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Suppose Q = length of the wave when t=0; then f(a) = 0, except when a is between the limits 0 and a. If therefore we consider a point P before the wave has reached it,

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the whole volume of the fluid which would be required to fill the hollow caused by the depression below the surface of equi. librium when t= 0. Hence we get

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a' being the horizontal co-ordinate of P, before the wave reaches P.

Also, let ac be the value of this co-ordinate after the wave has passed completely over P, then

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If 5 were wholly negative, or the wave were elevated above the surface of equilibrium, we should only have to write - V for V, and thus

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We see therefore, in this case, that the particles of the fluid by the transit of the wave are transferred forwards in the direction of the wave's motion, and permanently deposited at rest in a new place at some distance from their original position, and that the extent of the transference is sensibly equal throughout the whole depth. These waves are called by Mr Russel, positive ones, and this result agrees with his experiments, vide p. 423. If however $ were positive, or the wave wholly depressed, it follows from our formula, that the transit of the fluid particles would be in the opposite direction. The experimental investigation of those waves, called by Mr Russel, negative ones, has not yet been completed, p. 445, and tle last result cannot there fore be compared with experiment.

The value which we have obtained analytically for the extent over which the fuid particles are transferred, suggests a simple physical reason for the fact. For previous to the transit of a positive wave over any particle P, a volume of luid behind P, and equal to V, is elevated above the surface of equilibrium. During the transit, this descends within the surface of equilibrium, and most therefore force the fluid about P forward through the epace

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admitting as an experimental fact, that after the transit of the wave the fluid particles always remain absolutely at rest.

Mr Russel, p. 425, is inclined to infer from his experiments, that the velocity of the Great Primary Wave is that due to gravity acting through a height equal to the depth of the centre of gravity of the transverse section of the channel below the surface of the fluid. When this section is a triangle of which one side is vertical, as in channel (H), p. 443, the ordinary Theory of Fluid Motion may be applied with extreme facility. For if we take the lowest edge of the horizontal channel as the axis of 2, and the axis of z vertical and directed upwards, the general equations for small oscillations in this case become

............... (4),

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a being the angle which the inclined side of the channel makes with the vertical.

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The first of these conditions is due to the vertical side, and the second to the inclined one, since at these extreme limits the fluid particles must move along the sides.

Now from what has been shown in our memoir, it is clear that we may satisfy the equation (B) and the two conditions just given, by

$=$+$, (9* +2°) ..................... (0), $ and $, being two such functions of x and t only that

Da + 4$, .....................(C). It now only remains to satisfy the condition due to the upper surface. Let therefore

(=2- $8. be the equation of this surface. Then the formula (A) of our paper before cited gives

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or neglecting (disturbance)

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c being the vertical depth of the fluid in equilibrium.

Also at the upper surface p= 0, therefore by continuing to neglect (disturbance)" (4) gives

O=g5+ (when z=c).
Hence, by eliminating $, we get

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=9 + (when %=c),

which by (o) becomes, when we neglect terms of the order and a compared with those retained,

0 = 2gob. + diety Or eliminating $, by means of (C),

The particular integral of which belonging to the wave that proceeds in the direction of a positive is

$.=(-11/). and hence the velocity of propagation of the wave is

v=V ......................... (D).

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Dr Russel gives go as the velocity, but at the same time remarks, that in consequence of the attraction of the sides of the canal fixing a portion of the fluid in its lower angle, we ought in employing any formula to calculate for an effective depth in place of the real one, p. 442. Instead of adopting this method, let us compare the formula (D) given by the common Theory of Fluid Motion, with Mr Russel's experiments. And as in our theory we have considered those waves only in which the elevation above the surface of equilibrium is very small compared with the depth c, it will be necessary to select those waves in which this condition is nearly satisfied. I have therefore taken from the Table, p. 443, all the waves. in which

and have supposed g=324 feet: the results are given below.

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THE MOTION OF WAVES IN CANALS. 279 A more perfect agreement with theory than this could scarcely be expected. Had the formula 2go = v been used, the errors would have been much greater.

The theory of the motion of waves in a deep sea, taking the most simple case, in which the oscillations follow the law of the cycloidal pendulum, and considering the depth as infinite, is extremely easy, and may be thus exhibited.

Take the plane (az) perpendicular to the ridge of one of the waves supposed to extend indefinitely in the direction of the axis y, and let the velocities of the fluid particles be independent of the co-ordinate y. Then if we conceive the axis x to be directed vertically downwards, and the plane (wy) to coincide with the surface of the sea in equilibrium, we have generally,

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The condition due to the upper surface, found as before, is

organiseren en From what precedes, it will be clear that we have now only to satisfy the second of the general equations in conjunction with the condition just given. This may be effected most conveniently by taking

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by which the general equation is immediately satisfied, and the condition due to the surface gives

where 1 is ovidently the length of a wave. Hence, the velocity of these waves varies as vā, agreeably to what Newton asserts. Bat the velocity assigned by the correct theory exceeds Newton's value in the ratio Voto V2, or of 5 to 4 nearly,

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