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.
One of the earliest notices of the application of lenses to lighthouses is that recorded by Smeaton in 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 lights of 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 the notice of 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 the focus, 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 of reflection, this result is produced by the light being thrown back from a surface so formed as to make all the rays to proceed in one and the same required direction. In the case of refraction, on the other hand, the rays pass through the refracting medium, and are bent or refracted from their natural course into that which is desired.
The celebrated Buffon, 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 of Buffon regarding the construction of large burning glasses, was first executed, with tolerable success, about the year 1780, by the Abbé Rochon; 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 Messrs Cookson of 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 to Condorcet, who, in his Eloge de Buffon, published so far back as 1773, enumerates the advantages to be derived from this method. Sir David Brewster also described this mode of building lenses in 1811, in the Edinburgh Encyclopædia; and in 1822, the late eminent Fresnel, unacquainted with the suggestions of Condorcet or the description by Sir David Brewster, explained, with many ingenious and interesting details, the same mode of constructing those instruments. To Fresnel belongs the additional merit of having first followed up his invention, by the construction of a lens and, in conjunction with MM. Arago and Mathieu, 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 by Condorcet are,—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. Both Buffon and Condorcet, 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 in Buffon’s lens. It deserves notice, that the first lenses constructed for Fresnel by M. Soleil had their zones polygonal, so that the surfaces were not annular, a form which Fresnel considered 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 from striæ. 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 of annular lenses, 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, Mr William 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, the difference between the greatest and least values being only 0·00109.
A ray of light, in passing obliquely from 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 of Refraction. refraction has 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 be bent or broken at 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 are three in 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 the refractive 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 of refraction less than the angle of incidence; and those angles are measured with reference to 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 of incidence is less than the angle of refraction. 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
The diagram (fig. 47) will serve to render those laws more intelligible. Let a ray of light a O meet a surface of water n m at O, it will be immediately bent into the direction O a′; and if, from the centre O, we describe any circle, and draw a line b O b′, perpendicular to nm; then ab and a′ b′, perpendiculars drawn to the normal bb′, from the points a and a′ where the circle cuts the incident and refracted rays, will be the sines of the angle of incidence b O a, and of the angle of refraction b′ O a′, and the ratio of those sines to each other, or b ab′ a′ will be the relative index of refraction for the two media.
4. It may perhaps be added, for convenience, as a fourth law, 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 reflection ensues. 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 return inwards into the glass, being reflected at 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 C b, its incidence, be less than 41° 49′, it will be refracted in some direction C f; but if this angle be greater than 41° 49′, as O C′ b′, the ray will be reflected back into the glass in the direction C′ O′.
Fig. 48.
Refraction and reflection
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 of incidence and refraction, in the manner above described, as to the truth of the following general propositions resulting from those laws:—
1. A ray of light passing through a plate of some diaphanous substance such as glass, with parallel surfaces, suffers no change of direction, but emerges in a line parallel to its original path, merely suffering a displacement, 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 which a b is a normal to the plate, whose surfaces x x and x′ x′ are parallel, r r r r shews the path of the ray, r r the displacement, and r′ the apparent point of origin resulting from its altered direction.
Fig. 49.
Refraction and displacement
2. When a ray passes through a triangular prism a b c, the inclination of the faces a c and c b causes the emergent ray r′ to be bent towards a 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
3. When parallel rays fall on a concave lens, they will, at their emergence, be divergent. The section of the diaphanous body a b c d may be regarded as composed of innumerable frusta of prisms, having their apices directed towards the centre line x 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 the figure) 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
4. Lastly, when divergent rays fall on a convex lens a b, from a point f, 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 section a b of the convex lens to be composed of innumerable frusta of prisms, arranged with their bases towards 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
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 a case of the double convex having one side of an infinite radius. 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 the radiant and the focus.
Fig. 53.
Plano-convex lens
Let Q q be a section of a lens, and f A r its 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 (see fig. 53), then the principal focus f is that point where the rays from r 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, the principal focus f is the point whence the rays of light, proceeding in their naturally divergent course, fall on the inner surface Q A q of the lens, and are so changed by refraction there and at the outer face, that they finally emerge parallel to the optic axis in the directions Q r, 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 the principal 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 illustrious Fresnel, who seems to have devoted such minute attention to every detail of 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 - 1 in which r is the radius of curvature, and m is 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 the principal focus; and when the luminous body reaches the principal focus in front of 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 point more distant 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, led Condorcet, 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 to Plate 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 the optical 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 of Descartes, 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 for Fresnel by Soleil, the zones were united by means of small dowels or joggles of 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 in Plate 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 by Fresnel, to determine the elements of the compound lens,[56] in the calculation of which 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 Mr Edward Sang, and (more recently) to Mr William Swan, for their kind advice on this part of the subject.
Fig. 54.
Central disc
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 segment b a c, resting on a parallelogram b c d e, whose depth b d or c e is 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 by Fresnel and 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
Referring to fig. 55, we have, for obtaining the radius of the central disc, the following formulæ, in which
r = AB, half the aperture of the lens
r′ = AB′
φ = AF, the focal distance
t′ = A a, the thickness of the lens at the vertex
t″ = B b, 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:
tan i′ = rφ
sin e = sin i′μ
r′ = r - t″. tan e
tan i = r′φ
sin ε = sin iμ
ρ″ = rμ sin e √μ² - 2 μ cos e + 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 lens
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,
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 to fig. 57):
Let FB′ b x be the course of a ray refracted in the direction b x parallel to the axis A x′. 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
Let C be the centre of curvature (see fig. 57)
α = AC b the angle of emergence
η = B′ b C 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′ B b
AB = r
AB′ = r′
B b = t″ the thickness of the lens at the edge
AF = φ the focal distance.
Then tan i′ = rφ; sin e = sin i′μ whence b b′ = t″ tan e becomes known.
Now, since BB′ = b b′ nearly, AB′ = AB - b b′ or r′ = r - t″ tan e.
From this is obtained the angle of incidence i, and the first angle of refraction ε; for tan i = r′φ and sin ε = sin iμ.
Next B′ b C = B b C - B b B′ or η = α - ε
and sin α = μ sin η = μ sin (α - ε)
from which, sin α cos ε - cos α sin ε = sin α μ
whence sin α (cos ε - 1 μ) = cos α sin ε; and
sin² α (cos² ε - 2 cos εμ + 1μ²) = cos² α sin² ε = (1 - sin² α) sin² ε = sin² ε - sin² α sin² ε
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 ε μ² - 2 μ cos ε + 1
and sin α = μ sin ε √1 - 2 μ cos ε + μ²
Next, since b C = a′ b sin AC b = rsin α, putting C b = ρ″, and substituting we have ρ″ = rμ sin ε √μ² - 2 μ cos ε + 1
and, taking for the radius of curvature, the mean of ρ′ and ρ″ the values calculated for the central and marginal rays, we have finally ρ = ρ′ + ρ″ 2
Fig. 58.
Refraction of ring
I come next to the second case, which concerns the calculation of the elements of a concentric ring. The section a b c d e (fig. 58) of one of those rings includes a mixtilinear triangle a b e, and a rectangle b c e d, the thickness b c being 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
The radius of curvature of the zone may be calculated by the following formulæ, in which (see fig. 59)
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 = b C = m C
φ = focal distance AF
t = thickness of the joint B b
t″ = B b
μ = refractive index of the glass
i₁ = BFA
i₂ = EFA
Then tan i′₁ = r₁φ; tan i′₂ = r₂φ
sin e₁ = sin i′₁μ; sin e₂ = sin i′₂μ
r′₁ = r₁ - t″ sin e₁; r′₂ = r₂ - t″ sin e₂
tan i₁ = r′₁φ; tan i₂ = r′₂φ
sin ε = sin i₁μ; sin ε′ = sin i₂μ
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 is Fresnel’s value of the radius of curvature.[58]
[58] The following steps will conduct us to this expression:
Fig. 60.
Refraction of ring
Let B b f E (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′ b x, FE′m x′ the course of the extreme rays which are transmitted through the zone (and the latter of which passes from E′ to e through a portion of the zone or lens in contact with that under consideration). Then putting
AB = r₁; AB′ = r′₁; C b = ρ
AE = r₂; AE′ = r′₂; B b = t″; BE = r₁ - r₂ = l
ε = the first angle of refraction b B′ k
η = the second angle of refraction B′ b C
ε′ = the first angle of refraction e E k′
η′ = the second angle of refraction e m C
α = the angle of emergence b C q
α′ = the angle of emergence m C q
i′₁ = BFA; i′₂ = EFA; i₁ = B′FA; i₂ = E′FA
e₁ = B b B′; e₂ = E e E′.
Proceeding exactly as in the case of the central lens we shall have
tan i′₁ = BAAF = r₁φ; tan i′₂ = EAAF = r₂φ
sin e₁ = sin i′₁μ; sin e₂ = sin i′₂μ
r′₁ = r₁ - t″ sin e₁; r′₂ = r₂ - t″ sin e₂
tan i₁ = r′₁φ; tan i₂ = r′₂φ
sin ε = sin i₁μ; sin ε′ = sin i′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 angle b C m = α - α′ from which (since the triangle b m C is isosceles) b m C = 90° - ¹⁄₂ (α - α′); also, in the triangle b m e, the angle b m e = b m C - e m C = 90° - ¹⁄₂ (α - α′) - η and b e m = k′ e E′ = 90° - ε′
We have therefore in the triangle b m e
b m = b e sin b e m sin b m e = l cos ε′cos {η + ¹⁄₂ (α - α′)}
and in b m C
b C = b m sin b m C sin b C m = l cos ε′ cos ¹⁄₂ (α - α′) (cos (η + ¹⁄₂ (α - α′)) sin (α - α′)
= l cos 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, putting b C = ρ
ρ = l cos 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 (see fig. 60 below), by the equations:
| CG | = | ρ. sin α - a b | = | ρ. sin α - r₁ |
| CQ | = | ρ. cos α - q Q | = | ρ. cos α - t″ |
The elements of each successive zone are determined in the same manner. The annular lens of the first order of lights in Fresnel’s system 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]
[59]
Testing Lenses. The tests generally applied for examining the lenses used in Lighthouses, is to find the position of the conjugate focus behind the lens, due to a given position of a lamp in front of 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 RF r which 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 to q, another point in the lens on the other side of its axis, and parallel to it draw A r from the centre of the lens, cutting the plane of the principal focus in r. Join q r, in which line the image will lie; and hence the intersection of OR and q r, 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
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
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
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 Q q (see fig. 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, or vice 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 as the angle which the flame subtends at the principal focus of the lens, the maximum of which, produced at the vertex of Fresnel’s great lens by the lamp of four concentric wicks, is about 5° 9′.[60]
[60] This will be easily seen by examining the annexed figure (64), in which Q q represents the lens. A its centre, F the principal focus, b F and b′ F the radius of the flame; then is the angle b A b′ equal to the maximum divergence of the lens. Sin b AF = b FAF = sin b′ AF = Rad. of flameFocal distance; and twice b AF = the whole divergence at A. Then for the divergence at the margin of the lens, or at any other point, we have, FQ = √AQ² + AF² and Q x = √QF² + F x²; and for any angle at Q, we have sin FQ x = F xFQ.
Fig. 64.
Refraction of Fresnel lens
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 a mean focal distance whereby 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 the quotient of the surface of the lens divided by the surface of the flame as 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 in Plates XIII. and XIV.,[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 the lamp, 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; but Fresnel at 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.
