then giving the constants a, b, c the values which belong to the particular refracting substance, we have PM = y representing the rays which are reflected at the angle of incidence pOy out of every 100 rays incident.

The center c of the hyperbola is a point which is easily found from the equation; for we have (y − a) (r + b − x) = c3, the equation of the rectangular hyperbola referred to the asymptotes, and the asymptote CE is distant the quantity a from Ox, whilst the other CD is distant b beyond the parallel line to Oy through A,

Now OM=x is r sin i, therefore

the rays reflected of every 100 incident = a +

c2

br (1-sini)

By marking the observed quantities of the reflected light upon diagrams and searching for the numbers which best represented the experiments, the author concluded that the following were very nearly the values for the kinds of crown, plate, and flint glass which he used.

We obtain the number of rays reflected at the perpendicular

c2

incidence, by putting i = 0, to be a + ; so that the reflexion r+b

at a perpendicular incidence depends upon the properties which determine the values of the whole of the three constants a, b and c.

The number of rays reflected at the highest incidence is

c2

found, by putting i=90°, to be a + and the three constants are again involved in the value.

REFLECTED AT POLISHED SURFACES.

37

This last is never equal to 100 the number supposed incident, as Fresnel's hypothetical formula gives it; as will be seen in the table of the values of the reflexion at different incidences at page 114, Part I.

ART. 22. The physical properties on which the values of the constants depend are still unknown to us, and it will be exceedingly interesting to find if the light which enters the medium at the highest incidence should be found for other substances to be a function of the specific heat for equal bulks, as the author found for the above glasses, together with speculum metal and steel, and given at page 115, Part I.

ART. 23. In the original papers in Brewster's Edinburgh Journal of Science the investigation of the reflective power of the second surfaces of the above kinds of glass are given; and after allowing for the quantity reflected at the first surface and absorbed by the glass, it was found to be the same as at the first surface, and is therefore represented by the above formula.

ART. 24. The reflexion of highly polished metals is of an entirely different character to that of glass; as shewn at page 115, Part I. The quantity reflected of every 100 rays incident being set off in a geometrical construction like that for glass (fig. 20), they are found to be the ordinates for a straight line, as QPR, figure 21; or y representing the number of rays reflected of 100 incident, we have the equation,

y = = ax + b.

For speculum metal highly polished so that when placed near the flame of a candle the surface could not be seen, the following were found for the values of a and b, when

x = 100 sini,

a = trig. tang. 355o•12',

b = 72.3

For highly polished hard steel the values were found to be for a nearly the same as for speculum metal, but for b a value nearly 10 less, or b = 62·3, nearly.

ART. 25. It is still desirable to determine the intensity of the light reflected by glass when the beam of incident light is polarized in any given plane. The formula above given, where the intensity is represented by the ordinate of an hyperbola, may be put into a form which will most probably be found to agree with experiments. These experiments may be made without difficulty with the comparative photometer, by remembering that we may use an analyzer in place of a polarizer for one of the two reflexions which are to be compared.

Ordinary light being considered as made up of two portions polarized in any two planes at right angles to each other, at a perpendicular incidence and at the highest incidence they will be equals; and again, when the beam is considered to consist of one half polarized in the plane of incidence and the other half polarized in the perpendicular plane, then this latter will give no intensity for the reflected beam when incident at the polarizing angle upon the reflecting surface. These data suffice to put the formula for the ordinate to the hyperbola into the following form for polarized light.

The incident beam being supposed to consist of 100 rays polarized in a plane inclined at an angle to the plane of incidence, and i= the angle of incidence, then

The intensity of the reflected beam

member that the constants a, b, c2 are only approximately deter

mined from experiments.

CHAPTER IV.

ON THE INTERFERENCE OF ORDINARY LIGHT.

IN the fundamental experiment of interferences for which we are indebted to M. Fresnel, the light which has radiated around a luminous point, is made by two plane mirrors slightly inclined, or by a very obtuse-angled prism, to radiate as if coming from two points near together, and the two beams thus produced on meeting, being in a fit state, interfere, producing a series of bright and dark bars, as described in Part I. page 48.

ART. 26. PROP. To find the value of the luminiferous interval from the experiment of the two mirrors slightly inclined.

Let a, b be the the points in fig. 22, from which the light radiates after reflexion by the two mirrors; it was shewn, in Part I. page 52, how the distance ab is determined experimentally, and therefore may be supposed to be known. Again, the distance from a and b of any point p, where the luminiferous surfaces meet and interfere, can be measured, and will therefore be known: also by means of a micrometer and eye-piece, the. distance from p to q, where the interference is again of the same character as at p, can be measured in the field of view, and will be therefore again known. Now r being the succeeding place. of like interference with p, we have prλ, and on account of its smallness compared with pq we may consider the triangle pqr as rectilineal, and the angle pqr equals the angle apb, since the luminiferous surfaces are at right angles to the rays respectively; and the triangles pqr, apb are similar;

We see that the places of like interference occur at equal distances m very nearly, and the breadths of the bright and dark bars are proportional to λ, and therefore are narrowest for violet and broadest for red light. The middle bright bars are in this manner coloured in ordinary light, red on their outer edge and violet on their inner edge, and the colours are symmetrical on each side of the line which bisects perpendicularly the distance ab.

By this symmetrical arrangement of the colours we are enabled to judge which is the middle bar, where the rays have travelled over equal spaces along the bisecting line in figure 22. In normal circumstances when the sun is high in the heavens and the atmosphere without cloud or haze, this central bar is. seen black, contrary to the undulatory theory, with the colours of the bars symmetrical on each side of it. The author, to form a luminous point in an unobjectionable manner, used a convex mirror of speculum metal, so that the rays from the virtual image of the sun had not been in a previous state of interference. near a caustic.

ART. 27. PROP. From the observed phenomena of Newton's transmitted rings, to investigate the formula for the intensity when beams of different brightness interfere.

It was explained in Part I. page 73, how Newton's transmitted rings being produced by interfering beams of very dif

*Phil. Mag. for May, 1840, page 380.