DIOPTRIC[50]SYSTEM OF LIGHTS.

DIOPTRIC[50]SYSTEM OF LIGHTS.[50]Most probably directly derived from the Greek διόπτρον, an optical instrument with holes for looking through, whose name is a compound of διὰ, through, and ὄπτομαι,I see.

[50]Most probably directly derived from the Greek διόπτρον, an optical instrument with holes for looking through, whose name is a compound of διὰ, through, and ὄπτομαι,I see.

One of the earliest notices of the application of lenses to lighthouses is that recorded bySmeatonin his Narrative of the Eddystone Lighthouse, where he mentions a London optician, who, in 1759, proposed grinding the glass of the lantern to a radius of seven feet six inches; but the description is too vague to admit of even a conjecture regarding the proposed arrangement of the apparatus. About the middle of the last century, however, lenses were actually tried in several lighthouses in the south of England, and in particular at the South Foreland in the year 1752; but their imperfect figure and the quantity of light absorbed by the glass, which was of impure quality and of considerable thickness, rendered their effect so much inferior to that of the parabolic reflectors then in use, that after trying some strange combinations of lenses and reflectors, the former were finally abandoned. Lenses were also tried at the lightsof Portland, Hill of Howth, and Waterford, by Mr Thomas Rogers, a glass manufacturer in London; who possessed, it is said, the art of blowing mirrors of glass, “and by a new method silvered over the convex side without quicksilver.”[51]

[51]Hutchinson’s Practical Seamanship, p. 200. See also thenoticeof the spherical mirrors made by Messrs François and Letourneau of Paris in a subsequent part of this volume.

The object to be attained by the use of lenses in a Lighthouse is, of course, identical with that which is answered by employing reflectors; and both instruments effect the same end by different means, collecting the rays which diverge from a point called thefocus, and projecting them forward in a beam, whose axis coincides with the produced axis of the instrument. We have already seen that, in the case ofreflection, this result is produced by the light beingthrown backfrom a surface so formed as to make all the rays to proceed in one and the same required direction. In the case ofrefraction, on the other hand, the rays pass through the refracting medium, and arebentorrefractedfrom their natural course into that which is desired.

The celebratedBuffon, to prevent the great absorption of light by the thickness of the material, which would necessarily result from giving to a lens of great dimensions a figure continuously spherical, proposed to grind out of a solid piece of glass, a lens in steps or concentric zones. This suggestion ofBuffonregarding the construction of large burning glasses, was first executed, with tolerable success, about the year 1780, by the AbbéRochon; but such are the difficulties attending the process of working a solid piece of glass into the necessary form, that it is believed the only other instrument ever constructed in this manner, is that which was made by MessrsCooksonof Newcastle-upon-Tyne, for the Commissioners of Northern Lighthouses.

The merit of having first suggested the building of lenses in separate pieces, seems to be due toCondorcet, who, in hisEloge de Buffon, published so far back as 1773, enumerates the advantages to be derived from this method. SirDavid Brewsteralso describedthis mode of building lenses in 1811, in theEdinburgh Encyclopædia; and in 1822, the late eminentFresnel, unacquainted with the suggestions ofCondorcetor the description by SirDavid Brewster, explained, with many ingenious and interesting details, the same mode of constructing those instruments. ToFresnelbelongs the additional merit of having first followed up his invention, by the construction of a lens and, in conjunction with MM.AragoandMathieu, of placing a powerful lamp in its focus, and indeed of finally applying it to the practical purposes of a Lighthouse.

The great advantages which attend the mode of construction proposed byCondorcetare,—the ease of execution, by which a more perfect figure may be given to each zone and spherical aberration in a great measure corrected, and the power of forming a lens of larger dimensions than could easily be made from a solid piece. BothBuffonandCondorcet, however, chiefly speak of reducing the thickness of the material, and do not seem to have thought of determining the radius and centre of the curvature of the generating arcs of each zone, having contented themselves with simply depressing the spherical surface in separate portions.Fresnel, on the other hand, determined those centres, which constantly recede from the vertex of the lens in proportion as the zones to which they refer are removed from its centre; and the surfaces of the zones of the annular lens, consequently, are not parts of concentric spheres, as inBuffon’slens. It deserves notice, that the first lenses constructed forFresnelby M.Soleilhad their zones polygonal, so that the surfaces were not annular, a form whichFresnelconsidered less accommodated to the ordinary resources of the optician. He also, with his habitual penetration, preferred the plano-convex to the double-convex form, as more easily executed.[52]After mature consideration, he finally adopted crown glass, which, notwithstanding its greenish colour, he preferred to flint glass, as being more free fromstriæ. All his calculations were made in reference to an index of refraction of 1·51, which he had verified by repeated experiments, conducted with that patience and accuracy for which, amidst his higher qualities, he was so remarkably distinguished.[53]The instruments have received the name ofannularlenses, from the figure of the surface of the zones.