For intercepting the upper portion of the light, Fresnel employed 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 (see Plates XIII. and XIV.) 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, Fresnel was guided by a consideration of 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
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. Fresnel adopted 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 in Plates XV. and XVI.) for intercepting the light which escapes above as well as below the dioptric belt in fixed lights. Those curved mirrors are, strictly speaking, generated (see fig. 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 by screws, 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 Lighthouse to 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 from p and P, meeting in O, and draw N e parallel to a tangent of the curve, or to p P, then P O or p O is the radius required. Now, we have similar triangles P p d and N e n, and P H and p h are (proximate) ordinates; hence we have the following analogies:—
P d ∶ P p ∷ PH ∶ PN
N e ∶ N n ∷ PH ∶ PN
Fig. 66.
Finding radius and centre of circle
Hence compounding those ratios (in which P d = N n nearly)
| N e ∶ P p | ∷ | PH² ∶ PN² |
| also N e ∶ P p | ∷ | NO ∶ PO, |
(for O P p and N o e are similar triangles)
| PH² ∶ PN² ∷ NO ∶ OP, | |
| then PN²- PH² = HN² | |
| and PO - NO = NP, | |
| therefore | HN² ∶ PN² ∷ NP ∶ PO, |
| and finally, PO = PN³ HN². |
| Then put FP = HC = FN = ρ; HN | = | ρ - z; then as FP² - FH² = PH² = ρ² - z² |
| PN² = PH² + HN² | = | (ρ² - z²) + (ρ² - 2 ρ z + z²) |
| = | 2ρ² - 2 ρ z | |
| PN | = | √2 ρ (ρ - z) |
| Therefore PO | = | √{2 ρ (ρ - z)}³ (ρ - z)² |
| = | √{2 ρ (ρ - z)}³ (ρ - z)⁴ | |
| and finally, PO | = | 2√2 √ρ³ ρ - z |
Fig. 67.
Finding versed sine of curvature
To find the versed sine of the curvature (which may be useful in the examination of the mirrors by a mould) we may proceed (see fig. 67) to
put AG = f; BE = C; AC = R
then BG² = AG. GD
4 BG² = BE² = 4 AG. GD
C² = 4 f. (2 R - f)
C² = 8 f R - 4 f²
From which equation,
2 f - 2 R = ± √4 R² - C² = -2 R + C²4 R - C⁴64 R³ &c.
2 f = C²4 R - C⁴64 R³
f = C²8 R - C⁴128 R³.
Fig. 68.
Mirror workmanship test installation
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 M m (fig. 68); a = the distance of a light, f, which is arbitrarily placed in front of the mirror; and b = the distance of a moveable screen S, on which the rays reflected from the mirror may converge in a focus. We must find the distance b, at which, with any given distance a, such convergence should take place.
f M′ = a
SM′ = b
OM′ = R
Then (because f MS is bisected by OM, and for points near the vertex of the mirror at M′)
| SM′ ∶ f M′ ∷ SO ∶ O f | |
| or | b ∶ a ∷ R - b ∶ a - R |
| a b - R a = R b - a b. |
From which b = R a 2 a - 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 a clinometer, 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
Cylindric Refractors for Fixed Lights. Having once contemplated the possibility of illuminating Lighthouses by dioptric means, Fresnel quickly 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 small in 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 light equally brilliant in 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 rays in one plane coincident with the line of the section and 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, that Fresnel determined to substitute for it a vertical polygon, composed of what have been improperly called cylindric 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 of Fresnel, 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, that Fresnel invented the distinction for lights, which he calls a 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 made True 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 successfully completed by Messrs Cookson of 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 an annular lens, 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 Messrs Cookson. I also, at a subsequent period, greatly improved 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.
Fig. 70.
Compound Fresnel belt
Such an apparatus is shewn in Plate 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 Monsieur François 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 of totally reflecting prisms, even in lights of the first order or largest dimensions. In this attempt I was much encouraged by the singular liberality of Mr Leonor 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 brother Augustin 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. Mr Fresnel agreed 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 about forty rings 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 attending the formation of the Catadioptric 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 Dr Ritchie of 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 Messrs Cookson of 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 of Bougeur, 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 least twice as 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, are necessarily 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 grinding annular surfaces 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 Messrs Cookson at 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 throwing all the light below the 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’s Table afresh, with results differing from his only in one or two instances, to an amount whose angular value does not exceed more than 2″. The Table in the Appendix contains 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. Fresnel announced, 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 in Plates XVII. and XVIII. 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 appended Table have been determined; and this, I fear, I cannot do without considerable prolixity of detail. Referring to Plates XV., XVI., XVII., and XVIII., in which F shews the flame, RR, the refractors, and MRM and MRM, the spaces through which the light would escape uselessly above and below the 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 refracting them 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
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 of total 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 of total 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 be totally reflected from 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 following conditions:—
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 (see fig. 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 horizontal directions 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
Referring to fig. 72, which shews the first and second zone of the upper series, we have
Tan LCF = FLCL;
and if we make
| the known angle, | SCF = α |
| OCF = ξ = the complement of BCF = the angle of incidence for FC. | |
| DCΟ = γ = angle of refraction. | |
| LCF = θ = (HCF - 90°) = (2 α - 90°) | |
| SCD = (α + γ - ξ) | |
| And | m = the index of refraction for crown glass, |
we obtain the means of determining the angles γ and ξ in two equations, which are based upon the relation between the angles of incidence and refraction, and on the interdependence of the various angles about C. These primary equations are:
sin ξ = m. sin γ
and
[66]γ = 2 ξ - θ (making 2 α - 90° = θ)
[66] The truth of the first of these equations (sin ξ = m. sin γ) which merely expresses the ratio of the sines of the angles of incidence and refraction is obvious; but owing to the great number of small angles about C, a little consideration may be required to enable one to perceive the truth of the second. I therefore subjoin the steps by which I reached it. It is obvious (see fig. 72), that as ACH and BCF are equal, the line SC bisecting HCF must bisect ACB. But the production of AC clearly gives SCD opposite and equal to ACW and SCD is by construction = (α - ξ + γ) = (α + γ - ξ), and, therefore, ACB, which is twice ACW or SCD = (2 α + 2 γ - 2 ξ). Now, by construction OC is a normal to the refracting surface CB and its production C g gives AC g = γ. But γ = ACB - g CB = (2 α + 2 γ - 2 ξ) - g CB = (2 α + 2 γ - 2 ξ) - 90°, hence
| γ | = | {2 α + 2 γ - 2 ξ} - 90°, |
| and γ - 2 γ | = | -γ = -2 ξ + (2 α - 90°) by transposition, and finally changing signs, we have as above: |
| γ | = | 2 ξ - (2 α - 90°) |
| = | 2 ξ - θ. |
Eliminating γ between these two equations we obtain:
sin ξ = m. sin (2 ξ - θ)
an expression, which, after various transformations of circular functions, assumes the form
sin⁴ ξ - 1m sin θ. sin³ ξ + (1 4 m² - 1). sin² ξ + 12 m sin θ. sin ξ + ¹⁄₄ sin² θ = 0[67]
[67] This expression is equivalent to that of M. Fresnel, but owing to a simplification in the fractional coefficients, it is not literally the same. I was led to it by the following steps, starting from the original equation sin ξ = m sin (2 ξ - θ)
| sin ξ | = | m sin (2 ξ - θ) |
| = | m{sin 2 ξ. cos θ - cos 2 ξ sin θ} | |
| = | m cos θ. sin 2 ξ - m sin θ. cos 2 ξ | |
| = | m cos θ. 2 sin ξ. cos ξ - m sin θ. {1 - 2 sin² ξ} | |
| = | 2 m cos θ. sin ξ. cos ξ - m sin θ + 2 m sin θ. sin² ξ. |
Therefore, m sin θ + sin ξ - 2 m sin θ sin² ξ = 2 m cos θ. sin ξ. cos ξ.
Then:
m² sin² θ + 2 m sin θ. sin ξ - 4 m² sin² θ sin² ξ + sin² ξ - 4 m sin θ. sin³ ξ + 4 m² sin² θ. sin⁴ ξ = 4 m² cos² θ sin² ξ (1 - sin² ξ) = 4 m². cos² θ. sin² ξ - 4 m² cos² θ sin⁴ ξ.
Hence we have:
m² sin² θ + 2 m sin θ. sin ξ + (1 - 4 m²). sin² ξ - 4 m sin θ. sin³ ξ + 4 m² sin⁴ ξ = 0
Then dividing by 4 m² and arranging according to powers of ξ, we have as above:
sin⁴ ξ - 1 m sin θ. sin³ ξ + (1 4 m² - 1). sin² ξ + 12 m. sin θ. sin ξ + ¹⁄₄ sin² θ = 0
The solution of this equation, which is of the fourth degree, is somewhat tedious; but as the root, which will satisfy the optical conditions of the question, must be the sine of an angle, and necessarily lies between zero and unity; and as the protraction, if conducted with due care in the manner already described, affords the means of at once assuming a probable value of ξ not very distant from the truth, the labour of the calculation, in this particular case, is not quite so great as might be expected. But notwithstanding all the abridgments of which the particular case admits, a considerable amount of labour is required, and a corresponding risk of error incurred, in merely introducing the numerical values into the equation preparatory to its solution; and any other method requiring less arithmetical operation, is, of course, greatly to be preferred. I therefore willingly adopted the suggestion of a friend, the benefit of whose advice I have on many occasions experienced, and made use of the following ordinary method of approximating to the root of the equation.
If the equation sin ξ - m sin (2 - θ) = 0 (see page 274) be regarded as an expression for the error, when the true value of ξ which would satisfy the equation has been introduced into its first member, we may consider any error in the value of ξ as expressed by the equation:
sin ξ - m. sin (2 ξ - θ) = ε
and differentiating this expression we have:
| d ε | = | cos ξ. d ξ - 2 m cos (2 ξ - θ). d ξ |
| = | {cos ξ - 2 m cos (2 ξ - θ)}. d ξ |
Then dividing by the differential coefficient we obtain
d ξ = d ε cos ξ - 2 m cos (2 ξ - θ)
But when ξ becomes ξ + d ξ, ε will also become ε + d ε; but
ε + d ε = 0
therefore d ε = -ε
hence by substitution we have
d ξ = -ε cos ξ - 2 m cos (2 ξ - θ)
d ξ = -{sin ξ - m sin (2 ξ - θ)} cos ξ - 2 m cos (2 ξ - θ)
d ξ = -sin ξ + m sin (2 ξ - θ) cos ξ - 2 m cos (2ξ - θ)
By substituting, therefore, in this last equation the known values of m and θ, and the assumed value of ξ, a correction is obtained, which being applied to ξ and the same process repeated, new corrections may be found until the value of d ξ falls within the limits of error, which may be considered safe in the particular case. I need hardly say, that where so great a body of flame is employed as in the lights of the first order, these limits are soon passed, more especially as one soon acquires by a little experience the means of guessing a value of ξ not very far from the truth. It is this method I have employed in calculating the appended tables of the zones, in which I have on all occasions, though, perhaps, with needless exactness, pushed my angular determinations to seconds.
Having in this manner determined the angles of BCF, the obtuse angle BCA of the generating triangle of the zone is easily and directly deduced by the following expression, which results from the obvious relations existing among the known angles about C; and we have (see fig. 73),
BCA = 90° + γ = 90° + 2 ξ - θ.
We next proceed to consider the form of BA, the reflecting side of the zone, which is a point of the greatest consequence, as an error in the inclination of any part of its surface is doubled in the resulting direction of the reflected rays. The conditions of the question require, that every ray EW, after reflection at the surface AB, shall, like WI, be parallel to the first ray, which is reflected in the direction BC, and after a second refraction at C, emerges horizontally in CH. But, let us trace backwards the rays as they emerge in their horizontal directions IK, and it is obvious that if BA be made a straight line, then will every ray EW meet the first refracting side BC at the same angle, and there suffering the same refraction, they will go on parallel to each other, and never meet in the focus F. This convergence to F, which is a necessary condition of the problem, may, however, be produced by a curvature of AB, such that all the rays shall have a degree of convergence before falling on BC, sufficient to cause them to be finally refracted, so as to meet in F. On this account, they will occupy less space in passing through BC, than they did in passing through AC; and thus BC will be shorter than AC by some quantity which shall give to that part of AB which is at B the amount of downward inclination required for causing the ray BF finally to converge to F; and the line joining B and A must be a curve, every point of which has its tangent inclined so as to serve the same purpose.
