Chapter IX (5)

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.

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.

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.

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

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_

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).

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.

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]

[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.

~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.

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 ...

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.