Dispensing with mathematical formulae, the several conditions may be illustrated as follows:
Fig. 278.
In the diagrams (Fig. [278]), O P1 P2 P3 , etc. represents a radius, on which P1 , P2 , P3 , are the points attained by the outer border of the tubular shell after as many entire consecutive revolutions. And P1′, P2′, P3′, are the points similarly intersected by the inner border; OP ⁄ OP′ being always = λ, which is the ratio of growth, or “cutting-down factor.” Then, obviously, when O P1 is less than O P2′ the whorls will be separated by an interspace (a); (2) when O P1 = O P2′ they will be in contact (b), and (3) when O P1 is greater than O P2′ 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 λ be very large, that is to say if O P1 be greater, not only than O P2′ but also than O P3′, O P4′, 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 λ. (5) There is also a final case, not easily distinguishable externally from (4), where P′ 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 λ are easily ascertained.
Fig 279.
In Fig. [279] we have portions of two successive whorls, whose corresponding points on the same radius vector (as R and R′) are, therefore, at a distance apart corresponding to 2π. Let r and r′ refer to the inner, and R, R′ to the outer sides of the two whorls. Then, if we consider
R = a eθ cot α ,
it follows that