Note 189, pp. [138], [153], [156]. Reflection and Refraction. Let P C p, fig. 48, be perpendicular to a surface of glass or water A B. When a ray of light, passing through the air, falls on this surface in any direction I C, part of it is reflected in the direction C S, and the other part is bent at C, and passes through the glass or water in the direction C R. I C is called the incident ray, and I C P the angle of incidence; C S is the reflected ray, and P C S the angle of reflection; C R is the refracted ray, and p C R the angle of refraction. The plane passing through S C and I C is the plane of reflection, and the plane passing through I C and C R is the plane of refraction. In ordinary cases, C I, C S, C R, are all in the same plane. We see the surface by means of the reflected light, which would otherwise be invisible. Whatever the reflecting surface may be, and however obliquely the light may fall upon it, the angle of reflection is always equal to the angle of incidence. Thus I C, Iʹ C, being rays incident on the surface at C, they will be reflected into C S, C Sʹ, so that the angle S C P will be equal to the angle I C P, and Sʹ C P equal to Iʹ C P. That is by no means the case with the refracted rays. The incident rays I C, Iʹ C, are bent at C towards the perpendicular, in the direction C R, C Rʹ; and the law of refraction is such, that the sine of the angle of incidence has a constant ratio to the sine of the angle of refraction; that is to say, the number expressing the length of I m, the sine of I C P, divided by the number expressing the length of R n, the sine of R C p, is the same for all the rays of light that can fall upon the surface of any one substance, and is called its index of refraction. Though the index of refraction be the same for any one substance, it is not the same for all substances. For water it is 1·336; for crown-glass it is 1·535; for flint-glass, 1·6; for diamond, 2·487; and for chromate of lead it is 3, which substance has a higher refractive power than any other known. Light falling perpendicularly on a surface passes through it without being refracted. If the light be now supposed to pass from a dense into a rare medium, as from glass or water into air, then R C, Rʹ C, become the incident rays; and in this case the refracted rays, C I, C Iʹ, are bent from the perpendicular instead of towards it. When the incidence is very oblique, as r C, the light never passes into the air at all, but it is totally reflected in the direction C rʹ, so that the angle p C r is equal to p C rʹ; that frequently happens at the second surface of glass. When a ray I C falls from air upon a piece of glass A B, it is in general refracted at each surface. At C it is bent towards the perpendicular, and at R from it, and the ray emerges parallel to I C; but, when the ray is very oblique to the second surface, it is totally reflected. An object seen by total reflection is nearly as vivid as when seen by direct vision, because no part of the light is refracted. When light falls upon a plate of crown-glass, at an angle of 4° 32ʹ counted from the surface, the glass reflects 4 times more light than it transmits. At an angle of 7° 1ʹ the reflected light is double of the transmitted; at an angle of 11° 8ʹ the light reflected is equal to that transmitted; at 17° 17ʹ the reflected is equal to 1⁄2 the transmitted light; at 26° 38ʹ it is equal to 1⁄4, the variation, according to Arago, being as the square of the cosine.
Note 189, [p. 154]. Atmospheric refraction. Let a b, a b, &c., fig. 49, be strata, or extremely thin layers, of the atmosphere, which increase in density towards m n, the surface of the earth. A ray coming from a star meeting the surface of the atmosphere at S would be refracted at the surface of each layer, and would consequently move in the curved line S v v v A; and as an object is seen in the direction of the ray that meets the eye, the star, which really is in the direction A S, would seem to a person at A to be in s. So that refraction, which always acts in a vertical direction, raises objects above their true place. For that reason, a body at Sʹ, below the horizon H A O, would be raised, and would be seen in sʹ. The sun is frequently visible by refraction after he is set, or before he is risen. There is no refraction in the zenith at Z. It increases all the way to the horizon, where it is greatest, the variation being proportional to the tangent of the angles Z A S, Z A Sʹ, the distances of the bodies S Sʹ from the zenith. The more obliquely the rays fall, the greater the refraction.
Fig. 49.
Fig. 50.
Note 190, [p. 154]. Bradley’s method of ascertaining the amount of refraction. Let Z, fig. 50, be the zenith or point immediately above an observer at A; let H O be his horizon, and P the pole of the equinoctial A Q. Hence P A Q is a right angle. A star as near to the pole as s would appear to revolve about it, in consequence of the rotation of the earth. At noon, for example, it would be at s above the pole, and at midnight it would be in sʹ below it. The sum of the true zenith distances, Z A s, Z A sʹ, is equal to twice the angle Z A P. Again, S and Sʹ being the sun at his greatest distances from the equinoctial A Q when in the solstices, the sum of his true zenith distances, Z A S, Z A Sʹ, is equal to twice the angle Z A Q. Consequently, the four true zenith distances, when added together, are equal to twice the right angle Q A P; that is, they are equal to 180°. But the observed or apparent zenith distances are less than the true on account of refraction; therefore the sum of the four apparent zenith distances is less than 180° by the whole amount of the four refractions.
Note 191, [p. 155]. Terrestrial refraction. Let C, fig. 51, be the centre of the earth, A an observer at its surface, A H his horizon, and B some distant point, as the top of a hill. Let the arc B A be the path of a ray coming from B to A; E B, E A, tangents to its extremities; and A G, B F, perpendicular to C B. However high the hill B may be, it is nothing when compared with C A, the radius of the earth; consequently, A B differs so little from A D that the angles A E B and A C B are supplementary to one another; that is, the two taken together are equal to 180°. A C B is called the horizontal angle. Now B A H is the real height of B, and E A H its apparent height; hence refraction raises the object B, by the angle E A B, above its real place. Again, the real depression of A, when viewed from B, is F B A, whereas its apparent depression is F B E, so E B A is due to refraction. The angle F B A is equal to the sum of the angles B A H and A C B; that is, the true elevation is equal to the true depression and the horizontal angle. But the true elevation is equal to the apparent elevation diminished by the refraction; and the true depression is equal to the apparent depression increased by refraction. Hence twice the refraction is equal to the horizontal angle augmented by the difference between the apparent elevation and the apparent depression.
Fig. 51.