a*? Cornell llntuprattg Sltbrarg BOUGHT WITH THE INCOME OF THE SAGE ENDOWMENT FUND THE GIFT OF MiMirtj 10. Sage 1891 .ft,.La.a !B .i.g..fe.. it xii]AUt 1 93°« Cornell University Library QB 145.C19 1899 The elements of practical astronomy, 3 1924 012 500 777 The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924012500777 THE ELEMENTS PRACTICAL ASTRONOMY r hg?y^^> THE ELEMENTS PRACTICAL ASTRONOMY BY W. W. CAMPBELL ASTRONOMER IN THE LICK OBSERVATORY SECOND EDITION, REVISED AND ENLARGED THE MACMILLAN COMPANY LONDON: MACMILLAN & CO., Ltd. 1899 All rights reserved COPYKIGHT, 1891, By THE REGISTER PUBLISHING CO. Copyright, 1899, By THE MACMILLAN COMPANY. D.E Nortoocti ^rcss J. S. Cushing & Co. — Berwick St Smith Norwood Muss. U.S.A. PREFACE TO THE SECOND EDITION My experience in presenting the elements of practical astronomy to rather large classes of students in the University of Michigan led me to the conclusion that the extensive treatises on the subject could not be used satisfactorily, except in special cases. Brief lecture notes were employed in preference. Arrangements were made with a local publisher that the notes should be written out in full and printed, almost exclusively it was sup- posed, for use in my own classes. The process of en- largement had just begun when the call to my present position was accepted. The completion of the manu- script in the midst of new and pressing duties was extremely difficult ; the text and the details of the treat- ment lacked the harmony which can come only from a leisurely development of the subject. Nevertheless, the first edition has been used in a great many colleges and universities whose astronomical departments are of the highest character. This is my reason for carefully revis- ing and slightly enlarging the book for a second edition. A word concerning the limitations of the book may not be out of place. The field of practical astronomy has become very extensive, embracing essentially all the work carried on in our astronomical observatories. It VI PREFACE includes the photographic charting of the stars, the spec- troscopic determination of stellar motions, the determi- nation of solar parallax from heliometer observations of the asteroids, the construction of empirical formulae and tables for computing atmospheric refraction, and scores of other operations of equally high character. These, however, can best be described as special problems, re- quiring prolonged efforts on the part of professional astronomers ; in fact, the solution of a . single problem often severely taxes the combined resources of a number of leading observatories. While it is evident that a discussion of the methods employed in solving special problems must be looked for in special treatises and in the journals, yet these methods are all developed from the elements of astronomy, of physics, and of the other related sciences. It is intended that this book shall contain the elements of practical astronomy, with numer- ous applications to the problems first requiring solution. It is believed that the methods of observing employed are illustrations of the best modern practice. The methods of reduction are intended to be exact to the extent that none of the value and precision of the obser- vations will be sacrificed in the computations ; further refinement would be superfluous, and misleading to the inexperienced. The demonstrations are direct and fun- damental, except in the case of refraction. The scien- tific basis of the subject of refraction is largely physical, and the astronomical superstructure is almost wholly empirical. For these reasons, the proper proportions of the subject with reference to the rest of the book seem to be preserved by the insertion of the final formulae. PREFACE Vll An attempt has been made to give credit for methods which have not yet found their way into general practice. The illustrations of modern instruments are from cuts kindly furnished by the makers, viz.: those for Fig- ures 15, 20, 21 and 24 by Fauth & Co., Washington ; those for Figures 14, 17 and 26 by C. L. Berger & Sons, Boston; that for Figure 10 by the Keuffel & Esser Co., New York ; and that for Figure 27 by War- ner & Swasey, Cleveland. Figure 25, of a Repsold meridian circle, is copied with permission from Baron A., v. Schweiger-Lerchenfeld's Atlas der Mimmelshunde (Vienna). The author is indebted to his colleagues, Professors » Schaeberle, Tucker and Hussey, and Mr. Perrine, for valuable suggestions and assistance. W. W. CAMPBELL. Lick Observatory, University of California, January, 1899. CONTENTS CHAPTER I PAGE The Celestial Sphere 1 Definitions 2 Systems of coordinates 7 Transformation of coordinates 8 Distance between two stars 14 CHAPTER H Time — Definitions 15 Conversion of time 18 CHAPTER HI Correction op Observations — Form and dimensions of the earth 23 Parallax — Definitions 25 In zenith distance 26 In azimuth and zenith distance ...... 27 In right ascension and declination 31 Refraction 32 General laws — In zenith distance 33 In right ascension and declination 36 Dip of the horizon 36 Semidiameter — Of the moon 38 Contraction of semidiameters by refraction . . . .39 Aberration 40 Diurnal aberration in hour angle and declination . . 41 Diurnal aberration in azimuth and altitude .... 43 Sequence and degree of corrections 43 ix CONTENTS CHAPTER IV PAGE Precession — Nutation — Annual Aberration — Proper Motion . . . .44 Star places . . 45 Precession . 46 Annual precession . • 50 Proper motion . . . 54 Determination by the method of least squares . 58 Reduction to apparent place ....... 61 CHAPTER V Angle and Time Measurement — The vernier The reading microscope Eccentricity ....... The micrometer .... Determination of the value of a revolution The level . .... Determination of the value of a division The chronometer Eye and ear method of observing . The astronomical clock — The chronograph 65 67 69 70 72 79 81 83 85 CHAPTER VI The Sextant — Description General principles of the sextant . Methods of observing with the sextant Adjustments of the sextant . Corrections to sextant readings Determination of time . By equal altitudes of a fixed star By equal altitudes of the sun By a single altitude of a star . By a single altitude of the sun Determination of geographical latitude By a meridian altitude of a star or the By an altitude of a star . By circummeridian altitudes . Determination of geographical longitude By lunar distances .... 91 92 94 96 101 101 102 106 107 109 109 110 112 115 115 CONTENTS XI CHAPTER VII The Transit Instrument — Description Definition of instrumental constants . General equations Determination of the wire intervals Determination of the level constant Determination of the collimation constant Determination of the azimuth constant Meridian mark, or mire Adjustments Determination of time .... Reduction by the method of least squares Correction for flexure .... Personal equation . . . . Determination of geographical longitude By transportation of chronometers By the electric telegraph By the heliotrope .... By moon culminations . PAGE 122 127 129 132 134 137 142 143 144 146 152 157 157 159 159 160 163 164 CHAPTER VIII The Zenith Telescope 167 Determination of geographical latitude by Talcott's method . 167 Combination of results by the method of least squares . . 173 CHAPTER IX The Meridian Circle — Description Determination of right ascension . Determination of flexure Errors of graduation .... Determination of declination 175 177 181 183 187 CHAPTER X Astronomical Azimuth .... Azimuth by a circumpolar star near elongation Azimuth by Polaris observed at any hour angle 191 193 199 CONTENTS CHAPTER XI PAGE The Surveyor's Transit 203 Determination of time 203 By equal altitudes of a star 203 By a single altitude of a star 205 By a single altitude of the sun 207 Determination of geographical latitude 209 By a meridian altitude of a star 209 By a meridian altitude of the sun ...... 210 Determination of Azimuth 211 CHAPTER XII The Equatorial — Description .... Adjustments .... ... Magnifying power Field of view . . Determination of apparent place of an object By the method of micrometer transits . By the method of direct micrometer measurement Determination of position angle and distance The ring micrometer . .... 212 214 220 221 223 223 229 233 236 APPENDICES A. Hints on computing 241 B. Interpolation formulae 244 C. Combination and comparison of observations . . . 247 D. Objects for the telescope 251 Table I. Pulkowa refraction tables 254 Table II. Pulkowa mean refractions 257 Table III. Reduction to the meridian, or to elongation . . 258 Index 261 PRACTICAL ASTRONOMY CHAPTER I DEFINITIONS — SYSTEMS OF COORDINATES— TRANS- FORMATION OF COORDINATES 1. The heavenly bodies appear to us as if they were situated on the surface of a sphere of indefinitely great radius, whose center is at the point of observation. Their directions from us are constantly changing. They all appear to move from east to west at such a rate as to make one complete revolution in about twenty-four hours. This is due to the diurnal rotation of the earth. The sun appears to move eastward among the stars at such a rate as to make one revolution per year. This is caused by the annual revolution of the earth around the sun. The moon and the various planets have motions characteristic of the orbits which they describe. Measurements with instruments of precision enable us to detect other motions which, we shall see later, are conveniently divided into two classes : those due to parallax, refraction, and diurnal aberration, which depend upon the observer's geographi- cal position; and those due to precession, nutation, annual aberration, and proper motion, which are independent of the observer's position. From data furnished by systematic observations it has been shown that these motions occur in accordance with well-defined physical laws. It is therefore possible to compute the position of a celestial object for any given 2 PRACTICAL ASTRONOMY instant. A table giving at equal intervals of time the places of a body as affected by the second class of motions mentioned above, is called an ephemeris of the body. The astronomical annuals* furnish accurate ephemerides of the principal celestial objects several years in advance. If an observer knows his position on the earth, he can, from data furnished by the ephemerides, compute the direction of a starf at any instant. Conversely, by observing the directions of the stars with suitable, instru- ments, he can determine the time and his geographical position. It is with this converse problem that we are principally concerned. DEFINITIONS • 2. The sphere on whose surface the stars appear to be situated is called the celestial sphere. Any plane passing through the point of observation cuts the celestial sphere in a great circle. Since the radius is indefinitely great, all parallel planes whose distances apart are finite cut the sphere in the same great circle. In order to determine the position of a point on the sphere and express the relation existing between two or more points, the circles, lines, points and terms defined below are in current use. The horizon is the great circle of the sphere whose plane passes through the point of observation and is perpen- dicular to the plumb-line. The produced plumb-line, or vertical line, cuts the sphere above in the zenith and below in the nadir. The * The principal annuals are the American Ephemeris and Nautical Almanac, the Berliner Astronomisches Jahrbuch, the (British) Nautical Almanac, and the Connaissance des Temps. Unless otherwise specified we shall refer to the first of these, and call it the American Ephemeris, or the Ephemeris. t For convenience we shall use star, point or body to denote any- celestial object. DEFINITIONS 3 zenith and nadir are the poles of the horizon, and all great circles passing through them are called vertical circles. ^, The points of the horizon directly south, west, north and east of the observer are called, respectively, the south, west, north and east points. The meridian is the vertical circle which passes through the south and north points. The prime vertical is the vertical circle which passes through the east and west points. The altitude of a point is its distance from the horizon, measured on the vertical circle passing through the point. Distances above the horizon are + ; below, — . The alti- tudes of all points on the sphere are included between 0° and + 90°, and 0° and - 90°. Instead of the alti- tude, it is frequently convenient to use the zenith distance, which is the distance of the point from the zenith, meas- ured on the vertical circle of the point. It is the com- plement of the altitude. The zenith distances of all points on the sphere lie between 0° and + 180°. The azimuth of a point is the arc of the horizon inter- cepted between the vertical circle of the point and some fixed point assumed as origin. With astronomers it is customary to reckon azimuth from the south point around to the west through 360°. Surveyors frequently reckon from the north point. The celestial equator is the great circle of the sphere whose plane is perpendicular to the earth's axis. It therefore coincides with or is parallel to the terrestrial equator. The earth's axis produced is the axis of the celestial sphere. It cuts the sphere in the north and south poles of the equator. We shall for brevity call them the north and south poles. All great circles passing through the north and south poles are called hour circles. The hour circle passing through the zenith coincides with the meridian. 