In the first case the two radii must be equal in order to satisfy (i), and the lens ceases to be of any optical

possibility. Our conclusion therefore is that a single lens

cannot be achromatic.

CHAPTER V.

THE DETERMINATION OF THE FOCI AND OF THE PRINCIPAL POINTS OF A SYSTEM OF LENSES. THE NODAL POINTS.

132. To determine theoretically the positions of the Foci and of the Principal Points of a lens or system of lenses. Conjugate Foci lying upon the axis are connected with one another by the relation

where x, x' are the distances of the conjugate points from the Principal Points H, H' respectively.

Let us take any point 0 on the axis of the system, and use it as an origin from which all our distances may be reckoned; and let F, F', H, H' be the distances from it of the Principal Foci and the Principal Points respectively.

The above relation may now be written in the form

where, are the distances from O of any pair of conjugate points.

We have shown, in Art. 113, how to determine the position of the point conjugate to a given one. If therefore we take any four points on the axis, and determine the positions of the four points respectively conjugate to them, we get four simultaneous values of and '. These, when substituted successively in (ii), give us four independent equations for the determination of the four unknown quantities F, F', H, H'. Thus are found the positions of the Principal Foci and also of the Principal Points.

133. To determine experimentally the positions of the Foci of a lens or system of lenses.

We will suppose that, in the following figure, A is a micrometer or a frame holding two spider lines erossing one another, a stand supporting a telescope, and that B supports a cylinder enclosing the system of lenses. Moreover A, B, and C are supposed to be moveable by the hand or by means of screws to and fro along the graduated bar MN, and also to be so adjusted that the micrometer, the lenses, and the telescope have the same axis (fig. 22).

Let the telescope be turned first towards a distant object, and then accurately focused. The rays from the distant object are approximately parallel, and the image will be formed at the Principal Focus of the telescope.

When this has been done, the telescope must be placed on the stand C so that the micrometer A may be viewed through the system of lenses; the micrometer must then be moved to and fro along the graduated bar until the image of it, seen through the system of lenses and the telescope, becomes clear and distinct. Now this image is seen through a telescope which has been focused upon a distant object, hence we know that the image of the micrometer can be distinct only when the rays that fall upon the object glass of the telescope are parallel to the axis. Consequently the rays that emerge from the cylinder B must be parallel to the axis. Therefore the micrometer A must be at one of the Principal Foci of the lens-system. The Focus is thus determined in position.

G

We have still to measure its distance from the nearest

surface of the system. This might be done by moving the micrometer along the graduated bar until it came into contact with the surface, and then taking the difference of the readings in the two positions given by the scale. There is however a practical difficulty in ascertaining the exact moment of contact, and this method consequently leads to an unsatisfactory result. The distance between the micrometer and the nearest surface may be measured more accurately by a simple optical contrivance.

For this purpose let the telescope be focused upon some near object whose distance is greater than that which we have to measure; and let the telescope be removed to the other end of the bar so that the micrometer may be between it and the lens-system. If when this has been done the telescope be moved along the graduated bar until first the micrometer, and then the dust on the face of the lens be in focus successively; and if the scale be read for these two positions of the telescope, the difference between the readings will give us the distance between the micrometer and the face of the lens-system with tolerable accuracy. The micrometer being at a focus of the system, we thus get the distance of the focus from the surface nearest to it.

134. To determine the positions of the Principal Points when the Foci are known.

If d and d be the distances from the Foci of any two conjugate points on the axis, and if ƒ be the distance of a Focus from the corresponding Principal Point, we have

dd' = ƒ".

It has been shown how d and d' may be determined; hence the above equation enables us to determine ƒ, and consequently the positions of H and H'.

135. In order that the method described above may be a directly practical one, it is necessary that the lens-system should be a convex lens or equivalent to a

convex lens. by it.

Otherwise no real images will be formed

If the lens-system itself gives a real image, the method can be applied at once. But if it does not, it can be made to do so by combining with it a known convex lens of sufficient power. The method may then be applied to the joint-system, and by making allowance for the effect of the known convex lens, the proper result for the original system can be deduced from the one thus obtained.

136. The Principal Points, introduced by Gauss, have been supplemented by two other points which Listing introduced and called Nodal Points. They are principally of importance when the extreme media are not the same. This is found to be the case in the human eye.

The Nodal Points are situated upon the axis of the lens-system, and are conjugate to one another. Their distinguishing property is that an incident ray through one will produce an emergent ray in a parallel direction passing through the other.

When the extreme media are the same, we have seen that this is a property of the Principal Points. Hence in this case the Principal Points and the Nodal Points coincide.

When there is only one refracting surface we might call its centre of curvature the Nodal Point, for we know that an incident ray which passes through the centre of curvature crosses the surface without deviation.

137. To determine the positions of the Nodal Points.

Let H, H', F, F be the Principal Points and Foci of a lens-system, and let T be any point on the Focal Plane through F (fig. 23).

We know that a ray through T parallel to the axis, and meeting the Principal Planes at a, a' respectively, will on emergence pass through F". Moreover, since T is a point in a focal plane, its conjugate is on a plane at an infinite distance, and therefore all rays from it will on