Fig. 73.
Schematic reflecting side of lens
To trace tangents to this curve, is therefore the next step in the process. The direction of the first tangent AZ depends upon very simple considerations; and all that is necessary to be done is to draw a line AU (fig. 73), parallel to BC (which is the parallel to the direction of the reflected rays), and forming an angle CAU, which is, of course, equal to the inclination of the extreme rays refracted by CB at C, with rays reflected from the arc which we have yet to trace. The line AX bisecting this angle, must therefore be a normal to the reflecting surface at A, and AB drawn perpendicular to AX, is consequently a tangent to the reflecting arc.
We must next find the direction of the second tangent Z b, which must be so inclined that the ray F b will, after refraction at b, be reflected into the direction, b C; but as the rigorous determination of this is difficult, I shall describe two approximations suggested to me by M. Leonor Fresnel. The first method is based upon assuming the inclination of the ray refracted at b to the ray refracted at C as equal to:
b FCm
(in which expression, m is the refractive index of the glass); a supposition which obviously differs very little from the truth, as small arcs may be assumed as nearly equal to their sines. Now, it will be recollected, that the rays refracted at C and b, must be reflected at A and b, in a direction parallel to C b, and therefore the inclination of the reflecting surfaces, or that which should be formed by the tangents ZA and Z b, being half that of the incident rays, is, according to the assumption, equal to b FC2 m, which may be expressed by ¹⁄₃ b FC, m being equal to 1·51. But as the inclination of the two radii AX and BX is equal to the inclination of the tangents of the reflecting surfaces to which they are normals, we obtain for the excess B β of the secant of the reflecting arc over its radius the following expression:
Bβ = ¹⁄₂ AB. tan ¹⁄₃ BFC.[68]
The value of B b gives, of course, the direction of the second tangent Z b (which must be equal in length to AZ), whence we easily deduce the chord of the reflecting side A b.
[68] The following steps will shew the mode of obtaining this expression: Suppose (fig. 74, on opposite page) F n to be a ray incident on the surface BC very near b or B (which, although exaggerated in the figure for more easy reference, are close together), and let this ray F n be refracted in the direction n O, and draw n n′ parallel to CA, the ray which is refracted at C, then will n′ n O = m. b FC = ²⁄₃ b FC. But the tangent AZ should make with the tangent b Z an angle equal ¹⁄₃ b FC, or one-half the inclination of the rays refracted at b and C, which are afterwards by the agency of those tangents, to be reflected in the directions parallel to b C and to each other. Hence we have AX b (which is the inclination of the normals to those tangents),
or AX b = BZ b = b FC 2 m = b FC 3 nearly.
Fig. 74.
Explanation of formula
But putting AXB (fig. 73, p. 277) for AX b, and BFC for b FC, a supposition which may be safely made when the differences are so small, and founding upon the analogy
AX ∶ AB ∷ R ∶ tan AXB, we have BA = AX. tan AXB = AX. tan ¹⁄₃ BFC. Then
| AB² | = | B β (B β + 2 AX) |
| = | B β² + 2 B β. AX |
and neglecting B β², which is very small, we have:
BA² = B β. 2 AX nearly,
hence B β = BA²2 AX
But as above AX = BAtan ¹⁄₃ BFC
and substituting this value of AX we obtain:
B β = BA² 2 BAtan ¹⁄₃ BFC
hence we have, as in the text,
B β = ¹⁄₂ BA. tan ¹⁄₃ BFC
The second mode proposed by M. Fresnel, and that which I found most convenient in practice, consists in forming successive hypotheses as to the length of the side BC, and tracing the path of the incident ray FB, which being refracted at B, so as to make with the normal BK an angle = BKY = y′, and finally reflected in the direction BC, must make the angle YBZ = MBC. I shall describe it as follows: In the annexed figure (fig. 75) MBZ is a tangent to the reflecting surface at B, and KBF is the angle of incidence of the ray BF before its refraction at B. If KBF = x´, and the angle of incidence of FC = ECF = x, we have BFC (which is the inclination of those rays to each other, and must be equal to the difference of their angles of incidence to the same surface) = x - x´, whence knowing x, we easily find a value of x´ corresponding to the length of BC. Then for finding the angle of refraction KBY = y´ we have:
sin y´ = sin x´m
Fig. 75.
Refraction in lens
Now, if FB be refracted, so as to make with the reflecting side an angle equal to ZBY, it must (if the position of B be rightly chosen), be reflected so as to follow BC, thus making MBC = YBZ, and calling each of these angles = μ, we have the right angle NBZ made up of μ + y´ + NBK. But NBK clearly equals μ, because it is the inclination of the normals to BC and BZ, and hence y´ + 2 μ = 90°. This, therefore, forms a crucial test for the length of BC. I may only remark, that we already know the numerical value of y; and that of μ is easily found, for μ = CBA + ABM = CBA + BAM = CBA + (MAC - BAC) = CBA + ¹⁄₂ (180° - υ) - BAC. Thus knowing μ and y´, we have only to see whether
(y´ - 2 μ) - 90° = 0
We have now only to find the length of the radius AX or b X (see fig. 73, p. 277), which will describe the reflecting surface or arc AZ b, and to determine the position of its centre X. We already know the values of y′ and y, the angles of refraction of C and b, and their difference y - y′ gives us the inclination of the rays which are to be reflected (into directions parallel to C b) at b and at A. This quantity is, of course, double the inclination of tangents to the reflecting surface AZ and b Z, and of their normals AX and b X. Again, we have the chord line
A b = AC. sin AC b sin (b CA);
and, as above,
AX b = ¹⁄₂ y - y′ = φ
And AX = b X = ρ = A b. sin ¹⁄₂ (180° - φ) sin φ = ¹⁄₂ A b. cosec ¹⁄₂ φ.
And, lastly, for the co-ordinates to X, the centre of curvature for the reflecting arc, we have
| OX | = | ρ. sin OAX |
| and OA | = | ρ. cos OAX.[69] |
[69] The angle OAX is easily found, as will be seen by referring to fig. 73, p. 277; for, AH being horizontal by construction and AO vertical, HAO = 90°; and HAC and CAU being both known, we have
OAX = 90° - (HAU + UAX) = 90° - (HAU + ¹⁄₂ CAU).
Fig. 76.
Finding the position of two of three apices
The positions of the apices A and B of the angles of the zones are also easily found in reference to the focus, and are given in the Table in the Appendix. In fig. 76 we may, in reference to the known position of C, find that of A or B, by simply adding the quantities AH, HC, and BK, to C y or C x, and by deducting CK from C y; while it is obvious that those quantities are respectively proportional to the length of the known sides AC and BC, modified by the inclination of those sides with the horizon. Hence we have AH = AC. sin ACH; HC = AC. cos ACH; BK = BC. sin BCK; and CK = BC. cos BCK.
In the process of grinding the zones, it is found convenient for the workman to give a curved form to the refracting sides BC and AC, the one being made convex and the other concave, so that both being ground to the same radius, the convergence of the rays produced by the first shall be neutralized by the divergence caused by the second. By this arrangement we have three points given in space from which, with given radii, to describe a curvilinear triangle whose revolution round the vertical axis of the system generates the zone required. Co-ordinates to those two centres of curvature for the surfaces AC and BC were determined in reference to arris A of each zone, and will be found in the Appendix. The mode of finding those co-ordinates is, of course, similar to that already given; and, the radii being assumed at 4000 millimètres, the co-ordinates are respectively proportional to the sine and cosine of the inclination of the radius at A to the vertical line, which inclination depends upon the relations of known angles around A and C.
The section ABC (fig. 71, p. 271) of the first zone being thus determined, we proceed by fixing the point C₂ of the second zone, which is at the intersection of the horizon GAG₂ with the ray FBC₂ passing through B. This arrangement prevents any loss of light between the adjacent zones. The calculation of the elements of the second and of every following zone, is precisely similar to that of the first.
The mode of grinding the zones I shall not notice here; but shall refer the more curious reader to the Appendix, in which I have given the details of the process followed by M. Theodore Letourneau, who now manufactures the apparatus for the Northern Lights Board, in the room of M. François Soleil, who is engaged at St Petersburg in the same work. I accordingly proceed to consider what mode should be followed in Testing of Zones. testing the accuracy of the zones. For this purpose, various expedients suggested themselves, such as the application of gauges in the form of a radius, having at one end a plate with a triangular space cut through it, equal and similar to the cross section of the zone. The horizontal motion of this arm would, of course, detect the inaccuracies of the successive sections of the inclosed zone. The application of such a gauge, however, seemed difficult, and in order to test the form of the zones, I satisfied myself with using callipers (similar to the sliding rules used by shoemakers) for measuring the length of the sides of the zone, and a goniometer for the angles, which is represented in the figure (fig. 77), in which ABC represents the prism, with one angle inclosed between the arms AC and AB, moveable round a centre O, and RR the graduated limb. This instrument is inconvenient and defective, as the convexity of the sides AB and BC of the zone requires some skill in getting the arms to be tangents to them.
Fig. 77.
Goniometer
Fig. 78.
Spirit level and cross-hair telescope
A practical test, however, yet remained to be made of the zones when fixed in the brass frames (shewn at Plate XVIII.), and assembled around the common focus of the system, by measuring the final inaccuracy in the path of the rays emergent from them. I have successfully used the following mode. Having mounted the frames containing the zones on a carriage revolving round a small flame placed truly in the common focus, I carefully marked with a piece of soap the centre of the emergent surface of each zone; and having attached to a vertical rod of metal a telescope, provided with a spirit-level and cross-hairs (for cutting the centre of the image of the flame reflected through the zone) in such a manner as to be capable of sliding on the rod, I observed the cutting of the centre of the flame by the cross-hairs. In the case of any aberration from a normal emergence of the central ray, I had thus the means of at once determining its amount and direction. The telescope was moved up or down, and its vertical inclination was varied until the axis of the instrument coincided with the direction of the ray emergent from the centre of each zone, which was made to circulate round the flame, the observer noting any change in the position of the reflected image of the flame, and causing an attendant to mark the zones in which the change occurred, that they might again be subjected to separate examination of the same kind, by adjusting the telescope to the error of each. The vertical inclination of the telescope and the consequent aberration of the ray, was then measured by a graduated arc, with an adjusting spirit-level, moved by a rack and pinion. The accompanying figure (fig. 78) shews the arrangement just described. E is the small flame in the focus; ABC is the zone; TT is the telescope; and R a graduated limb, on which is read the angular deviation θ of the axis of the telescope from the horizon. In the figure, the ray emergent from the centre of AC is shewn dipping below the true level, to which the line TC is supposed to be parallel. I have succeeded by this method in detecting the inaccurate position of some of the zones in the frame; and the error has been reduced by carefully resetting them, so as to diminish considerably the error of a great proportion of the emergent rays. Another mode, and that which, owing to its convenience, was chiefly employed in preference to that just described, was to measure the vertical inclination (given in the Table in the Appendix), of each surface of the zone, and more especially the reflecting surface, by means of the instrument, shewn in figures 79 and 80, after the zones were fixed in their place. The figure (No. 79) shews the mode of gauging the reflecting side AB of a zone of the upper series; and the second (No. 80) shews the position of the instrument in gauging the reflecting side AB of a zone of the under series. In those figures, L is a spirit-level; R, a graduated limb for reading the angular deviation from the true inclination of the tangents to each surface; and SS are studs which rest on the convex surfaces AB and BC of the zones, so as to make the ruler parallel to the tangents of those sides. I have only to add, that I have restricted the error, in the position of the reflecting side of the zones, to 50′ as an extreme limit; and I have invariably endeavoured, in altering the position of the zone in the frame, to throw any error on the side of safety, by causing the rays to dip below the horizon, rather than to rise above it.[70]
Fig. 79.
