Now consider the "epipodites." They are well shown in _Calymene_ in the specimens illustrated on plate 27, figure 11 (1918), and plate 3, figure 3 (1881), and less clearly in one or two others. Slices 22 (pl. 27, fig. 12, 1918) and 80 (our fig. 12) show what is called the same organ in Ceraurus. It will be noted that all of these slices are cut in the same way, that is, more or less parallel to the under surface of the head, or, at any rate, on a plane parallel to a plane which would be tangent to the axial portion of the coiled shell. The sections which show the spirals best are those which are cut by a plane perpendicular to the long axis of the body. If one were to attempt to cut an enrolled _Triarthrus_ in such a way as to get a section showing the length of the setæ, one would not cut a section perpendicular to the axis of the animal, nor, in fact, would he cut one parallel to the ventral plane, but it is obvious that in this latter type of section he would stand a better chance of finding a part of the plane of the exopodite coincident with the plane of his section than in the former. And that seems to be what has happened in these sections of _Calymene_ and _Ceraurus_. If the exopodites were preserved, transverse sections were bound to cut across many sets of fringes, and the resultant slice would show transverse sections of the setæ as a series of overlapping spots. A few fortunately located sections in a more nearly horizontal plane might cut the setæ and occasionally the shaft of one or more exopodites in the longitudinal plane, and the resulting effect would produce the so-called "epipodites." A careful study has shown that no one of these epipodites is complete, and they do not have the palmate form shown in Walcott's figures. And the last and most important argument against the spiral appendages is that certain slices, of both _Calymene_ and _Ceraurus_, show definitely exopodites of exactly the type found in other trilobites. These are discussed later in the detailed description of the various slices. If these series of spots are interpreted on the basis of the known structure of _Triarthrus_, they are of course a series of sections through the setæ of the exopodites. It will be shown in Part IV that these setæ are not circular in section, but flattened, in _Cryptolithus_ even blade-like, and that they overlap one another. A section across them would give the same general appearance as, for instance, that shown in figures 4, 6, 9, and 10 of Walcott's plate 3 (1881).