system; and the points A,, A,, ...A, in which the axis meets the successive surfaces may be called the vertices of the surfaces.

The surfaces may be of any number, of any degree of curvature, and at any distance apart; moreover, they may have their convex surfaces turned either way.

We will now suppose the spaces between each two consecutive surfaces to be occupied by homogeneous, not doubly refracting, media, of known refractive indices. The media may be all different, or two or more may be alike. But two similar media should not be adjacent to one another, for the effect would be the same as if the dividing surface were not there.

3. If we consider a particular case, and suppose the number of surfaces to be four, and the successive media to be air, glass, air, glass, and air, we so have the telescope in its simplest form, with one eyepiece and one object glass.

4. If now a ray of light proceed from a luminous point L, and cross the successive surfaces at the points P1, P P3, ... P, its course will be bent at each of these points, but in consequence of our hypothesis concerning the nature of the media, its course between any two of the points will be a straight line. The path of the ray may therefore be represented by the broken line LPPP,... PL' (fig. 2).

FIC.2.

3

1

The end we aim at is the determination of a relation between the lines LP, and PL', so that when we know the initial path of a ray entering the system, we may at once be able to ascertain its final path on leaving it.

5. The general system includes also the case in which one or more of the surfaces are reflective. In order to make our results applicable to it, we have simply to consider the particular surface which is reflective, as if it were the boundary between two media whose refractive indices are and -μ, respectively; μ, being the refractive index of the medium immediately preceding the surface which reflects.

μ

6. With regard to the rays themselves, we shall consider those only which are inclined at very small angles to the axis of the system, and which cross the surfaces at points very near the vertices.

If P be the point of incidence of a ray, A the vertex, and C the centre of curvature of the surface, it will follow that if the arc PA be small, so also will be the angle PCA (fig. 3).

FIG. 3.

We shall suppose that the angle PCA is so small that all powers of its circular measure, higher than the second, may be neglected. Consequently, to this degree of approximation, we may consider the sine, tangent and circular measure of the angle PCA to be equal to one another.

7. In the first place, we will investigate formulæ connecting the position of the incident ray, with that of the ray after refraction, at one surface only.

8. To find the angle between the incident ray and the refracted ray, after crossing one surface.

Let C be the centre of curvature of any refracting surface, and A the vertex. Suppose XPX' to be the course of a ray which crosses the surface at the point P; and let the incident

ray and the refracted ray, produced if necessary, meet the axis at the points X, X' respectively (fig. 4).

In this, and in all subsequent investigations, we will consider distances measured from left to right to be positive, and all distances measured from right to left to be negative. The above figure is so drawn that AC and AX', measured from A are positive, and AX negative.

At the point P, the ray is bent through the angle XPR, the point R being in X'P produced. We will call this angle the Deviation at P, and will denote it by 8.

Let the normal CP be produced to C', and let

9. We may put the expression for the deviation in another form.

Let the arc AP=h, and denote the distances from A of X, X', and C, by – u, v, and r respectively; we then get

10. If rays of light proceed from a luminous point X on the axis, they will, after crossing a refracting surface, meet again in one and the same point X', which also lies upon the axis. To find the relation between the positions of X and X'. With the same notation as before, we have (fig. 4)

μsin C'PX=μ, sin CPX';

therefore usin (PXA + PCA) = μ, sin (PCA – PX'A);

In this formula o, μ, and r are quantities which depend

only upon the nature of the media, and upon the curvature of

the separating surface; they are the same whatever be the ray which we may consider. Hence we see, that since the formula gives only one value of v corresponding to a particular value of u, it follows that all the rays, which proceed from any point X on the axis, will after refraction meet again in a point X', also on the axis; the relative positions of these two points being given by equation (1).

Conversely, for a particular value of v we get from (1) only one value for u. Hence the rays, which after crossing a refracting surface meet together in a point X' on the axis, must before incidence have proceeded from one and the same point X, also lying upon the axis.

Again, if we consider X' instead of X as the luminous point, or the origin of the rays, and that the rays travel from right to left, it is clear that after refraction at the surface they will all meet together in the point X.

Consequently the point X bears the same relation to X', when X' is the origin of rays, that X' bears to X, when the origin of rays is at X.

Two points such as X and X', which are related to one another in this way, are called Conjugate Points.

11. There are now two particular cases to be considered : (1) If the point X is at an infinite distance from A, we have u∞, and the incident rays are all parallel to the axis. The equation connecting u and v being

(2) If the point X' be at an infinite distance from A, we have v; and the rays after refraction are parallel to the axis.

The corresponding value of u is given by the equation