The lenses, as seen in a vertical section of the eye of Asaphus expansus, (pl. I fig. 12), are columnar prisms, like the pillars of basalt, attaining a length of 0,2 mm and at the point where the eye joins the test of 0,3 mm. At their interior extremity they have a breadth of 0,066 mm. On that point the surface is convex and at the exterior surface plane. They are closely packed and in a transverse section resemble a pavement of regular hexaeders. But they also assume other shapes and become rhombs or even quadrates, as seen in a specimen of Asaphus fallax (pl. I fig. 18), where the hexaeders and quadrates lie side by side without transitional forms. As a rule the lenses become more and more irregular in the vicinity of the surrounding frame or near the suture, nearly blotted out, as it were, and without any definite border line mingled with the confused, spongy mass that like a belt or a frame surrounds they eye in Asaphus and is sharply limited from the other part of the free cheek. This remarkable zone which is almost only present amongst the Asaphidæ (Asaphus, Megalaspis, Ptychopyge, Isotelus) retains in a confused manner somewhat of the prismatic structure of the eye as shown in the section (Pl. I, fig. 11, b). The eye of Bumastus also is environed by a similar zone, with a structure like that of the eye (Pl. II fig 35, 41). In an undetermined species of Asaphus the lenses, although somewhat apart, are of an elongated hexaedric outline, which passes into a regular circular one farther away and on the surface of the eye they are slightly convex (Pl. I figs 27-29). In other genera belonging to this group the shape of the lenses are like those of Asaphus, so for instance in Illænus (I. chiron and I. Esmarki) and in Niobe. In Dysplanus centrotus they are shorter and broad, and their interior or lower surface strongly convex. It is likewise so in Nileus, where Nileus armadillo has an exceedingly thick exterior integument above the lenses. Such an integument has in a still higher degree increased in Bumastus sulcatus so as to exceed in thickness the stratum of the corneal prisms and it may in fact be doubted if the eyes of this species ever were able to function as visual organs. Proetus nearly resembles Bumastus in the thickness of the integument covering the prismatic lenses, which are interiorly convex, with a diameter of 0,03 mm. In all genera belonging to this group a horizontal section gives the image of the hexaeders as in Asaphus with some change to squares or rhombs.