Subgenus 1. _Apocubus_, Haeckel. _Definition._--Basal ring without descending feet. {993}1. _Acrocubus octopylus_, n. sp. (Pl. 82, fig. 9). Mitral ring somewhat smaller than the basal ring; both rings rhombic, with curved outlines. Sagittal ring elliptical, with six pairs of nodulate protuberances. Four columellæ curved. Nodal points without radial spines. _Dimensions._--Height of the frontal ring 0.12, breadth 0.18. _Habitat._--Western Tropical Pacific, Station 225, depth 4475 fathoms. 2. _Acrocubus tesseralis_, n. sp. Mitral ring of the same size as the basal ring; both rings square, smooth, with straight outlines. Sagittal ring also square, smooth. Four columellæ straight. Nodal points without radial spines. The shell has the form of a regular geometrical cube, the edges of which are represented by the rings. _Dimensions._--Height of the frontal ring 0.1, breadth 0.11. _Habitat._--Tropical Atlantic, Station 348, depth 2450 fathoms. Subgenus 2. _Dipocubus_, Haeckel. _Definition._--Basal ring with two descending feet. 3. _Acrocubus brachiatus_, n. sp. Mitral ring smaller than the basal, both rings square, with thick straight rods. Sagittal ring tuberculate, square, equatorial part thinner. Four columellæ curved, the two lateral ones prolonged downwards into two vertical parallel straight feet of half their length. _Dimensions._--Diameter of the cube 0.12; length of the feet 0.06. _Habitat._--North Pacific, Station 244, depth 2900 fathoms. 4. _Acrocubus amphistylus_, n. sp. Mitral and basal rings of nearly equal size, square, with thick curved rods. Sagittal ring elliptical, in the equatorial part thinner. Frontal ring tuberculate. Four columellæ curved, the two lateral ones prolonged downwards into two vertical, tuberculate feet of half their length. _Dimensions._--Diameter of the cube 0.13; length of the feet 0.07. _Habitat._--Central Pacific, Station 268, depth 2900 fathoms. 5. _Acrocubus arcuatus_, n. sp. (Pl. 93, fig. 15). Mitral and basal rings, as well as the sagittal ring, of the same form as in the preceding nearly allied species. The two lateral columellæ are prolonged not only downwards into two short {994}vertical feet, but also upwards into two curved bows, which are united in a vertical ascending horn, arising from the apical pole of the sagittal ring. _Dimensions._--Diameter of the cube 0.14; total height 0.21. _Habitat._--Central Pacific, Station 265, depth 2900 fathoms. Subgenus 3. _Tripocubus_, Haeckel. _Definition._--Basal ring with three descending feet (one odd caudal and two paired lateral). 6. _Acrocubus lasanum_, n. sp. Mitral ring smaller than the basal, both rings square, with curved rods. Sagittal ring ovate, prolonged upwards into an occipital simple horn, downwards into a caudal foot. Two lateral columellæ prolonged downwards into two pectoral feet. All three feet conical, about as long as the height of the cube.