In constructing this micrometer for astronomical purposes, the semilenses may be made to move only along a portion of the axis 0 /, particularly if the in strument is intended to measure the diameters of the sun and moon, or any series of angles within given limits. By increasing the focal itngth of the semi lenses, or by diminishing the distance between their centres, the angles may be made to vary with any de gree of slowness, and of course each unit of the scale will correspond to a very small portion of the whole angle. The accuracy and magnitude of the scale, in deed, may be increased without limit ; but it is com pletely unnecessary to carry this any farther than till the error of the scale is less than the probable error of observation.
Let us now examine the theory of this micrometer, and endeavour to ascertain the nature of the scale for measuring the variations of the angle. For this pur pose, let LL, Fig. 1, be the object-glass, which forms an inverted image, In n, of the object MN, and let the semilenses AB, having their centres at an invariable distance, be interposed between the object-glass and its principal focus, in such a manner, that their cen tres are equidistant from the axis Of. Now, it is ob vious, that the size of the image In n is proportional to the size of the object ; and, as the angle subtend ed by MN depends upon its size, the magnitude of the image in n may, in the case of small angles, be as sumed as a measure of the angle subtended by MN. As the rays which proceed from the point M are all converged to in by means of the lens LL alone, the ray b A. which passes through the centre of the semi lens A, must of course have the direction b in; and, as it suffers no refraction in passing through the cen tre of A, it will proceed in the same direction b A In after emerging from the semilens, and will cross the axis at F. For the same reasons the ray c B, pro ceeding from N, and passing through the centre of B, will cross the axis at F, as it advances to n. If the distance of F from A and B happens to be equal to the focal length of the lenses A and B, when combin ed with LL, distinct images of M and N will be form ed at F, and they will appear to touch one another ; and the line in n. being the size of the image that would have been formed by the lens LL alone, will be a measure of the angle subtended by the points M, N. If the point F, where the lines A 771, B n cross the axis, should not happen to coincide with the focus of the lenses A, B, when combined with LL, then let this focus be at F', nearer A and B than F. Draw the lines Al' in, BF' n, then it is obvious, that if the angle subtended by MN were enlarged, so as to be represented by n' in', instead of n in, or so that the lens LL alone would form an image of it equal to n' m', the point of intersection F would coincide with the focus F ; so that, in every position of the lenses A, B, with respect to LL, the points INI, N may always be made to subtend such an angle, that when they are placed before the telescope, the points F, F' will coincide, and consequently the images of the points M, N will be distinctly formed at F', and will be in contact.
Whenever this happens, the space n' in will be a mea sure of the angle thus subtended by MN. Ilence, it follows, that whatever be the position of the semi lenses A, 13, on the axis Of, the rays b A, c B, which pass through the centres of the semilenses, will cross the axis at some point F, corresponding with the focus of rays diverging from M, N, and will mark out the size of the image n' m', and consequently the relative magnitude of the angle subtended by the two points M, N.
From the equality of the vertical angles AF' B, n' F' in', and the parallelism of the !hies AB, a' in', we shall have n' m': AB =./. F': GF'; and F'=b, and considering that b being the focal F b length of the semilenses, we have n' : AB=b F-Fb and consequently ni = AB 4- Now, call ing AB=2, F=I0, and 6=1, 2, 3, successively, we shall obtain faom which it appears, that when b is in arithmetical progression, the angle n' in' varies at the same rate, and consequently the scale which measures the varia tions of the angle, subtended by the centres of the two images, is a scale of equal parts.
This instrument undergoes a very singular change, when constructed as in Plate CCCLXXVI Fig. 3, so that the semilenses are outermost and immoveable, while another lens, LL, is made to move along the axis Gf. In this case, a double image is formed as before, but the angle subtended by the centres of the images never suffers any change during the motion of the lens LL along the axis of the telescope. If the two images are in contact when the lens LL is close to the semilen ses, they will continue in contact in every other position of LL ; but the magnitude of the images is constantly increasing during the motion of LL towards f, the prin cipal focus of the semilenses. The reason of this remark able property will be understood from Fig. 3, where M, N, are two objects placed at such an angle, that the rays passing through the centres A, B, of the semi lenses, cross the axis at F, the focus of the combined lenses for rays divergent from M and N In this ease, distinct images of M and N will be formod at F, and will consequently be in contact If the lens 1.L is re moved to the position L' L'. the rays 111 m, N n, which are incident upon it at the points nt and a, having the t.same degree of convergency as before, will be refract ed to the focus of the combined lenses for rays di verging from MN. Two distinct images of the ob ject will therefore be formed at F', and these images will still be in contact. In like manner, it may be shewn, that whatever be the position of the lens LL between G and f, the rays NI f, N f, will cross the axis at a point coincident with the focus of the combined lenses, and will there form two images always in con tact. Hence it follows, that though the magnifying power of the instrument is constantly changing with the position of the lens LL, yet the angle subtended by the centres of the two images never suffers the least variation.