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lent to that marked (A) in our paper on Sound; and the general equations in -^ and w being the same, we may immediately obtain, the intensity of the reflected or refracted waves, by merely writing in the simple formulae contained in that paper,
A — i and A, = 1 for light polarized in the plane of incidence j
or A =• -. and A, = —t for light polarized perpendicular to the
plane of incidence.
Aa an example, we will here deduce the intensity of the refracted wave for both kinds of light.
Representing, therefore, the parts of 10 and w, due to the disturbances in the Incident Reflected and Refracted waves by
f(ax + by + ot), F(-ax + by+ ct), and / (a,x + by + ct)
respectively, and resuming the first of our expressions (7) in the paper on Sound, y\z.—
we get for light polarised in the plane of incidence, where
A = A, = 1,
/" i + «~7+^-~ «n (*. + *)'
which agrees with the value given in Airy's Tracts, p. 356*.
For light polarized perpendicular to the plane of incidence
we have A = -, and A, =—-t. If, therefore, we here represent the
parts of ifr and ^r, due to the same disturbances by f, -Fand /,
/'_ 2 sing, cos fl 2
f1 ~ 7" cot0, ~ sin 0 cos B,' coaflsinfl"
77+coT5 cos 0, sin 0,
[» Airy, «W wp. p. 109.]
Also, if D be the disturbance of the incident wave in its own plane, and Dt the like disturbance in tlie refracted wave, we have by first equation of (31),
D sin 0 = w= ¥ = ?>/ (ax + by + ct),
and Z>, sin 6, = ut = -y! ~ ty, (ax + by+ ct),
retaining in -$• the part due to the incident wave only.
which agrees with the formula in use. (Vide Airy's Tracto, p. 858*.)
In our preceding paper, the two media have been supposed to terminate abruptly at their surface of junction, which would not be true of the luminiferous ether, unless the radius* of the sphere of sensible action of the molecular forces was exceedingly small compared with \, the length of a wave of light.
In order, therefore, to form an eslimate of the effect which would be produced by a continuous though rapid change of state of the ethereal medium in the immediate vicinity of the surface of junction, we will resume the conditions (29), which belong to light polarized in a plane perpendicular to that of Beflexion, viz.
and instead of supposing the index of refraction to change suddenly from 0 to /t, we will conceive it to pass through the
[* Airy, ubi tup. p. 110.]
regular series of gradations,
/*»» /*!> r*,i /*• ^n?
T being the common thickness of each of these successive media.
Then it is clear we should have to replace the last system by
But it is evident from the form of the equations on the right side of system (33), that the total effect due to the last terms of their second members will be far less when n ia great, than that due to the corresponding term in the second equation of system (29)*. If, therefore, we reject these second terms, and conceive the common interval T so small that the result due to the first terms may not differ very sensibly from that which would be produced by a single refraction, we should have to replace the system (29) by (30), and the intensity of the reflected wave would then agree with the law assigned by Fresnel. In virtue of this law, however highly refracting any substance may be, homogeneous light will always be completely polarized at a certain angle of incidence; and Sir David Brewster states
* In fact, in the system (33) each of the last terms willy in consequence of the factors (ft,* - it*), Sec. be quantities of the order -5 compered with the last term of (19'), and as their number is only n, their joint effect will be a quantity of the
order - compared with that of the term just mentioned.
that this is the case with diamond at the proper angle. Bat the phenomena observed by Professor Airy appears to him entirely inconsistent with this result (Vide Camb. Fhil. Trans., Vol. IV. p. 423); what immediately precedes seems to render it probable that considerable differences in this respect may be doe to slight changes in the reflecting surface.