A single individual of _Acidaspis trentonensis_, obtained from the same locality and horizon as the specimens of _Triarthrus_ and _Cryptolithus_, when cleaned from the ventral side shows a number of poorly preserved endopodites which seem very similar in shape and position to those of _Triarthrus_. One endopodite on the right side of the head and the first five on the right side of the thorax are the best shown. All are slender, are directed first forward at an angle of about 45 with the axis, then, except in the case of the cephalic appendage, turn backward on a gentle curve and extend a little distance beyond the margin of the test, but not as far as the tips of the lateral spines of the thoracic segments. The individual segments of the endopodites can not be seen clearly enough to make any measurements. On the fourth and fifth endopodites of the thorax, some of the segments seem to be broad and triangular as in _Triarthrus_. All that can be seen indicates that _Acidaspis_ had appendages entirely similar to those of _Triarthrus_, but perhaps not quite so long, as they seem not to have projected beyond the limits of the lateral spines. There are no traces of antennules nor, unfortunately, of exopodites. _Measurements:_ Length 8 mm. Walcott (1881, p. 206) stated that his sections had shown the presence in this species of legs "both cephalic and thoracic" and also the "spiral branchiæ." His specimens were from the Trenton at Trenton Falls, New York. The Appendages of Cryptolithus. =Cryptolithus tessellatus= Green. (pl. 6, fig. 7; pls. 7-9; text figs. 20, 25, 45, 46.) (See also Part IV.) Illustrated: Beecher, Amer. Jour. Sci., vol. 49, 1895, pl. 3. When Professor Beecher wrote his short article on the "Structure and Appendages of _Trinucleus_" (1895), he had only three specimens showing appendages. In his later work he cleaned several more, so that there are now thirteen specimens of _Trinucleus_ = _Cryptolithus_ available for study, though some of these do not show much detail. In his last and unpublished study, Beecher devoted the major part of his attention to this genus, and summarized his findings in the drawings which he himself made of the best individuals (text figs. 45, 46). Valiant (1901) stated that he had found a _Trinucleus_ with antennæ in the Frankfort shale south of Rome, New York. The specimen has not been figured. None of the specimens shows much more of the appendages of the cephalon than, the hypostoma and the antennules, so that we are still in ignorance about the mouth parts.