If we present either surface of a negative or concave lens to a beam of rays parallel to the axis of the lens, we shall be unable to find a real image of the object from which the parallel rays emanated ; no real focus will be formed by the lens.

Again, if we bring the object nearer to the lens, until it is in close proximity with it, the lens will still be found incapable of forming a real image.

It is evident, then, that negative lenses, used alone, will be quite useless for photographic purposes.

Our object will be to examine the manner in which rays of light are affected by negative lenses, with a view, subsequently, of using them in conjunction with positive lenses, in such a manner that real images can be formed.

Let us first make an experiment with a negative lens to see its effect upon a beam of parallel rays, and acquaint ourselves with the meaning of a "virtual" focus, by determining its approximate focal length in a practical manner.

Take a thin disc of any opaque material, such as cardboard, of the same diameter as the lens under examination ; at equal distances from the centre of the disc, and rather nearer the edge than the centre, make two small perforations a, b ; measure their distance apart. If we now place this disc in contact with the lens, and present the disc to the sun's rays, we shall find that the pencils passing through a and b will be divergent after passing through the lens. These divergent pencils can be traced as small discs of light upon a screen held behind the lens; if the screen be made to occupy the position where the distance between these two discs a' and b' is just double the distance between a and b, the distance c f' between the disc and the screen is a rough measure of the focal length of the lens, or is equal to c f. The direction of the pencils a a' and b b' appears to originate at f. f is the "virtual" focus of the lens, and is one of the "focal points" of the lens, f' being the other.

In negative as well as in positive lenses there are two " principal"

Fig.30.

points; but here again it will be sufficient for our purpose to consider them as coinciding in one "optical centre."

From the determination of the position of these three elements in a negative lens, we know : (1) That any ray meeting a lens in a direction parallel to the axis emerges from the lens as though it came from the focal point situated on the same side of the lens as the incident ray; (2) that any ray passing through the optical centre of the lens proceeds in a straight line.

We can then, as before, plot a negative lens on paper and indicate it by the principal plane passing through the optical centre of the lens, with its two focal points equally distant on either side.

Before proceeding, let us again refer to Fig. 30 to see what it teaches us. We have noticed that parallel rays diverge from the lens as though they proceeded from the focal point on the same side. This fact read alone is unimportant to our investigation ; we must grasp and remember the converse: that rays (a' a, b' b) converging towards the focal point f of the negative lens on the further side leave the lens in a parallel direction; a focus would be formed by the negative lens at infinity. This is an important, though not practical, conclusion.

Let us now consider the case of an object placed nearer to the lens, but distant some multiple of the focal length (greater than unity).

Fig.31.

Let o o represent the axis passing through c, the optical centre of the lens, and f and f' the focal points (cf = cf/.)

We will now determine the position of the virtual image of the object a c b. First consider a ray a l meeting the lens parallel to the axis. This ray after refraction takes the course f a' l, as though the ray emanated from the focal point f. Now take a second ray from a passing through the centre of the lens c; this ray in passing through c cuts the first (virtual) ray at a!, and determines the position of its virtual image. In the same manner we can determine the position of the virtual image of b, at b'; a' b' is the virtual image of a b, is erect, and lies between the focal point f and the centre of the lens.

To examine the converse: suppose convergent rays on the left of the lens are proceeding to form a real image a' b' c', and we interpose the negative lens so that a'c' b' would fall between c and f as shown in the figure, the negative lens will now form an enlarged image of a' d b' at a b c.

In Fig. 31 we have supposed the object acb placed at a distance equal to five times the focal length of the lens, and have seen that the virtual image is formed on the same side as the object.

The relation between the size of object and image is found as follows :

Divide the distance of the object from the lens, by the focal length of the lens, and add one; this is the magnification. (See Notes.)

In the case before us the object is distant from the lens five times its focal length, hence the "magnification " is 5 +1, or 6, or the virtual image is one-sixth the size (linear) of the object. Further, it is evident that the distance of the object from the lens c c is in this same relation to the distance of the image c c'; so that c c' is one-sixth of the distance c c.

Again interpret the converse, for this is the chief interpretation we use:

If convergent rays on the left of the lens are proceeding to form a real image at a' c' b', and we interpose a negative lens of known focal length, and find that a real image is formed at acb, we know that the distance of acb from the lens, divided by the focal length of the negative lens, plus one, gives us the magnification of the image a' c' b' occurring at a c b.