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277. IV. By construction we have

by fimilarity of triangles therefore

RK: RP m:n PF: RK=CF: CR PF: PRmCF: nCR mPRX CF CRX PF therefore mPR: nCR=PF; CF and

and

mPR-nCR: mPR=PF-CF: PF ultimately mVR—nCR: mVR=VC: VF. This is a very general optical theorem, and affords an eafy method for computing the focal diftance of refracted rays. For this purpose let VR, the diftance of the radiant point, be expreffed by the symbol r, the distance of the focus of refracted rays by the fymbol f, and the radius of the spherical furface by a; we have

mr-nr-a: mra:f, and

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mr-nr-a m-nrX nd

In its application due attention must be paid to the qualities of r and a, whether they be pofitive or negative, according to the conditions of laft corollary.

278. V. If Q (Plate 252, fig. 8.) be the focus of parallel rays coming from the opposite fide, we fhall have RQ: QC=RV: VF. For draw Cq parallel to PF, cutting RP in q: the Rq: qC=RP: PF. Now is the focus of the parallel rays FP, Cq. And when the point P ultimately coincides with the point V. q muft coincide with Q, and we have RQ: QC RV: VF. This is the most general optical theorem, and is equally applicable to lenfes, or even to a combination of them, as to fimple furfaces. It is alfo applicable to reflections, with this difference, that Q is to be affumed the focus of parallel rays coming the fame way with the incident rays. It affords the moft compendious method of computing fymbolically and arithmetically the focal distances in all cafes.

RV2

279. VI. We have alfo Rq: RP=RV: RF, and ultimately for central rays RQ: RV=RV: RF, and RF = This propofition is true in lenRQ fes and mirrors, but not in fingle refracting fubftances.

280. VII. Alfo Rq: RC=RP: RF, and ultiRVXRC mately RQ: RV=RC: RF, and RF= RQ N. B. These four points Q, V, C, F, either lie all one way from R, or two of them forward and two backward.

281. VIII. Also, making O the principal focus of rays coming the fame way, we have Rq: qC Co: oF, and ultimately RQ: Qe=cO: OF, and QCX Co

OF= and therefore reciprocally proRQ portional to RO, becaufe QCX co is a conftant quantity. Thele corollaries or theorems give us a variety of methods for finding the focus of refracted rays, or the other points related to them; and each formula contains 4 points, of which any three being given, the 4th may be found. Per. haps the laft is the moft fimple, as the quantity oc XcQ is always negative, because o and Q are

on different fides.

282. IX. From this conftruction we may alfo derive a very eafy and expeditious method of

drawing many refracted rays. Draw through the centre C (fig. 15, 16.) a line to the point of incidence P, and a line CA parallel to the incident ray RP. Take VO to VC as the fine of incidence to the fine of refraction, and about A, with the radius VO, describe an arch of a circle cutting PC produced in B. Join AB; and PF parallel to AB is the refracted ray. When the incident light is parallel to RC, the point A coincides with V, and a circle defcribed round V with the diftance VO will cut the lines PC, pC, &c. in the points Bb. The demonftration is evident. Having thus determined the focal diftance of refracted rays, it will be proper to point out a little more particularly its relation to its conjugate focus of incident rays. We shall confider the 4 cafes of light incident on the convex or concave furface of a denfer or a rarer medium.

283. I. Let light moving in air fall on the con14.) Let us fuppofe it tending to a point beyond vex furface of glass. (Plate CCLII, fig. 5. to fig. the glafs infinitely diftant. It will be collected to its principal focus o beyond the vertex V. Now is at a great diftance beyond the furface. The let the incident light converge a little, fo that R focus of refracted rays F will be a little within O or nearer to V. As the incident rays are made to converge more and more, the point R comes nearwith a much flower motion, being always fituated er to V, and the point F alfo approaches it, but between O and C till it is overtaken by R at the centre C, when the incident light is perpendicular to the furface in every point, and therefore fuffers no refraction. As R has overtaken F at C, it now paffes it, and is again overtaken by it at V. Now the point R is on the fide from which the light comes, that is, the rays diverge from R. After refraction they will diverge from F a little without R; and as R recedes farther from V, F tion, till, when R comes to Q, F has gone to an recedes ftill farther, and with an accelerated moinfinite diftance, or the refracted rays are parallel. When R ftill recedes, F now appears on the other fide, or beyond V; and as R recedes back to an infinite diftance, F has come to O: and this com. pletes the series of variations, the motion of F during the whole changes of fituation being in the fame direction with the motion of R.