4 PRACTICAL ASTRONOMY The declination of a point is its distance from the equator, measured on the hour circle passing through the point. Distances north are + ; south, — . The declina- tions of all points on the sphere are included between 0° and + 90°, and 0° and - 90°. Instead of the declination, it is sometimes convenient to use the north polar distance, which is the distance of I a point from the north pole, measured on the hour circle of the point. It is therefore the complement of the \ declination. The north polar distances of all points lie \between 0° and + 180°. v The hour angle of a point is- the arc of the equator intercepted between the meridian, or south point of the equator, and the hour circle passing through the point. In practice, however, it is customary to consider the hour angle as the equivalent angle at the north pole between the meridian and hour circle. It is reckoned from the meridian around to the west through 24 hours, or 360°. The ecliptic is the great circle of the sphere formed by the plane of the earth's orbit; or, it is the great circle described by the apparent annual motion of the sun. It intersects the equator in two points called the equinoxes. The vernal equinox is that point through which the sun appears to pass in going from the south to the north side of the equator (about March 20). The autumnal equinox is that point through which the sun appears to pass in going from the north to the south side of the equator (about Sept. 22). The solstices are the points of the ecliptic 90° from the equinoxes. The sun is in the summer solstice about June 21 ; in the winter solstice about Dec. 21. The equinoctial colure is the hour circle passing through the equinoxes. The solstitial colure is the hour circle passing through the solstices. The angle between the equator and ecliptic is called the obliquity of the ecliptic. DEFINITIONS O The right ascension of a point is the arc of the celestial equator intercepted between the vernal equinox and the hour circle of the point. It is measured from the vernal equinox toward the east through 24 hours, or 360°. The sidereal time at any point of observation is equal to the right ascension of the observer's meridian. It is like- wise equal to the hour angle of the vernal equinox. Great circles perpendicular to the ecliptic are called latitude circles. The latitude of a point is its distance from the ecliptic, measured on the latitude circle passing through the point. Distances north are + ; south, — . The latitudes of all points on the sphere are included between 0° and + 90°, and 0° and - 90°. The longitude of a point is the arc of the ecliptic inter- cepted between the vernal equinox and the latitude circle of the point. It is measured from the vernal equinox toward the east through 360°. The position of an observer on the earth's surface is denned by his geographical latitude and longitude. The geographical latitude of a place is the declination of the zenith of the place. It is also equal to the altitude of the north pole. Latitudes of places north of the equator are + ; south, — . The geographical longitude of a place is the arc of the equator intercepted between the meridian of the place and the meridian of some other place assumed as origin. It is customary to reckon longitudes west (+) and east ( — ) from the meridian of Greenwich, through 12 hours, or 180°. The preceding definitions are illustrated by Fig. 1. The celestial sphere is orthogonally projected on the plane of the horizon, SWNE. The zenith Z is projected on the point of observation. NZS is the meridian ; EZW the prime vertical; WVQE the equator; VLBV the ecliptic ; P the north pole ; P' the north pole of the > 6 PRACTICAL ASTRONOMY ecliptic; Fthe vernal equinox; V the autumnal equi- nox; VP the equinoctial colure; OPP' the solstitial colure; B C=PP' = BVC*= the obliquity of the ecliptic. Let be any point on the sphere ; then ZOA is its vertical circle; MOP its hour circle; LOP' its latitude circle. The position of the point is defined by the following arcs, called spherical coordinates,: 'N p p / s*' / i /> - \ — >«A f * / 1 / i — ' V\\ / / / V ~f / v\ 1 / / J-^i "'"' ¥1 / A 7 ,S* \ —" "**" A V 'If 1 / 1 / \ z ^' w \>'' 1 r sF-- ____ X ~^Ve J VV^''///^'i \ M Q~"^ ~ — ^ i—- — """'^ E B Fig. 1 AO = Altitude, A, ZO = Zenith distance, z, 54 = SZA = Azimuth, A, MO = Declination, 8, PO = North polar distance, P, QM=QPM = Hour angle, *, VM = Right ascension, a, VQ = VPQ = Sidereal time, 0, LO = Latitude, /3, VL — Longitude, A., PP' = BVC = Obliquity of the ecliptic, c. * To he exact, we should say that the angle BVC is measured by the arc BO and by the arc PP' ; but in the operations of practical astronomy the distinction is seldom made. SYSTEMS OF COORDINATES ■ 7 3. It will be observed that the horizon, equator and ecliptic are of fundamental importance. They are called primary circles. Vertical circles, hour circles and lati- tude circles, which are respectively perpendicular to them, are called secondary circles. Two spherical coordinates, one measured on a primary circle, the other on its sec- ondary, are necessary and sufficient to determine com- pletely the direction of a point ; and from the definitions just given, to meet the requirements of astronomical work, we formulate four tjiS < -„*ERENCE Coordinates System Primary Secondary Primary Secondary I II . HI IV Horizon Equator Equator Ecliptic Vertical circle Hour circle Hour circle Latitude circle Azimuth Hour angle Right ascension Longitude Altitude Declination Declination Latitude The altitude, azimuth and hour angle of a star are con- tinually changing. They are functions of the time and the observer's position. Hence they are adapted to the determinations of time, azimuth and geographical latitude and longitude. Right ascension and declination are nearly independent of the observer's position, and vary with the time. They are largely used for recording the relative positions of stars, and in ephemerides. Latitude and longitude are also nearly independent of the observer's position, but are employed almost exclusively in theoreti- cal astronomy. In the solution of many problems of practical astron- omy, it is required that the coordinates of a point in one system be transformed into the corresponding coordinates in another system. PBACTICAL ASTRONOMY 9*'h TRANSFORMATION OF COORDINATES 4. Given the altitude and azimuth of a star, required its declination and hour angle. This transformation is effected by solving the spherical triangle PZO, Fig. 1, whose vertices are at the pole, the star, and the zenith. Three parts of this triangle are known . ZO the zenith distance or complement of the given altitude, PZO the supplement of the given azimuth, and PZ the complement of the given latitude ; from which, by the ..methods of Spherical' Trigonometry, we can find PO the complement of the required declination, and ZPO the required hour angle. For any spherical triangle ABC we hare [Chauvenet' s Sph. Trig., § 114] the general equations cos a = cos b cos c + sin b sin c cos A, (1) sin a cos B = cos b sin c — sin b cos c cos A, (2) sin a sin B = sin b sin A. (3) To adapt these equations to the triangle POZ, let A = PZ.O = 180° - A, a = PO = 90° - 8, b = ZO = 90° - h, B = ZPO = t, c =PZ = 90° - $. Then (1), (2) and (3) become sin 8 = sin h sin — cos h cos cos A~ cos 8 cos t = sin h cos + cos h sin cos A f cos 8 sin t = cos h sin A, which enable us to find S and t. If h be replaced by its equivalent, 90° — z, become -» sin 8 = cos # z sin — sin z cos <£ cos A , cos 8 cos t = cos 2 cos + sin z"sm $ cos A , cos 8 sin t = sin z sin A . ® ® these ® CD These equations are not adapted to logarithmic compu- tations (unless addition and subtraction logarithmic tables are employed), and they will be further transformed. TRANSFORMATION OP COORDINATES Let m be a positive abstract quantity, and M an angle such that m sin M= sin z cos A, (10) m cos ilf = cos z, (11) which conditions may always be satisfied [Chauvenet's Plane Trig., § 174]. Substituting these in (7), (8) and (9), they become sin 8 = m sin ( - M), COS 8 COS t = 111 cos ( — M), cos 8 sin« = sinz sin A. From these and (10) and (11) there result tan M = tan z-cos A, . . tan A sin ilf tan t = - (12) (13) (14) cos ((£ - M)' tan 8 = tan (<£ — M ) cos i, which completely effect the transformation. The com- putations are partially checked by (9). The quadrant of M is determined by (10) and (11). t is greater or less than 180° according as A is greater or less than 180°, since both terminate on the same side of the meridian. The quadrant of 8 is fixed by (14). Example. At Ann Arbor, 1891 March 13 the altitude of Regulus is + 32° 10' 15".4, and the azimuth is 283° 5' 6". 4. Find the declination and hour angle. [For instructions in the art of computing, see Appendix A.] + 42° 16' 48" .0 (Amer. Ephem., p. 482) z 57 49 44 .6 tan (0 - M) 9.616914 A I A 283 5 6 .4 tanz 0.201331 cos .A 9.354873 M 19° 47' 40".l 4> 42 16 48 .0 tan^4 0.633702„ sin M 9.529753 sec (<£ — m) 0.034339 tseat 0.197794 B t 302° 22' 54".0 t '20» 9">3i».6 cos* 9.728805 8 +12°29'56".4 Proof sinz 9.927608 sin^4 9.988575, cosec t 0.073401 B sec 8 0.010417 logl 0.000001 / 10 PRACTICAL ASTRONOMY 5. Given the declination and hour angle of a star, required its azimuth and zenith distance. In the general equations (1), (2) and (3) let b = 90° - 8, c = 90° - , A = t, B = 180°-4, a = z; and they become cos z = sin 8 sin + cos 8 cos <£ cos t, (15) sin z cos A = — sin 8 cos + cos 8 sin <£ cost, (16) sinzsin^T= cos 8 sin t. (17) To transform them for logarithmic computation, put (18) (19) Whence (20) (21) (22) which effect the transformation. (17) furnishes a partial check on the computations. Example. At Ann Arbor, 1891 March 13, when the hour angle of Regulus is 20 A 9 m 31 s . 6, what are the azimuth and zenith distance ? 8 + 12° 29' 56".4 (Amer. Ephem., p. 332) t 302 22 54 .0 tan 8 9.345719 ^ tan ( - N) 9.556204 cos* 9.728805 cos .4 P.354873 N 22° 29' 7".9 z 57°49'44".6 42 16 48 .0 Proof tan* 0.197794„ cos 8 9.989583 cosiV 9.965661 sin* 9.926599„ cosec(<£-iV) 0.470247 cosec4 0.011425 B tan J. 0.633702„ cosecz 0.072392 A 283° 5' 6" .4 y~) log 1 9.! vk siniV = sin 8, cosN~ = cos 8 cos t. taniV = tan 8 cosi tan A = tan t cos N sin ( -N) tan z = tan ( -N) cos A ' TRANSFORMATION OP COORDINATES 11 6. The angle POZ, Fig. 1, between the hour and vertical circles of a star, is called the star's parallactic angle. Let q represent it. To find the parallactic angle when z, A, and — sin z cos cos A, (23) cos 8 cos q = sin z sin <£ + cos z cos cos A, (24) cos 8 sin q = sin A cos <£. (25) Assume ks'mK = sin <£, (26) k cos if = cos cos -4, (27) and we obtain tan^ = ^S_^, (28) cos A v ' tan^costf cos (K — z) v ' The quadrant of j is determined by (25) and (29). To find the parallactic angle and zenith distance when S, t, and $ are given, we have, from (1), (2) and (3); cos z = sin 8 sin + cos 8 cos cos t; (30) sin z cos q = cos 8 sin — sin 8 cos cos t, (31) sin z sin 5 = sin t cos tf>. (32) Assume / sin L = cos <£ cos t, (33) IcosL = sin <£, (34) and we obtain tan L = cot <£ cos *, (35) +„„ „ * an t sin Z ,„„. tan q = , (36) * cos(S+£) k ' tanz = c°t(8 + £), cos q (37) The computatipns may be partially checked by (32). The values of q obtained from the data of § 4 and § 5 are equal to each other, and to 312° 25' 33".5. 12 PRACTICAL ASTRONOMY 7. Given the declination and zenith distance of a star, required its hour angle. If a, b and c are the sides and A an angle of a spherical triangle, we have [ Chauvenefs Sph. Trig., § 18] tan i A = ± J sin( f -b)siu(s-c) , \ sin s sin (s — a) sin s sin (s — a) in which s = | (a + b + c). If in this we substitute from triangle POZ A = t, a = z, b = 90° - 8, c = 90° - 4>, it reduces to tan It = ±A ^ K 2 + O ~ 8 >J si n ^ z ~ <* ~ 8 >]. (38) \cosi[2 + (^ + 8)]cosi[2-(^ + 8)] Similarly, it can be shown that sin 1 1 = ± J «" H' + (*" «)] Bin j['-(*-8 JI (39) \ cos d> cos 8 cos cos 8 To determine the quadrant of t it must appear from the data of the problem whether the star is west or east of the meridian. If it is west, J t is in the first quadrant ; if east, ^ t is in the second. Applications of formula (38) may be found in §§ 81 and 82. 8. Given the hour angle of a star, required its right ascension, and vice versa; the sidereal time in both cases being known. In Fig. 1, for any star we have VM = right ascension of star = a, MQ = hour angle of star = t, VQ, = sidereal time = 6. Then =(+tr= £r a = 0-t, (40) 4-^o, w and .U-t-e t = 6-a, (41) which effect the transformations. TRANSFORMATION OP COORDINATES 13 Applications of (40) and (41) are numerous throughout the book. 9. Given the right ascension and declination of a star, required its longitude and latitude, and vice versa. The transformation formulae are obtained by applying the general equations (1), (2) and (3) to the triangle POP', Fig. 1, in which OP = 90° - 8, OP' = 90° - p, OPP' = 90° + a, OP'P = 90" - A, PP' = obliquity of ecliptic = e. In order to adapt the resulting equations to logarithmic * computation, assume /sihF = sinS, (42) / cos F = cos 8 sin a, (43) and we shall obtain . ta,nF = ^^, (44) sin a Un\= C0S ( F - £ ) tma , (45) cos-F tan ji = tan (F - e) sin A. (46) The computations may be partially checked by the equation cos 8 sin a sec F cos (F — . e) cosec A sec /8 = 1, (47) which is derived without difficulty from the transforma- tion formulae. Example. The coordinates of Begulus on 1891 March 13 are a = 150° 38' 43".5, 8 = + 12° 29' 56" .4. What are the corresponding longitude and latitude ? The necessary value of e, furnished by the American Ephemeris, page 278, is 23° 27' 16". 0. The resulting coordinates are A = 148° 19' 3". 1, £ = + 0° 27' 40".5. 14 PRACTICAL ASTRONOMY 10. Given the right ascensions and declinations of two stars, required the distance between them. Let the coordinates of the stars be a', 8', and a", 8", and d the required distance. In the spherical triangle whose vertices are at the two stars and the pole, the sides are 90° — 8', 90° - 8" and d, and the angle at the pole is a" — a'. Let B' represent the angle opposite 90° — 8' . If in (1), (2) and (3) we put a = d, B = B', b = 90° - 8', c = 90° - 8", A = o" - a', they become cos d = sin S' sin 8" + cos 8' cos 8" cos (a" - a'), (48) sin d cos B' = sin 8' cos 8" - cos 8' sin 8" cos (a" - a'), (49) sin d sin B' = cos 8' sin (a'' — a'). (50) If d can be determined from its cosine with sufficient precision, (48) will give the required distance; otherwise it should be determined from the tangent. If we assume g sin G = cos 8' cos (a'' — a'), (51) g cos G = sin 8', (52) we shall find that tan G = cot 8' cos (a" - a'), (53) tan B = tap (*" ~ a '> si " G , (54) cos (8" + (?) v ' tanrf= cot (8" + g) . (55) cos B' (50) furnishes a partial check on the computations. An application of these formulae may be found in § 75, (c). CHAPTER II TIME 11. The passage of any point of the celestial sphere across the meridian of an observer is called the transit, or culmination, or meridian passage of that point. In one rotation of the sphere about its axis, every point of the sphere is twice on the meridian; once at upper culmina- tion (above the pole), and once at lower culmination (below the pole). For an observer in the northern hemi- sphere, a star whose north polar distance is less than the latitude is constantly above the horizon, and both culmi- nations are visible; a star whose south polar distance is less than the latitude is constantly below the horizon, and both culminations are invisible; and a star between these limits is visible at upper culmination, but invisible at the lower. For an observer in the southern hemisphere the first two cases are reversed. Three systems of time are required in the operations- of practical astronomy: sidereal, apparent (or true) solar and mean solar. A sidereal day is the interval of time between two suc- cessive transits of the true vernal equinox over the same meridian. The sidereal time at any instant is the hour angle of the vernal equinox at thatf instant. It is 0* m 0' when the vernal equinox is on the meridian — this instant is called sidereal noon — and is reckoned through 24 hours. The sidereal time is also equal to the right ascension of the observer's meridian, since the right ascension of the meridian is equal to the hour angle of the vernal equinox. 