Fig. 11.—Principal Centre of Lens.
Optical Centre of a Lens.—Secondary Axes.—Let O and O′ (Fig. 11) be the centres of the two spherical surfaces of a lens. Draw any two parallel radii, O I, O′ E, to meet these surfaces, and let the joining line I E represent a ray passing through the lens. This ray makes equal angles with the normals at I and E, since these latter are parallel by construction; hence the incident and emergent rays S I, E R also make equal angles with the normals, and are therefore parallel. In fact, if tangent planes (indicated by the dotted lines in the figure) are drawn at I and E, the whole course of the ray S I E R will be the same as if it had passed through a plate bounded by these planes.
Let C be the point in which the line I E cuts the principal axis, and let R, R′ denote the radii of the two spherical surfaces. Then from the similarity of the triangles O C I, O′ C E, we have (O C)/(C O′) = R′/R; which shows that the point C divides the line of centres O O′ in a definite ratio depending only on the radii. Every ray whose direction on emergence is parallel to its direction before entering the lens, must pass through the point C in traversing the lens; and conversely, every ray which in its course through the lens traverses the point C, has parallel directions at incidence and emergence. The point C which possesses this remarkable property is called the centre, or optical centre, of the lens.
This diagram may also be taken to prove my former proposition, that the convex lens is practically a form of two prisms combined.
Fig. 12.—Conjugate Foci, one Real, the other Virtual.
Conjugate Foci, one Real, one Virtual.—When two foci are on the same side of the lens, one (the most distant of the two) must be virtual. For example, in [Fig. 12], if S, S′ are a pair of conjugate foci, one of them S being between the principal focus F and the lens, rays sent to the lens at a luminous point at S, will, after emergence, diverge as if from S′; and rays coming from the other side of the lens, if they converge to S′ before incidence, will in reality be made to meet in S. As S moves towards the lens, S′ moves in the same direction more rapidly; and they become coincident at the surface of the lens.
Formation of Real Images.—Let A B ([Fig. 13]) be an object in front of a lens, at a distance less than the principal focal length. It will have a real image on the other side of the lens. To determine the position of the image by construction, draw through any point A of the object a line parallel to the principal axis, meeting the lens in A′. The ray represented by this line will, after refraction, pass through the principal focus, F, and its intersection with the secondary axis, A O, determines the position of a, the focus conjugate to A. We can in like manner determine the position of b, the focus conjugate to B, another point of the object; and the joining line a b will then be the magnified image of the line A B. It is evident that if a b were the object, A B would be the image.
