[59]Testing Lenses.The tests generally applied for examining the lenses used in Lighthouses, is to find the position of the conjugate focusbehindthe lens, due to a given position of a lamp infrontof it. This test depends on the following considerations:—Draw a line from an object O in front of a lens, to any point Q in the lens; and from A, the centre of the lens, draw AR parallel to OQ, and cutting a line RFrwhich passes through the principal focus F, at right angles to the axis of the lens; then join the points Q and R, and produce the line joining them: I, the image of O must be in that line. In the same way, draw a line from O toq, another point in the lens on the other side of its axis, and parallel to it draw Arfrom the centre of the lens, cutting the plane of the principal focus inr. Joinqr, in which line the image will lie; and hence the intersection of OR andqr, in I, will be the point in which the image of O is formed, or will be the conjugate focus of the lens due to the distance OA. This mode will serve to give the distance of the conjugate focus of a lens (neglecting its thickness) for rays falling on its surface at any angle.Fig. 61.Testing a lensWe shall suppose QA (fig. 61) to represent the half of a lens, and remembering the conditions described in reference to the last figure, we shall at once perceive the truth of the following analogy(fig. 62):—Fig. 62.Lens testingOA ∶ AF ∷ AQ ∶ FR ∷ AI ∶ FI, and putting OA = δ, AI = φ′, and AF = φ, we have δ ∶ φ ∷ φ′ ∶ φ′ - φ, and, consequently, δ φ′ - δ φ = φ φ′; and hence the following equations, which express the relations subsisting between the principal focus of the lens and the distance of any object and its corresponding image:1st, To find the principal focal distance of a lens from the measured position of its object and its image refracted through it, we have, φ =δ φ′δ + φ′.2d, For the distance of the object, when that of the image is known, we have, δ =φ φ′φ′ - φ.3d, For the position of the image, when that of the object is known, we have, φ′ =δ φδ - φ.Fig. 63.Lens testingIn testing lenses, of course, it is this last equation which we use, because the value of φ or the principal focus is always known, and is that whose accuracy we wish to try, while δ may be chosen within certain limits at will. I have found that the best mode of proceeding is the following:—In front of the lens Qq(seefig. 63) firmly fixed on a frame, place a lamp at O at the distance of about 50 yards. Calculate the value of φ′ due to 50 yards, which in this case is equal to AF′, OA being equal to δ; and move a screen of white paper backwards and forwards until you receive on it the smallest image that can be formed, which is at the point where the cones of converging and diverging rays meet. The image will always increase in size whether you approach nearer to the lens or recede farther from it, according as you pass from the converging into the diverging cone of rays, orvice versa; and hence the intermediate point is easily found by a very little practice. The distance from the centre of the lens to the face of the screen, which must be adjusted so as to be at right angles to a line joining the centre of the lens and the lamp, is then measured; and its agreement with the calculated length of φ′, is an indication of the accuracy of the workmanship of the lens. When the measured distance is greater than the calculated φ′, we know that the lens is too flat; and it is on this side the error generally falls. On the other hand, when φ′ is greater than the measured distance, we know that the lens has too great convexity. I have only to add, that an error of ¹⁄₆₀ on the value of φ′ may be safely admitted in Lighthouse lenses; but I have had many instruments made by M.François Soleil, whose error fell below ¹⁄₈₀ of φ′. Owing probably to the mode of grinding, the surfaces of all the lenses I have yet examined are somewhat too flat.
[59]
Testing Lenses.The tests generally applied for examining the lenses used in Lighthouses, is to find the position of the conjugate focusbehindthe lens, due to a given position of a lamp infrontof it. This test depends on the following considerations:—Draw a line from an object O in front of a lens, to any point Q in the lens; and from A, the centre of the lens, draw AR parallel to OQ, and cutting a line RFrwhich passes through the principal focus F, at right angles to the axis of the lens; then join the points Q and R, and produce the line joining them: I, the image of O must be in that line. In the same way, draw a line from O toq, another point in the lens on the other side of its axis, and parallel to it draw Arfrom the centre of the lens, cutting the plane of the principal focus inr. Joinqr, in which line the image will lie; and hence the intersection of OR andqr, in I, will be the point in which the image of O is formed, or will be the conjugate focus of the lens due to the distance OA. This mode will serve to give the distance of the conjugate focus of a lens (neglecting its thickness) for rays falling on its surface at any angle.
Fig. 61.Testing a lens
Fig. 61.
We shall suppose QA (fig. 61) to represent the half of a lens, and remembering the conditions described in reference to the last figure, we shall at once perceive the truth of the following analogy(fig. 62):—
Fig. 62.Lens testing
Fig. 62.
