Branchial appendages.--The branchiæ have required more time and labor to determine their true structure than any of the appendages yet discovered. They were first regarded as small tubes arranged side by side, like the teeth in a rake; then as setiferous appendages, and finally as elongate ribbon-like spirals and bands attached to the side of the thoracic cavity, the epipodite being a so-called branchial arm. All of these parts are now known to belong to the respiratory system, but from their somewhat complex structure, and the various curious forms assumed by the parts when broken up and distorted, it was a long time before their relations were determined. The respiratory system is formed of two series of appendages, as found beneath the thorax. The first is a series of branchiæ attached to the basal joints of the legs, and the second, the branchial arms, or epipodites. The branchiæ, as found in _Calymene_, _Ceraurus_, and _Acidaspis_, have three forms. In the first they bifurcate a short distance from the attachment to the basal joint of the leg, and extend outward and downward as two simple, slender tubes, or ribbon-like filaments. In the second form they bifurcate in the same mariner, but the two branches are spirals. These two forms occur in the same individual but, as a rule, the more simple ribbon-like branchia is found in the smaller or younger specimens, and the spiral form in the adult.... The spiral branchiæ of Ceraurus are usually larger and coarser than those of _Calymene_. The third type of the branchiæ [consists of rather long straight ribbons arranged in a digitate manner on a broad basal joint]. As far as yet known, this is confined to the anterior segments of the thorax. The epipodite or branchial arm was attached to the basal joints of the thoracic legs and formed of two or more joints. This has been called a branchial arm, not that it carried a branchia, but on account of its relation to the respiratory system. It is regarded as an arm or paddle, that, kept in constant motion, produced a current of water circulating among the branchiæ gathered close beneath the dorsal shell. . . . Of the modification the respiratory apparatus underwent beneath the pygidium, we have no evidence. In his latest publication (1918, pp. 147-153, pls. 26-28, 33), Walcott has reviewed his earlier work on _Calymene_ and _Ceraurus_, and presented a new restoration of the former. The coxopodites are now interpreted as being similar to those of _Triarthrus_ and Neolenus, but the exopodites are still held to be spiral and the setiferous organs labelled as epipodites rather than exopodites. Comparison of the Appendages of Calymene and Ceraurus with those of Triarthrus. As one may see by reading the above quotations from Doctor Walcott's descriptions, he found certain branchial organs in _Ceraurus_ and _Calymene_ which have not been found in other trilobites but otherwise the essential features of the appendages of all are in agreement.