Lens gauge
Fig. 80.
Lens gauge
[70] In connection with the use of the clinometer, I determined the inclinations of the tangents or chords of the three curve surfaces AB, BC, and AC of each zone with NP, the axis of the system, by means of the obvious relations of the known angles about C, A, and B. Those inclinations (fig. 81) are shewn by the angles BNO, BON, and CPF; and are given in the Table of the Zones in the Appendix.
Fig. 81.
Schematic of cords of curved surfaces
Framing of Zones. The mode of framing the greater zones is shewn in Plate XVIII. and is nearly the same as that used for the Small Harbour Light apparatus of the fourth order (Plate XIX.). The chief difference consists in the diagonal framing, which I adopted for supporting the cupola of 13 zones, which, from its great weight, could not be safely made to rest on the dioptric belt below. That frame is seen in Plates XVII. and XVIII. and is in accordance with the mode of jointing the refractors already described. This system has now been rendered still more complete by the adoption of lanterns composed of diagonal framework, afterwards described and shewn at Plate XXVI.
Mechanical Lamp. We have next to consider the great Lamp, to the proper distribution of whose light, the whole of the apparatus, above described, is applied. Fresnel immediately perceived the necessity of combining with the dioptric instruments which he had invented, a burner capable of producing a large volume of flame; and the rapidity with which he matured his notions on this subject and at once produced an instrument admirably adapted for the end he had in view, affords one of the many proofs of that happy union of practical with theoretical talent, for which he was so distinguished. Fresnel himself has modestly attributed much of the merit of the invention of this Lamp to M. Arago; but that gentleman, with great candour, gives the whole credit to his deceased friend, in a notice regarding lighthouses, which appeared in the Annuaire du Bureau des Longitudes of 1831. The lamp has four concentric burners, which are defended from the action of the excessive heat, produced by their united flames, by means of a superabundant supply of oil, which is thrown up from a cistern below by a clockwork movement and constantly overflows the wicks, as in the mechanical lamp of Carcel. A very tall chimney is found to be necessary, in order to supply fresh currents of air to each wick with sufficient rapidity to support the combustion. The carbonisation of the wicks, however, is by no means so rapid as might be expected, and it is even found that after they have suffered a good deal, the flame is not sensibly diminished, as the great heat evolved from the mass of flame, promotes the rising of the oil in the cotton. I have seen the large lamp at the Tour de Corduan burn for seven hours without being snuffed or even having the wicks raised; and, in the Scotch Lighthouses, it has often, with Colza oil, maintained, untouched, a full flame for no less a period than seventeen hours.
Fig. 82.
Plan of burner
Fig. 83.
Section of burner
Fig. 84.
Burner with chimney and damper
The annexed diagrams will give a perfect idea of the nature of the concentric burner. The first (fig. 82) shews a plan of a burner of four concentric wicks. The intervals which separate the wicks from each other and allow the currents of air to pass, diminish a little in width as they recede from the centre. The next (fig. 83) shews a section of this burner. C, C′, C″, C‴ are the rack-handles for raising or depressing each wick; AB is the horizontal duct which leads the oil to the four wicks; L, L, L, are small plates of tin by which the burners are soldered to each other, and which are so placed as not to hinder the free passage of the air; P is a clamping screw, which keeps at its proper level the gallery R, R, which carries the chimney. The last figure (No. 84) shews the burner with its glass chimney and damper. E is the glass chimney; F is a sheet-iron cylinder, which serves to give it a greater length, and has a small damper D, capable of being turned by a handle, for regulating the currents of air; and B is the pipe which supplies the oil to the wicks. The only risk in using this lamp arises from the liability to occasional derangement of its leathern valves that force the oil by means of clockwork; and several of the lights on the French coast, and more especially the Corduan, have been extinguished by the failure of the lamp for a few minutes, an accident which has never happened, and scarcely can occur with the fountain lamps which illuminate the reflectors. To prevent the occurrence of such accidents, and to render their consequences less serious, various precautions have been resorted to. Amongst others, an alarum is attached to the lamp, consisting of a small cup pierced in the bottom, which receives part of the overflowing oil from the wicks, and is capable, when full, of balancing a weight placed at the opposite end of a lever. The moment the machinery stops, the cup ceases to receive the supply of oil, and, the remainder running out at the bottom, the equilibrium of the lever is destroyed, so that it falls and disengages a spring which rings a bell sufficiently loud to waken the keeper should he chance to be asleep. It may justly be questioned whether this alarum would not prove a temptation to the keepers to relax in their watchfulness and fall asleep; and I have, in all the lamps of the dioptric lights on the Scotch coast, adopted the converse mode of causing the bell to cease when the clockwork stops. There is another precaution of more importance, which consists of having always at hand in the light-room a spare lamp, trimmed and adjusted to the height for the focus, which may be substituted for the other in case of accident. It ought to be noticed, however, that it takes about twenty minutes from the time of applying the light to the wicks to bring the flame to its full strength, which, in order to produce its best effect, should stand at the height of nearly four inches (10cm.). The inconveniences attending this lamp have led to several attempts to improve it; and, amongst others, M. Delaveleye has proposed to substitute a pump having a metallic piston, in place of the leathern valves, which require constant care, and must be frequently renewed. A lamp was constructed in this manner by M. Lepaute, and tried at Corduan; but was afterwards discontinued until some further improvements could be made upon it. It has lately been much improved by M. Wagner, an ingenious artist whom M. Fresnel employed to carry some of his improvements into effect. In the dioptric lights on the Scotch coast, a common lamp, with a large wick, is kept constantly ready for lighting; and, in the event of the sudden extinction of the mechanical lamp by the failure of the valves, it is only necessary to unscrew and remove its burner, and put the reserve-lamp in its place. The height of this lamp is so arranged, that its flame is in the focus of the lenses, when the lamp is placed on the ring which supports the burner of the mechanical lamp; and as its flame, though not very brilliant, has a considerable volume, it will answer the purpose of maintaining the light in a tolerably efficient state for a short time, until the light-keepers have time to repair the valves of the mechanical lamp. Only three occasions for the use of this reserve-lamp have yet occurred.
Height of the flame of the Mechanical Lamp.The most advantageous heights for the flames in dioptric lights are as follows:—
| Inches. | ||||
|---|---|---|---|---|
| 1st Order, | 10 to 11 | centimètres | = | 3·94 to 4·33 |
| 2d Order, | 8 to 9 | ...... | = | 3·15 to 3·54 |
| 3d Order, | 7 to 8 | ...... | = | 2·76 to 3·15 |
Those heights of flame can be obtained only by a careful adjustment of the heights of the wicks and the relative levels of the shoulder of the glass-chimney and the burner, together with a due proportion for the area of the opening of the iron-damper which surmounts it. The wicks must be gradually raised during the first hours of burning to the level of 7 millimètres (0·27 inch) above the burner, a height which they may only very rarely and but slightly exceed. By raising the shoulder of the glass-chimney the volume of the flame is increased; but, after a certain height is exceeded, the flame, on the other hand, becomes reddish, and its brilliancy is diminished. The height of the flame is decreased, and it becomes whiter by lowering the chimney. The chimney is lowered or raised by simply turning to the right or to the left the cylindric glass-holder in which it rests (see Plate XXV.). In regulating the flame, however, recourse is most frequently had to the use of the damper, by enlarging the opening of which the flame falls and becomes whiter and purer; while by diminishing its aperture, the contrary effect is produced. The area of the opening depends on the inclination of a circular disc capable of turning, vertically through a quadrant, on a slender axle of wire, which is commanded by the light-keeper by means of a fine cord which hangs from it to the table below. When the disc (see fig. 84, p. 287) is in a horizontal plane the chimney is shut, when in a vertical plane it is open; and each intermediate inclination increases or decreases the aperture.
I need scarcely add, that in order to produce the proper effect of a system of lenses or refractors, the vertical axis of the flame should Position of flame in reference to focus of apparatus. coincide with their common axis; and it is further necessary, in order to bring the best portion of the flame into a suitable position with reference to the apparatus, that the top of the burner should be quite level, and should stand below the plane of the focus in the following proportions, viz.:—
| For | 1st order, | 28 | millimètres | = | 1·10 | inches. |
| ... | 2d order, | 26 | ... | = | 1·02 | ... |
| ... | 3d order, | 24 | ... | = | 0·95 | ... |
Fig. 85.
Spherical segment spirit level
For the purpose of placing the lamp in the centre of the apparatus, a plumbet with a sharp point suspended in the axis of the apparatus, is used to indicate, by its apex, the place for the centre of the burner. The lamp is then raised or lowered as required by means of four adjusting screws Q at the bottom of its pedestal (Plate XX.); and the top of the burner is made horizontal by a spirit-level, the most convenient form of which is that of the spherical segment, which acts in every azimuth. Its application to this purpose is due, I believe, to M. Letourneau, the successor of M. François in the construction of dioptric apparatus at Paris. This level is shewn in the annexed figure (fig. 85), in which a b is the brass frame containing the level, and O the air-bubble; and e shews circles of equal altitudes engraved on the glass. After the first application of this level, the adjustment of the burner as to its central position is carefully repeated by means of a centre gauge (shewn at fig. 90, p. 295), with reference to the vertex of each lens, or to many points on the internal surface of the refractors; and being found correct, the level is again applied to the top of the burner, to detect any deviation from horizontality that may have occurred during the process of adjusting it to the axis.
The lamp is subject to derangement, chiefly from the stiffness of the clack-valves for want of regular cleaning, bursting of the leathern valves of the oil-box, stiffness of the regulator, and the wearing of the bevelled gearing which gives motion to the connecting-rod that works the valves of the oil-pumps.
Working of the Pumps of the Lamp. The pumps of the lamp should raise, in a given time, four times the quantity of oil actually consumed by burning during that time. Their hourly produce should be,
| lb. avoirdupois. | ||
|---|---|---|
| For the lamp with | four wicks, | 6·615 |
| ...... | three wicks, | 4·410 |
| ...... | two wicks, | 1·675 |
This surplus of three times what is burned is necessary to prevent the wick from being carbonised too quickly; and it has been found quite sufficient for that purpose. The discharge from the pumps is, of course, regulated by changes in the angle of the fans of the regulator, or in the amount of the moving weight.
Care must be taken, in preparing the leathern valves of the pump-box or chamber, shewn in Plate XXII., that they be neither too flaccid from largeness nor too tense from smallness; and also that, after being fitted, they draw no air. To remove the old valves and replace them by fresh ones, is a very simple process, more especially when a proper die or mould is used, which at once cuts the kid-leather, of which the valves are formed, to the required size and squeezes them into the proper shape. In Plates XX., XXI., XXII., XXIII., XXIV., and XXV., the most minute details are given as to the clockwork, pumps, burners, and flame of the great lamp.[71]
[71] See also M. Leonor Fresnel’s Instructions sur l’organisation et la surveillance du service des Phares et Fanaux de France. Paris, 1842, pp. 12, 13, 14, and 15.