OA ∶ AF ∷ AQ ∶ FR ∷ AI ∶ FI, and putting OA = δ, AI = φ′, and AF = φ, we have δ ∶ φ ∷ φ′ ∶ φ′ - φ, and, consequently, δ φ′ - δ φ = φ φ′; and hence the following equations, which express the relations subsisting between the principal focus of the lens and the distance of any object and its corresponding image:
1st, To find the principal focal distance of a lens from the measured position of its object and its image refracted through it, we have, φ =δ φ′δ + φ′.
2d, For the distance of the object, when that of the image is known, we have, δ =φ φ′φ′ - φ.
3d, For the position of the image, when that of the object is known, we have, φ′ =δ φδ - φ.
Fig. 63.Lens testing
Fig. 63.
In testing lenses, of course, it is this last equation which we use, because the value of φ or the principal focus is always known, and is that whose accuracy we wish to try, while δ may be chosen within certain limits at will. I have found that the best mode of proceeding is the following:—In front of the lens Qq(seefig. 63) firmly fixed on a frame, place a lamp at O at the distance of about 50 yards. Calculate the value of φ′ due to 50 yards, which in this case is equal to AF′, OA being equal to δ; and move a screen of white paper backwards and forwards until you receive on it the smallest image that can be formed, which is at the point where the cones of converging and diverging rays meet. The image will always increase in size whether you approach nearer to the lens or recede farther from it, according as you pass from the converging into the diverging cone of rays, orvice versa; and hence the intermediate point is easily found by a very little practice. The distance from the centre of the lens to the face of the screen, which must be adjusted so as to be at right angles to a line joining the centre of the lens and the lamp, is then measured; and its agreement with the calculated length of φ′, is an indication of the accuracy of the workmanship of the lens. When the measured distance is greater than the calculated φ′, we know that the lens is too flat; and it is on this side the error generally falls. On the other hand, when φ′ is greater than the measured distance, we know that the lens has too great convexity. I have only to add, that an error of ¹⁄₆₀ on the value of φ′ may be safely admitted in Lighthouse lenses; but I have had many instruments made by M.François Soleil, whose error fell below ¹⁄₈₀ of φ′. Owing probably to the mode of grinding, the surfaces of all the lenses I have yet examined are somewhat too flat.
In the combination of lenses with the flame of a lamp, similar considerations must influence us in making the necessary arrangements, as in the case of reflectors. We have already seen that the size of the flame and its distance from the surface of reflecting instruments have an important practical bearing on the utility of the instrument,Divergence of Annular Lenses.and that the divergence of the resultant beam materially affects its fitness for the purpose of a Lighthouse. So also, in the case of the lens, unless the diameter of the flame of the lamp has to the focal distance of the instrument a relation such as may cause an appreciable divergence of the rays refracted through it, it could not be usefully applied to a Lighthouse; for, without this, the light would be in sight during so short a time, that the seaman would have much difficulty in observing it. To determine the amount of this divergence of the refracted beam, therefore, is a matter of great practical importance, and I shall briefly point out the conditions which regulate its amount, as they are nearly identical with those which determine the divergence of a paraboloïdal mirror illuminated by a lamp in its focus. The divergence, in the case of lenses, may be described asthe angle which the flame subtends at the principal focus of the lens, the maximum of which, produced at the vertex ofFresnel’sgreat lens by the lamp of four concentric wicks, is about 5° 9′.[60]
[60]This will be easily seen by examining the annexedfigure (64), in which Qqrepresents the lens. A its centre, F the principal focus,bF andb′F the radius of the flame; then is the anglebAb′equal to the maximum divergence of the lens. SinbAF =bFAF= sinb′AF =Rad. of flameFocal distance; and twicebAF = the whole divergence at A. Then for the divergence at the margin of the lens, or at any other point, we have, FQ = √AQ² + AF²and Qx= √QF² + Fx²; and for any angle at Q, we have sin FQx=FxFQ.Fig. 64.Refraction of Fresnel lens
[60]This will be easily seen by examining the annexedfigure (64), in which Qqrepresents the lens. A its centre, F the principal focus,bF andb′F the radius of the flame; then is the anglebAb′equal to the maximum divergence of the lens. SinbAF =bFAF= sinb′AF =Rad. of flameFocal distance; and twicebAF = the whole divergence at A. Then for the divergence at the margin of the lens, or at any other point, we have, FQ = √AQ² + AF²and Qx= √QF² + Fx²; and for any angle at Q, we have sin FQx=FxFQ.
Fig. 64.Refraction of Fresnel lens
Fig. 64.