15 16 PRACTICAL ASTRONOMY It follows, then, that any star will be at upper culmina- tion at the instant when the sidereal time is equal to the star's right ascension; and at lower culmination when the sidereal time differs 12 hours from the star's right ascen- sion. The rotation of the earth on its axis is perfectly- uniform; but owing to precession and nutation the vernal equinox has a minute and irregular motion to the west (amounting on the average to 0".126 per day): so that a sidereal day does not correspond exactly to one rotation of the earth, nor is its length absolutely uniform, but it is sensibly so. An apparent (or true) solar day is the interval of time between two successive upper transits of the sun over the same meridian. The>hour angle of the sun at any instant is the apparent time at that instant. It is reckoned from 0" m s at noon — called apparent noon — -through 24 hours. But the apparent day varies greatly in length, for two reasons, viz. : First, — The earth moves in an ellipse with a variable velocity. Hence the sun's (apparent) eastward motion (in longitude) is variable. Second, — The sun's (apparent) motion is in the ecliptic. Hence the sun's motion in right ascension and hour angle is variable, and a clock cannot be rated to keep apparent time. A convenient solar time is obtained in this way: Assume an imaginary body to move in the ecliptic with a uniform angular velocity such that it and the sun pass through perigee at the same instant. Assume a second imaginary body to move in the equator with a uniform angular velocity such that the two will pass through the vernal equinox at the same instant. The second body is called the mean sun. A mean solar day is the interval of time between two successive upper transits of the mean sun over the same meridian. The hour angle of the mean sun is the mean TIME 17 time. It is reckoned from A 0™0 S at noon — called mean noon — through 24 hours. The difference between the apparent and mean time is called the equation of time. Its value is given in the American Ephemeris for the instants of Greenwich appar- ent and mean noon and Washington apparent noon, whence its value may be obtained for any other instant by interpolation. The astronomical solar day begins at noon, whereas the day popularly used — called the civil day — begins at (the preceding) midnight. Thus, Feb. 1, 10 A a.m., civil reck- oning, is Jan. 31 d 22* astronomical mean time. 12. The interval of time between two successive pas- sages of the mean sun through the mean vernal equinox — called a tropical year — was for the year 1800, accord- ing to Bessel, 365.24222 * mean solar days. The number of sidereal days in this interval is 366.24222, since in that interval of time the mean sun moves eastward through about 360°, and therefore the vernal equinox dur- ing the year makes one more 'transit over any given merid- ian than the sun. Thus we have 365.24222 mean days = 366.24222 sidereal days. Whence 24* mean time = 24* 3™ 56».555 sidereal time, 24* sidereal time = 23 56 4.091 mean time. From these equations it is found that the gain of side- real time on mean time in one mean hour is 9". 8565; and in one sidereal hour, 8 s . 8296. These are the amounts by which the right ascension of the mean sun increases in one mean and one sidereal hour, respectively. * The length of the tropical year is diminishing at the rate of about 0".6 per century. This is due to the fact that the mean vernal equinox is moving westward at an accelerated rate, as will be seen later, from the last of equations (120). 18 PRACTICAL ASTRONOMY CONVERSION OF TIME 13. In nearly every problem of practical astronomy it is necessary to convert the time at one place into the cor- responding time at another place, or to convert the time in one system into the corresponding time in another sys- tem. By means of the data furnished in the Ephemeris this is readily done. 14. To convert the time at one place into the correspond- ing time at another. Since every epoch of time is defined by an hour angle, the difference of time at two places is the difference of the two corresponding hour angles ; and that is equal to the difference of the longitudes of the two places. There- fore, if the difference of longitude be added to the time at the western place the sum is the corresponding time at the eastern. If it be subtracted from the time at the east- ern place the result is the time at the western. Example 1. The Ann Arbor mean time is 1891 March 10 d 21" 10 m 54 s .70. What is the corresponding Greenwich mean time ? Ann Arbor mean time, 1891 March 10* 21" 10™ 54«.70 Longitude Ann Arbor, Amer. Ephem., p. 482, + 5 34 55 .14 Greenwich mean time 1891 March 11 2 45 49.84- Example 2. The Washington sidereal time is 0" 23™ 17M0. What is the corresponding Ann Arbor sidereal time? Washington sidereal time, 0" 23 m 17«.10 Difference of longitude, Amer. Ephem., p. 482, 26 43 .10 Ann Arbor sidereal time, 23 56 34 .00 Example 3. The Ann Arbor apparent time is 1891 March 20 d 21" 58™ 19U7. What is the Berlin apparent, time at the same instant ? Ann Arbor apparent time, 1891 March 20* 21 s 58 TO 19«.17 Difference of longitude, Q 28 30 .05 Berlin apparent time, 1891 March 21 4 26 49 .22 CONVERSION OF TIME ' 19 15. To convert apparent time at any place into mean time, and vice versa. The equation of time at the given instant is required. When this is applied with the proper sign to the one, it gives the other. If apparent time is given, convert it into Greenwich apparent time, and take the equation of time from page I of the given month in the Ephemeris. If mean time is given, convert it into Greenwich mean time, and take the equation of time from page II of the month. In taking these and other data from the Ephemeris, care must be exercised in making the interpolations. Thus, let it be required to determine the equation of time at Greenwich apparent time 1891 Feb. 24 d 10 A . Its value for apparent noon is +13™ 25 s . 52, and the difference for one hour at noon of that day is s . 381. The difference for one hour at noon the next day is Oi.406. The hourly difference is therefore variable, but we may assume the second difference to be constant. The change in the equa- tion during the 10 hours after noon is ten times the average hourly change for the 10 hours ; that is, since the second difference is constant, ten times the hourly change at the middle period, or at 5 hours after noon. The average hourly change is s . 386, and the desired equation of time is + 13™ 25'.52 - 10 x (K386 = + 13™ 21».66. Example. The Berlin mean time is 1891 Feb. 28 d 0" ll m 20 s .60. What is the apparent time ? ^j"0 *' Berlin mean time, 1891 Feb. 28* 0* 11™ 2CK60 Longitude Berlin, - 4 53 34.91 Greenwich mean time, Feb. 27 23 18 , This is 23*. 30 after Gr. mean noon Feb. p, or 0\70 before noon Feb. 28. In this and similar easels the inter- polation should be made for the interval before noon. Equation of time, Gr. mean noon, Feb. 28, - 12"> 44".52 Change before noon, 0.70 x 0'.473, .33 Equation of time, . J- 12 44 .85 ^ >0 Berlin apparent time, 1891 Feb. 27* 23* 58 35 .75 («■ , \/ .1 % w v „ u N v /:> 7 ? 20 PRACTICAL ASTRONOMY 16. To convert a mean time interval into the equivalent sidereal interval, and vice versa. In § 12 it is shown that sidereal time gains 9 s . 8565 on mean time in one mean hour. The corresponding gain for any number of hours, minutes, and seconds, is tabulated in Table III of the appendix to the American Ephemeris. If this gain be added to the mean time inter- val, the sum is = geographical latitude* of = OBQ, ' = geocentric latitude of = OAQ, p = radius of earth at O = A 0, <£' — <£ = reduction to geocentric latitude = A OB, x, y = rectangular coordinates of = AC, CO. FlG - 2 * It frequently happens, especially in mountainous regions, that the plumh-llne is not normal to the theoretical ellipsoidal surface of the earth, 23 24 PEACTICAL ASTRONOMY From a discussion of all available observations, Bessel found a = 3962.802 miles, b = 3949.555 miles; and therefore, for the eccentricity of a meridional section, QPQ'P', e = 0.0816967. 21. Given the geographical latitude of a point on the earth's surface, required the corresponding geocentric lati- tude. The equation of the ellipse (Fig. 2) is *+£ = !• ( 56 > a 2 b 2 Differentiating, and substituting tan' = — 2 tan <£ = (1 - e 2 ) tan . ~~ (57) The reduction to the geocentric latitude, <£' — , can be expressed in terms of <£>. If the equation tan x = p tan y, which is identical in form with (57), be developed in series it becomes [ Chauvenet's Plane Trig., § 254] x — y = q sin 2y + ^g 2 sin4y + Jg 8 sin 6y + •••, in which p-1 q = <- * p + l owing to the fact that the local irregularities of surface and of density be- come appreciable. In such cases, the zenith determined by the plumb- line will not coincide with the theoretical zenith. Consequently the latitude and longitude, determined astronomically, will differ from the latitude and longitude determined geodetically. The geodesist has to deal with both systems, but the astronomer uses only the former. J PARALLAX 25 Substituting from (57) the values corresponding to x, y, and p, and dividing by sin 1" in order to express the result in seconds of arc, we obtain the practically rigorous formula <£'-<£ = - 690".65 sin 2 <£ + 1".16 sin i . (58) 22. To find the radius of the earth for a given latitude. Substituting x = p cos $' and y = p sin <£' in (56) and eliminating b by (57), we obtain °Vc -£2*4 (59) I cos ( — ') COS <£' In using this equation make a = 1, since the equatorial radius is taken as the unit. The values of ' — and of p for the positions of the principal observatories are given on pp. 482-485 of the American Ephemeris for 1891. Formulae (58) and (59) give the correct values of P *' P 0° 0".0 50° 6".7 10 . 1 -5 60 7 .6 20 3 .0 70 •8 .3 30 4 .4 80 8 .7 40 5 .7 90 8 .8 For refined observations (61) is not sufficiently exact, and recourse must be had to formulae which consider the earth as a spheroid. 26. divert the true zenith, distance and azimuth of a star, required its apparent zenith distance and azimuth, the earth being regarded as a spheroid. 28 PBACTICAL ASTRONOMY Let the star be referred to a system of rectangular axes whose origin is at the point of observation, the positive axis of x being directed to the south point, the positive axis of y to the west point and the positive axis of z to the zenith. Let X', T, Z' = the rectangular coordinates of the star, A' = the star's distance from the observer, A' = its apparent azimuth, ».' = its apparent zenith distance. Then X' = A' sin z' cos A', Y' = A' sin 2' sin A', Z' = A' cos 2'. Again, let the star be referred to a second system of rectangular axes parallel to the first, the origin being at the center of the earth. Let X, Y, Z = the rectangular coordinates of the star, A = the star's distance from the origin, A = its true azimuth, z = its true zenith distance. Then X = A sin 2 cos A, Y = A sin z sin A , Z = A cos z. Let the coordinates of the point of observation referred to the second system be X", Y", Z". From Fig. 2 it is seen that X" = p sin (<£ - <£'), Y" = 0, Z" = p cos (<£ - <£'). Now X' = X- X", Y'=Y~ Y", Z> = Z - Z", and therefore A' sin z' cos A' = A sin z cos A - p sin (<£ - <£'), A' sin z' sin A' = A sin 2 sin A, A' cos z' = A cos 2 - p cos (<£ - '). (65) PARALLAX 29 These equations completely determine A', z' and A\ and therefore the parallax z' — z and A! — A. It is better, however, to transform them so that the parallax can be computed directly. For this purpose, divide the equations through by A and put also substitute from (60), a being unity, sin 7r = — , A and we have /sin z' cos A' = sin z cos A — p sin ir sin ( — <£'), (66) /sinz' sin.4' = sinzsin.4, (67) /cosz' =cosz — psin7rcos ( — — ') sin .4, (69) /sinz' cos (A' — A) = sinz — p sin ir sin ( — ^>') cos A. (70) Putting m _ p sin ir sin ( — ') tan y. (75) This combined with (68) gives 30 PRACTICAL ASTRONOMY fain (z! — z) = p sin w cos ( <£ — <£') - — * — ~ ^ , (76) cosy /cos (z' - z) = 1 - p sin 7T cos (<£ - <£') cos (z ~ ?) . (77) Assume „ _ p BJIHT COB ( -z) = "sin(z-y) , (79) 1 — n cos (z — y) Formulae (71) and (72) rigorously determine the par- allax in azimuth, and (74), (78) and (79) the parallax in zenith distance. We may abbreviate the computation by writing (74) in the form y = (<(>- ') cos A, (80) which is in all cases sufficiently exact. 27. Given the apparent zenith distance and azimuth of a body, required its true zenith distance and azimuth, the earth being regarded as a spheroid. From (68) and (75) we obtain sin (z 1 - z) = P si " * cos (4> ~ ') sin (z' - y) ^ cosy for which, since (f> — ' and 7 are small angles, we can write sin (z' — z) = p sin ir sin (z' — y), (81) in which 7 is given without sensible error by y = (cj> - ') cos A' . (82) We obtain from (66) and (67) sin (4' - A) = PSmirsin(^-4>')BinA' , gg) sinz v J in which the value of z is found by the solution of (81). Formulas (82), (81) and (83) completely solve the prob- lem. For all known bodies save the moon we may write PARALLAX 31 z 1 - z = pirsin(z' — y), (84) A' — A = p tr sin (

