Studies in Animal Locomotion I

STUDIES IN ANIMAL LOCOMOTION I. THE MOVEMENT OF FISH WITH SPECIAL REFERENCE TO THE EEL

BY J. GRAY. King's College, Cambridge.

(From the Laboratory of Experimental Zoology, Cambridge.)

(Received 20th October, 1932.)

(With Four Plates and Eleven Text-figures)

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. Since all propellers operate by driving astern a volume of water, the reaction from which compensates the surface resistance of the moving object, the initial problem of the fish's movements consists in demonstrating that the fish moves its body in such a way as to drive water away from its surface in a backward direction. All inanimate propellers belong to one of three types: (1) The jet propellers—as exemplified by all reaction turbines which project a current of water from a nozzle. The reaction caused by the water moves the nozzle in a direction opposite to that of the movements of the water. (2) Paddles—whereby a backward thrust is exerted on the water, parallel to the direction of motion of the paddle and at right angles to the surface of the paddle. The paddle can only be submerged during one-half of its complete movement, or it must be capable of being rotated about an axis at right angles to its line of motion, in order that no appreciable thrust is exerted during the period which follows the effective phase of the movement. (3) Screws—the theory 1 The present paper deals only with the propulsive properties of the bodies 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. Studies in Animal Locomotion 89 of screw propulsion is essentially that of an inclined plate—which, by motion through the water, generates a force at right angles to its surface (see Fig. 1). This force (P) has a component (T) at right angles to the direction of motion of the plate, which tends to move the plate along a line at right angles to its original direction of motion. The mechanism of propulsion of a typical fish does not conform to the design of a jet or a paddle, and since all screws operate by means of a true rotary movement, the possibility of a screw is, at first sight, excluded. The object of this paper is to consider the motion of a fish's body and to compare the underlying mechanism with that of a typical screw propeller. During the whole of the work, an attempt has been made to record the form and position of the fish at known intervals of time by photo-

Fig. 1. AB is a cross-section of the blade of a screw moving along cd in the direction of the large arrow. A force P is generated at right angles to AB. This has a component T which tends to move the screw in the direction of a. graphic means. An experimental tank was set up in the field of a timed cinemato- graph camera, so that the position of the fish could be determined by means of a graduated field placed immediately underneath the fish. The method of recording the interval between successive photographs has been described elsewhere (Gray, 1930). I am greatly indebted to Mr J. E. Harris for his valuable help in the preparation of these photographic records. I. OBSERVATIONS ON THE MOVEMENTS OF FISH. As observed by the human eye, the motions of various types of fish appear to vary considerably from one species to another. The most conspicuous features of a moving eel—as are seen in the photographs taken by Marey (1894)—are the waves of curvature which pass along the length of the body from head to tail. In the dogfish (and, still more, the mackerel and whiting), the presence of such waves is less obvious, and the visible movements appear to be due to transverse strokes 90 J. GRAY executed by the posterior end of the body across the axis of motion. It can be seen from the photographs reproduced in Figs. 2-10 (Pis. I—IV), however, that in all these cases waves of curvature pass along the body alternately on the two sides, but that they differ in the various fish in certain important characteristics. Firstly, their speed of propagation along the body varies greatly. In the examples illustrated the approximate speeds of the waves and the rates of movement of the fish are as shown in Table I. Secondly, the form of the waves differs. In the reversing eel (Fig. 10) the amplitude of the waves is very large, and is of approximately the same value as their wave-length. In Ammodytes (Fig. 7) and the mackerel (Fig. 5), the relative amplitude is very much smaller, while the dogfish, glass-eel, butterfish, and rockling occupy intermediate positions. Thirdly, when the fish are swimming at a steady rate, the frequency of the waves per second varies in the different species. In the examples illustrated, the approximate number of waves passing down each side of the body are shown in Table II. Fourthly, the amplitude of the waves is always greatest at the posterior end of the body, but the variation between the amplitude of the head and tail varies very greatly in different types. In the small eel the amplitude of the movements of the head is relatively very much greater than those of the mackerel or whiting. Table I.

