and the sum of the squares of the velocities is (a2 + b2) cos2 ↓ + (a2 + c2) cos2 e sin3 y + (b2 + c2) sin2 e sin2 ↓ = a2 (cos2 + cos2 e sin2 y) + b2 (cos2 ↓ + sin2 e sin3 †) + c2 sin2 y = a2 (cos2 € + sin2 e cos2 y) + b2 (sin2 e + cos2 e cos2 ↓) + c2 sin2 y which is the same expression as we deduced for the sum of the squares of the velocities in (20). 25. Having then results coinciding with those of M. Fresnel, I shall pursue the subject no further. The formula which I have given for the value of the difference of the squares of the velocities of the two vibrations, is a very elegant and useful one. Whether it had ever before been deduced from theory, or not, I cannot tell. Mr Herschel states that it has long been established by experiment. The only analogous one which I can find, is that of M. Fresnel, viz., "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 of the crystal." M. Fresnel also defines "optic axes" as those in which the rays travel when their velocity is the same for both. I have preferred to retain the name of optic axes to those directions which are normals to the directions of waves which move with a common velocity perpendicular to their own front; and it is very evident that these are the optic axes of experiment. I wish to add that, as far as I am aware, M. Fresnel's law, beautiful as it undoubtedly is, appears to me utterly incapable of being tested by experiment; so far as I can see, it requires a connexion with the index of refraction in order to apply experiment at all, and the index of refraction depends only on the wave. It must however be observed, that the older experimenters always use the word ray, but the slightest examination is sufficient to convince us that they mean, what we now call wave. It is not then a matter of surprize, that modern writers should in some cases confound the two; and this particular formula has been differently enunciated by different writers. Thus Mr M'Cullagh, in the Transactions of the Royal Irish Academy, enunciates Fresnel's proposition as follows: "The difference of the squares of the reciprocals of the velocities of the two rays having a common direction in the crystal, is proportional to the product of the sines of the angles which that direction makes with the optic axes." Mr Airy gives the following: "The difference between the reciprocals of the squares of the velocities of the two rays is proportional to the product of the sines of the two angles made by the front of the wave with the two circular sections, or to the product of the sines of the angles made by the normal to the front with the two optic axes." The latter is, I have no doubt, incorrect. Having then, in some instances, contradictory statements of the nature of the theory, I have, probably, here misled in some points. With respect to the mechanical part to which I object, all statements, which I have seen, coincide. 26. I refrain from making any extended application of the subject, but will only trouble you with one case, which I adduce on account of its great importance. The explanation of the lemniscates in biaxal crystals depends on the difference of the retardation of two vibrations which have a common normal to their front. The usual method of proceeding has been to find the retardation for uniaxal crystals, and from the circumstance of the retardation in that case being proportional to the difference of the squares of the velocities of the two waves, the same is true of the difference of the squares of the velocities of the two rays in biaxal crystals, and then, finally, to assume the difference of the reciprocals of the squares of the velocities of the rays to vary as the product of the sines of the angle made by the normal with the optic axes. The expression which I have given above is remarkably elegant, and is evidently the one on which the differences of the refractions of the different rays depends, whilst M. Fresnel's formula is not susceptible, as far as I know, of any application, except in those numerous instances where, being incorrectly adopted, it still gives a result nearly correct. This arises from the difference of the reciprocals of the squares of the velocities of the rays varying as the difference of the squares of the velocities of the vibrations parallel to their fronts. 27. To apply the formula to the particular case in question: Let T be the thickness of a plate of a biaxal crystal cut perpendicularly to the greatest or least axis of elasticity; V the velocity in air; v, v those of the vibrations perpendicular to their front, the incidence being nearly perpendicular; p, 'the angles which the perpendiculars to the fronts of our waves before and after incidence make with the normal: Then the retardation of this wave may be easily shewn (Airy's Tracts, p. 376.) to equal Now (a2 - c2) is a quantity be omitted, v'v (v' + v) small quantity, hence, if the square of such a v'v (v' + v) = 2a3 ; and difference of retardation becomes TV 28. In conclusion, the principal point in which the present view of the subject differs from those which have gone before, is in the fact of the non-existence of a normal vibratory force, or, in other words, that there is no resolved part of the force perpendicular to the front of the wave. The greatest utility of this view of the subject will appear when we shall consider the effect which takes place at the confines of two media, for it is evident that in resolving our vibrations at the point of change, we shall be obliged to consider the whole resolved part as lying in the plane of the front of the new wave. The complete discussion of this point, however, involves considerable difficulty, and I must delay it for the present, hoping shortly to make it the subject of a separate communication. 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 waves. 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 |