284. II. Let the light moving in air fall on the concave furface of glass; and let us begin with parallel incident rays, conceiving, as before, R to lie beyond the glass at an infinite diftance. The refracted rays will move as if they came from the principal focus O, lying on that fide of the glafs from which the light comes. As the incident rays are made gradually more converging, and the point of convergence R comes towards the glafs, the conjugate focus F moves backward from O; the refracted rays growing lefs and lefs diverging, till the point R comes to Q, the principal focus on the other fide. The refracted rays are now parallel, or F has retreated to an infinite diftance. The incident light converging ftill more, or R coming between Q and V, F will appear on the other fide, or beyond the furface, or within the glafs, and will approach it with a retarded motion, and finally overtake R at the furface of the glafs. Let R continue its motion backwards (for

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it has all the while been moving backwards, or in a direction contrary to that of the light); that is, let R now be a radiant point, moving backwards from the surface of the giafs. F will at first be without it, but will be overtaken by it at the centre C, when the rays will fuffer no refraction. R ftill receding, will get without F; and while R recedes to an infinite distance, F will recede to O, and the feries will be completed.

285. Ille Let the light moving in glafs fall on the convex furface of air; that is, let it come out of the concave furface of glass, and let the incident rays be parallel, or tending to R, infinitely diftant; they will be difperfed by refraction from the principal focus O within the glafs. As they are not more converging, R comes near, and F retreats backwards, till R comes to Q, the princi, pal focus without the glafs, and the refracted rays are parallel. R ftill coming nearer, F now ap. /pears before the glafs, overtakes R at the centre C, and is again overtaken by it at V. R now becoming a radiant point within the glafs, F follows it backwards, and arrives at O, when R has receded to an infinite distance, and the series is completed.

286. IV. Let the incident light, moving in glass, fall on the concave surface of air, or come out of the convex furface of glafs. Let it tend to a point R at an infinite diftance without the glass. The refracted rays will converge to O, the principal focus without the glafs. As the incident light is made more converging, R comes towards the glafs, while F, fetting out from v, alfo approaches the glass, and R overtakes it at the furface V. R now becomes a radiant point within the glafs, receding backwards from the furface. Frecedes flower at first, but overtakes R at the centre C, and paffes it with an accelerated motion to an infinite diftance; while R retreats to Q, the principal focus within the glafs. R ftill retreating, F appears before the glafs; and while R retreats to an infinite diftance, F comes to V, and the feries is completed.

IV. Of GLASSES.

287. Glafs for optical purposes may be ground into nine different fhapes. Glaffes cut into five of those shapes are called lenses, which, together with their axes, are described under DIOPTRICs, Se&. 1, § 15, 16. The other 4 are,

288. 1. A plane glass, which is flat on both fides, and of equal thickness in all its parts, as EF, Pl. CCLIII, fig. 1.

289. 2. A flat plano-convex, whofe convex fide is ground into feveral little flat surfaces, as A, fig. 2. 290. 3. A prifm, which has three flat fides, and when viewed end wife appears like an equilateral triangle, as B.

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291. 4. A concavo-convex glass, as C, which has hitherto received no name, and is feldom, if ever, made use of in optical inftruments.

292 A ray of light Gh (fig. 1.) falling perpendi. cularly on a plane glafs EF, will pass through the glafs in the fame direction hi and go out of it into the air in the fame ftraight line iH. A ray of light AB falling obliquely on a plane glafs, wil go out of the glafs in the fame direction, but not in the same straight line: for in touching the glafs,

it will be refracted in the line BC; and in leaving the glafs, it will be refracted in the line CD.