[52]The plano-convex lens, with its curved side towards the parallel rays, is also a form producing small spherical aberration, a circumstance which may also have influenced his choice.[53]My friend, MrWilliam Swan, carefully examined, by his new and ingenious method, described in the Edinburgh New Philosophical Journal, January 1844, several specimens of the St Gobain glass (which is now used in the manufacture of the lenses), and found its refractive index to be 1·51793, thedifferencebetween the greatest and least values being only 0·00109.

[52]The plano-convex lens, with its curved side towards the parallel rays, is also a form producing small spherical aberration, a circumstance which may also have influenced his choice.

[53]My friend, MrWilliam Swan, carefully examined, by his new and ingenious method, described in the Edinburgh New Philosophical Journal, January 1844, several specimens of the St Gobain glass (which is now used in the manufacture of the lenses), and found its refractive index to be 1·51793, thedifferencebetween the greatest and least values being only 0·00109.

A ray of light, in passingobliquelyfrom one transparent body into another of different density, experiences at the point of the intersection of the common surface of the two planes, a sudden change of direction, to which the name ofRefraction.refractionhas naturally been given, in connection with the most familiar instance of the phenomenon, which is exhibited by a straight ruler with one half plunged into a basin of water while the other remains in the air. The ruler no longer appears straight, but seems to bebentorbrokenat the point where it enters the water. It may not be out of place to call attention to the laws which regulate the change of direction in the incident light, which arethreein number.

1. Incidence and refraction, in uncrystallized media of homogeneous structure such as glass, always occur in a plane perpendicular to that of the refracting surface.

2. In the same substances, the angle formed with the perpendicular by the ray at its entering the surface of the second medium, has to the angle which it makes with the normal after it has entered the surface, such a relation, that their sines have a fixed ratio, which is called therefractive index. When a ray falls normally on the surface of any substance, it suffers no refraction.

3. The effect of passing from a rare to a dense medium, as from air into water or glass, is to make the angle ofrefractionless than the angle ofincidence; and those angles are measured with referenceto a normal to the plane which separates the media at the point of incidence. The converse phenomenon, of course, takes place in the passage from a dense to a rare medium, in which case the angle ofincidenceis less than the angle ofrefraction. To this rule there are a few exceptions; for there are certain combustible bodies, such as diamond, whose refractive powers are much greater than other substances of equal density.

Fig. 47.Refraction explained

Fig. 47.

The diagram (fig. 47) will serve to render those laws more intelligible. Let a ray of lightaO meet a surface of watern mat O, it will be immediately bent into the direction Oa′; and if, from the centre O, we describe any circle, and draw a linebOb′, perpendicular tonm; thenabanda′ b′, perpendiculars drawn to the normalbb′, from the pointsaanda′where the circle cuts the incident and refracted rays, will be the sines of the angle of incidencebOa, and of the angle of refractionb′Oa′, and the ratio of those sines to each other, orb ab′ a′will be the relative index ofrefractionfor the two media.

4. It may perhaps be added, for convenience, as afourthlaw, deducible from the others, that since rays passing from a dense into a rare medium, have their angle of refraction greater than the angle of incidence, there must be some angle of incidence whose corresponding angle of refraction is a right angle; beyond which no refraction can take place, because there is no angle whose sine can be greater than the radius. In such circumstances,total reflectionensues. For common glass, whose index of refraction is 1·5, we have (in the case of emergent rays) sine of incidence =sine of refraction1·5; but, as no sine can exceed radius or unity, the angle of incidence must be limited to 41° 49′; beyond which total reflection will take place, and the light will returninwardsinto the glass, beingreflectedat its surface.

Thus, if a ray proceed from a point O (fig. 48), within a piece of glass, to a point C, at its surface A B; and if O Cb, its incidence, be less than 41° 49′, it will berefractedin some direction Cf; but if this angle be greater than 41° 49′, as O C′b′, the ray will bereflectedback into the glass in the direction C′ O′.

