The alæ are originally developed at the period of the metamorphosis, as slight lateral protuberances in the upper part of the compartments; from being overlapped, and therefore not exposed to external influences, and from growing (as in the case of the radii) against rectangular indentations or furrows in the adjoining compartments, they generally assume an extremely different appearance from the parietes, and might naturally be thought to have a very different nature. But the alæ in all cases (as is obvious in Pachylasma) are nothing but the protuberant lateral edges of the compartments, rendered thin and modified during growth. In order that the margins of the alæ should be received in an indentation, the upper internal surfaces of the walls of the recipient compartments are thickened all round, excepting where they receive the alæ. This thickened, upper, internal portion of the walls or shell, together with the alæ themselves, form the part called the sheath. The sheath sometimes blends insensibly into the lower parts of the compartments, and then perhaps it would not be thought to be a distinct element; but often it is abruptly separated by an overhanging edge (see Pl. 9, fig. 5 _b_, 9 _b_; Pl. 20, fig. 1 _b_; Pl. 25, fig. 1, K′) from the lower part, and then the sheath greatly complicates the internal appearance, but not the essential structure of the shell. The sheath acts beautifully, like an internal hoop, in strengthening the shell round the orifice, where it is naturally weaker than at the lower or basal end, where it adheres to the surface of attachment: in the upper part of the shell, moreover, the sutures between the compartments do not go straight through, but owing to the alæ projecting and being overlapped, are extremely oblique; or the joints, in the language of carpenters, may be said to be broken.