ing on assumptions in some instances confessedly empirical, in others dependent on analogies, or hypotheses not free from doubt, and at any rate little connected into a system. 4. This investigation, whose questionable points are so fairly stated, and ably grappled with, by Mr. Airy in his tract on the Undulatory Theory (1831, art. 128 et seq.), has been since pursued on different principles by M. Cauchy, and especially by the late Prof. Maccullagh in his memoir "On the Laws of Crystalline Reflexion and Refraction" (Mem. Roy. Irish Acad. vol. xviii. 1838), whose views have been ably but briefly expounded by Dr. Lloyd in his Lectures on the Wave Theory (part 2. p. 30, 1841). More recently, Mr. Power has investigated the subject by a systematic analysis, directed to other objects, but including an important element in these deductions ("On Absorption of Rays," &c., Phil. Trans. 1854, part 1). 5. But among these distinguished philosophers there exists considerable diversity, and even contradiction of views. Nor, so far as I am aware, has the subject been so discussed as to enable us to trace the source of these discrepancies, or fairly to estimate the claims of the opposing theories, or the force of the experimental results which bear upon them. Thus it seems highly desirable, that questions affecting so fundamental a part of the undulatory theory should be cleared up and placed on an unassailable basis. Having long ago thrown aside some investigations on the subject, in which I was then engaged, I have of late had my attention recalled to the question, and have thus been induced to revise and extend those investigations, in the hope of contributing towards the settlement of the points involved, or at any rate of putting the whole discussion before the student in a connected point of view; with which object I have been led to commence ab initio, so that those who have only an elementary acquaintance with the theory may be enabled to follow the deductions without difficulty, and may here be furnished with that systematic elucidation which is not, as far as I am aware, to be found in any existing publication. Theoretical Views. 6. The formulas for the amplitudes of the incident, reflected, and refracted rays, as given by Prof. Maccullagh and later writers, though closely corresponding with those of Fresnel, and fulfilling generally the same conditions, yet differ from them in certain cases as to the sign, and in others as to the values of the expressions. 7. But the main point of difference and difficulty consists in this: Fresnel investigates two sets of formulas; one set (H) for the respective rays deduced on the supposition that the vibrations are perpendicular to the plane of incidence, another set (K) on the supposition that they are parallel to that plane. Now those of Maccullagh, which correspond closely to Fresnel's first set (H), are deduced on the contrary supposition of vibrations parallel to incidence, while those corresponding to Fresnel's second set (K) are for vibrations perpendicular to that plane. 8. In either investigation the formulas (K) are those which represent evanescence of the light at the polarizing angle, while the formulas (H) represent brightness at that incidence. But when a ray vanishes at the polarizing angle, we know that its plane, of this second incidence, must be perpendicular to that of its first incidence or original polarization. Hence, according as the vibrations (K) may be parallel or perpendicular to this second plane of incidence, they must be respectively perpendicular or parallel to the first or plane of polarization. The question thus reduces itself to whether, in polarized light in general, the vibrations are parallel or perpendicular to the plane of polarization. 9. M. Cauchy, in an earlier paper (Mem. Instit. vol. x. p. 304), had inferred with Maccullagh, from dynamical views, that the vibrations are parallel to the plane of polarization. But in a later memoir (Bull. Math. July 1830) he deduces formulas corresponding to Fresnel's on the hypothesis of vibrations perpendicular to the plane of polarization, and even more formally renounces his earlier opinion and returns to that of Fresnel. He also connects similar equations with higher dynamical principles in the Nouv. Exercices Math. (liv. 7). Synopsis of Formulas referred to. 10. Fresnel's formulas for vibrations perpendicular to the plane of incidence (h being the amplitude of the incident, h' of the reflected, and h, of the refracted rays, and dividing by h, i being the angle of incidence, r of refraction), are 11. Here we may observe, in the numerator of h 12. It is also desirable to notice, that these expressions are the same as those given in Mr. Airy's Tract, § 129, under the slightly different form in which they directly result from the peculiar process there pursued, viz. writing sin (ri) and -tan (r-i). Also the numerator of k' is positive for all values of (¿—r), which is necessarily less than 90°, while the denominator becomes ∞ at (i+r)=90°, which, according to Brewster's law, is the polarizing angle, and for greater values continues negative. 13. Prof. Maccullagh's formulas are, for vibrations parallel to the plane of incidence, 14. Comparing these formulas with Fresnel's (distinguished by using roman letters), we may observe from (11), h'=-h', h=h,μ, Here also k, undergoes the same change of sign at the incidence of complete polarization. Densities and Vibrating Masses. 15. In deducing these formulas, it is in all cases necessary to express the ratio of the masses of æther simultaneously vibrating without and within the medium; and the differences in the respective formulas are mainly dependent on the very opposite suppositions made by the several philosophers as to the density of the æther in different media,-Fresnel supposing it more dense within the denser medium; Cauchy, less dense; and Maccullagh, equally dense in all media. The last-named writer has argued that refraction cannot be dependent on the density of the æther as such. He especially observes, that "in doubly-refracting crystals, the density, being independent of the direction, could not be conceived to vary with the refractive index" (p. 39). And Prof. Stokes has observed, that in the vibrations of æther, "diminution of velocity seems capable of being accounted for on several distinct hypotheses." 16. The expressions for the masses of æther vibrating in the same time without and within the denser medium, are obtained on these different suppositions as to the density of the æther, as follows: v If v be the velocity of the incident ray, v, that of the refracted, and the index μ= μπ then at a perpendicular incidence, the simultaneously vibrating masses will be simply If the densities be 8, 8,, then, according to the view of Fresnel, 8, 8, and = sin2r 1 sin2 i = με and we must multiply in this ratio, which gives If 8=8,, according to the view of Maccullagh, 17. In either case, for oblique incidences we must multiply by the rectangular breadth of the rays on the same base or section of the surface, which will be as cosi: cos r, or 18. It may here be observed, that if we admit equal densities, we must nevertheless suppose some retarding power in the æther within the denser medium. It is still conceivable that this may follow the same law as that of increased density, and that thus Fresnel's formula might still apply. Or again, this reduces itself to the condition, that for perpendicular incidence we should have (m) 1 which might be simply the original condition, without involving the division by μ2, as above, and might be dependent directly on some hypothesis assumed as to the constitution of the æther. 19. Expressing Fresnel's values by roman letters, and comparing with Maccullagh's, we have 21. Mr. Power, taking a for the distances of the molecules without, and a, within, the medium, obtains what is equivalent to but having avoided any assumption of the law of refraction at the outset, he deduces (§§ 18, 28) the value which seems irreconcileable with the admitted principle μ= unless by supposing &=8,, which would agree with Maccullagh's` view. Or if we could have with Fresnel's view; or if &s=μ2, with that of diminished density. But as neither of these suppositions seem reconcileable with admitted principles, it will not be material to discuss them further. Equivalent Vibrations. 22. As to the general nature of the vibratory forces concerned, it will be on all hands admitted that the vibratory force of the |