542 



THE LOGARITHMIC SPIRAL 



[CH. 



Dispensing with mathematical formulae, the several conditions 

 may be illustrated as follows : 



In the diagrams (Fig. 278), OPiP^Ps, etc. represents a radius, 

 on which P^, P2, P3, 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 ; OP /OP' being always = A, which is the ratio of growth, 

 or "cutting-down factor." Then, obviously, when OPi is less 

 than OP2 the whorls will be separated by an interspace {a); 

 (2) when OP^ = OP2 they will be in contact (6), and (3) when 

 OPi is greater than OP2 there will a greater or less extent of 



Pa- («) 



Fig. 278. 



overlapping, that. is to say of concealment of the surfaces of the 

 earlier 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 OP^ be greater, not 

 only than OP 2 but also than OP^', OP^' , 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. mnbilicatus ; 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 externally from (4), where P' hes on 



