the equations of an infinite number of ellipses which, in general, do not belong to the same curve surface. If, however, we cause each ellipsis to turn 90° in its own plane, the whole system will belong to an ellipsoid, as may be thus shewn: Let (xyz) be the co-ordinates of any point p in its original position, and (x'y'z') the co-ordinates of the point p' which would coincide with p when the ellipsis is turned 90' in its own plane. Then ∞2 + 1/2 + x2=x”2 + y22 + z'3, 12 12 since the distance from the origin O is unaltered, 0=ax' + by' + cz', since the plane is the same, 0=xx'+yy +zz', since pOp' = 90°. The two last equations give But Hence the last of the equations (9) becomes w3 = Lx3 + My'3 +Nz2. x2 + y2 + 2'2 = w3 {(cy — bz)2 + (az — cx)2 + (bx − ay)2} = w2 {(b2 + n2) x2 -† (c2 + a2) y2 + (b2 + c2) x2 — 2 (bcyz+abxy + acxz} = w2 {{a3 +b3 + c3) (x2 + y2 + z3) − (ax + by + cz)2} = w3 (x2 + y2 + z3) = x2 + y3 + z3. Therefore @2 = 1, and our equation finally becomes. 1 = Lx2 + My'2 + N1⁄23. (10). We thus see that if we conceive a section made in the ellipsoid to which the equation (10) belongs, by a plane passing through its centre and parallel to the wave's front, this section, when turned 90 degrees in its own plane, will coincide with a similar section of the ellipsoid to which the equation (8) belongs, and which gives the directions of the disturbance that will cause a plane wave to propagate itself without subdivision, and the velocity of propagation parallel to its own front. The change of position here made in the elliptical section, is evidently equivalent to supposing the actual disturbances of the ethereal particles to be parallel to the plane usually 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 4, 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 (yz); then a = 1, b=0, c = 0, and the equation (8) of our ellipsoid becomes 1 = t + A (iểu tỷ +}+M+ Ng; and that of a section by a plane through its centre parallel to the wave's front, will be 1 = (A + N) y2 + (A + M ) z2 : and hence, by what precedes, the velocities of propagation of our two polarized waves will be JA+N. The disturbance being parallel to the axis of y, √A + M. to the axis of z. Similarly, if the plane of the wave's front is parallel to the plane (xz), the wave-velocities are JB+N. The disturbance being parallel to the axis, Or if the plane of the wave's front is parallel to (xy), the velocities are JC+ M. The disturbance being parallel to x, JC + L. y. Fresnel supposes that the wave-velocity depends on the direction of the disturbance only, and is independent of the position of the wave's front. Instead of assuming this to be generally true, let us merely suppose it holds good for these three principal waves. Then we shall have N+A=C+L, M+A-B+L, and B+N=C+M; or we may write A-LB-M C-N=v (suppose). Thus our equation (8) becomes since a2 + b2 + c2 = 1, 1 = μ (ax+by+cz)2 + v (.x2 + y2 + z2) + (La2 + Mb2 + Nc2) (x2 + y2+ z2) + L cy - bz)2 + M (az − cx)2 + N (bx – ay)2, But the two last lines of this formula easily reduce to (M + N) x2 + (N + L) y2 + (L + M) z2 - + L {a2x2 — (by + cz)2} + M {b3y3 — (ax + cz)2} And hence our last equation becomes 1 = (v + M + N) x2 + (v +N+L) y2 + (v +L+M) z2 +μ (ax+by+cz)' + L{a2x2 − (by + cz)2} + M (b2y2-- (ax + cz)'} + N {c2z2— (ax + by)?} . (11). In consequence of the condition which was satisfid in forming the equation (8), it is evident that two of its semi-axes are in a plane parallel to the wave's front, and of which the equation is 0 = ax + by + cz ..(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 ) x2 + (v + L + N} y2 + (v + £+M ) z2 0 = ax + 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 plane of polarization. Let us again consider the system as quite free from extraneous pressure and take the most general value of 4, containing twenty-one coefficients. Then, if to abridge, we make -- 6, = (§3) § 2+ n2) n2 + (5) 52 + 2 (nŠ) n5 + 2 (ES) ES+2 (En) En + (a3) a3 + (B2) Ba + (y3) y2+ 2 (By) By + 2 (ay) ay + 2 (aß) áß + 2 (a§) a§ + 2 (BE) B§ + 2 (vE) v§ + 2 (an) an + 2 (Bn) Bn + 2 (yn) vn + 2 (a5) a$ + 2 (BC) B5 + 3 (v$) v$, where (F), (a), &c. are the twenty-one coefficients which enter into Suppose now the equation to the front of a wave is 0 = ax + by + cz Then, by what was before observed, the right side of the equation of the ellipsoid, which gives the directions of disturbance of the three polarized waves and their respective velocities. will be had from 4, by changing u, v, and w into ∞, y, and 2; also d d dx' du' d dz and into a, b, and c. We shall thus get Provided 1 = Ax2 + By2 + Cz2 + 2 Dys + 2Exz + 2 Fxy. ▲ = (5o) a2 + (Bo) c2 + (y3) b2 + 2 (By) bc + 2 (§P) ac + 2 (§y) ab, + (an) ở + (as) ở + (3n) + (25) E=(ES) ac + (8) ac + (ay) b2 + (oß) bo + (3y) ab + {B§) a2 + (BS) &2 + (a§) ab + (ys) be, F = (§n) ab + (y1) ab + (aß) c2 + (ay) bc + (By) ac + (y§) a2 + (yn) b* + (a§) ac + (Bn) bc. 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 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 = (40) ac+E (c2 - a) + Fbc- Dab, 0 = (B − C) bc + D (e* - b2) + Fac - Eab. |