Fig. 48.Refraction and reflection

Fig. 48.

The material hitherto employed in the construction of lighthouse apparatus is crown glass, which, although it possesses a lower refractive power than flint glass and has, besides, a slightly greenish tinge, offers the great practical advantages of being more easily obtained of homogeneous quality; and, being less subject to deterioration from atmospheric influences, it is peculiarly suitable for use in the exposed situations generally occupied by Lighthouses. The refractive index of crown glass, as already noticed, is about 1·5.

Any one may easily satisfy himself by a careful protraction of the angles ofincidenceandrefraction, in the manner above described, as to the truth of the following general propositions resulting from thoselaws:—

1. A ray of light passing through a plate of some diaphanous substance such as glass, with parallel surfaces, suffers no change ofdirection, but emerges in a line parallel to its original path, merely suffering adisplacement, depending on the obliquity of the incident ray, and the refractive power and thickness of the plate. The effect of this displacement is merely to give the ray an apparent point of origin different from the true one. This will be easily understood by the diagram (fig. 49), in whicha bis a normal to the plate, whose surfacesx xandx′ x′are parallel,r r r rshews the path of the ray,r rthe displacement, andr′the apparent point of origin resulting from its altereddirection.

Fig. 49.Refraction and displacement

Fig. 49.

2. When a ray passes through a triangular prisma b c, the inclination of the facesa candc bcauses the emergent rayr′to be bent towardsa b, the base of the prism, in a measure depending on the inclination of the sides of the prism and the obliquity of the incident ray to the first surface.

Fig. 50.Prism

Fig. 50.

3. When parallel rays fall on a concave lens, they will, at their emergence, be divergent. The section of the diaphanous bodya b c dmay be regarded as composed of innumerable frusta of prisms, having their apices directed towards the centre linex r; and the rays which pass through the centre, being normal to the surface, will be unchanged in their direction, while all the others will (as shewn in thefigure) suffer a change of direction, increasing with their distance from the centre, owing to the increasing inclination of the surfaces of the lens as they recede from its axis.

Fig. 51.Concave lens

Fig. 51.

4. Lastly, when divergent rays fall on a convex lensa b, from a pointf, called the principal focus, they are made parallel at their emergence; while,conversely, parallel rays which fall on the lens are united in that point.[54]This effect, which is the opposite of that caused by the concave lens, may be explained in a similar manner, by conceiving the sectiona bof the convex lens to be composed of innumerable frusta of prisms, arranged with theirbasestowards the centre of the lens.

[54]It is, of course, to be understood that only rays incident near the axis of the lens are refracted accurately to a focus.

Fig. 52.Convex lens

Fig. 52.

Now, it is obvious, that we can derive no assistance, in economising the rays of a lamp for Lighthouse purposes, from concave lenses, whose property is to increase the dispersion of the rays incident on them. With concave lenses, therefore, we have no concern; and we shall confine ourselves to the consideration of the convex or converging lenses.

The lens always used in Lighthouses is (for reasons already noticed) plano-convex, and differs from the last only by having a plane and a curve surface, instead of two curve surfaces, whose radii are on opposite sides of the lens. The plano-convex is generally regarded, by writers on optics, as acaseof the double convex having one side of aninfiniteradius. Both forms cause parallel rays to converge to a focus.

We commence with a general view of the relations which exist between the position of theradiantand the focus.

Fig. 53.Plano-convex lens

Fig. 53.

Let Qqbe a section of a lens, andfArits optical axis, or the line in which a ray of light passes unchanged in its direction through the lens, from its being normal to both surfaces, whether the lens be double-convex as above, or plano-convex (seefig. 53), then theprincipal focus fis that point where the rays fromr r r, which fall parallel to the optic axis on the outer face of the lens, meet after refraction at the two faces,—or, to speak more in the language of the art which is under consideration, theprincipal focus fis the point whence the rays of light, proceeding in their naturally divergent course, fall on the inner surface Q Aqof the lens, and are so changed by refraction there and at the outer face, that they finally emerge parallel to theoptic axisin the directions Qr,q r. The position of this point depends partly on the refractive power of the substance of which the lens is composed and partly on the curvature of the surface or surfaces which bound it.

