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FOCAL LENGTH OF LENSES ; SCALE OF IMAGE ; CONJUGATE POINTS 6o. Conjugate Points. When a point R' is the image of a point R (Fig. 44) formed by an optical system, the point R is also the image of R' (principle of reversibility of light rays') ; the points of such a pair are two conjugate points of the optical system considered. Various formulae and simple graphical constructions enable us, when the focal length of a lens and the positions of the nodal points or foci (§§ 44 and 45) are known, to determine the position of the image of a point the position of which is known.

In order to avoid the complication of curva ture of the field, we shall consider, in what follows, only points on the optical axis, the images of which are therefore also on the optical axis.

The power of a lens (convergent power or con vergence) is the reciprocal x/F of the focal length ; when the focal length is measured in metres the power of the system is expressed in diopters. Thus, for example, a lens of 020 m. (8 in.) focal length has a power of i/o-2o, or 5 The effect of an optical system is to add its con vergence (or subtract, in the case of a divergent system, i.e. of negative vergence) to that of the light beams passing through it. Now a point at a distance p from the nodal point of incidence sends to the system a divergent beam of which the negative vergence is itp (this is also some times known as the proximity, (optical) of the point considered), which can also be expressed in diopters. If the convergence (positive vergence) 1/F of the system is greater than the negative vergence of the beam i/p, the emergent beam will have a convergence (i/F diopters. The proximity 1/pi of the image (p' being measured from the nodal point of emergence) is then given by = ',IF - ',fp or 3-1F which translates literally the mechanism of the alteration of the waves of light made by the optical system in question If, instead of considering, as above, the ultra nodal distances (reckoned from the nodal points), we consider the ultra-focal distances d and d' (reckoned from the foci), we shall obtain the law of conjugate points in the form given by Newton, which is often very advantageous, d X d'-----= or, in ordinary language, the product of the ultra-focal distances of a point and of its image is equal to the product of the focal length by itself.'

61. Among the different methods of graphic ally representing the law of conjugate points, the following (Lissajous, 1870) enables us to account for all the practical consequences of this rela tionship at first sight. Construct a square (Fig. 45) NFMF', of which the sides are equal to the focal length of the lens considered, and produce the sides NF and NF' to X and Y respectively. From the origin N mark off NR on NX equal to the ultra-nodal distance (p) of the point object (FR is thus the extra focal distance d), join RM and produce it to meet NY in R'. The length NR' is equal to the ultra nodal distance p' of the point-image and F'R' is the ultra-focal distance d'. If now RI?' is rotated about M, its intersections (produced if necessary) with NX, NY correspond to two conjugate points.' It is seen that as R moves away from the lens, R' moves nearer to it, and vice versa. When R moves to infinity, the straight line MR becomes parallel to NX, and R' coincides with P. Inversely, if R approaches F, the straight line RAI becomes parallel to NY and consequently R' recedes to infinity.

already mentioned (§ 44) that the nodal points are conjugate.

62. Relations between the Size of Object and Image. If we consider a lens without appreciable distortion or curvature of the field (which would obviously not be the case with the meniscus represented in Fig. 46), we know that the images If the point R approaches closer to the lens than the focal length, e.g. to the position marked T, the straight line MT no longer meets NY, but its prolongation NY', in T', corresponding to a virtual image (§ 43).