424 W A V E T H E 11 5. Propagation of Waves in General. It has been shown under OPTICS that a system of rays, how ever many reflexions or refractions they may have undergone, are always normal to a curtain surface, or rather system of surfaces. From our present point of view these surfaces are to be regarded as wave-surfaces, that is, surfaces of constant phase. It is evident that, so long as the radius of curvature is very large in comparison with , each small part of a wave-surface propagates itself just as an infinite plane wave coincident with the tangent plane would do. If we start at time t with a given surface, the corresponding wave- surface at time t + dt is to be found by prolonging every normal by the length Vdt, where V denotes the velocity of propagation at the place in question. If the medium be uniform, so that V is constant, the new surface is parallel to the old one, and this property is re tained however many short intervals of time be considered in suc cession. A wave-surface thus propagates itself normally, and the corresponding parts of successive surfaces are those which lie upon the same normal. In this sense the normal may be regarded as a ray, but the idea must not be pushed to streams of light limited to pass through small apertures. The manner in which the phase is determined by the length of the ray, and the conditions under which energy may be regarded as travelling along a ray, will be better treated under the head of Huygens s principle, and the theory of shadows ( 10). Format s From the law of propagation, according to which the wave- prin- surfaces are always as far advanced as possible, it follows that the ciple. course of a ray is that for which the time, represented byyV Vs, is a minimum. This is Format s principle of least time. Since the refractive index (/x) varies as V 1 , we may takeyjuds as the measure of the retardation between one wave surface and another ; and it is the same along whichever ray it may be measured. Law of The principle is.tfp.ds is a minimum along a ray lends itself magnify- readily to the investigation of optical laws. As an example, we ing. will consider the very important theory of magnifying power. Let A ,B be two points upon a wave-surface before the light enters the object-glass of a telescope, A, 15 the corresponding points upon a wave-surface after emergence from the eye-piece, both surfaces being plane. The value otf^ds is the same along the ray A A as along B B ; and, if from any cause B be slightly retarded relatively to A , then B will be retarded to the same amount relatively to A. Suppose now that the retardation in question is due to a small rotation (6) of the wave-surface A B about an axis in its own plane perpendicular to AB. The retardation of B relatively to A is then A B .0 ; and in like manner, if <p bo the corresponding rotation of AB, the retardation is A B.0. Since these retardations are the same, we have or the magnifying power is equal to the ratio of the widths of the strcitiii of light before and after passing the telescope. Prisms. The magnifying power is not necessarily the same in all direc tions. Consider the case of a prism arranged as for spectrum work. Passage through the prism does not alter the vertical width of the stream of light; hence there is no magnifying power in this direction. What happens in a horizontal direction de pends upon circumstances. A single prism in the position of minimum deviation does not alter the horizontal width of the beam. The same is true of a sequence of any number of prisms each in the position of minimum deviation, or of the com bination called by Thollon a couple, when the deviation is the least that can be obtained by rotating the couple as a rigid system, although a further diminution might be arrived at by violating this tie. In all these cases there is neither horizontal nor vertical magnification, and the instrument behaves as a telescope of power unity. If, however, a prism be so placed that the angle of emergence differs from the angle of incidence, the horizontal width of the beam undergoes a change. If the emergence be nearly grazing, there will be a high magnifying power in the horizontal direction ; and, whatever may be the character of the system of prisms, the horizontal magnifying power is represented by the ratio of widths. Brewster suggested that, by combining tvvo prisms with refracting edges at right angles, it would be possible to secure equal magnifying power in the two directions, and thus to imitate the action of an ordinary telescope. Appar- The theory of magnifying power is intimately connected with ent that of apparent brightness. By the use of a telescope in regarding bright- a bright body, such, for example, as the moon, there is aconcentra- ness. tion of light upon the pupil in proportion to the ratio of the area of the object-glass to that of the pupil. 