It would be quite beyond the scope of these Notes to attempt to present the subject of refraction at spherical surfaces before the reader’s view in a rigorous or systematic manner, and thus to advance, step by step, to the practical application of refracting instruments, as a means of directing and economising the light in a Pharos. This would involve the repetition, in a less elegant form, of what is to be found in all the works on optics; and instead of this, I am content to refer, where needful, to those works, and shall confine myself simply to what concerns Lighthouse lenses and their use. It would also be superfluous to determine the position of the principal focus of a plano-convex lens, in terms of the refractive index and radius of curvature,[55]as it can be very accurately found in practice by exposing the instrument to the sun, in such a manner that his rays may fall upon it in a direction parallel to its axis. The point of union between the converging and diverging cones of rays (where the spectrum is smallest and brightest), which is theprincipal focus, is easily found by moving a screen behind the lens, farther from or nearer to it as may be required. The path of the Lighthouse optician, moreover, generally lies in the opposite direction; and his duty is not so much to find the focal distance of a ready-made lens, as to find the best form of a lens for the various circumstances of a particular Pharos, whose diameter, in some measure, determines the focal distance of the instruments to be employed. All, however, that I shall really have to do is to give an account of what has been done by the late illustriousFresnel, who seems to have devoted such minute attention to every detailof the Dioptric apparatus, that he has foreseen and provided for every case that occurs in the practice of Lighthouse illumination. His brother, Mons.Leonor Fresnel, who succeeded him in the charge of the Lighthouses of France, has, with the greatest liberality, put me in possession of the various formulæ used by his lamented predecessor, in determining the elements of those instruments which have so greatly improved the lighthouses of modern days.

[55]F =rm- 1in whichris the radius of curvature, andmis the refractive index.—Coddington’s Optics, Chap. VIII. If the radiant be brought near the lens, so as to cast divergent rays on its surface, then the conjugate focus will recede behind theprincipal focus; and when the luminous body reaches theprincipalfocusin frontof the lens, the rays will emerge from its posterior surface in a direction parallel to its axis. If it be brought still nearer the lens, the rays would emerge as a divergent cone. Hence converging lenses can only collect rays into a focus, when they proceed from some pointmoredistant than the principal focus.

Spherical lenses, like spherical mirrors, collect truly into the focus those rays only which are incident near the axis; and it is, therefore, of the greatest importance to employ only a small segment of any sphere as a lens. The experience of this fact, among other considerations, ledCondorcet, as already noticed, to suggest the building of lenses in separate pieces.Fresnel, however, was the first who actually constructed a lens on that principle; and he has subdivided, with such judgment, the surface of the lens into a centre lens and concentric annular bands and has so carefully determined the elements of curvature for each, that no farther improvement is likely to be made in their construction. For the drawings of the great lens, I have to refer toPlate XII., which also contains a tabular view of the elements of its various parts. The central disc of the lens, which is employed in lights of the first order, and whose focal distance is 920 millimètres, or 36·22 inches, is about 11 inches in diameter; and the annular rings which surround it vary slightly in breadth from 2³⁄₄ to 1¹⁄₄ inches. The breadth of any zone or ring is, within certain limits, a matter of choice, it being desirable, however, that no part of the lens should be much thicker than the rest, as well for the purpose of avoiding inconvenient projections on its surface, as to permit the rays to pass through the whole of the lens with nearly equal loss by absorption. The objects to be attained in the polyzonal or compound lens, are chiefly, as above noticed, to correct the excessive aberration produced by refraction through a hemisphere or great segment, whose edge would make the parallel rays falling on its curve surface converge to a point much nearer the lens than the principal focus, as determined for rays near theoptical axis, and to avoid the increase of material, which would not only add to the weight of the instrument and the expense of its construction, but would greatly diminish by absorption the amount of transmitted light. Various modes of removing similar inconveniences in telescopic lenses have been devised; and the suggestions ofDescartes, as to combinations of hyperbolic and elliptic surfaces with plane and spherical ones, more especially fulfil the whole conditions of the case; but the excessive difficulty which must attend grinding and polishing those surfaces, has hitherto deprived us of the advantages which would result from the use of telescopic lenses entirely free from spherical aberration. In Lighthouse lenses, where so near an approach to accurate convergence to a single focus is unnecessary, every purpose is answered by the partial correction of aberration which may be obtained, by determining an average radius of curvature for the central disc, and for each successive belt or ring, as you recede from the vertex of the lens. In the lenses originally constructed forFresnelbySoleil, the zones were united by means of smalldowelsorjogglesof copper, passing from the one zone into the other; but the greater exactness of the workmanship now attained, has rendered it safe to dispense with those fixtures; and the compound lens is now held together solely by a metallic frame and the close union between the concentric faces of the rings, which, however, are in contact with each other at surfaces of only ¹⁄₄ inch in depth, as shewn inPlate XII.It is remarkable, that an instrument, having about 1300 square inches of surface, and weighing 109 lb., and which is composed of so many parts, should be held together by so slender a bond as two narrow strips of polished glass, united by a thin film of cement.

