PROP. 10.-When diverging or converging rays are incident nearly perpendicularly upon a spherical reflector, the distance of the focus of incident rays from the principal focus, measured along the axis of the pencil, is to the distance of the principal focus from the centre, as this distance is to the distance of the principal focus from the geometrical focus of reflected rays.

Let ACB, fig. 11, be the spherical reflector, whose centre is E; Q the focus of the rays; QA, QC, two rays of the pencil, of which QC passes through the centre E, and is therefore reflected in the direction CQ; joiu EA, and, in the plane QACE, make the angle EA9 equal to the angle EAQ; then the ray QA will be reflected in the direction Aq.

Draw DA parallel to QC, and make the angle EAe equal to the angle EAD; bisect EC in F. Then, since the DAE= the EAe, and the ZQAE= the EAq, the DAQ, or its equal AQe, is equal to the ZeAq; also, the geA is common to the two triangles AQe, Aqe; therefore they are similar, and Qe: eA :: eA: eq; or, since eA =¿E QE E eE: eq. Now let the arc AC be dimmished without limit, or the ray QA be incident nearly perpendicularly, then e coincides with F; and the limit of the intersections of CQ and Aq, is determined by the proportion QF : FE :: FE : Fq.

The diagram, fig. 11, is constructed for the case in which diverging rays are incident upon a concave spherical surface, and the same demonstration is applicable when the incident rays converge, as is represented in fig. 12.

If the lines DA, QA, EA, eA, qA, be produced, the figures serve for those cases in which the rays are incident upon the convex surface. Cor. 1. If q be the focus of incident, Q will be the focus of reflected rays; and Q and q are called conjugate foci.

Cor. 2. If the distance QF be very great when compared with FE, Fq is very small when compared with it. Thus, if the rays diverge from a point in the sun's disc, and fall upon a reflector whose radius does not exceed a few feet, F and q may, for all practical purposes, be considered as coincident.

Cor. 3. When Q coincides with E, all the rays are incident perpendicu larly upon the reflector, and therefore they are reflected perpendicularly, or q coincides with E.

Cor. 4. The point e bisects the secant of the arc AC.

Cor. 5. Since Qe: eE:: eE: eq, by composition or division, Qe: QE :: eE: Eq; alternately, Qe: eE:: QE: Eq; and, when QA is incident nearly perpendicularly, QF : FE :: QE Eq.

Cor. 6. Since EA bisects the angle QAq (or PAq), QA : Aq :: QE : Eq (Euc 3 6); and, when QA is incident nearly perpendicularly, QC: Cq:: QE: Eq. That is, the distances of the conjugate foci from the centre, are proportional to their distances from the surface.

Cor. 7. Since QE : Eq :: QF : FE, and QE : Eq :: QC : Cq, e have, ultimately, QF FE :: QC

Cq.

Cor. 8. As the arc AC decreases, Eq, the distance of the intersection of the reflected ray, and the axis from the centre decreases, unless Q coincide with E, or lie between E and e.

For, Qe eE:: QE: En, and as AC decreases Ee decreases; therefore, when Qis in eE produced, the terms of the ratio of greater inequality, Qe: Ee, are equally diminished, and that ratio, or its equal QE Eq, increases; and, since QE is invariable, Eq decreases.

When Qis in Ee produced, as AC decreases Qe increases, and Ee decreases; therefore the ratio of Qe: Ee, or of QE Eq increases; and consequently, as before, Eq decreases.

But when Q lies between E and e, as AC decreases the terms of a ratio of less inequality Qe: Ee are equally diminished; therefore that ratio, or its equal QE: Eq, decreases; and since QE is invariable, Eq increases. When Q coincides with E, q also coincides with it, whatever be the magnitude of the arc AC.

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PROP. 11.-The conjugate foci, Q and q, lie on the same side of the principal focus; they move in opposite directions, and meet at the centre and surface of the reflector.

Since QF FE :: FE: Fq, we have QFX Fq =FE2; that is, Q and 9 are so situated that the rectangle under QF and Fq is invariable. Also, when Q coincides with E, q coincides with it (fig. 13); in this case then, QF and Fq are measured in the same direction from F; and, since their rectangle is invariable, they must always be measured in the same direction.

