When there are atexossive reflections, the inclination of the plane of the first to that of the last reflection is the deviation in plane, the last ray being then in a different plane from the first, the first kind of deviation, or that of direction, is the angle between the first ray pro duced beyond the Incident point (which is the course the ray would have pursued it unrefiected), and a parallel to the last ray drawn from the same incident point.
When light is reflected by two parallel planes there will be neither deviation in plane nor in direction ; more generally there will be no deviation in plane when the first incident ray is in a plane perpendicular to the intersection of the two reflecting planes.
In the latter case, where both reflections take place in tho same plane, let us consider the amount of the deviation in direction.
Let ...BCD represent the course of the ray reflected at a and c by the reflecting planes E B, EC. Let B a be the first ray produced, and d the parallel to the final ray ; theu the angle a s d to the deviation. \Viten the other two rays are at opposite aides of the intermediate ray a c (jig. I), then the deviation of an from A a is the difference of the two deviations at a and e, or twice the angle no P—twice the angle c a a, that is, 2 L r, ur double the Inclination of the mirrors. But when A 5, OD, ora at the same side of a c (jig. 2), the total deviation is the sun of the deviations at a and C, or twice the angle it o+ twice the angle x c as, which is the /Lame as 30tr -2 L It This is a re-cutrant angle alien i is acute, and therefore we may then substitute for it the corre sponding natural angle 2 L r. Hence the deviation is double the inclination of the mirrors when Dente, and double its supplement when obtuse. This property is turned to excellent use in I lealley'a sextant. lu general, alien there are any number of reflections in one plane, the total deviation is the sum of the deviations at each reflection, giving negative signs to those where the rapt are turned in a contrary way to the find reflection ; this stun is independent of the first angle of incidence when this number of planes is even.
With the exception of this ease, the ray will deviate not only in direction but in plane. The following mullet may family be proved to be true of the course of a ray reflected any number of times at any number of mirrors which are parallel to the same line :—The inclina tion of the ray to this above line will be constant throughout its course, and the projections of the rays on a plane perpendicular to the mirrors will obey the law of reflection of rays ouinciding with those projections.
When light diverging from any luminous point fells on a plane reflecting surface, it will after reflection diverge accurately from a point similarly situated at the opposite aide of the mirror. Let a ho the luminous point, DE the mirror, draw BA perpendicular to the mirror, and produce it until A8=AS; let an be an incident ray, join an, and produce it to e, then it is evident that L sna= L aim= L CBE. Now ac, being in the normal plane BAB, and making, with the normal nr, an angle cnr equal to the angle ear of incidence, must therefore be the reflected ray. The position of a being independent of that of a, the point of incidence, it follows that every other reflected ray be will diverge from the same point. Thus the reflected light will appear to an eye c as if proceeding from a point a behind the mirror similarly situated with B.
Hence if any body en be placed before a mirror De, the light which emanates from r will appear after reflection to proceed from the similarly situated point p behind the mirror, and thus an linage pe exactly similar to the body taa will be seen by looking at the mirror; the common looking.glase is a familiar example.
If we seek generally the nature of a surface by which light con verging to or diverging from one given poiat may after reflection diverge from or converge to another, it will be simplest to seek first the plane curve possessing the same property, then the surface generated by the revolution of this curve round an axis passing through the two given points will evidently be of the nature required.
Let r, r' be the radii vectored drawn from a point of the curve to the given points, one will correspond to an incident, the other to a reflected ray ; and lets be an arc of the curve measured from a fixed point to that of incidence, then ± is the sine of the angle of inci dence, using the upper or lower sine according as r increases or diminishes with a; hence we must have 0, whence r-Fer= const. Taking the upper sign we have an ellipse, or with the lower an hyperbola, of which the two fixed points are the foci : hence the prolate spheroid and hyperboloid are the surfaces sought. But if the incident light fall in parallel rays, and is reflected to one point, take the axis of x through this point iu the direction of the rays, the sine of dr' d.e incidence is then + --a-, whence 7; ± — as de 0, r' ± = const., which is the equation to a parabola having the given point for focus ; therefore the paraboloid of revolution is the required surface.