I now proceed to the formulæ employed byFresnel, to determine the elements of the compound lens,[56]in the calculation ofwhich two cases occur, viz., the central disc and a concentric ring. The focal distance of the lens and the refractive index of the glass are the principal data from which we start.

[56]It may be proper to mention that, while the formulæ given in the text are those of M.Fresnel, I am responsible for the investigations in the Notes; I have, at the same time, much pleasure in acknowledging my obligations, at various times (about ten years ago), to MrEdward Sang, and (more recently) to MrWilliam Swan, for their kind advice on this part of the subject.

Fig. 54.Central disc

Fig. 54.

I begin with the case of the central disc or lens round which the annular rings are arranged. Its principal section is a mixtilinear figure (fig. 54) composed of a segmentbac, resting on a parallelogrambcde, whose depthbdorceis determined by the strength which is required for the joints which unite the various portions of the lens. Those particulars have, as I already stated, been determined with so much judgment byFresneland the dimensions of the lenses so varied to suit the case of various lights, that nothing in this respect remains to be done by others.

Fig. 55.Refraction by central disc

Fig. 55.

Referring tofig. 55, we have, for obtaining the radius of the central disc, the following formulæ, in which

r= AB, half the aperture of the lensr′= AB′φ = AF, the focal distancet′= Aa, the thickness of the lens at the vertext″= Bb, the thickness of the jointμ = the index of refractionρ = the radius of curvature.

r= AB, half the aperture of the lens

r′= AB′

φ = AF, the focal distance

t′= Aa, the thickness of the lens at the vertex

t″= Bb, the thickness of the joint

μ = the index of refraction

ρ = the radius of curvature.

Then for the radius of curvature near the axis we have:

ρ′ = (μ - 1)(φ +t′μ)

and for that near the margin we have:

tani′=rφsine=sini′μr′=r-t″. tanetani=r′φsin ε =siniμ

ρ″ =rμ sine√μ² - 2 μ cose+ 1and, finally ρ =ρ′ + ρ″2[57]

ρ″ =rμ sine√μ² - 2 μ cose+ 1

and, finally ρ =ρ′ + ρ″2[57]

[57]The following steps lead to the formulæ given in the text. Let APQB (fig. 56) represent a section of the central lens by a plane passing through its axis AF; F the focus for incident rays; and FQPH the path of a ray refracted finally in the direction PH, parallel to the axis. Let C be the centre of curvature, then PC is a normal to the curve at P; and, producing PQ to meet the axis in G, we have G the focus of the rays, after refraction at the surface BQ.Fig. 56.Refraction of central lensThen μ =sin PCGsin GPC=PGCG; and also μ =sin QFGsin QGF=QGQFNow, as P approaches A, we have ultimately PG = AG, QG = BG, and QF = BF;Therefore, putting AG = θ and AC = ρ′μ =AGCG=θθ - ρ′; μ =BGBF=θ -t′φ,from which μ θ - μ ρ′ = θ; and μ φ = θ -t′and eliminating θ, we have μ² φ + μ ρ′ = μ φ +t′, whence, as above, ρ′ = (μ - 1)(φ +t′μ.)But as this value of the radius of curvature, as already stated, is calculated for rays near the axis, it would produce a notable aberration for rays incident on the margin of the lens. In order, therefore, to avoid the effects of aberration as much as possible, a second radius of curvature must be calculated, so that rays incident on the margin of the lens may be refracted in a direction parallel to the axis. This second value of the radius is called ρ″ in the text, and is found as follows (referring tofig. 57):Let FB′bxbe the course of a ray refracted in the directionbxparallel to the axis Ax′. This ray meets the surface AB in the point B′, whose position may be found approximately by tracing the path of the ray FB, on the supposition that the surface of the refracting medium is produced in the directions AB,a′b′.Fig. 57.Refraction in central lensLet C be the centre of curvature (seefig. 57)α = ACbthe angle of emergenceη = B′bC the second angle of refractionε = B b B′ the first angle of refractioni= B′FA the first angle of incidencei′= BFAe=b′BbAB =rAB′ =r′Bb=t″the thickness of the lens at the edgeAF = φ the focal distance.Then tani′=rφ; sine=sini′μwhencebb′=t″tanebecomes known.Now, since BB′ =bb′nearly, AB′ = AB -bb′orr′=r-t″tane.From this is obtained the angle of incidencei, and the first angle of refraction ε; for tani=r′φand sin ε =siniμ.Next B′bC = BbC - BbB′ or η = α - εandsin α = μ sin η = μ sin (α - ε)from which, sin α cos ε - cos α sin ε =sin αμwhence sin α(cos ε -1μ)= cos α sin ε; andsin² α(cos² ε -2 cos εμ+1μ²)= cos² α sin² ε = (1 - sin² α) sin² ε = sin² ε - sin² α sin² εThen transposing we havesin² α{(cos² ε + sin² ε) -2 cos εμ+1μ²}= sin² εand because (cos² ε + sin² ε) = 1 we have, by dividing,sin² α =sin² ε{1 -2 cos εμ+1μ²}=μ² sin² εμ² - 2 μ cos ε + 1and sin α =μ sin ε√1 - 2 μ cos ε + μ²Next, sincebC =a′bsin ACb=rsin α, putting Cb=ρ″, and substituting we haveρ″=rμ sin ε√μ² - 2 μ cos ε + 1and, taking for the radius of curvature, the mean ofρ′andρ″the values calculated for the central and marginal rays, we have finally ρ =ρ′ + ρ″2

