Page:Encyclopædia Britannica, Ninth Edition, v. 17.djvu/862

OPTICS OPTICS, GEOMETRICAL. The subject of optics is so extensive that some subdivision of it is convenient if not necessary. Under the head of LIGHT will be found a general sketch accompanied by certain developments. The wave theory and those branches of the subject which are best expounded in connexion with it are reserved for treat ment in a later volume. The object of the present paper is to give some account of what is generally called geo metrical optics, a theoretical structure based upon the laws of reflexion and refraction. We shall, however, find it advisable not to exclude altogether the conceptions of the wave theory, for on certain most important and practical questions no conclusions can be drawn without the use of facts which are scarcely otherwise interpretable. Indeed it is not to be denied that the too rigid separation of optics into geometrical and physical has done a good deal of harm, much that is essential to a pro- ^^ ^^ per comprehension of the subject hav ing fallen between the two stools. Systems of Rays in General. In the investigation of this subject a few preli- J) minary propositions FV , will be useful. If a ray AB (fig. 1) travelling in a homogeneous medium suffer reflexion at a plane or curved surface BD, the total path between any two points A, C on the ray is a minimum, i.e., AB + BC is less along the actual path than it would be if the point B were slightly varied. For a variation of B in a direction perpendicular to the plane of reflexion (that of the diagram) the truth of this statement is at once evident. For a small variation BB in the plane of reflexion we see that the difference AB - AB is equal to the projection of BB upon AB, and that the difference CB - CB is equal to the projection of BB upon BC. These projections are equal, since by the law of re flexion AB and BC are equally inclined to BB , and thus the variation of the total path, AB + B C - (AB + BC), vanishes. A corresponding proposition holds good in the case of refraction. If we multiply the distances travelled in the first and second media respectively by the refractive indices appropriate to the media, the quantity so obtained is a minimum for the actual path of the ray from any point to any other. It is sufficient to consider the case of a varia tion of the point of passage in the plane of refraction. In the first medium (fig. 2) /* AB - yuAB = /^BB cos ABD, and in the second medium //CB - //CB = //BB cos CBD. The whole variation of the quantity in question is therefore BB (M cos ABD - fji cos CBD). Now by the law of refrac tion the sines of the angles of incidence and refraction are in the ratio // : ft, and accordingly fj. cos ABD - // cos CBD = 0. In whichever direction, therefore, the point of tran sition be varied, the varia tion of the quantity under consideration is zero. It is evident that the second pro position includes the first, since in the case of reflexion the two media are the same. The principle of the superposition of variations now allows us to make an important extension. If the quantity, which we may denote by 2yus, be a minimum for separate variations of all the points of passage between contiguous media, it is also a minimum even when simultaneous varia tions are admitted. However many times a ray may be reflected or refracted at the surfaces of various media, the actual path of the ray between any two points of its course makes 2/^s a minimum. Even if the variations of refrac tive index be gradual instead of sudden, the same principle holds good, and the actual path of the ray makes f/j-ds, as it would now be written, a minimum. The principle itself, though here deduced from the laws of reflexion and refraction, is an immediate consequence of the fundamental suppositions of the wave -theory of light, and if we are prepared to adopt this point of view we may conversely deduce the laws of reflexion and re fraction from the principle. The refractive index fj. is in versely proportional to the velocity of propagation, and the principle simply asserts that in passing from any point to any other the light follows the shortest course, that is, the course of earliest arrival. If two points be such that rays issuing from one of them, and ranging through a finite angle, converge to the other after any number of reflexions and refractions, the value of 2/*s from one focus to the other must be the same for all the rays. Thus, in order to condense rays issu ing from one point S upon a second point H by a single reflexion (fig. 3), the reflecting surface must be such that SP + HP = const., i.e., must be an ellipsoid of revolution with S and H for foci. Again, if it be required to effect the same operation by a single refraction at the surface of a medium whose index is fj., we see that the surface (fig. 4) must be such that SP + ya HP = const. If S be at an in- Fi S- 4. finite distance, i.e., if the incident rays be parallel, the surface is an ellipsoid of revolution with H for focus, and of eccentricity p.~ l (/*>!). Another important proposition, obvious from the point of view of the wave-theory, but here requiring an independ ent proof, was enunciated by Malus. It asserts that a system of rays, emanating originally from a point, retains always the property of being normal to a surface, whatever reflexions or refractions it may undergo in traversing singly- refracting media. Suppose that ABCDE, A B C D E . . . (fig. 5) are rays- originally normal to a surface AA , which undergo reflexions or refractions at BB , CC , &c. On every ray take points E, E , &c., such that 2//s is the same along the courses AE, A E , &c. We shall prove that the rays in the final medium are normal to the surface EE . For by hypothesis S/MS along ABCDE is the same as along Fig. 5. A B C D E , and, by the property proved above to attach to every ray, 2/^s reckoned along the neighbouring hypo-