That Q and q move in opposite directions may thus be proved: the rectangle QFX Fq is invariable; and therefore as one of these quantities increases, the other decreases; also, Q and 9 lie the same way from the fixed point F, they must therefore move in opposite directions.

Having given the place of Q, and FE the focal length of the reflector, to determine the place of the conjugate focus q, we must take QF : FE :: FE : Fq, and measure FQ and Fq in the same direction from F.

Thus, when Q, the focus of incident rays, is further from the reflector than E, and on the same side of it, FQ is greater than FE, therefore FE is greater than Fq; or q, the focus of reflected rays, lies between F and E.

When Q is between E and F, q lies the other way from E; and whilst Q moves from E to F, 2 moves in the opposite direction from E an infinite distance.

When Q is between F and C, QF is less than FE or FC; therefore FC is less than Fq; and, since FQ and Fq are measured in the same diTection from F, q is on the convex side of the reflector.-Fig 14,

When Q coincides with C, QF is equal to FC; therefore FC is equal to Fq, or q coincides with C.

When converging rays are incident upon the concave surface of the reflector, QF is greater than FC; therefore FC is greater than Fq; or q lies between F and C.

Cor. 1. A concave spherical reflector lessens the divergency, or increases the convergency of all pencils of rays incident nearly perpendicularly upon it.

For, the rays diverge from a point farther from the reflector than the principal focus, they are made to converge.

If they diverge from F, they are reflected parallel to CE.

If the focus of incidence lie between F and C, q is on the other side of the surface, or the rays diverge after reflection; and because QF: FE :: QC: Cq, and QF is less than FE, QC is less than Cq; also, the subtense AC is common; therefore the angle contained between the incident rays QA, QC, is greater than the angle contained between the reflected rays AP, CQ; or the reflected rays diverge less than the incident rays If converging rays fall upon the refleetor, QF (fig. 12, p. 748) is greater than FE, therefore QC is greater than Cq; or the reflected rays converge to a focus nearer to the reflector than the focus of incident rays, and their convergency is increased.

Cor. 2. In the same manner it may be shown, that a convex spherical reflector increases the divergency, or diminishes the convergency, of all rays incident nearly perpendicularly upon it.

Parallel rays may be made to converge or diverge accurately, by means of a parabolic reflector.-See Fig. 15.

Let ACB be a parabola, by a revolution of which about its axis QC, a parabolic reflector is generated'; take F the focus; let DA, which is parallel to QC, be a ray of light incident upon the concave side of this reflector, and join AF. Draw TAE in the plane DAF, and, touching the paraboloid in A. Then since the angle TAD is equal to the angle EAF, from the nature of the parabola, the ray DA will be reflected in the direc tion AF. In the same manner it may be shown, that any other ray, parallel to QC, will be reflected to F; and therefore the reflected rays converge accurately to this point.

If DA, FA, be produced, it is manifest that rays, incident upon the convex surface of the paraboloid, parallel to the axis, will, after reflection, diverge accurately from F..

The advantage, however, of a parabolic reflector is not so great as might, at first, be expected; for, if the pencil be inclined to the axis of the parabola, the rays will not be made to converge or diverge accurately; and the greater this inclination is, the greater will the error become.

Cor. If F be the focus of incidence, the rays will be reflected parallel to the axis.

Diverging or converging rays may be made to converge or diverge accurately, by a reflector in the form of a spheroid; and to diverge or converge accurately, by one in the form of an hyperboloid.—See Figs. 16 and 17.

Let F and D be the foci of the conic section, by the revolution of which, about its axis, the reflecting surface is formed; F the focus of incident rays, then D will be the focus of reflected rays.

For, let FA be an incident ray, join DA, and produce it to d, draw TAE in the plane DAF, and touching the reflector in A ; then the angle EAF is equal to the angle DAT, in the ellipse, and to dAT in the hyper

boja; therefore AD is the reflected ray in the fortner case, and Ad in the latter; thus D is the focus of reflected rays.-Fig. 17.

If FA be produced to f, the figures serve for the cases in which rays are incident upon the convex surfaces.