293. Lemma. There is a certain point E (Plate 253. fig. 3, to 6.) within every double convex or double concave lens, through which every ray that paffes will have its incident and emergent parts QA, aq parallel to each other: but in a pla no-convex or plano-concave lens, that point È is removed to the vertex of the concave or convex furface; and in a menifcus, and in that other concavo-convex lens, it is removed a little way out of them, and lies next to the furface which has the greatest curvature. For iet REr be the axis of the lens joining the centres Rr of its surfaces A a. Draw any two of their femidiameters RA, ra parallel to each other, and join the points A, a, and the line Aa will cut the axis in the point E above defcribed. For the triangles REA, rEa, being equiangular, RE will be to Er in the given ratio of the femidiameters RA, ra; and confe quently the point E is invariable in the fame lens, Now fuppofing a ray to pass both ways along the line Aa, it, being equally inclined to the perpendiculars to furfaces, will be equally bent, and contrary wife in going out of the lens; fo that its emergent parts AQ, aq will be parallel. Now any of thefe lenfes will become plano-convex or planoconcave, by conceiving one of the femidiameters RA, ra to become infinite, and confequently to become parallel to the axis of the lens, and then the other femidiameter will coincide with the axis; and fo the points A E or a E will coincide. Q. E. D.

294. Carol. Hence when a pencil of rays talls almoft perpendicularly upon any lens, whofe thick nefs is inconfiderable, the courfe of the ray, which paffes through E, above defcribed, may be taken for a ftraight line paffing through the centre of the lens, without fenfible error in fenfible things. For it is manifeft from the length of Aa and from the quantity of the refractions at its extremities, that the perpendicular diftance of AQ, aq when produced, will be diminished both as the thickness of the lens and the obliquity of the ray is diminished.

295. PROPOSITION I. To find the focus of parallel rays falling almoft perpendicularly upon any given lens. Let E (Plate 253. fig. 7, to 12.) be the centre of the lens, R and r the centres of its furfaces, Rr its axis, gEG, a line parallel to the incident rays upon the surface B, whose centre is R. Parallel to gE draw a femidiameter BR, in which produced let V be the focus of the rays after their firft refraction at the furface B, and joining Vr, let it cut gE produced in G, and G will be the focus of the rays that emerge from the lens. For fince V is alfo the focus of the rays incident upon the fecond furface A, the emergent rays must have their focus in fome point of that ray which paffes ftraight through this furface; that is, in the line Vr, drawn through its centre r: and fince the whole courfe of another ray is reckoned a ftraight line gEG, its interfection G with Vr determines the focus of them all. Q. E. D.

296. Corol. 1. When the incident rays are parallel to the axis rR, the focal diftance EF is equal to EG. For let the incident rays that were parallel to gE be gradually more incli ned to the axis till they become parallel to it; and their first and fecond focuses V and G will A a a 2

defcribe

defcribe circular arches VT and GF whofe centres are R and E. For the line RV is invariable; being in proportion to RB in a given ratio of the lefler of the fines of incidence and refraction to their difference by a former propofition; confequently the line EG is alfo invariable, being in proportion to the given line RV in the given ratio of rE to rR, because the triangles EGr, RVr are equiangular.

297. Corol. 2. The laft proportion gives the following rule for finding the focal diftance of any thin lens. As Rr, the interval between the centres of the furfaces, is to rE, the femidiameter of the fecond furface, fo is RV to RT, the continua. tion of the first femidiameter to the first focus, to EG or EF, the focal distance of the lens; which, according as the lens is thicker or thinner in the middle than at its edges, mult lie on the fame fide as the emergent rays, or on the oppofite fide.

298. Corol. 3. Hence when ray's fall parallel on both fides of any lens, the focal diftances EF Ef are equal. For let rt be the continuation of the femidiameter Er to the first focus t of rays failing parallel upon the furface A; and the fame rule that gave R to E as RT to EF, gives alfo R to RE as rt to Ef. Whence Ef and EF are equal, becaufe the rectangles under rE, RT and alfo under RE, rt are equal. For rE is to rt and alfo RE to RT in the fame given ratio.

199. Carol. 4. Hence in particular in a doubleconvex or double-concave lens made of glafs, it is as the fum of their femidiameters (or in a menifcus as their difference) to either of them, fo is double the other, to the focal distance of the glafs. For the continuations RT rt are feverally double their femi-diameters; because in glafs ET is to TR and alfo Et to tr as 3 to 2.

