This shows that for the actual path, μAP+ 'PB is a minimum.

The previous theorem is a particular case of this; we have only to put μ-μ to deduce it from the more general theorem.

Next, suppose that the ray of light in its passage from A to B undergoes any number of refractions or reflexions. Let p be the length of the path in any medium whose refractive index is μ. Then it has been shown that Σup is a minimum for separate variations of the points of incidence between consecutive media; and therefore by the principle of superposition of small variations, it will be a minimum when simultaneous variations are admitted. The actual path, therefore, makes up a minimum between any two points.

75. Another important proposition, enunciated by Malus, easily follows from the preceding.

Any system of rays originally normal to a surface, will always retain the property of being normal to a surface after any number of reflexions or refractions.

EE

Let ABCDE, A'B'C'D'E' ... be a series of rays normal to a surface at A, which undergo any number of refractions and reflexions. Measure off along these rays distances to E, E'......, such that Σup is the same along each ray; then we shall show that the rays are finally normal to a surface EE. Join A'B and E'D. Then Zup along ABCDE is the same as along A'B'C'D'E'. But by what has been shown above for any ray and its consecutive it follows that up along A'BCDE' is the same as along A'B'C'D'E', and

Take away the

therefore the same as along ABCDE. common parts; then if μ, μ belong to the initial and final media, there remains the equation, μA'B + μ'DE' =μAB+μ'DE. But, since AB is normal to the surface AA', A'BAB ultimately, and therefore, DE' = DE; that is, EE' is perpendicular to DE. The same may be proved for every point E' near E, and thus the surface EE' near E is perpendicular to the ray DE, and by similar reasoning to every other ray of the system.

76. A system of rays which can be cut at right angles by a surface, we shall call an orthotomic system.

A system of rays diverging from a point, or such that by any combination of mirrors or refracting surfaces they can be made to meet in a point, is clearly orthotomic; for a sphere whose centre is the point through which all the rays pass, will cut them all at right angles.

If a system of rays diverging from a point converge to another point after

any number of reflexions

and refractions, the values of Σup taken from one point to the other will be the same for all rays. Thus, in order to condense rays issuing from one point S, on a second point H, by means

H

P

of a single reflexion at a curved surface, we choose our surface such that SP+ PH may be the same for all paths, and therefore the surface must be an ellipsoid of revolution whose foci are S and H.

If the rays are parallel, the point S will be at infinity, and the surface is a paraboloid of revolution whose axis is parallel to the common direction of the rays.

Next, let us find the form of the surface which will refract to a point H all the rays proceeding from a

point S. Let μ, μ be the refractive indices of the

H

media; then if P be any point of the surface, the surface must be such that

where c is a constant.

μSP +μ'HP = c,

Hence the surface is formed by the revolution of a Cartesian oval of which S and H are foci. The theory of the Cartesian oval may be found in Williamson's Differential Calculus, Appendix.

As a particular case suppose the rays parallel, so that S is at infinity. Draw a plane MX perpendicular to the rays, and let any ray be produced to meet this surface in M. Then

But μSP+ PM is also constant. Choose the plane MX so that this constant quantity may be equal to c; then 'HP = μPM, and therefore the surface is formed by the revolution of a conic whose focus is H and directrix MX, about its major axis.

77. When the orthotomic surface is a surface of revolution about an axis, all the rays intersect the axis, and we may first consider the rays in any meridian plane and afterwards suppose this plane system of rays to be revolved about the axis.

The rays in any meridian plane are a series of normals to a curve. Consecutive rays will intersect each other in points lying on a curve which is called the evolute of the given curve, and each ray will touch this evolute. The evolute which is touched by all the rays is called a caustic curve; and the surface formed by its revolution about the axis a caustic surface. A caustic curve of such a symmetrical system as we are considering always has a cusp on the axis.

When the system is revolved about the axis consecutive rays along the circle traced out by a point will meet on the axis, and therefore the axis may be considered as a second caustic surface. At points on a caustic surface the rays are closer together than at other points, and therefore if the pencil be exhibited on a screen, points on the caustic surface will appear brighter than the rest.

78. The character of a limited pencil of rays is shown in the figure; BAB' is the orthogonal surface, F is the cusp of the caustic curve.

If the pencil be received on a screen perpendicular to the axis, the nature of the caustic surface can be shown by examining the bright patch of light on the screen as the screen is moved from DD' towards F. At DD', there will be a circular patch of light with a brighter ring round its outer edge, and as the screen is moved along, this ring will gradually contract. As soon as C

is reached, the other part of the caustic surface is shown,

B

M

F

B

and a bright spot is developed in the centre. When the screen is at EE′ the circle of light reaches its minimum; this circle is called the least circle of aberration. When this position is passed, the outer boundary expands again though the bright ring still contracts. Beyond F, no part of the screen is specially illuminated.

If any ray BCE meet the axis in C, then FC is called the longitudinal aberration of the ray.

79. The caustic by reflexion at a circle may be found by elementary geometry in two cases, first, when the incident rays are parallel, and secondly, when they diverge from a point on the circumference of the circle.

When the incident rays are parallel, the caustic is an epicycloid formed by the rolling of one circle upon another of twice its radius.

For from the centre C of the reflecting circle, draw the radius CA parallel to the incident rays; then the caustic is symmetrical with regard to the line CA. Let SP be any one of the incident rays, reflected by the circle at the point P in the direction PQ. Join CP; then by the law of reflexion, CP will bisect the angle SPQ. With centre C and a radius equal to half the radius of the