Instead of the difference of right ascensions, the dis tance of the planet from the sun's limb, in lines paral lel to the equator, may be inure accurately observed thus : Separate tile segments, and turn about the wires and micrometer, so as to make both images V, v, Plate CCCLX X V. Fig. 17, run along HR, or so that the two intersections I, T of the sun's image may pass C c at the same time. Then bring the planet's and suit's limbs into contact, as at V, and do the same for the other limb of the sun, and half the difference gives the distance of the centre of the planet from the mid dle of the chord on the sun's disc parallel to the equa tor, or the difference of the right ascensions of their centres, allowing for the motion of the planet in the interval of the observations, without any regard to the error of adjustment, for the same reason as before. For kf you take any point in the chord of a circle, half the difference of the two segments is manifestly the distance of the point from the middle of the chord ; and as the planet runs along 1111, (Plate CCCLXXV. Fig. 18,) the chord is parallel to the equator.
In like manner, the distances of their limbs may be measured in lines perpendicular to the equator, by bringing the micrometer into the position already de scribed, and instead of bringing V to HR, separate the segments till the northernmost limbs coincide as at V ; and in the same manner make their southernmost images to coincide, and half the difference of the two measures, allowing for the planet's motion, gives the difference of the declinations of their centres.
Hence the true place of a planet in the sun's disc may at any time of its transit be found ; and conse quently the nearest approach to the centre and the time of ecliptic conjunction may be deduced, although the middle should not be observed." Phil. Trans. 1771.
In using the divided object-glass micrometer, it has been found in practice to give different measures of the same angle at different times. This error arises from an alteration in the focus of the eye of the observer; for if the eye is not suited to see distinctly the inter section of the two pencils PF, QF, Fig. 13, at the point F, the two images will either appear separate or overlap one another, according as the eye is adapted to a nearer or a more distant object than F. In order to remedy this inconvenience, Dr Maskelyne contrived, in the year 1776, the prismatic micrometer, which we shall describe in his own words.
" Let ACB (Plate CCCLXXV. Fig. 19.) represent the object-glass, and d the eye-glass of a telescope, and PR a prism placed to intercept part of the rays coming from an object, suppose the sun, before they fall on the object-glass. The rays EE, proceeding front the east ern limb of the sun, and refracted through the object glass ACB without passing through the prism, will form the corresponding point of the sun's image at e ; and the rays \V\V proceeding in like manner from the western limb of the sun, will be refracted to form the correspondent point of the sun's image at w. But the rays 2E, 2E, 2W, 2W, proceeding in like manner from the eastern and western limbs of the sun, and falling on the prism PR, and thence refracted to the object-glass ACB, will, after refraction through it, form the corres pondent points of the sun's image at 2e, 2w. Let the refraction of the prism be equal to the sun's apparent dia meter: in this case, at whatever distance the prism be placed beyond the object-glass, the two images of the Gun, we, 2w 2e will touch one another externally at the point e2w; for the rays 2W, 2W, proceeding from the western limb of the sun, being inclined to the rays EE, proceeding from the eastern limb in the angle of the sun's apparent diameter, will, after suffering a refrac tion in passing through the prism equal to the sun's apparent diameter, emerge from the prism, and fall upon the object-glass parallel to the rays Eli, and con sequently will have their focus 2w coincident with the focus e of the rays EE, and therefore the two images of the sun, we, 2w2e, will touch one another externally at the point e2w, and the instrument will measure the angle EC2W, and that only.
But if the prism be placed within the telescope, the angle measured by the instrument will be to the re fraction of the prism as the distance of the, prism from the focus of the object-glass is to the focal ;'isiance ut the object-glass ; or if two prisms be used to form the two images, with their refracting angles placed con trary ways, as represented in Plate CCCLXXV. Fig. 20, and 21. the angle measured will be to the sum of the refractions of the prisms as the distance of the prisms from the focus of the object-glass is to the focal distance of the object-glass. For let ACB (Fig. 20.) represent the object-glass, and d the eye-glass of a telescope, and PR, RS, two prisms interposed between them, with their refracting angles turned contrary ways, and the common sections of their refracting planes touching one another at R. The rays proceeding from an object, suppose the sun, will be disposed, by the refraction of the ob ject-glass, to form an image of the sun at the focus ; but part of them falling on one prism, and part on the other, will be thereby refracted contrary ways, so as to form two equal images Tv e, 2w2e, which, if the re fractions of the prisms be of proper quantities, will touch one another externally at the point e2w. Let ECN be the axis of the pencil of rays EE proceeding from the sun's eastern limb ; and WCO the axis of the pencil of rays \V\V proceeding from the sun's western and the point N the place where the image of the sun's eastern limb would be formed, and the point 0 where that of the western limb would be formed, were not the rays diverted from their course by the re fractions of the prisms. But, by this means, part of the rays EE, which were proceeding to N, falling on the prism PR, will be refracted, to form an image of the sun's eastern limb at e, while others of the rays EE, which fall on the prism RS, will be refracted, to form an image of the sun's eastern limb at 2e. In like manner, part of the rays \V\V, which were proceeding to form an image of the sun's western limb at 0, fall ing on the prism RS, will be refracted, to form an image of the sun's western limb at 2w coincident with e, the point of the image correspondent to the sun's eastern limb : while others of the rays WW, which fall on the prism PR, will be refracted to form the image of the sun's western limb at w. The two images w e, 2w2e, are supposed to touch one another externally at the point e2w. The ray EFR, which belongs to the axis ECN, and is refracted by the prism PR to e, undergoes the refraction NRe, which (because small angles are proportional to their sines, and the sine of NR e is equal to the sine of its supplement NRC) is to NCR as NC or C e is to NR or R e. In like manner, the ray WGR, which belongs to the axis \VCO, and is refracted by the prism RS to 2w or e, undergoes the refraction OR e, which is to OC e as OC or C e is to RO or R e thereby e, by composition, ORN, the sum of the refractions OR e, NR e, is to OCN, the sum of the angles OC e, NC e, or the sun's apparent diameter, as C e to R e ; that is, as the focal distance of the ob ject-glass to the distance of the prisms from the focus of the object-glass.