guelin, instead of being an objection againft this theory, afford an argument in its favour..... ton, would have made him expect differences quite anomalous in the difperfive powers of different transparent bodies; at the same time that they would have afforded to his fagacious mind the strongeft arguments for the actual emiffion of 263. We are quite unacquainted with the law of action of bodies on light, that is, with the variation of the 34tenfity of the attractions and repulfions exerted at different diftances. All that we can fay is, that from the experiments and obfervations of Grimaldi, Newton, and others, light is deflected towards a body, or is attracted by it, at fome diftances, and repelled at others, and this with a variable intenfity. The action may be extremely different, both in extent and force, in different bodies, and change by a very different law with the fame change of diftance. amidst all this variety, there is a certain fimilarity arifing from the joint action of many particles, which fhould be noticed, because it tends both to explain the fimilarity obferved in the refractions of light, and alfo its connection with the phenomena of reflection. But, 260. VI. Thofe philofophers who maintain the theory of undulation, are under the neceflity of connecting the difperfive, powers ot bodies with their mean refractive powers. Mr. EULER has at-light from the luminous body. tempted to deduce a neceffary difference in the 262. Having thus eftablished the obferved law velocity of the rays of different colours from the of refraction on mechanical principles, fhowing it different frequency of the undulations, which he to be a neceffary confequence of the known acaffigns as the caufe of their different colorific tion of bodies on light, we proceed to trace its powers. His reafoning on this fubject is of the mathematical confequences through the various moft delicate nature, but unintelligible to fuch as cafes in which it may be exhibited to our obferare not completely masters of the infinitefimal cal- . vation. These constitute that part of the matheculus of partial differences, and unfatisfactory.tomatical branch of optical science which is called fuch as áre able to go through its intricacies. It DIOPTRICS. is contradicted by fact. Confident, however, of his analyfis, he gave a deaf ear to all that was told him of Mr DOLLOND'S improvements on telescopes, and afferted, that they could not be such as were related; for an increase of mean refraction must always be accompanied with a determined increase of difperfion. Newton had faid the fame thing, being misled by a limited view of his own principles; but the difperfion affigned by him was different from that affigned by Euler. The difpute between Euler and Dollond was confined to the decifion of this question only; and when some glaffes made by a German chemift at St Petersburgh convinced Euler that his determination was erroneous, he had not the candour to give up the principle which had forced him to this determination of the difperfion, but immediately introduced a new theory of the achromatic telescopes of Dollond: a theory which took the artifts out of the tract marked out by mathematicians, and in which they had made confiderable advances, and led them into another path, propofing maxims of conftruction hitherto untried, and inconfiftent with real improvements which they had already made. The leading principle in this theory is to arrange the different ultimate images of a point, which arife either from the errors of a spherical figure or different refrangibility, in a ftraight line paffing through the centre of the eye. The theory itfeif is fpecious, but false; and indeed, by admitting any difperfive power, whatever may be the mean refraction, all the phyfical doctrines in his Nova Theoria Lucis et Colorum are overlooked, and therefore never once mentioned. although the effects of Mr Zeiher's glafs are taken notice of as inconfiftent with that mechanical propofition of Newton's which occafioned the whole dispute between Euler and Dollond. 261. The late difcoveries in chemistry afford very diftinct proofs, that light is not exempted from the laws of chemical action, and that it is fufceptible of chemical combination. The changes produced by the fun's light on vegetable colours fhow the neceffity of illumination to produce the green fecula; and the aromatic oils of plants, the irritability of their leaves by the action of light, the curious effects of it on the mineral acids, on manganese, and the calces of bifmuth and lead, and the imbibition and fubfequent emiffion of it by phofphorefcent bodies, are ftrong proofs of its chemical affinities, quite inexplicable on the theory of undulations. All these confiderations, had they been known to Sir Ifaac New 264. The law of variation in the joint action of many particles adjoining to the furface of a refracting medium, is extremely different from that of a single particle; but when this last is known, the other may be found out. We fhall illuftrate this by a very fimple cafe. Let DE (fig 9. pl. CCLI.) be the surface of a medium, and suppose that the action of a particle of the medium on a particle of light extends to the diftance EA, and that it is proportional to the ordinates ED, Ff, Gg, Hb, &c. of the line Ab CgfD; that is, that the action of the particle E of the medium on a particle of light in F, is to its action on a particle in Has Ff to Hh, and that it is attracted at F, but repelled at H, as expreffed by the fituation of the ordinates with refpt to the abfciffa. In the line AE produced to B, make EB, Ex, Ex, Ey, Ep, &c. refpectively equal to EA, EH, EC, EG, EF, &c. It is evident that a