In the combination of lenses with the flame of a lamp, similar considerations must influence us in making the necessary arrangements, as in the case of reflectors. We have already seen that the size of the flame and its distance from the surface of reflecting instruments have an important practical bearing on the utility of the instrument, Divergence of Annular Lenses. and that the divergence of the resultant beam materially affects its fitness for the purpose of a Lighthouse. So also, in the case of the lens, unless the diameter of the flame of the lamp has to the focal distance of the instrument a relation such as may cause an appreciable divergence of the rays refracted through it, it could not be usefully applied to a Lighthouse; for, without this, the light would be in sight during so short a time, that the seaman would have much difficulty in observing it. To determine the amount of this divergence of the refracted beam, therefore, is a matter of great practical importance, and I shall briefly point out the conditions which regulate its amount, as they are nearly identical with those which determine the divergence of a paraboloïdal mirror illuminated by a lamp in its focus. The divergence, in the case of lenses, may be described as the angle which the flame subtends at the principal focus of the lens, the maximum of which, produced at the vertex of Fresnel’s great lens by the lamp of four concentric wicks, is about 5° 9′.[60]
[60] This will be easily seen by examining the annexed [figure (64)], in which Q q represents the lens. A its centre, F the principal focus, b F and b′ F the radius of the flame; then is the angle b A b′ equal to the maximum divergence of the lens. Sin b AF = b FAF = sin b′ AF = Rad. of flameFocal distance; and twice b AF = the whole divergence at A. Then for the divergence at the margin of the lens, or at any other point, we have, FQ = √AQ² + AF² and Q x = √QF² + F x²; and for any angle at Q, we have sin FQ x = F xFQ.
Fig. 64.
Illuminating Power of Lenses. On the subject of the illuminating power of the lenses, it seems enough to say, that the same general principle regulates the estimate as in reflectors. Owing to the square form of the lens, however, there is a greater difficulty in finding a mean focal distance whereby to correct our estimate of the angle subtended by the light, so as to equate the varying distance of the several parts of the surface; but, practically, we shall not greatly err if we consider the quotient of the surface of the lens divided by the surface of the flame as the increased power of illumination by the use of the lens. The illuminating effect of the great lens, as measured at moderate distances, has generally been taken at 3000 Argand flames, the value of the great flame in its focus being about 16, thus giving its increasing power as nearly equal to 180. The more perfect lenses have produced a considerably greater effect.
The application of lenses to Lighthouses is so obvious as scarcely to admit of farther explanation than simply to state, Arrangement of the Lenses in a Lighthouse. that those instruments are arranged round a lamp placed in their centre, and on the level of the focal plane in the manner shewn in [Plates XIII.] and [XIV.],[61] so as to form by their union a right octagonal hollow prism, circulating round the flame which is fixed in the centre, and shewing to a distant observer successive flashes or blazes of light, whenever they cross a line joining his eye and the lamp, in a manner similar to that already noticed in describing the action of the mirrors. The chief difference in the effect consists in the greater intensity and shorter duration of the blaze produced by the lens; which latter quantity is, of course, proportional to the divergence of the resultant beam. Each lens subtends a central horizontal pyramid of light of about 46° of inclination, beyond which limits the lenticular action could not be advantageously pushed, owing to the extreme obliquity of the incidence of light; but Fresnel at once conceived the idea of pressing into the service of the mariner, by means of two very simple expedients, the light which would otherwise have uselessly escaped above and below the lenses.
[61] The Plates shew the nature of the mechanical power which gives movement to the lenses. It consists of a clockwork movement driven by a weight which sets in motion a plate bearing brackets that carry the lenses. All this, however, can be seen from the Plates; and I am unwilling to expend time in a detailed explanation of what is obvious by inspection.
For intercepting the upper portion of the light, Fresnel employed eight smaller lenses of 500 mm. focal distance (19·68 inches) inclined inwards towards the lamp, which is also their common focus and thus forming, by their union, Pyramidal Lenses and Mirrors. a frustum of a hollow octagonal pyramid of 50° of inclination. The light falling on those lenses is formed into eight beams parallel to the axis of the smaller lenses, and rising upwards at an angle of 50° inclination. Above them are ranged eight plane mirrors, so inclined (see [Plates XIII.] and [XIV.]) as to project the beams transmitted by the small lenses in the horizontal direction, so as finally to increase the effect of the light. In placing those upper lenses, it is generally thought advisable to give their axis an horizontal deviation of 7° or 8° from that of the great lenses and in the direction contrary to that of the revolution of the frame which carries the lenticular apparatus. By this arrangement, the flashes of the smaller lenses precede that of the large ones, and thus tend to correct the chief practical defect of revolving lenticular lights by prolonging the bright periods. The elements of the subsidiary lenses depend upon the very same principles, and are calculated by the same formulæ as those given for the great lenses. In fixing the focal distance and inclination of those subsidiary lenses, Fresnel was guided by a consideration of the necessity for keeping them sufficiently high to prevent interference with the free access to the lamp. He also restricted their dimensions within very moderate limits, so as to avoid too great weight. The focal distance is the same as that for lenses of the third order of lights.
Fig. 65.