The numerous new genera discovered in the rich collection of the Challenger, and mainly in the wonderful Radiolarian ooze of the Central Pacific (Stations 263 to 274) throw a new light upon the complicated affinities of the whole group, and manifest a far greater richness in developmental variations and admirable morphological productions than Bütschli could suppose. The following description, however, must remain very incomplete, since the exact and thorough study of all individual forms, in the same manner as {1017}Bütschli had done in the case of a small number of species, would require several years. I can therefore only give the basis of further researches. In the majority of #Spyroidea# the three essential elements of the Nassellarian skeleton are easily recognisable, viz.:--A, the vertical sagittal ring of the #Stephoidea#; B, the basal tripod of the #Plectoidea#, and C, the latticed cephalis of the #Cyrtellaria#. These three elements are constantly so combined that the primary sagittal ring bisects the cavity of the lattice-shell and produces an external, longitudinal, more or less distinct constriction, separating the lateral inflated halves of the bilocular cephalis; while the three divergent feet of the basal tripod descend from the cortinar or basal plate of the cephalis. The number of the feet is often augmented; they are rarely wanting (probably reduced). The sagittal ring is constantly present, though its relation to the shell-wall exhibits many modifications. The bilocular form of the fenestrated cephalis, with its sagittal constriction, is characteristic of all true #Spyroidea#, and separates them from the #Botryodea# (with multilobate cephalis) on the one hand, and from the #Cyrtoidea# (with simple cephalis) on the other. There are, however, some intermediate forms between these three suborders of #Cyrtellaria#, which show a direct transition to one another. Almost constantly the length of the three dimensive axes in the cephalis of the #Spyroidea# is so different that the lateral or frontal axis is the longest, the sagittal or dorso-ventral axis the shortest, and the principal or longitudinal axis intermediate between them. As in all dipleuric or bilaterally symmetrical forms, the two poles of the frontal axis are equal, whilst the two poles of the sagittal and of the principal axis are more or less different. In only a few forms of #Spyroidea# these differences are difficult to make out; usually the dorsal side is distinctly different from the ventral, and the apical side from the basal. A perfect knowledge of the dipleuric shell requires therefore accurate observation from all six sides.