Fig. 460,461, 462.

*Art. 420. 423.

Fig. 460.

Fig. 463.

a Art. 459

b Art. 418,

423.

*Euc. III. 35

Euc. VI. 23.

Fig. 464.

d Art. 421 demonftr.

pendicularly upon its plane furface; and confequently are only refracted by the fpherical furface; the pofition of the refracted rays that fell upon the circumference of the lens is determined as in the next cafe.

482. Cafe 4. While Cremains in the denfer medium, tranfpofe the focus A into the rarer, and draw the tangents AD, AD to the refracting circle DBD and joyn CD, CD upon which as diameters defcribe the femicircles CED, CED towards the denfer medium, and infcribe in them the lines CE, CE, taken in proportion to the whole fine CD, as the fine of refraction to the fine of incidence. And the legs of the cauftick beginning from E* in the direction DE, DE will approach towards the axis AC, till they meet in the principal focus F; provided CA be greater than Ca; or will have the fame pofitions as in the first cafe.

483. The focus A being in the rarer medium, let the circle DBD be continued quite round, and let it cut the cauftick in F and F, the axis AC in G and any other ray AB Fin b; and while the ray AB is carried with an angular motion round about A from the axis AC towards the tangent AD, the arch Gb will first increase till it equals the arch GF, and then will decrease again till it equals the arch Gi, cut off by the last refracted ray DEI. This is manifeft from the motion of the refracted ray BbF while it touches the convexity of the cauftick in F, provided the focus A be fo remote from the furface that the last refracted ray DEi may converge towards the axis AC.

484. Caufticks made by refractions at other curves are alfo determinable by the 423d article. For inftance, the imaginary cauftick AFK made by refractions at an equiangular spiral AHB is also an equiangular spiral, their common pole A being the focus of incident rays. For fuppofing what has already been faid of this curve; from the center C of a circle of equal curvity to it at B, let down the fines of incidence and refraction CD, CE upon the incident and refracted rays AB and BFI produced backwards. And because D coincides with the focus A of incident rays, E does alfo coincide with F the focus of refracted ones. Joyn AE or AF, and because the angular points A and E of the right angles CAB, CEB are in a femicircumference whofe diameter is CB, the angles CBA, CEA ftanding upon the fame chord CA are equal and being fubftracted from the right angles CBG, CEI, the remaining angles ABG, AEI, which the lines AB, AE or AF do make with the curves, are every where equal; which is the property of the fpiral. So that this cauftick-fpiral differs from the other in pofition only.

C

485. To find the length of any caustick by refractions, imagine the reflected ray BF produced, to be unlaped like a string from the convexity of the cauftick Fo; then because the figures bdn B, CDEB are fimilard, the increment b of the incident ray AB, is every where to the decrement by of the ftring BFq, in the given ratio of the fine CD to the fine CE;

(for

(for which put n to m;) and therefore when AB, BFcome into a new pofition AB, Bo, the fum of the increments of AB, that is Aß-AB, is to the fum of the decrements of the ftring, that is BF Fo-Bo, as n to m.

m

m

Whence × AB-AB=BF÷Fq¬ßq, or × AB- AB+BQ →

BF Fo.

rays com

486. To find the points of any cauftick made by two fucceffive refrac- Fig. 465. tions, let the ray BFb which touches the cauftick EFF in F, (made as before by the first refraction of the rays), meet with another refracting curve Gb F, or with the fame curve continued; and let it be refracted at b into the line bd; in which let hd be the focal distance of other ing parallel to Bb, and in bB let be be the focal distance of other coming parallel to db; then fince F is the focus of incident rays upon the curve bG, fay as Fc:cb:: bd: dk, and placing dk as ufual, the point k will be the focus of a flender pencil after both refractions, or a point of the second caustick KFk; whofe points may also be found by art. 434 and 436, without finding the points F of the first caustick.