1 But the apparent bright ness remains unaltered, the apparent superficial magnitude of the object being changed in precisely the same proportion, in accord ance with the law just established. These fundamental propositions were proved a long while since by Cotes and Smith ; and a complete exposition of them, from the point of view of geometrical optics, is to be found in Smith s treatise. 2 6. Waves Approximately Plane or Spherical. A plane wave of course remains plane after reflexion from a truly plane surface ; but any irregularities in the surface impress themselves upon the wave. In the simplest case, that of perpen dicular incidence, the irregularities are doubled, any depressed portion of the surface giving rise to a retardation in the wave front of twice its own amount. It is assumed that the lateral dimen sions of the depressed or elevated parts are large multiples of the wave-length ; otherwise the assimilation of the various parts to plane waves is not legitimate. In like manner, if a plane wave passes perpendicularly through a parallel plate of refracting material, a small elevation t at any part of one of the surfaces introduces a retardation (/j.- l)t in the corre sponding part of the wave-surface. An error in a glass surface is thus of only one-quarter of the importance of an equal error in a reflecting surface. Further, if a plate, otherwise true, be distorted by bending, the errors introduced at the two surfaces are approxi mately opposite, and neutralize one another. 3 In practical applications it is of importance to recognize the Symme- effects of a small departure of the wave-surface from its ideal plane trical or spherical form. Let the surface be referred to a system of rect- aberra- angular coordinates, the axis of z being normal at the centre of tion. the section of the beam, and the origin being the point of contact of the tangent plane. If, as happens in many cases, the surface be one of symmetry round OZ, the equation of the surface may be represented approximately by z = r"/2p + A) A + (1), in which p is the radius of curvature, or focal length, and r" = x 1 + y 2 . If the surface be truly spherical, A = l/8p 3 , and any deviation of A from this value indicates ordinary symmetrical spherical aberration. If, however, the surface be not symmetrical, we may have to Unsym- encounter aberration of a lower order of small quantities, and metrical therefore presumably of higher importance. By taking the axis of aberra- x and y coincident with the directions of principal curvature at 0, tion. we may write the equation of the surface 1 It is hero assumed that the object-glass is large enough to- fill the whole of the pupil with light; also that the glasses are perfectly transparent, and that there is no loss of light by reflexion. For theoretical purposes the latter re quirement may be satisfied by supposing the transition between one optical mei-ium and uiunher to be jrr.ulual in all cases. p, p being the principal radii of curvature, or focal lengths. The most important example of unsynnnetrical aberration is in the spectroscope, where (if the faces of the prisms may be regarded as at any rate surfaces of revolution) the wave-surface may by suitable adjustments be rendered symmetrical with respect to the horizontal plane y= 0. This plane may then be regarded as primary, p being the primary focal length, at which distance the spectrum is formed. Under these circumstances and 5 may be omitted from (2), which thus takes the form The constants a and 7 in (3) may be interpreted in terms of the differential coefficients of the principal radii of curvature. By the usual formula the radius of curvature at the point x of the inter section of (3) with the plane y = is approximately p(l-6apa.-). Since 7/ = is a principal plane throughout, this radius of curvature is a principal radius of the surface ; so that, denoting it by p, we have 1 dp-* a = 6dx < 4 > Again, in the neighbourhood of the origin, the approximate value of the product of the principal curvatures is ^ + ^. P P Thus d( }= -- < P.- d ^^^ + ^ PP J P P p -p P P whence by (4) __ I PP The equation of the normal at the point x, y, ~ is C-= = _ _Lr _______ ___JJI#_ - 1 p - 1 x + 3 etc- + 72/- p ~ l y + 2y.?y ~ Smith, Compleat System of Optics, Cambridge, 1738. The reader may be referred to a paper entitled "Notes, chiefly Historical, on some Fundamental Pro positions in Optics" (Phil. May., June 1886), in which some account is given of Smith s work, and its relation to modern investigations. 3 On tills principle Grubb has explained the observation that the effects of bending stress are nearly as prejudicial in the case of thick object-glasses as in
the ca>u of thin ones.