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.
[Illustration: Fig. 53.]
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]
_r_
F = -------
_m_ - 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.
[Illustration: Fig. 54.]
I begin with the case of the central disc or lens round which the
annular rings are arranged. Its principal section is a mixtilinear
figure (fig. 54) composed of a 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.
[Illustration: Fig. 55.]
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:
( _t′_)
ρ′ = (μ - 1)(φ + ----)
( μ )
and for that near the margin we have:
r
tan _i′_ = -
φ
sin _i′_
sin _e_ = --------
μ
_r′_ = _r_ - _t″_ . tan _e_
_r′_
tan _i_ = ----
φ
sin _i_
sin ε = -------
μ
_r_
ρ″ = ---------√(μ² - 2 μ cos _e_ + 1)
μ sin _e_
and, finally
ρ′ + ρ″[57]
ρ = -------
2
[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.
[Illustration: Fig. 56.]
Then
sin PCG PG
μ = ------- = --;
sin GPC CG
and also
sin QFG QG
μ = ------- = --
sin QGF QF
Now, as P approaches A, we have ultimately PG = AG, QG = BG, and QF =
BF;
Therefore, putting AG = θ and AC = ρ′
AG = θ BG θ - _t′_
μ = -- = ------; μ = -- = --------,
CG = θ - ρ′ BF 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,
( _t′_)
ρ′ = (μ - 1)(φ + ----)
( 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′_.
[Illustration: Fig. 57.]
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
_r_ sin _i′_
tan _i′_ = ---; sin _e_ = --------
φ 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
_r′_ sin _i_
tan _i_ = ---- and sin ε = -------.
φ μ
Next B′ _b_ C = B _b_ C - B _b_ B′ or η = α - ε and sin α = μ sin η =
μ sin (α - ε)
sin α
from which, sin α cos ε - cos α sin ε = -----
μ
( 1)
whence sin α (cos ε - -) = cos α sin ε; and
( μ)
( 2 cos ε 1)
sin² α (cos² ε - ------- + --) = cos² α sin² ε =
( μ μ²)
(1 - sin² α) sin² ε = sin² ε - sin² α sin² ε
Then transposing we have
{ 2 cos ε 1}
sin² α {(cos² ε + sin² ε) - ------- + --} = sin² ε
{ μ μ²}
and because (cos² ε + sin² ε) = 1 we have, by dividing,
sin² ε μ² sin² ε
sin² α = ------------------ = ------------------
{ 2 cos ε 1} μ² - 2 μ cos ε + 1
{1 - ------- + --}
{ μ μ²}
and
μ sin ε
sin α = ---------------------
√(1 - 2 μ cos ε + μ²)
Next, since
_a′_ _b_ _r_
_b_ C = ---------- = -----,
sin AC _b_ sin α
putting C _b_ = _ρ″_, and substituting we have
_r_
_ρ″_ = ------- √(μ² - 2 μ cos ε + 1)
μ sin ε
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
[Illustration: Fig. 58.]
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.
[Illustration: Fig. 59.]
The radius of curvature of the zone may be calculated by the following
formulæ, in which (see fig. 59)
_r_₁ = AB the distance of the outer margin of the zone from the axis of
the lens
_r_₂ = AE the distance of the inner margin from the axis
_l_ = BE the breadth of the zone = _r_₁ - _r_₂
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
_i_₁ = BFA
_i_₂ = EFA
_r_₁ _r_₂
Then tan _i′_₁ = ----; tan _i′_₂ = ----
φ φ
sin _i′_₁ sin _i′_₂
sin _e_₁ = ---------; sin _e_₂ = ---------
_μ_ _μ_
_r′_₁ = _r_₁ - _t″_ sin _e_₁; _r′_₂ = _r_₂ - _t″_ sin _e_₂
_r′_₁ _r′_₂
tan _i_₁ = -----; tan _i_₂ = -----
φ φ
sin _i_₁ sin _i_₂
sin ε = --------; sin ε′ = --------
μ μ
μ sin ε
sin α = ---------------------;
√(μ² - 2 μ cos ε + 1)
μ sin ε′
sin α′ = ---------------------; η = α′ - ε′
√(μ² - 2 μ cos ε′ + 1)
2 cos ε′
and lastly ρ = --------------------------------------
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:
[Illustration: Fig. 60.]