Illuminating Power of Lenses.On the subject of the illuminating power of the lenses, it seems enough to say, that the same general principle regulates the estimate as in reflectors. Owing to the square form of the lens, however, there is a greater difficulty in finding amean focal distancewhereby to correct our estimate of the angle subtended by the light, so as to equate the varying distance of the several parts of the surface; but, practically, we shall not greatly err if we consider thequotient of the surface of the lens divided by the surface of the flameas the increased power of illumination by the use of the lens. The illuminating effect of the great lens, as measured at moderate distances,has generally been taken at 3000 Argand flames, the value of the great flame in its focus being about 16, thus giving its increasing power as nearly equal to 180. The more perfect lenses have produced a considerably greater effect.
The application of lenses to Lighthouses is so obvious as scarcely to admit of farther explanation than simply to state,Arrangement of the Lenses in a Lighthouse.that those instruments are arranged round a lamp placed in their centre, and on the level of the focal plane in the manner shewn inPlates XIII.andXIV.,[61]so as to form by their union a right octagonal hollow prism, circulating round the flame which is fixed in the centre, and shewing to a distant observer successive flashes or blazes of light, whenever they cross a line joining his eye and thelamp, in a manner similar to that already noticed in describing the action of the mirrors. The chief difference in the effect consists in the greater intensity and shorter duration of the blaze produced by the lens; which latter quantity is, of course, proportional to the divergence of the resultant beam. Each lens subtends a central horizontal pyramid of light of about 46° of inclination, beyond which limits the lenticular action could not be advantageously pushed, owing to the extreme obliquity of the incidence of light; butFresnelat once conceived the idea of pressing into the service of the mariner, by means of two very simple expedients, the light which would otherwise have uselessly escaped above and below the lenses.
[61]The Plates shew the nature of the mechanical power which gives movement to the lenses. It consists of a clockwork movement driven by a weight which sets in motion a plate bearing brackets that carry the lenses. All this, however, can be seen from the Plates; and I am unwilling to expend time in a detailed explanation of what is obvious by inspection.
[61]The Plates shew the nature of the mechanical power which gives movement to the lenses. It consists of a clockwork movement driven by a weight which sets in motion a plate bearing brackets that carry the lenses. All this, however, can be seen from the Plates; and I am unwilling to expend time in a detailed explanation of what is obvious by inspection.
For intercepting the upper portion of the light,Fresnelemployed eight smaller lenses of 500 mm. focal distance (19·68 inches) inclined inwards towards the lamp, which is also their common focus and thus forming, by their union,Pyramidal Lenses and Mirrors.a frustum of a hollow octagonal pyramid of 50° of inclination. The light falling on those lenses is formed into eight beams parallel to the axis of the smaller lenses, and rising upwards at an angle of 50° inclination. Above them are ranged eight plane mirrors, so inclined (seePlates XIII.andXIV.) as to project the beams transmitted by the small lenses in the horizontal direction, so as finally to increase the effect of the light. In placing those upper lenses, it is generally thought advisable to give their axis an horizontal deviation of 7° or 8° from that of the great lenses and in the direction contrary to that of the revolution of the frame which carries the lenticular apparatus. By this arrangement, the flashes of the smaller lenses precede that of the large ones, and thus tend to correct the chief practical defect of revolving lenticular lights by prolonging the bright periods. The elements of the subsidiary lenses depend upon the very same principles, and are calculated by the same formulæ as those given for the great lenses. In fixing the focal distance and inclination of those subsidiary lenses,Fresnelwas guided by a considerationof the necessity for keeping them sufficiently high to prevent interference with the free access to the lamp. He also restricted their dimensions within very moderate limits, so as to avoid too great weight. The focal distance is the same as that for lenses of the third order of lights.
Fig. 65.Curved mirror
Fig. 65.
