[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 (seefig. 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 Cggives ACg= γ. But γ = ACB -gCB = (2 α + 2 γ - 2 ξ) -gCB = (2 α + 2 γ - 2 ξ) - 90°, henceγ={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.
[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 (seefig. 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 Cggives ACg= γ. But γ = ACB -gCB = (2 α + 2 γ - 2 ξ) -gCB = (2 α + 2 γ - 2 ξ) - 90°, hence
Eliminating γ between these two equations we obtain:
sin ξ =m. sin (2 ξ - θ)
sin ξ =m. sin (2 ξ - θ)
an expression, which, after various transformations of circular functions, assumes the form
sin⁴ ξ -1msin θ . sin³ ξ +(14m²- 1). sin² ξ +12msin θ . sin ξ + ¹⁄₄ sin² θ = 0[67]
sin⁴ ξ -1msin θ . sin³ ξ +(14m²- 1). sin² ξ +12msin θ . sin ξ + ¹⁄₄ sin² θ = 0[67]
[67]This expression is equivalent to that of M. Fresnel, but owing to a simplification in the fractional coefficients, it is notliterallythe same. I was led to it by the following steps, starting from the original equation sin ξ =msin (2 ξ - θ)sin ξ=msin (2 ξ - θ)=m{sin 2 ξ . cos θ - cos 2 ξ sin θ}=mcos θ . sin 2 ξ -msin θ . cos 2 ξ=mcos θ . 2 sin ξ . cos ξ -msin θ . {1 - 2 sin² ξ}=2mcos θ . sin ξ . cos ξ -msin θ + 2msin θ . sin² ξ.Therefore,msin θ + sin ξ - 2msin θ sin² ξ = 2mcos θ . sin ξ . cos ξ.Then:m² sin² θ + 2msin θ . sin ξ - 4m² sin² θ sin² ξ + sin² ξ - 4msin θ . sin³ ξ + 4m² sin² θ . sin⁴ ξ = 4m² cos² θ sin² ξ (1 - sin² ξ) = 4m² . cos² θ . sin² ξ - 4m² cos² θ sin⁴ ξ.Hence we have:m² sin² θ + 2msin θ . sin ξ + (1 - 4m²) . sin² ξ - 4msin θ . sin³ ξ + 4m² sin⁴ ξ = 0Then dividing by 4m² and arranging according to powers of ξ, we have as above:sin⁴ ξ -1msin θ . sin³ ξ +(14m²- 1). sin² ξ +12m. sin θ . sin ξ + ¹⁄₄ sin² θ = 0
[67]This expression is equivalent to that of M. Fresnel, but owing to a simplification in the fractional coefficients, it is notliterallythe same. I was led to it by the following steps, starting from the original equation sin ξ =msin (2 ξ - θ)
Therefore,msin θ + sin ξ - 2msin θ sin² ξ = 2mcos θ . sin ξ . cos ξ.
Then:
m² sin² θ + 2msin θ . sin ξ - 4m² sin² θ sin² ξ + sin² ξ - 4msin θ . sin³ ξ + 4m² sin² θ . sin⁴ ξ = 4m² cos² θ sin² ξ (1 - sin² ξ) = 4m² . cos² θ . sin² ξ - 4m² cos² θ sin⁴ ξ.
