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 *-* + *,(/+«') («)» $ and 0, being two such functions of x and t only that •-£+«* to It now only remains to satisfy the condition due to the upper surface. Let therefore be the equation of this surface. Then the formula (A) of our paper before cited gives •-2-I-22 «*•—+0. or neglecting (disturbance)2 c being the vertical depth of the fluid in equilibrium. Also at the upper surface p — 0, therefore by continuing to neglect (disturbance)1 (.A) gives 0 = $£+^! (when e-c). Hence, by eliminating £, we get °^fz + ^(whea»-o), which by (c) becomes, when we neglect terms of the order y* and z* compared with those retained, Or eliminating ^, by means of (C), v~ de 2*d?" • The particular integral of which belonging to the wave that proceeds in the direction of x positive is w(*Vf). and hence the velocity of propagation of the wave is «' = \/f (i* Mr ftussel gives kf -i- 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 au 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 thosa 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 f<20' and have supposed g = 32£ feet: the results are given below. A more perfect agreement with theory than this could scarcely be expected. Had the formula . /-£- = v been used, the errors would have been much greater. The theory of the motion of wares 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 (xz) perpendicular to the ridge of one of the wave3 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 z to be directed vertically downwards, and the plane {xy) to coincide with the surface of the sea in equilibrium, we have generally, 1 dx** dz*' The condition due to the upper surface, found as before, is 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 <f> = Be x sin — (»'« —a), by which the general equation is immediately satisfied, and the condition due to the surface gives 2tt where X is evidently the length of a wave. Hence, the velocity of these waves varies as Vx, agreeably to what Newton asserts. Bat the velocity assigned by the correct theory exceeds Newton's value in the ratio Vw to V2, or of 5 to i nearly. What immediately precedes is not given as new, but merely ga account of the extreme simplicity of the analysis employed. -^v7e shall, moreover, be able thence to deduce a singular consequence which has not before been noticed, that I am aware of. Let (a, b, c) be the co-ordinates of any particle P of the fluid when in equilibrium. Then, since --* 2tt _ - H\ -%' 2tt . , . and the general formuhe (2) give d<P U -± . 2ir,., . da v \ v , dfb H -sx 27T., , z = c+ -r- =c + — e A cos—-(vt — a), Hence, fa and therefore any particle P revolves continually in a circular orbit, of which the radius is II -*« v round the point which it would occupy in a state of equilibrium. The radius of this circle, and consequently the agitation of the fluid particles, decreases very rapidly as the depth c increases, and much more rapidly for short than long waves, agreeably to observation. Moreover, the direction of the rotation is such, that in the upper part of the circle the point P moves in the direction of the motion of the wave. Hence, as in the propagation of the Great Primary Wave, the actual motion of the fluid particles is direct where the surface of the fluid rises above that of equilibrium, and retrograde in the contrary case. |