(b) Rays of light starting in all directions from a point A and travelling onward for a definite length of time, reach a surface σ, whose tangent plane at a point B is conjugate, in the medium surrounding B, with the last element of the ray AB.
(c) If all rays issuing from A are concentrated at a point B, the integral ∫u-1ds has the same value for each of them.
(d) In case (b) the variation of the integral caused by an infinitely small displacement q of B, the point A remaining fixed, is given by δ∫u−1 ds = q cos θ/vB. Here θ is the angle between the displacement q and the normal to the surface σ, in the direction of propagation, vB the velocity of a plane wave tangent to this surface.
In the case of isotropic bodies, for which the relation (8) holds, we recover the theorems concerning the integral ∫µds which we have deduced from the emission theory (§ 5).
10. Further General Theorems.—(a) Let V1 and V2 be two planes in a system of isotropic bodies, let rectangular axes of coordinates be chosen in each of these planes, and let x1, y1 be the coordinates of a point A in V1, and x2, y2 those of a point B in V2. The integral ∫µds, taken for the ray between A and B, is a function of x1, y1, x2, y2 and, if ξ1 denotes either x1 or y1, and ξ2 either x2 or y2, we shall have
| ∂2 | ∫ µ ds = | ∂2 | ∫ µ ds. |
| ∂ξ1 ∂ξ2 | ∂ξ2 ∂ξ1 |
On both sides of this equation the first differentiation may be performed by means of the formula (3). The second differentiation admits of a geometrical interpretation, and the formula may finally be employed for proving the following theorem:
Let ω1 be the solid angle of an infinitely thin pencil of rays issuing from A and intersecting the plane V2 in an element σ2 at the point B. Similarly, let ω2 be the solid angle of a pencil starting from B and falling on the element σ1 of the plane V1 at the point A. Then, denoting by µ1 and µ2 the indices of refraction of the matter at the points A and B, by θ1 and θ2 the sharp angles which the ray AB at its extremities makes with the normals to V1 and V2, we have
µ12 σ1 ω1 cos θ1 = µ22 σ2 ω2 cos θ2.
(b) There is a second theorem that is expressed by exactly the same formula, if we understand by σ1 and σ2 elements of surface that are related to each other as an object and its optical image—by ω1, ω2 the infinitely small openings, at the beginning and the end of its course, of a pencil of rays issuing from a point A of σ1 and coming together at the corresponding point B of σ2, and by θ1, θ2 the sharp angles which one of the rays makes with the normals to σ1 and σ2. The proof may be based upon the first theorem. It suffices to consider the section σ of the pencil by some intermediate plane, and a bundle of rays starting from the points of σ1 and reaching those of σ2 after having all passed through a point of that section σ.