ON THE POLARIZATION OF DIFFRACTED LIGHT. [From the Philosophical Magazine, XIII, 1857, pp. 159–61: ON considering the recent interesting experimental researches of M. Holtzmann on this subject*, I am induced to make the following remarks†. In the more common phenomena of diffraction, in which the angle of diffraction is but small, we know that the character of the diffracting edge, and the nature of the body by which the light is obstructed, are matters of indifference. This was made the object of special experimental investigation by Fresnel; and its truth is further confirmed by the wonderful accordance which he found between the results of the most careful measurements and the predictions of a theory in which it is assumed that the office of the opaque body is merely to stop a portion of the incident light. But when diffraction is produced by a fine grating, the angle of diffraction is no longer restricted to be small; and it becomes an open question whether the precise circumstances of the diffraction may not have to be taken into account, and not merely the form and dimensions of the apertures through which the light passes. If so, the problem becomes one of extreme complexity. In my memoir on the Dynamical Theory of Diffraction, published in the ninth volume of the Cambridge Philosophical Transactions, I investigated the problem on the hypothesis that in diffraction at a large angle, as we know to be the case in diffraction at a small one, the office of the opaque body is merely to stop a portion of the incident light. I distinctly stated this as a hypothesis, and I always regarded it as rather precarious. I was guided by the following consideration. Let AB [* Pogg. Ann. 1856; also Phil. Mag. xIII, 1857, pp. 125-9.] [ Cf. also the author's remarks of date 1882, ante Vol. 11, p. 327, in which Lorenz's experiments of 1860, confirming his own results, are referred to.] be the section of a transparent interval by the plane of diffraction, supposing for simplicity the diffraction to take place in air or in a homogeneous medium, and not at the confines of two different media; let AB = b; let B be the angle of diffraction, and the wave-length in the medium. Supposing the light to be incident perpendicularly on the grating, the difference of phase of the secondary waves which started from A, B, respectively, will be determined by the length of path b sin ẞ within the medium. In experiment this will usually be a considerable multiple of λ. In the line AB take two points, A', B', equidistant from A, B, respectively, and comprising between them as large a multiple as possible of a cosec B. If we suppose the influence of the opaque body insensible at the distance AA' or BB' from A or B, the secondary waves which start from all points in the interval A'B' will neutralize each other by interference, so that the whole effect will be due to the secondary waves which start from AA' and BB'. Suppose the angle ẞ to belong to the brightest part of a "spectrum of the first class" (Fraunhofer); then AA' + BB′ = 1⁄2λ cosec ß, λ referring to mean rays, so that AA' or BB' is only equal to cosec B. If, for example, ß=30°, AA' is only equal to λ. At such very small distances it may well be doubted whether the influence of the opaque body may not have to be taken into account. When diffraction takes place at the confines of two different media, suppose air and glass, the problem is still further complicated. We may, however, apply the theory to which reference has been made on the two extreme suppositions, first, that the diffraction takes place wholly in the first, secondly, that it takes place wholly in the second medium. The results of my own experiments were very fairly represented by theory, the vibrations being supposed perpendicular to the plane of polarization, provided the diffraction be conceived to take place in the first medium, or in other words, just before the light reaches the grating; but they would not at all fit the hypothesis of vibrations parallel to the plane of polarization. I put forth some considerations, founded on probable reasoning, to show that the supposition of diffraction taking place in the first medium was in accordance with the physical circumstances of the case. So decided was the result obtained, that it seemed to me a strong argument in favour of the hypothesis that the vibrations are perpendicular to the plane of polarization, though I still felt the necessity of repeating the experiments under varied circum stances. But since the appearance of M. Holtzmann's researches the state of the question is changed. I have no reason to doubt the correctness of his results, while on the other hand the result I myself obtained was far too decided to be passed by. The conclusion which, in the present state of the question, seems to me most probable is, that the polarization of light diffracted at a large angle is, in fact, influenced by the nature of the diffracting body. The subject demands a much more extensive experimental investigation, in which the circumstances of diffraction shall be varied as much as possible. I hope to have leisure to undertake such an investigation: meanwhile it would be premature to offer any decided opinion. It seems to me, however, worthy of attentive consideration, whether a glass grating may not offer a fairer experiment for the decision of the question as to the direction of vibration in polarized light than a smoke grating, inasmuch as in the former we have to do with an uninterrupted medium, glass, the surface of which is merely rendered irregular, whereas in the latter the problem is complicated by the existence of two distinct media, glass and