and q in that plane; then the resolved parts of the velocity v/μ2 of the ether within a refracting medium will be p/m2, q/μ3. Let us first consider the effect of the velocity p. It is easy to see that, as far as regards this resolved part of the velocity of the ether, the directions of the refracted and reflected waves will be the same as if the ether were at rest. Let BAC (fig. 1) be the intersection of the refracting surface and the plane P; DAE a normal to the refracting surface; AF, AG, AH normals to the incident, reflected and refracted waves. Hence AF, AG, AH will be in the plane P, and Draw Gg, Hh perpendicular to the plane P, and in the direction of the resolved part p of the velocity of the ether, and Ff in the opposite direction; and take Ρ μ 2 : V, and Gg = Ff, and join A with f, g and h. Then fA, Ag, Ah will be the directions of the incident, reflected and refracted rays. Draw FD, HE perpendicular to DE, and join fD, hE. Then ƒDF, hEH will be the inclinations of the planes fAD, hAE to the plane P. Now and sin FAD = μ sin HAE; therefore tan FDf=tan HEh, and therefore the refracted ray Ah lies in the plane of incidence fAD. It is easy to see that the same is true of the reflected ray Ag. Also gAD =ƒAD; and the angles fAD, hAE are sensibly equal to FAD, HAE respectively, and we therefore have without sensible error, sin fADμ sin hAE. Hence the laws of reflexion and refraction are not sensibly affected by the velocity p. Let us now consider the effect of the velocity q. As far as depends on this velocity, the incident, reflected and refracted rays will all be in the plane P. Let AH, AK, AL be the intersections of the plane P with the incident, reflected and refracted waves. Let,, ' be the inclinations of these waves to the refracting surface; let NA be the direction of the resolved part q of the velocity of the ether, and let the angle NAC = a. The resolved part of q in a direction perpendicular to AH is q sin (a). Hence the wave AH travels with the velocity V+q sin (y+a); and consequently the line of its intersection with the refracting surface travels along AB with the velocity cosec {V+q sin (y+a)}. Observing that qu2 is the velocity of the ether within the refracting medium, and Vμ the velocity of propagation of light, we shall find in a similar manner that the lines of intersection of the refracting surface with the reflected and refracted waves travel along AB with velocities But since the incident, reflected and refracted waves intersect the refracting surface in the same line, we must have μ sin ↓, { V+q sin (↓ + a)} = sin ↓ {V+q sin (†, − a)}) ...(A) sin (y+a)'S Draw HS perpendicular to AH, ST parallel to NA, take ST: HS :: q: V, and join HT. Then HT is the direction of the incident ray; and denoting the angles of incidence, reflexion and refraction by 4, 4,, ', we have = = On substituting these values in equations (A), and observing that in the terms multiplied by q we may put 6-4, μ sin'd'= sin, the small terms destroy each other, and we have sin o, sin 4, μsin 'sin . Hence the laws of reflexion and refraction at the surface of a refracting medium will not be affected by the motion of the ether. In the preceding investigation it has been supposed that the refraction is out of vacuum into a refracting medium. But the result is the same in the general case of refraction out of one medium into another, and reflexion at the common surface. For all the preceding reasoning applies to this case if we merely substitute p/p, q/p" for p, q, V/p' for V, and μ/p' for u, u' being the refractive index of the first medium. Of course refraction out of a medium into vacuum is included as a particular case. 12 It follows from the theory just explained, that the light coming from any star will behave in all cases of reflexion and ordinary refraction precisely as it would if the star were situated in the place which it appears to occupy in consequence of aberration, and the earth were at rest. It is, of course, immaterial whether the star is observed with an ordinary telescope, or with a telescope having its tube filled with fluid. It follows also that terrestrial objects are referred to their true places. All these results would follow immediately from the theory of aberration which I proposed in the July number of this Magazine; nor have I been able to obtain any result, admitting of being compared with experiment, which would be different according to which theory we adopted. This affords a curious instance of two totally different theories running parallel to each other in the explanation of phenomena. I do not suppose that many would be disposed to maintain Fresnel's theory, when it is shewn that it may be dispensed with, inasmuch as we would not be disposed to believe, without good evidence, that the ether moved quite freely through the solid mass of the earth. Still it would have been satisfactory, if it had been possible, to have put the two theories to the test of some decisive experiment. [From the Cambridge and Dublin Mathematical Journal, Vol. I. p. 183 (May, 1846).] ON A FORMULA FOR DETERMINING THE OPTICAL CONSTANTS OF DOUBLY REFRACTING CRYSTALS. In order to explain the object of this formula, it will be necessary to allude to the common method of determining the optical constants. Two plane faces of the crystal are selected, which are parallel to one of the axes of elasticity; or if such do not present themselves, they are obtained artificially by grinding. A pencil of light is transmitted across these faces in a plane perpendicular to them both, as in the case of an ordinary prism. This pencil is by refraction separated into two, of which one is polarized in the plane of incidence, and follows the ordinary law of refraction, while the other is polarized in a plane perpendicular to the plane of incidence, and follows a different law. It will be convenient to call these pencils respectively the ordinary and the extraordinary, in the case of biaxal, as well as uniaxal crystals. The minimum deviation of the ordinary pencil is then observed, and one of the optical constants, namely that which relates to the axis of elasticity parallel to the refracting edge, is thus determined by the same formula which applies to ordinary media. This formula will also give one of the other constants, by means of the observation of the minimum deviation of the extraordinary pencil, in the particular case in which one of the principal planes of the crystal bisects the angle between the refracting planes: but if this condition be not fulfilled it will be necessary to employ either two or three prisms, according as the crystal is uniaxal or biaxal, to determine all the constants. The extraordinary pencil, however, need not in any case be rejected, provided only a formula be obtained connecting the minimum deviation observed |