Owing to the necessary arrangements of a lantern, only a very small portion of those rays, which escape from below the lenses, can be rendered available for the purposes of a Lighthouse; and any attempt to subject it to lenticular action, so as to add it to the periodic flashes, would have led to a most inconvenient complication of the apparatus.Fresneladopted the more natural and simple course of transmitting it to the horizon in the form of flat rings of light, or rather of divergent pencils, directed to various points of the horizon.Curved Mirrors.This he effected by means of small curved mirrors, disposed in tiers, one above another, like the leaves of a Venetian blind—an arrangement which he also adopted (shewn inPlates XV.andXVI.) for intercepting the light which escapes above as well as below the dioptric belt in fixed lights. Those curved mirrors are, strictly speaking, generated (seefig. 65) by portions, such as a b, of parabolas, having their foci coincident with F, the common flame of the system. In practice, however, they are formed as portions of a curved surface, ground by the radius of the circle, which osculates the given parabolic segment.[62]The mirrors are plates of glass, silvered on the back and set in flat cases of sheet-brass. They are suspended on a circular frame byscrews, which are attached to the backs of the brass cases, and which afford the means of adjusting them to their true inclination, so that they may reflect objects on the horizon of the Lighthouseto an observer’s eye, placed in the common focus of the system.[63]
[62]To find the radius and centre of a circle, which shall osculate a given parabola, whose focus is in F, draw the normals to the curve frompand P, meeting in O, and draw Neparallel to a tangent of the curve, or topP, then P O orpO is the radius required. Now, we have similar triangles Ppdand Nen, and P H andphare (proximate) ordinates; hence we have the followinganalogies:—Pd∶ Pp∷ PH ∶ PNNe∶ Nn∷ PH ∶ PNFig. 66.Finding radius and centre of circleHence compounding those ratios (in which Pd= Nnnearly)Ne∶ Pp∷PH² ∶ PN²also Ne∶ Pp∷NO ∶ PO,(for O Ppand Noeare similar triangles)PH² ∶ PN² ∷ NO ∶ OP,then PN²- PH² = HN²and PO - NO = NP,thereforeHN² ∶ PN² ∷ NP ∶ PO,and finally, PO =PN³HN².Then put FP = HC = FN = ρ; HN=ρ -z; then as FP² - FH² = PH² = ρ² -z²PN² = PH² + HN²=(ρ² -z²) + (ρ² - 2 ρz+z²)=2ρ² - 2 ρzPN=√2 ρ (ρ -z)Therefore PO=√{2 ρ (ρ -z)}³(ρ -z)²=√{2 ρ (ρ -z)}³(ρ -z)⁴and finally, PO=2√2√ρ³ρ -zFig. 67.Finding versed sine of curvatureTo find the versed sine of the curvature (which may be useful in the examination of the mirrors by a mould) we may proceed (seefig. 67) toput AG =f; BE = C; AC = Rthen BG² = AG . GD4 BG² = BE² = 4 AG . GDC² = 4f. (2 R -f)C² = 8fR - 4f²From which equation,2f- 2 R = ± √4 R² - C²= -2 R +C²4 R-C⁴64 R³&c.2f=C²4 R-C⁴64 R³f=C²8 R-C⁴128 R³.Fig. 68.Mirror workmanship test installationIn order to test the accuracy of the workmanship of the mirrors, recourse must again be had, as in the case of the lenses and parabolic mirrors, to the formula of conjugate foci, in which we shall call R = the radius of curvature of the mirror Mm(fig. 68);a= the distance of a light,f, which is arbitrarily placed in front of the mirror; andb= the distance of a moveable screen S, on which the rays reflected from the mirror may converge in a focus. We must find the distanceb, at which, with any given distancea, such convergence should take place.fM′ =aSM′ =bOM′ = RThen (becausefMS is bisected by OM, and for points near the vertex of the mirror at M′)SM′ ∶fM′ ∷ SO ∶ Oforb∶a∷ R -b∶a- Rab- Ra= Rb-ab.From whichb=Ra2a- R, the distance required, in which an error of ¹⁄₃₀ (of its whole length) may be safely admitted.[63]At such times when the horizon cannot be seen, the mirror may be placed, by means of aclinometer, with a spirit-level, set to the proper angle, which may be easily mechanically determined as follows: Draw a line from the focus F through the point O, where the centre of the mirror is to be, producing it beyond that point to a convenient distance at I; through O draw HOH, parallel to the horizon FH; bisect IOH by MOM, which coincides with a tangent to the mirror at its centre O; and MOH is the angle required to be laid off, or its complement.Fig. 69.Determination of angle
[62]To find the radius and centre of a circle, which shall osculate a given parabola, whose focus is in F, draw the normals to the curve frompand P, meeting in O, and draw Neparallel to a tangent of the curve, or topP, then P O orpO is the radius required. Now, we have similar triangles Ppdand Nen, and P H andphare (proximate) ordinates; hence we have the followinganalogies:—
Pd∶ Pp∷ PH ∶ PNNe∶ Nn∷ PH ∶ PN
Pd∶ Pp∷ PH ∶ PN
Ne∶ Nn∷ PH ∶ PN
Fig. 66.Finding radius and centre of circle
Fig. 66.
Hence compounding those ratios (in which Pd= Nnnearly)
(for O Ppand Noeare similar triangles)
Fig. 67.Finding versed sine of curvature
Fig. 67.
