The suborder of #Spyroidea# is here divided into four different families. In the first and original family, the Zygospyrida, the shell is represented by the bilocular cephalis only. The second family, Tholospyrida, is distinguished by a fenestrated cupola or galea, a hemispherical or conical dome, arising from the upper or coryphal face of the cephalis. In the third family, Phormospyrida, a thorax is developed, or a fenestrated basal chamber, arising from the lower or basal face of the cephalis. The fourth family, Androspyrida, combines the characters of the second and the third family, a galea arising from the upper and a thorax from the lower face of the cephalis. The bilocular cephalis, as the most important and constant part of the skeleton in all #Spyroidea#, requires a further general consideration. Its lattice-work, very variable in the numerous species, is usually more or less different on the six sides of the nut-shaped shell. Its most important part is the horizontal basal plate or lower face, and the sagittal ring arising from it in the vertical median plane. This part of the shell corresponds exactly to the skeleton of the Semantida, and exhibits the same modifications {1018}as in that family; it is therefore probable that the majority of the #Spyroidea# (if not the whole group) have been derived directly from the Semantida by further development of network from the fenestrated ring of _Clathrocircus_. Whilst in this latter Semantid the two lateral sides of the central capsule remain naked, in the #Spyroidea# they become enclosed and covered by lattice-work, which arises on the right and left from the ring. In some forms, however, as in the _Ceratospyris acuminata_ of Hertwig, and in my Perispyrida, the frontal ring also of the Coronida and the mitral ring of the Tympanida appear as the initial parts of a complete latticed cephalis.