LENSES. A portion of transparent matter hounded by two spherical surfaces and sym metrical about the line joining their centres is called a 'lens;' this line of symmetry is called the 'axis.' A homoeent•ic pencil of homogeneous rays from any point on the axis gives rise after two refractions to another homocentric pencil with its vertex on the axis. If the souree of rays is 0. at a distance n from a thin lens. the image o' will he at a distance y on the opposite side of the lens, where n and y are connected by the relation 1 1 1 v f f being a constant depending on the radii of the two surfaces of the lens and its index of refrac tion for the particular rays. 0 and 0' are 'con jugate foci.' If 0' is the real image of 0. then 0 will he the image of 0' as a source of rays. (If in this formula, on substituting for n and f their values. v is a negative quantity, 0' is on the same side of the lens as 0.) There are two classes of lenses; for one, f is positive; and for the other, f is negative.
(I) f is essentially positive.—As a special case let 0 be at an infinite distance. i.e. u =x,and all the rays from 0 are lines parallel to the axis. Therefore v = f. a positive quantity; and tr is on the opposite side of the lens at a distance f. This point is called the 'principal focus' on that side; and all rays on the other side parallel to the axis pass through this focus after refraction through the lens; for this reason a lens of this kind is called a 'converging' one. Similarly. if v = f. r = ; i.e. all rays passing through a point on the axis at a distance f from the lens emerge on the other side of the lens parallel to the axis. There are tlius two principal foci at equal distances from the lens on its two sides.. Again, any ray through the point where the axis cuts the lens has its direction unaltered, be cause at this point, which is called the 'centre' of the lens, the two surfaces of the lens are parallel and close together. assuming that the lens is thin. These principles enable one to trace at least three rays leaving a point. and thus to find its image and to draw images for any point or for any object as formed by such a lens. Drawings are given for a few special eases: It is evident from the formula that if u < f. v <0 and the image is virtual as shown in Fig.
S. In this case the image is magnified. The plane perpendicular to the axis at a principal faces is called a 'focal plane.' If an object lies in this plane, as shown in Fig. 9, the image of each of its points lies off at infinity, as ap pears from the fact that the rays are parallel after leaving the lens. Conversely. parallel rays falling upon a lens for which f is positive eon verge after passing through the lens to that point in the focal phine through which passes that one of the parallel rays which cuts the lens at its middle point or centre.
(2) f is essentially negative.—As a special ease let 0 he at an infinite distance, i.e. u= co and all the rays from 0 are lines parallel to the axis.
Therefore c = f, as a negative quantity: and 0' is on the same side of the lens as ti at a distance f from the lens. This point is called a 'principal focus;' and there is evidently another one on the other side of the lens at the same distance from it. Therefore all rays parallel to the axis on one side of the lens dircrge alter passing through the lens as if from a point at a distance f from the lens on the same side as were the parallel rays. For this reason a lens of this kind is called a 'diverging' one. Similarly. a ray on one side of the lens pointed toward the principal focus on the farther side emerges from the lens parallel to the axis. Further, a ray the centre of the lens remains parallel to itself. These principles enable one to trace at least three rays leaving a point not on the axis and thus to find its image, and to draw images of any object. Drawings are given of a few special eases.
It is evident from the general formula that r will be positive, i.e. there will be a real image if u is negative and numerically le.s than j. i.e. if 0 is on the opposite side of the lens from that on which the rays enter, and between the lens and the principal focus. This is the ease when the entering rays are converging toward the point 0, as shown in Fig. 11 ; under these con ditions 0 would be called a 'virtual' source. Again. as is seen from Fig. 12. parallel rays falling on the lens emerge as if coining from that point in the focal plane on the same side as that from which the rays come where the ray through the centre of the lens meets the plane.