I have represented the body of the fish as forced into two curves when swimming, as there are never less than two. These I designate the cephalic (_d_) and caudal (_c_) curves, from their respective positions. In the long-bodied fishes, such as the eels, the number of the curves is increased, but in every case the curves occur in pairs, and are complementary. The cephalic and caudal curves not only complement each other, but they act as fulcra for each other, the cephalic curve, with the water seized by it, forming the _point d’appui_ for the caudal one, and _vice versâ_. The fish in swimming lashes its tail from side to side, precisely as an oar is lashed from side to side in sculling. It therefore describes a figure-of-8 track in the water (_e_ _f_ _g_ _h_ _i_ _j_ _k_ _l_ of fig. 32). During each sweep or lateral movement the tail is both extended and flexed. It is extended and its curve reduced when it approaches the line _a b_ of fig. 32, and flexed, and a new curve formed, when it recedes from the line in question. The tail is effective as a propeller both during flexion and extension, so that, strictly speaking, the tail has no back or non-effective stroke. The terms effective and non-effective employed by authors are applicable only in a comparative and restricted sense; the tail always operating, but being a less effective propeller, when in the act of being flexed or curved, than when in the act of being extended or straightened. By always directing the concavity of the tail (_s_ and _t_) towards the axis of motion (_a b_) during extension, and its convexity (_c_ and _v_) away from the axis of motion (_a b_) during flexion, the fish exerts a maximum of propelling power with a minimum of slip. In extension of the tail the caudal curve (_s_) is reduced as the tail travels _towards_ the line _a b_. In flexion a new curve (_v_) is formed as the tail travels _from_ the line _a b_. While the tail travels from _s_ in the direction _t_, the head travels from _d_ in the direction _w_. There is therefore a period, momentary it must be, when both the cephalic and caudal curves are reduced, and the body of the fish is straight, and free to advance without impediment. The different degrees of resistance experienced by the tail in describing its figure-of-8 movements, are represented by the different-sized curves _e f_, _g h_, _i j_, and _k l_ of fig. 32, p. 68. The curves _e f_ indicate the resistance experienced by the tail during flexion, when it is being carried away from and to the right of the line _a b_. The curves _g h_ indicate the resistance experienced by the tail when it is extended and carried towards the line _a b_. This constitutes a half vibration or oscillation of the tail. The curves _i j_ indicate the resistance experienced by the tail when it is a second time flexed and carried away from and to the left of the line _a b_. The curves _k l_ indicate the resistance experienced by the tail when it is a second