Hence we have:
m² sin² θ + 2msin θ . sin ξ + (1 - 4m²) . sin² ξ - 4msin θ . sin³ ξ + 4m² sin⁴ ξ = 0
Then dividing by 4m² and arranging according to powers of ξ, we have as above:
sin⁴ ξ -1msin θ . sin³ ξ +(14m²- 1). sin² ξ +12m. sin θ . sin ξ + ¹⁄₄ sin² θ = 0
The solution of this equation, which is of the fourth degree, is somewhat tedious; but as the root, which will satisfy the optical conditions of the question, must be the sine of an angle, and necessarily lies betweenzeroandunity; 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 equationsin ξ -msin (2 - θ) = 0(seepage 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
sin ξ -m. sin (2 ξ - θ) = ε
and differentiating this expression we have:
Then dividing by the differential coefficient we obtain
dξ =dεcos ξ - 2mcos (2 ξ - θ)But when ξ becomes ξ +dξ, ε will also become ε +dε; butε +dε = 0thereforedε = -εhence by substitution we havedξ =-εcos ξ - 2mcos (2 ξ - θ)dξ=-{sin ξ -msin (2 ξ - θ)}cos ξ - 2mcos (2 ξ - θ)dξ =-sin ξ +msin (2 ξ - θ)cos ξ - 2mcos (2ξ - θ)
dξ =dεcos ξ - 2mcos (2 ξ - θ)
But when ξ becomes ξ +dξ, ε will also become ε +dε; but
ε +dε = 0
thereforedε = -ε
hence by substitution we have
dξ =-εcos ξ - 2mcos (2 ξ - θ)
dξ=-{sin ξ -msin (2 ξ - θ)}cos ξ - 2mcos (2 ξ - θ)
dξ =-sin ξ +msin (2 ξ - θ)cos ξ - 2mcos (2ξ - θ)
By substituting, therefore, in this last equation the known values ofmand 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 ofdξ 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 appendedtablesof the zones, in which I have on all occasions, though, perhaps, with needless exactness, pushed my angular determinations toseconds.
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 (seefig. 73),
BCA = 90° + γ = 90° + 2 ξ - θ.
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 theresulting 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 occupylessspace in passing through BC, than they did in passing through AC; and thus BC will beshorterthan AC by some quantity which shall give to that part of AB which is at B the amount ofdownwardinclination required for causing the ray BF finally to converge to F; and the line joining B and A must be a curve, every point of which has its tangent inclined so as to serve the same purpose.
Fig. 73.Schematic reflecting side of lens
Fig. 73.
To trace tangents to this curve, is therefore the next step in the process. The direction of the first tangent AZ depends upon very simple considerations; and all that is necessary to be done is to draw a line AU (fig. 73), parallel to BC (which is the parallel to the direction of the reflected rays), and forming an angle CAU, which is, of course, equal to the inclination of the extreme rays refracted by CB at C, with raysreflected 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 Zb, which must be so inclined that the ray Fbwill, after refraction atb, be reflected into the direction,bC; 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 atbto the ray refracted at C as equal to:
bFCm
bFCm
(in which expression,mis 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 andb, must be reflected at A andb, in a direction parallel to Cb, and therefore the inclination of the reflecting surfaces, or that which should be formed by the tangents ZA and Zb, being half that of the incident rays, is, according to the assumption, equal tobFC2m, which may be expressed by ¹⁄₃bFC,mbeing equal to 1·51. But as the inclination of the two radii AX and BX is equal to the inclination of the tangents of the reflecting surfaces to which they are normals, we obtain for the excess B β of the secant of the reflecting arc over its radius the following expression:
Bβ = ¹⁄₂ AB . tan ¹⁄₃ BFC.[68]
Bβ = ¹⁄₂ AB . tan ¹⁄₃ BFC.[68]
The value of Bbgives, of course, the direction of the second tangent Zb(which must be equal in length to AZ), whence we easily deduce the chord of the reflecting side Ab.