' cos ( (86) tan d>' cos i (a — a') ,„_. tan y = -^ ±-i— '-, (87) cos [0 - J (a + a')] ' tan (S - 8') = P sin tt sin <£' sin (y - 8) sin y — p sin ir sin <£' cos (y — 8) (88) These rigorously determine the parallax in right ascen- sion, a — a', and the parallax in declination, 8 — S', when the geocentric coordinates are the known quantities. If the apparent coordinates a', 8' and t' have been obtained by observation, and a, 8 and t are unknown, we substitute a', 8' and £' for a, 8 and t in the second members, and solve. The resulting approximate values of the parallax furnish nearly correct values of a, 8 and t. Employing these in a second solution of the equations we shall obtain sufficiently exact values of the parallax. V 32 PRACTICAL ASTRONOMY 29. For all known bodies except the moon the values of ir, a - a' and 8 - 8' will be very small, and we may write (86), (87) and (88), without sensible error, in the form 8". 8 p cos ' sin t , on . a - a' = ^ r i > (89) A • cos 8 tany= ^ (90) cos ( S 5' ^ 8 "- 8 P si " ft' si " ( y ~ 8 ) , (9i) A • siu y in which A is expressed in terms of the astronomical unit of distance. These formulae will determine the parallax satisfactorily also if t and 8 are replaced by t' and 8', for which case an application of them will be found in § 155. At the fixed observatories it is customary to construct tables which greatly facilitate the computation of paral- laxes. The equations (89) and (91) may be written (a - a') A = 8".8 p cos ' sin I' sec 8', (92) (8 - 8') A = 8".8 p sin ' cosec y sin (y - 8'). (93) The second members of these equations are the parallaxes in right ascension and declination of an imaginary body at distance unity when observed, at a given station (/>, <£'), in the direction t', 8' . They are called parallax factors. Their values are generally computed and tabulated, at a given observatory, for every 10™ of hour angle and every degree of declination. When a body is observed at any hour angle and declination, the corresponding parallax factors may be obtained by interpolation from the tables. The parallaxes, a— a' and 8 — 8', may then be determined by dividing the parallax factors by the distance A of the body, as will be seen from (92) and (93). An application of these formulae will be found in § 154. REFRACTION 30. It is shown in Optics that when a ray of light passes obliquely from one transparent medium into an- KEFKACTION 33 other of greater density, it is refracted from its original direction according to the following laws : (a) The incident ray, the normal to the surface which separates the two media at the point of incidence, and the refracted ray, lie in the same plane. (b) The sines of the angles of incidence and refraction are inversely as the indices of refraction of the two media. A ray of light coming from a star to an observer is assumed to travel in a straight line until it reaches the upper limit of the earth's atmosphere. It then passes continually from a rarer to a denser medium until it reaches the earth's surface. If we regard the earth as a sphere, it follows from (a) and (b~) that the path of the ray is a curve whose direction constantly approaches the center of the earth. Let Fig. 4 represent a section of the earth and its atmos- phere made by a vertical plane passing through the point of observation and a star S. The path of the ray, Sab . .n . . 0, lies wholly in this plane and is concave towards the earth. The apparent direction of the star is OS', a tangent to the curve at the point of observa- tion. The true direction is that of a straight line joining and S. The difference of these directions is the refraction. It appears that refraction increases the altitude, and decreases the zenith distance, of a star, but in general does not affect its azimuth.* Fig. i * It is known that appreciable deviations in the azimuth are sometimes produced by refraction, especially in observations made very near the hori- zon ; but as they are due to abnormal and unknown arrangements of the strata of air, there is unfortunately no direct method of eliminating them. 34 PRACTICAL ASTRONOMY The amount of the refraction depends upon the density of the air, which is a function of the atmospheric pressure and temperature. Our knowledge of the state of the atmosphere is very imperfect. The theory of refraction is complex and tedious, refraction tables to be reliable must be largely empirical, and we shall not attempt an investigation of the subject. The Pulkowa Refraction Tables given in the Appendix, Table I, are based on the formula r = fita,nz(BTy y ^a', (94) in which z is the apparent zenith distance, /x, A, \, and a- are functions of the apparent zenith distance, B depends on the reading of the barometer, T depends on the temperature of the column of mercury as indicated by the attached* ther- mometer, 7 depends on the temperature of the atmosphere as indicated by the external thermometer, i depends on the time of the year, and r is the refraction in seconds of arc. For logarithmic computation (94) takes the form logr = log/i + logtanz + 4 (logi? + log T) + A. logy + £log + cos z cos <£ cos A )dz, which reduces by means of (23) to dS = — cos q dz. (98) Differentiating (30), regarding z, S and t as variables, we obtain — sin zdz— (cos 8 sin — sin 8 cos <£ cos i) dS — cos 8 cos sin t dt, which by (31), (32) and (98) reduces to cos 8 dt = sin q dz. (99) But from (41) dt = - da. Making this substitution and replacing dz by the refraction r, (98) and (99) become dS = - r cos q, (100) da = — r sin q sec 8. (101) These corrections reduce from the apparent to the true values of a and S. If the true place is given and the apparent place is required, the signs of the corrections must be reversed. To compute r we must know z. If z and A are given, q is determined by (29); if t and S are known, q and z are determined by (36) and (37). Si DIP OF THE HORIZON 32. At sea the altitudes of celestial objects are meas- ured from the visible sea horizon. This is below the true DIP OP THE HORIZON 37 horizon by an amount depending on the elevation of the observer's eye above the surface of the sea. Let Fig. 5 represent a section of the earth made by a vertical plane passing through the eye of an observer at 0. OS 1 is a line in the visible horizon, OS is the corresponding line in the true horizon, and SOS' is the dip of the horizon. Let x = the height of the eye above the ■water in feet = OA, a = the radius of the earth in feet = AC, D = the dip of the horizon = HOH' = OCB. We may write tan D ■- OB _ CB~ V2 ax + x 2 Fig. 5 -JW) (102) But — is a very small quantity and may be neglected. Tan D may be replaced by D tan 1". The apparent dip is affected by refraction. The amount of this refraction is uncertain, but an approximate value of the true dip is obtained by multiplying the apparent dip by the factor 0.92. The mean value of a is 20,888,625 feet. Introduc- ing these quantities in (102) it reduces to D = 59" y/x in feet, (103) by which amount the measured altitude must be decreased. A convenient rule, much used by navigators, follows approximately from (103), thus : The dip in minutes of arc is expressed by the square root of the number of feet that the observer's eye is above the water. To illustrate, if the observer's eye is 30 feet above the water, the dip is very nearly 5'. 5. The dip must in all cases be subtracted from the ob- served altitude in order to obtain the true altitude. 38 PRACTICAL ASTRONOMY SEMIDIAMETBR 33. When we observe a celestial body having a well- defined disk, as in the case of the sun and moon, the measurements are made with reference to some point on the limb, and the position of the center is obtained by correcting the observation for the angular semidiameter of the body. The geocentric semidiameters of the sun, moon and major planets are tabulated in the Ephemeris. The appar- ent semidiameter of the moon, however, is appreciably different for different altitudes, on account of its nearness to the earth, and its value must be determined. 34. To find the apparent semidiameter of the moon. Let Fig. 6 represent a section made by a plane passing through the observer 0, the center of the moon M, and the center of the earth 0, the earth being considered a sphere.* Let S = the moon's geocentric semidiameter = MCB, S' = the moon's apparent semidiameter = AOM, A = the distance of the moon's center FlG - 6 from the earth's center = CM, A' = the distance of the moon's center from the observer = OM, ir = the equatorial horizontal parallax of the moon, p = the parallax in zenith distance = OM C, z = the moon's true zenith distance = ZCM, z' = the moon's apparent zenith distance = ZOM. Then we can write gin^ = A = sin iL±£l __ cosssinp sm S A' smz r 8 i n z * The maximum error produced by neglecting the eccentricity of the l meridian even in the case of the moon never exceeds 0".06. SEMIDIAMETEK _ 39 From (61) sin p = sin ir sin z' ; (104) therefore sin S' = sin S ( cos p + sin 7r cos z sln 2 ). (105) \ sin z I (104) and (105) furnish very nearly an exact solution of the problem. For our purpose, and for all ordinary observations, we can write S' = S (1 + sin it cos z). (106) 35. To find the contraction of any semidiameter of the sun or moon, produced by refraction. The apparent disk of the sun or moon is not circular, since the refraction for the lower limb is greater than for the center, and that for the center is greater than for the upper limb. It will be sufficiently exact to assume the disk to be an ellipse whose center coincides with the center of the sun or moon. The contraction of the vertical semidiameter is found by taking the difference of the refractions for the center and the upper or lower limb. The contraction of the horizontal semidiameter for all zenith distances less than 85° is very nearly constant and equal to about 0".25. For our purpose it may be neg- lected, and we shall not investigate the subject. The contraction of any semidiameter making an angle q with the vertical semidiameter is readily obtained from the properties of the ellipse. Thus let a = the horizontal semidiameter, b = the vertical semidiameter, S" = the inclined semidiameter, and we have a? I* S" sin q = x, S" cos q = y ; whence S" = j ab ==■ (107) v a 2 cos 2 q + i 2 sin 2 q 40 PRACTICAL ASTRONOMY ABERRATION 36. The observed direction of a star differs from its true direction in consequence of the motion of the ob- server in space. The ratio of the velocity of light to the velocity of the observer is finite, and a telescope changes its position appreciably while a ray of light is passing from the objective to the eyepiece. In Fig. 7 let be the center of the objective and E the center of the eyepiece of a telescope at the instant when a ray from a star S reaches the point 0. If OE' represent the direction and velocity of the ray, and AB rep- resent the direction of the ob- server's motion and EE' his velocity, the telescope will be in the position O'E' when the ray reaches E'. While the true direc- tion of the star is E' 0, the apparent direction is E' 0'. The change in the apparent direction, OE' 0', is called the aberration. The star is apparently displaced toward that point of the celestial sphere which the observer is momen- tarily approaching. To find the amount of this displace- ment let (Fig. 7) y = BE'O — the angle between the true direction of the star and the line of the observer's motion, ■/ = BE'O' = the angle between the apparent direction of the star and the line of the observer's motion, dy = y — ■/ = the correction for aberration in the plane of the star and the observer's motion, V = OE' = the velocity o£ light, v =EE' = the velocity of the observer. ABERRATION 41 Then from the triangle EOE' we have sin (-y — /) = sin dy = — sin -/. But d, C'C = dt, CO - CO' = d&, and we can write sin 8 = sin y sin ('■•) (HI) (110) and (111) give by differentiation, t, 8 and 7 being variables, — sin t sin 8 d8 + cos t cos Sdt = sin y rfy, — cos Z sin 8 tf 8 — sin t cos 8 rf< = cos y cos «o dy. Eliminating dS and then eft, we obtain cos 8 dt = (sin y cos t — cos y sin i cos a>) dy, sin 8 g?8 = — (sin y sin t + cos y cos t cos a>) dy, which, by means of (109), (110) and (111), reduce to dt= cost sec 8 -^L, (112) smy dS = - sin t sin 8 -&- (113) smy The value of the factor — — is given by (108) from sin 7 the known values of v and V. For an observer at the earth's equator it is 0".31 ; in latitude it is 0".31 cos(/>. Substituting this value in (112) and (113) we obtain dt = + 0".31 cos sin t sin 8, (H5) which are the corrections to be applied to the observed hour angle and declination. When the star is observed on the meridian, t = 0, and (114) and (115) become dt = 0".31 cos <£ sec 8, (116) d8 = . (117) SEQUENCE AND DEGREE OE CORRECTIONS 43 39. To find the diurnal aberration in azimuth and alti- tude. The problem is identical with that in § 38 save that the horizon is the plane of reference, instead of the equator. If in (114) and (115) we replace t by A and 8 by h we obtain the desired corrections, dA = + 0".31 cos cos A sec h, (118) dh = — .31 cos sin A sin h, (H9) which are the corrections to be applied to the observed azimuth and altitude. SEQUENCE AND DEGREE OF CORRECTIONS 40. In applying the corrections considered in this chap- ter it is necessary that a proper sequence be followed. In all cases the refraction must be applied first, its amount being obtained by the methods of § 30 and § 31. Except in a few cases the diurnal aberration may be neglected. Observations on the sun or moon refer to points on the limb. They must be reduced to the center. In the case of the moon the reduction is made by formulae (106) and (107); of the sun, by (107). The parallax is now determined by the methods of §§ 25-29. It is wholly inappreciable for the stars. The degree of refinement to which these corrections should be carried, can be stated only in a general way. Usually it is sufficient to compute the corrections to one order of units lower than that to which the observations have been made. Thus, in reducing an observation made with a sextant reading to 10", the corrections should be computed to the nearest second. If the mean of a large number of sextant readings is employed, it is advisable to carry the corrections to tenths of a second ; and simi- larly in other cases. CHAPTER IV PRECESSION — NUTATION — ANNUAL ABERRATION — PROPER MOTION 41. In the preceding chapter we considered the correc- tions necessary to be applied to observed coordinates in order to reduce them to the center of the earth. We shall now consider the corrections which must be applied to the apparent geocentric coordinates. While the relative positions of the fixed stars change very slowly, — and in most cases no change at all has been detected, — their apparent coordinates are continually varying. These variations are divided into two general classes, secular and periodic. Secular variations are very slow and nearly regular changes covering long periods of time ; so that for a few years, and in some cases for centuries, they may be re- garded as proportional to the time. Periodic variations are changes which pass quickly from one extreme value to another, so that they cannot be treated as proportional to the time except for very short intervals. The planes of the ecliptic and equator are subject to slow motions, which give rise to variations in the obliquity of the ecliptic and in the positions of the equinoxes. The coordinates of the stars therefore undergo changes which do not arise from the motions of the stars themselves, but from a shifting of the planes of reference and the origin of coordinates. The forces producing these changes are variable, and while the variations of the coordinates are progressive, they are not uniform. They may be regarded 44 PRECESSION 45 as made up of two parts, viz.: a secular variation called precession, and a periodic variation called nutation. Owing to annual aberration [see § 37] the stars are not seen in their true positions, but are apparently dis- placed toward that point of the sphere which the earth is approaching, thus giving rise to periodic variations of their apparent coordinates. In the case of stars having proper motions, — that is, apparent individual motions due to motions of the stars themselves, and to the motion of the solar system in space, — their positions on the sphere change, and give rise to secular variations of the coordinates. 