Class : Agnatha. I. Subclass : CYCLOSTOMI. With three orders : (a) _Hyperoartia_ (Lampreys); (b) _Hyperotreti_ (Myxinoids); (c) _Cycliae_ (Palaeospondylus). II. Subclass : OSTRACODERMI. With four orders : (a) _Heterostraci_ (Coelolepidae, Psammosteidae, Drepanaspidae, Pteraspidae); (b) _Osteostraci_ (Cephalaspidae, Ateleaspidae, &c.); (c) Antiarchi (Asterolepidae, Pterichthys, Bothrolepis, &c.); (d) Anaspida (Birkeniidae). Class : Pisces. I. Subclass : ELASMOBRANCHII. With four orders : (a) _Pleuropterygii_ (Cladoselache); (b) _Ichthyotomi_ (Pleuracanthidae); (c) _Acanthodii_ (Diplacanthidae, and Acanthodidae); (d) _Selachii_ (divided from the structure of the vertebral centres into Asterospondyli and Tectospondyli). II. Subclass : HOLOCEPHALI. With one order : _Chimaeroidei._ III. Subclass : DIPNOI. With two orders : (a) _Sirenoidei_ (Lepidosiren, Ceratodus, Uronemidae, Ctenodontidae); (b) _Arthrodira_ (Homosteus, Coccosteus, Dinichthys). IV. Subclass : TELEOSTOMI. A. Order : _Crossopterygii._ With four suborders: (1) _Haplistia_ (Tarassius); (2) _Rhipidistia_ (Holoptychidae, Rhizodontidae, Osteolepidae); (3) _Actinistia_ (Coelacanthidae); (4) _Cladistia_ (Polypterus). B. Order : _Actinopterygii._ With about twenty suborders: (1) _Chondrostei_ (Palaeoniscidae, Platysomidae, Chondrosteidae, Sturgeons); (2) _Protospondyli_ (Semionotidae, Macrosemiidae, Pycnodontidae, Eugnathidae, Amiidae, Pachycormidae); (3) _Aetheospondyli_ (Aspidorhynchidae, Lepidosteidae); (4) _Isospondyli_ (Pholidophoridae, Osteoglossidae, Clupeidae, Leptolepidae, &c.); (5) _Plectospondyli_ (Cyprinidae, Characinidae); (6) _Nematognathi_; (7) _Apodes_; and the other Teleosteans. There are, however, grave objections to this system, which cannot be said to reflect the present state of our knowledge. In his masterly paper on the evolution of the Dipneusti, L. Dollo has conclusively shown that the importance of the autostyly on which the definition of the Holocephali from the Elasmobranchii or Selachii and of the Dipneusti from the Teleostomi rested, had been exaggerated, and that therefore the position assigned to these two groups in Günther's classification of 1880 still commended itself. Recent work on _Palaeospondylus_, on the Ostracoderms, and on the Arthrodira, throws great doubt on the propriety of the positions given to them in the above classification, and the rank assigned to the main divisions of the Teleostomi do not commend themselves to the writer of the present article, who would divide the fishes into three subclasses:--