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u, and y, the disturbances in the lower medium being expressed by similar formulæ in ¢, and ,
The two last equations of (17) give, since
d'=p?pi, t' = My); $ and $, being accented for a moment to distinguish between the particular values belonging to the plane (ye) and their more general values
$=elf and 0.=rt The correctness of these values will be cvident on referring to the Memoir, formulæ (20), (21), and recollecting that
b = a'ra'.
Hence the first equation gives, since x = 0,
And since we may differentiate or integrate the equations (17) relative to any variable except x, we get for the conditions requisite to complete the determination of y and yes,
(m? - 1)' d'y, }(when <=0)..... (29).
Or neglecting the term which is insensible except for highly refracting substances,
X = w ten andy (when == 0) ............(30),
*where y=" is the index of refraction.
These equations belong to light polarized in a plane perpendicular to that of incidence, and as $ and $, are insensible at sensible distances from the surface of junction of the two media, we have, except in the immediate vicinity of this sørface,
When light is polarized in the plane of incidence, the conditions at the surface of junction have been shewn to be
w = 20, 2 dw dw. } (when x = 0)................ (32).
dx dx J Since in these conditions we may differentiate or integrate relative to any of the independent variables except x, we see that the expressions (30) and (32) are reduced to a form equiva
• Though these equations have been obtained on the supposition that the vibrations of the media follow the law of the cycloidal pepdulum, yet as (6) has disappeared, they aro equally applicable for all plane waves whatever. In fact, instead of using the value
V=a, sin (a,x+by + ct), and corresponding values of the other quantities, we might have taken the infinite series
Vi=a, sin n (a,x + by + c), where a and n may have any series of values at will. But the last expression is the equivalent of an arbitrary function of
0,+ by+ct: Or the same equations might have been immediately obtained from (17), without introducing this consideration. The method in the text has been employed for the sake of the intermediate result (29), of which we shall afterwards make use.
lent to that marked (1) in our paper on Sound; and the general equations in y and w being the same, we may immediately obtain the intensity of the reflected or refracted waves, by merely writing in the simple formulæ contained in thal paper, A = 1 and A,= 1 for light polarized in the plane of inci.
dence; or A- and 4, = for light polarized perpendicular to the
plane of incidence. As an example, we will here deduce the intensity of the refracted wave for both kinds of light.
Representing, therefore, the parts of w and w, due to the disturbances in the Incident Reflected and Refracted waves by
f(ax + y + ot), F(- ax + by + ct), and f. (0,00 + by + ct) respectively, and resuming the first of our expressions (7) in the paper on Sound, viz.
we get for light polarized in the plane of incidence, where
A = 4, = 1,
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 4, 5 . If, therefore, we here represent the parts of th and f, due to the same disturbances by f, F and f., we get
2 sip & cos 6 2
cos , sin ,
· Also, if D be the disturbance of the incident wave in its own plane, and D, the like disturbance in the refracted wave, we have by first equation of (31),
retaining in f the part due to the incident wave only.
Di sin o f' _cos @
cos sin À 3 Itu
cos 6 sin 0
cos O sin
cos o ,, tan (0,-0)
cos o, * tan (@+0)}' cos 6, sin , which agrees with the formula in use. (Vide Airy's Tracks, p. 358*.)
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 1, the length of a wave of light.
In order, therefore, to form an estimate 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 Reflexion, viz.
por=pit, and on the way I love (x = 0) ... (29) ; and instead of supposing the index of refraction to change suddenly from 0 to M, we will conceive it to pass through the
[* Airy, ubi sup. p. 110.]
regular series of gradations,
Hos Mige Mega flog ...... .......... flemi 7 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 is 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 7 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 tight will always be completely polarized at a certain angle of incidence; and Sir David Brewster states
• In fact, in tho system (33) each of the last terms will, in consequence of the factors (w, ? – Mo?), &c. be quantities of the order compared with the last term of (39), 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.