similarly intersected by the inner border; OPã₤ãã₤OPÿ£¢ãý being always =ã₤ö£, which is the ratio of growth, or ãcutting-down factor.ã Then, obviously, when OãP ÿ£¢ 1 is less than OãP ÿ£¢ 2 ÿ£¢ãý the whorls will be separated by an interspace ( a ); (2) when OãP ÿ£¢ 1 =ã₤ OãP ÿ£¢ 2 ÿ£¢ãý they will be in contact ( b ), and (3) when OãP ÿ£¢ 1 is greater than OãP ÿ£¢ 2 ÿ£¢ãý there will a greater or less extent of 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 ö£ be very large, that is to say if OãP ÿ£¢ 1 be greater, not only than OãP ÿ£¢ 2 ÿ£¢ãý but also than OãP ÿ£¢ 3 ÿ£¢ãý, OãP ÿ£¢ 4 ÿ£¢ãý, 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 ãspecies,ã is constituted accordingly by a difference in the value of ö£. (5) There is also a final case, not easily distinguishable externally from (4), where Pÿ£¢ãý lies on {543} the opposite side of the radius vector to P , and is therefore imaginary. This final condition is exhibited in Argonauta.
The limiting values of ö£ are easily ascertained.
In Fig. 279 we have portions of two successive whorls, whose corôÙreôÙsponôÙding points on the same radius vector (as R and Rÿ£¢ãý) are, therefore, at a distance apart corôÙreôÙsponôÙding to 2ü. Let r and rÿ£¢ãý refer to the inner, and R, Rÿ£¢ãý to the outer sides of the two whorls. Then, if we consider
it follows that
and