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XV. Supplement to the Memoir on the Transmission of Light in Crystallized Media. By PHILIP KELLAND, B.A. Fellow and Tutor of Queens' College.

[Read May 1, 1837.]

(BIOT's LAW.)

1. IN the latter part of this Memoir, I make an application of the formula which I had before deduced, viz. “that the difference of the squares of the velocities of two waves having a common normal, in the direction of that normal, is proportional to the product of the sines of the angles made by it with the two optic axes of the crystal.”

As my object was merely to shew that it was a Theorem wanted for such considerations, I adopted all the approximations which I found in common use. On examining the subject more attentively, I find that some of them if allowable are superfluous, and that the same result is attained, by proceeding to work in a direct manner. I am not, it is true, quite sure that the authors of the investigations considered them as approximations; they make no remark to that effect, but assume at once that the ray and wave coincide.

2. In order to find the appearance presented on the transmission of polarized light through a plate of biaxal crystal, the most important point to be determined is, the difference of retardation of the two 2007068.

The want of a proposition, such as that which appears in (23), seems to have driven writers to adopt an approximative process of the following nature.

First, a ray is supposed nearly to coincide with a wave, and the theorem that the difference of the squares of the reciprocals of the velocities of the two rays is proportional to the product of the sines

of the angles which their common direction makes with the optic axes suggested (apparently) that the same Theorem approximately held when wave was put for ray, and normal to front for direction, &c., and thus a Theorem which is in no way connected with the result, does from the circumstance of its close analogy to the true one, give correct results, or nearly so.

3. Let BC be the direction of one ray in the crystal; BE a normal to its front; CG perpendicular to BA; p the angle of incidence; p' the angle which BE makes with the normal to the plane surface

of the crystal; BC makes 9 with the same; T the thickness of the plate. (Note at end.) o

Then if v be the velocity before incidence, v' the velocity perpendicular to the front after refraction, v sin o' v Tsing

and the ray has moved perpendicularly to its former front through a space

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T
= ...a cos(?-9),
since + = Bc, also –– is the velocity along Bc.
* cost; + 9 cos (9 p") is the velocity along y
... time of describing BC = Teo (6-8),
v' COS 6

therefore the space which the wave would describe in the same time in air, is

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and if v, be the velocity of the other wave, its retardation is
p. vi" . .
T{: 1 – ; sin p coso)

the angle of emergence being supposed the same for both.

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4. This hypothesis that the two waves are moving parallel to each other at emergence, is clearly not compatible with the hypothesis that they have the same normal within the crystal.

If v, be the velocity of the wave which has a common normal with that whose velocity is vi, we have

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= Tv {} - #} if p' be very small; 2 hence the hypothesis that the angle of incidence is small, reduces this case to the same form as the former, and we may in such circumstances consider the difference of the retardation as proportional to the difference between the two refractive indices.

5. In the applications of this formula, we must introduce the relations which are given by the constitution of the crystal determined by the passage of light through it. Such relations must, I conceive, depend on the refractive energies of the crystal in different directions.

Now the refractive energy has undoubtedly no connexion whatever with the velocities of transmission of the rays, since these velocities are merely nominal ones; that is, they are not estimated in the direction in which the effect is transmitted. Indeed, I do not suppose we have any notion of these velocities independent of theory, whilst the velocity of the wave is a physical motion, apart from the idea which is suggested by the expression.

I have been under the necessity of giving the term radial to M. Fresnel's axes, since they are not at all the same thing as the optic axes. M. Fresnel himself remarks, that “although the difference between them is very slight in almost all crystals, there are some where it becomes more sensible, and in which we must not confound the two.”

6. We are concerned only with waves which have a common direction in air, and must consequently assume that the difference of the velocities of the two corresponding refracted waves, is very nearly the same as the difference of the velocities of two waves which travel in the direction of one of them, omitting consequently the variation of velocity of one wave due to difference of its velocity from that of the other, or in other words, omitting the variation of the difference of the velocities, compared with that difference itself which is

perfectly allowable.

Let m', n' be the angles made by the direction, which we consider common to the two, with the optic axes.

a, 3 the angles made in air by the incident ray, with rays, which, when they enter the crystal, move in rays; the normals to which are

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