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What immediately precedes is not given as new, but merely an 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 P of the fluid when in equilibrium. Then, since
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 contrar case.
SUPPLEMENT TO A MEMOIR
ON THE REFLEXION AND REFRACTION
From the Transactions of the Cambridge Philosophical Society, 1839.
[Read May 6, 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. I 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 apon, these inferences, feeling persuaded that in researches like the present, little confidence is due to such consequences as are not supported by a rigorous analysis.
Supra, p. 348.
The principal object of this supplement has been to put the equations due to the surface of junction of two media, and belonging to light polarized perpendicular to the plane of incidence, under a more simple form. The resulting expressions have here been made to depend on those before given in our paper on Sound, and thus the determination of the intensities of the reflected and refracted waves becomes in every case a matter of extremne facility. As an example of the use of the new formulæ, the intensities of the refracted waves have been determined for both kinds of light : the consideration of which waves had inadvertently been omitted in a former communication.
Perhaps I may be permitted on the present occasion to state, that though I feel great confidence in the truth of the fundamental principle on which our reasonings concerning the vibrations of elastic media have been based, the same degree of confidence is by no means extended to those adventitious suppositions which have been introduced for the sake of simplifying the analysis.
Let us here resume the equations of the paper before mentioned, namely,
do dy do dyr,
(when x = 0)............ (17), g' di =gi di
se te dua where u and v, the disturbances in the upper medium parallel to the axes x and y, are given by