42. In order that we may define the positions of the ecliptic and equator at any instant, it will be convenient to adopt the positions of these planes at some epoch as fixed planes, to which their positions at any other instant may be referred. Let their positions at the beginning of the year 1800 be adopted as the mean ecliptic and equator at that instant. The true equator and ecliptic at any instant are the real equator and ecliptic at that instant. Their positions are affected by precession and nutation. The positions of the mean equator and ecliptic at any instant are the positions these circles would occupy at that instant if they were affected by precession, but not by nutation. The mean place of a star at any instant is its position referred to the mean equator and ecliptic of that instant. It is affected by precession and proper motion. The true place of a star is its position referred to the true equator and ecliptic. It is the mean place plus the variation due to nutation. The apparent place of a star is the position in which it would be seen by an observer (at the center of the earth). It is the true place plus the variation due to annual aberration. 46 PRACTICAL ASTRONOMY 43. In solving the problems considered in the following chapters we require to know the apparent right ascensions and declinations of the celestial objects at the instants when they are observed. The apparent places of the sun, moon, major planets, and several hundred of the brighter stars, are given in the Ephemeris at intervals such that their places for any instant may be obtained by interpola- tion. But occasionally it is desirable to employ stars not included in this list. If the mean places of these stars are given in the Ephemeris for the beginning of the year* they must be reduced, by means of the proper formulae, to the apparent places at the times of observation. If we observe stars which are not contained in the Ephemeris we must refer for their positions to the general Star Catalogues, which contain their mean places for the begin- ning of a certain year. These must be reduced to the corresponding mean places for the beginning of the year in which the observations are made, and thence to the apparent places as before. We shall now very briefly consider the matters essential to these reductions. PRECESSION 44. If from the figure of the earth we subtract a sphere whose radius is equal to the earth's polar radius, there will remain a shell of matter symmetrically situated with refer- ence to the equator. The attractions of the sun and moon on this shell tend to draw it into coincidence with the ecliptic. This tendency is resisted by the diurnal rotation of the earth. The combined effect of these forces is to shift the plane of the equator, without changing the obliquity of the ecliptic, in such a way that its intersec- tion with the ecliptic continually moves to the west. This causes a common annual increase in the longitudes of the * This does not refer to the ordinary or tropical year, but to the fictitious year, which begins at the instant when the sun's mean longitude is 280°. PRECESSION 47 stars, which is called the luni-solar precession. It affects the longitudes, right ascensions and declinations, but not the latitudes. The attractions of the other planets upon the earth tend to draw it out of the plane in which it is revolving around the sun. The effect is to shift the plane of the ecliptic in such a way that its intersection with the equator moves to the east. This causes a small annual decrease of the right ascensions of the stars, called the planetary precession. It affects the longitudes, latitudes and right ascensions, but not the declinations. The attractions of the planets produce a slight change in the obliquity of the ecliptic. Its annual effect upon the coordinates of the stars is combined with the luni-solar and planetary precession, the whole being called the general precession. 45. These motions are illustrated in Fig. 9. Let CV Q be the fixed or mean ecliptic at the beginning of the year 1800, UV Q the mean equa- tor, and V the mean equi- nox. By the action of the sun and moon in the time t the equator is shifted to the position QV V the vernal equinox moves from V to V v and V V 1 is the luni- solar precession in the in- terval t. By the attraction of the planets the ecliptic is *V a ^ v ' shifted to the position OV s , the vernal equinox moves from V x to V, and V 1 V is the planetary precession in the interval t. Let e = the mean obliquity of the ecliptic for 1800 =CV U, *! = the obliquity of the fixed ecliptic for 1800 + t = CV^Q, c = the mean obliquity of the ecliptic for 1800 + t =CVQ, Fig. 9 48 PRACTICAL ASTRONOMY ip = the luni-solar precession in the interval t = V a V v # = the planetary precession in the interval t =V^V, i/fj= the general precession in the interval t =CV — CV . The values of these quantities are, according to Struve and Peters, referred to the beginning of the year 1800, c = 23° 27' 54".22, 1 cj = € + 0".00000735 t% e = t - 0".4738 t - 0".0000014 fi, i/r = 50".3798 t - 0".0001084 * 2 , 9 = 0".15119 t - 0".00024186 fl, ft = 50".2411 ( + 0".0001134 ( 2 . (120) 46. Given the mean right ascension and declination (a, 8) of a star for any date 1800 + 1, required the mean right ascension and declination (a', £') for any other date 1800 + 1'. In Fig. 9 let CV 2 be the ecliptic of 1800, V X Q the mean equator of 1800 + £, and V 2 Q the mean equator of 1800 + *'. If we distinguish by accents the values given by (120) for the time t' we have Vjr t = y-i,, QV{V t =\W-t v QF 2 F 1 = £l '. Now let <2F ] = 90°-z, Q7 2 =90" + z', V r QV 2 = e, and we have [ Chauvenefs Sph. Trig., § 27] cos \ 6 sin \ (z' + z) = sin \ (if/ — if/) cos J (e/ + Cj), cos i cos i (z 1 + z) = cos i (if/ — ij/) cos £ (e/ — cj, sin J $ siii J (z' — z) = cos i (i/^ — i/() sin J (c/ — tj), sin £ 6 cos J (z' — z) = sin J (i/^ — ip) sin J (ej' + tj). But J (V — a) and J (e x ' — e x ) are very small arcs, and we can write tan i (z' + z) = tan J (f - i" = 7 l Qr i j=tf, SPP' = 90° -MQ=A, SP'P = 90° + M'Q = 180° - 4'. Substituting these in (2) and (3) we obtain cos 8' cos 4' = cos 8 cos A cos0 — sin 8 sin0, cos 8' sin^4' = cos 8 sin .4, from which we deduce cos8'sin(4'— A)= cos8sm4 sin0(tan8 + tan£0 cos^L), (125) cos 8' cos (A'-A) = cos 8- cos 8 cos A sin (tan 8+tan £ 6 cos A) ; (126) or, putting p = sin0(tanS + tan$0cos4), (127) we have ta,n(A'-A)= P sinA — / 12 8) 1 — p cos A From the triangle SPP' we can also obtain [ Chauvenetfs Sph. Trig., § 22] tan K 8'-8) = tan**^±^ (129) Having determined e v yjr, ,% e/, i/r' and #' from (120), z, a' and 6 from (121), (122) and (123), and A from (124), we obtain a' from (127), (128) and (124), and 8' from (129). 50 PRACTICAL ASTRONOMY Example. The mean place of Polaris for 1755.0 was a = 0" 43» 42U1, 8 = + 87° 59' 41".ll ; neglecting proper motion what will be its mean place for 1900.0? In this case t = - 45 and t' = + 100, and we find, from (120), *-, 23° 27' 54".23488 £ i 23° 27' 54". 29350 ,/, -; 37 47 .31 * ' + 1 23 56 . 90 # - 7 .29 ■& + 12 . 70 and therefore *(«i' + 23° 27' 54".26 Kf-iM + 1 52 .10 i(«i'-«i) + .02931 tani(f -"/-) 8.248163 sin A 9.3125989 COS i (£j' + £j) 9.962513 log jo 9.6050849 *(*'+*) 0° 55' 50".14 cos .4 9.9906400 logi(«i'- e i) 8.467016 log^j cos A 9.5957249 cot 4 (f - i/O 1.751837 Sub* 0.2176761 cosec£(e/ + «i) 0.399910 tan (A' - A) 9.1353599 log $ (;' - 2) 0.618763 A'- A 7° 46' 36".67 W-z) 4".16 A' 19 37 47 .01 z' 0° 55' 54".30 a' = A' + z'-$' 20 33 28 .61 z 55 45 .98 a' 1" 22™ 13' .91 sinKf -"Z') 8.248095 sin \ (t/ + £;) 9.600090 A' + A 31° 28' 57".35 40 0° 24' 14".16 tan \6 7.8481943 a 10 55 31 .65 cosl(4' + J) 9.9833991 A =a + z + & 11 51 10 .34 sec J (.4' -A) 0.0010009 tan 4 7.848194 tanj(8' -8) 7.8325943 cos A 9.990640 K»'-8) 0° 23' 22".85 tan J cos A 7.838834 8' -8 46 45 .70 tan S 1.4557773 8' 88 46 26 .81 Add* 0.0001049 sin 6 8.1492027 logp 9.6050849 47. Require •d the annual precession in right ascension and declination at , any time 1800 + t. * Zech's Tafeln der Additions- und Subtractions-Logariihmen are used here. PRECESSION 51 The precession for one year being small we can put, in (124) and (125), without sensible error, 8' = 8, sin (A' - A ) = (A' - A ) sin 1", sin A = sin a, sin 6 tan J 6 = 0, sin 6 = 6 sin 1", and obtain A' -A = a'-a + (#'-#)-(z' + z) = 0sinatan8. (130) For (121) and (123) we may write Z 1 + Z = (if/ 1 — )/f) COS £j, 6 =(ip' — iff) sin e r Substituting these in (130) and dividing by £'— £, we obtain _ t T. cos c, h i £ sin £, sin a tan 8. «' - * t' - 1 t' -t t' -t Similarly, from (129) we can obtain 8' _ 8 M - d, ■ _ 5C X sln £ cos a . a - t ? -t ' In order to express the rate of change in a and 8 at the instant 1800 + t we must let £' — £ become very small. Passing to the limit we have ^ = #cos£,-^+^sin ei sinatan8 ) dt dt dt dt dS dib ■ — = —£ sin £, cos a. dt dt If we let dt equal one year, and put dib d$ dib . m = -J- cos e, , « = — t- sin e,, <2« ' dt dt ' we obtain for the annual precession at 1800 + £ — = m + nsinatan8, (131) dt — = ncosa. (132) From (120) we find ^ cos £l = (50".3798 - 0" .0002168 t) cos £l dt = 46".2135 - 0" .0001989 t, ^ = 0".1512 - 0" .0004837 1 ; 52 PRACTICAL ASTRONOMY and therefore m = 46".0623 + 0".0002849 1, n = 2O".O607 - O".00O0863 1. (133) (134) Except for stars near the poles and for long intervals of time, formulae (131) and (132) are very convenient for computing the whole precession between two dates. Thus if it is required to determine the precession in a and 8 from 1800 + t to 1800 + t,' we first obtain approximate values of a and 8 for the middle date 1800 + \ (t + £'). Using these values we then compute the annual precession for this date, which is approximately the average annual precession for the interval t' — t, and thence the whole precession by multiplying this by t' — t. It is convenient to have the values of m and n given by (133) and (134) tabulated as follows : Date A™ log^n log n 1750 3 '.06987 0.126348 1.302439 1760 3 .07006 0.126330 1.302421 1770 3 .07025 0.126311 1.302402 1780 3 .07044 0.126292 1.302383 1790 3 .07063 0.126274 1.302365 1800 3 .07082 0.126255 1.302346 1810 3 .07101 0.126236 1.302327 1820 3 .07120 0.126218 1.302309 1830 3 .07139 0.126199 1.302290 1840 3 .07158 0.126180 1.302271 1850 3 .07177 0.126162 1.302253 1860 3 .07196 0.126143 1.302234 1870 3 .07215 0.126124 1.302215 1880 3 .07234 0.126106 1.302197 1890 3 .07253 0.126087 1.302178 1900 3 .07272 0.126068 1.302159 1910 3 .07291 0.126050 1.302141 1920 3 .07310 0.126031 1.302122 1930 3 .07329 0.126012 1.302103 1940 3 .07348 0.125994 1.302085 PRECESSION 53 Example. The mean place of /3 Ononis for 1850.0 was a = 5» 7 M 19«.856, 8 = - 8° 22' 44".74 ; neglecting proper motion, find its mean place for 1900.0. Using the values of m and n for the middle date 1875.0, and « and 8 for 1850.0, we may obtain very nearly the annual precession in a and 8 for 1862.5 from (131) and (132). log An .126115 sin a 9 .988429 tan 8 9 .168186, log 9 .282730, number ■ - 0U9175 A'" 3 .07224 da 2 .88049 log n 1.302206 cos a 9.357543 ^ 4".568 dt dt The approximate coordinates of the star for 1875.0 are therefore a = 5» 8» 31'.87, 8 = 8° 20' 50".5. Using these values we have log ^ n .126115 sin a 9 .988955 tan 8 9 .166514„ log 9 .281584 n number - 0M0124 log n 1.302206 i«i 3 .07224 cos a 9.347705 — 2 .88100 — 4" .46592 dt dt These are very nearly the exact values of the annual pre- cession for 1875.0, and the mean place for 1900.0 is there- fore a' = 5» 9™ 43'.906, 8' = - 8° 19' 1".44, which is practically identical with that given by the rigor- ous method of § 46. In many star catalogues the annual precession in a and 8 is given for each star for the epoch of the catalogue, by means of which the approximate place of the star for the 54 PRACTICAL ASTRONOMY middle time is found at once, and the first approximation made above is avoided. PROPER MOTION 48. The proper motion of a star has already been defined to be an apparent motion of the star itself on the surface of the sphere. It is assumed to take place in the arc of a great circle, and to be uniform. The proper motions in right ascension and declination are the components of this motion in and perpendicular to the equator. They are variable since the equator is a moving circle, and it must be specified to which equator they refer. When a star's place is required to be known very accu- rately, its position should be taken from as many catalogues as possible. In order that the data thus obtained may be properly combined, a thorough knowledge of the subject of proper motion is essential. 49. Given the observed mean places («, 8) of a star for 1800 + t and («', 8') for 1800 + t', required the annual proper motion. Starting from the first observed place and computing the precession for the interval t' — t by the methods of § 46 or § 47, let the resulting place for 1800 + t' be k v B v The discrepancies a' — a x and B 1 — Sj are due to proper motion, and the annual proper motion for the interval is d a ' = a '~ a l , d8' = ^lA (135) i' -t t' — t referred to the equator of 1800 + t'. Starting from the second observed place, computing the precession for the interval t — t', and applying it to «' and 8', let the resulting place for 1800 + t be « 2 , 8 2 . The annual proper motion for the interval is tfS = ^A, (136) t 1 t-t' referred to the equator of 1800 + t PROPER MOTION 55 Example. The mean places of Polaris for 1755.0 and 1900.0 given in NewcomVs Standard Stars are for 1755.0, a = 0» 43" 42M1, 8 = + 87° 59' 41".ll, for 1900.0, a' = l 22 33.76, 8' = + 88 46 26 .66; determine the proper motion referred to the equator of 1900.0. By applying the precession to the place for 1755.0 the place for 1900.0 was found to be, § 46, u L = 1* 22" 13*.91, 8 1 = + 88° 46' 26".81, and therefore, by (135), the annual proper motion of Polaris referred to the equator of 1900.0 is da' = + 0M369, do" = - 0".00103. 