XI] OF SHELLS GENERALLY 799 



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

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



Dispensing with mathematical formulae, the several conditions 

 may be illustrated as follows : 



In the diagrams (Fig. 385), OF-^F^F^, etc. represents a radius, 

 on which P^, F^, Pg are the points attained by the outer border 

 of the tubular shell after as many entire consecutive revolutions. 

 And Pi', P2', P3' are the points similarly intersected by the inner 

 border ; OFjOF' being always = A, which is the ratio of growth, 

 or ' ' cutting-down factor. ' ' Then, obviously, ( 1 ) when OP^ is less than 



Fig. 385. 



OP2' the whorls will be separated by an interspace (a); (2) when 

 OF^ = OP2' they will be in contact (6), and (3) when OF^ is greater 

 than OF^ there will be a greater or less extent of overlapping, 

 that is to say of concealment of the surfaces of the earher by the 

 later whorls (c). And as a further case (4), it is plain that if A be 

 very large, that is to say if OF-^ be greater, not only than OF^ 

 but also than OP3', OP4', etc., we shall have complete, or all but 

 complete, concealment by the last formed whorl of the whole of 

 its predecessors. This latter condition is completely attained in 

 Nautilus pompilius, and approached, though not quite attained, in 

 N. umbilicatus] and the difference between these two forms, or 

 "species," is constituted accordingly by a difference in the value 

 of A. (5) There is also a final case, not easily distinguishable 



