In _Protympanium_ (Pl. 93, fig. 14), the simplest and most primitive form of Tympanida, the connection of the two horizontal rings is effected only by the dorsal and ventral rods of the sagittal ring forming two parallel more or less vertical "columellæ." In all other Protympanida accessory columellæ are developed between the latter. In the greater number we find two secondary columellæ in the lateral plane, being the lateral {989}segments of an incomplete frontal ring (_Acrocubus_, Pl. 93, figs. 15-17). Between the latter and the former there are sometimes developed four tertiary columellæ, as segments of two crossed incomplete diagonal rings (_Tympanidium_, Pl. 94, figs. 1, 18). At other times we find six columellæ; here probably the frontal ring is cleft into halves on each side (_Tympaniscus_, Pl. 94, figs. 4-7). Some genera are distinguished by the development of an equatorial ring, or a third horizontal ring (between the upper mitral and the lower basal ring); this equatorial ring is either complete (_Microcubus_ Pl. 94, figs. 8-10) or incomplete, developed only laterally (_Octotympanum_, Pl. 94, figs. 2, 3). The genus _Toxarium_ (Pl. 93, figs. 18-20) is distinguished by the development of accessory pairs of bows in the frontal plane, upper or galear bows and lower or thoracal bows, remarkable as beginnings of the accessory joints, which we distinguish in many #Spyroidea# and #Cyrtoidea# as "galea and thorax." The second subfamily of Tympanida are the Paratympanida, in which the two horizontal rings become closed by lattice-work. The numerous columellæ connecting the two rings are here either divergent (_Paratympanum_, with two unequal rings, Pl. 94, fig. 14) or parallel and vertical (_Lithotympanum_, with two equal rings, Pl. 83, fig. 1). The shell of the latter assumes the form of a drum. A third small subfamily, Dystympanida, is represented by a single genus only, _Dystympanium_ (Pl. 94, figs. 15, 16). Here only the upper or mitral ring is closed by lattice-work, whilst the lower or basal ring exhibits quite a simple opening, a large "basal gate" which has probably arisen by loss of the basilar rod of the sagittal ring. These curious forms exhibit a remarkable resemblance to some Monocyrtida, and also to the _Dictyocha_ (PHÆODARIA).