Fig. 56.Refraction of central lens

Fig. 56.

Then μ =sin PCGsin GPC=PGCG; and also μ =sin QFGsin QGF=QGQF

Now, as P approaches A, we have ultimately PG = AG, QG = BG, and QF = BF;

Therefore, putting AG = θ and AC = ρ′μ =AGCG=θθ - ρ′; μ =BGBF=θ -t′φ,

Therefore, putting AG = θ and AC = ρ′

μ =AGCG=θθ - ρ′; μ =BGBF=θ -t′φ,

from which μ θ - μ ρ′ = θ; and μ φ = θ -t′and eliminating θ, we have μ² φ + μ ρ′ = μ φ +t′, whence, as above, ρ′ = (μ - 1)(φ +t′μ.)

But as this value of the radius of curvature, as already stated, is calculated for rays near the axis, it would produce a notable aberration for rays incident on the margin of the lens. In order, therefore, to avoid the effects of aberration as much as possible, a second radius of curvature must be calculated, so that rays incident on the margin of the lens may be refracted in a direction parallel to the axis. This second value of the radius is called ρ″ in the text, and is found as follows (referring tofig. 57):

Let FB′bxbe the course of a ray refracted in the directionbxparallel to the axis Ax′. This ray meets the surface AB in the point B′, whose position may be found approximately by tracing the path of the ray FB, on the supposition that the surface of the refracting medium is produced in the directions AB,a′b′.

Fig. 57.Refraction in central lens

Fig. 57.

Let C be the centre of curvature (seefig. 57)

α = ACbthe angle of emergence

η = B′bC the second angle of refraction

ε = B b B′ the first angle of refraction

i= B′FA the first angle of incidence

i′= BFA

e=b′Bb

AB =r

AB′ =r′

Bb=t″the thickness of the lens at the edge

AF = φ the focal distance.

Then tani′=rφ; sine=sini′μwhencebb′=t″tanebecomes known.

Now, since BB′ =bb′nearly, AB′ = AB -bb′orr′=r-t″tane.

From this is obtained the angle of incidencei, and the first angle of refraction ε; for tani=r′φand sin ε =siniμ.

Then transposing we have

sin² α{(cos² ε + sin² ε) -2 cos εμ+1μ²}= sin² ε

and because (cos² ε + sin² ε) = 1 we have, by dividing,

sin² α =sin² ε{1 -2 cos εμ+1μ²}=μ² sin² εμ² - 2 μ cos ε + 1

and sin α =μ sin ε√1 - 2 μ cos ε + μ²

Next, sincebC =a′bsin ACb=rsin α, putting Cb=ρ″, and substituting we haveρ″=rμ sin ε√μ² - 2 μ cos ε + 1and, taking for the radius of curvature, the mean ofρ′andρ″the values calculated for the central and marginal rays, we have finally ρ =ρ′ + ρ″2

Fig. 58.Refraction of ring

Fig. 58.