The focal point for the lenses and refractors is in the centre of the flame and on the level of its brightest film, as shewn in Plate XXV. Choice of Focal Point for various parts of the apparatus. The choice of a focus for the zones naturally formed a most important practical consideration in their arrangement; and the judicious remarks of M. Leonor Fresnel on that subject, already noticed, would alone have induced me to discard my former calculations in favour of his. For the upper zones, M. Fresnel had adopted a point in the centre of the flame 10 millimètres above the focus of the lenses, so that all the light below that point necessarily falls between the horizon and the Lighthouse; but for the lower zones, it was necessary, owing to their arrangement for convenience in a cylindric form, to adopt a separate focus for each zone in the direction of the centre of gravity of that part of the flame which would light each zone. In this manner (fig. 86) the foci of the zones recede upwards from a to f in proportion to the depression of the zones a, b, c, d, e, f, so that the line joining each zone and its focus, must revolve as a radius vector round some point O between them. The details of this arrangement are shewn in Plate XVIII.; and are also given in the Table of the Catadioptric Zones in the Appendix.
Fig. 86.
Foci of individual lenses
Fig. 87.
Spherical mirror in dioptric light
In the arc next the land, in fixed lights, a great loss of light ensues from the escape of the rays uselessly in that direction. So far back as 1834, I suggested the Application of Spherical Mirrors to fixed Dioptric Lights. placing a segment of a spherical mirror, with its centre of curvature coincident with F the focus of the system, so that the luminous pyramid MFM, of which the mirror MM forms the base, might be thrown back through the focal point and finally refracted into such a direction as to contribute to the effect of the lens QAq in seaward and opposite arc. In the diagram (fig. 87), r r indicate rays proceeding directly from F; r′ r′ rays reflected from MM through F, and finally refracted at QA q; and r″ r″ is the beam compounded of both. In the best glass-silvered mirrors, this accession of light would amount to nearly half of the light incident on them. In such an arrangement, a considerable radius is desirable to decrease the amount of aberration produced by a large flame. In the case of revolving lights of the first order, the radius would, of course, be limited to somewhat less than three feet, which is the focal distance of the lenses, between which and the focus, the reflecting segment must be placed; but in fixed lights, the lantern is the limit of radius, so that a focal length of five feet ten inches may be obtained. M. François ground some beautiful mirrors of three feet radius, which were afterwards tinned by his successor, M. Letourneau, by a new process discovered by himself;[72] and that gentleman is at present engaged in the construction of reflecting spherical segments 1200mm. square (about 16 superficial feet), to a radius of 1770mm. (5 feet 10 inches), which subtend a vertical arc of about 40°.
[72] See notice of a similar process practised about the year 1750 by Mr Rogers of London, ante, p. 240.
Arrangement of Dioptric Apparatus. The arrangements of the dioptric apparatus in the lightroom will be more fully understood by referring to the Plates.
Plate XIII. shews an elevation of a revolving dioptric apparatus of the first order; F is the focal point, in which the flame is placed; L, L great annular lenses, forming by their union an octagonal prism, with the lamp in its axis, and projecting, in horizontal beams, the light which they receive from the focus; L′L′, the upper lenses, forming by their union a frustum of an octagonal pyramid of 50° of inclination, and having their foci coinciding in the point F. They parallelise the rays of light which pass over the lenses. M, M are plane mirrors, placed above the pyramidal lenses L′L′, and so inclined as to project the beams reflected from them in planes parallel to the horizon; Z, Z are the lower zones, first used at Skerryvore, in the room of the curved mirrors which were used at Corduan. The lower part of Plate XIII. shews the moveable framework which carries the lenses and mirrors, and the rollers on which it circulates, with the clockwork which gives motion to the whole. Plate XIV. is the plan of the apparatus shewn in Plate XIII.
Plate XV. shews a section of a fixed dioptric light of the first order. F is the focal point in which the flame is placed; R, R cylindric refractors, forming by their union a prism of thirty-two sides, or a true cylinder, with the lamp in its axis, and producing a zone of light of equal intensity in every point of the horizon; M, M, curved mirrors, ranged in tiers above and below the cylindric refractors, and having their foci coinciding in the point F; the effect of the mirrors increases the power of the light, by collecting and transmitting the rays which would otherwise pass above and below them, without increasing the effect of the light. Plate XVI. gives the elements of the curved mirrors MM of Plate XV.
After the details given of the nature of the catadioptric zones, all that is needful is briefly to refer to Plate XVII., in which ABC and A′B′C′ shew the upper and lower zones which supply the place of the mirrors shewn at M, M, in Plate XV.; while DEF shews the cylindric belt as lately improved, with the diagonal joints M, N, C; and X, X, represent the diagonal supports for the cupola ABC. This plate, in connection with the enlarged view of the same apparatus at Plate XVIII., affords a complete explanation of the arrangement of all the parts.
Arrangement of the Dioptric Apparatus in the Lightroom. For the purpose of arranging the various parts of the dioptric apparatus in their proper positions, three gauges are employed. The first (fig. 88) is for ascertaining that the lenses l, l, meet at the proper horizontal angle, so that their axes shall meet with the proper inclination in F the focus. This is done by means of two arms, whose projecting points r, r, r, r touch the backs of the lenses, while the graduated arc c indicates the inclination of l, l, to l, l, or the complement of that inclination at F.
Fig. 88.
Gauge
Fig. 89.
Clinometer
Again, for ascertaining the verticality of the main lenses, or for setting the subsidiary lenses or mirrors shewn in Plates XIII. and XIV., at the required angle of inclination, recourse is had to a clinometer (fig. 89) touching the back of the lens LL by means of studs at A, A, while the spirit-level S indicates, on the graduated limb, the amount of deviation from the vertical position of the instrument, whether accidental or intentional.
Fig. 90.
Gauge
Lastly, to test the true position of the lamp itself, with reference to the lenses thus properly arranged, we apply a radius or trainer (fig. 90) which fits into the centre burner at F, while its point A touches the centre of the lens l, l; at B is a graduated slide, which admits of the trainer being lengthened or shortened to suit various focal distances; and the spirit-level at c at once corrects any error in the length of the trainer arising from depression or elevation, and also serves to indicate the proper level for the burner which is noticed at page 290, in speaking of the lamp. The dotted line F c′ B′A′ shews the position of the trainer in reference to an adjacent lens.
The elegant apparatus invented by Augustin Fresnel for Harbour Lights, on the same principle as that just described for Sea Lights, is shewn beneath (fig. 91). It consists of thirteen rings of glass of various diameters arranged one above another, in an oval form. The five middle rings have an interior diameter of 11·81 inches (30cm.), and like those of the larger apparatus, refract equally over the horizontal plane of the focus the light which they receive from it. The other rings or prisms, five of which are upper and three lower, are ground and set in such a manner, that they project all the light derived from the focus in a direction parallel to the other rays by total reflection.
Fig. 91.
Fresnel's harbour light apparatus
The arrangements of this apparatus, which is distinguished by the addition of external refractors arranged vertically, will be more fully understood by a reference to fig. 91, which shews its section and plan. F is the focal point in which the flame is placed; r, r cylindric refractors, forming by their union a cylinder with a lamp in its axis, and producing a zone of light of equal intensity all round the horizon; x, x are catadioptric prismatic rings acting by total reflection, and giving out zones of light of equal intensity at every point of the horizon. The dotted lines shew the course traversed by the rays of light which proceed from the lamp, and are acted upon by the rings of glass. The letters r′ r′ shew the external prisms, having their axes at right angles to those of the principal bent prisms, composing the refractors at r, r, and revolving around them. This ingenious application of the property of crossed prism is already described at page 264.
When this apparatus is employed to light only a part of the horizon, the rings are discontinued on the side next the land, and room is thus obtained for using a common fountain lamp; but when the whole horizon is illuminated, the apparatus must inclose the flame on every side; and it has in that case been found most convenient to employ the hydrostatic lamp of Thilorier, which has a balance of sulphate of zinc in solution.
An instrument differing from this small apparatus only in size, has lately been introduced into the Lighthouses in France, and has also been adopted in Scotland for lights in narrow seas. It has the same number of rings of glass as the small apparatus, and of the same proportional dimensions. Its internal diameter, however, is 500 millimètres (about 19¹⁄₂ inches). Drawings of the smaller apparatus are given at Plate XIX., which also contains the radii and the centre of curvature for the rings of the central dioptric belt; while the following Table gives the elements of the eight prismatic zones (above and below the belt), with the co-ordinates to their centres of curvature, measured from the arris A of the outer or emergent surfaces, in whatever position the zone may lie on the lathe. The dimensions are in millimètres; but may be easily converted into imperial inches, in the manner described in the Table of the Great Zones, which will be found in the Appendix:—
| Number of Zone. | Radii of curvature. | Horizontal distance from the axis of the system. | Vertical distance from the outer arris. | Number of Zone. | Radii of curvature. | Horizontal distance from the axis of the system. | Vertical distance from the outer arris. |
|---|---|---|---|---|---|---|---|
| Reflecting Surfaces (Convex). | |||||||
| I. | 1094·5 | 723·5 | 833·78 | V. | 1222·9 | 523·6 | 1169·1 |
| II. | 1045·7 | 652·8 | 891·01 | VI. | 1249·4 | 544·0 | 1192·1 |
| III. | 1044·5 | 598·4 | 933·67 | VII. | 1150·4 | 593·5 | 1062·9 |
| IV. | 1087·3 | 551·9 | 1010·20 | VIII. | 1113·6 | 650·5 | 985·9 |
| Outer Refracting Surfaces (Concave). | |||||||
| I. | 1250·0 | 1290·7 | 303·19 | V. | 1250·00 | 1100·22 | 829·52 |
| II. | 1250·0 | 1274·2 | 445·89 | VI. | 1250·00 | 1112·74 | 821·16 |
| III. | 1250·0 | 1234·0 | 587·05 | VII. | 1250·00 | 1190·50 | 698·00 |
| IV. | 1250·0 | 1173·0 | 717·71 | VIII. | 1250·00 | 1251·90 | 557·22 |
| Inner Refracting Surfaces (Convex). | |||||||
| I. | 1250·0 | 150·67 | 1211·90 | V. | 1250·00 | 452·23 | 1184·20 |
| II. | 1250·0 | 228·19 | 1208·80 | VI. | 1250·00 | 453·07 | 1185·20 |
| III. | 1250·0 | 305·10 | 1203·35 | VII. | 1250·00 | 374·60 | 1196·00 |
| IV. | 1250·0 | 381·00 | 1195·30 | VIII. | 1250·00 | 294·35 | 1203·80 |
Power of Dioptric Instruments. The effect of an annular lens, in combination with the great lamp, may be estimated at moderate distances to be nearly equal to that of 3000 Argand flames of about an inch diameter; that of a cylindric refractor at about 250; and that of a curved mirror may perhaps on an average be assumed at about 10 Argand flames.
Orders of the French Lights. The dioptric lights used in France are divided into four orders, in relation to their power and range; but in regard to their characteristic appearances, this division does not apply, as, in each of the orders, lights of identically the same character may be found, differing only in the distance at which they can be seen, and in the expense of their maintenance. The four orders may be briefly described as follows:—
1st, Lights of the first order having an interior radius or focal distance of 36·22 inches (92cm.), and lighted by a lamp of four concentric wicks, consuming 570 gallons of oil per annum.
2d, Lights of the second order having an interior radius of 27·55 inches (70cm.), lighted by a lamp of three concentric wicks, consuming 384 gallons of oil per annum.
3d, Lights of the third order, lighted by a lamp of two concentric wicks, consuming 183 gallons of oil per annum. The instruments used in those lights are of two kinds, one having a focal distance of 19·68 inches (50cm.), and the other of 9·84 inches (25cm.).