particle of the medium at B will exert no action on the particle of light in E, and that the particles of the medium in xy E, will exert on it actions proportional to Hb, Gg, Ff, ED. Therefore, fuppofing the matter of the medium continuous, the whole action exerted by the row of particles EB will be represented by the area AhCDE; and the action of the particles between B and will be reprefented by the arca AhCƒF, and that of the particles between E and by the area FƒDE. Now let the particle of light be in F, and take Fo=AE. It is no less evident that the particle of light in F will be acted on by the particles in Eo alone, and that it will be acted on in the fame manner as a particle in E is acted on by the particles in B. Therefore the action of the whole row of particles EB on a par ticle in F will be represented by the area AhCƒF. And thus the action on a particle of light in any · point of AE will be represented by the area which lies beyond it. 265. But let us fuppofe the particles of light to be within the medium, as at , and make ed AE. It is again evident, that it is acted on by the particles of the medium between and d with a force reprefented by the area AhCDE, and in the opposite direction by the particles in Ep with a force represented by the area Ff DE. This balances an equal quantity of action, and there remains an action expreffed by the area A¿CƒF. Therefore, if an equal and fimilar line to AħCDE be defcribed on the abfciffa EB, the action of the medium on a particle of light in will be reprefented by the area ofuh B, lying beyond it. If we now draw a line AKLMRNPB, whofe ordinates CK, FQ, øR, &c. are, as the areas of the other curve, estimated from A and B; these ordinates will represent the whole forces which are exerted by the particles in EB, on a particle of light moving from A to B. This curve will cut the axis in points L, N, fo, that the ordinates drawn through them intercept areas of the first curve, which are equal on each fide of the axis; and in these points the particle of light sustains no action from the medium.. These points are very different from the fimilar points of the curve expreffing the action of a fingle particle. These laft are in the very places where the light fuftains the greatest repulfive action of the whole row of par. ticles. In the fame manner may a curve be conftructed, whofe ordinates exprefs the united action of the whole medium. From these observations we learn, in general, that a particle of light within the space of action is acted on with equal forces, and in the fame direction, when at equal diftances on each fide of the furface of the me dium. II. Of the FOCAL DISTANCE of RAYS REFRAC TED, by paffing out of one MEDIUM into another of DIFFERENT DENSITY, and through a PLANE SURFACE. \tions is to the greatest or least of them as the difference of the tangents to the greatest or leaft tangent..... 268, PROBLEM. Let two rays RV, RP diverge from, or converge to, a point R (Plate CCLII. figs. 1, 2, 3, 4.) and pafs through the plane furface PV, feparating two refracting mediums AB, of which let B be the most refracting, and let RV be perpendicular to the furface. It is required to determine the point of difperfion or convergence F, of the refracted rays VD, PE. Make VR to VG as the fine of refraction to the fine of incidence, and draw GIK parallel to the surface, cutting the incident ray in I. About the centre P, with the radius PI, describe an arch of a circle IF, cutting VR in F; draw PE tending from or towards F. We fay PE is the refracted ray, and F the point of difperfion or convergence of the rays RV, RP, or the conjugate focus to R. For fince GI and PV are parallel, and PF equal to PI, we have PF: PR-PI: PR,=VG : VR,=fin. incid. : fin. refr. But PF: PR fin. PRV: fin. PFV, and RRV is equal to the angle of incidence at P; therefore PFV is the correfponding angle of refraction, FPE is the refracted ray, and F the conjugate focus to R. 269. Corol. 1. If diverging or converging rays fall on the furface of a more refracting medium, they will diverge or converge lefs after refraction, F being farther from the furface than R. The contrary muft happen when the diverging or converging rays fall on the furface of a lefs refracting medium; because, in this cafe, F is nearer to the furface than R. 270. Corol. 2. Let Rp be another ray, more oblique than RP, the refracting point p being fat. ther from V, and let fpe be the refracted ray, determined by the fame conftruction. Because the arches FI, fi, are perpendicular to their radii, it is evident that they will converge to fome point within the angle RÍK, and therefore will not cross each other between F and I : therefore Rf will be greater than RF, as RF is greater than RG, for fimilar reasons. Hence it follows, that all the rays which tended from or towards R, and were from or converge to F, but will be diffused over incident on the whole of VPp, will not diverge the line GFf. This diffufion is called aberration from the focus, and is fo much greater as the rays are more oblique. No rays flowing from or tonearer to R than F is; but if the obliquity be inwards R will have point of concourse with RV confiderable, so that the ratio of RP to FP does of concourfe will not be fenfibly removed from G. not differ fenfibly from that of RV to FV, the point G is therefore ufually called the conjugate focus to R. It is the conjugate focus of an indefinitely flender pencil of rays falling perpendicularly on the furface. The conjugate focus of an oblique pencil, or even of two oblique rays, whofe difperfion on the surface is considerable, is of more difficult investigation. See Gravefande's Natural Philofophy for a very neat and elementary deterBV mination. 