crossing of the wires in the collimator, and the number of revolutions with the part of a revolution must be read. The same moveable wire must then be brought to the fixed wire in the transit telescope, and the number of revolutions with the part of a revolution again read. These movements and readings are to be repeated several times with the transit telescope in one position; and the like operations and readings must take place with that telescope in a reversed position: then the difference between the two means of the readings, reduced to seconds of time (corrected for the effects of diurnal aberration if thought necessary), is the error of collimation with respect to the middle wire of the transit telescope. The distance (in time) of this middle wire from the place of the imaginary mean wire is supposed to be known (art. 91.), and that distance must be added to or subtracted from the error just found in order to have the exact correction which, on account of imperfect collimation, is to be applied to the mean time of the observed transit of a star; it being understood that the star is in the equator, otherwise the correction must be divided by the cosine of the stars declination (art. 70.) before it is applied. 126. The Repeating Circle is a graduated instrument carrying two telescopes which are parallel to its plane, one on each side: these turn independently of each other on the centre of the circle; and the whole instrument is supported on a stand, the upper part of which is made to turn round in azimuth, carrying with it the circle and telescopes, and it has a joint which allows the plane of the circle to be placed in any position which may be required. When the zenith distances of celestial bodies are to be observed, the circle is placed in a vertical plane, and the correctness of its position is determined by a plumb line. Let ZHNO represent the circle in a vertical position in the plane of the meridian (for example), and let s be the place of a celestial body in the same plane. d 127. The telescope EF being clamped so that the index is at the zero of the graduations, suppose at E, let the circle be moved in its own plane about c till that telescope is directed to £ the objects; and then turn the telescope on the opposite face of the circle till, by the spirit-level attached to it, it is in a horizontal position, as HO. Next turn the circle half round in azimuth so that the telescope EF may lie in the position ef, the zero point H E N e now being at e on the circle, and the telescope HO remaining horizontal but its extremities being reversed. The circle being clamped, turn the telescope at ef about C till the object is again seen in the centre of its field; it will a second time have the position EF and the number of the graduation at E will express the value of the arc E Ne, or twice the zenith distance of s. Again turn the circle half round in azimuth, the telescope EF remaining clamped and HO continuing to be in a horizontal position; then EF will be a second time in the position ef. The circle remaining fixed turn the telescope at ef about c till it is a third time directed to s; then the number of the graduation at E will express four times the zenith distance of s, if it be supposed that s has not changed its altitude during the time of making the observations. This process may be repeated any number of times, even till the whole arc passed over by the index of the telescope EF contains several circumferences, and it will be necessary merely to read the number of the graduation shown by the index after the last repetition; for the whole number of degrees passed over by the index may then be found, and if this number be divided by twice the number of repetitions the result will be the correct value of the angle zcs. 128. In the above example, the telescope HO has only served, by its spirit level, to keep the telescope EF at the same angle with the horizon in the direct and reversed positions of the circle on the vertical axis of the stand. But when an angle is to be taken between two objects, as s and s' in a plane, which, passing through the eye, is parallel or oblique to the horizon; then, the circle being placed in that plane, let the telescope EF be clamped to it with the zero of the graduations at E, and let the circle be turned about c in its own plane till that telescope is directed to s': next turn the telescope HO by its independent motion till it is directed to s, and having made it fast to the circle, turn the latter with the telescope, till HO is directed to s', when EF will be in the position fe and the zero of the graduations will be at f. Now the circle being clamped, turn the telescope from the position fe to the position EF in the line cs; then the graduation at E will express the degrees in the arc Ef, or twice the required angle scs'. Again, the telescopes being clamped, turn the circle till the telescope EF is in the position cs' as at first; and, the circle being clamped, turn the telescope HO into the position cs: then this telescope being clamped, turn the whole circle till the same telescope is directed to s', and proceed as before. When the telescope EF is again brought to the position EF H or Cs, the graduation at E will express twice the value of Ef or four times the required angle scs'; and the above process may be repeated any number of times. 