rays

a Art. 423

487. Hence it will appear that a cauftick made by refractions through Fig. 465a circular fection of a cylinder or a great circle of a sphere, will have fuch a shape as is here reprefented. Each half of this cauftick on each fide of the axis ACK confifts of two arches Kk Fl and lki, that are convex towards one another and form a cufp at / within the circle. The arch Kk Fl of the second cauftick cuts the circle in the fame point Fas the first cauftick does. For by the proportion above, when the points F, h coincide, or when Fc equals cb, then hd and dk are also equal. The reafon of the cufp at /is this; that while bk is increafing, and then decreafing again, the point b is continually approaching towards G*. The arches KFl and Art. 483Iki are convex towards one another, because the emergent ray, while its point of contact k is moving from Kto F, to l and to i, cuts the axis CK in greater and greater angles, till at last it emerges at i in a tangent to the circle and to the cauftick too. When the focus of incident rays is nearer to the sphere than its focal distance, the fecondary caustick Fĺk will have two afymptotes like as the primary one; and their shapes will be much alike.

CHAP

Fig. 466 to 469.

488.

W

CHAPTER X.

Concerning the Rain-bow.

PROPOSITION I.

HEN a ray of light is refracted into a circle, and fucceffively reflected within it any given number of times before it emerges out of the circle by a fecond refraction; let the angle of refraction be multiplyed by the number of fucceffive reflections increased by an unite; and the excess of the refulting angle above the angle of incidence will be equal to half the angle contained under the incident and the emergent ray produced till they meet: that is, the excess abovementioned is equal to half that angle, under the incident and the emergent ray, in which the refracting circle lyes, when the number of reflections is odd; and is equal to half the other angle, under the fame rays, which is the complement of the former to two right angles, when the number of reflections is even.

For let ABCDE be a great circle of a sphere whofe center is O, and let an incident ray SA be refracted at A to B, and be reflected from B to Art. 183 &c. C; and at C let it either go out by refraction to G, or be reflected to D*; where let it either go out by refraction to H or be reflected to E; and fo on. And when the number of reflections is odd, a line OR drawn through the center O and the middlemoft point of reflection, will bifect the angle at R under the incident and the emergent ray produced: because the reflections and refractions on each fide of the line OK are equal in number and magnitude; the chords AB, BC, CD, DE described by the reflected ray being equal to one another. And for the fame reafon when the number of reflections is even, a line OT, drawn through the center perpendicular to the chord that joins the two middlemoft points of reAlection, will bifect one of the angles at T under the incident and the emergent ray produced; and a line TV, perpendicular to TO, will bifect the other angle under them, which is the complement of the former to two right ones. Hence the line TV is parallel to the middlemoft chord, because TO is perpendicular to them both. Draw a diameter PO2 parallel to the incident ray SAM, and let it cut the reflected rays BC, CD, DE produced, in ß, y, d, refpectively. Join OA, OB and in fig. 466 the fums of the three angles in each of the triangles OAB, OAR, are equal to one another; take away the common angle AOB, and the fum of the equal angles OAB, OBA in the first triangle, will be equal to the fum of the angles OAR, ORA in the second triangle. And by fubftracting the angle of incidence OAR or OAM from both fums, we have 20AB-ÕAM =ORA=B02, Hence in fig. 467. the angle STV or PRC, being an

external

external angle of the triangle OB3, equals OBC + BOQ=OAB →→ 20AB-0AM-30AB-OAM. Hence again in fig. 468 the angle SRO or POC, being an external angle of the triangle OCB, equals OCB → P&C=OAB → 30AB-OAM=40AB-0AM. Hence again in fig. 469 the angle STV or PyD, being an internal angle of the triangle COy, equals OCD-COy=50AB-OAM, throwing away two right angles. For COy=2 right angles - POC=2 right angles -40AB + OAM. And fo forward continually. Therefore if the number of fucceffive reflections increased by an unite be called m, it appears that mOAB - OAM equals half the angle under the incident and emergent rays, 2. E. D.

PROPOSITION IL

489. Things remaining as they were, let the angle of incidence increase from nothing till it becomes a right angle; and the angle under the incident and the emergent ray, after any given number of reflections called n, will firft increafe and then decrease again; and will be the greatest of all when the tangent of the angle of incidence, is to the tangent of the angle of refraction, as

n I to r.

For putting m=n+1, we had half the angle under an incident and Fig. 466 to the emergent ray equal to the excefs of mOAB above OAM*; which ex- 469. Árt. 448. cefs, when the angles OAB, OAM are very fmall, will alfo be but fmall; and will increafe fo long as the fucceffive increments of mOAB shall exceed the contemporary increments of OAM; and will decrease again when the fucceffive increments of m OAB are exceeded by the increments of OAM; and confequently will be the greatest of all when m times the least increment of OAB is equal to once the contemporary increment of OAM; that is when the leaft increment of the angle of incidence OAM is to the contemporary increment of the angle of refraction OAB, and confequently the tangent of incidence is to the tangent of refraction, as a Art. 421. m to 1. 2. E. D.

PROPOSITION III.

490. It is propofed to find two angles, whofe fines fhall be in a given ratio of Ito R, and whofe tangents fhall be in another given ratio of m to 1.

In

any given line CEDA, let CA be to CD as I to R, and CA to CE Fig. 470. as m to 1; with the center C and femidiameter CD defcribe an arch DB, cutting a circle ABE whofe diameter is AE, in B; draw AB F, and joining BC, the fine of the angle CBF will be to the fine of CAF as I to R; and the tangent of CBF to the tangent of CAF as m to 1; and confequently CBF, CAF are the angles required. For in the triangle CAB the fine of the angle CBA or CBF, is to the fine of CAF, as CA to C B* * Art. 2213

or.

or CD, as I to R by conftruction. Join BE and compleat the parallelogram CEBG; and CG produced will cut ABF at right angles in F, becaufe ABE is a right angle in the femicircle ABE. Therefore the lines FC, FG are tangents of the angles CBF, GBF or CAF to the radius BF; * Enc. VI. 2. and the tangent FC is to the tangent FG as FA to FB* or as CA to CE* or as m to 1 by conftruction. 2. E. D.

No. 297.

491. Corol. I. When parallel rays of the fun fall upon a spherical drop of rain, let the given ratio of I to Ŕ stand for the ratio of the fine of incidence to the fine of refraction; and let n be any given number of fucceffive reflections made by every ray before it emerges out of the drop, and let m=n; and by thefe propofitions it appears, that half the greateft angle which any of the emergent rays can make with the incident rays, is equal to mXang. CBF-CAF. For CBF and CAF or GBF are angles whofe fines are as I to R, and whofe tangents are as m to I; and confequently are the angles of incidence and refraction of that ray, whofe incident and emergent parts produced contain the greatest angle.

492. Corol. 2. The foregoing conftruction for determining the angle a Phil. Trans. CBF is Dr. Halley's, and Sir Ifaac Newton's rule for calculating it, is this that follows. Asymm-1XRR is to VTT-RR, fo is the tabular radius to the cofine of the angle of incidence CB F. Whence this angle and its fine are given by the tables, and from thence by the ratio of I to R the tabular fine of the angle of refraction and the angle it felf are also given. The rule may thus be demonftrated. We had CA: CB:: I: R and FA: FB:: m: 1. Hence CA2 = CB2, and AF2mm BF'; and fo

Euc. I. 47.

Defign.

II

RR

II

RR

II

RR

CB2 — mm BF2 = (CA2 — AF2 = FC2 * =) CB2 - BF2. Hence

CB2-CB2mm BF2 - BF2, and II- RRXCB2mm-IX RRXBF; and by refolving this equality into a proportion, and by extracting the roots, we have ymm—1×RR:√ÏI-RR::CB: BF:: radius: cofine ang. CBF.

PROPOSITION IV.

To explain the Phænomena of the Rain-bow.

493. Having premised fuch mathematical principles as are necessary for an exact computation of the apparent diameters and breadths of the Rain-bows, I will here fubjoin Sir Ifaac Newton's entire explication of the colours of the bows and of the manner in which they are formed; taking the liberty here and there of making a few additions to it; for the fake of fuch readers as may not be so skilful as those that he generally

writes to.