To find the versed sine of the curvature (which may be useful in the examination of the mirrors by a mould) we may proceed (seefig. 67) to
put AG =f; BE = C; AC = Rthen BG² = AG . GD4 BG² = BE² = 4 AG . GDC² = 4f. (2 R -f)C² = 8fR - 4f²
put AG =f; BE = C; AC = Rthen BG² = AG . GD4 BG² = BE² = 4 AG . GDC² = 4f. (2 R -f)C² = 8fR - 4f²
From which equation,
2f- 2 R = ± √4 R² - C²= -2 R +C²4 R-C⁴64 R³&c.2f=C²4 R-C⁴64 R³f=C²8 R-C⁴128 R³.
2f- 2 R = ± √4 R² - C²= -2 R +C²4 R-C⁴64 R³&c.2f=C²4 R-C⁴64 R³f=C²8 R-C⁴128 R³.
Fig. 68.Mirror workmanship test installation
Fig. 68.
In order to test the accuracy of the workmanship of the mirrors, recourse must again be had, as in the case of the lenses and parabolic mirrors, to the formula of conjugate foci, in which we shall call R = the radius of curvature of the mirror Mm(fig. 68);a= the distance of a light,f, which is arbitrarily placed in front of the mirror; andb= the distance of a moveable screen S, on which the rays reflected from the mirror may converge in a focus. We must find the distanceb, at which, with any given distancea, such convergence should take place.
fM′ =a
SM′ =b
OM′ = R
Then (becausefMS is bisected by OM, and for points near the vertex of the mirror at M′)
From whichb=Ra2a- R, the distance required, in which an error of ¹⁄₃₀ (of its whole length) may be safely admitted.
[63]At such times when the horizon cannot be seen, the mirror may be placed, by means of aclinometer, with a spirit-level, set to the proper angle, which may be easily mechanically determined as follows: Draw a line from the focus F through the point O, where the centre of the mirror is to be, producing it beyond that point to a convenient distance at I; through O draw HOH, parallel to the horizon FH; bisect IOH by MOM, which coincides with a tangent to the mirror at its centre O; and MOH is the angle required to be laid off, or its complement.
Fig. 69.Determination of angle
Fig. 69.
Cylindric Refractors for Fixed Lights.Having once contemplated the possibility of illuminating Lighthouses by dioptric means,Fresnelquickly perceived the advantage of employing for fixed lights a lamp placed in the centre of a polygonal hoop, consisting of a series of refractors,infinitely smallin their length and having their axes in planes parallel to the horizon. Such a continuation of vertical sections, by refracting the rays proceeding from the focus, only in the vertical direction, must distribute a zone of lightequally brilliantin every point of the horizon. This effect will be easily understood, by considering the middle vertical section of one of the great annular lenses, already described, abstractly from its relation to the rest of the instrument. It will readily be perceived that this section possesses the property of simply refracting the raysin one plane coincident with the line of the sectionand in a direction parallel to the horizon, and cannot collect the rays from either side of the vertical line; and if this section, by its revolution about a vertical axis, becomes the generating line of the enveloping hoop, above noticed, such a hoop will of course possess the property of refracting an equally diffused zone of light round the horizon. The difficulty, however, of forming this apparatus appeared so great, thatFresneldetermined to substitute for it a vertical polygon, composedof what have been improperly calledcylindric lenses, but which in reality are mixtilinear and horizontal prisms, distributing the light which they receive from the focus nearly equally over the horizontal sector which they subtend. This polygon has a sufficient number of sides to enable it to give, at the angle formed by the junction of two of them, a light not very much inferior to what is produced by one of the sides; and the upper and lower courses of curved mirrors are always so placed as partly to make up for the deficiency of the light at the angles. The effect sought for in a fixed light is thus obtained in a much more perfect manner, than by any combination of the parabolic mirrors used in the British Lighthouses.
Application of crossed prisms to cause occasional flashes.An ingenious modification of the fixed apparatus is also due to the inventive mind ofFresnel, who conceived the idea of placing one apparatus of this kind in front of another, with the axis of the cylindric pieces crossing each other at right angles. As those cylindric pieces have the property of refracting all the rays which they receive from the focus, in a direction perpendicular to the mixtilinear section which generates them, it is obvious that if two refracting media of this sort be arranged as above described, their joint action will unite the rays which come from their common focus into a beam, whose sectional area is equal to the overlapped surface of the two instruments, and that they will thus produce, although in a disadvantageous manner, the effect of an annular lens. It was by availing himself of this property of crossed prisms, thatFresnelinvented the distinction for lights, which he callsa fixed light varied by flashes; in which the flashes are caused by the revolution of cylindric refractors with vertical axes, ranged round the outside of the fixed light apparatus already described.