[68]The following steps will shew the mode of obtaining this expression: Suppose (fig. 74, on opposite page) Fnto be a ray incident on the surface BC very nearbor B (which, although exaggerated in the figure for more easy reference, are close together), and let this ray Fnbe refracted in the directionnO, and drawnn′parallel to CA, the ray which is refracted at C, then willn′nO =m.bFC = ²⁄₃bFC. But the tangent AZ should make with the tangentbZ an angle equal ¹⁄₃bFC, orone-halfthe inclination of the rays refracted atband C, which are afterwards by the agency of those tangents, to be reflected in the directions parallel tobC and to each other. Hence we have AXb(which is the inclination of the normals to those tangents),or AXb= BZb=bFC2m=bFC3nearly.Fig. 74.Explanation of formulaBut putting AXB (fig. 73, p. 277) for AXb, and BFC forbFC, a supposition which may be safely made when the differences are so small, and founding upon the analogyAX ∶ AB ∷ R ∶ tan AXB, we have BA = AX . tan AXB = AX . tan ¹⁄₃ BFC. ThenAB²=B β (B β + 2 AX)=B β² + 2 B β . AXand neglecting B β², which is very small, we have:BA² = B β . 2 AX nearly,hence B β =BA²2 AXBut as above AX =BAtan ¹⁄₃ BFCand substituting this value of AX we obtain:B β =BA²2BAtan ¹⁄₃ BFChence we have, as in the text,B β = ¹⁄₂ BA . tan ¹⁄₃ BFC
[68]The following steps will shew the mode of obtaining this expression: Suppose (fig. 74, on opposite page) Fnto be a ray incident on the surface BC very nearbor B (which, although exaggerated in the figure for more easy reference, are close together), and let this ray Fnbe refracted in the directionnO, and drawnn′parallel to CA, the ray which is refracted at C, then willn′nO =m.bFC = ²⁄₃bFC. But the tangent AZ should make with the tangentbZ an angle equal ¹⁄₃bFC, orone-halfthe inclination of the rays refracted atband C, which are afterwards by the agency of those tangents, to be reflected in the directions parallel tobC and to each other. Hence we have AXb(which is the inclination of the normals to those tangents),
or AXb= BZb=bFC2m=bFC3nearly.
or AXb= BZb=bFC2m=bFC3nearly.
Fig. 74.Explanation of formula
Fig. 74.
But putting AXB (fig. 73, p. 277) for AXb, and BFC forbFC, 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
and neglecting B β², which is very small, we have:
BA² = B β . 2 AX nearly,hence B β =BA²2 AXBut as above AX =BAtan ¹⁄₃ BFC
BA² = B β . 2 AX nearly,
hence B β =BA²2 AX
But as above AX =BAtan ¹⁄₃ BFC
and substituting this value of AX we obtain:
B β =BA²2BAtan ¹⁄₃ BFC
B β =BA²2BAtan ¹⁄₃ BFC
hence we have, as in the text,
B β = ¹⁄₂ BA . tan ¹⁄₃ BFC
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 describeit 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 knowingx, we easily find a value ofx´corresponding to the length of BC. Then for finding the angle of refraction KBY =y´we have:
siny´=sinx´m
siny´=sinx´m
Fig. 75.Refraction in lens
Fig. 75.
Now, if FB be refracted, so as to make with the reflecting side an angle equal to ZBY, it must (if the position of B be rightly chosen), be reflected so as to follow BC, thus making MBC = YBZ, and calling each of these angles = μ, we have the right angle NBZ made up of μ +y´+ NBK. But NBK clearly equals μ, because it is the inclination of the normals to BC and BZ, and hencey´+ 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 ofy; 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 andy´, we have only to see whether
(y´- 2 μ) - 90° = 0
(y´- 2 μ) - 90° = 0
We have now only to find the length of the radius AX orbX (seefig. 73, p. 277), which will describe the reflecting surface or arc AZb, and to determine the position of its centre X. We already know the values ofy′andy, the angles of refraction of Candb, and their differencey-y′gives us the inclination of the rays which are to be reflected (into directions parallel to Cb) atband at A. This quantity is, of course, double the inclination of tangents to the reflecting surface AZ andbZ, and of their normals AX andbX. Again, we have the chord line
Ab= AC .sin ACbsin (bCA);
Ab= AC .sin ACbsin (bCA);
and, as above,
AXb= ¹⁄₂y-y′= φAnd AX =bX = ρ =Ab. sin ¹⁄₂ (180° - φ)sin φ= ¹⁄₂ Ab. cosec ¹⁄₂ φ.
AXb= ¹⁄₂y-y′= φ
And AX =bX = ρ =Ab. sin ¹⁄₂ (180° - φ)sin φ= ¹⁄₂ Ab. cosec ¹⁄₂ φ.
And, lastly, for the co-ordinates to X, the centre of curvature for the reflecting arc, we have
[69]The angle OAX is easily found, as will be seen by referring tofig. 73, p. 277; for, AH being horizontal by construction and AO vertical, HAO = 90°; and HAC and CAU being both known, we haveOAX = 90° - (HAU + UAX) = 90° - (HAU + ¹⁄₂ CAU).
[69]The angle OAX is easily found, as will be seen by referring tofig. 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).