300. Gorol. 5. Hence it the femidiameters of the furfaces of the glafs be equal, it's focal diftance is equal to one of them; and is equal to the focal diftance of a plano-convex or plano concave g'afs whofe femidiameter is as fhort again. For confidering the plane furface as having an infinite femidiameter, the firft ratio of the laft mentioned proportion may be reckoned a ratio of equality.

301. PROP. II. The focus of incident rays upon a fingle furface, fphere, or lens being given, it is required to find the focus of the emergent rays? Let any point Q (fig. 1, to 6, PI CCLIV.) be the focus of incident rays upon a fpherical furface, lens, or fphere, whofe centre is E; and let other rays come parallel to the line QEq the contrary way to the given rays, and after refraction let them belong to a focus F; then taking Ef equal to EF in the lens or sphere, but equal to FC in the fingle furface, fay as QF to FE fo Ef to fq; and placing fg the contrary way from to that of FQ from F, the point q will be the focus of the refracted rays, without fenfible error; provided the point Q be not fo remote from the axis, nor the furfaces fo broad, as to cause any of the rays to fall too obliquely upon them. For with the centre E and femidiameters EF and Ef defcribe two arches FG, fg cutting any ray QA aq in G and g, and draw EG and Eg. Then fuppofing G to be a focus of incident rays (as GA), the emergent rays (as ogq) will be parallel to GE; and on the other hand fuppofing & another focus of incident rays (as go)

the emergent rays (as AGQ) will be parallel to gE. Therefore the triangles QGE, Egq are equiangular, and confequently QG is to GE as Eg to gq; that is, when the ray QAaq is the nearest to QEq, QF is to FE as Efto fg. Now when Q accedes to F and coincides with it, the emergent rays become parallel, that is q recedes to an infinite diftance; and confequently when Q paffes to the other fide of F, the focus q will also pafs through an infinite space from one fide of ƒ to the other fide of it. Q. E. D.

302. Corol. 1. In a fphere or lens the focus q may be found by this rule: As QF to QE fo QË to Qg, to be placed the fame way from Q as QF lies from Q. For let the incident and emergent rays QA qa be produced till they meet in e; and the triangles QGE, Qeq being equiangular, we have QG to QE as Qe to Q7; and when the angles of thefe triangles are vanishing, the point e will comcide with E; because in the sphere the triangle Aea is equiangular at the base Aa, and confequently Ae and ae will at laft become femidiameters of the fphere. In a lens the thickness Aa is inconfiderable. The focus may also be found by this rule;-QF: FE :: QE: Eq. for QG; GE :: QA: Ag. And then the rule formerly demonftrated for fingle surfaces holds good for the lenfes.

303. Corol. 2. In all cafes the distance fq varies reciprocally as FQ does; and they lie contrary. wife from f and F; because the rectangle or the fquare under EF and Ef, the middle terms in the foregoing proportions, is invariable. The principal focal diftance of a lens may not only be found by collecting the rays coming from the fun, confidered as parallel, but alfo (by means of this propofition) it may be found by the light of a candle or window. For, because Qq qA:: QE: EG, we have (when A coincides with E) Qq; qE= QE: EF; that is, the diftance obferved between the radiant object and its picture in the focus is to the diftance of the lens from the focus as the distance of the lens from the radiant is to its principal focal distance. Multiply therefore the dif tances of the lens from the radiant and focus, and divide the product by their fum.

304 Corol. 3. Convex lenfes of different shapes that have equal focal distances, when put into each others places, have equal powers upon any pencil of rays to refract them to the fame focus. Because the rules above-mentioned depend only upon the focal diftance of the lens, and not upon the proportion of the femidiameters of its furfaces.

305. Corol 4. The rule that was given for a fphere of an uniform denfity, will ferve alfo for finding the focus of a pencil of rays refracted through any number of concentric furfaces, which intercede uniform mediums of any different denfities. For when rays come parallel to any line drawn through the common centre of thefe mediums, and are refracted through them all, the diftance of their focus from that centre is invariable, as in an uniform fphere.

306. Gorol. 5. When the focufes Q q lie on the fame fide of the refracting surfaces, if the incident rays flow from Q, the refracted rays will also flow from q; and if the incident rays How towards Q, the refracted will alfo flow towards; and the con

trary

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