These project, generally abruptly, from the sides of the upper part of the compartments; they appear from the first growth of the shell; they are overlapped by the radius and by part of the wall of the adjoining compartment; they are thinner, and have, owing apparently to being overlapped, a very different aspect from the parietal portion; but they do not differ from it in essential nature. They are solid, that is, they are never permeated by pores; but their edges are generally crenated, and there is, in some cases, as in Chelonobia, sufficient evidence that these crenations answer to the horizontal septa on the edges of the radii (also often reduced to mere crenations), and consequently, likewise, to the longitudinal septa of the parietes. In Coronula the edge of each ala consists of a medial ridge, sending off denticulated septa on _both_ sides, and is therefore anomalous as compared with the alæ in other genera, but corresponds in structure with the similarly anomalous radius of Coronula. In order to allow of the growth of the edge of the ala, a fine thread of corium runs up the narrow furrow in which the edge is lodged, proceeding from the corium of the sack. In proportion as this thread runs up higher or lower, so are the summits of the alæ rendered, during growth, less or more oblique. _Structure of the Sheath._ As the compartments overlap each other, the edges of the alæ would have projected, and the inner surface of the orifice of the shell would not have been smooth and rounded, had not that part of each wall, which does not overlie an ala, been thickened so as to allow of the formation of a shoulder or indentation, against which the edge of the ala fits and abuts. The thickened portions, and the alæ themselves, together form the sheath, of which the use seems to be to strengthen, like a broad internal hoop, the upper part of the shell round the orifice, where naturally it is weak. The sheath is composed of successive, fine, shelly layers, which extend, as the shell is added to at the basal margin, lower and lower down on the inner surface of the walls. The lower edge of the sheath either simply projects a little inwards, or more commonly is formed into a sharp depending ridge, as represented in fig. 1, K′, Pl. 25. In some species of Pyrgoma (Pl. 13, fig. 2 _b_), the sheath reaches nearly to the bases of the compartments; and in Chelonobia (Pl. 14, fig. 4 _e_ _c_ _e_), the inner layer of shell surrounding the sack, which seems to correspond more nearly to the sheath than to the inner lamina of the walls, actually rests on the basal membrane. The opercular membrane is generally, but not invariably, attached only a little way above the lower edge of the sheath: at each exuviation, a new opercular membrane is formed, and is attached to the next lower zone of the sheath; the old membrane being cast off, but a circular slip of it is left, investing the last zone. Hence the whole upper part of the sheath above the opercular membrane,