Sidebilder
PDF
ePub

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; org 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, if 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 lic between F and C, q is on the other side of the surface, or the rays diverge after reflection; and because QF : F6 :: 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 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 iocident nearly perpendicularly upon it.

Purallel 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 wbich about its axis QC, a parabolic reflector is generated'; take F the focus; let DA, wbich 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, theray DA will be reflected in the direction 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 accsrately, 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 in. ciilent 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 irr the hyper

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

II FA be produced to f, the figures serve for the cases in which rays are incident upon the convex surfaces. Fig. 15. Fig. 16.

Fig. 17.

[merged small][merged small][ocr errors][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][ocr errors]

We may here remark, as in the preceding article, that if rays fall upon the reflector converging to, or diverying 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 froin any point in an object, and fall upon

the
eye,

excite a certain sensation in the mind, corresponding to whiclı, 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 evi

dent, 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 wbich 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 cor. rected by particular circumstances which have no place in naked vision, as the diminution of light, or the presevce of the reflecting surface, and we are sensible of the illusion ; but still the impression is made, and representation, or image of the olvject, from which the rays originally proceeded, is formed.

Thus, the rays which diverge from Q, after re Fig. 18. flection 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 visi. ble 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.

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 plıysical line is determined by tinding the images of all the points in the line; and 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.

C I

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 PŘ.

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

Also, from the similar triangles ABR, ADP, RB : AB :: PD : AD, and from the similar triangles A Br, ADp, AB : Br :: AD : Dp; er æquo, 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 io. clination 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, rЕ, cutting the reflector in C and D; then, considering the popil 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 tlic rays enter the eye as if they came from a real object.-Fi. 20. Fig. 19.

Fig. 20.

R

[ocr errors]

E

P

[merged small][merged small][ocr errors]

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 fainier 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 righ: line NEI perpendicular to AB or CD. Take FG=FE, KH = KG, FI=TI, &c. Also, take KL=KE, FV=FL, KN =KI, &c.

Then, the rays which diverge from Eand fall upon AB, will, after reflection, diverge from G, that is, G will be an image of E. Also, thesc rays, after reflection at AB, will fall upon CD as if they proceeded from a real object at G, and after reflection at CD they will diverge from H; that is, H will be an image of G, or a second image of E, &c. In the same manner, the rays which diverge from E, and fall apon CD, will form the images L, M, N, &c.

It is found, by experiment, that all the light incident upon any surface, however well polished, is not regularly reflected from it. A part is dispersed in all directions, and a very considerable portion enters the surface, and seems to be absorbed by the body. In the passage also of light through any uniform medium, some rays are continually dispersed, or absorbed ; and thus, as it is thrown backward and forward through the plate of air contained between the two reflectors, AB, CD, its quantity is diminished. On all these accounts, therefore, the succeeding images become gradually fainter, and, at length, wholly invisible.

Cor. If E move towards F, the images G, H, I, &c. move towards the reflectors, and L, M, N, &c. from them; thus the images L and H, M and I, respectively approach each other, and when E coincides with F, these pairs respectively coincide.

PROP. 14.-If an object be placed between two plane reflectors inclined to each other, the images formed will lie in the circumfer. ence of a circle, whose centre is the intersection of the two planes, and radius the distance of the object from that intersection.

Let AB, AC, fig. 22, be two plane reflectors, at the angle BAC, E an object placed between them. Draw EF perpendicular to AB, and produce it to G, making FG=EF; then the rays which diverge from E and fall upon AB, will, after reflection, diverge from G, or G will be an image of E. From G, draw GH perpendicular to AC, and produce it to I, making HI=GH, and I will be a second image of E, &c. Again, draw ELM perpendicular to AC, and make LM = EL; also, draw MNO perpendicular to AB, and make NO= MN, &c. and M, 0, &c. will be images of E, formed on the supposition that it is placed before AC. Let K, V; P, Q, be the other images, determined in the sam manner.

Fig. 22.
VOR

P
Fig. 21.

[ocr errors]

А

с

[merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small]

Then, since EF is equal to FG, and Ar common to the triangles AFG, AFE, and the angles at F are right angles, AG is equal to AE (Enc. 4. 1). lu the same manner it appears, that AI, AK, &c. AM, ÀO, AP, &c. are cqual to each other, and to AE, that is, all the images lie in the circumference of the circle EMIK whose centre is A and radius AE.

Cor. If the angle BAC be finite, the number of images is limited. For, BA and CA being produced to S and R, the rays which are reflec

[ocr errors]

ted by either surface, diverying from any point Q between S and R, will not meet the other reflector; that is, muinarc of Q will be iormed.

PROP. 15.- Haring girer the inclination of two plane reflectors, and the situation of an object between them, to find the num. ber of images.

It appears from the construction in the last proposition, that the lines EG, MO, IK, PQ, &r. are paralici, as also EM, GI, OP, K, &c. Hence it follous, that the arcs EG, MI. OK, PV, &c. are equal; as also, thie arcs EM, GO, IP, KQ, &c. Let BC=n, EB=b, EC=c; then the arc EG = 26, EM= 2c, EO= EG + GO= EG + EM= 26+20 = 2a, EK= EO + OK=EO + EG = 2a + 2b, EOQ = EK + KQ = EK + EM = 2a + 26 +2c=4c, &c. Thus, there is one series of images, formed by the reflections at AB, whosc distances from E, mea. sured along the circular arc EOR, arc 26, 2a + 2b, 4a + 2b, ..... 28

- 2a + 26 (2na — 2c), where u is the number of images, this series will be continued as long as 2na – 2a + 2b, or 2na - 20 is less than the are EOR, or 180° + b, and conscquevtly n, the number of iniages in this

181 +6+2 series, is that whole number which is next inferior to

or to

2a 180 tate. There is also a scroud series of images, formed by reflec

24 tions at the same surface, whose distances from E are 2a, 4R, 6a,

2ma, continued as long as 2ma is less than 180 + b, anıl therr. fure m, the pumber of these images, is that whule uumber which is next

180 +b inferior to

2a In the same manner, the number of images formed by reflectiolis at the surface AC, is found by taking the whole numbers next infirm lo 180 tatb 180+

and 2a

2a Cor. 1. If a be a measure of 180, the number of images formed will

360 be For, if a be contained an even number of times in 180, or 2a be a

180 measure of 180, the number of images in each series is ; and the

2a

180 360 number upon the whole is 4 X

If a be contained an old

2a number of times in 180, 2n is a measure of 180 ta, or 180

180-at

180) the number of images is

180 + a
+

+
20
2a
24

22 60

[ocr errors]

ai and

180 +1

Cor. 2. When a is a mcasure of 180, two injagcs coincide.

[merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][ocr errors][merged small]
« ForrigeFortsett »