A. The number of joints into which the shell is divided by transverse constrictions, serves here for the distinction of four primary groups or suborders of the #Cyrtoidea#, viz., (1) Monocyrtida with one joint; (2) Dicyrtida with two joints; (3) Tricyrtida with three joints; and (4) Stichocyrtida with four or more joints. In my Prodromus (1881, p. 426) I divided the latter group into Tetracyrtida (with four joints), and Stichocyrtida (with five or more joints); but these two groups may be united, since the fourth and all the succeeding joints are of rather indifferent shape and of little morphological value. The three first joints, however, are usually very different and possess a high morphological importance, so that we distinguish the first joint as cephalis, the second as thorax, and the third as abdomen. The uppermost transverse constriction, which separates the two first joints, cephalis and thorax, is the collar stricture and is usually caused by an internal fenestrated septum, the cortinar septum. The second constriction, which separates the second and third joints (thorax and abdomen) is called the lumbar constriction. The following constrictions (in the Stichocyrtida) are indifferent and of little morphological interest, and require therefore no peculiar designation. {1129}B. The radial structure, indicated by radial apophyses arising from the shell, offers three principal differences, according to which the whole group of #Cyrtoidea# may be divided into three large groups or sections, viz., (1) Pilocyrtida, or Cyrtoidea triradiata, with three radial apophyses; (2) Astrocyrtida, or Cyrtoidea multiradiata, with numerous radial apophyses (four to nine or more); and (3) Corocyrtida, or Cyrtoidea eradiata, without external radial apophyses. The majority of #Cyrtoidea# are Pilocyrtida, with three radial apophyses, which are probably homologous to the three primary feet of the #Plectoidea# and of _Cortina_ (therefore "cortinar feet"). The Astrocyrtida, or the #Cyrtoidea# with a variable number of radial apophyses (at least four to six) may be derived from the Pilocyrtida by interpolation of secondary or interradial apophyses between the three primary or perradial apophyses. The Corocyrtida, however, or the #Cyrtoidea# without external radial apophyses, may have originated by reduction and loss of the latter, either from the Pilocyrtida or from the Astrocyrtida. C. The shape of the basal mouth in the #Cyrtoidea# exhibits two essential differences only, viz., (1) the terminal mouth of the shell is a simple wide opening in the Stomocyrtida, or (2) the terminal mouth is closed by a lattice-plate, in the Clistocyrtida. As these two different cases occur in all the twelve families, which we have distinguished according to the differences in the number of joints and in the radial structure, we get altogether twenty-four subfamilies which are synoptically arranged in the following table:--