129. The reflecting octant, first brought into use by Mr. Hadley, as well as the reflecting sextant, quintant and circle, are now from their portability, and the convenience of being used when held in the hand, generally employed for the purpose of obtaining by observation the angular altitude of a celestial body above the horizon, or the angular distance between the moon and the sun, or a star. These instruments are, however, so well known as to render a minute description of them unnecessary; and therefore only a brief explanation of their nature and adjustments will be given. H S K B M E D D' The subjoined figure represents the usual sextant; and the optical principle on which all such instruments are constructed is the same. Each carries on its surface, which is supposed to coincide with the plane of the paper, a mirror at A, and at B a glass which is in part transparent, and in part a mirror, the line of division being parallel to the plane of the instrument: both the glasses are perpendicular to the plane of the instrument; but B is fixed, and A is capable of being turned on an axis perpendicular to that plane by the motion of an index bar AC. 130. The arch OP of the instrument is graduated, and when the plane of the mirror A passes through AO, the index upon it is, or should be, at the zero of the graduations, and the glasses at A and B should be parallel to one another. In this state, if a pencil of light coming from a remote object at s fall upon A, it will be reflected to the quicksilvered part of B, and from thence be reflected in the direction BE parallel to SA; therefore if the eye of the observer be in the line BE, a pencil of light coming from the same object s will pass through the transparent part of B and enter the eye in the same direction as the reflected pencil; and the direct and reflected images of s will appear to coincide; the distance of s being great enough to render the angle at s between the directions of the pencils insensible. But if the index bar be placed in some other position as AC, so that a pencil of light from a remote object as s' may fall on A and be reflected to B, from whence it may be again reflected in the direction BE; then if the axis s'A of the pencil be produced to meet B E, as in E, the angle at E between S'A thus produced, and the axis of a pencil of light coming through the transparent part of в from an object in the direction EB produced, will be equal to twice the angle AFB, at which the mirrors are inclined to each other, or to twice the equal angle OAF. The graduations on the arch or are numbered so as to express, from o to the index near C, twice the value of any angle, as O AF; and hence the graduation read on that arch expresses the value of the angle between the ray S'A produced and BE, that is, the angle at E subtended by an arc corresponding to S'H in the heavens between the object seen directly through B, and by reflection from a and B. 131. The eye of the observer being necessarily situated at an aperture in a plate at D, or at one end D' of a telescope, the angle subtended at the eye by the arc s'H, is S'D'H; and the angle ES'D', which is evidently equal to the difference between the angles at E and D', is called the parallax of the instrument. This angle is, of course, insensible when the object at s' is a celestial body. 132. In order to prove that the angle s'EH is equal to twice the angle AFB or OAF, produce FB to K, and FA to M: then because the exterior angle of a triangle is equal to the sum of the interior and opposite angles (Euc. 32. 1.), the angle ABK BAF + AFB; whence 2 ABK = 2BAF +2AFB: but the angle of incidence being (by optics) equal to the angle of reflection, hence but ABK = DBF or KBH, and BAF ABH BAE + AEB; MAS' or EAF; BAE: therefore from the preceding equation, AEB2AFB, or = 20AF; that is, the angle subtended by s' H is equal to twice the inclination of the mirrors. 133. If therefore EH were a horizontal line, the angle S'EH, or 20AF, that is, the angle read on the arch OP would be the altitude of the celestial body at s' above the visible horizon. When an altitude is observed at sea, the instrument is held so that the line EH is a tangent to the earth's surface at the edge of the sea, and consequently it is depressed below a line passing through the eye of the observer perpendicularly to the earth's diameter at the ship by a small angle which is called the dip (art. 164.) of the horizon. On land the instrument is held so that the line EII may pass through the image of the sun when seen by reflection in quicksilver; and then the observed angle is equal to twice the altitude of the celestial body above the horizon. For imagine a vertical plane to pass through s' and E, and through the surface of the quicksilver, cutting the latter in MP; then the axis s'N of a pencil of light being reflected at N to the eye of the spectator at E, the reflected image of s' will appear to be at H in the line EN produced; but the angle s'NP will by optics be equal to ENM or to E P N M the vertically opposite angle HNP; therefore s'N H2S' NP. The line EN being very short compared with the distance of s' from the spectator, the angle s'EN may be considered without error as equal to s'N H, or double the apparent altitude. In taking the altitude of the sun or moon above the sealine, it is customary to move the index till the lower edge, or limb, of the celestial body is a tangent to that line, because the contact of the limb with the sea-line can be more correctly distinguished than the coincidence of the centre of the disk with that line; in this case, there is obtained the altitude of the lower limb, and to this must be added the angle subtended by the visible semidiameter of the luminary, in order to have the altitude of its centre. On land, where an artificial horizon of quicksilver must be used, it is customary, for a like reason, to move the index till the upper or lower edge of the disk seen in the quicksilvered part of the horizon glass B is made to coincide with the lower or upper edge of the disk seen in the artificial horizon; and thus there is obtained twice the altitude of the upper or lower limb: the angular measure of the semidiameter must consequently be subtracted from, or added to half the angle obtained from the observation, in order to have the altitude of the centre. 134. A reflecting circle cannot measure an angle much exceeding the greatest which may be measured by a sextant (about 120 degrees); but it has an advantage over the latter instrument, since, by means of its three indexes, the value of the observed angle may be read on as many different parts of the circumference, and thus the errors of the graduation are diminished; the angle between two fixed objects may also be observed twice, by turning the mirror A in contrary directions, and thus an error in its position may be eliminated. |