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a plane wave to propagate itself without subdivision, and the velocity of propagation parallel to its own front. The change of position here inade in the elliptical section, is evidently equivalent to supposing the actual disturbances of the ethereal particles to be parallel to the plane nsually denominated the plane of polarization.

This hypothesis, at first advanced by M. Cauchy, has since been adopted by several philosophers; and it seems worthy of remark, that if we suppose an elastic medium free from all extraneous pressure, we have merely to suppose it so constituted that two of the wave-disturbances shall be accurately in the wave's front, agreeably to Fresnel's fundamental hypothesis, thence to deduce his general construction for the propagation of waves in biaxal crystals. In fact, we shall afterwards prove that the function ,, which in its most general form contains twenty-one coefficients, is, in consequence of this hypothesis, reduced to one containing only seven coefficients; and that, from this last form of our function, we obtain for the directions of the disturbance and velocities of propagation precisely the same values as given by Fresnel's construction.

The above supposes, that in a state of equilibrium every part of the medium is quite free from pressure. When this is not the case, A, B, and C will no longer vanish in the equation (8). In the first place, conceive the plane of the wave's front parallel to the plane (ya); then a= 1, b = 0, c= l, and the equation (8) of our ellipsoid becomes

1 = uz* + A (x2 + y +z+) + Mz? + Vy'; and that of a section by a plane through its centre parallel to the wave's front, will be

1= (A + N) y' + (A + M) z: and hence, by what precedes, the relocities of propagation of our two polarized waves will be

VA+N. The disturbance being parallel to the axis of y, WA+M1........

............ to the axis of z.

Similarly, if the plane of the wave's front is parallel to the plane (az), the wave-velocities are

JB + N. The disturbance being parallel to the axis ir,

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...................................... to the axis 2,

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are in a plane parallel to the wave's front, and of which the equation is

0= ax + by + c2 .......... ............ (12); the same therefore will be true for the ellipsoid whose equation is (11), as this is only a particular case of the former. But the section of the last ellipsoid by the plane (12) is evidently given by 1 =(V+M+N) + (y +L+N) y + (v + L+M)

.. (12, 1) 0 = 0x -+ by + cz

By what precedes, the two axes of this elliptical section will give the two directions of disturbance which will cause a wave to be propagated without subdivision, and the velocity of propagation of each wave will be inversely as the corresponding semi-axes of the section: which agrees with Fresnel's construction, supposing, as he has done, the actual direction of the disturbance of the particles of the ether is perpendicular to the plarie of polarization.

Let us again consider the system as quite free from extraneous pressure and take the most general value of $, containing twenty-one coefficients. Then, if to abridge, we make

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But if the directions of two of the disturbances are rigorously in the front of a wave, a plane parallel to this front passing through the centre of the ellipsoid, and whose equation is

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must contain two of the semi-axes of this ellipsoid ; and therefore a system of chords perpendicular to the plane will be bisected by it, and hence we get

0= (A - C)ac + B (d-a") + Fbc Dab,
0 = (B-C) bc + D (CK - b*) + Fac – Eab.

Substituting in these the values of A, B, &c., before given, we shall obtain the fourteen relations following between the coefficients of $,, viz.

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+20 let

ry}
+ 22 {(dz tiempo en terug na regte ...12),
and hence we get for the equation of the corresponding ellipsoid,

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+ 2 P(cx - az) (ay - bx) + 2 Q(tz-cy) (ay-bx) + 2 R (bz cy) (ca - az)...(13)

But if in equation (8) and corresponding function (A), we suppose A = 0, B= (). and C =0, and then refer the equation to axes taken arbitrarily in space, we shail thus introduce three

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