Velocity of wave Velocity o cm. per sec. cm. per

Gla9s eel (Anguilla vulgaris) 6-2 4-0 Butterfish (Centronotus gunnellus) I7-S 117 Whiting (Gadus merlangtit) 25-0 168 Dogfish (Acantlrias vulgaris) 55 29 Mackerel (Scot/tber tcombrus) 77 42-5 Ammodytes (A. lanceolatut) 160 80

Table II. Waves per min.

Glass-eel 93 Butterfish 120 Whiting 120 Dogfish 54 Mackerel 170 Ammodytes 120

The movement of the muscular waves along an eel's body was recorded photographically by Marey (1894), who made no attempt to define the mechanical principles which are responsible for the forward movement of the whole fish. These principles have been considered by Breder (1926), whose description of "anguilli- form" movement is as follows: "The forward motion is certainly attained by the pressure of the fish's body against the water in the following manner. The mechanical forces brought to bear on the water are diagonally backwards (from the posterior surfaces of each of the curves of the body). As these are distributed symetrically about the line of progression, a forward resultant of reaction follows, for pressure Studies in Animal Locomotion 91 from a moving plane is always at right angles to its surface." That the fish's body exerts a pressure on the water at right angles to its own surface is in accordance with the analysis given later in this paper, but Breder goes on to state that " It might be objected that as the eel is moving ahead there is likewise adverse pressure diagonally forward from the anterior sides of these backwardly moving waves. The truth of this is evident and it simply makes it necessary for the fish to pass these waves posteriorly at a rate considerably faster than it expects to move forward The speed of the waves moving backward must exceed that of the forward motion of the animal as a whole. If the two speeds just equalled each other it would mean that any point on a wave such as its crest would be stationary with reference to the sea- bottom; but as one is dependent on the other this is obviously impossible." The mechanical principles involved by this explanation are by no means clear, for it is certain that the propulsive thrust of the moving body is due to the fact that each part of the body is executing a series of transverse movements. Although these movements can be expressed in terms of longitudinally moving waves of contraction, the principles of propulsion of a fish are much more readily derived from a study of the transverse movements of each section of the body than from a direct investigation of the propagated waves of contraction. In the present paper an attempt will be made to investigate the propulsive effect of those transverse movements which are induced in the various parts of the body by a series of muscular contractions which are of such a nature as to produce the phenomenon of a propagated wave. The movements of the body can be considered in two ways. Firstly, it is possible, from a series of instantaneous photographs taken at equal and known intervals of time, to plot the position in space of any particular point on the surface of the body. Secondly, it is possible to consider the movements executed by one part of the body relative to any other part, and not to fixed points in the environment of the fish. By combining these two sets of observations it is possible to form an idea of the way in which the contractions of the muscles induce changes in the relative position of the parts of the body, one to another, which are such as enable the fish, as a whole, to transmit to the water a backward momentum equal and opposite to that of the frictional forces which oppose the motion of the fish through the water. Owing to the well-defined nature of its muscular waves, attention may be con- centrated on the small glass-eel (Anguilla vulgaris), about 7 cm. in length, shown in Fig. 2; the same type of analysis can be applied to other forms, but for various reasons it is convenient to defer this until later. Fig. 12 A-C shows the track of the head, the middle point of the body, and the tip of the tail of an Anguilla whose form during motion is shown in Fig. 12 D. It will be noticed that the successive positions of each point lie along a sinusoidal curve whose " pitch " or wave-length is the same in all cases, namely 3-2 cm., and that this is less than the "pitch" or wave-length of the waves which characterise the body itself, viz. 4-7 cm. It can also be seen that the amplitude (to) of the waves is least at the head and greatest at the tip of the tail. The axis of motion (ab) of the fish is shown in the figure, and it can be seen that if a line (cd) is drawn at right angles to 92 J. GRAY this axis at points where the track of the point on the body cuts the axis of motion, then the angle (8°) between the track of the fish and this line cd (the transverse axis of movement) becomes progressively less as the tip of the tail of the fish is ap- proached. If we now examine (Fig. 13) the angle (6m) made by any part of the body of the fish and the line cd as this particular part crosses the line ab (i.e. crosses the axis of forward movement), it can be seen from Fig. 13 that this angle also decreases

