The second or lateral girdle is commonly not in direct connection with the medullary shell, or only by some scattered radial beams (mainly in the principal axis). This lateral guide arises by prolongation of both wings of the transverse girdle in the lateral plane, so that from both sides (right and left) they become united on the poles of the principal axis. The minor axis of the elliptical lateral ring (thus formed) is therefore the major axis of the foregoing (transverse) elliptical ring; the major axes of both are perpendicular one to another. The major axis of the lateral ring is the principal (or longitudinal) axis of the whole body.
The third or sagittal girdle becomes developed from the second almost in the same manner as the second from the first. On both poles of the principal axis two latticed wings arise from the lateral girdle, growing further in the direction of an elliptical ring, which represents the perimeter of the sagittal plane or median plane. These wings are already mentioned by J. Müller as "prominent roofs, protecting the gates of the Tetrapyle-shell." If these roofs grow towards the equatorial plane of the shell and become united in pairs on the poles of the sagittal axis, the third girdle becomes complete. R. Hertwig supposes that the minor axis of this sagittal girdle is constantly at the same time the major axis of the lateral girdle, but this is not always the case. Very often the size of both these girdles is nearly the same, or one is not much larger than the other. In this case the principal axis of the body is the major axis of the second as well as of the third girdle.
The characteristic "gates" of the Pylonida, or the large wide openings in their cortical shell, remaining between the crossed latticed girdles, are in general roundish, sometimes nearly circular, commonly more elliptical, kidney-shaped or semilunar, their special form varying much according to the different form of the girdles. The narrowest part of each girdle, or its "isthmus," in the case of the halves of the transverse girdle is commonly at their origin from the medullary shell, in the case of the halves of the lateral girdle at the poles of the principal axis, and in the case of the halves of the sagittal girdle at the poles of the sagittal axis. The number of the gates is quite constant in the different genera. If only one girdle (the transverse) be developed, we find only two large gates, between the two wings on the poles of the principal axis (in Monozonium, Amphipyle, Amphipylonium). In all other cases there are four gates (determining the original name "Tetrapyle"), as well if only two or if all three girdles be completed. If two girdles be complete (in Dizonium, Tetrapyle, Tetrapylonium) the four gates lie opposite in pairs on the sagittal faces (two anterior and two posterior gates), and are limited by the transverse and lateral girdles. If all three girdles be complete (in Trizonium, Pylonium, Pylozonium) the four gates lie opposite in pairs on the lateral faces (two dorsal and two ventral gates), and are limited by the sagittal and the transverse girdles. If we turn the shell through an angle of 90°, we have the same aspect as in the former group. A sagittal septum sometimes becomes developed, beginning with two polar beams, rising from the poles of the principal axis. If these polar beams become branched and connected with the middle part of the lateral girdle, we get a latticed vertical septum, which divides the four gates of Tetrapyle into eight gates, Octopyle.
The lattice-work of the Pylonida is commonly very variable and irregular, with roundish meshes of very unequal size, therefore without value in the determination of the species. Commonly the outside of the shell is thorny, and often distinguished by larger radial spines, symmetrically disposed. We can separate these into two groups; "dimensive" spines, lying in one of the three dimensive axes (principal, transverse, or sagittal), and "diagonal" spines, lying crossed in pairs in diagonal axes. Among these latter eight diagonal wing-spines, which arise from the lateral edges of the four gates, are particularly remarkable; they are not only characteristic of Tetrapyle octacantha, but also of a large number of other Pylonida, and form the starting-point for many specific forms.
The shell of the Pylonida is characterised by extraordinary variability and great inclination to individual abnormalities, formation of varieties and transitions into other families, hence derived, as Larnacida, Tholonida, Lithelida, &c.
