The purpose of this paper is to examine the evidence afforded by the chimaeroids and fossil holocephalans on the origin and evolution of the Holocephali, and, in particular, on the hypothesis recently proposed by Ørvig of a relationship between chimaeroids and ptyctodont arthrodires. Members of all the fossil holocephalan groups which are known by moderately complete specimens except the petalodonts and edestids are described, particular attention being paid to the histology of the dermal skeleton. Three species from the English Lower Lias,
Squaloraja polyspondyla
Agassiz,
Myriacanthus paradoxus
Agassiz and
Metopacanthus granulatus
(Agassiz) (a species previously placed in
Myriacanthus
) are described in detail. From the British Lower Carboniferous
Deltoptychius armigerus
(Traquair), previously placed in
Oracanthus
, is described in detail and two new species,
Deltoptychius moythomasi
sp.nov. and
Menaspacanthus armaghensis
gen. and sp.nov., are described on fragmentary material. The genus
Oracanthus
Agassiz is reviewed: it is shown that the type species is probably acanthodian but that many species have been wrongly included in the genus. There is found to be strong evidence that the chimaeroids are derived, through the Jurassic myriacanthids and allied forms, from a Palaeozoic group which includes the Permian
Menaspis
and whose most primitive known genus is
Deltoptychius
. The most striking feature of
Deltoptychius
is the dermal armour of dentine-like tissue on the head which includes a complete head shield bearing a pair of sensory canals. This armour is of a peculiar type which is not homologous with the armour of the arthrodires, and appears to be confined to the Chimaeriformes. It is suggested that
Helodus
is not directly related to the Chimaeriformes. These conclusions are embodied in a new classification of the Holocephali in which
Helodus
is made the type of a new order Helodontiformes and in which the Chimaeriformes is divided into four sub-orders, Chimaeroidei, Squalorajoidei, Myriacanthoidei and Menaspoidei, with new families being made in the last two sub-orders. On the basis of the information obtained from the systematic part, the origin and evolution of the various characters of chimaeroids are discussed. The ethmoid canal appears to be a recent acquisition which occurs only in Chimaeroidei. The tritors on chimaeroid tooth plates are another recent adaptation. There is no convincing evidence that the teeth of Holocephali were primitively numerous, as they are in
Helodus
, petalodonts and edestids: this ‘selachian’ type of dentition is quite possibly secondarily derived from a 1chimaeroid ’ type of dentition. The dorsal fin spine of the chimaeroids appears to be a recent adaptation which first appears in the myriacanthoids and has arisen independently, being unrelated to the fin spine of
Helodus
. A series of changes has taken place in the structure of the fin spine in Chimaeriformes which is similar to the changes which have occurred in the fin spines of sharks. The scales of
Deltoptychius
are cyclomorial: the change from a cyclomorial to a placoid (synchronomorial) scale has taken place independently among elasmobranchs at least three times: in the selachians, in the edestids, and in the Chimaeriformes. The scales of
Helodus
are synchronomorial: this is the only Palaeozoic elasmobranch in which this condition has yet been found. The crescentic calcifications surrounding the sensory canals of chimaeroids are shown to be modified scales. The pre-pelvic tenacula of male chimaeroids have evolved from groups of enlarged scales without skeletal support which occur in myriacanthoids. It is suggested that the frontal clasper of male chimaeroids has evolved from paired structures, defensive in function, which occurred in both sexes in Palaeozoic Chimaeriformes. The dermal armour of Chimaeriformes is formed by fusion of scales. It has undergone a series of changes in histological structure which is similar to that undergone by the dorsal fin spine, but which took place much earlier in time. The evidence for a relationship between Holocephali and the selachians or arthrodires is reviewed. It is concluded that there is evidence of relationship between holocephalans and arthrodires, but that the resemblances between the two groups are in general features only. Mainly because of the structure of the exoskeleton in
Deltoptychius
, a relationship with the ptyctodonts is very unlikely. The holocephalans seem to be closest to the most primitive arthrodires (rhenanids and stensioellids), suggesting that the two groups share a common ancestor but are not directly related.
VI.—On Professor Hull's Carboniferous Classification