50. Given the proper motion (da, dB) referred to the equator of 1800 + t, required the corresponding proper motion (da', d8'~) referred to the equator of 1800 + t', and vice versa. When the star S (Fig. 9) moves on the surface of the sphere it causes variations in all the parts of the triangle SP'P, except P'P The solution of the present problem requires a knowledge of the relations existing between these variations. If in a spherical triangle ABC we suppose all the parts except a to vary, we can write [ Ohauvenefs Sph. Trig., § 153, (286) and (287)] sin c rfB = sin Adb — sin a cos A sin B oosec AdC, dc = cos A db + sin a sin B dC Substituting in these, from § 46, a = PP' = 6, db = d(SP) = - dS, dc = d(SP>) = - dtif, dB = d(SP'P) = tf(180° - A 1 ) = - da', dC = d(SPP') = dA = da, and putting y for A, we obtain cos 8' da' = sin y dS + cos 8 cos y da, (137) do' = cos y dB — cos 8 sin yda, (1 3 &) 56 PRACTICAL ASTRONOMY in which 7 is determined by sin y = sin 6 sin A sec 8' = sin 6 sin A 1 sec 8, (139) cos y = (cos 6 — sin 8 sin 8') sec 8 sec 8'. (140) These determine the proper motion for 1800 + t' in terms of that for 1800 + t. From (137) and (138) we obtain cos 8 da = cos 8' cos y da' — sin y dS', (141) dS = cos 8' sin y da' + cos y d8', (H2) which determine the proper motion for 1800 + t in terms of that for 1800 + t>. Example. The proper motion of Polaris referred to the equator of 1900.0 is da' = + 0U369 = + 2".0535, rfS' = - 0".00103. Deduce the proper motion referred to the equator of 1755.0. Using and A from § 46 and &' = + 88° 46' 26".66 we find, from (139) and (140), sin y = 9.131493, cos y = 9.995984, and therefore from (141) and (142) we obtain da = + 1".2480 = + (K0832, d8 = + 0".00493. 51. Given the proper motion (da, dK) and the mean place (a, 8) of a star for the epoch 1800 + t, required its mean place (a', S') for the epoch 1800 + t'. The proper motion for the whole interval t' — t is -first computed and applied to the mean place for 1800 + 1. With the resulting values of a and B, which we shall de- note by etj and S v the precession is computed and applied to a x and S v The result is the star's mean place for 1800 + 1'. If the proper motion (da 1 , d8') is given for the epoch 1800 + t ! , we first compute the precession, using a and 8, and then apply the proper motion for the interval. PEOPBE MOTIOK 57 Example 1. Given the mean place and proper motion of Polaris for 1755.0, a = s 43»» 42».ll, 8 = + 87° 59' 41". 11, + V> a = a + Aa ; then equations (144) take the form (* - Lick Observatory, by Al- van Clark & Sons. The wires are in the box S. To render them visible at night, they are illumi- nated by the lamp L, sup- ported by the framework' JQRPV (designed by Burnham), and counter- balanced by IHF. The graduated position circle XY remains fixed with reference to the telescope, whereas the micrometer box (and the illuminating apparatus) may be ro- tated about the line of sight so as to place the wires in any direction. Their direction will be indicated by the circle ( readings at X and Y. ^ Further, by turning the screw JEE' the microme- ter box, with the entire system of wires, can be moved in a direction parallel to the micrometer screw. The system of wires may therefore be given motions Fig. 12 72 PRACTICAL ASTRONOMY of rotation and translation, to place them in any desired position. The light from the lamp shines in the direction TO, but at the intersection of the tubes TO and NM there is a diagonal mirror which reflects the light in the direction NV into the box. The mirror may be rotated by rotating a disk at 0, thereby varying the intensity of the illumina- tion. If electric lighting is available, the oil lamp L should be replaced by a small incandescent lamp, as the latter has many practical advantages. The distance between the fixed and the movable wires, or the distance between two positions of the movable wire, is indicated by the readings of the graduated micrometer heads A and B: A indicating the whole number of revolu- tions of the screw and B the fractional parts of a revolu- tion. To convert the readings into arc, the value of one revolution of the screw must be known. 61. The angular value of a revolution of the micrometer screw depends upon the pitch of the screw and the focal length of the telescope. It may be found in several ways. (a) By the methods described above, measure with the micrometer any known angle, and divide the number of seconds in the angle by the corresponding number of revo- lutions of the screw. If the distance between two stars is measured, the true distance must be corrected for refraction. A method com- monly employed consists in measuring the difference of declination of two selected stars in the Pleiades when that group is near the meridian. The positions of these stars are very accurately known, pairs of almost any desired distance can be selected, and the correction for refraction is simple. Example. The difference of declination of d Ursce Majo- ris and Grroombridge 1564 was measured with the movable THE MICROMETER 73 wire of the micrometer of the. zenith telescope of the Detroit Observatory, when on the meridian, 1891 March 28. Barom. 29.206 inches, Att. Therm. 58°.0 F., Ext. Therm. 37°.5 F. Find the value of a revolution of the screw. The zenith level was read immediately after bisect- ing each star, in order to correct for any change in the pointing of the telescope. The value of one division of the level is 2".74. Star W. Apparent 5 Level Micrometer iL s d Ursce Maj. Gr. 1564. 9 s 25™ 9 33 + 70° 18' 46".5 + 69 44 12 .7 40.4 41.3 18.7 19.5 48.566 2.598 The correction for level (see § 62) is 0.85d=2".3, by which amount the measured distance must be increased, or the difference of the declinations decreased. The differ- ence of the refractions for the two stars is 0".7, by which amount the difference of the declinations must be decreased. The corrected difference of the declinations is 34' 30". 8. Therefore the value of one revolution of the screw is 45" .05. ______ — ■ — ._ ^ — 'Jl') A more accurate value is obtained by observations on one of the close circumpolar stars. The telescope is directed so that the star is just entering the field, and will be carried through the center by its diurnal motion. The micrometer is revolved so that the micrometer wire will be perpendicular to the diurnal motion of the star when it passes through the center of the field. The wire is set just in advance of the star, the time of transit of the star over it is noted, and the micrometer is read. The wire is moved forward one revolution, or a part of a revolution, and the transit observed as before. In this way the observations are carried nearly across the field. 74 PRACTICAL ASTRONOMY In Fig. 13, let P be the pole, JSP the observer's meridian, SS' the diurnal path of a star, AS the position of the micrometer wire when at the center of the field and coincident with an hour circle PM, and BS' (parallel to AS) any other position of the wire. Now let m Q be the micrometer reading, t the hour angle, and T the sidereal time when the star is at S, and let m, t and T be the corresponding quanti- ties when the star is at *S", and let R be the value of one revolution of the screw. Through S' pass an arc of a great circle S' C perpendicular to AS. Then in the triangle CS'P, right-angled at C, we have CS' = (m - m )R, S'P = 90° - 8, and we can write CPS' = t-t B = T- T ; sin [(m — m )iJ] = sin ( T — T ) cos 8 ; or, since (m — m^R is always a small angle, (m - m )R = sin (T - T ) Similarly, for another observation, (m! -m )R= sin(2 T l '- T ) cos 8 sin 1"' cos 8 sin 1"' (159) Combining these to eliminate the zero point, (»' - m)R = sin (7" - T ) 4^| - sin (T - T ) 4^|> (160) sin J. siii x from which the value of R is obtained. The micrometer readings are supposed to increase with the time. The times of transit are supposed to be noted by means of a sidereal time-piece. If its rate [§ 64] is large it must be allowed for. If a mean time-piece is used the intervals T— T must be converted into sidereal intervals. The resulting value of R is slightly in error on account of refraction, since the star is observed at unequal zenith THE MICROMETER 75 distances. But the effect of refraction is inappreciable if the observations are made near the meridian. The method is therefore advantageous for a meridian instrument with a micrometer in right ascension, the star being observed at upper or lower culmination. However, any variations in the azimuth or level constants of the instrument during the progress of the observations introduce errors in the results. If a and b are the values of these constants at the beginning of the series of transits and a' and V their values at the close of the series, it can be shown from the theory of the transit instrument [Chapter VII], that the distance between the first and last positions of the wires has been decreased by the quantity (a' - a) sin (<£ T 8) + (6' - b) cos (<£ =F 8), (161) which divided by the corresponding difference of the micrometer readings is the correction to the value of one revolution of the screw. The lower signs are for lower culmination. The azimuth constants are determined by observing suit- able pairs of stars before and after the series of micrometer transits is taken, according to the methods described later. The level constants are determined by the method of § 62. The variations of azimuth and level may be considered to be uniform and proportional to the time. For a meridian instrument properly mounted the variation of the azimuth may be neglected without a sacrifice of accuracy. Example. Polaris was observed at lower culmination at Ann Arbor, 1891 March 28, to determine the value of a revolution of the micrometer screw of the transit instru- ment. The micrometer was set at every three-tenths of a revolution, and one hundred and fifty transits observed. The times were noted by means of a sidereal chronometer which was 16 m 30 8 .6 slow. The position of Polaris was a = 1 u 17 ™ 4 6 ..o, -8 = + 88° 43' 40".25, 76 PRACTICAL ASTRONOMY American Ephemeris, p. 304 ; and therefore the chronom- eter time of lower culmination was T — 13 A V" lb' A. A few of the observations and their reduction are given below. Each printed observation is the mean of three consecutive original observations. m T T- -T T-To sin (T- T,,) (m — m )R 7.8 12*23" 12«.3 -38" 3».l — 9° 30' 46' .5 9.218194„ - 756".83 8.7 25 17 .3 35 58.1 8 59 31 .5 9.193953„ 715 .75 9.6 27 19.7 33 55.7 8 28 55 .5 9.168793„ 675 .46 10.5 29 22.0 31 53.4 7 58 21 .0 9.142070„ 635 .15 11.4 31 22 .0 29 53 .4 7 28 21 .0 9.11411U 595 .55 20.4 51 42.3 9 33.1 2 23 16 .5 8.619771™ 190 .80 21.3 53 43 .3 7 32 .1 1 53 1 .5 8.516822„ 150 .53 22.2 55 46.0 5 29.4 1 22 21 .0 8.379348„ 109 .69 23.1 57 47 .0 3 28.4 52 6 .0 8.180547n 69 .40 24.0 59 50 .0 — 1 25 .4 -0 21 21 .0 7.793121„ - 28 .44 25.8 13 3 54.0 + 2 38.6 + 39 39 .0 8.061960 + 52 .82 26.7 5 58.3 4 42.9 1 10 43 .5 8.313268 94 .21 27.6 7 59.5 6 44.1 1 41 1 .5 8.468092 134 .55 28.5 10 .0 8 44.6 2 11 9 .0 8.581389 174 .66 29.4 12 3.7 10 48 3 4 42 4 .5 8.673281 215 .82 38.4 32 24.3 31 8.9 7 47 13 .5 9.131914 620 .48 39.3 34 28.3 33 12 .9 8 18 13 .5 9.159630 661 .20 40.2 36 32 .2 35 16 .8 8 49 12 .0 9.185629 702 .16 41.1 38 34.0 37 18.6 9 19 39 .0 9.209722 742 .21 42.0 13 40 37 .0 + 39 21.6 + 9 50 24 .0 9.232735 + 782 .60 Subtracting the 1st from the 11th, the 2d from the 12th, etc., we have m' — m (to' — m)R R V „2 18.0 809".65 44".981 — 0".065 0.0042 18.0 809 .96 44 .998 -0 .048 0.0023 18.0 810 .01 45 .001 -0 .045 0.0020 18.0 809 .81 44 .989 -0 .057 0.0032 18.0 811 .37 45 .076 + .030 0.0009 18.0 811 .28 45 .071 + .025 0.0006 18.0 811 .73 45 .096 + .050 0.0025 18.0 811 .85 45 .103 + .057 0.0032 18.0 811 .61 45 .089 + .043 0.0018 18.0 811 .04 45 .058 + .012 0.0001 R = i5 .046 Probable error* = ± 0.674-J- 2« 2 = 0.0208 0208 10X9 = ±0".010. * See Appendix C, § 1. THE MICROMETER 77 By reducing the whole series of transits the value of B and its probable error were found to be R = 45" .059 ± 0".006. From the level readings b = + 5".17, V = + 7".05; and from observations for azimuth on /3 Cas&iopeice and 4 H. Draeonis, and on 9 Bootis and 36 H. Cassiopeice, a^= —9". 15, a' = — 8".79. Substituting these in (161) and dividing by 46, the difference of the first and last micrometer read- ings of the original series, we have as a correction to B, - 0".017, and therefore R = 45". 042 ± 0".006. There is an indication from the individual results for B that its value increases as the micrometer readings in- crease. This irregularity should be fully investigated by further observations, and allowed for in refined observa- tions if it proves to be real. The value of a revolution is affected by changes of tem- perature. To determine the rate of change, observations should be made on several nights at widely different tem- peratures. If B is the value of a revolution at the temper- ature t, B the value at the temperature 50°, and x the correction to B for a rise of 1° in temperature, each night's observations furnish an equation of the form R = R + (t- 50°) x. (162) The solution of these equations by the method of least squares gives the most probable values of B and x, and therefore of B. (c) If the micrometer is designed for the measurement of zenith distances, the micrometer wire being horizontal, the observations are made at the time of the star's greatest western or eastern elongation. This occurs when the ver- tical circle of the star is tangent to its diurnal circle. At this time the micrometer wire is parallel to the star's hour circle. If m Q , t and T refer to the instant of greatest 78 PRACTICAL ASTRONOMY elongation and m, t and T to any other instant, the formula (159) is applicable to this case. At the instant of greatest elongation the parallactic angle ZOP, Fig. 1, is 90° for western and 270° for eastern elongation, and we can write cos t = tan cot 8, cos z = sin cosec S, T = a + t - (163) Set the telescope at the zenith distance z when the star is just entering the instrument. Note the time of transit over the micrometer wire ; and, as before, carry the ob- servations nearly across the field. Any change in the zenith distance of the telescope during the progress of the observations will affect the resulting value of R. The amount of the change will be indicated by the zenith level and can be allowed for. The level should be read after each transit is observed. If l is the level reading, i.e. the reading of the level scale for the middle of the bubble, at the time T , I the level reading at the time T, and d the value in arc of a division of the level, we have (m - m ) R = ± sin (T - T ) ^A + (7 - l )d ; (164) (»' - m )R =± sin (T - TJ^+Q' - yd and for another observation Whence (m'-m) R = ±sin ( T - T )^1 T sin(r _ r o )^+ {V -l)d, (165) sin J. sin J. in which the lower sign is for eastern elongation. The micrometer readings are supposed to increase with the time for western elongations, and the level readings to increase towards the north. The resulting value of M must be corrected for refrac- tion. From the values of z and M, the zenith distances corresponding to the first and last observations can be obtained, and thence the refractions. The difference of the refractions divided