Having been directed by the Commissioners of the Northern Lighthouses to convert the fixed catoptric light of the Isle of May, into a dioptric light of the first order, I proposed, that an attempt should be madeTrue Cylindric form given to the Refractors and other improvements in their Construction.to form a true cylindric, instead of a polygonal belt for the refracting part of the apparatus; and this task was successfullycompleted by MessrsCooksonof Newcastle in the year 1836. The disadvantage of the polygon lies in the excess of the radius of the circumscribing circle over that of the inscribed circle, which occasions an unequal distribution of light between its angles and the centre of each of its sides; and this fault can only be fully remedied by constructing a cylindric belt, whose generating line is the middle mixtilinear section of anannularlens, revolving about a vertical axis passing through its principal focus. This is, in fact, the only form which can possibly produce an equal diffusion of the incident light over every part of the horizon.
I at first imagined that the whole hoop of refractors might be built between two metallic rings, connecting them to each other solely by the means employed in cementing the pieces of the annular lenses; but a little consideration convinced me that this construction would make it necessary to build the zone at the lighthouse itself, and would thus greatly increase the risk of fracture. I was therefore reluctantly induced to divide the whole cylinder into ten arcs, each of which being set in a metallic frame, might be capable of being moved separately. The chance of any error in the figure of the instrument has thus a probability of being confined within narrower limits; whilst the rectification of any defective part becomes at the same time more easy. One other variation from the mode of construction at first contemplated for the Isle of May refractors, was forced upon me by the repeated failures which occurred in attempting to form the middle zone in one piece; and it was at length found necessary to divide this belt by a line passing through the horizontal plane of the focus. Such a division of the central zone, however, was not attended with any appreciable loss of light, as the entire coincidence of the junction of the two pieces with the horizontal plane of the focus, confines the interception of the light to the fine joint at which they are cemented. With the exception of those trifling changes, the idea at first entertained of the construction of the instrument was fully realised at the manufactory of MessrsCookson. I also, at a subsequent period, greatlyimproved the arrangement of this apparatus, by giving to the metallic frames which contain the prisms, a rhomboidal,[64]instead of a rectangular form. The junction of the frames being thus inclined from the perpendicular, do not in any azimuth intercept the light throughout the whole height of the refracting belt, but the interception is confined to a small rhomboidal space, whose area is inversely proportional to the sine of the angle of inclination; and if the helical joints be formed between the opposite angles of the old rectangular frames, the amount of intercepted light becomes absolutely equal in every azimuth.[65]
[64]The form would not be exactly rhomboidal, but would be a portion of a flat helix intercepted between two planes, cutting the enveloped cylinder at right angles to its axis.[65]See my Report on the Refractors of the Isle of May Light, 8th October 1836.
[64]The form would not be exactly rhomboidal, but would be a portion of a flat helix intercepted between two planes, cutting the enveloped cylinder at right angles to its axis.
[65]See my Report on the Refractors of the Isle of May Light, 8th October 1836.
Fig. 70.Compound Fresnel belt
Fig. 70.
Such an apparatus is shewn inPlate XVII.; and the accompanying diagram (fig. 70) shews an elevation ABCD, a section BD, and a plan ABD, of a single pannel of this improved compound belt. AC and BD are the diagonal joints above described. Time and perseverance, and the patience and skill of MonsieurFrançois Soleil, whom I urged to undertake the task, were at length crowned with success; and I had the satisfaction at last of seeing a fixed light apparatus, having its form truly cylindric, and its central belt in one piece, while the joints were inclined to the horizon at such an angle as to render the light perfectly equal in every azimuth.
