32. When this paper was read, drawings were exhibited of two of the prisms whose elements are shown in the preceding table one with an index of refraction 1.5, another with an index 16, and giving deviations of 120° and 135° respectively. For these I think it desirable to substitute, as more instructive, fig. 4, which represents a series of prisms with an index 15 for the extreme red rays, giving deviations from 90° to 120° for the central rays, and reflecting totally all the light incident from a flame subtending at the successive prisms angles varying from about 9° to about 6°. The prisms are constructed, as shown in the figure, to project horizontally the rays from a focal point near the back of the flame, while nearly all the rest of the incident light is sent forth in directions more or less inclined downwards, so as to reach the surface of the sea. The following table exhibits the elements employed in the calculation of each prism, with the computed values of the angle i, and of the maximum angles of incidence of the central and excentric rays. These have been computed by means of the formulæ of article 30, and verified by those of article 28. It is well, perhaps, to note that the new method of constructing monesoptric prisms is here put to a severe test, owing to the large angles which the flame has been assumed to subtend at the prisms. Nevertheless, it appears that even for prisms giving so great a deviation as 120,° and in circumstances more unfavourable than any likely to occur in practice, the maximum angle of incidence does not exceed 47°; while, for central rays, it is less than 41°-a limit over-passed in some cases by prisms of Fresnel's construction.†

*

*The angles i are not seen in the figure, because in practice it will be always desirable and often essential, to deprive the prisms of superfluous glass, by cutting off the angle ACB, fig. 1, by a line joining those points of the faces AC, BC, between which and the angle C no rays are incident; and the figure represents the prisms thus truncated. In the figure the central rays, emerging at the uppermost angles of the first, second, and third prisms, have been accidently omitted, and should be represented by dotted lines parallel to Ox. A Fresnel prism, for which 1.54, and a 29° 14′ 22′′, has 1: 42° 43′ 39".-Renaud sur l'éclairage des Côtes de France, p. 412.

μ

=

=

33. To illustrate the practical application of the results which have now been obtained, I have designed the holophote represented in fig. 5, in which a series of Fresnel prisms is continued up to such a point as to give a deviation at the last prism of 120°. The prisms giving deviations less than 90° have been laid down for an index of refraction 1.5 by means of Mr Balfour's refraction protractor, while the remaining prisms have by the same means been constructed for an index 1.6.

The prisms in the diagram giving deviations exceeding 100° or thereby, are to be understood as representing either Fresnel prisms, or monesoptric prisms of the new construction, like those shown in fig. 4, according to the index of refraction of the glass of which they may be formed. Thus, with an index 15, we could not obtain by means of a Fresnel prism a deviation of 120°. The last prism of the holophote may then be either a Fresnel prism of heavy flint glass, or one of the new monesoptric prisms with an index of 1.5, and, as shown in article 31, an angle varying from 53° 30', possibly to 29°, according to the magnitude of the luminary as compared with its distance from the prism. The light, which is not intercepted by the Fresnel polyzonal lens and the series of prisms, is supposed to be reflected back through the flame by a spherical mirror of metal or glass, according to the designs of Mr Thomas Stevenson.*

Had I been acquainted with Mr Chance's construction of Mr Stevenson's dioptric mirror, I should have availed myself of it; but the designs, fig. 5 and fig. 16, were completed before I had seen his paper on lighthouse apparatus, in which the improved dioptric mirror is described.

But in Mr Stevenson's holophote the Fresnel prisms are employed for deviations only up to 90°, and the reflector, for what may be termed the "back light," is therefore hemispherical. With the new construction, a segment less than a hemisphere, varying with the greatest deviation obtained by the prisms, and just able to catch the light which escapes behind them, will be sufficient.

34. According to the index of refraction of the glass employed in making the prisms, if they be of Fresnel's construction, the maximum deviation of the rays of light will vary; and supposing that in every case the series of prisms is carried as far back as possible, the ratio of the quantity of light incident on the prisms and lenticular apparatus to that incident on the back mirror, will in like manner vary. The following tables exhibit in different cases the portion of light which is thus emitted by the holophote after undergoing simply transmission through a single lens, lenticular zone, or totally-reflecting prism, and the other portion which undergoes two successive actions, being first reflected back through the flame, and then transmitted through a lens, lenticular zone, or prism. I have also added a column of numbers to indicate the portion of light saved in each case from successive transmission through two optical agents, as compared with the performance of a holophote in which the back mirror is a complete hemisphere.

The relation of the portion of light incident on the lenticular and prismatic apparatus to that incident on the back mirror, is computed as follows.

Light being supposed to be radiated equally in all directions, and the quantity of light incident on the surface of a sphere concentric with the luminary being assumed as unity, then if a be the angle AOx, fig. 1, for the hindmost prism, the quantity of light incident on the prisms and lenticular apparatus will be

versin a

Holophotes with Fresnel Prisms giving the greatest attainable Deviations of Central Rays.

Holophotes with new Monesoptric Prisms, giving advantageous Angles of Incidence on the hindmost Prism, with Excentric Rays for which ΑΦ' = 5° 11'.

On New Forms of Lighthouse Apparatus. Part II. By WILLIAM SWAN, F.R.S.E., F.R.S.S.A, &c.*

35. We shall have occasion to make use of the following problem:

Given the directions of the incident and refracted rays at a given point, to find the normal to the refracting surface.

Let AB, BC, Fig. 6, be the incident and refracted rays, it is required to determine BN, the normal to the refracting surface at the point B.

AB, BC, being given, the angle DBC, contained by BC, and AB produced, or the deviation of the ray of light by refraction, is known.

* Read before the Society 27th January 1868.