by the difference of the first and THE LEVEL 79 last micrometer readings is the amount by which the value of R must be decreased. If both R and d are unknown a close approximation to the value of R is obtained by neglecting the term (I 1 — V)d. With this value of R the value of d is computed (§ 63) and substituted in (165), and the corrected value of R obtained. A second approximation to the value of d will rarely be required. THE LEVEL ( ' 62. The spirit level consists of a sealed glass tube, ground on the upper interior surface to the arc of a circle of large radius, and nearly filled with alcohol or ether. The bubble of air occupying the space not filled by the liquid is always at the highest point of the curve. There- fore a change in the relative elevations of the ends of the tube causes a motion of the bubble, the amount of which is read from a scale marked on the surface of the glass. The level is adapted to the determination of the angle which a nearly horizontal line makes with the horizon, or the very small angle moved over by a telescope. The level tube is mounted and attached to astronomical instruments in various ways, but there is one general method of using it. Let the divisions of the scale be num- bered in both directions from zero at the center, and let d be the angular value of one division. If the level be placed on a truly horizontal line — say, for convenience, an east and west line — the center of the bubble will not be at zero, owing to the non-adjustment of the level. If the cen- ter is x divisions from the zero, the error of the level is dx. Let the level be placed on a line inclined to the hori- zon at an angle b, and let the reading of the west end of the bubble be w and the east end e. Then the elevation of the west end of the line is given by b = \(w — e)d =F dx. 80 PRACTICAL ASTRONOMY Now let the level be reversed in direction and let the read- ing of the west end be w' and the east end e'. Then b = l(w' - e')d ± dx. Combining these values of b we have b = l[(w + w')-(e + e'))d; (166) from which it appears that the error of the level is elimi- nated by reversing. A positive value of b will indicate that the west end of the line is higher than the east end. Whenever it is possible the level should be reaid several times, the same number of readings being made in each position, — level direct and level reversed, — care being taken to remove the level from its bearings after each reading is made. Example. The inclination of the axis of a transit instru- ment is required from the following level readings, the value of one division being 1".88. IV e Direct 14.1 9.7 •2.W 53.4 Reversed 12.6 11.1 Se 41.8 Reversed 12.7 11.1 *8) + 11.6 Direct 14.0 9.9 + 1.45 Sum 53.4 41.8 The axis makes an angle + 1.45 d = + 2".73 with the horizon, the west end being higher than the east. In case the zero of the scale is at one end of the tube and the numbers increase continuously to the other, — which is a better system, — we can show that J = £[(w + e) - (»' + e'XR (167) in which the readings w and e for level direct correspond to that position of the level for which the readings increase toward the west. * It must be noticed that there are two complete observations for de- termining 6, and hence the divisor is 8 instead of 4. Reversed w' 16.3 95.3 e' 39.4 111.6 w' 16.4 8) -16.3 e' 39.5 - 2.037 Sum 111.6 'le — 2.037 a = -5".59 with the THE LEVEL 81 Example. Find the inclination of the axis of a transit instrument from the following level readings, the value of one division being 2". 743. Direct w 35.4 e 12.4 w 35.3 e 12.2 Sum 95.3 The axis makes an horizon, the west end being lower than the east. 63. The value of one division of the level is determined best by means of a level-trier. This consists of a hori- zontal bar supported at one end by two bearings and at the other by a vertical micrometer screw. The level is placed on the bar and the readings of the micrometer and bubble are noted. The screw is now turned and the bubble moves to a new position. The readings of the micrometer and bubble are again noted. The angle moved over by the bar is known from the length of the bar, the pitch ^of the screw and the difference of the micrometer readings ; whence the angular value of one division of the level may be obtained. If possible, the determination of the value of a division should be made after the level tube is fixed in its final mounting, rather than before. The essential principles of the level-trier are well illus- trated by Fig. 14. Fig. 14 In the absence of a level-trier, an accurate determination of the value of a division can be obtained by means of any telescope provided with a micrometer in zenith distance. To illustrate, let an equatorial be directed upon a distant 82 PRACTICAL ASTRONOMY terrestrial mark directly north or south of it, and adjust the micrometer wire to parallelism with the horizon. Mount the level upon the telescope so that the vertical plane pass- ing through the axis of the level tube is parallel to the line of sight, and so that the bubble is at one end of the scale. The mark is bisected by the micrometer wire, and the level and micrometer readings noted. The instrument is then turned through an angle such that the bubble moves to the other end of the scale. The mark is again bisected by the wire, and the level and micrometer read- ings noted as before. The difference of the level readings corresponds to the difference of the micrometer readings, whence the value of one division of the level can be obtained from the known value of a revolution of the micrometer screw. In general, such observations are best made on an overcast day. Example. The following observations were made Feb- ruary 19, 1891, to determine the value of a division of the striding level of the Detroit Observatory transit instru- ment, the telescope being directed to a distant mark. The value of one revolution of the screw is 45". 042. Find the value of a division of the level. Level Micrometer Differences d n £ Level Micrometer 20.9 2.0 1.8 20.6 1.1 20.0 20.2 1.4 17.019 17.791 17.773 16.969 18.9 18.8 0.772 0.804 0.0408 R 0.0428 R The mean of eighteen observations gave d = 0.0417 B ± 0.0004 B = 1".878 ± 0".018. The level tube should be thoroughly tested for irregu- larity of curvature before using. If different portions of a. THE CHRONOMETER 83 level give sensibly different values for a division of the scale, it should not be used in refined observations. The value of a division should also be determined at two or more very different temperatures in order that a tem- perature correction may be introduced if necessary. A level should be adjusted by the vertical adjusting screws so that the bubble will stand near the center of the tube when the level is placed on a horizontal line. It should be adjusted by the horizontal screws so that the axis of the tube will be parallel to the line whose inclina- tion is to be measured. This adjustment is tested by revolving the level slightly about its bearings. If the readings are different when the level is equally displaced in opposite directions from the vertical pla^e through its bearings, the adjustment is not perfect, v / THE CHRONOMETER 64. A chronometer is a large and carefully constructed watch which is "compensated" so that changes of tem- perature have very little effect on the time in which the balance-wheel vibrates. It is a very accurate time-piece when properly handled, comparing favorably with the astronomical clock, and being portable is adapted to field work and navigation. The chronometer correction is the amount which must be added to the reading of the chronometer face to obtain the correct time. It is + when the chronometer is slow. The chronometer rate is the daily increase of the chronometer correction. It is + when the chronometer is losing. It is not necessary that the correction and rate be small, though it is convenient to have the rate less than ± 5 s a day. The test of a good time-piece lies in the uniformity of its rate. The correction is generally allowed to increase indefinitely. The chronometer correction is obtained by observations on the celestial objects, or by comparison with a time-piece whose correction is known. If 84 PRACTICAL ASTRONOMY AT = the chronometer correction at a time T , AT = the chronometer correction at a time T, 8T = the chronometer rate, we determine the rate per unit of time by 8T= AT ~^ T Q - (168) Conversely, if the rate and the correction at the instant T are known, the correction at the instant T is given by AT=AT + 8T(T-T ). (169) JHzample. The correction to chronometer T. S. & J. D. Negus, no. 721, was + 16 m 19 s .5 at Ann Arbor mean time 1891 March 25 d 11", and + 16™ 55 s .6 at 1891 April ¥ IP- Find the daily rate, and the correction at 1891 March 28* 13". From (168) we find the daily rate oT= + 3 S .61. Sub- stituting this and T= March 28 d 13 A in (169) we find AT = + 16™ 19«.5 + 3».61 x 3.08 = + 16™ 30-.6. The above equations are true only when the rate is con- stant for the interval T — T . Such constancy can be assumed for an interval of a few days in the case of the best chronometers ; but when great accuracy is required the interval between observations for determining chro- nometer correction should be as small as possible. When several chronometers are employed, the correction to one is obtained by observation ; and to the others, by comparison with the first. If two chronometers which keep the same kind of time are compared, it will generally happen that they do not beat together. The fraction of a second by which one beats later than the other can be esti- mated after some practice to within s .l or S .2, so that the correction can be obtained to that degree of accuracy by this method. When a sidereal chronoineter is compared with a mean THE CHRONOMETER 85 time chronometer the degree of accuracy is higher. If the chronometers tick half seconds, the beats of the two will coincide once in every 183 s , since in this interval sidereal time gains S .5 oh mean solar. The ear is capable of esti- mating the coincidence of the beats within S .02 or S .03. When the coincidence occurs the observer notes the times indicated by the two chronometers. The correction to the one being known, a satisfactory value of the correction to the other is readily obtained. When a chronograph [§ 68] is at hand and the chronom- eters are provided with break-circuits (or make-circuits), the comparisons are most conveniently and accurately made by placing the two chronometers in the chronograph circuit. The beats are recorded on the chronograph sheet and the distance between them can be measured very accurately by means of a scale. It is convenient to use a sidereal chronometer when making observations on the stars, planets, comets, etc., and a mean time chronometer when making observations on the sun. 65. The observer should be able to "carry the beat" of the chronometer ; that is, mentally to count the successive seconds from the tick of the chronometer without look- ing at it. An experienced observer will carry the beat^ for several minutes, estimate the times of transits of a > star over several wires (or other similar phenomena) to tenths of seconds, and write them on a slip of paper with- out taking his eye from the telescope : then, still carrying the beat, he will look at the chronometer face to verify Ms ,' count. This is called the "eye and ear method" of observ- / ing, and it is very important that every observer should be j able to employ it with accuracy and perfect ease. 66. To obtain the best results from a chronometer the following precepts should be rigidly observed : (a) It should be wound at regular intervals. If it re- 86 PRACTICAL ASTRONOMY quires winding daily it should always be wound at the same hour of the day; otherwise an unused part of the spring is brought into action and a change of rate results. (5) The hands should not be moved forward oftener than is necessary, and they should not be moved backward. (c) A chronometer on shipboard should be allowed to swing freely in its gimbals, so that it may always take a horizontal position; but when carried about on land it should be clamped so as to avoid the violent oscillations due to the sudden motions it receives. (d) It should be kept in a dry place; as nearly at a uniform temperature as possible ; away from magnetic in- fluences ; and when at rest should always be in the same position with respect to the points of the compass. (e) All quick motions should be avoided: in particular, it should never be rotated rapidly about its vertical axis. (/) In out-of-door use it should be protected from the direct rays of the sun. 67. The astronomical clock is a finely constructed clock whose pendulum is compensated for changes of tempera- ture. Its rate is in general more uniform than that of a chronometer. It is one of the fixed instruments of an observatory, and to that extent the remarks concerning the chronometer are applicable to it. THE CHRONOGRAPH 68. The chronograph is a mechanical device for recording the instant when an observation is made. A sheet of paper on which the record is to be made is wrapped around a metallic cylinder which is caused to rotate once per minute by means of clock-work. A pen is attached to the armature of an electro-magnet in such a way as to press its point on the moving paper. The magnet is carried slowly along the cylinder by a screw, so that the pen traces a continuous spiral on the paper. The electro-magnet is placed in an 88 PRACTICAL ASTRONOMY electric circuit which passes through the chronometer or clock (or, better, through a relay connected with the time- piece), in such a way that the circuit is broken for an instant at the beginning of every second, or every other second. At each of these instants the electro-magnet releases the armature carrying the pen, the pen .moves laterally for the moment, and in this way the spiral is graduated by notches to seconds of time. One notch is usually omitted at the beginning of each minute, to assist in identifying the seconds. One of the circuit wires passes through a signal-key held in the observer's hand. When a star, for example, is being observed he presses the key at the exact instant when the star is crossing a wire, thus breaking the circuit and making the record on the chrono- graph sheet. The beats of the chronometer being recorded on the sheet, the chronometer time when the key was pressed can be read from the sheet by means of a scale with great accuracy, and at the observer's leisure. When the chronograph is first set in motion the observer records in his note-book the hour, minute, and second cor- responding to a certain marked notch on the sheet, which serves as a reference point in identifying all the notches on the sheet. In