266. LEMMA. The indefinitely small variation of the angle of incidence is to the fimultaneous variation of the angle of refraction as the tangent of incidence is to the tangent of refraction; or, the cotemporaneous variations of the angles of incidence and refraction are proportional to the tangents of these angles. Let RVF, rVf(pl. CCLI. fig. 10.) be the progrefs of the rays refracted at V (the angle rXR being confidered as in its nafcent or evanefcent ftate), and VC perpendicular to the refracting furface VA. From C draw CD; CB, perpendicular to the incident and refracted rays RV, VF, cutting rV, Vfin♪ and B, and let Ca, Cb be perpendicular to rV, Vf. Because the ratio, their fimultaneous variations are in the fame conftant ratio. Now the angle RVr is to the anB3 ᎠᎴ BC gle EVƒin the ratio of BV DV DC fines of incidence and refraction are in a conftant to Dy; that is, of to ; that is, of .fin. incid. to fin. refr. cof. refr.; that is, of tan. incid to tan refr. 271. As optical ABERRATION is the chief caufe of the imperfection of optical inftruments, and as the only method of removing this imperfection is to diminish this aberration, or correct it by a sub fequent fequent aberration in the oppofite direction, we fubjoin a fundamental and very fimple propofition, which will apply to all important cafes. This is the determination, of the focus of an infinitely flender pencil of oblique rays RP, Rp. "Retaining the former conftruction for the ray PF, (fig. 1.) fuppofe the other ray Rp infinitely near to RP. Draw PS perpendicular to PV, and Rr perpendicular to RP, and make Pr: PSVR VF. On Pr defcribe the femicircle rRP, and on PS the femicircle SP, cutting the refracted ray PF in e, draw pr, pS, pe." It follows from the lemma that if be the focus of refracted rays, the varia tion Pep of the angle of refraction is to the correfponding variation PRp of the angle of incidence as the tangent of the angle of refraction VFP to the tangent of the angle of incidence VRP. Now Pp may be confidered as coinciding with the arch of the femicircles. Therefore the angles PRp, Prp are equal, as also the angles Pop, PSp. But PSP is to Prp as Pr to PS; that is, as VR to VE; that is, as the cotangent of the angle of incidence to the cotangent of the angle of refraction; that is, as the tangent of the angle of refraction to the tangent of the angle of incidence. Therefore the point is the focus. III. Of REFRACTION by SPHERICAL SURFACES. 272. GENERAL PROBLEM. To find the focus of refracted rays, the focus of incident rays being given? Let PV (pl. CCLII, from fig. 5. to 14.) be a spherical furface whofe centre is C, and let the incident light diverge from or converge to R. 273. Solution. Draw the ray RC through the centre, cutting the furface in the point V, which we denominate the vertex, while RC is called the axis. This ray paffes on without refraction, becaufe it coincides with the perpendicular to the furface. Let RP be another incident ray which is refracted at P, draw the radius PC. In RP make RE to RP as the fine of incidence m to the all the rays which flow from it are made to di- 275. II. If the incident ray R'P (fig. 5.) is pa mn m m-n ༈ ན་ 276. III. In this cafe, too, we have the focal diftance of central parallel rays reckoned from m XVC. For fince PO is ultithe vertex=; mately VO, we have m: n=VO: CO, and m-n : m=VO-CO: VO,=VC : CO, and VO= XVC. This is called the principal focal fine of refraction; and about the centre R, with diftance of parallel rays. Also CO, the principal the diftance RE, defcribe the circle EK, cutting PC in K: draw RK and PF parallel to it, cutting focal distance reckoned from the centre, = m-n the axis in F. PF is the refracted ray, and F is the focus; for the triangles PCF, KCR are fimi- XVC.-N. B. When m is lefs than n, m-n is a lar, and the angles at P and K are equal. Allo negative quantity. In applying fymbols to this RK is equal to RE, and RPD is the angle of in computation of the focal diftances, thofe lines are cidence. Now m: n=RK: RP, fin. DPR: fin. to be, accounted pofitive which lie from their beRKP,=fin. DPR : fin. CPF. Therefore CPF is ginnings, that is, from the vertex, or the centre, or the angle of refraction correfponding to the angle the radiant point, in the direction of the incident Thus when rays diverge from R on the rays. ray, convex furface of a medium, VR is accounted negative and VC pofitive. If the light paffes out of air into glafs, m is greater than n; but if it paffes out of glafs into air, is less than 7. If, therefore, parallel rays fall on the convex furface of glafs out of air, in which cafe m; n=3: 2 very nearly, we have for the principal focal distance 3_VC, or + 3VC. But if it pafs out of glass 3 -2 into the convex furface of air, we have VO 2 -VC, or -2VC; that is, the focus O will be in the fame fide of the furface with the incident light. In like manner, we fhall have for thefe two cafes CO=+2VC and -3VC. of incidence RPD, and PF is the refracted and F the focus. Q. E. D.. ск 274. Cor. I. CK: CP=CR: CF, and CF CPX CR Now CPX CR is a conftant quantity; and therefore CF is reciprocally as CK, which evidently varies with a variation of the arch VP. Hence it follows, that all the rays flowing from R are not collected at the conjugate focus F. The ultimate fituation of the point F, as the point P gradually approaches to, and at laft coincides with, V, is called the conjugate focus of central rays; and the diftance between this focus and the focus of a lateral ray is called the aberration of that ray, 2rifing from the fpherical figure. There are, however, two fituations of the point R fuch, that VOL. XVI PART II. 23 A a a 277. 1 |