Abstract
1. The paired fins of fishes are largely concerned with the production of vertical forces, and thus principally affect the pitching (rising and diving) equilibrium. 2. In the sharks the pelvic fins increase to a small extent the static stability for pitching movements. Nevertheless, the relatively large area and forward position of the pectoral fins preponderates over the influence of the pelvics on the pitching stability, so that the contribution of the latter is very small. This is borne out by amputation experiments. 3. In the bony fishes, the development of the actinopterygian fin leads to a much greater mobility of the fins. In consequence, the pelvic fins of the bony fishes exhibit a considerable adaptive radiation. 4. In the percoid fishes the use of the pectoral fins as brakes produces a lift as well as a drag force. It is shown that the neutralization of this lift force by a downward force produced by the pelvic fins necessitates the forward migration of the latter. If this migration did not take place, the fish would either tilt upwards or rise bodily during the stop. 5. The dynamical basis for this migration of the paired fins is considered in an approximate mathematical treatment of the equilibrium during the stop. 6. This hypothesis is confirmed by amputation experiments, and also by the occurrence of a convergent fin migration in the Coelacanthidae. The absence of the forward pelvic fin migration in flying fishes also affords indirect support. 7. There is no evidence to suggest that the pelvic fins can function as bilge keels, though they may be used actively to produce a rolling moment.

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