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 = _r_₁; AB′ = _r′_₁; C _b_ = ρ
AE = _r_₂; AE′ = _r′_₂; B _b_ = _t″_; BE = _r_₁ - _r_₂ = _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; _i_₁ = B′FA; _i_₂ = E′FA
_e_₁ = B _b_ B′; _e_₂ = E _e_ E′.
Proceeding exactly as in the case of the central lens we shall have
BA _r_₁ EA _r_₂
tan _i′_₁ = -- = ----; tan _i′_₂ = -- = ----
AF φ AF φ
sin _i′_₁ sin _i′_₂
sin _e_₁ = ---------; sin _e_₂ = ---------
μ μ
_r′_₁ = _r_₁ - _t″_ sin _e_₁; _r′_₂ = _r_₂ - _t″_ sin _e_₂
_r′_₁ _r′_₂
tan _i_₁ = -----; tan _i_₂ = -----
φ φ
sin _i_₁ sin _i′_₂
sin ε = --------; sin ε′ = ---------
μ μ
μ sin ε μ sin ε′
sin α = ---------------------; and sin α′ = ----------------------
√(μ² - 2 μ cos ε - 1) √(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_ _e_ sin _b_ _e_ _m_ _l_ cos ε′
_b_ _m_ = ----------------------- = ---------------------
sin _b_ _m_ _e_ cos {η + ¹⁄₂ (α - α′)}
and in _b_ _m_ C
_b_ _m_ sin _b_ _m_ C _l_ cos ε′ cos ¹⁄₂ (α - α′)
_b_ C = --------------------- = ----------------------------------
sin _b_ C _m_ 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
_l_ cos ε′
ρ = ---------------------------------------
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 - _r_₁
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.
[Illustration: Fig. 61.]
We shall suppose QA (fig. 61) to represent the half of a lens, and
remembering the conditions described in reference to the last figure,
we shall at once perceive the truth of the following analogy (fig.
62):--
[Illustration: Fig. 62.]
OA ∶ AF ∷ AQ ∶ FR ∷ AI ∶ FI, and putting OA = δ, AI = φ′, and AF
= φ, we have δ ∶ φ ∷ φ′ ∶ φ′ - φ, and, consequently, δ φ′ - δ φ =
φ φ′; and hence the following equations, which express the relations
subsisting between the principal focus of the lens and the distance
of any object and its corresponding image:
1_st_, To find the principal focal distance of a lens from the
measured position of its object and its image refracted through it,
we have,
δ φ′
φ = ------.
δ + φ′
2_d_, For the distance of the object, when that of the image is
known, we have,
φ φ′
δ = ------.
φ′ - φ
3_d_, For the position of the image, when that of the object is
known, we have,
δ φ
φ′ = -----.
δ - φ
[Illustration: Fig. 63.]
In testing lenses, of course, it is this last equation which we use,
because the value of φ or the principal focus is always known, and
is that whose accuracy we wish to try, while δ may be chosen within
certain limits at will. I have found that the best mode of proceeding
is the following:--In front of the lens 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.
~Divergence of Annular Lenses.~
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,
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.
_b_ F Rad. of flame
Sin _b_ AF = ----- = sin _b′_ AF = --------------;
AF Focal 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 _x_²); and for any angle at Q, we have
F _x_
sin FQ _x_ = -----.
FQ
[Illustration: Fig. 64.]
~Illuminating Power of Lenses.~
On the subject of the illuminating power of the lenses, it seems
enough to say, that the same general principle regulates the estimate
as in reflectors. Owing to the square form of the lens, however,
there is a greater difficulty in finding 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.
~Arrangement of the Lenses in a Lighthouse.~
The application of lenses to Lighthouses is so obvious as scarcely
to admit of farther explanation than simply to state, 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.
~Pyramidal Lenses and Mirrors.~
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, 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.
