Chapter IX (4)

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

The diagram (fig. 47) will serve to render those laws more intelligible. Let a ray of light _a_ O meet a surface of water _n m_ at O, it will be immediately bent into the direction O _a′_; and if, from the centre O, we describe any circle, and draw a line _b_ O _b′_, perpendicular to _nm_; then _ab_ and _a′ b′_, perpendiculars drawn to the normal _bb′_, from the points _a_ and _a′_ where the circle cuts the incident and refracted rays, will be the sines of the angle of incidence _b_ O _a_, and of the angle of refraction _b′_ O _a′_, and the ratio of those sines to each other, or

_b a_
-------
_b′ a′_

will be the relative index of _refraction_ for the two media.

4. It may perhaps be added, for convenience, as a _fourth_ law, deducible from the others, that since rays passing from a dense into a rare medium, have their angle of refraction greater than the angle of incidence, there must be some angle of incidence whose corresponding angle of refraction is a right angle; beyond which no refraction can take place, because there is no angle whose sine can be greater than the radius. In such circumstances, _total reflection_ ensues. For common glass, whose index of refraction is 1·5, we have (in the case of emergent rays) sine of

sine of refraction
incidence = ------------------;
1·5

but, as no sine can exceed radius or unity, the angle of incidence must be limited to 41° 49′; beyond which total reflection will take place, and the light will return _inwards_ into the glass, being _reflected_ at its surface.

Thus, if a ray proceed from a point O (fig. 48), within a piece of glass, to a point C, at its surface A B; and if O C _b_, its incidence, be less than 41° 49′, it will be _refracted_ in some direction C _f_; but if this angle be greater than 41° 49′, as O C′ _b′_, the ray will be _reflected_ back into the glass in the direction C′ O′.

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

Any one may easily satisfy himself by a careful protraction of the angles of _incidence_ and _refraction_, in the manner above described, as to the truth of the following general propositions resulting from those laws:--

1. A ray of light passing through a plate of some diaphanous substance such as glass, with parallel surfaces, suffers no change of _direction_, but emerges in a line parallel to its original path, merely suffering a _displacement_, depending on the obliquity of the incident ray, and the refractive power and thickness of the plate. The effect of this displacement is merely to give the ray an apparent point of origin different from the true one. This will be easily understood by the diagram (fig. 49), in which _a b_ is a normal to the plate, whose surfaces _x x_ and _x′ x′_ are parallel, _r r r r_ shews the path of the ray, _r r_ the displacement, and _r′_ the apparent point of origin resulting from its altered _direction_.

2. When a ray passes through a triangular prism _a b c_, the inclination of the faces _a c_ and _c b_ causes the emergent ray _r′_ to be bent towards _a b_, the base of the prism, in a measure depending on the inclination of the sides of the prism and the obliquity of the incident ray to the first surface.

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

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

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

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

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

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

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.

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.

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.

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′_.

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

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.

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:

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.

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):--

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

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

~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.~

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

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_)

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

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.

~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.

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.~