The gradual variation of the magnifying power, which is thus essential to the construction of the in strument, may be effected by different contrivances,—by changing the distance between the two parts of the achromatic eyepiece ; by separating one or more of the lenses of the compound object glass ; or by making a convex, a concave, or a meniscus lens, move along the axis the telescope, between the object glass and its prin cipal focus.
The last of these contrivances, which is, for many reasons, preferable to any of the other two, is represent ed in Plate CCCLXXV. Fig. 8. where 0 is the object glass, whose principal focus is at f, and L the separate lens, which is moveable between 0 and / The parallel rays R, R, converging to f, after refraction by the object glass 0, are intercepted by the lens L, and made to converge to a point F, where they form an image of the object from which they proceed. The fo cal distance of the object glass 0 has therefore been diminished by the interposition of the lens L, and conse quently the magnifying power of the telescope, and the angle subtended by a pair of fixed wires in the eye piece, have suffered a corresponding change. When the lens is at 1, in contact with the object glass, the fo cus of parallel rays will be about ep; the magnifying power will be the least possible, and the angle of the wires will be a maximum; and when the lens is at so that its distance from 0 is equal to Of, the focus of parallel rays will be at f;—the magnifying power will 'tie the greatest possible, and the angle of the wires will be a minimum. When the lens L has any intermediate position between 1 and l', the magnifying power and the angle of the wires have an intermediate value, which depends upon the distance of the lens from the object glass. Hence it appears, that the scale which measures these variations in the angle of the wires, may always be equal to the focal length of the object glass; and it may be shewn in the following manner, that it is a scale of equal parts, the changes upon the angle being always proportional to the variation in the position of the moveable lens.
The point f being that to which the rays incident upon L always converge, we shall have, by the princi ples of optics, F-I-Lf: F=Lf:LF, F being equal to the focal length of the lens L. Now it is obvious, that the magnitude of the image formed at F, alter refrac tion through both the lenses, will be to the magnitude of the image formed at f by the object glass 0, (or by both lenses when L is at l',) as LF is to Lf; for the image formed atf is the virtual object from which the image at F is formed, and the magnitude of the image is always to the magnitude of the object directly as their respective distances from the lens. Hence the
magnifying power of the telescope, when the lens L is in these two positions, is in the ratio of LF to Lf, con sequently the angle subtended by the wires, which must always be inversely as the magnifying power, will be as Lf to LF By making Lf=b, the preceding formula becomes F-Fb :F=b : LF. Hence LF= Then calling A the least angle subtended by the wires, or the angle which they subtend when the lens L is at I', and cc the angle which they subtend when the lens is at L or in any other position, we have A : re=LF : L f, that is F b A A : cc,=— : b, and angle for any dis F4- b F tance b. Calling P the greatest magnifying power, and 9r the magnifying power for any distance 6, we shall PF have P : : and power for any distance b. Making A=20, F=10, and b=0, 1, 2, 3, 4, successively, we obtain from these two formula the results in the following table.
Hence it appears, that when the different values of b are in arithmetical progression, the angle a of the wires varies at the same rate, and therefore the scale which measures these angular variations is a scale of equal parts. The magnifying power, however, does not vary with equal differences, and consequently a scale for measuring its variations, if any scale were wanted, is not a scale of equal parts.
Having thus ascertained the nature of the scale, we shall now proceed to point out the method of constructing it. It is obvious that the length of the scale is arbi trary, and may be made equal either to the whole fo cal length 0 f of the object glass, or to any portion of it. If the lens L moves along the whole length of the axis Of, the angle subtended by the wires can be va ried to a greater degree than if the lens moves only along a portion of the axis; but as this advantage may be obtained by a contrivance hereafter to be described, it will be found more convenient for astronom:cal pur poses to make the lens moveable only along a part of the axis, as from L towards!.