XI] OF SHELLS GENERALLY 541 



increase as we pass from the centre of the coil outwards, and 

 consequently for the values of the angle a to diminish. The case 

 is precisely comparable to that of a cone with slightly curving 

 sides : in which, that is to say, there is a slight 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 rolled 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 rapidly will its inner border tend to 

 encroach on the outer border of the preceding whorl. 



But it is also plain that the greater be the apical angle of the 

 cone, and the broader, consequently, the cone itself be, the greater 

 difference will there be between the total lengths of its inner and 

 outer border, under given conditions of flexure. And, since the 

 inner and outer borders are describing precisely the same spiral 

 about the pole, it is plain that 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 earlier 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 



the equation to the inner one will be 



or r' = fle(^-^)cota^ 



and j8 may then be called the angle of retardation, to which the 

 inner curve is subject by virtue of its slower rate of growth. 