~Curved Mirrors.~
[Illustration: Fig. 65.]
Owing to the necessary arrangements of a lantern, only a very small
portion of those rays, which escape from below the lenses, can be
rendered available for the purposes of a Lighthouse; and any attempt
to subject it to lenticular action, so as to add it to the periodic
flashes, would have led to a most inconvenient complication of the
apparatus. 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.
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
[Illustration: Fig. 66.]
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,
PN³
PO = ---.
HN²
Then put FP = HC = FN = ρ; HN = ρ - _z_; then as FP² - FH² = PH² = ρ²
- _z_²
PN² = PH² + HN² = (ρ² - _z_²) + (ρ² - 2 ρ _z_ + _z_²)
= 2 ρ² - 2 ρ _z_
PN = √(2 ρ (ρ - _z_))
Therefore
√{2 ρ (ρ - _z_)}³
PO = ----------------
(ρ - _z_)²
√({2 ρ (ρ - _z_)}³)
= ------------------
(ρ - _z_)⁴
( ρ³ )
and finally, PO = 2 √2 √(-------)
(ρ - _z_)
[Illustration: Fig. 67.]
To find the versed sine of the curvature (which may be useful in the
examination of the mirrors by a mould) we may proceed (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 _f_²
From which equation,
C² C⁴
2 _f_ - 2 R = ± √(4 R² - C²) = --- - ----- &c.
4 R 64 R³
C² C⁴
2 _f_ = --- - -----
4 R 64 R³
C² C⁴
_f_ = --- - -------.
8 R 128 R³
[Illustration: Fig. 68.]
In order to test the accuracy of the workmanship of the mirrors,
recourse must again be had, as in the case of the lenses and
parabolic mirrors, to the formula of conjugate foci, in which we
shall call R = the radius of curvature of the mirror 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_.
R _a_
From which _b_ = ---------,
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.
[Illustration: Fig. 69.]
~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.
~True Cylindric form given to the Refractors and other improvements
in their Construction.~
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 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.
[Illustration: Fig. 70.]
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.
~Catadioptric Zones.~
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, 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.
[Illustration: Fig. 71.]
The problem proposed is, therefore, the determination of the elements
and position of a triangle ABC, which, by its revolution about a
vertical axis, passing through the focus of a system of annular
lenses or refractors in F, would generate a ring or zone capable of
transmitting in an horizontal direction by means 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.
[Illustration: Fig. 72.]
Referring to fig. 72, which shews the _first_ and _second_ zone of the
upper series, we have
FL
Tan LCF = --;
CL
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
1 ( 1 )
sin⁴ ξ - --- sin θ . sin³ ξ + (------ - 1) . sin² ξ +
_m_ (4 _m_² )
1
------ sin θ . sin ξ + ¹⁄₄ sin² θ = 0[67]
2 _m_
[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:
_m_² sin² θ + 2 _m_ sin θ . sin ξ - 4 _m_² sin² θ sin² ξ + sin² ξ -
4 _m_ sin θ . sin³ ξ + 4 _m_² sin² θ . sin⁴ ξ =
4 _m_² cos² θ sin² ξ (1 - sin² ξ) = 4 _m_² . cos² θ . sin² ξ - 4 _m_²
cos² θ sin⁴ ξ.
Hence we have:
_m_² sin² θ + 2 _m_ sin θ . sin ξ + (1 - 4 _m_²) . sin² ξ -
4 _m_ sin θ . sin³ ξ + 4 _m_² sin⁴ ξ = 0
Then dividing by 4 _m_² and arranging according to powers of ξ, we
have as above:
1 ( 1 ) 1
sin⁴ ξ - --- sin θ . sin³ ξ + (------ - 1) . sin² ξ + ----- . sin θ .
_m_ (4 _m_² ) 2 _m_
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_ ε
_d_ ξ = ---------------------------
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
-ε
_d_ ξ = ---------------------------
cos ξ - 2 _m_ cos (2 ξ - θ)
-{sin ξ - _m_ sin (2 ξ - θ)}
= ----------------------------
cos ξ - 2 _m_ cos (2 ξ - θ)
-sin ξ + _m_ sin (2 ξ - θ)
_d_ ξ = ---------------------------
cos ξ - 2 _m_ cos (2 ξ - 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.
