and we say that the sine of the angle of incidence is to the sine of the angle of refraction as the rapidity of propagation of light in the first medium is to the rapidity of propagation of light in the second medium. The relation of the rapidities of propagation from one medium to the other is constant. It is given the sign n, and called the index of

Hence n indicates how much more or less rapidly light is propagated in the second than in the first medium, and how much the sine of the angle of refraction is smaller or greater than that of the angle of incidence.

The absolute index of refraction is that which is found when light passes from a vacuum into a given medium.

The relative index of refraction is that which is found when light passes from atmospheric air into another medium.

The difference of rapidity of propagation is, moreover, not very great between a vacuum and atmospheric air at 0° (centigrade), or 32 degrees Fahrenheit, and 760 millimetres of barometric pressure. They are to each other as 1 to 1,000,294. Hence we may neglect their difference, and regard light passing from the air into another medium as coming from a vacuum.

Letting v represent the speed of light in a vacuum, v′ in a medium I, v" in a medium II, the absolute index of refraction of the medium I will be

That is to say, that the product of the sine of the angle of incidence and the index of refraction of the first medium is equal to the product of the sine of the angle of refraction and the index of the second medium.

v'

When the first medium, M', in which the luminous point is situated, is less refractive than the second, M", the rapidity (v) of the former is greater than that (") of the second; n = becomes greater than 1, and sin i greater than sin r. Hence the refracted ray BC (Fig. 3) approaches the perpendicular PP. The converse is the case when the light passes from the second into the first medium. We then have = v n

The index of refraction of the second medium is thus the reciprocal value of the index of the first one. The refracted ray CD (Fig. 3) recedes from the perpendicular P' P' in the same proportion that the ray BC approached it. From this it follows that, when a dioptric medium is limited by plane and parallel surfaces which separate it from the same medium on both sides, the incident rays are so refracted that they leave the medium in a direction parallel to their primitive one. They are only more or less displaced laterally. See the rays AB and CD (Fig. 3).

The case is the same for an infinite number of media of different indices of refraction, separated by plane and parallel surfaces, provided that, after having passed through all of them, the luminous ray re-enter the first medium: it will then be parallel to its primitive direction.

We take, as measure of the refractive power of a substance, the deviation to which a ray of light coming from the air (or rather from a vacuum) is subject when it enters this substance, i.e., the relation between the sine of the angle of incidence and the sine of the angle

of refraction, when light passes from a vacuum into this substance. This is, in other words, the absolute index of refraction. It is called simply the index of refraction.

The following are the absolute indices of refraction for certain substances:

Let us take water as an example. The table shows that the index of refraction of water is 1:33 or . This signifies that when a luminous ray penetrates a surface of water at an angle i, the sine of this angle, divided by the sine of the angle of refraction, gives the quotient 1:33.

Letting 10° equal the angle of incidence, what is the angle of re

1 The indices of refraction of the crystalline are those found by Woinow in a person sixteen years of age. The others are taken from Wundt's Physique médicale, translated by Monoyer, and from Pouillet-Müller's Lehrbuch der Physik.

fraction? We may solve the problem either by calculation or by construction. Let us first calculate. We write

sin log 0.1158-17° 30′ 10′′

Dividing 10° by 7° 30′ 10′′ we obtain in fact 1:33.

Let us solve the problem again by construction, which is no less instructive. The number 1.33 is equivalent to, and if we say that the sine of the angle of incidence is to that of the angle of refraction as 4 is to 3, this signifies that the second is always less by than the first.

In Figure 4, let A B represent the incident ray which meets the surface S S in B, at an angle A B P, P P' being the vertical erected at B. In order to find first its sine, we place one point of a pair of dividers at B, and describe a circle of any diameter around this point as a centre. From the point D, where the circle intersects the ray A B, we draw a perpendicular D E to BP. This is the sine of the angle of incidence. Now divide this length ED into four equal parts. Measure off from B toward F, a distance equal to three of these parts. Draw from this point a line parallel to B P, and the point where it intersects the circle will mark the point through

DE

which the refracted ray B C should pass. Indeed G H is the sine of the angle of refraction C B P', for G H =F P= of E D.

The task may be rendered still simpler if we commence by measuring off from B, upon the horizontal, four divisions, BI, to the right in our examples, and three divisions of equal value, BF, toward the left. From the extremity of this line, to the right, erect a vertical line until it intersects the ray at D; at the left let fall the vertical F G.

A circle whose centre is B, and having a radius BD, will indicate, at G, the point through which the refracted ray passes.

There are still several other methods of constructing the refracted ray, but we do not wish to waste time in details.

LIMIT ANGLE-TOTAL REFLECTION.

The greatest value that the angle of incidence can have is evidently 90°, which is the case when light passes along the refracting surface in a direction parallel to the plane of that surface. The sine of 90° is 1.

The value which results, from this, for the angle of refraction r, is called the limit angle. This is evidently the greatest value that the angle of refraction can have.

0.75 corresponds to the sine of 48° 35'. This is the limit angle for

water.

If we consider the matter from another point of view, we may say that, a luminous point being in the water, all rays emanating from it, up to and including those which form with the surface an angle of 48° 35', emerge from the water. The latter The latter rays will pass along the surface of the water. But rays forming with the surface an angle