I come next to thesecondcase, which concerns the calculation of the elements of a concentric ring. The sectionabcde(fig. 58) of one of those rings includes a mixtilinear triangleabe, and a rectanglebced, the thicknessbcbeing the same as that of the edge of the central disc; and the elements to be determined are the radius of the curve surface, and the position of the centre of curvature, with reference to the vertex of the lens.

Fig. 59.Refraction of ring

Fig. 59.

The radius of curvature of the zone may be calculated by the following formulæ, in which (seefig. 59)

r₁ = AB the distance of the outer margin of the zone from the axis of the lensr₂ = AE the distance of the inner margin from the axisl= BE the breadth of the zone =r₁ -r₂ρ = the radius of curvature =bC =mCφ = focal distance AFt= thickness of the joint Bbt″= Bbμ = refractive index of the glassi₁ = BFAi₂ = EFA

r₁ = AB the distance of the outer margin of the zone from the axis of the lens

r₂ = AE the distance of the inner margin from the axis

l= BE the breadth of the zone =r₁ -r₂

ρ = the radius of curvature =bC =mC

φ = focal distance AF

t= thickness of the joint Bb

t″= Bb

μ = refractive index of the glass

i₁ = BFA

i₂ = EFA

Then tani′₁ =r₁φ; tani′₂ =r₂φsine₁ =sini′₁μ; sine₂ =sini′₂μr′₁ =r₁ -t″sine₁;r′₂ =r₂ -t″sine₂tani₁ =r′₁φ; tani₂ =r′₂φsin ε =sini₁μ; sin ε′ =sini₂μsin α =μ sin ε√μ² - 2 μ cos ε + 1;sin α′ =μ sin ε′√μ² - 2 μ cos ε′ + 1; η = α′ - ε′and lastly ρ =2 cos ε′2 cos {η + ¹⁄₂(α - α′)} sin ¹⁄₂ (α - α′)

Then tani′₁ =r₁φ; tani′₂ =r₂φ

sine₁ =sini′₁μ; sine₂ =sini′₂μ

r′₁ =r₁ -t″sine₁;r′₂ =r₂ -t″sine₂

tani₁ =r′₁φ; tani₂ =r′₂φ

sin ε =sini₁μ; sin ε′ =sini₂μ

sin α =μ sin ε√μ² - 2 μ cos ε + 1;

sin α′ =μ sin ε′√μ² - 2 μ cos ε′ + 1; η = α′ - ε′

and lastly ρ =2 cos ε′2 cos {η + ¹⁄₂(α - α′)} sin ¹⁄₂ (α - α′)

which isFresnel’svalue of the radius of curvature.[58]

[58]The following steps will conduct us to this expression:Fig. 60.Refraction of ringLet BbfE (fig. 60) represent the section of a zone by a plane passing through the axis of the lens AF, C the centre of curvature, F the radiant point, and FB′bx, FE′mx′the course of the extreme rays which are transmitted through the zone (and the latter of which passes from E′ toethrough a portion of the zone or lens in contact with that under consideration). Then puttingAB =r₁; AB′ =r′₁; Cb= ρAE =r₂; AE′ =r′₂; Bb=t″; BE =r₁ -r₂ =lε = the first angle of refractionbB′kη = the second angle of refraction B′bCε′ = the first angle of refractioneEk′η′ = the second angle of refractionemCα = the angle of emergencebCqα′ = the angle of emergencemCqi′₁ = BFA;i′₂ = EFA;i₁ = B′FA;i₂ = E′FAe₁ = BbB′;e₂ = EeE′.Proceeding exactly as in the case of the central lens we shall havetani′₁ =BAAF=r₁φ; tani′₂ =EAAF=r₂φsine₁ =sini′₁μ; sine₂ =sini′₂μr′₁ =r₁ -t″sine₁;r′₂ =r₂ -t″sine₂tani₁ =r′₁φ; tani₂ =r′₂φsin ε =sini₁μ; sin ε′ =sini′₂μsin α =μ sin ε√μ² - 2 μ cos ε - 1; and sin α′ =μ sin ε′√μ² - 2 μ cos ε′ + 1Now, the anglebCm= α - α′ from which (since the trianglebmC is isosceles)bmC = 90° - ¹⁄₂ (α - α′); also, in the trianglebme, the anglebme=bmC -emC = 90° - ¹⁄₂ (α - α′) - η andbem=k′eE′ = 90° - ε′We have therefore in the trianglebmebm=besinbemsinbme=lcos ε′cos {η + ¹⁄₂ (α - α′)}and inbmCbC=bmsinbmCsinbCm=lcos ε′ cos ¹⁄₂ (α - α′)(cos (η + ¹⁄₂ (α - α′)) sin (α - α′)=lcos ε′ cos ¹⁄₂ (α - α′)cos {η + ¹⁄₂ (α - α′)} 2 sin ¹⁄₂ (α - α′) cos ¹⁄₂ (α - α′)from which, puttingbC = ρρ =lcos ε′2 cos {η + ¹⁄₂ (α - α′)} sin ¹⁄₂ (α - α′)