soot, placed alternately. I call the layer of soot a medium, for though no light can pass through any sensible thickness of it, we must not conclude from that that it is without influence on the light which passes excessively close to it. I have not mentioned the effect of oblique refraction in the experiments of M. Holtzmann, because if it were allowed for, the character of the results obtained would remain unchanged, the magnitude of the observed effect would only be somewhat diminished. ON THE DISCONTINUITY OF ARBITRARY CONSTANTS WHICH APPEAR IN DIVERGENT DEVELOPMENTS. [From the Transactions of the Cambridge Philosophical Society, Vol. x, pp. 106–128. Read May 11, 1857.] [Abstract. Proc. Camb. Phil. Soc. Vol. 1, pp. 181–2.] IN a paper "On the Numerical Calculation of a class of Definite Integrals and Infinite Series" printed in the ninth volume of the Cambridge Philosophical Transactions, the author succeeded in putting the integral under a form which admits of extremely easy numerical calculation when m is large, whether positive or negative. The integral is obtained in the first instance under the form of circular functions for m positive, or an exponential for m negative, multiplied by series according to descending powers of m. These series, which are at first convergent, though ultimately divergent, have arbitrary constants as coefficients, the determination of which is all that remains to complete the process. From the nature of the series, which are applicable only when m is large, or when it is an imaginary quantity with a large modulus, the passage from a large positive to a large negative value of m cannot be made through zero, but only by making m imaginary and altering its amplitude by π. The author succeeded in determining directly the arbitrary constants for m positive, but not for m negative. It was found that if, in the analytical expression applicable in the case of m positive, were written for m, the result would become correct on throwing away the part involving an exponential with a positive index. There was nothing however to show à priori that this process was legitimate, nor, if it were, at what value of the amplitude of m a change in the analytical expression ought to be made, although the occurrence of radicals in the descending and ultimately divergent series, which did not occur in ascending convergent series by which the function might always be expressed, showed that some change analogous to the change of sign of a radical ought to be made in passing through some values of the amplitude of the variable m. The method which the author applied to this function is of very general application, but is subject throughout to the same difficulty. In the present paper the author has resumed the subject, and has pointed out the character by which the liability to discontinuity in the arbitrary constants may be ascertained, which consists in this, that the terms of an associated divergent series come to be regularly positive. It is thus found that, notwithstanding the discontinuity, the complete integrals, by means of divergent series, of the differential equations which the functions treated of satisfy, are expressed in such a manner as to involve only as many constants as correspond to the degree of the equation. unknown Divergent series are usually divided into two classes, according as the terms are regularly positive, or alternately positive and negative. But according to the view here taken, series of the former kind appear as singularities of the general case of divergent series proceeding according to powers of an imaginary variable, as indeterminate forms in passing through which a discontinuity of analytical expression takes place, analogous to a change of sign of a radical. In a paper "On the Numerical Calculation of a class of Definite Integrals and Infinite Series," printed in the ninth volume of the Transactions of this Society, I succeeded in developing the integral [cos (w3 - mw) dw in a form which admits of extremely easy 2 numerical calculation when m is large, whether positive or negative, or even moderately large. The method there followed is of very general application to a class of functions which frequently occur in physical problems. Some other examples of its use are given in the same paper; and I was enabled by the application of it to solve the problem of the motion of the fluid surrounding a pendulum of the form of a long cylinder, when the internal friction of the fluid is taken into account*. These functions admit of expansion, according to ascending powers of the variables, in series which are always convergent, and which may be regarded as defining the functions for all values of the variable real or imaginary, though the actual numerical calculation would involve a labour increasing indefinitely with the magnitude of the variable. They satisfy certain linear differential equations, which indeed frequently are what present themselves in the first instance, the series, multiplied by arbitrary constants, being merely their integrals. In my former paper, to which the present may be regarded as a supplement, I have employed these equations to obtain integrals in the form of descending series multiplied by exponentials. These integrals, when once the arbitrary constants are determined, are exceedingly convenient for numerical calculation when the variable is large, notwithstanding that the series involved in them, though at first rapidly convergent, become ultimately rapidly divergent. * Cambridge Philosophical Transactions, Vol. IX, Part II. [Ante, Vol. II, p. 329. Further papers on this subject of dates 1868, 1889, 1902, are reprinted infra.] |