^ A more perfect agreement with theory than this could scarcely j$m ejected. Had the formula fj*-^ — » been used, the errors vroold. have been much greater. The theory of the motion of waves in a deep sea, taking the most simple case, in which the oscillations follow the law of the cycloidal pendulum, and considering the depth as infinite, is extremely easy, and may be thus exhibited. Take the plane (xz) perpendicular to the ridge of one of the waves supposed to extend indefinitely in the direction of the axis' y, and let the velocities of the fluid particles be independent of the co-ordinate y. Then if we conceive the axis z to be directed vertically downwards, and the plane (xy) to coincide with the surface of the sea in equilibrium, we have generally, <7*-£ = #, * p dt' 4* d*$ The condition due to the upper surface, found as before, is ° 9 dz df 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~ * * sin ^ (»'<—«), by which the general equation is immediately satisfied, and the condition due to the surface gives 2ir , . /oft, where \ is evidently the length of a wave. Hence, the velocity of these waves varies as *J\, agreeably to what Newton asserts. Bat the velocity assigned by the correct theory exceeds Newton's value m the ratio >Jw to Via, or of 5 to i nearly. What immediately precedes is not given as new, but merely on account of the extreme simplicity of the analysis employed, We 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 Pof the fluid when in equilibrium. Then, since tf> = He x ' sin -r- (v't — x); A. and the general formula (2) give , d<t> U . 2w, „ , x = a+ -r- = a re * sin — (vt — a), da v \ v" d<t> Hiv,,d , s=c + —- = c + -7{ A cos— (vt — a), dc v X v Hence, («-a)'+(.-*)»-(3?«~*),' and therefore any particle P revolves continually in a circular orbit, of which the radius is 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. SUPPLEMENT TO A MEMOIR ON THE REFLEXION AND REFRACTION From the Traiuactiom of the Cambridge Philosophical Society, 1839. SUPPLEMENT TO A MEMOIR ON THE REFLEXION AND REFRACTION OF LIGHT. In a paper which the Society did me the honour to publish some time ago*, I endeavoured to determine the laws of Reflexion and Refraction of a plane wave at the surface of separation of two elastic media, supposing this surface perfectly plane, and both media to terminate there abruptly: neglecting also all extraneous forces, whether due to the action of the solid particles of transparent bodies on the elastic medium, which is supposed to pervade their interstices, or to extraneous pressures. 1 am inclined to think that in the case of non-crystallized bodies the latter cause would not alter the form of our results in the slightest degree; and possibly there would be some difficulty in submitting the effects of the former to calculation. Moreover, should the radius of the sphere of sensible action of the molecular forces bear any finite ratio to \, the length of a wave of light, as some philosophers have supposed, in order to explain the phenomena of dispersion, instead of an abrupt termination of our two media we should have a continuous though rapid change of state of the ethereal medium in the immediate vicinity of their surface of separation. And I have here endeavoured to shew, by probable reasoning, that the effect of such a change would be to diminish greatly the quantity of light reflected at the polarizing angle, even for highly refracting substances: supposing the light polarized perpendicular to the plane of incidence. The same reasoning would go to prove that in this case the quantity of the reflected light would depend greatly on minute changes in the state of the reflecting surface. I have on the present occasion merely noticed, but not insisted upon, these inferences, feeling persuaded that in researches like the present, little confidence is due to such consequences as are not supported tjy a rigorous analysis. * Supra, p. 243. |