Shell smooth, campanulate, nearly hemispherical. Pores subregular circular, about as broad as the bars. Horn conical, half as long as the shell. Peristome scarcely constricted, with six conical divergent feet, as long as the horn. _Dimensions._--Shell diameter 0.04 to 0.05, length of the feet 0.02. _Habitat._--Fossil in tertiary rocks of Barbados and North America (Richmond, Virginia). 2. _Halicalyptra campanula_, n. sp. _Podocyrtis spinosa_, Bury, 1862, Polycystins of Barbados, pl. xvii. fig. 3. Shell smooth, campanulate. Pores irregular, roundish, about as broad as the bars. Horn large conical, about as long as the shell. Peristome not constricted, with six large divergent feet, which are as long as the shell, lamellar, with dentate edges. _Dimensions._--Shell diameter 0.07, length of the feet 0.07. _Habitat._--Fossil in Barbados. 3. _Halicalyptra ampulla_, n. sp. _Petalospyris foveolata_, var., Bury, 1862, Polycystins of Barbados, pl. ix. fig. 1. Shell smooth, subspherical, inflate. Pores regular, circular, twice as broad as the bars. Horn slender conical, about as long as the shell. Peristome constricted, with six divergent cylindrical, irregularly curved feet, twice to three times as long as the shell. _Dimensions._--Shell diameter 0.12, feet 0.2 to 0.3 long. _Habitat._--Central Pacific, Station 265, depth 2900 fathoms; also fossil in Barbados. 4. _Halicalyptra novena_, n. sp. Shell smooth, campanulate. Pores irregular, roundish. Horn slender conical, about as long as the shell. Peristome slightly constricted, with nine slender divergent feet of the same length. (Very similar to _Petalospyris foveolata_, Ehrenberg, but without internal columella and ring, with perfectly simple shell-cavity and open mouth.) _Dimensions._--Shell 0.09 diameter, feet 0.1 long. _Habitat._--Fossil in Barbados. 5. _Halicalyptra petalospyris_, n. sp. (Pl. 98, fig. 12). Shell smooth, campanulate. Pores small, circular, irregularly scattered. Horn stout, pyramidal, about as long as the shell. Peristome scarcely constricted, with fifteen to twenty broad {1170}lamellar or lanceolate feet, which are twice to three times as long as the shell, parallel and vertical. (Very similar to _Petalospyris flabellum_, &c., Ehrenberg, 1875, _loc. cit._, Taf. xxii. figs. 6-8, but without any trace of sagittal ring or columella.) _Dimensions._--Shell 0.05 to 0.08 diameter, feet 0.15 to 0.25 long. _Habitat._--Central Pacific, Stations 266 to 268, depth 2700 to 2900 fathoms. Subgenus 2. _Echinocalpis_, Haeckel, 1881, Prodromus, p. 427. _Definition._--Shell spiny, covered with numerous spines or thorns (besides the feet and the apical horn). 6. _Halicalyptra spinosa_, n. sp.