4th, Lights of the fourth order, or harbour-lights, having an internal radius of 5·9 inches (15cm.), and lighted by a lamp of one wick, or Argand burner, consuming 48 gallons of oil per annum. This apparatus is, as already noticed, now more generally used of a larger scale, having a focal distance of 9·84 inches (25cm.), and a lamp of two concentric wicks, consuming about 130 gallons of oil per annum.[73]
[73] An apparatus of 0m.·185cm. was recently added to the list of French lights under the name of the Fifth order; while that of 25cm. radius has been called the Fourth, and that of 15cm. radius is styled the Sixth order. Those minute subdivisions I consider to be unnecessary.
Distinctions of the Dioptric Lights. Those four orders are not intended as distinctions; but are characteristic of the power and range of lights, which render them suitable for different localities on the coast, according to the distance at which they can be seen. This division, therefore, is analogous to that which separates our lights into sea-lights, secondary lights, and harbour-lights, terms which are used to designate the power and position, and not the appearance of the lights to which they are applied.
Each of the above orders is susceptible of certain combinations, which produce various appearances, and constitute the distinctions used for dioptric lights; but the following are those which have been actually employed as the most useful in practice:—
The first order contains, 1st, Lights producing, once in every minute, a great flash, preceded by a smaller one, by the revolution of eight great lenses and eight smaller ones combined with eight mirrors; 2d, Lights flashing once in every half minute, and composed of sixteen half lenses. Those lights may have the subsidiary parts simply catoptric, or diacatoptric; and, 3d, Fixed lights, composed of a combination of cylindric pieces, with curved mirrors or catadioptric zones ranged in tiers above and below them.
The second order comprises revolving lights with sixteen or twelve lenses, which make flashes every half minute; and fixed lights varied by flashes once in every four minutes, an effect which, as already noticed, is produced by the revolution of exterior cylindrical pieces.
The third order (larger diameter) contains common fixed lights, and fixed lights varied by flashes once in every four minutes.
The third order (smaller diameter) contains fixed lights, varied by flashes once in three minutes.
The fourth order has fixed lights varied by flashes once in every three minutes, and fixed lights of the common kind. It has been thought necessary to change the term “fixed lights varied by flashes,” for “fixed light with short eclipses,” because it has been found that, at certain distances, a momentary eclipse precedes the flash.
These distinctions depend upon the periods of revolution, rather than upon the characteristic appearance of the light; and therefore seems less calculated to strike the eye of a seaman, than those employed on the coasts of Great Britain and Ireland. In conformity with this system, and in consideration of the great loss of light which results from the application of coloured media, all distinctions based upon colour have been discarded in the French lights.
The distinctions are, in fact, only four in number, viz.: Fixed; Fixed, varied by flashes;[74] Revolving, with flashes once a minute; and Revolving, with flashes every half minute. To those might be added, Revolving, with bright periods once in two minutes, and perhaps Flashing once in five seconds (as introduced by me at the Little Ross, but I cannot say with such complete success as would induce me to recommend its general adoption). My own experience would also lead me to reject the distinction called “Fixed, varied by flashes,” which I do not consider as possessing a marked or efficient character.
[74] The “Feu fixe, varié par des éclats,” or “Feu fixe, à courtes éclipses,” of Fresnel.
Comparison of Dioptric and Catoptric Apparatus for Revolving Lights. Having thus fully described the nature of the catoptric and dioptric modes of illuminating lighthouses, I shall next compare the merits of both systems, with a view to determine their eligibility in revolving or in fixed lights.
Repeated experiments were made at Gullan-hill, which is distant from Edinburgh about fifteen miles, during the winters of 1832 and 1833, under the inspection of the Commissioners of Northern Lights, the result of which was, that the light of one of the great annular lenses used in the revolving lights of the first order, was equal to the united effect of about eight of the large reflectors employed in the revolving lights on the Scotch coast. It may be said, however, that the dia-catoptric[75] combination of pyramidal lenses and plane mirrors of Corduan, adds the power of more than two reflectors to the effect of the great lens; but it ought to be remembered that in the French lights, this additional power is used only to compensate for one of the defects of the system by lengthening the duration of the flash, and therefore contributes, if at all, only in a very indirect manner, to render the light visible to the mariner at a greater distance. M. Fresnel found, from the smaller divergence of the lens, that the eclipses were too long and the bright periods of the revolution too short; and he therefore determined to adopt the horizontal deviation of 7° for the upper lenses, with a view to remedy this defect. Assuming, therefore, that it were required to increase the number of reflectors in a revolving light of three sides, so as to render it equal in power to a dioptric revolving light of the first order, it would be necessary to place eight reflectors on each face, so that the greatest number of reflectors required for this purpose may be taken at twenty-four. M. Fresnel has stated the expenditure of oil in the lamp of four concentric wicks at 750 grammes of colza oil per hour; and it is found by experience at the Isle of May and Inchkeith, that the quantity of spermaceti oil consumed by the great lamp, is equal to that burned by from fourteen to sixteen of the Argand lamps used in the Scotch lights. It therefore follows that, by dioptric means, the consumption of oil necessary for between fourteen and sixteen reflectors, will produce a light as powerful as that which would require the oil of twenty-four reflectors in the catoptric system of Scotland; and, consequently, that there is an excess of oil equal to that consumed by ten reflectors, or 400 gallons in the year, against the Scotch system. But in order fully to compare the economy of producing two revolving lights of equal power by those two methods, it will be necessary to take into the calculation the interest of the first outlay in establishing them.
[75] I use this word to designate the arrangement of pyramidal lenses and plane mirrors, by which the light is first refracted, and then reflected.
The expense of fitting up a revolving light with twenty-four reflectors, ranged on three faces, may be estimated at L.1298, and the annual maintenance, including the interest of the first cost of the apparatus, may be calculated at L.418, 8s. 4d. The fitting up a revolving light with eight lenses and the dia-catoptric accessory apparatus, may be estimated at L.1459, and the annual maintenance at L.354, 10s. 4d. It therefore follows, that to establish and afterwards maintain a catoptric light of the kind called revolving white, with a frame of three faces, each equal in power to a face of the dioptric light of Corduan, an annual outlay of L.63, 18s. more would be required for the reflecting light than for the lens light; while for a light of the kind called revolving red and white, whose frame has four faces, at least thirty-six reflectors would be required in order to make the light even approach an equality to that of Corduan; and the catoptric light would in that case cost L.225 more than the dioptric light.
The effect produced by burning an equal quantity of oil, in revolving lights on either system, may be estimated as follows:—In a revolving light, like that of Skerryvore, having eight sides, each lighting with its greatest power a horizontal sector of 4°, we have 32° (or units) of the horizon illuminated with the full power of 3200 Argand flames, and consequently an aggregate effect of 102,400 flames, produced by burning the oil required for sixteen reflectors; while in a catoptric apparatus, like that of the old light at Inchkeith, having seven sides of one reflector, each lighting with its greatest power a sector of 4°·25′, we have nearly 31° (or units) of the horizon illuminated with the full power of 400 Argand flames, and consequently an aggregate effect of 12,400 flames as the result of burning the oil required for seven reflectors. Hence, the effect of burning the same quantity of oil in revolving lights on either system, will be represented respectively by 16 7 12,400 = 28,343 for the catoptric, contrasted with 102,400 for the dioptric light; or, in other words, revolving lights on the dioptric principle use the oil more economically than those on the catoptric plan, nearly in the ratio of 3·6 to 1.
Comparison of Catoptric and Dioptric Apparatus for Fixed Lights. I shall now speak of fixed lights, to which the dioptric method is peculiarly well adapted. The effect produced by the consumption of a gallon of oil in a fixed light, with twenty-six reflectors, which is the smallest number that can be properly employed, may be estimated as follows:—The mean effect of the light spread over the horizontal sector, subtended by one reflector, as deduced from measurements made at each horizontal degree, by the method of shadows, is equal to 174 unassisted Argand burners. If, then, this quantity be multiplied by 360 degrees, we shall obtain an aggregate effect of 62,640, which, divided by 1040 (the number of gallons burned during a year in twenty-six reflectors), would give 60 Argand flames for the effect of the light maintained throughout the year by the combustion of a gallon of oil. On the other hand, the power of a catadioptric light of the first order, like that lately established at Girdleness, may be estimated thus:—The mean effect of the light produced by the joint effect of both the dioptric and catadioptric parts of a fixed light apparatus, may be valued at 450 Argand flames, which, multiplied by 360 degrees, gives an aggregate of 162,000; and if this quantity be divided by 570 (the number of gallons burned by the great lamp in a year), we shall have about 284 Argand flames for the effect of the light produced by the combustion of a gallon of oil. It would thus appear that in fixed lights, the French apparatus, as lately improved, produces, as the average effect of the combustion of the same quantity of oil over the whole horizon, upwards of four times the amount of light that is obtained by the catoptric mode; although, in certain directions, opposite the axis of each reflector, the catoptric light be fully 50 per centum more powerful than the dioptric light.
But the great superiority of the dioptric method chiefly rests upon its perfect fulfilment of an important condition required in a fixed light, by distributing the rays equally in every point of the horizon. In the event of the whole horizon not requiring to be illuminated, the dioptric light would lose a part of its superiority in economy, and when half the horizon only is lighted, it would be more expensive than the reflected light; but the greater power and more equal distribution of the light, may be considered of so great importance, as far to outweigh the difference of expense. In the latter case, too, an additional power (as noticed p. 293) can be given to the dioptric light, by placing at the landward side of the lightroom, spherical mirrors with their centres in the focus of the refracting apparatus.[76] The luminous cones, or pyramids of which such reflectors would form the bases, instead of passing off uselessly to the land, would thus be thrown back through the focal point, and finally refracted, so as to increase the effect of the light seaward, by nearly one-third of the light which would otherwise be lost.
[76] A similar arrangement can also be made in revolving lights by making the radius of the mirrors somewhat less than that of the inscribed circle of the octagon bounded by the lenses, so that they may circulate freely round the backs of the mirrors. The shortness of the radius of the reflecting surface would, of course, increase the divergence of the beam of light refracted through the lenses, as the flame would, in this case, subtend a greater angle at the face of the mirrors.
The expense of establishing a fixed light composed of twenty-six reflectors, may be estimated at L.950, and its annual maintenance, including interest on the first cost of the apparatus, may be reckoned at L.425, 10s.: and the expense of fitting up a fixed light on the dioptric principle with catadioptric zones is L.1511, while its annual maintenance may be taken at L.285, 6s. 4d. It thus appears that the annual expenditure of the dioptric fixed light is L.140, 3s. 8d. less than that of a fixed light composed of twenty-six reflectors; while the average effect, equally diffused over the horizon, is four times greater.
The comparative views already given of the catoptric and dioptric modes of illuminating lighthouses, demonstrate that the latter produces more powerful lights by the combustion of the same quantity of oil; while it is obvious that the catoptric system insures a more certain exhibition of the light, from the fountain-lamps being less liable to derangement than the mechanical lamps used in dioptric lights. The balance, therefore, of real advantages or disadvantages, and, consequently, the propriety of adopting the one or the other system, involves a mixed question, not susceptible of a very precise solution, and leaving room for different decisions, according to the value which may be set upon obtaining a cheaper and better light, on the one hand, as contrasted, on the other, with less certainty in its exhibition.
Summary of considerations as to the fitness of the two systems for Revolving Lights. A few general considerations, serving briefly to recapitulate the arguments for and against the two systems, may not be out of place. And, first, regarding the fitness of dioptric instruments for revolving lights, it appears from the details above given,—
1st, That by placing eight reflectors on each face of a revolving frame, a light may be obtained as brilliant as that derived from the great annular lens; and that, in the case of a frame of three sides, the excess of expense by the reflecting mode, would be L.63, 18s.; and in the case of a frame of four sides, the excess would amount to L.225.
2d, That for burning oil economically in revolving lighthouses, which illuminate every point of the horizon successively, the lens is more advantageous than the reflector in the ratio of 3·6 to 1.