We may here remark, as in the preceding article, that if rays fall upon the reflector converging to, or diverging from, 'any other point than one of the foci, they will not converge or diverge accurately after reflection.

ON IMAGES FORMED BY REFLECTION.

The rays of light which diverge from any point in an object, and fall upon the eye, excite a certain sensation in the mind, corresponding to which, as we know by experience, there exists an external substance in the place from which the rays proceed; and, whenever the same impression is made upon the organ of vision, we expect to find a similar object, and in a similar situation. It is also evident, that, if the rays belonging to any pencil, after reflection or refraction, converge to, or diverge from, a point, they will fall upon the eye, placed in a proper situation, as if they came from a real object; and therefore the mind, insensible of the change which the rays may have undergone in their passage, will conclude that there is a real object corresponding to that impression.

In some cases, indeed, chiefly in reflections, the judgment is corrected by particular circumstances which have no place in naked vision, as the diminution of light, or the presence of the reflecting surface, and we are sensible of the illusion; but still the impression is made, and representation, or image of the object, from which the rays originally proceeded, is formed.

Thus, the rays which diverge from Q, after reflection at the plane surface ACB, enter an eye, placed at E, as if they came from q, or q is the image of Q.-Fig. 18.

If then the rays, which diverge from any visible point in an object, fall upon a reflecting or refracting surface, the focus of the reflected or refracted rays is the image of that point.

Fig. 18.

C

The image is said to be real or imaginary, according as the foci of the rays by which it is formed are real or imaginary.

The image of a physical line is determined by finding the images of all the points in the line; aud of a surface, by finding the images of all the lines in the surface, or into which we may sudpose the surface to be divided.

PROP. 12.-The image of a straight line, formed by a plane reflector, is a straight line, on the other side of the reflector; the image and object are equally distant from, and equally inclined to, the reflecting plane; and they are equal to each other.-See Fig. 19. Let PR be a straight line, placed before the plane reflector AB; produce RP, if necessary, till it meets the surface in A, draw RBr at right angles to AB, and make Br equal to RB; join Ar, and from P draw PDp perpendicular to AB, meeting Ar in p, then will pr be the image of PR.

Since RBr is perpendicular to AB, and Br is equal to BR, r is the image of R.

Also, from the similar triangles ABR, ADP, RB : AB :: PD : AD, and from the similar triangles ABr, ADP, AB: Br :: AD: Dp; ex aquo, RB Br :: PD: Dp, and since RB is equal to Br, PD is equal to Dp, or p is the image of P. In the same manner it may be shown, that the image of every other point in PQR is the corresponding point in pqr; that is, pr is the whole image of PR.

Again, since BR is equal to Br, and AB common to the two triangles ABR, ABr, and also the angles at B are right angles, the angles of inclination RAB, BAr are equal, and AR is equal to Ar. In the same manner, AP is equal to Ap; therefore PR is equal to pr.

Cor. 1. If the object PR be parallel to the reflector, the image pr will also be parallel to it.

Cor. 2. If PR be a curve, pr will be a curve, similar and equal to PR, and similarly situated on the other side of the reflector.

Cor. 3. Whatever be the form of the object, the image will be similar and equal to it. For, the image of every line in the object is an equal and corresponding line, equally inclined to, and equally distant from, the reflector.

Cor. 4. Let pr be the image of PR, and suppose an eye to be placed at E, join pE, rE, cutting the reflector in C and D; then, considering the pupil as a point, the image will be seen in the part CD of the reflector, and it will subtend the angle CED at the eye, because all the rays enter the eye as if they came from a real object.-Fig. 20.

Cor. 5. When PR is parallel to AB, and E is situated in PR, CD is the half of pr, or PR.

For, in this case, pr is parallel to AB, and therefore CD: pr:: ED : Er 1: 2.

PROP. 13. When an object is placed between two parallel plane reflectors, a row of images is formed which are gradually fainter as they are more remote, and at length they become invisible.-Fig. 21.

Let AB, CD, be two plane reflectors, parallel to each other; E an object placed between them; through E draw the indefinite right line NEI perpendicular to AB or CD. Take FGFE, KH KG, FI = TH, &e. Also, take KL KE, FM = FL, KN = KM, &c.