The loss of light by reflection at the surface of the most perfect mirrors, and the perishable nature of the material composing their polish, induced me, so far back as 1835, in a Report on the Light of Inchkeith, which had just been altered to the dioptric system, to propose the substitution oftotally reflectingprisms, even in lights of the first order or largest dimensions. In this attempt I was much encouraged by the singular liberality of MrLeonor Fresnel, to whose friendship (as I have often, with much pleasure, acknowledged) I owe all that I know of dioptric Lighthouses. He not only freely communicated to me the method pursued by his distinguished brotherAugustin Fresnel, in determining the forms of the zones of the small apparatus, introduced by him into the Harbour Lights of France, and his own mode of rigorously solving some of the preliminary questions involved in the computations; but put me in possession of various important suggestions, which substantially embrace the whole subject. Another friend also helped me, by pointing out certain less direct methods of determining some of the elements, which greatly abridged the labours of computation. MrFresnelagreed with me in anticipating a considerable increase of the light derived from the accessory part of the apparatus; but he expressed his opinion, that in order to prevent great absorption, the rings should not greatly exceed those of the small apparatus in their sectional area. This would have required aboutfortyrings to intercept the same quantity of light acted upon by the curved mirrors; and, although the difficulties of grinding were somewhat similar to those which had already been encountered in forming the cylindric belt for the Isle of May apparatus, there were also some special difficulties attendingthe formation of theCatadioptric Zones.catadioptric zones, which appeared so formidable as to deter me by the expense of grinding so many zones, and led me to think of adopting flint glass. Considerable masses, of a very pure and homogeneous appearance, had been shewn to me by the late DrRitchieof the London University, who calculated upon the uniform and permanent success of his process; but, whatever foundation there might have been for this hope, it was removed by his death, which occurred soon afterwards, and I was forced to return to the idea of using crown glass. In order, therefore, to enable me to estimate more correctly the advantage of the zones, I procured from MessrsCooksonof Newcastle, an average specimen of crown glass, of the thickness of 40 mm. (about 1¹⁄₂ inch), which is the distance traversed by the ray between its immergence into and its emergence out of the zones of the small apparatus; and having had it carefully polished, with both faces parallel, I found, as the result of numerous trials, conducted with every precaution I could think of, that the loss of light due to the transmission through it, was somewhat less than ²⁄₇ths of the incident light. According to the experiments ofBougeur, the loss by the two refractions may be assumed at ¹⁄₂₀th; so that we could not sensibly err in concluding that the whole loss due to the transmission of the light through the zones would not much exceed ²⁄₇ths of the incident light. In the lights of the first order, the loss by reflection from the surface of the mirrors, and by the escape of light through the interstices which separate them, is not less than ²⁄₃ds of the light incident on that part of the apparatus. On the most moderate expectation, therefore, which this proportion seemed to warrant, it appeared that, without any allowance for imperfections in the figure of the zones, at leasttwiceas much light would be transmitted through the zones as can be reflected by the mirrors. The prospect even of a part of this increase being obtained without the expenditure of more oil, seemed too important to be readily renounced, more especially when it was considered that the fixed lights, to which it chiefly applies, arenecessarily much feebler than the revolving lights, as well as more numerous and more expensive. So many motives pressed me to the work, that I commenced my labours (during my leisure hours while engaged at the Skerryvore), and computed Tables of the Elements of 45 zones, whose lesser sides were 40 millimètres in length, which were printed in 1840. In 1841, in consequence of having seen at Paris specimens of purer crown glass, I printed other Tables from computations of larger zones, which I had made in 1838, but had discarded as unsuited to the inferior quality of English glass, whose absorption rendered the use of smaller dimensions of the zone imperative. In the first Table, I had adopted the form of isosceles triangles, to avoid the difficulty of grindingannularsurfaces with radii of great length (which I found required to be nearly 30 feet), but in the second Table, I adopted a suggestion, conveyed to me in a letter from M.Leonor Fresnel, by giving each zone the form of an oblique triangle whose base is the chord of the circle which osculates the surface of the reflecting side of the zone. Some attempts were made by MessrsCooksonat Newcastle to execute the largest of the zones; but the forms differed so widely from the dimensions assigned in the Table, that I had begun to despair of success. About this time, I received a communication from M.Fresnel, pointing out several inaccuracies in my Tables, and more especially directing my attention to the disadvantage of choosing, for the focus of the upper series of zones, a high part of the flame, as I had done with the view of throwingallthe lightbelowthe horizon, so that none might be lost. He, at the same time, informed me of the success of M.François Soleil, in executing zones for the smaller apparatus, known by the name of the Third Order; and put me in possession of the results of his computations of large zones of the First Order, suited to the greatly improved quality of the crown glass of St Gobain, with an invitation, before I should adopt his dimensions, to verify his calculations. This I willingly undertook, and computed the elements of the zones in M.Fresnel’sTable afresh, with results differing from his only in one ortwo instances, to an amount whose angular value does not exceed more than 2″. The Table in theAppendixcontains the result of my calculations, which are verifications of those of M.Fresnel. The subject of the zones has thus been very fully weighed; and it is most satisfactory to think that complete success has attended the perseverance and ardour of M.François Soleil, who at once boldly undertook to furnish for the Skerryvore Lighthouse the first catadioptric apparatus ever constructed on so magnificent a scale. On the 23d December 1843, M.Fresnelannounced, in a letter to me, the complete success which had attended a trial of the apparatus at the Royal Observatory at Paris, whereby it appeared that the illuminating effect of the cupola of zones, was to that of the seven upper tiers of mirrors of the first order, as 140 to 87. Nothing can be more beautiful than an entire apparatus for a fixed light of the first order, such as that shewn inPlates XVII.andXVIII.It consists of a central belt of refractors, forming a hollow cylinder 6 feet in diameter, and 30 inches high; below it are six triangular rings of glass, ranged in a cylindrical form, and above a crown of thirteen rings of glass, forming by their union a hollow cage, composed of polished glass, 10 feet high and 6 feet in diameter! I know no work of art more beautiful or creditable to the boldness, ardour, intelligence, and zeal of the artist.