[Illustration: Fig. 73.]
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_ FC
------
_m_
(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_ FC
------,
2 _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:
B β = 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),
_b_ FC _b_ FC
or AX _b_ = BZ _b_ = ------ = ------ nearly.
2 _m_ 3
[Illustration: Fig. 74.]
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,
BA²
hence B β = ----
2 AX
BA
But as above AX = -------------
tan (¹⁄₃ BFC)
and substituting this value of AX we obtain:
BA²
B β = -----------------
( BA )
(2 -------------)
( tan (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 _x´_
sin _y´_ = --------
_m_
[Illustration: Fig. 75.]
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
sin (AC _b_)
A _b_ = AC . ------------;
sin (_b_ CA)
and, as above,
AX _b_ = ¹⁄₂ (_y_ - _y′_) = φ
sin (¹⁄₂ (180° - φ))
And AX = _b_ X = ρ = A _b_ . ------------------- =
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).
[Illustration: Fig. 76.]
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.
~Testing of Zones.~
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 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.
[Illustration: Fig. 77.]
[Illustration: Fig. 78.]
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]
[Illustration: Fig. 79.]
[Illustration: Fig. 80.]
[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.
[Illustration: Fig. 81.]
~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.
[Illustration: Fig. 82.]
[Illustration: Fig. 83.]
[Illustration: Fig. 84.]
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 (10^{cm.}). 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.
~Position of flame in reference to focus of apparatus.~
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
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 ...
[Illustration: Fig. 85.]
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.
~Choice of Focal Point for various parts of the apparatus.~
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.
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.
[Illustration: Fig. 86.]
~Application of Spherical Mirrors to fixed Dioptric Lights.~
[Illustration: Fig. 87.]
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 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 QA _q_ 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 1200^{mm.} square
(about 16 superficial feet), to a radius of 1770^{mm.} (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.
[Illustration: Fig. 88.]
[Illustration: Fig. 89.]
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.
[Illustration: Fig. 90.]
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 (30^{cm.}), 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_.
[Illustration: Fig. 91.]
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:--
+------+-------+--------+--------++------+-------+--------+--------+
| | | Hori- | || | | Hori- | |
| | | zontal | || | | zontal | |
| | |distance|Vertical|| | |distance|Vertical|
| | Radii |from the|distance|| | Radii |from the|distance|
|Number| of | axis of|from the||Number| of | axis of|from the|
| of | curva-| the | outer || of | curva-| the | outer |
| Zone.| ture. | system.| arris. || Zone.| ture. | system.| 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:--
1_st_, Lights of the first order having an interior radius or focal
distance of 36·22 inches (92^{cm.}), and lighted by a lamp of four
concentric wicks, consuming 570 gallons of oil per annum.
2_d_, Lights of the second order having an interior radius of 27·55
inches (70^{cm.}), lighted by a lamp of three concentric wicks,
consuming 384 gallons of oil per annum.
3_d_, 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 (50^{cm.}), and the other of 9·84 inches (25^{cm.}).
4_th_, Lights of the fourth order, or harbour-lights, having an
internal radius of 5·9 inches (15^{cm.}), 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 (25^{cm.}), and a lamp of
two concentric wicks, consuming about 130 gallons of oil per annum.[73]
[73] An apparatus of 0^{m.}·185^{cm.} was recently added to the list
of French lights under the name of the _Fifth_ order; while that of
25^{cm.} radius has been called the _Fourth_, and that of 15^{cm.}
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, 1_st_, 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;
2_d_, 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, 3_d_, 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,--
1_st_, 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.
2_d_, 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.
3_d_, 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.
4_th_, 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.
5_th_, 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:--
1_st_, 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.
2_d_, 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.
3_d_, 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_.
4_th_, 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.
5_th_, 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.
6_th_, 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.
7_th_, 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_.
~Captain BASIL HALL’s proposal for producing the appearance of Fixed
Lights by rapid movement.~
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;
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.