7.98 THE EQUIANGULAR SPIRAL [ch. 



that of a cone with shghtly curving sides : in which, that is to say, 

 there is a sHght acceleration of growth in a transverse as compared 

 with the longitudinal direction. 



In a tubular spiral, whether plane or hehcoid, the consecutive 

 whorls may either be (1) isolated and remote from one another; 

 or (2) they may precisely meet, so that the outer border of one 

 and the inner border of the next just coincide; or (3) they may 

 overlap, the vector plane of each outer whorl cutting that of its 

 immediate predecessor or predecessors. 



Looking, as we have done, upon the spiral shell as being essentially 

 a cone roUed up*, it is plain that, for a given spiral angle, intersection 

 or non-intersection of the successive whorls will depend upon the 

 apical angle of the original cone. For the wider the cone, the more 

 will its inner border tend to encroach on the preceding whorl. But 

 it is also plain that the greater the apical angle of the cone, and the 

 broader, consequently, the cone itself, the greater difference will 

 there be between the total lengths of its inner and outer borders. 

 And, since the inner and outer borders are describing precisely 

 the same spiral about the pole, we may consider the inner border 

 as being retarded in growth as compared with the outer, and as 

 being always identical with a smaller and earUer part of the latter. 



If A be the ratio of growth between the outer and the inner curve, 

 then, the outer curve being represented by 



r = ae^ootcc^ 



the equation to the inner one will >^e 



/ = aAe^cota^ 



or r' = ae^^-y^^^^"", 



* To speak of a cone "rolling up," and becoming a nautiloid spiral by doing so, 

 is a rough and non-mathematical description; nor is it easy to see how a cone of 

 wide angle could roll up, and yet remain a cone. But if (i) the centre of a sphere 

 move along a straight line and its radius keep proportional to the distance the 

 centre has moved, the sphere generates as its envelope a circular cone of which 

 the straight line is the axis; and so, similarly, if (ii) the centre of a sphere move 

 along an equiangular spiral and its radius keep proportional to the arc-distance 

 along the spiral back to the pole, the sphere generates as its envelope a self-similar 

 shell-surface, or nautiloid spiral. 



