These pressures, which are expressed by partial differential coeffi cients relatively to x, y, and z, in the coordinate axes, are subtracted from the accelerative forces arising from the attraction of the earth, and the perturbations exercised by the sun or moon, by which the molecule would be made to recede from that origin; and the differences in the directions of the axes are represented by LY, and d In these equations of motion the partial differential coefficients representing the pressures are transformed into others depending on the distance of the molecule from the centre of the earth, and on its latitude and longitude; while the perturbations of the sun or moon in the directions of the coordivate axes are expressed in terms of the right ascension and declination of the disturbing body, and also of the distances of the latter from the particle disturbed and from the centre of the earth. The result is that the expression for the altitude of a molecule of water above the mean level, in consequence of the perturba tion produced by the sun or moon, consists of three parts (` Mee. Cd1.', lib. iv., c. 1) ; the first does not depend on the rotation of the earth, and indicates a tide which goes through its changes in a long period ; it may consequently be disregarded. The second depends on that rotation and on the hour angle of the disturbing body : it indicates the diurnal tides, or those which take place when the celestial bodies are on or near the meridian, above the horizon ; and which follow one another at intervals of twenty-four hours for the sun, and about 24h. 50m. for the moon. The third depends on an angle equal to the double of that on which the second depends; and consequently it represents the semi diurnal tide.
But the subject of waves and tides has been treated in conformity to the theory of undulations by Mr. Airy, the astronomer royal, in a valuable essay originally published in the' Encyclopedia Metropolitana the investigations, though admitting of general application, are par titularly adapted to the phenomena of tides in rivers and arms of the sea; and they are conducted by an analysis within the reach of persons acquainted with the ordinary processes of the differential and integral calculus.
As in the theory of La Place, there is formed an " equation of con tiniiity," which is founded on the equality of a rectangular parallelo piped of water at rest, to the oblique parallelopiped formed, when the water is in a state of disturbance, by the new positions of the eight ArtieIra constituting the angular points of the former parallelopiped.
But, as the water is supposed to be in a rectangular canal, the coot of the parallelopiped in the direction of the breadth of the canal Is supposed to be constant; and therefore it is sufficient to assume the equality of the parallelograms which form a side of each in the direction of the length of the canal.
The canal being of uniform depth, the "equation of continuity" is expresses' by T — *1 — 7; (between 0 and y) where x and y are respectively the horizontal and vertical 000rdirustes of a particle of fluid, and where x and Y are respectively the horizontal and vertical displacements of the particle by the action of the dis turbing forces: the equation expresses a relation between those coordinates and the disturbances or displacements.
An equation of the pressure experienced by any particle from the forces which act upon it is next found in the following manner:— Let p represent the pressure in every direction on the lower part of a disturbed molecule of water in consequence of the height or weight of the nlament of particles above it : then, the vertical coordinate of the particle being y' or y +r, suppose in the element tit of time the vertical coordinate to become y'+ by' (the vertical height of the filament above the molecule in that position being increased by the general rising of the wave), the pressure on the upper part of the molecule will be dp greater than before, and may be represented by p cause a.Y quently the molecule may be supposed to be pressed downwards by a p d force represented by -3-7 By'. Now, in order to render the expression "It for the hydrostatical pressure homologous to that which is employed for the force of gravity, it must be censidered as accelerative, or as a motive-power divided by the mass; and therefore the accelerative dp pressure downwards becomes which being added to g, representing y dp the force of gravity and supposed to be constant, there arises --7 g dy for the whole acceleration of the molecule downwards : hence there is obtained the equation cry dp (17.3 This equation being integrated between the limits for the bottom of the molecule and the top of the wave, gives the hydrostatical force by which a vertical filament of water descends, or that by which it is carried forward horizontally.
Let the slender column of water above the molecule have a hori zontal breadth equal to it in the direction of x; then the horizontal pressure in front, by which the column is forced backwards, will exceed the pressure by which it is carried forwards by a force repre sented by d p dh, or by an acceleration represented by ; therefore thehorizontalaccelerationforwardais— dp : if extraneous forces, as dx the attraction of the sun or moon on the molecule, and the effects of friction, be together represented by when estimated in the direction dp of x, there arises the expression *xi for the whole acceleration forwards; then the "equation of motion" becomes d:x dp 0 " 17 d which gives relations between the terms x, r,y, and 1. This " equation of equal pressure" and the " equation of continuity" con stitute the theory of the motion of fluids in canals of uniform breadth. The general equation representing the disturbance or displacement of a particle of water is the same as that which expresses the die turbauce of a particle of light in the undulatory theory ; and, in order to Indicate oscillatory motion, both tho horizontal and vertical 'displacements are represented by terms containing the sines or cosines of angles depending on the time 1.