OAX = 90° - (HAU + UAX) = 90° - (HAU + ¹⁄₂ CAU).
Fig. 76.Finding the position of two of three apices
Fig. 76.
The positions of the apices A and B of the angles of the zones are also easily found in reference to the focus, and are given in theTablein the Appendix. Infig. 76we 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 Cyor Cx, and by deducting CK from Cy; 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 forthe 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 theAppendix. The mode of finding those co-ordinates is, of course, similar to that already given; and, the radii being assumed at 4000 millimètres, the co-ordinates are respectively proportional to the sine and cosine of the inclination of the radius at A to the vertical line, which inclination depends upon the relations of known angles around A and C.
The section ABC (fig. 71, p. 271) of the first zone being thus determined, we proceed by fixing the point C₂ of the second zone, which is at the intersection of the horizon GAG₂ with the ray FBC₂ passing through B. This arrangement prevents any loss of light between the adjacent zones. The calculation of the elements of the second and of every following zone, is precisely similar to that of the first.
The mode of grinding the zones I shall not notice here; but shall refer the more curious reader to theAppendix, 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 inTesting of Zones.testing the accuracy of the zones. For this purpose, various expedients suggested themselves, such as the application of gauges in the form of a radius, having at one end a plate with a triangular space cut through it, equal and similar to the cross section of the zone. The horizontal motion of this arm would, of course, detect the inaccuracies of thesuccessive sections of the inclosed zone. The application of such a gauge, however, seemed difficult, and in order to test theformof the zones, I satisfied myself with using callipers (similar to the sliding rules used by shoemakers) for measuring thelengthof the sides of the zone, and a goniometer for the angles, which is represented in the figure (fig. 77), in which ABC represents the prism, with one angle inclosed between the arms AC and AB, moveable round a centre O, and RR the graduated limb. This instrument is inconvenient and defective, as the convexity of the sides AB and BC of the zone requires some skill in getting the arms to be tangents to them.
Fig. 77.Goniometer
Fig. 77.
Fig. 78.Spirit level and cross-hair telescope
Fig. 78.
A practical test, however, yet remained to be made of the zones when fixed in the brass frames (shewn atPlate 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 emergentsurfaceof 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 anattendant 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 thetrue 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 theTablein the Appendix), of each surface of the zone,and more especially the reflecting surface, by means of the instrument, shewn infigures 79and80, 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 todipbelow the horizon, rather than to rise above it.[70]
Fig. 79.Lens gauge
Fig. 79.
Fig. 80.Lens gauge
Fig. 80.
[70]In connection with the use of the clinometer, I determined the inclinations of the tangents or chords of the three curve surfaces AB, BC, and AC of each zone with NP, the axis of the system, by means of the obvious relations of the known angles about C, A, and B. Those inclinations (fig. 81) are shewn by the angles BNO, BON, and CPF; and are given in theTable of the Zonesin the Appendix.Fig. 81.Schematic of cords of curved surfaces
[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 theTable of the Zonesin the Appendix.
Fig. 81.Schematic of cords of curved surfaces
Fig. 81.
Framing of Zones.The mode of framing the greater zones is shewn inPlate 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 inPlates XVII.andXVIII.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 atPlate 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.Fresnelimmediately 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.Fresnelhimself 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 theAnnuaire du Bureau des Longitudesof 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 heatevolved from the mass of flame, promotes the rising of the oil in the cotton. I have seen the large lamp at the Tour de Corduan burn for seven hours without being snuffed or even having the wicks raised; and, in the Scotch Lighthouses, it has often, with Colza oil, maintained, untouched, a full flame for no less a period than seventeen hours.
Fig. 82.Plan of burnerFig. 83.Section of burner
Fig. 82.Plan of burner
Fig. 82.Plan of burner
Fig. 82.
Fig. 83.Section of burner
Fig. 83.Section of burner
Fig. 83.
Fig. 84.Burner with chimney and damper
Fig. 84.