I must now endeavour to trace the various steps by which the elements of the zones given in the appendedTablehave been determined; and this, I fear, I cannot do without considerable prolixity of detail. Referring toPlates XV.,XVI.,XVII., andXVIII., in which F shews the flame, RR, the refractors, and MRM and MRM, the spaces through which the light would escape uselesslyaboveandbelowthe lens, but for the corrective action of the mirrors MM, which project the rays falling on them to the horizon, I have to observe that a similar effect is obtained, but in a more perfect manner, by means of the zones ABC and A₂B₂C₂ (fig. 71, on page 271), whose action on the divergent rays of the lamp causes the rays FC, FB and FC₂, FB₂ to emerge horizontally, by refractingthem at the inner surfaces BC, B₂C₂, reflecting them at AB, A₂B₂, and a second time refracting them at AC, A₂C₂.
Fig. 71.Details behind the tabulated data
Fig. 71.
The problem proposed is, therefore, the determination of the elements and position of a triangle ABC, which, by its revolution about a vertical axis, passing through the focus of a system of annular lenses or refractors in F, would generate a ring or zone capable of transmitting in an horizontal direction by means oftotal reflection, the light incident upon its inner side BC from a lamp placed in the point F. The conditions of the question are based upon the well-known laws oftotal reflection, and require that all the rays coming from the focus F shall be so refracted at entering the surface BC, as to meet the side BA at such an angle, that instead of passing out they shall betotally reflectedfrom it, and passing onwards to the side CA shall, after a second refraction at that surface, finally emerge from the zone in an horizontal direction. For the solution of this problem, we have given the positions of F the focus, of the apex C of the generating triangle of the zone, the length of the side BC or CA, and the refractive index of the glass. The form of the zone must then be such as to fulfil the followingconditions:—
1. The extreme ray FB must suffer refraction and reflection at B, and pass to C, where being a second time refracted, it must follow the horizontal direction CH.
2. The other extreme ray FC must be refracted in C and passing to A, must at the point be reflected, and a second time reflected, so as to follow the horizontal course AG (seefig. 72, on opposite page).
These two propositions involve other two in the form of corollaries.
1. That every intermediate ray proceeding from F, and falling upon BC in any point E, between B and C, must, after refraction at the surface BC in E into the direction EW, be so reflected at W from AB into the direction WI, that being parallel to BC, it shall, after a second refraction in I, at the surface AC, emerge horizontally in the line IK.
And, 2. That the paths of the two extreme rays must therefore trace the position of the generating triangle of the zone.
To these considerations it may be added, that as the angles BCH and FCA are each of them solely due to the refraction at C, as their common cause, they must be equal to each other, and BCA being common to both, the remaining angle ACH = the remaining angle BCF.
We naturally begin by the consideration of the lowest ray FC, whose path being traced gives the direction of the two refracting sides BC and AC, leaving only the direction of the reflecting side BA to be determined. I shall not now explain the reason for neglecting entirely the consideration of the reflecting side at present, as I could not do so without anticipating what must be more fully discussed in the sequel; but I may content myself with stating, that as the positions of BC and AC depend upon the direction of the incident ray FC, and on the refractive index of the glass, this part of the investigation may be carried on apart from any interference with the reflecting side.
As we know the relation existing between the angles of incidence and refraction, we might determine the relative positions of the sides AC and BC, by means of successive corrections obtained by protraction, tracing the paths of the rays from the horizontaldirections backwards through the zones to the focus. This method, however, depends entirely upon accurate protraction, and is therefore unsatisfactory as a final determination, or if employed for any other purpose than that of affording a rough approximation to the value of the angle, a knowledge of which may occasionally save trouble in the employment of more exact means of determination. I have not, however, on any occasion employed this process, as I found that a little practice enabled me to make my first estimation very near the truth. I shall therefore at once proceed to give a view of the reasoning employed in the investigation.
Fig. 72.Details behind the tabulated data
Fig. 72.
Referring tofig. 72, which shews thefirstandsecondzone of the upper series, we have
Tan LCF =FLCL;
Tan LCF =FLCL;
and if we make
we obtain the means of determining the angles γ and ξ in two equations, which are based upon the relation between the angles of incidenceand refraction, and on the interdependence of the various angles about C. These primary equations are:
sin ξ =m. sin γand[66]γ = 2 ξ - θ (making 2 α - 90° = θ)
sin ξ =m. sin γand[66]γ = 2 ξ - θ (making 2 α - 90° = θ)