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Transactions of the Boyal Society of South Africa. 
case of the Achirus, for it could still perform a valvular function in con- 
junction with the operculum in front of it, or it may well have been the 
case that in the primitive vertebrate with its increased number of gill slits y 
not only the opercular flap of the hyoid arch became enlarged to cover some 
of the gill clefts, but that this happened also in two of the branchial arches. 
It need not, however, necessarily find any form of useful activity to justify 
its continued existence. 
It may be significant in connection with this suggested opercular origin 
of the paired limbs that in the lower vertebrates, such as Cephalochordata 
and Cyclostomata, in which no opercular flaps are developed, though there 
are branchial skeletal elements, there is no indication of the existence of 
paired limbs at any time in their development or life-history. 
The origin of paired limbs from such opercula is not a new suggestion, as 
it is part at least of the hypothesis associated w T ith the name of Gegenbaur, 
who suggests that the limb-girdle and pterygia are derived from a visceral 
arch and its rays, and (2) the pinna or distal part from the flaps or external 
free part of the gill septum. He, however, did not apparently entertain 
the idea that one of these parts of his theory might be correct while the 
other might be w r rong. Another theory (Graham Kerr) is that external gills 
were in primitive vertebrates developed on each visceral arch and that these 
gave rise to the paired limbs. This theory is not inconsistent with 
Gegenbaur's suggestion that the girdles are derived from visceral arches. 
A third theory (Balfour) is that the paired limb is derived, not from any 
external modification of the gill septum, but from an epidernal fold which 
in primitive vertebrates is supposed to have occurred along each side of the 
body. It seems to have been generally taken for granted throughout the 
history of the discussion that this is inconsistent with the derivation of the 
supporting structures of the limbs from visceral arches, but it is not quite 
evident that it is necessarily so. It is true that in the elaboration of the lateral 
fold theory it has been suggested that the differentiation into limb rudiments 
began at the anterior and posterior regions of the folds, but this is not a 
necessary corollary of the hypothesis, for the differentiation may have 
commenced at the anterior extremity in association wdth a disappearing gill 
arch, and, on the movement backwards of this combined structure (epidermal 
fold and supporting visceral arch), the process may have been repeated to 
give rise to the anterior pair of limbs. 
On the other hand, while all three theories look for an external epidermal 
organ as the origin (in part at. least) of paired fins, the derivation of the 
supporting structures (girdles and radials) from gill arches is not bound up 
with this origin. Any of the three views in this limited sense are consistent 
with the origin of these elements de novo. Thus even Gegenbaur's or Graham 
Kerr's hypotheses may be correct so far as the limb proper is concerned, and 
yet open to question as regards the origin of its supporting structures. 