Head " Mid Point "Tip of Tail A B C D Fig. 12. A-C. The paths followed by (i) the head, (ii) the middle of the body, (iii) the tip of the tail of a young Anguilla (glass-eel). Note that the amplitude is greatest at the posterior end of the body, and that the wave-length of each track (A) is less than that of the curve of the animal's body (D). from the head to the tail; since the pitch of the body is greater than the pitch of the curve of movement, it follows that the angle (8m) made by the body with the transverse axis (cd) must be greater than the angle (80) between the path of motion and the transverse axis of movement. The difference between these two angles (8m — 8p) is of fundamental importance and will be called the angle of attack; it is designated by the symbol a. Similar curves to those shown in Figs. 12 and 13 can be constructed for other types of fish with similar results except that in most fish the amplitude of the transverse movements of the head are very small compared with Studies in Animal Locomotion 93 those of the tail. It can be seen in Figs. 2-10 that not only are the wave crests travelling along the body of the fish but they are also travelling backwards with reference to the environment. To define the movements of a point on the body relative to other parts of the body it is necessary to adopt two fixed axes of reference. One of these is provided by the axis of forward movement (ab), for this is also the axis about which each point of the body is moving in a transverse direction relative to any other point. It is not so easy to obtain a fixed transverse axis. The ideal procedure would be to plot the position of each point on the body with reference to a transverse axis which is

< a

Fig. 13. Tracings from enlarged photographs of Anguilla showing that the angle between the body and the axis of forward movement (ab) decreases from the anterior to the posterior end of the body.

moving forward with the fish at a velocity equal to the average forward velocity of the fish. This can be done within small limits of error if successive photographs are enlarged and then superimposed on each other in such a way that the tip of the head lies along the same transverse axis and if the longitudinal axis of motion (ab) of each photograph is superimposed on that of the others. This has been done in Figs. 14 and 15, which represent a fish whose waves are moving down the body in the normal way, but whose body is unable to progress forwards. It can be seen that during each phase of its motion, any given point forms part of a segment1 of the body (Fig. 14) which is inclined with its leading surface (i.e. the surface towards the direction of 1 The term "segment" is not used in its strict morphological sense. fl J. GRAY transverse movement) turned towards the hinder end of the body. Thus in Fig. 14 the segment XY is travelling from the right side of the axis of movement towards the left side and its leading surface is facing backwards and towards the left. Con-

Fig- IS- Fig. 14. Enlarged drawings of a young Anguilla arranged to show the movements of short segments of the body during the passage of the complete wave past the segments. Note that the segment XY is travelling from right to left and is directed backwards and to the left. The segment X1Y, is travelling from left to right and is directed backward and towards the right. Note also that the tip of the tail is moving in a figure of 8 curve. Fig. 15 A. Tracings of left side of a butterfish showing the passage of a wave, and the corresponding positions of the tail (1—7). The dotted line shows the longitudinal axis of motion. Fig .15 B. Shows the relative transverse velocity of the tail at different phases of its motion. Note that the velocity is greatest when the tail is crossing the axis of forward movement. versely, the segment X1 Yy shows the corresponding movement of a segment from left to right, and the leading surface is facing backwards and towards the right. It will be noted that as the segment XY is passing from right to left, the segment forms part of a wave whose crest is travelling down the right side of the body, and that as Studies in Animal Locomotion 95

X1 yx moves from left to right it is part of a wave travelling down the left side of the body. Although the movements executed by each segment of the fish closely resemble the movements of the blade of an oar when sculled from the back of a boat, the body of the fish exhibits certain peculiar features of considerable theoretical importance. (1) The speed at which a segment moves along its transverse path is not uniform. When displaced to its maximum extent from the longitudinal axis of movement (ab), the segment is moving very slowly; as it crosses the axis of longitudinal movement it is travelling at its maximum speed. Fig. 15 B shows the position of a segment at equal periods of time, and it can be seen that the speed of its movement varies inversely with its displacement from the longitudinal axis of movement; during the phase of movement towards this axis the segment is accelerating, and during the phase of movement away from this axis the segment is decelerating. (2) During each phase of movement the angle (9) between the body segment and the longitudinal axis of movement is changing. It is greatest at the extreme positions and is least as the segment is crossing the line (ab). Towards the end of each phase, the segment is parallel to the axis of longitudinal movement, but as it begins to move towards this axis, the segment becomes more and more inclined backwards—after crossing the axis the process is reversed until the segment again points directly forward; finally it becomes inclined in the opposite direction as it begins to move backwards towards the axis (ab). The important point to notice is that the angle 6 is least when the segment is crossing the line ab—i.e. when it is travelling at its maximum transverse speed. (3) It is only when the segment is near to the axis ab that its leading surface is bounded by a plane—in all other positions the surface is curved. As the segment approaches the axis ab the leading surface is bounded by a curve which is concave towards the direction of movement; after passing the axis (ab) the leading surface is convex towards the direction of movement. (4) If a point is marked on the surface (e.g. the base of the tail-fin in Fig. 15 B), it is found to travel in a figure of 8 curve relative to the head. The transverse axis of the figure of 8 is at right angles to the axis ab. The greater the amplitudes of transverse movement relative to the pitch or wave-length of the curve of the body, the more marked are the figures of 8 (see footnote, p. 97). The figure of 8 movement can be expressed in another way, namely, when the whole fish is travelling forward at a constant average velocity, the forward velocity of segments at the extreme positions of transverse displacement is rather greater than the average velocity of the whole fish and the forward velocity of segments which are crossing the axis of motion is rather less than this average value. When the fish is in motion, the path traced out in space by any given point on the body represents, of course, the locus of a point travelling along a figure of 8 curve which is endowed with a forward velocity equal to the average forward speed of the fish. The angle between the body of the fish and its path of motion is of 96 J. GRAY vital significance when we consider the propulsive properties of the body; the effect of the figure of 8 motion is, as is seen in Fig. 16, to increase this angle when a segment of the body is crossing the axis of longitudinal motion, and to decrease it as it approaches the extreme positions of its transverse displacement. The figure of 8 motion of the tail of a slowly moving sturgeon was observed by Pettigrew (1873), who implied that the movement is a physiological adaptation to efficient propulsion. It can, however, be shown that it is the inevitable result of the propagation of a wave of curvature along an inextensible body.

Fig. 16. If AB is a segment travelling transversely along the dotted line a,^ and endowed with a velocity of 1 cm. per unit of time towards the right, then the track of the mid-point of the segment is shown by the dotted curve to the right of the figure. If AB travels along the figure of 8 (0,0,0, ft, 6,6,) and is endowed with the same transverse velocity as before, then the track of the mid-point is shown by the full curve on the right of the figure. Note that the angle ot, is greater than the angle at,. The larger the amplitude of the figure of 8 the greater is the increase effected in ot at the mid- point of the transverse movement. [The reverse is the case at the extreme positions.]

Fig. 17 shows an inextensible string along which is passing a wave of curvature of constant form. If one end of the string be allowed to move along the axis (cd), and the position of any given point on the string be marked in respect to this axis and the longitudinal axis (ab), it can be seen that except at stated intervals (of half- wave-lengths from the constrained end of the string) the path traced out by a fixed point along the string is a figure of 8. The greater be the relative amplitude to the wave-length of the waves the more marked is the figure of 8 movement. Since the relative amplitude at the tail end of Studies in Animal Locomotion 97 a fish is greater than at any other point, the tail exhibits a more marked figure of 8 than does any other point1. The observed movements of the fish's body can be summarised as follows: (1) All parts of the fish's body which are in transverse motion have their leading surfaces directed backwards and towards the direction of transverse movement, but the angle of inclination is most pronounced when the segment is crossing the axis of longitudinal motion, and at this point the segment of the body is travelling at its maximum speed. (2) Each point on the body is travelling along a figure of 8 curve relative to a transverse line which is moving forward at the average forward velocity of the whole fish. In other words, a segment when moving across the axis of longitudinal motion is travelling backwards relative to a segment which has reached the extreme position of lateral displacement. The track of any point on the body (relative to the earth) is a sinusoidal curve whose pitch or wave-length is less than that of a curve 1 2 .3 4. 5 6 7 8 9 Cr

Fig. 17. The dotted lines show the loci of points situated at i, 5, 10, 175 cm. from the end of an inextensible string along which a sine wave is moving. Note that each point travels on a figure of 8 curve unless it is situated at or about one-half of a wave-length from the front end of the string. The larger numerals show successive positions of the crest of wave; the smaller numerals show the corresponding positions of the selected points. which defines the body of the fish. There is therefore a definite angle between the surface of the fish and its path of motion. The movements executed by the body of a fish are closely analogous to those exhibited by a flexible but elastic body, one end of which is made to vibrate along a transverse axis. The validity of this analogy will be discussed elsewhere. 1 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 —r-, where OJ is the amplitude of the sine wave and A is the wave-length. 98 J. GRAY It now remains to be shown that (i) the movements of the body are such as to drive the fish forwards through the water, and (2) the movements of the body are the direct effect of a series of waves of muscular contraction which start at the anterior end of the body and pass backwards towards the tail. The latter consideration will be discussed in a later paper.

II. THE BODY AS A PROPELLER. Towards the extreme positions of each transverse cycle the velocity, form and inclination of the body are rapidly changing and although it is possible to see in an empirical way their general propulsive significance, these changes render impossible an analysis of the effect of a complete cycle on the distribution of the surrounding water. This distribution may be affected by the acceleration as well as by the velocity of the body, and for this reason it is convenient to consider in the firstinstanc e the forces acting on a segment as the latter passes the axis of forward movement, i.e. when it is inclined to this axis to its maximum extent, and when the velocity of its transverse movement is greatest and reasonably constant.

Fig. 18 a. Fig. 186. Fig. 18 a. The dotted line shows the locus of the point O on the segment AB when the latter is travelling transversely from left to right. The position and inclination of the segment at various phases of its movement are shown at 1-7. ab, longitudinal axis of movement, cd transverse axis of movement. The fish is depicted as stationary. Fig. 186. Diagram showing the position of the segment AB of a stationary fish in respect to its own direction of motion and to the longitudinal axis ab. The segment is travelling along OE with a velocity equal to OE. The angle BOE = a, and is the angle of attack. The angle BOd is the angle (6m) at which the segment is inclined to the axis cd. The movement of water relative to the segment is at first EO; after meeting the body, a finite volume of water flows along OA with a velocity OH. The effect of the water on the body is to endow the latter with a velocity GO relative to the water. Studies in Animal Locomotion 99

Symbols used. a = the angle between the surface of the body and its direction of motion. 9 = the angle between the surface of the body and a line drawn at right angles to the longitudinal axis of movement (ab) of the fish. 9m = the value of 9 for any particular point on the body when this point is crossing the axis ab. ab = the longitudinal axis movement of the fish. cd = an axis at right angles to the longitudinal axis of movement, i.e. an axis of transverse movement. A*= the pitch or wave-length of the muscular waves on the body. OJ = the amplitude of the muscular waves on the body. P = the pressure normal to the surface of the body. F = the factional force tangential to the surface of the body. T •= the forward component of the force P. D = the transverse component of the force P. X = the transverse component of the force F. Y = the forward component of the force F. V = the velocity of forward movement of the fish. Vg = the transverse velocity of any given segment. VR = the resultant velocity of any given segment and the surrounding water. As long as the fish does not move forward, the direction of movement (relative to the earth) is to the side and backwards (Fig. 18); but when the fish is travelling forwards the direction of motion of AB is to the side and forwards. During this movement water is displaced, and the water displaced at the leading surface (AB in Fig. 19) must either flow along the surface of the body or over the dorsal and under the ventral fins. Since the body presents a relatively flat surface to the water, some at least of the water must be deflected along the surface of the body in a backward direction, i.e. in the direction OA. Let the mass of water so deflected be m gm. ^Z per sec. If the original velocity of this water axis ab. The segment AB is moving along the ((relativ e to the bodyy) is Frcm. ppe r sec. (( = OE, ^^^^ ^^t^th^^t e ?J%?J% Fig. 19), this Can be resolved into two com- inclination (0m) is FOd. The normal pressure f the the w at er is ponents, one (OF) being parallel to the body ° D ^J °" . f proportional to r ' v ' 1 ,11 OP and the fnctional force is proportional and the other (EF) normal to the body. to OF-OH. ioo J. GRAY After encountering the body, the component EF is lost in respect to m gm. of water per sec., hence there has been a loss of normal momentum by the water of m , EF gm./cm. per sec.: this momentum must be gained by the body and it represents the pressure of the water on the body in a direction at right angles to its surface. Now EF = Vr sin a (where a. is the angle between the body and its direction of motion), hence the normal pressure of the body (P) on the water is m Vr sin a and the component of this force along the axis of longitudinal

A K /A N

Jo?'

\1 M N Jh

A' KN I,K N / t /

// / a/' 1 A M A M A M L 4 L 5 L 6 Fig. 20. Showing the effect of an increase in forward speed on the value of the angle a. r, fish stationary; the forward velocity of the segment has a negative value of MN. 2, forward velocity of fish = backward velocity of segment. 3 and 4, forward velocity giving reduced but positive values for a. 5, forward velocity giving zero value for a. 6, forward velocity giving negative value for a, giving negative thrust. AIN —- forward velocity of segment; LAf -= trans, velocity of segment; MK = the pitch of the segment which is a function of the wave length of the muscular waves; cd = trans, axis of movement. movement is the corresponding propulsive thrust. As long as the body moves at an angle to its own path of motion, there must therefore be a tendency for the fish to move bodily through the water. As soon as the fish begins to move, however, two events occur: (i) the angle a diminishes, and (2) frictional forces are generated at its surface. The diminution of the angle a is seen in Fig. 20, where it can be seen that there is a value for the rate of forward progression which is such that a = 0°, and at this point the propulsive thrust must be zero; at the moment it may be noted that the faster is the forward speed of the fish the smaller is the angle of attack and the smaller is the normal pressure of the fish against the water. Studies in Animal Locomotion 101 As long as the fish is in motion, its movements will be resisted by factional forces which are due to the disturbances set up in the water in the neighbourhood of the body. These forces act along all surfaces and their direction is tangential to the direction of motion of the surface, so that when water is moving past the body of the fish, the velocity of the water is reduced. Thus in Fig. 19 if FO be the relative tangential velocity of m gm. of water struck by the fish per sec., and if OH be the relative velocity of this water after passing over the segment, then the water has lost momentum equivalent to m (FO-OH) gm./cm. per sec. in the direction of OH. This momentum is gained by the body and represents the frictional force (F) acting in the same direction. Thus the net effect of moving a segment of the body through the water at an angle to its own direction of motion is to impress on the body two forces—one normal to the surface (P)1 and the other tangential to the surface (F). The longitudinal resultant of these two forces represents, if the present K

analysis is correct, the net propulsive thrust which drives the fish against the resistance of the water. If forward motion at a uniform velocity is to take place, the resultant forces acting on the segment of the body when measured in any direction must be zero. The conditions under which this will occur can be seen by resolving the forces P and F along the longitudinal (ab) and transverse (cd) axes of movement respectively. Thus in Fig. 21 let LM = Vo = the velocity at which the segment AB is travelling, along the' transverse axis of movement. Let MN = V = the velocity of forward movement of the fish. Then the segment inclined at an angle 9 to LM is travelling a 2 along LN with a velocity LN = Vr — VFo + V ,* and the angle between the body 1 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 (0) 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. 102 J. GRAY and its path of motion is a. The force P (which depends on the value of a, see p. ioo) and the force F can both be resolved along ab and cd. The propulsive thrust along ab is T - Y and the resisting force along cd is D + X. Now if these forces are to have no resultant they must be compensated by equal and opposite forces operating on the segment. These latter forces are (i) the force exerted by the muscles, (ii) the resistance exerted by other parts of the body. Whether or not a structure (e.g. the dorsal and ventral fins, the skull, etc.) is exerting a propulsive thrust there will always be a frictional force at its surface, and as such structures may be moving in any forward direction it follows that the frictional forces can always be resolved into components acting along ab or cd. Uniform motion will ensue, therefore, when the forces along cd are collectively equal and opposite to the force exerted by the muscles, and when the forces acting along ab are equal and opposite to the longitudinal components of all the forces developed by other parts of the fish which resist the forward motion of the segment. It follows that when a fish (which is initially at rest) begins to move its body in such a way that the leading surface is inclined backwards at an angle to the axis (ab) it will move forward with increasing velocity until the angle (a) between the leading surfaces and their direction of motion is reduced to such a value that the net propulsive thrust is exactly equal and opposite to the effect of the frictional forces acting on the body. It will be remembered that the value of 6 varies for different phases of the movement of a single segment (see Fig. 14) and for different regions of the body (Fig. 14). For regions lying towards the middle of the body of the fish shown in Fig. 14 the value of 8m (when the segment is passing the longitudinal axis of movement) is about 500. As the segment moves away from this axis the value increases to 900, and as this increase occurs, the transverse velocity falls. It is clear from Fig. 21 that a rapid fall probably occurs in the thrust and in the work done as the value of 8 increases—and for high values, a very weak and very inefficient thrust remains. It has also been shown that the value of 8 varies for different regions along the body of the fish. The thrust and therefore the useful work done by a segment thus depend on its position in the body as well as on the particular phase of its own cycle, so that the total thrust exerted by the whole fish represents the sum of the thrusts exerted by all the segments of the body, all of which have different values of 8 and may have different values of a. Before examining these phenomena in greater detail it is convenient to consider the relationship which exists between the values of the angles 8 and a on the one hand and the form and properties of the muscular waves, which pass from one segment to another, on the other. This will be done in a subsequent paper. The analysis given above assumes that the relative velocity of the body and the surrounding water is the resultant of the transverse velocity of the body and its forward velocity through the water; in other words, that the water which encounters the body is at rest relative to the earth. It is unlikely that this condition is strictly fulfilled, since the anterior regions of a fish such as that of a mackerel may influence the rate of flow of the water past the segments lying more posteriorly; in this case the velocity with which these segments encounter the water may not be Studies in Animal Locomotion 103 simply the resultant of the transverse and longitudinal motions of the fish itself. At present there is no means of determining the exact flow of water past the fish, and it is necessary to assume that disturbances of this type are comparatively small.

III. SUMMARY. 1. The waves of muscular contraction which pass along the body of a swimming eel occur also in other fish. The waves vary greatly in speed of propagation, amplitude and frequency. The speed of propagation of the waves is too low to be controlled by the rate of conduction of a simple nervous impulse. 2. The movements executed by a localised area on the surface of the body are such that each area moves in a direction transverse to the line of forward movement. During these movements the leading surface of the body is inclined backwards towards the tail and at an angle to the path of motion of the area concerned. The angle of inclination and the angle made with the path of motion vary with (a) different regions of the body, and (b) with different phases in the motion of each region. 3. Each point on the body travels in a horizontal figure of 8 relative to a transverse axis which is moving forward at the same average velocity as the whole fish. A segment of the body at the mid-point of its transverse motion is travelling forwards at a rate slightly less than that of a segment at the extreme position of its transverse movements. These movements are the mechanical result of the inextensibility of the body, and they effect significant changes in the angle between the surface of the body and its direction of movement. 4. The movements of each part of the body are shown to be such as to generate a forward thrust which drives the fish forwards against the resistance of the water. The magnitude of the forward thrust depends among other things on (a) the angle which the surface of the fish makes with its own path of motion, and (b) on the angle between the surface of the fish and the axis of forward movement of the whole fish, (c) on the velocity of transverse movement of the body. 5. The propulsive properties of each segment of the body are greatest as the segment is crossing the axis of forward movement.

