At present we shall only consider the path of rays of light which pass through the prism in a plane perpendicular to both its faces, and therefore perpendicular to the edge of the prism; we shall call such a plane a principal section of the prism.

When a ray of light passes through a prism which is more highly refractive than the surrounding medium, the deviation is, in all cases, from the refracting angle towards the thicker part of the prism.

Let PQRS be the course of a ray of light through a prism in a principal section QOR. Draw the normals at Qand R meeting in L. There are three cases to be considered, according as the triangle OQR is acute angled, or contains a right angle or an obtuse angle.

A

S

R

P

In the first case the rays PQ and RS lie on the sides of the normals away from the vertex, and therefore the deviations both at ingress and egress will be away from the edge of the prism.

In the second case let one of the angles of the triangle OQR be a right angle; at that point of incidence there

will be no deviation and at the other point of incidence the deviation is away from the vertex.

S

R

In the last case, one of the angles, ORQ, is obtuse, the

R

other angle, OQR, being acute. Then the ray SR lies on the side of the normal towards the vertex, so that the corresponding deviation is towards the vertex, while at Q the deviation is away from the vertex. But the angle of refraction at Q is greater than that at R, the former being the exterior angle of the triangle QRL and the latter an interior angle. Hence the deviation at Q is greater than that at R, so that on the whole the deviation is away from the vertex.

If the prism be less highly refractive than the surrounding medium, all these effects are reversed.

22. This theorem may also be proved by comparing the action of a prism with that of a plate.

When a ray of light passes through a plate bounded by two parallel faces, it emerges parallel to its original direction. Let PQRS be the path of a ray through such a plate bounded by the faces AB, CD. Let RN be the normal at the second face.

H

R

N

N'

B

Now suppose the second face turned about R towards AB, in such a way as to make a prism whose edge is

perpendicular to the plane of the ray. Let RN' be the new position of the normal to the second face and RS' the emergent ray. Then, as may be seen from the figure, the angle of incidence at the second face is increased; hence the deviation at the second face is increased. The ray is therefore deviated towards the thicker part of the prism.

Similarly, if the second face CD be turned in the opposite direction, the deviation at the second face will be diminished and the same result will follow.

23. Let PQRS be the path of a ray through a prism whose edge is at O, and whose refracting angle is .

N

M

R

S

Draw the normals at Q and R, LQM and LRN respectively, meeting in L.

Leto, o be the angles of incidence and refraction at Q, and let, be the angles of emergence and incidence at R, respectively. We shall consider and as positive when they are measured from the normal towards the thicker part of the prism, so that ' and 'will be positive when they are measured from the normals towards the vertex. In the figure o, ☀', †, y' are all positive.

Also, the angles at the base of the triangle OQR are respectively-p', and ', hence

or

This result is also true when the triangle OQR is in this case one of the angles p' or 'would be

obtuse; negative.

If y be the critical angle for the medium, then o' and can never be greater than y. If, therefore, the refracting angle of the prism be greater than 2y, no ray can pass through the prism. If be greater than y, d' and ' must always both be positive.

し

Let D be the whole deviation of the ray as it passes through the prism. Then at the first refraction the ray is deviated through an angle -', and at the second refraction it is further deviated through an angle ↓ — y'. Therefore

or

D= &+ √

l.....

.(3).

The whole theory of the path of a ray of light in a principal section through a prism is contained in the equations (1), (2) and (3).

24. The deviation is a minimum when the ray of light passes symmetrically through the prism.

Let be the value of for this symmetrical path, and let gradually increase from 4. Then ' and ' increase and decrease, respectively, by equal increments by virtue of equation (2); hence, since o' becomes greater than, the deviation at the first face increases faster than that at the second face diminishes, so that on the whole the total deviation increases. The same result is