some forms of the chronograph the circuit is made by pressing the key, but the break-circuit is preferable. The chronographic method is generally preferable to the eye and ear method because it relieves the mind from carrying the beat and making the record, thus allowing greater care to be given to other parts of the observation, and because more observations can be made in a given time. But in the case of transit observations of slowly-moving stars, or of very faint objects, and in many forms of microm- eter observations, the eye and ear method is at least as satisfactory as the chronographic method. A very common form of chronograph is illustrated in Fig. 15. <0 CHAPTER VI THE SEXTANT 69. The sextant is an instrument especially adapted to he determination of time, latitude and longitude when ixtreme accuracy is not required, as in navigation and ixploration. It con- ists essentially of a >rass frame ADO, ?ig. 16, bearing a graduated arc AC, a elescope EF, whose ine of sight is paral- el to the plane of he graduated arc, nd the mirrors S nd D, whose planes re perpendicular to he plane of the arc. ?he mirror D, called be index-glass, is xed to the index- rm DB, which revolves about D at the center of the arc, nd which carries a vernier at B. The mirror H, called lie horizon-glass, is attached to the frame. The lower alf of it is silvered, the upper half is left clear. Figure 17 illustrates a form of sextant commonly em- loyed. The special parts already described in connec- !on with Fig. 16 will be recognized without difficulty, 'he telescope is mounted on an adjustable standard so 89 Fig. 16 90 PRACTICAL ASTRONOMY that its distance from the frame of the sextant may be varied by turning the screw-head at the lower end of the standard. Colored or neutral-tint glasses are mounted in front of the index and horizon glasses. They can be rotated into the paths of the sun's rays to protect the eyes while observing that body. A dense neutral-tint glass may also be screwed Fig. 17 over the eyepiece for the same purpose. The telescope may be replaced by others of different magnifying power, or by one with a larger object-glass for observing stars, — shown in the foreground of the cut. The index-arm car- rying the vernier is furnished with a clamp and slow motion for setting accurately to any desired reading. 70. To illustrate the method of using a sextant and the principles involved, let it be required to measure the angle SJSS' between the stars S and S'. The instrument is held in the hand and the telescope directed to the star S. The ray SE passes through the unsilvered part of H and forms THE SEXTANT 91 a direct image of the star at the focus F. The sextant is revolved about the line of sight until its plane passes through the other star *S". The index-arm is then moved until the reflected image of *S" is brought into the field and nearly in coincidence with the direct image of S. The index-arm is clamped and the two images brought into perfect coincidence by turning the slow-motion or tangent screw. If the instrument is perfectly constructed and adjusted, the required angle is given at once by the circle reading. The ray of light S'D, which forms the reflected image at F, traverses the path S' D-DH-HF, being re- flected by the two mirrors D and H. When the direct and reflected images coincide, the angle between the stars is twice the angle between the mirrors. That is, 8FS' = 2HLD, since the angle SES' = 180° - EDH - EHD = 180° - 2 HDL - 2 (LHD - 90°) = 2 (180° - HDL - LHD) = 2 HLD. If A is the position of the zero of the vernier when the two mirrors are parallel, and B its position when the two images coincide, we have SES' = 2HLD = 2ADB = 2AB. (170) It thus appears that to enable us to read the required angle directly from the circle, the circle reading must be twice the corresponding arc. Thus, the 120° line is really only 60° from the 0° line [or a sextant, hence the name]. An improved form of the sextant is known as the Pistor and Martins (Berlin) prismatic sextant, in which the hori- zon glass is replaced by a totally reflecting prism, occupying a somewhat different position on the frame of the instru- ment. Among other advantages of the prismatic form it can be used for measuring angles up to 180° and even 92 PRACTICAL ASTRONOMY greater ; whereas with the common form, the angle is lim- ited to about 140°. Again, the graduated arc is sometimes a complete circle, in which case the index arm is extended over a diameter of the circle and carries a vernier on each extremity. Such an instrument is called a reflecting circle or prismatic circle according as a horizon -glass or prism is used. Its chief advantage lies in the fact that the eccentricity is eliminated by the use of two verniers 180° apart. 71. In order to obtain good results with the sextant, the instrument must be accurately adjusted, and the tele- scope focused; the direct and reflected images should be about equally bright; and several complete observations should be made, the mean of all being used. The images are made equally bright by moving the tele- scope from or toward the frame, so as to utilize more or less of the light passing through the transparent part of the horizon-glass, or by placing colored-glass shades in front of the index-glass. In measuring the angular distance between two stars, the images of the stars are brought into exact coincidence in the middle of the field of view. In measuring the dis- tance of the moon from a star, the star is brought into coincidence with that point of the moon's bright limb which lies in the great circle joining the star and the center of the moon. The measured distance is then in- creased or decreased by the moon's semidiameter [§§ 33, 34, 35]. In the case of the sun and moon the images of the nearest limbs are made to coincide, and the measured distance is increased by the semidiameters of both objects, as before. Results obtained in this way, when corrected for any instrumental errors, are the apparent distances be- tween the objects. The sextant is also used for measuring the apparent alti- tudes of the heavenly bodies. At sea the telescope is THE SEXTANT 93 directed to that point of the horizon which is below the object. The reflected image is brought into contact with the horizon line. When the instrument is vibrated slightly about the line of sight the image should describe a curve tangent to the horizon. The sextant reading corrected for instrumental errors and the dip of the horizon [§ 32] is the apparent altitude. If the object is the sun, the lower or upper limb is made tangent to the horizon ; if the moon, the bright limb ; and the sextant readings must be further corrected for semidiameter. For observing altitudes on land an artificial horizon is used. This is a shallow basin of mercury over which is placed a roof, made of two plates of glass set at right angles to each other in a frame, to protect the mercury from agitation by air currents. The mercury forms a very perfect horizontal mirror which reflects the rays of light from the star. If the observer places his eye at some point in a reflected ray, he will see an image of the star in the mercury, whose angle of depression below the horizon is equal to the altitude of the star above the horizon. If then he directs the telescope to the image in the mercury, and brings the two images into coincidence as before, the sextant reading corrected for instrumental errors is double the apparent altitude of the star. The sun's altitude is measured by making the two images tangent externally. The corrected sextant reading is double the altitude of the lower or upper limb, according as the nearest or far- thest limbs of the sun and its image in the mercury are observed. The double altitudes of stars near the meridian are changing slowly, and the images are brought into contact by means of the slow-motion screw as before. But the double altitudes of stars at a distance from the meridian are changing rapidly, and another method is used. To illustrate, suppose the sun is observed for time when it is east of the meridian, and the altitude therefore increasing. 94 PRACTICAL ASTRONOMY The upper limb is observed first. The two images are brought into the field and the index moved forward until the sextant reading is from 10' to 20' greater than the double altitude of the upper limb, and the instrument is clamped. The images are now slightly separated, but they are approaching. When they become tangent, the ob- server notes the time on the chronometer and reads the circle. The index is again moved forward from 10' to 20' and the contact observed as before. In this way, four or five observations are made. The double diameter of the sun is about 64', and for observing the lower limb the index is quickly moved backward about 45'. The two images now overlap, but they are separating, and the time is noted when they become tangent. Moving the index forward as before, four or five observations are made on the lower limb. If the sun is observed west of the meridian, the altitudes of the lower limb should be measured first. 72. The faces of the glass in the horizon roof should be perfectly parallel. If they are prismatic the observed alti- tudes are erroneous. The error is eliminated by observing one-half of a set of altitudes with the roof in one position and the other half with the roof in the reversed position, and taking the mean of all. Likewise, the glass screens in front of the index and horizon glasses must have parallel faces. The surface of the mercury can be freed from impurities by adding a little tin-foil. The amalgam which forms can be drawn to one side of the basin by means of a card, leav- ing a perfectly bright surface. ADJUSTMENTS OP THE SEXTANT 73. (a) The index-glass. Place the sextant on a table, unscrew the telescope and set it in a vertical position on the graduated arc. Place the eye near the index-glass ADJUSTMENTS OF THE SEXTANT 95 and move the index-arm toward the telescope until the telescope and its image in the mirror are seen very nearly in coincidence. Their corresponding outlines will be par- allel if the index-glass is perpendicular to the plane of the arc. If they are not parallel, the glass is removed and one of the points against which it rests is filed down the proper amount. The axis of the telescope is here assumed to be perpendicular to the plane of the end on which it rests. This can be tested by rotating the telescope about its axis and noticing whether the angle between the tube and its image varies. The telescope should be set at the mean of the two positions which give the maximum and minimum values of this angle.* (6) The horizon-glass. The index-glass having been adjusted, the telescope is directed to a star and the index- arm is brought near the zero of the arc. If the horizon- glass is parallel to the index-glass the reflected image will pass through the direct image when the index-arm is moved slowly to and fro. If it passes on either side of the direct image the horizon-glass needs adjustment. This is done by turning the screws provided for the purpose. (e) The telescope. Two parallel wires are placed in the telescope tube. These are made parallel to the plane of the sextant by revolving the tube containing them. The line of sight is the line joining a point midway between these wires and the center of the object glass. This should be parallel to the plane of the sextant. To test the adjust- ment, select two well-defined objects about 120° apart, and bring the two images into coincidence on one of the side wires, and then move the sextant so as to bring the images on the other wire. If the images still coincide, the line of sight needs no adjustment. If the images are separated, the collar which holds the telescope is shifted by means of screws until the adjustment is satisfactory. * This method was proposed by Professor J. M. Schaeberle : The Sidereal Messenger, May, 1888. 96 PRACTICAL ASTRONOMY CORRECTIONS TO SEXTANT READINGS 74. The index correction. It is seen from (170) that all angles measured with the sextant are reckoned from A, the point where the zero of the vernier falls when the two mirrors are parallel ; whereas the circle readings are measured from 0°. The index correction is the reading 0° (or 360°) minus the reading at A. Let it be represented by I. The value of I can be reduced to zero by rotating slightly the horizon-glass by means of screws provided for that purpose. But this adjustment is very liable to derangement, and it is customary to determine I every time the sextant is used and apply it to all the sextant readings. (a) To determine I for correcting stellar observations, point the telescope to a star and bring the direct and reflected images into coincidence. Let the sextant reading be R. The index correction is given by / = 0° - ie. (171) Example. Determine I from the following readings : 359° 56' 0" 359° 55' 50" 56 10 56 10 55 50 55 55 The mean of the six readings is 359° 55' 59".2, and therefore I = 360° - 359° 55' 59".2 = + 4' 0".8. (5) For reducing solar observations, point the telescope to the sun and bring the direct and reflected images exter- nally tangent to each other and read the circle. Then move the reflected image over the direct image until they are again externally tangent, and read the circle. Let the readings in the two positions be R l and R 2 , R 1 being the greater. The reading when the two images coincide is ^(-Rj + i? 2 ), and the index correction is given by I = 3&P-i(R 1 + R 2 ). (172) CORRECTIONS TO SEXTANT READINGS 97 The observed semidiameter of the sun is given by S = $(R 1 -R 2 ). (173) To eliminate the effect of refraction the horizontal semi- diameter should be measured. Example. Find I and S from the following readings on the sun made Thursday, 1891 April 23. 360° 28' 35" 359° 24' 50" 40 40 45 45 45 45 40 50 Means 360 28 41 .0 359 24 46 .0 / = 360° - 359° 56' 43".5 = + 3' 16".5. S = \{1° 3' 55".0) = 15' 58".7. From the American Ephemeris, p. 56, S = 15' 56".3. 75. Correction for eccentricity. The arc of a sextant being short, the eccentricity cannot be eliminated by means of two verniers 180° apart, and it must be investi- gated. This can be done by comparing several angles measured with the sextant with their known values ob- tained in some other way. Thus in Fig. 11 the sextant reading is twice the arc AM. The true value of the angle is obtained by correcting the reading at M for eccentricity, and correcting the position of A for eccentricity and index error. The true reading at M is given by (157). The true reading at the zero point A is given by R„ = — I + e" sin r). The true value of the angle is R ^ R = M + e" sin (y + j?) - e" sin v + I. (174) But B — E is the known value of the angle ; let d repre- sent it. