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certain fixed amount further back in its path than is its neighbour on one side, and therefore exactly as . much more forward than is its neighbour on the other side.

(1) shows the condition of the row of points at the moment when the extreme point on the left is beginning its swing from left to right, which, in accordance with the direction of the arrow in the figure, we may call its forward swing. The equidistant vertical straight lines fix the extent of vibration for each oscillating point. The constant amount of retardation between successive points is, in the instance here selected, one-eighth of the path traversed by each point during the period of a complete oscillation. Thus, proceeding from left to right along the line (1), we have the first point beginning a forward swing, the second, third, fourth and fifth points entering respectively on the fourth, third, second, and first quarters of a backward swing, and the sixth, seventh, eighth and ninth points on the fourth, third, second, and first quarters of a forward swing.

Since the ninth point is just beginning a forward swing, its situation is exactly the same as that of the first point. Beyond this point, therefore, we have only repetitions of the state of things between the first and ninth points. The row (1) is therefore

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made up of a series of groups, or cycles, of the same number of points arranged in the same manner throughout. Two such cycles, included by the large brackets A and B, are shown in (1). Each cycle is divided by the small brackets a, a and b, b' into

two parts. In a and b the distances between successive points are less than, and in a and b greater than, the corresponding distances when the points occupied their undisturbed positions, as in (0). The cycles correspond to complete waves on the surface of water, the shortened and elongated portions of each cycle answering to the crest, and trough of which each water-wave consists.

(2) shows the state of the row of points when an interval of time equal to one eighth of the period of a complete point-vibration has elapsed from the moment shown in (1). The wave A has here moved forward into the position indicated by the dotted lines.

The following rows (3,) (4), (5), &c., show the state of things when two-eighths, three-eighths, four-eighths, &c., of a vibration-period has elapsed since (1). In each, the wave A moves forward one step further.

In (9), a whole vibration-period has elapsed since (1). Accordingly every oscillating point has performed one complete vibration, and returned to the position it held in (1). The wave A, meanwhile, has travelled constantly forward so as to be, in (9), where B was in (1), i.e. to have advanced by one whole wave-length. The proposition proved for. waves due to transverse vibrations in § 8 is thus

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shown to hold good likewise for waves due to longitudinal vibrations.

16. In the waves shown in Fig. 14, the points in the bracket a are mutually equidistant, as are also those in the bracket b. This is due to the fact that, in the case there represented, the oscillating points move uniformly, i. e. with equal velocity, throughout their paths. If we take other modes of vibration, we shall find that this equidistance no longer exists. Fig. 15 shows three distinct modes of vibration with the wave resulting from each, on the plan of (1) Fig. 14. The extent of vibration, and length of wave, are the same in the three cases.

In (I) the points move quickest at the middle and slowest at the ends, of their paths; in (II) fastest at the ends, and slowest in the middle; in (III) slowest at the left end, and fastest at the right.

The shortest distance separating any two points contained in a is, in (I), that between 7 and 8; in (II), that between 8 and 9; in (III), that between 5 and 6. The corresponding greatest distances are, in (I), between 2 and 3; in (II), between 1 and 2; in (III), between 4 and 5. The remaining points

likewise exhibit differences of relative distance in

the three cases. Thus, the positions of greatest

shortening, and greatest lengthening, occupy different situations in the wave, and the interme

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diate variations between them proceed according to different laws, when the modes of point-vibration are different. The more points we lay down in their proper positions in a and b, the less abrupt will be the changes of distance between successive points. By indefinitely increasing the number of vibrating points, we should ultimately arrive at a state of things in which perfectly continuous changes of shortening and lengthening intervened between the positions of maximum shortening and maximum lengthening in the same wave.

17. Let us now replace our row of indefinitely numerous points by the slenderest filament of some material whose parts (like those of an elastic string) admit of being compressed, or dilated, at pleasure.

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