1. The anatomy of Helodus simplex Ag. is described.
2. The skull is found to be holostylic, and to have many characters in common with the skull of the Holocephali, but in some respects is less specialized.
3. The pectoral fins, with their long metapterygium, small propterygium, and fused anterior radials, resemble very closely those of the Holocephali.
4. The pelvic and unpaired fins, and general body shape are found to resemble those of the Holocephali.
5. It is concluded that the Cochliodonts are almost certainly closely related to the ancestors of the Holocephali, and the relatively unspecialized condition of the teeth gives support to the view that the holostylic condition of the jaws is primitive for the group. It is suggested that all the Bradyodonts were holostylic, that the hyomandibular may never have been suspensory, and that they may have diverged from the true Selachii before the hyomandibular played a part in the jaw suspension. The difficulties which this view involves are appreciated.
6. The condition of the ethmoid region of Helodus supports the view of the nature of the ethmoidal canal put forward by De Beer and Moy-Thomas (1935) rather than that of Allis (1917, 1926, 1934).