We come now to the explanation of ordinary reflection and refraction on the undulatory theory, according to the principles laid down by Huygens. We shall in the first instance, for the take of simplicity, suppose the reflecting surface to be plane, and the incident waves to be plane likewise. Let the plane of the paper be the plane of incidence, and therefore perpendicular at the same time to the reflecting surface and to the planes of the waves. Let it cut the reflecting surface along A a ; and let it N, r 4. n s, lying in the plane of the paper, be three normals to the incident waves, representing therefore the courses of incident rays. If there were no obstacle, the wave at one instant at N would, after the lapse of time required to describe the perpendicular distance of is plane from s, come into the position represented in section by 8 q n, perpendicular to n s, r Q q, and n N 71. In order to • examine the effect of the inter, option, imagine each portion of the wave as it arrives at N 8 to become a centre of disturbance, from whence diverge hemispherical waves into the two media respectively, with the velocity appropriate to each. Consider for the present the first medium only. The times which have elapsed since the wave, now supposed to be at s, was at the points N, 2,- - -, are those required to describe N a, Q q, - • - and therefore the radii N n', q', • • -, of the hemi spherical waves diverged from N, fa- -, are equal to sn,oq - Through s draw a plane perpendicular to the plane of t he paper, touch ing in a' the hemisphere whose centre is at N. It is evident that the point re" will lie in the plane of the paper, and that the plane s n' will touch all the secondary waves, and therefore will be the front of the reflected wave. Moreover N n', which is perpendicular to this plane, will represent the reflected ray corresponding to the incident ray of at N. Hence the reflected ray lies in the plane of incidence ; and on account of the equality of the triangles s N n, a N a', the angles s ar n, s N n', which are the complements of the angles of incidence and reflection, are equal, and therefore angles of incidence and reflection are themselves equal.
Consider now the second medium. Everything will be the same as before, except that the radii N q', of the secondary waves diverging from N, Q, instead of being equal, will only be proportional, to N n, q q, bearing to them the ratio of r' to v, where r is the velo city of propagation in the second medium. The semicircles in which the hemispheres diverging from the various points of NS are cut by the plane of the paper will form a system of curves for which s is a centre of similitude ; and it will be readily seen that if through s be drawn a plane perpendicular to the plane of the paper, and touching in n' the hemisphere diverging from N, it will be the envelope of the secondary waves in the second medium, the point n' will lie in the plane of the paper, and 77 77' will be the course of the ray refracted at which will therefore lie hi the plane of incidence. Also the angles N s n, N s re, which are the complements of the angles a N n, 8 N a', are equal to the angled of incidence and refraction respectively, and sine of incidence : eine of refraction :: N a : N s : : Nn N n' a ratio which is constant, that is,independcnt of the angle of incidence; which gives the law of refraction.
These laws may be easily extended to the general cue in which the incident waves and the reflecting or refracting surface are of any form. For let P q in either of the above figures represent a ray incident at. q,. and therefore normal to the incident wave, and through Q imagine two tangent planes drawn, one to the incident wave, and the other to the reflecting or refracting surface. Confining our attention to the secondary waves which start in either medium from the immediate neighbourhood of the point Q, if we suppose them to start when a wave coinciding with the tangent plane to the wave, instead of the actual wave itself, arrives at the surface, and again, if we replace the actual surface by the tangent plane to it drawn throughshall only commit an error on the position of the centre aud magnitude of the radius of a secondary wave which is a small quantity of the second order, the distance of its centre from the point q being deemed a small quantity of the first order. Hence the line of intersection of any two such secondary waves will only be rendered erroneous by a small quantity of the first order, which vanishes in the limit, and therefore the point of ultimate inter section of the secondary waves will not be affected. Hence the laws of reflection and refraction will remain the Annie as before, the normal to the surface at the point of incidence taking the place of the perpen dicular to the plane.
This mode of conceiving of reflection and refraction [shows at once that if the point of incidence be slightly varied, and a ray starting from a point in the Incident, and reaching a point in the reflected or refracted ray, be supposed to follow the varied instead of the actual course, tra velling in each medium with the velocity appropriate to it, the time of transit will be ultimately unchanged. This law may be readily extended to any number of reflections and refractions. In a great many cases the time of passage is less along the actual than along the varied course, In which case the proposition becomes equivalent to Format's law of swiftest propagation. The time may, however, be a maximum in place of a minimum, or neither a maximum nor a minimum. Thus, if a ray emanate from a point A, and after reflection at the point r reach the point a, the plane A r a will be perpendicular to the tangent plane at r, to which A P, n r, will be equally inclined ; and if A, it, be made the foci of a prolate spheroid of revolution, the magnitude of which is increased until it. passes through the point r, it will there touch the reflecting surface. If now this surface touch the 'There'd externally at the point r, the sum Ae+ rn will be a minimum when the point of incidence is slightly varied along the reflecting surface ; but if it touch the spheroid internally, the sum will be a maximum ; and if it both touch and cut the spheroid, so as to be external to it in some directions from r and internal in others, the sum will be neither a maximum nor a minimum.