nifying the movements transmitted from P" to P by means of a pantograph CaP"a'Pb'P"bC, the size will be considerably reduced. If the two coupled lozenges of the pantograph have sides of length 1 and L respectively, it will only be necessary to satisfy the relationship F2 L +1 (2) Another linkage (G. Koenigs, x900) is formed of an articulated lozenge P'AP"B (Fig. 51) and two equal rods AC, CB, pivoted at their (3) Let (Fig. 5rA) the points 0,1 on the optical axis of the lens C be the intersections of the copyholder and plateholder. From these points joint. If we represent by m the common length of the four sides of the lozenge and by n that of the connecting rods, the lengths d = CP' and di= CP" (which, by reason of symmetry, are obviously in a straight line) will always be such that dd' =- — We then only need to give to the constant value of this product the square of the focal length in order to get the required linkage, but in these circumstances the jointed lozenge would. generally be of very great dimensions ; by mag draw towards the lens, distances equal to the focal length/. The points A and B thus obtained must be at distances from C such that CA . CB =P. Using AB as a diameter draw a circle with a centre M and through C draw the chord. DD' perpendicular to AB. Elementary geo metry teaches that CA . CB = CD . CD', hence CD = f. The rays MA, MB, MD are evidently equal and, conversely, by this last equation the respective distances of the conjugate points 0G. IC can be definitely determined (P. R. Burchall, 1933). Among the methods of linking based on this principle may be mentioned (A. Bonnetain, 1934) a system of three racks engaging in M on one and the same toothed wheel.
() Finally, there are numerous arrangements based more or less directly on the hyperbolic cam (G. Pizzighelli, 1889). For instance, a table T on which the optical centre C is fixed can slide on two rails RR (Fig. 52) perpendicular to the object plane P. Movement of the table T is communicated by means of racks and pinions to a table T', but in a direction at right angles. A slot in T' in the form of a rectangular
hyperbola acts on a stud F' which is constrained. to move in an axial slot in T. Any plane per pendicular to the optical axis and containing F' will be conjugate to P.
An obstacle to the employment of these devices in practice is the difficulty of obtaining delivery of a series of lenses of exactly equal focal length, so that it is impossible to make these apparatus in quantity. A number of linkage arrangements have been brought out in the last few years which have an adjustment for com pensating for slight variations in the focal length.
70. Combination of Lenses or Optical Systems. It sometimes happens that another system, convergent or divergent, has to be added to a lens, and it is desirable to be able to determine the focal length of the combination, the optical axes of the different components being assumed to coincide (centred system).
For thin lenses in contact the law that the power (§ 6o) of the system is the sum of the powers of the components may be considered exact, it being understood that negative powers (corre sponding with divergent lenses) are to be subtracted.
Calling f and f' the focal lengths of the components and F that of the resultant system, then 1/F = 1/1 'If' In general, however, this rule is not applicable, and account must be taken of the spacing of the combination, i.e. the separation between the nodal point of emergence of the first system and the nodal point of incidence of the Referring the reader to a treatise on optics for the proof, we shall limit ourselves to formulating the rule. Calling e the separation as defined above, the resultant focal length is given by 1/F = + — elf' The resultant focus is at a distance D from the posterior focus of the second system, equal to D e (f f') These rudiments will have an application to the case of lenses in which focussing is effected by varying the separation of their components (§ 108), sets of lenses (§ 112), supplementary lenses (convergent or divergent') and telephoto lenses (§ 109).