Let us now proceed to determine the elements of a lens necessary for assigning the position and 7nagnitude of the image of any object. They are termed the "four cardinal points" of a lens:* the two "principal"

* See paper on " The Measurement of Lenses," by Prof. S. P. Thompson, Society of Arts Journal, Nov. 1891.

points - these may usually be taken to coincide in one "optical centre " for our present purpose - and the two "focal" points.

I. To determine the position of the "optical centre," and the two "principal points" of a lens. - Draw any two parallel radii o r, o' r' (of the spherical surfaces) and join r r', meeting the axis of the lens in c. The point c is called the centre, or optical centre, of the lens.

Any ray of light which, in its passage through the substance of the lens passes through c, will emerge parallel to its original direction, because the lens will act upon such a ray as a plate of glass with parallel faces. In Fig. 14 (3) we have the same figuring as in Fig. 11 ; the direction of a' b' alters in passing through the thickness of the plate, or in passing through the centre of the substance of the lens, but the ray emerges parallel to its original direction.

Fig.14. In Fig. 14 (2) if we produce a' b' to meet the axis in n1 ,and d' c' to meet the axis in n2, these two new points, n1 and N2, are called the "principal" or "nodal" points of the lens.

They have the property, which is evident from the figure, that any ray of light proceeding from any direction towards one of these points passes out of the lens as though it had passed through the other. (See Notes to Chapter II (The Formation Of Images By The "Pinhole Camera," And Its Perspective Drawing).)

We ascertained from Fig. 11 that if the parallel plate is very thin the displacement of the ray passing through it will be very small. The same reasoning will apply to a very thin lens. If we neglect the thickness of the plate the ray a' b' will continue in a straight line, and no displacement will occur. Similarly, in Fig. 14 (2) and (3), the thinner the lens becomes, the less will the ray be displaced in passing through it, and the nearer will N2 become to n2, until, if we consider the lens of negligible thickness, n1 and n3 will coincide (in the centre c), and any ray proceeding from any direction towards the centre c will pass through it in a straight line.

We see then that in thick lenses there are in reality two principal points (see Notes); but for the purposes of our discussion it will be less effort to the reader to consider lenses as very thin, and that these two principal points coincide in one principal point or centre.

FiG.15. Rays of light proceeding from any direction towards the centre of the lens pass through the centre without deviation.

We define a plane passing through this centre, perpendicular to the axis, as the principal plane of the lens.

II. To determine the two focal points of a lens. - If a beam of parallel rays, emanating from a very distant luminous point, meets one surface of a positive lens, in a direction parallel to its axis, the rays will, after passing through the lens, meet in a point f. This point is called the principal focus, or principal "focal point," of the lens, and its distance from the centre c of the lens is called the focal length of the lens.

Similarly, if we present the second surface of the lens to the incident parallel rays in the opposite direction, they will meet in a point f'.

This point is the second principal "focal point," and its distance from the centre of the lens c is again a measure of the focal length of the lens.

No real image can be formed at any position nearer to the centre of the lens than f or f'. It is evident that rays from a luminous object placed at either f or f' would, after passing through the lens, emerge parallel; if the object were nearer to c, the rays would diverge after passing through the lens.

f and f' are the two focal points, and possess the property that any ray meeting the lens in a direction parallel to the axis of the lens passes through the focal point on the further side of the lens.

Fig.16. Every lens (or combination of lenses) may be defined as possessing a centre and two focal points.

These elements are sufficient for us to determine the position and magnitude of the image of any object.

We need not now trouble to draw lenses, but indicate them by the principal plane passing through the centre of the lens, and by setting off the positions of the focal points on the axis.

Let o o represent the axis passing through the lens c; and f and F' the focal points (cf=cf').

Let us first determine the position of the image of an object a b c graphically.

From the object a draw a line (or take a ray) parallel to the axis a l, meeting the lens at l. This ray, after refraction, must pass through the focal point f in the direction l f a'. Now take another ray passing from a through the centre of the lens c. This ray passes in a straight line through c and meets the first ray (which took the direction l f a') in the point a', forming the image of a at a'.

Similarly two rays from c, one meeting the lens parallel to the axis, and the other passing through the centre, will meet in c', forming the image of c at c'.

We have thus determined the position of the image. Next as to its magnitude. If we compare Fig. 3, illustrating the formation of the image by the pinhole camera, with Fig. 16, we observe that the rays passing through the centre of the lens correspond with those passing through the pinhole, and the proportion existing between the sizes of object and image is identical with the relation between the distances of object and image from the centre of the lens. The magnitude of the image is to that of the object as cb : c b'.