ABSTRACT
When a body is moving in water it encounters a resistance in the direction of its motion, and consequently the body must be supplied with energy if motion is to occur at a uniform speed. A study of the mechanism of propulsion of a fish falls therefore into two parts, (1) a study of the forces resisting motion through the water, and (2) a study of the mechanism whereby the fish utilises the energy liberated by its muscles for overcoming the forces of resistance. To some extent these two aspects of the problem are interdependent and involve considerable hydrodynamical difficulties, but in the present paper an attempt will be made to show that the movements of a fish’s body1 are such as to generate forces capable of opposing the forces of resistance whatever be the nature or magnitude of the latter. The problem was attacked two centuries ago by Borelli and by Pettigrew in 1873 since then comparatively little attention has been devoted to the subject except by Breder (1926), whose results will be considered later.
The present paper deals only with the propulsive properties of the bodiet of a selected number of fish whose appendages play little or no part in the propulsion when the fish are moving at reasonable speeds. The propulsive properties of the caudal fin will be considered in a subsequent paper.
The term “segment” is not used in its strict morphological sense.
It might be objected (see Fig. 17) that if the tail were to lie at a point equal to a multiple of half a wave-length from the head, it should travel approximately in a straight line. It must be remembered, however, that this is only the case under two purely artificial conditions, (a) when the amplitude of the movements is the same along the whole length of the body, (b) when the axis of reference is such that the head is moving along a straight transverse line. If it were possible to refer the movements to an axis which is moving forward at the same average forward velocity of the whole fish (i.e. if the propagation of the wave involved no displacement of the centre of gravity of the system), every point of the body would appear to travel in a figure of 8, the horizontal amplitude of which would vary directly with a power of the amplitude of the metachronal wave. If the wave have the form of a sine curve and the amplitude be small, it can be shown that the longitudinal amplitude of the figure of 8 curve is 
where ω is the amplitude of the sine wave and λ is the wave-length.
The existence of a force normal to the surface of the body, and the reduction of the longitudinal component produced by an increase in the angle of inclination (8) of the body, was pointed out by Breder (1926).
This is the flow of the water relative to the body if the water is stationary in respect to the earth.
