and condensing, or divergent and dispersing; the term positive is

sometimes applied to the former, and the term negative to the latter. Convergent lenses transform a parallel pencil into a converging one, and increase the convergence, and diminish the divergence o any pencil. Divergent lenses, on the other hand, transform a parallel pencil into a diverging one, and diminish the convergence, and increase the divergence of any pencil. In convergent lenses the first principal focal distance is positive and the second principal focal distance negative; in divergent lenses the converse holds. ('lghe four forms of lenses are interpretable by means of equation I0 .

f: f1f27l

in-Ii{n(r2-ni-l-Ein-Ill

(EU) If § 1 br<}}poi itive and rg pigative. This typ? iixcalled bicocpvex g. 9, I . e rst principa ocus is in ront o the ens, an the second principal focus behind the lens, and the two principal points Off lf lf

1 2 3 4 5 6-FIG.

9.

are inside the lens. The order of the cardinal points is therefore FS1HH'S2F'. The lens is convergent so lon as the thickness is less than n(r, -rz)/(nel). The special case wiien one of the radii is infinite, in other words, when one of the bounding surfaces is plane is shown in fig. 9, 2. Such a collective lens is termed plana-convex. As d increases, F and H move to the right and F' and H' to the left. If d=n(r1-12)/(n-I), the focal length is infinite, i.e. the lens is telescopic. If the thickness be greater than n(n-rg)/(n-1), the lens is dispersive, and the order of the cardinal points is HFS, S¢F'H'.

(2) If rl is negative and 12 positive. This type is called biconcave (fig. 9, 4). Such lenses are dispersive for all thicknesses. If d increases, the radii remaining constant, the focal len ths diminish. It is seen from the equations giving the distances of the cardinal points from the vertices that the first principal focus F is alwa s behind 51, and the second principal focus F' always infront of i, and that the principal points are within the lens, H' always followixgg H. If one of the radii becomes infinite, the lens is plana-concave (~ »5»

%3?If the radii are both positive. These lenses are called convexoconcave. Two cases occur according as r2>r1, or <n. (a) If r2>r, , we obtain the mensicus (fig. 9, 3). Such lenses are always collective; and the order of the cardinal points is FHH'F'. Since sr and sn are always negative, the object-side cardinal points are always in front of the lens. H' can take up different positions. Since sH'=-dr2/R= -dn/l11(f2-f|)+d(n-1)}, sn' is greater or less than d, i.e. H' is either in front of or inside the lens, according as d<or> in-n(r2-11) }/(n- I). . (b) If 12<r1 the lens is dispersive so long as d<n(r, -rz)/(n-1). H is always behind S1 and H' behind S2, since sa and sn' are always positive. The focus F is always behind S1 and F' in front of Si. If the thickness be small, the order of the cardinal points is F'HH'F; a dispersive lens of this type is shown in fig. 9, 6. As the thickness increases, H, H' and F move to the right, F more rapidly than H, and H more rapidly than H'; F', on the other hand, moves to the left. As with biconvex lenses, a telescopic lens, having all the cardinal oints at infinity, results when d=n(r1-rg)/(nf-I). If d>n(r1-r2)fin-1), f is positive and the lens is collective. The cardinal oints are in the same order as in the mensicus, viz. FHH'F'; anti) the relation of the principal points to the vertices is also the same as in the mensicus. (4) If r, and r2 are both negative. This case is reduced to (3) above, by assuming a change in the direction of the light, or, in other words, by interchanging the object- and image-spaces. The six forms shown in fig. 9 are all used in optical constructions. It may be stated fairly generally that lenses which are thicker at the middle are collective, while those which are thinnest at the middle are dispersive.

Diferent Positions of Object and I mage.—The principal points are always near the surfaces limiting the lens, and consequently the lens divides the d-irect

pencil containing

the axis into two

parts. The, object

can be either in

front of or behind

the lens as in fig. IO.

If the object point

be in front of the

lens, and if it be realized by rays passing from it, it is called real. If, on the other hand, the object be behind the lens, it is called yirtual; it does not actually exist, and can only be realized as an image. 7

~- D .

1" T

0 Q -

x a L ¢

FIG. Io.

4-25

When we speak of “ object-points, " it is always understood that the rays from the object traverse the first surface of the lens before meeting the second. In the same way, images may be either real or virtual. If the image bebehind the second surface, it is real, and can be intercepted on a screen. If, however, it be in front of the lens, it is visible

to an eye placed

behind the lens,

although the rays do "

¢

not actually inter- ' 0' 6'~~" r

sect, but only appear "

to do so, but the

image cannot be in- ' “ 1 *

tercepted ona screen FIG. II.

behind the lens.

inch an image is saidto beg virtual. These relations are shown in g. 11. ~ .

By referring to the equations given above, it is seen that a thin conver ent lens produces both real and virtual images of real objects, but only a real image of a virtual object, whilst a divergent lens produces a virtual image of a real object and both real and virtual images of a virtual object. The construction of a real image of a 0|

Q:

ffi ' H I ',

0 F' 0 1 F

I A .

FIG. 12.,

real object by a conver ent lens is shown in fig. 3; and that of a. virtual image of a real oimject by a divergent lens in fig. 12. The optical centre of a lens is a point such that, for any ray which passes through it, the incident and emergent rays are parallel. The idea of the optical centre was originally due to Ir Harris (Treatise on. Optics, 1775); it is not properly a cardinal oint, although it has several interesting properties. In fig. I3, let 8113; and CZPQ be two parallel radii of a biconvex lens. *Join P, P2 and let 01131 and OgP¢ o,

Fi

CQ- V

s, CN

H

I I

|

FIG. 13.

be incident and emergent rays which have P1P¢ for the path through the lens. Then if M be the intersection of P1P2 with the axis, we have angle C1P1M=angle C2P2M; these two angles are-for a ray travelling in the direction O1P1P2O2-the angles of emergence and of incidence respectively. From the similar triangles CZPQM and C, P1l/I we have

C1MfC2M=C1P1Z C2P2=1'1JT2. (II)

Such rays as P, P, therefore divide the distance GCR in the ratio of the radii, i.e. at the fixed point M, the optical centre. Calling 51M =S1, SQM =.$'2, tl'l€1'1 C181 =C1M =C1M“S1M, Sl1'lC€ C151 =1'1, C1M = f1'i'S1, al'1Cl similarly CQM =f2+S2. Also S132 =S1M ='S1M-SZM, i.e. d=s1-sg. Then by using equation (11) we have sl =1'1d/(1-rg) and s2=r2d/(11-12), and hence $1/s2=r1/12. The vertex distances of the optical centre are therefore in the ratio of the radii.

The values of sl and 52 show that the optical centre of a biconvex or biconcave lens is in the interior of the lens, that in a plano-convex or plano-concave lens it is at the vertex of the curved surface, and in a concavo-convex lens outside the lens. The Wave-theory Derivation of the Focal Length.-The formulae above have been derived by means of geometrical rays. We here give an account of Lord Rayleigh's wave-theory' derivation of the focal length of a convex lens in termsof the aperture, thickness and refractive index' (Phil. Mag. 1879 (5) 8, p. 480; 1885, 20,5