The annexed diagrams will give a perfect idea of the nature of the concentric burner. The first (fig. 82) shews a plan of a burner of four concentric wicks. The intervals which separate the wicks from each other and allow the currents of air to pass, diminish a little in width as they recede from the centre. The next (fig. 83) shews a section of this burner. C, C′, C″, C‴ are the rack-handles for raising or depressing each wick; AB is the horizontal duct which leads the oil to the four wicks; L, L, L, are small plates of tin by which the burners are soldered to each other, and which are so placed as not to hinder the free passage of the air; P is a clamping screw, which keeps at its proper level the gallery R, R, which carries the chimney. The last figure (No. 84) shews the burner with its glass chimney and damper. E is the glass chimney; F is a sheet-iron cylinder, which serves to give it a greater length, and has a small damper D, capable of being turned by a handle, for regulating the currents of air; and Bis the pipe which supplies the oil to the wicks. The only risk in using this lamp arises from the liability to occasional derangement of its leathern valves that force the oil by means of clockwork; and several of the lights on the French coast, and more especially the Corduan, have been extinguished by the failure of the lamp for a few minutes, an accident which has never happened, and scarcely can occur with the fountain lamps which illuminate the reflectors. To prevent the occurrence of such accidents, and to render their consequences less serious, various precautions have been resorted to. Amongst others, an alarum is attached to the lamp, consisting of a small cup pierced in the bottom, which receives part of the overflowing oil from the wicks, and is capable, when full, of balancing a weight placed at the opposite end of a lever. The moment the machinery stops, the cup ceases to receive the supply of oil, and, the remainder running out at the bottom, the equilibrium of the lever is destroyed, so that it falls and disengages a spring which rings a bell sufficiently loud to waken the keeper should he chance to be asleep. It may justly be questioned whether this alarum would not prove a temptation to the keepers to relax in their watchfulness and fall asleep; and I have, in all the lamps of the dioptric lights on the Scotch coast, adopted the converse mode of causing the bell to cease when the clockwork stops. There is another precaution of more importance, which consists of having always at hand in the light-room a spare lamp, trimmed and adjusted to the height for the focus, which may be substituted for the other in case of accident. It ought to be noticed, however, that it takes about twenty minutes from the time of applying the light to the wicks to bring the flame to its full strength, which, in order to produce its best effect, should stand at the height of nearly four inches (10cm.). The inconveniences attending this lamp have led to several attempts to improve it; and, amongst others, M.Delaveleyehas 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 thismanner 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.Fresnelemployed 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 asfollows:—
Those heights of flame can be obtained only by a careful adjustment of the heights of the wicks and the relative levels of theshoulderof 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 itsbrilliancy 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 cylindricglass-holderin which it rests (seePlate 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 (seefig. 84, p. 287) is in a horizontal plane the chimney is shut, when in a vertical plane it is open; and each intermediate inclination increases or decreases the aperture.
I need scarcely add, that in order to produce the proper effect of a system of lenses or refractors, the vertical axis of the flame shouldPosition of flame in reference to focus of apparatus.coincide with their common axis; and it is further necessary, in order to bring the best portion of the flame into a suitable position with reference to the apparatus, that the top of the burner should be quite level, and should standbelowthe plane of the focus in the following proportions,viz.:—
Fig. 85.Spherical segment spirit level
Fig. 85.
For the purpose of placing the lamp in the centre of the apparatus, a plumbet with a sharp point suspended in the axis of the apparatus, is used to indicate, by its apex, the place for the centre of the burner. The lamp is then raised or lowered as required by means of four adjusting screws Q at the bottom of its pedestal (Plate XX.); and the top of the burner is made horizontal by a spirit-level, the most convenient form of which is that of the spherical segment, which acts in every azimuth. Its application tothis purpose is due, I believe, to M.Letourneau, the successor of M.Françoisin the construction of dioptric apparatus at Paris. This level is shewn in the annexed figure (fig. 85), in whichabis the brass frame containing the level, and O the air-bubble; andeshews 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 atfig. 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,fourtimes the quantity of oil actually consumed by burning during that time. Their hourly produce should be,
This surplus ofthreetimes 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 inPlate 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 valvesand 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. InPlates XX.,XXI.,XXII.,XXIII.,XXIV., andXXV., the most minute details are given as to the clockwork, pumps, burners, and flame of the great lamp.[71]