3d, That the divergence of the rays from the lens being less than from the reflector, it becomes difficult to produce, by lenses, the appearance which characterises the catoptric revolving lights, already so well known to British mariners; and any change of existing lights which would, of course, affect their appearance, must, therefore, involve many grave practical objections which would not at all apply to the case of new lights.
4th, That the uncertainty in the management of the lamp renders it more difficult to maintain the revolving dioptric lights without risk of extinction, an accident which has several times occurred at Corduan and other lighthouses both in France and elsewhere.
5th, That the extinction of one lamp in a revolving catoptric light is not only less probable, but leads to much less serious consequences than the extinction of the single lamp in a dioptric light; because, in the first case, the evil is limited to diminishing the power of one face by an eighth part; whilst, in the second, the whole horizon is totally deprived of light. The extinction of a lamp, therefore, in a dioptric light, leads to evils which may be considered infinitely great in comparison with the consequences which attend the same accident in a catoptric light.
Summary of considerations as to the fitness of the two systems for Fixed Lights. In comparing the fixed dioptric, and the fixed catoptric apparatus, the results may be summed up under the following heads:—
1st, It is impossible, by means of any practicable combination of paraboloïdal reflectors, to distribute round the horizon a zone of light of exactly equal intensity; while this may be easily effected, by dioptric means, in the manner already described. In other words, the qualities required in fixed lights cannot be so fully obtained by reflectors as by refractors.
2d, The average light produced in every azimuth by burning one gallon of oil in Argand lamps, with reflectors, is only about one-fourth of that produced by burning the same quantity in the dioptric apparatus; and the annual expenditure is L.140, 3s. 8d. less for the entire dioptric light than for the catoptric light.
3d, The characteristic appearance of the fixed reflecting light in any one azimuth would not be changed by the adoption of the dioptric method, although its increased mean power would render it visible at a greater distance in almost every direction; the only exception being in the azimuths opposite the axis of each reflector, where the catoptric light has an excess of power equal to about 50 per centum.
4th, From the equal distribution of the rays, the dioptric light would be observed at equal distances in every point of the horizon; an effect which cannot be fully attained by any practicable combination of paraboloïdal reflectors.
5th, The inconveniences arising from the uncertainty which attends the use of the mechanical lamp, are not perhaps so much felt in a fixed as in a revolving light, because the greater simplicity of the apparatus admits of easier access to it, in case of accident.
6th, But the extinction of a lamp in a catoptric light, leaves only one twenty-sixth part of the horizon without the benefit of the light, and the chance of accident arising to vessels from it, may, therefore, be considered as incalculably less than the danger resulting from the extinction of the single lamp of the dioptric light, which deprives the whole horizon of light.
7th, There may also, in certain situations, be some risk arising from irregularity in the distances at which the same fixed catoptric light can be seen in the different azimuths. This defect, of course, does not exist in the dioptric light.
Advantages and disadvantages of both systems under certain circumstances. There can be little doubt, that the more fully the system of Fresnel is understood, the more certainly will it be preferred to the catoptric system of illuminating lighthouses, at least in those countries where this important branch of administration is conducted with the care and solicitude which it deserves. It must not, however, be imagined, that there are no circumstances in which the catoptric system is not absolutely preferable to illumination by means of lenses. We have hitherto attended only to horizontal divergence and its effects, and this is unquestionably the more important view; but the consideration of vertical divergence must not be altogether overlooked. Now, while it is obvious that vertical divergence, at least above the horizon, involves a total loss of the light which escapes uselessly upwards into space, it is no less true, that if the sheet of light which reaches the most distant horizon of the lighthouse, however brilliant, were as thin as the absence of all vertical divergence would imply, it would be practically useless; and some measure of dispersion in the arc below the horizon is therefore absolutely indispensable to constitute a really useful light. In the reflector, the greatest vertical divergence below the horizontal plane of the focus is 16° 8′, and that of the lens is about 4° 30′.
Let us consider for a moment the bearing of those facts upon the application of the two modes of illumination to special circumstances. The powerful beam of light transmitted by the lens, peculiarly fits that instrument for the great sea-lights which are intended to warn the mariner of his approach to a distant coast which he first makes on an over-sea voyage; and the deficiency of its divergence, whether horizontal or vertical, is not practically felt as an inconvenience in lights of that character, which seldom require to serve the double purpose of being visible at a great distance, and at the same time of acting as guides for dangers near the shore. For such purposes, the lens applies the light much more economically than the reflector, because, while the duration of its least divergent beam is nearly equal to that of the reflector, it is eight times more powerful. A revolving system of eight lenses illuminates an horizontal arc of 32° with this bright beam. The reflector, on the other hand, spreads the light over a larger arc of the horizon; and, while its least divergent beam is much less powerful than that of the lens, the light which is shed over its extreme arc is so feeble as to be practically of little use in lights of extensive range, even during clear weather. When a lighthouse is placed on a very high headland, however, the deficiency of divergence in the vertical direction is often found to be productive of some practical inconvenience; but this defect may be partially remedied by giving to the lenses a slight inclination outwards from the vertical plane of the focus, so as to cause the most brilliant portion of the emergent beam to reach the visible horizon which is due to the height of the lantern. It may be observed, also, that a lantern at the height of 150 feet, which, taking into account the common height of the observer’s eye at sea, commands a range of upwards of 20 English miles, is sufficient for all the ordinary purposes of the navigator, and that the intermediate space is practically easily illuminated, even to within a mile of the lighthouse, by means of a slight inclination of the subsidiary mirrors, even where the light from the principal part of the apparatus passes over the seaman’s head. For the purpose of leading lights, in narrow channels, on the other hand, and for the illumination of certain narrow seas, there can be no doubt that reflectors are much more suitable and convenient. In such cases, the amount of vertical divergence below the horizon, forms an important element in the question, because it is absolutely necessary that the mariner should keep sight of the lights even when he is very near them; while there is not the same call for a very powerful beam which exists in the case of sea-lights. Yet even in narrow seas, where low towers, corresponding to the extent of the range of the light, are used, but where it is, at the same time, needful to illuminate the whole or the greater part of the horizon, the use of dioptric instruments will be found almost unavoidable, especially in fixed lights, as well from their equalizing the distribution of the light in every azimuth, as from their much greater economy in situations where a large annual expenditure would often be disproportionate to the revenue at disposal. In such places, where certain peculiarities of the situation require the combination of a light equally diffused over the greater portion of the horizon, along with a greater vertical divergence in certain azimuths, than dioptric instruments afford, I have found it convenient and economical to add to the fixed refracting apparatus a single paraboloïdal reflector, in order to produce the desired effect, instead of adapting the whole light to the more expensive plan for the sake of meeting the wants of a single narrow sector of its range. In other cases, where the whole horizon is to be illuminated, and great vertical divergence is at the same time desirable, a slight elevation of the burner, at the expense, no doubt, of a small loss of light, is sometimes resorted to, and is found to produce, with good effect, the requisite depression of the emergent rays.
In certain situations, where a great range and, consequently, a powerful light must be combined with tolerably powerful illumination in the immediate vicinity of the lighthouse, we might, perhaps, advantageously adopt a variation of the form and dimensions of the mirrors employed, so as to resemble those formerly employed at the Tour de Corduan, which were of considerably larger surface and longer focal distance than those which are used in Britain. If such a form were adopted, the power of the light for the purpose of the distant range would be increased; and I would propose to compensate for the deficiency of divergence consequent on a long focal distance, by placing a second burner in some position between the parameter and the vertex, and slightly elevated above the axis of the instrument, so as to throw the greater portion of the beam resulting from this second burner below the horizontal plane of the focus. Such an expedient is no doubt somewhat clumsy and would at the same time involve the consumption of twice the quantity of oil used in an ordinary catoptric light; but I can still conceive it to be preferable, in certain situations, to the use of lenses alone.
Thus it appears that we must not too absolutely conclude against one, or in favour of the other mode of illumination for lighthouses; but, as in every other department of the arts, we shall find the necessity of patiently weighing all the circumstances of each particular case that comes before us, before selecting that instrument, or combination of instruments, which appears most suitable.
Distinctions of the Dioptric Lights, and the application of coloured media. The mode of distinguishing lights in the system of Fresnel, depends more upon their magnitude and the measured interval of the time of their revolution, than upon their appearance; and no other very marked distinctions, except Fixed and Revolving, have been successfully attempted in France. As above stated, I consider the distinction of the fixed light varied by flashes, to possess an appearance too slightly differing from that of a revolving light, to admit of its being safely adopted in situations where revolving lights are near. The trial which I made at the Little Ross, in the Solway Frith, of producing, by means of lenses, a light flashing once in five seconds of time, although successful so far as mere distinction is concerned, has several practical defects, arising from the shortness of the duration of the flashes, compared with the powerful effect of the fixed part of the apparatus, which I consider sufficient to prevent its adoption in future, especially considering that a much more marked appearance can be produced, by means of reflectors, as has been done at the Buchanness in Aberdeenshire, and the Rhinns of Islay in Argyllshire. Coloured media have never, so far as I know, been applied to Dioptric apparatus, except in the case of the Maplin Light at the mouth of the Thames, and Cromarty Point Light at the entrance to the Cromarty Frith, and in both instances successfully. The enormous loss of light, however, amounting to no less than 0·80 of the whole incident rays, forms a great bar to the adoption of colour as a distinction; and any means which could tend to lessen that absorption, and at the same time produce the characteristic appearance, would be most valuable. I have tried some glasses of a pink tinge, prepared by M. Letourneau of Paris, whose absorption does not exceed 0·57 of the incident rays; but the appearance of the light, at a distance, is much less marked than that produced by the glasses used in Britain.[77] Such deficiency of characteristic colour might lead to serious consequences, as the transmission of white rays, through a hazy atmosphere, too often produces, by absorption, a reddish tinge of the light, for which the less marked appearance given by the paler media might be easily mistaken. This colouring power of absorption is so well known, that red lights are seldom used except in direct contrast with white ones; but, on a coast so thickly studded with Lighthouses as that of Britain, the number of distinctions is insufficient to supply all our wants, so that we are sometimes reluctantly compelled to adopt a single red light in some situation of lesser importance, or which, from some local circumstances and the appearance of the lights which must be seen by the mariner before passing it, is not likely to be mistaken. The great loss of light by coloured media causes the red beam, in a revolving light, to be seen at a shorter distance than the white; and it is conceivable that, in certain circumstances, this might lead the mariner to mistake a red and white light for a white light revolving at half the velocity. Such a mistake might perhaps prove dangerous; but the lights are generally so situated that there is ample time for the mariner, after first discovering the red light, and thus correcting any mistake, to shape his course accordingly. All other coloured media except red have been found useless as distinctions for any lights of extensive range, and fail to be efficient, owing to the necessity of absorbing almost all the light before a marked appearance can be obtained. In a few pier or ferry lights, green and blue media have been tried, and found available at the distance of a few cables’ lengths.
[77] See page 229, ante.
It seems to be a natural consequence of the physical distribution of light, that fixed lights, which illuminate the whole horizon, should be less powerful than revolving lights which have their effect concentrated within narrow sectors of the horizon. Any attempt to increase the power of fixed lights is, therefore, worthy of attention; Captain Basil Hall’s proposal for producing the appearance of Fixed Lights by rapid movement. and when the late Captain Basil Hall proposed a plan for effecting this object, it received, as it deserved, the full consideration of the Lighthouse Board, who authorised me to repeat Captain Hall’s experiments, and verify his results by observations made at a considerable distance. As some interesting phenomena of irradiation were evolved in the course of those trials, I think it right to give some account of the results which were obtained, as they bear upon various questions connected with the practical arrangements of Lighthouses, under certain circumstances.