The central capsule in all Pylonida, in which I could observe it, was a true lentellipsis (or a "triaxial ellipsoid" in the geometrical sense, with the three unequal isopolar axes of the "rhombic octahedron"). In the living Pylonida it is commonly coloured pink or scarlet. During growth its dimensions are probably more or less changed, and perhaps the axes alternate. Regarding the relation of the central capsule to the skeleton, we can distinguish two different groups, quite as in the nearly allied Larnacida. In the Haplozonaria (as also in the Larnacillida) the central capsule encloses only the central chamber and is enveloped by the first system of girdles, whereas in the Diplozonaria and Triplozonaria (as in the Larnacalpida) that "trizonal shell" becomes enclosed (as the "medullary shell") in the central capsule, which is now enveloped by the second system of girdles as the "cortical shell."
Synopsis of the Genera of Pylonida.
III. Subfamily Haplozonaria. One system of girdles. Medullary shell simple, spherical or lentelliptical; cortical shell simple, with one, two, or three girdles. |  | Cortical shell only with one latticed (transverse) girdle, | 279. Monozonium. | ||
| Cortical shell with two perfect girdles (transverse and lateral), | 280. Dizonium. | ||||
| Cortical shell with three perfect girdles (transverse, lateral, and sagittal | 281. Trizonium. | ||||
III. Subfamily Diplozonaria. Two systems of concentric girdles. Medullary shell trizonal, with three perfect girdles; cortical shell simple, with one, two or three girdles. |  | Cortical shell only with one perfect (transverse) girdle, | 282. Amphipyle. | ||
| Cortical shell with two perfect girdles (the transverse and lateral). |  | Four gates simple, | 283. Tetrapyle. | ||
| Four gates bisected by a sagittal septum, | 284. Octopyle. | ||||
| Cortical shell with three perfect girdles (transverse, lateral, and sagittal), | 285. Pylonium. | ||||
III. Subfamily Triplozonaria. Three systems of concentric girdles. Medullary shell trizonal, with three perfect girdles, quite as the inner cortical shell; outer cortical shell with one, two, or three girdles. | | Outer cortical shell (third system) only with one perfect (transverse) girdle, | 286. Amphipylonium. | ||
| Outer cortical shell with two perfect girdles (transverse and lateral), | 287. Tetrapylonium. | ||||
| Outer cortical shell with three perfect girdles (transverse, lateral, and sagittal), | 288. Pylozonium. | ||||
| I. Subfamily Haplozonaria. One system of girdles. Medullary shell simple, spherical or lentelliptical; cortical shell simple, with one, two, or three girdles. | ||||||
| Cortical shell only with one latticed (transverse) girdle, | ||||||
| 279. Monozonium. | ||||||
| Cortical shell with two perfect girdles (transverse and lateral), | ||||||
| 280. Dizonium. | ||||||
| Cortical shell with three perfect girdles (transverse, lateral, and sagittal | ||||||
| 281. Trizonium. | ||||||
| II. Subfamily Diplozonaria. Two systems of concentric girdles. Medullary shell trizonal, with three perfect girdles; cortical shell simple, with one, two or three girdles. | ||||||
| Cortical shell only with one perfect (transverse) girdle, | ||||||
| 282. Amphipyle. | ||||||
| Cortical shell with two perfect girdles (the transverse and lateral). | ||||||
| Four gates simple, | ||||||
| 283. Tetrapyle. | ||||||
| Four gates bisected by a sagittal septum, | ||||||
| 284. Octopyle. | ||||||
| Cortical shell with three perfect girdles (transverse, lateral, and sagittal), | ||||||
| 285. Pylonium. | ||||||
| III. Subfamily Triplozonaria. Three systems of concentric girdles. Medullary shell trizonal, with three perfect girdles, quite as the inner cortical shell; outer cortical shell with one, two, or three girdles. | ||||||
| Outer cortical shell (third system) only with one perfect (transverse) girdle, | ||||||
| 286. Amphipylonium. | ||||||
| Outer cortical shell with two perfect girdles (transverse and lateral), | ||||||
| 287. Tetrapylonium. | ||||||
| Outer cortical shell with three perfect girdles (transverse, lateral, and sagittal), | ||||||
| 288. Pylozonium. | ||||||