*** Central spine solitary and hooked. brunneus (53). **** Central spines more than one and none of them hooked. Centrals 2: radials 6 to 20. scheerii* (40), robustispinus (41), recurvatus (42), scolymoides* (48). Centrals 3: radials 6 to 40. scheerii* (40), scolymoides* (48), echinus* (50), conoideus (54), neo-mexicanus* (59), arizonicus* (60). Centrals 4 or 5: radials 6 to 40. scheerii* (40), scolymoides* (48), echinus* (50), conoideus (54), tuberculosus* (56), viviparus* (57), radiosus* (58), neo-mexicanus* (59). arizonicus* (60), macromeris* (64). Centrals 6 or 7: radials 12 to 40. potsii* (55), tuberculosus*(56), viviparus* (57), neo-mexicanus* (59), arizonicus* (60), chloranthus (62). Centrals 8 to 14: radials 12 to 40 or more. potsii* (55), tuberculosus* (56), viviparus* (57), neo-mexicanus* (59), deserti* (61), chloranthus* (62), alversoni* (63). GEOGRAPHICAL DISTRIBUTION It is only possible to deal with the forms that occur within the borders of the United States, as even individual stations of common Mexican forms are little if at all known. These United States forms represent a northern extension of an abundant Mexican display. The group EUMAMILLARIA, containing twelve of the thirty-one forms defined as occurring north of the Rio Grande, makes the feeblest extension northward, at no place being found far from the boundary, and all the twelve are Mexican forms which extend but slightly into the United States. Only five of the forms are found east of the Pecos: heyderi, the most widely distributed EUMAMILLARIA, extending from the southeastern border of Texas westward along the whole Mexican boundary except in California; hemisphaericus, extending through southern Texas and southern New Mexico; meiacanthus, also along the Mexican border of Texas and New Mexico; texanus, a low ground form of the Rio Grande Valley, extending from the mouth of the river to El Paso, and suggesting a connection with the West Indian stellatus; and sphaericus, another low ground valley form of similar range, but apparently only extending up the Rio Grande to the region of Eagle Pass. The Pecos forms the eastern boundary of five other EUMAMILLARIA forms: micromeris, extending northward from Coahuila and Chihuahua, apparently only in the mountains between the Pecos and El Paso; wrightii, of similar narrow northward extension, but ranging further northward on the high plains of the Upper Pecos in New Mexico; denudatus, also with a narrow northward extension west of the Pecos; lasiacanthus, extending from Chihuahua with a northern limit between the Pecos and Arizona; and grahami, a Sonoran type which has spread between the Pecos and southeastern California.