In revolving lights on the dioptric principle, the annular lens of Fresnel, as formerly stated, is employed. This instrument, as the reader already knows, possesses the property of projecting to the horizon, in the form of one pencil or beam, all the light which falls on its inner surface from a lamp placed in its principal focus. The consequence of this action is, that when several lenses are so arranged as to form a right prism which circulates round a lamp placed in the common focus, a distant observer receives from each lens, as its axis crosses his line of vision, a bright flash, which is succeeded by total darkness, when one of the dark spaces intermediate between the lenses passes over his eye; and this succession of bright flashes alternating with dark intervals, produces the characteristic appearance of a revolving light.
The fixed light, on the other hand, presents to the eye a steady and unchanging appearance; and the chief object to be obtained in its construction, is to unite the greatest brilliancy with an equal distribution of the light in every direction. The condition of perfect distribution, as already said, is most rigorously fulfilled by the use of refracting zones or belts, which form, by their union, a cylinder enveloping the flame placed in its centre, and possess the property of refracting the light in the vertical direction only, without affecting its natural divergence horizontally. The light from the focus which is incident on the inner surface of the belt is therefore projected forwards in the shape of a flat ring of equal brilliancy all round the horizon.
This repetition may seem needless, but it is hoped it will be found useful in rendering intelligible the following outline of the plan proposed by Captain Hall for the improvement of fixed lights, and the account of the trials that were made with that object in view.
The familiar experiment of whirling a burning stick quickly round the head, so as to produce a ribbon of light, proves the possibility of causing a continuous impression on the retina by intermittent images succeeding each other with a certain rapidity. From the moderate velocity at which this continuity of impression is obtained, we should be warranted in concluding, a priori, that the time required to make an impression on the retina is considerably less than the duration of the impression itself; for the continuity of effect must, of course, be caused by fresh impulses succeeding each other before the preceding ones have entirely faded. If it were otherwise, and the time required to make the impression were equal to the duration of the sensation, it would obviously be impossible to obtain a series of impulses so close or continuous in their effects as to run into and overlap each other, and thus throw out the intervals of darkness; because the same velocity which would tend to shorten the dark intervals, would also curtail the bright flashes, and thus prevent their acting on the eye long enough to cause an impression. Accordingly, we find that the duration of an impression is in reality much greater than the time required for producing the effect on the retina. It is stated by Professor Wheatstone, in the London Transactions for 1834, that only about one millionth part of a second is required for making a distinct impression on the eye; and it appears, from a statement made by Lamé, at p. 425 of his Cours de Physique, that M. Plateau found that an impression on the retina preserved its intensity unabated during one hundredth of a second, so that, however small those times may be in themselves, the one is yet 10,000 greater than the other.
It has been ascertained by direct experiment,[78] that the eye can receive a fresh impression before the preceding one has faded; and, indeed, if this were impossible, absolute continuity of impression from any succession of impulses, however rapid, would seem to be unattainable; and the approach to perfect continuity would be inversely as the time required to make an impression.
[78] Lamé, Cours de Physique, p. 424. “L’impression peut subsister encore lorsque la suivante a lieu.”
Effects of rapid motion on the power of Lights. From this property which bright bodies passing rapidly before the eye possess of communicating a continuous impression to the sense of sight, Captain Hall conceived the idea, not merely of obtaining all the effects of a fixed light, by causing a system of lenses to revolve with such a velocity as to produce a continuous impression, but, at the same time, of obtaining a much more brilliant appearance, by the compensating influence of the bright flashes, which he expected would produce impulses sufficiently powerful and durable to make the deficiency of light in the dark spaces almost imperceptible. The mean effect of the whole series of changes would, he imagined, be thus greatly superior to that which can be obtained from the same quantity of light equally distributed, as in fixed lights, over the whole horizon. Now this expectation, if it be considered solely in reference to the physical distribution of the light, involves various difficulties. The quantity of light subjected to instrumental action is the same whether we employ the refracting zones at present used in fixed dioptric lights, or attempt to obtain continuity of effect by the rapid revolution of lenses; and the only difference in the action of those two arrangements is this, that while the zones distribute the light equally over the whole horizon, or rather do not interfere with its natural horizontal distribution, the effect of the proposed method is to collect the light into pencils, which are made to revolve with such rapidity, that the impression from each pencil succeeds the preceding one in time to prevent a sensible occurrence of darkness. To expect that the mean effect of the light, so applied, should be greater than when it is left to its natural horizontal divergence, certainly appears at first to involve something approaching to a contradiction of physical laws. In both cases, the same quantity of light is acted upon by the instrument; and in either case, any one observer will receive an impression similar and equal to that received by any other stationed at a different part of the horizon; so that, unless we imagine that there is some loss of light peculiar to one of the methods, we are, in the physical view of the question, shut up to the conclusion, that the impressions received by each class of observers must be of equal intensity. In other words, the same quantity of light is by both methods employed to convey a continuous impression to the senses of spectators in every direction, and in both methods equality of distribution is effected, since it does not at all consist with our hypothesis, that any one observer in the same class should receive more or less than his equal share of the light. Then, as to the probability of the loss of light, it seems natural to expect that this should occur in connection with the revolving system, because the velocity is an extraneous circumstance, by no means necessary to an equal distribution of the light, which can, as we already know, be more naturally and at the same time perfectly, attained by the use of the zones.
On the other hand, it must not be forgotten, that although the effect of both methods is to give each part of the horizon an equal share of light, there is yet this difference between them, that while the light from the zones is equally intense at every instant of time, that evolved by the rapidly circulating lenses is constantly passing through every phase between total darkness and the brightest flash of the lens; and this difference, taken in connection with some curious physiological observations regarding the sensibility of the retina, gives considerable countenance to the expectation on which Captain Hall’s ingenious expedient is based. The fact which has already been noticed, and which the beautiful experiments of M. Plateau and Professor Wheatstone have of late rendered more precise, that the duration of an impression on the retina is not only appreciable, but is much greater than the time required to cause it, seems to encourage us in expecting, that while the velocity required to produce continuity of effect would not be found so great as to interfere with the formation of a full impression, the duration of the impulse from each flash would remain unaltered, and the dark intervals which do not excite the retina would, at the same time, be shortened, and that, therefore, we might in this manner obtain an effect on the senses exceeding the brilliancy of a steady light distributed equally in every direction by the ordinary method. Some persons, indeed, who have speculated on this subject, seem even to be of opinion, that, so far from the whole effect of the series of continuous impressions being weakened by a blending of the dark with the bright intervals, the eye would in reality be stimulated by the contrast of light and darkness, so as thereby to receive a more complete and durable impulse from the light. It is obvious, however, that this question regarding the probable effect to be anticipated from a revolution so rapid as to cause a continuous impression, could only have been satisfactorily answered by an appeal to experiment.
In experimenting on this subject, I used the apparatus formerly employed by Captain Hall. It consisted of an octagonal frame, which carried eight of the discs that compose the central part of Fresnel’s compound lens, and was susceptible of being revolved slowly or quickly at pleasure, by means of a crank-handle and some intermediate gearing. The experiments were nearly identical with those made by Captain Hall, who contrasted the effect of a single lens at rest, or moving very slowly, with that produced by the eight lenses, revolving with such velocity as to cause an apparently continuous impression on the eye. To this experiment I added that of comparing the beam thrown out by the central portion of a cylindric refractor, such as is used at the fixed light of the Isle of May, with the continuous impression obtained by the rapid revolution of the lenses. Captain Hall made all his comparisons at the short distance of 100 yards; and in order to obtain some measure of the intensity, he viewed the lights through plates of coloured glass until the luminous discs became invisible to the eye. I repeated those experiments at Gullan, under similar circumstances, but with very different results. I shall not, however, enter upon the discussion of those differences here, although they are susceptible of explanation, and are corroborative of the conclusions at which I arrived, by comparing the lights from a distance of 14 miles; but shall briefly notice the more important results which were obtained by the distant view. They are as follows:—
1. The flash of the lens revolving slowly was very much larger than that of the rapidly revolving series; and this decrease of size in the luminous object presented to the eye, became more marked as the rate of revolution was accelerated, so that, at the velocity of eight or ten flashes in a second, the naked eye could hardly detect it, and only a few of the observers saw it; while the steady light from the fixed refractor was distinctly visible.
2. There was also a marked falling off in the brilliancy of the rapid flashes as compared with that of the slow ones; but this effect was by no means so striking as the decrease of volume.
3. Continuity of impression was not attained at the rate of five flashes in a second, but each flash appeared to be distinctly separated by an interval of darkness; and even when the nearest approach to continuity was made, by the recurrence of eight or ten flashes in a second, the light still presented a twinkling appearance, which was well contrasted with the steady and unchanging effect of the cylindric refractor.
4. The light of the cylindric refractor was, as already stated, steady and unchanging, and of much larger volume than the rapidly revolving flashes. It did not, however, appear so brilliant as the flashes of the quickly revolving lenses, more especially at the lower rate of five flashes in a second.
5. When viewed through a telescope, the difference of volume between the light of the cylindric refractor and that produced by the lenses at their greatest velocity was very striking. The former presented a large diffuse object of inferior brilliancy, while the latter exhibited a sharp pin-point of brilliant light.
Upon a careful consideration of these facts it appears warrantable to draw the following general conclusions:—
1. That our expectations as to the effects of light, when distributed according to the law of its natural horizontal divergence, are supported by observed facts as to the visibility of such lights, contrasted with those whose continuity of effect is produced by collecting the whole light into bright pencils, and causing them to revolve with great velocity.
2. It appears that this deficiency of visibility seems to be chiefly due to a want of volume in the luminous object, and also, although in a less degree, to a loss of intensity, both of which defects appear to increase in proportion as the motion of the luminous object is accelerated.
3. That this deficiency of volume is the most remarkable optical phenomenon connected with the rapid motion of luminous bodies, and that it appears to be directly proportional to the velocity of their passage over the eye.
4. That there is reason to suspect that the visibility of distant lights depends on the volume of the impression in a greater degree than has perhaps been generally imagined.
5. That, as the size and intensity of the radiants causing these various impressions to a distant observer were the same, the volume of the light and, consequently, cæteris paribus, its visibility, are, within certain limits, proportionate to the time during which the object is present to the eye.
Such appear to be the general conclusions which those experiments warrant us in drawing; and the practical result, in so far as lighthouses are concerned, is sufficient to discourage us from attempting to improve the visibility of fixed lights in the manner proposed by Captain Hall, even supposing the practical difficulties connected with the great centrifugal force generated by the rapid revolution of the lenses to be less than they really are.
Connection of the experiments with Irradiation. This decrease in the volume of the luminous object caused by the rapid motion of the lights is interesting, from its apparent connection with the curious phenomenon of irradiation. When luminous bodies, such as the lights of distant lamps, are seen by night, they appear much larger than they would do by day; and this effect is said to be produced by irradiation. M. Plateau, in his elaborate essay on this subject, after a careful examination of all the theories of irradiation, states it to be his opinion, that the most probable mode of accounting for the various observed phenomena of irradiation is to suppose, that, in the case of a night-view, the excitement caused by light is propagated over the retina beyond the limits of the day-image of the object, owing to the increased stimulus produced by the contrast of light and darkness; and he also lays it down as a law confirmed by numerous experiments, that irradiation increases with the duration of the observation. It appears, therefore, not unreasonable to conjecture, that the deficiency of volume observed during the rapid revolution of the lenses may have been caused by the light being present to the eye so short a time, that the retina was not stimulated in a degree sufficient to produce the amount of irradiation required for causing a large visual object. When, indeed, the statement of M. Plateau, that irradiation is proportional to the duration of the observation, is taken in connection with the observed fact, that the volume of the light decreased as the motion of the lenses was accelerated, it seems almost impossible to avoid connecting together the two phenomena as cause and effect.