beam, in the circumstances represented in Fig. 705, is turned aside some 40° or 50° from its original course.1

Since the rays which traverse a prism are bent away from the edge, the object from which they proceed will appear, to an observer looking through the prism, to be more nearly in the direction of the

edge than it really is. If, for example, he looks at the flame of a candle through a prism placed so that the edge which corresponds to the refracting angle is at the top (Fig. 706), the apparent place of the flame will be above its true place.

995. Formulæ for Refraction through Prisms. Minimum Deviation. -Let SI (Fig. 707) be an incident ray in the plane of a principal section A B C of a prism. Let i be the angle of incidence SIN, and

1 The phenomena here described are complicated in practice by the unequal refrangibility of rays of different colours (Chap. lxxii.). The complication may be avoided by employing homogeneous light, of which a spirit-lamp, with common salt sprinkled on the wick, affords a nearly perfect example.

REFRACTION THROUGH PRISM.

1007

r the angle of refraction MII'. Then, denoting the index of refraction by μ, we have sin i=μ sin r. In like manner, putting r' for the angle of internal incidence on the second face II' M, and i for the angle of external refraction N'I'R, we have sin i'=μ sin r'.

The deviation produced at I is i-r, and that at I' is '-', so that the total deviation, which is the acute angle D contained between the rays SI, RI', when produced to meet at o, is

But if we drop a perpendicular from the angular point A on the ray II', it will divide the refracting angle BAC into two parts, of which that on the left will be equal to r, and that on the right to r', since the angle contained between two lines is equal to that contained between their perpendiculars. We have therefore A=r+r', and by substitution in the above equation

When the path of the ray through the prism II' makes equal angles with the two faces, the whole course of the ray is symmetrical with respect to a plane bisecting the refracting angle, so that we have

This last result is of great practical importance, as it enables us to

calculate the index of refraction μ from measurements of the refracting angle A of the prism, and of the deviation D which occurs when the ray passes symmetrically.

When a beam of sunlight in a dark room is transmitted through a prism, it will be found, on rotating the prism about its axis, that there is a certain mean position which gives smaller deviation of the transmitted light than positions on either side of it; and that, when the prism is in this position, a small rotation of it has no sensible effect on the amount of deviation. The position determined experimentally by these conditions, and known as the position of minimum deviation, is the position in which the ray passes symmetrically. 996. Construction for Deviation.-The following geometrical con

N

N

B'

B

Fig. 708.

General Construction for Deviation.

=

draw a radius OA of the smaller circle to represent the direction of the incident ray, and let NAB be the direction of the normal to the surface at the point of incidence, so that OAN is the angle of incidence. Join O B. Then OBN is the angle of refraction, since sin O AN OB index of refraction; hence OB is parallel to the resin OBN OA fracted ray. If the incidence is from dense to rare, we must draw OB to represent the incident ray, make OBN equal to the angle of incidence, and join O A. In either case the angle AOB is the deviation, and it evidently increases with the angle of incidence OA N, attaining its greatest value when this angle (O A N" in the figure) is a right angle, in which case the angle of refraction O B" N" is the critical angle.

2. To find the deviation in refraction through a prism, describe two concentric circular arcs as before (Fig. 709), the ratio of their radii being the index of refraction. Draw the radius OA of the smaller circle to represent the incident ray, NB to represent the

MINIMUM DEVIATION FOR PRISM.

1009

normal at the first surface, BN'the normal at the second surface. Then O B represents the direction of the ray in the prism, OA' the direction of the emergent ray, and AO A' is accordingly the total deviation. In fact we have

Again, the deviation AOA', being the angle at the centre of a circle, is measured by the arc A A', which subtends it. To obtain the minimum deviation, we must so arrange matters that the angle ABA' being given (=angle of prism), the arc A A' shall be a minimum. Let A B A', a B a' (Fig. 710), be two consecutive positions,

N

A a

Α

B

Fig. 703.-Application to Prism.

B

Fig. 710.-Proof of Minimum Deviation.

BA' and Ba' being greater than BA and B a. Then, since the small angles A Ba, A' B a' are equal, it is obvious, for a double reason, that the small arc A' a' is greater than A a, and hence the whole arc a a' is greater than A. A'. The deviation is therefore increased by altering the position in such a way as to make B A and BA' depart further from equality, and is a minimum when they are equal.

997. Conjugate Foci for Minimum Deviation.-When the angle of incidence is nearly that corresponding to minimum deviation, a small change in this angle has no sensible effect on the amount of deviation.

Hence a small pencil of rays sent in this direction from a luminous point, and incident near the refracting edge, will emerge with their divergence sensibly unaltered, so that if produced backwards they

would meet in a virtual focus at the same distance (but of course not in the same direction) as the point from which they came.

In like manner, if a small pencil of rays converging towards a point, are turned aside by interposing the edge of a prism in the position of minimum deviation, they will on emergence converge to another point at the same distance. We may therefore assert that, neglecting the thickness of a prism, conjugate foci are at the same distance from it, and on the same side, when the deviation is a minimum.

998. Double Refraction. Thus far we have been treating of what is called single refraction. We have assumed that to each given incident ray there corresponds only one refracted ray. This is true when the refraction is into a liquid, or into well-annealed glass, or into a crystal belonging to the cubic system.

On the other hand, when an incident ray is refracted into a crystal of any other than the cubic system, or into glass which is unequally stretched or compressed in different directions; for example, into unannealed glass, it gives rise in general to two refracted rays which take different paths; and this phenomenon is called double

refraction. Attention was first called to it in 1670 by Bartholin, who observed it in the case of Icelandspar, and its laws for this substance were accurately determined by Huygens.

999. Phenomena of Double Refraction in Iceland-spar. - Iceland-spar or calcspar is a form of crystallized carbonate of lime, and is found in large quantity in the country from

which it derives its name. It is usually found in rhombohedral form, as represented in Figs. 711, 712.

To observe the phenomenon of double refraction, a piece of the spar may be laid on a page of a printed book. All the letters seen through it will appear double, as in Fig. 712; and the depth of their blackness is considerably less than that of the originals, except where the two images overlap.