the spiral; and the spirals so pro- 

 duced will include, as special or limiting cases, the circle and the 

 straight hne. 



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 "outer edge") 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, 



