second focus, H'HH" the refracting surface or the principal plane, and K the nodal point. The image of the point L' must be formed somewhere on the axis XY. If we have merely the direction of

another ray emanating from L', the image of L' will be given by the intersection of this line with the axis.

Let us take a ray, L'H', which passes through the focal plane at O, and through the principal plane at H'. Supposing that the point O itself be luminous, the rays emitted by it will be parallel to each other after having passed through the surface, and, moreover, parallel to the ray of direction O K P drawn through the nodal point K. The ray OH' may evidently be regarded as coming likewise from 0. Hence it will be parallel to O P after having passed through the surface, and will be directed toward Q. The ray H' Q meets the principal axis at L". Hence L" is the image of L'.

Something analogous to this will occur for the rays given off from the point A'. But, in order not to render the figure perplexing, we

have drawn another one (Fig. 18) for the construction of the image of the point A'.

The ray A'K passes without deviation through the surface, because it is directed toward K. It indicates consequently the line A'K T on which the image of A' is situated. Another ray emanating from A', A'H', is parallel to the axis X Y. All rays which are parallel to this axis pass through the second principal focus p". Hence H'p"R is the direction that the ray A'H' will take after its passage through the surface. Continuing its course, it meets the ray A'T at A". It is here, then, that the image of A' is formed.

The same point may be found by still another way, by means of the ray A'S (Fig. 18), which passes through the first principal focus p'. All rays emanating from are parallel to the axis XY, after Hence p'S (or A'S) will continue

having passed through the surface. its course parallel to the principal axis, and will also meet A'T at the point A".

A" L" is therefore the image of A' L'.

The same thing would evidently have been produced if we had chosen an object extending below the axis; its image would be produced on the other side of the axis above L" A".

PASSAGE OF LIGHT THROUGH A SYSTEM MADE UP OF SEVERAL REFRACTING SURFACES.

When, after having passed through one surface, the light encounters a second, then a third, a fourth, &c., all of which separate media having different indices of refraction, the problem relative to the place where the image is formed, and its size, becomes much more complicated.

It is easy to conceive that the image formed by the first surface becomes the object for the second surface, and that this is true whether the image be actually formed, or whether the rays be intercepted by the second surface, before their reunion. The image for the second becomes the object for the third surface, and so forth.

Performing successively the calculations for all the refractive surfaces and media, we obtain a final image. This image, formed by the last surface, is the image of the object after the passage of its rays through the entire system.

These calculations might become extremely complicated, especially in an apparatus like the eye, where we have to do with a considerable number of refracting surfaces. As illustrating this, it suffices to recall the surfaces of the different layers of the crystalline.

Very fortunately, these calculations have been considerably reduced by the eminent mathematician GAUSS. He has found, for the course of light through a centred system of whatever number of surfaces, comparatively simple and highly demonstrative formulæ. They have since been completed by MOBIUS. 2

These savans have demonstrated that there exists for every dioptric system, composed of whatever number of centred spherical surfaces, ie., whose centres are all situated on the same axis, three pairs of cardinal points, likewise situated on the axis and comparable with those that we have found for a single refracting surface; and two pairs of planes passed through four of these points perpendicularly to the axis.

The six cardinal points are the two principal foci, the two principal points, and the two nodal points.

The FIRST PRINCIPAL FOCUS is the point at which the incident rays must cross each other in order that the rays which have passed through the system, i.e., the emergent rays, may be parallel to the principal axis.

The SECOND PRINCIPAL FOCUS is the point where the emergent rays cross each other when the incident rays have been parallel to the principal axis.

The PRINCIPAL POINTS are characterised by the following property: When an incident ray, prolonged if necessary, passes through the first principal point, the corresponding emergent ray, or its prolongation, passes through the second principal point, but the incident is not parallel to the emergent ray. The principal points are the images of one another.

The NODAL POINTS are two points of the principal axis, so situated that every ray which, before being refracted, is directed toward the first of them, seems, after its refraction, to come from the second one, and takes a direction parallel to that which it had at first. These two parallel rays are called lines of direction, and act, in the combined system, the same part as the line passing through the nodal point of a single refracting surface.

The second nodal point is the image of the first, and vice versá.

The distance between the two principal points is equal to that which separates the two nodal points.

The two pairs of planes are-the two FOCAL PLANES, passed through the principal foci, and the two PRINCIPAL PLANES, passed through the principal points, perpendicular to the principal axis.

1 Gauss, Dioptrische Untersuchungen in Verhandlungen der k. Gesellschaft der Wissenschaften zu Göttingen, Bd. i., 1840.

2 Mobius, Kurze Darstellung der Haupteigenschaften eines Systems von Linsengläsern (Crelles Journal, Bd. v., p. 113).

The FOCAL PLANES have the same property as those of a single refracting surface. The rays given off from a point of the first focal plane are, after refraction, parallel to each other and to the rays of direction.

The rays that were parallel before refraction are focused in a point of the second focal plane. This point is likewise indicated by the rays of direction.

The PRINCIPAL PLANES are defined as follows: If a parallel to the principal axis be drawn through the point at which an incident ray, or its prolongation, pierces the first principal plane, the point where this line pierces the second principal plane is in the course of the corresponding emergent ray or its prolongation. In other words, the directions of any incident ray, and of the corresponding emergent ray, pierce the first and second principal planes in two points situated on the same side of, and at the same distance from, the principal axis of the system.

The second principal plane is the optical image of the first, and vice versa. These are the only two conjugate images which have the same size and direction. The two principal planes of the composite system correspond to the single principal plane of the simple system (having but one refracting surface).

The FIRST PRINCIPAL FOCAL DISTANCE is the interval which separates the first principal focus from the first principal point.

The SECOND PRINCIPAL FOCAL DISTANCE is the interval which separates the second principal point from the second principal focus.

Let XY (Fig. 19) be the principal axis;

and " the first and second foci;

H' and H" the first and second principal points;
K' and K" the first and second nodal points.
'H' is the first principal focal distance = F';

" H" is the second principal focal distance = F".

C

The distance (G') from the first focus to the first nodal point is equal to the second principal focal distance, and the distance (G") from the second nodal point to the second focus is equal to the first principal focal distance:

'K' = p′′ H′′ = G′ = F′′

'H' "K" F=G",

=

=

if, as for the single refracting surface, we designate by F' and F" the distances from the foci to the principal points, and by G' and G′′ those from the foci to the nodal point.

From this it follows that the respective distance, of each principal point and the nodal point of the same kind, is equal to the difference of the two focal distances:

H' K'=H" K" = F" - F,

and that, moreover, the distance between the two principal points is the same as that between the two nodal points:

H' H" = K' K".

Finally, the two principal focal distances are to each other as the indices of refraction of the first and last media:

if n"" be the index of refraction of the last,-of the fourth medium, for instance.

If, then, the last medium be of the same nature as the first, and we have n'=n"" (as in most optical instruments, but not in the eye), the two principal focal distances are equal, and the principal points coincide with the nodal points.

As soon as we know the cardinal points of a dioptric system, it is easy to calculate or construct, for any object, the image of such object as formed by the system in question. Hence, when one has to deal with a dioptric system, it is essential at the outset to ascertain the location of its cardinal points.

TWO SURFACES SEPARATING TWO REFRACTIVE MEDIA.

LENSES.

Let us first take the simplest case-that of two surfaces separating one medium from another. This is the case with lenses in air or the crystalline of the eye, supposed to be homogeneous, since the aqueous humor and the vitreous body have, as we shall see, the same index of refraction.