1. Typical undulatory progression over a rigid environment depends on three fundamental factors :

(i) Internal bending couples change the lateral curvature of each region of the body to that previously characteristic of the region lying immediately anterior to itself.

(ii) The phase of lateral bending varies along the length of the animal's body.

(iii) The presence of external restraints prevents all regions of the body from moving along any path other than one tangential to their own circumference of curvature.

2. The magnitude of the forward tangential thrust imparted to the body depends on (a) the magnitude of the internally generated bending couples, and (b) the form of the waves. If friction operates on the surfaces of external restraint the thrust also depends on the coefficient of lateral friction and on the position of the restraints.

3. From a mechanical point of view, an undulating organism (irrespective of its size and internal structure) can be regarded either as a series of curved levers or as a series of inclined planes.

4. The general principles of undulatory swimming are the same as for a terrestrial glide, except for the fact that each element of the body must possess a component of motion normal to its surface if it is to contribute towards the propulsion of the animal; this type of motion can only occur when the waves move backwards relative to the ground. The animal cannot move forward as fast as the waves are propagated over the body.

5. The propulsive powers of three-dimensional waves are limited to the extent to which the organism is restrained by external forces from spinning about its own longitudinal axis. Otherwise the principles of progression are the same as for two-dimensional waves: the resultant of all the forces acting normally to the body is equal but opposite to that of all tangential forces.

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