Let us suppose, therefore, that when the object glass 0 is 36 inches in focal length, 10 inches will be a con venient length for the scale, and that the telescope is constructed so that the lens L can move freely through that space reckoned from f, the next thing to be deter mined is the focal length of the L. It is evident that a lens of 6 inches focal length will produce a much greater diminution of magnifying power, and conse quently a much greater increase upon the angle of the wires in moving from f to L than a lens of greater fo cal length ; so that the value of the whole scale in mi nutes or seconds, or the increase in the angle occasion ed by the motion of the lens from/ to L, must be in versely as the focal length of the moveable lens. If the angle of the wires is 26 minutes, for example, and if the magnifying power of the telescope is diminish ed from 40 to 30 by the motion of the lens from f to L; then when the lens is at L, the angle of the wires will be 3 40", for 30 : 40=26' : 34' 40". Hence we have a scale of 10 inches to measure 26'-34' 40", or 8' 40", and therefore every tenth of an inch on the scale will be equal to 5".2.
If we employ a lens of much greater focal length, so as to diminish the magnifying power only from 40 to 35, and if the angle of the wires is 29 minutes ; then when the lens is at L, the angle of the wires will be 33' 9" nearly, for 35 : 40=29': 33' 9". And hence we have a scale of 10 inches to measure 9", or 4' 9", consequently every tenth of an inch on the scale corresponds to 3".3. From this it will be mani fest, that the accuracy of the scale is increased by in creasing the focal length of the moveable lens.
The two preceding examples are suited to a micro meter for measuring the diameters of the sun and moon at their various distances from the earth ; but, in order to show the resources of the principle on which the in strument is founded, we shall take another cy.ample, better adapted to this purpose.
Let us suppose that the pair of fixe,1 wires subtends only an angle of 40", for the purpose of measuring the distance between double stars, or the diameters of some of the smaller planets, that the telescope magnifies 300 times, and that the lens L in its m Ann f:.utn f to L. through a space of 10 inches, diminishes the power of the instrument to 240 ; then when the lens is at L, the angle of the wires will be 50", for 240 : 300= 40" : 50". Hence we have a scale of ten inches to measure 40" 50", or 10", so that every inch of the scale corresponds to 1", and every tenth of an inch to 6"'.) Fro al this it follows, that the accuracy of the scale increases as the angle subtended by the fixed wires diminishes.
If it should he found convenient to make each divi sion of the scale correspond to a greater variation in the angle than in any of the examples which we have given, it would then he molter to 111,11“; use of a vernier for subdividing the units of the scale.
In order to show more clearly the method of com pleting the scale, we have represented a telescope fur nished with a micrometer, in Plate CCC1.XXV. 9, tt here AB is the principal tube, with the object glass at B ; CD a secondary tube, at. the right hand extremity of which is fixed the lens L, (Fig. 8.) which is moved backwards and forwards with the tube, by the milled head F ; and E the eye-piece, which is ad justed to distinct vision by the milled head G. The small index i projecting from the principal tube below A, and furnished with a vernier scale if necessary, points out the divisions on the scale. Let it now be required to construct the scale for the 2d Example, where the lens L, by moving from f to L, changes the power of the telescope from 40 to 35. Having moved the tube CD as far out as possible, by the milled head F, mark the point of at n to which the index i points, and this will be theibeginning or zero of the scale. Adjust tha eye•tube E to distinct vision, and find by ex periment* the angle subtended by the fixed wires: Let this angle be 29 minutes. Move the tube CD as far in as possible by means of the nut F, till the index i points to m, and mark this as the other extremity of the scale. Let the eye•tube E be again adjusted to distinct vision, and the angle subtended by the wires again determined experimentally ; and let this angle be now 33' 9". In order to find the point of the scale corresponding to 33', say, As 4'9", the value of the whole scale is to 10 inches, the length of the scale, so is 9" to 36 hundredths of an inch, which, being set from in to o, will mark out the point o as corresponding to 33'. The space o n being divided into four parts for minutes, and each minute into as many divisions as possible, the micrometer will be ready for use. If great accuracy is required, every unit of the scale might be determined experimentally, by any of the methods mentioned in the preceding note. The instrument thus constructed, is capable of measur ing angles only between 29' and 33' 9", and is therefore peculiarly fitted for determining the diameters of the sun and moon. Its range, however, could easily have been extended, by lengthening the tube CD, or by em ploying a moveable lens, of smaller focal length;—or instead of one pair of wires, we might use several pairs, as AB, CD, a b, c d, cep, (Plate CCCLXXV. Fig.