Subglobose, 10 cm. in diameter, simple: tubercles long (15 to 20 mm.) and spreading, with woolly axils: radial spines 3, rigid and recurved, 5 mm. long; central spines 3, very stout and much recurved, 20 to 30 mm. long, alternating with the radials; all ashy colored and often twisted: flower and fruit unknown.--Type in Herb. Coulter. The few spines, with the very short radials alternating with the very long and stout centrals, furnish a striking character. Occasionally one of the centrals is wanting. 2.Cactus acanthophlegmus (Lehm.) Kuntze, Rev. Gen. Pl. 260 (1891). Mamillaria acanthophlegma Lehm. Delect. Sem. Hamb. (1833) Subglobose with a deeply depressed vertex, or becoming cylindrical, 3 to 8.5 cm. in diameter: tubercles sharply quadrangular-conical, with densely woolly axils: radial spines 15 to 30, white, very slender (bristly) and radiant, sometimes coarse capillary, 4 to 7 mm. long, interwoven with those of neighboring tubercles and so covering the whole plant; central spines 2 to 4, robust and straight, erect or divergent, whitish or reddish, black-tipped, 5 to 6.5 mm. long: flowers reddish, 1 to 2 cm. broad: fruit unknown. Type unknown. From Coahuila and San Luis Potosi to Oaxaca. Fl. May. Specimens examined: Coahuila (Poselger of 1856; Pringle 3116 of 1890): San Luis Potosi (Eschanzier of 1891). The central spines are quite variable in number and arrangement. In case there are two they are vertically placed and are either erect and parallel or widely divergent. Even three centrals may occur in the same vertical plane; but more usually the three or four centrals are arranged about a center and are widely divergent. The tubercles are apt to persist and to become naked and corky with age. The axillary wool and the capillary radials are also apt to be more or less persistent, thus giving the whole plant a woolly appearance. 3.Cactus brandegei, sp. nov. Cylindrical: tubercles sharply quadrangular-conical, 6 to 8 mm. long, with densely woolly axils: radial spines about 10, slender and rigid, whitish with dusky tips, spreading but not radiant, 7 to 10 mm. long; central spines 3 or 4, stouter and slightly longer, erect-spreading (sometimes slightly curved), reddish-brown below, becoming blackish above: flowers small (scarcely longer than the tubercle?): fruit unknown. Type in Herb. Calif. Acad. San Jorge, Lower California. Fl. April. Specimens examined: Lower California (Brandegee of 1889, at San Jorge). The species has somewhat the spine characters of C. palmeri, but the sharply quadrangular and longer tubercles with axillary wool free from bristles suggest a very different affinity. 4.Cactus densispinus, sp. nov.