REFERENCES. BREDER, C. M. (1926). Zoologica, 4, 159. GRAY, J. (1930). Proc. Roy. Soc. B, 107, 313. MAREY, E. J. (1894). he Movement. Paris. PETTIGREW, J. B. (1873). Animal Locomotion. London.

EXPLANATION OF PLATES. PLATE I. Fig. 2. Successive positions of a young eel (Anguilla vulgaris) (7 cm. long) during a period of 1 sec. The photographs were taken at 009 sec. intervals. The side of each square is 1 in. The passage of the muscular waves is marked by black dots and crosses. The dark line represents the pigmented mid-dorsal line of the transparent animal. Note the well-defined curvature of the body. 104 J- GRAY Fig. 3. Successive positions of a butterfish (C. gunnelltu) in i sec. The photographs were taken at 0-05 sec. intervals. The side of each square ia 3 in. Note the 9mall amplitude of transverse movement of the head. Fig.4. Successive positions of a butterfish in J sec. The photographs were taken at 005 sec. intervals. The side of each square is 1 in. The passage of the waves is marked by dots or by crosses. Note that the tail is almost at right angles to the path of motion of the fish when it is crossing the longitudinal axis in photographs 1 and 10. PLATE II. Fig. 5. Successive positions of a mackerel (Scomber tcombrui) within a period of 0035 sec. The interval between each photograph was 005 sec. and the grid shown has 3 in. squares. The grid has been inked over in the photograph—note the disturbance of the water in the neighbourhood of the fish. Note also the rapid rate of propagation of the muscular waves and the high forward speed of the fish. Fig. 6. Successive positions of a whiting (Gadus merlangus) within a period of 05 sec. Interval between each photograph 005 sec. Scale 3 in. The wave crests are marked by dots. Note that the pitch angle of the tail is greater than that of the butterfish (Fig. 4), but less than that of the mackerel (Fig. 5) or of Ammodytes (Fig. 7). Fig. 7. Successive positions of a sand-eel (Ammodytes lanccolatus) within a period of 05 sec. The interval between each photograph is approx. 005 sec. Scale 3 in. Note the relatively high pitch of the body as compared to the eel shown in Fig. 2. Note also the much greater forward velocity in com- parison to Fig. 2. The forked appearance of the tail is due to the shadow cast on the bottom of the tank. PLATE III. Fig. 8. Dogfish. Note the large amplitude of the movements of the body and tail. A wave crest is marked by a black dot in photographs 3-8. Interval between the photographs 010 sec. Scale 3 in. Fig. 9. Rockling (Onos). Note that the transverse movements are almost completely confined to the tail. Note that the angle of inclination (9) of the body is distinctly steeper than in the dogfish and that the frequency of the movements is higher. Interval between the photographs 005 sec. Scale 3 in. PLATE IV. Fig. 10. An eel moving backwards. In 10 sec. the fish has moved back about 3 in. Note the passage of the waves from the tail towards the head of the fish: note also the large amplitude of the waves. Interval between each photograph approx. o-1 sec. Fig. 11. A young glass-eel which is stationary and yet exhibits curvature of the body. Compare with photograph 4 in Fig. 2. The form of the waves is approximately the same as when the waves are moving and the fish is in motion. Total period 05 sec. Scale 1 in. JOURNAL OF EXPERIMENTAL ISIOLOGY, X, i- PLATE I.

1 2 31 41 51 61 71 8 9 10' 11

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5 i 6 ii 7 i8, 9 10 11

Fig. 4»

GRAY—STUDIES IN ANIMAL LOCOMOTION (pp. 88—104).

JOURNAL OF EXPERIMENTAL RIOLOGY, X, i. PLATE II.

Fig. 7.

GRAY—STUDIES IN ANIMAL LOCOMOTION (pp. 88—104).

JOURNAL OF EXPERIMENTAL BIOLOGY, X, i PLATE III.

GRAY—STUDIES IN ANIMAL LOCOMOTION (pp. 88—104). JOURNAL OF EXPERIMENTAL BIOLOGY, X, i. PLATE IV.

Fig. 10.

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GRAY—STUDIES IN ANIMAL LOCOMOTION (pp. 88—104).