[58]The following steps will conduct us to this expression:

Fig. 60.Refraction of ring

Fig. 60.

Let BbfE (fig. 60) represent the section of a zone by a plane passing through the axis of the lens AF, C the centre of curvature, F the radiant point, and FB′bx, FE′mx′the course of the extreme rays which are transmitted through the zone (and the latter of which passes from E′ toethrough a portion of the zone or lens in contact with that under consideration). Then putting

AB =r₁; AB′ =r′₁; Cb= ρAE =r₂; AE′ =r′₂; Bb=t″; BE =r₁ -r₂ =lε = the first angle of refractionbB′kη = the second angle of refraction B′bCε′ = the first angle of refractioneEk′η′ = the second angle of refractionemCα = the angle of emergencebCqα′ = the angle of emergencemCqi′₁ = BFA;i′₂ = EFA;i₁ = B′FA;i₂ = E′FAe₁ = BbB′;e₂ = EeE′.

AB =r₁; AB′ =r′₁; Cb= ρ

AE =r₂; AE′ =r′₂; Bb=t″; BE =r₁ -r₂ =l

ε = the first angle of refractionbB′k

η = the second angle of refraction B′bC

ε′ = the first angle of refractioneEk′

η′ = the second angle of refractionemC

α = the angle of emergencebCq

α′ = the angle of emergencemCq

i′₁ = BFA;i′₂ = EFA;i₁ = B′FA;i₂ = E′FA

e₁ = BbB′;e₂ = EeE′.

Proceeding exactly as in the case of the central lens we shall have

tani′₁ =BAAF=r₁φ; tani′₂ =EAAF=r₂φsine₁ =sini′₁μ; sine₂ =sini′₂μr′₁ =r₁ -t″sine₁;r′₂ =r₂ -t″sine₂tani₁ =r′₁φ; tani₂ =r′₂φsin ε =sini₁μ; sin ε′ =sini′₂μsin α =μ sin ε√μ² - 2 μ cos ε - 1; and sin α′ =μ sin ε′√μ² - 2 μ cos ε′ + 1

tani′₁ =BAAF=r₁φ; tani′₂ =EAAF=r₂φ

sine₁ =sini′₁μ; sine₂ =sini′₂μ

r′₁ =r₁ -t″sine₁;r′₂ =r₂ -t″sine₂

tani₁ =r′₁φ; tani₂ =r′₂φ

sin ε =sini₁μ; sin ε′ =sini′₂μ

sin α =μ sin ε√μ² - 2 μ cos ε - 1; and sin α′ =μ sin ε′√μ² - 2 μ cos ε′ + 1

Now, the anglebCm= α - α′ from which (since the trianglebmC is isosceles)bmC = 90° - ¹⁄₂ (α - α′); also, in the trianglebme, the anglebme=bmC -emC = 90° - ¹⁄₂ (α - α′) - η andbem=k′eE′ = 90° - ε′

We have therefore in the trianglebme

bm=besinbemsinbme=lcos ε′cos {η + ¹⁄₂ (α - α′)}

and inbmC

from which, puttingbC = ρ

ρ =lcos ε′2 cos {η + ¹⁄₂ (α - α′)} sin ¹⁄₂ (α - α′)

Lastly, the position of C the centre of curvature for a ring is easily determined by two co-ordinates in reference to their origin, A, which is the vertex of the lens (seefig. 60below), by the equations:

The elements of each successive zone are determined in the same manner. The annular lens of the first order of lights inFresnel’ssystem consists, as already stated, of a central disc 11 inches in diameter, and 10 concentric rings, all of which have a common principal focus, where the rays of the sun meet after passing through the lens. With such accuracy are those rings and the disc ground and placed relatively to each other, that the position of the actual conjugate focus of the entire surface of the compound lens, differs in a very small degree from that obtained by calculation in the manner described below.[59]

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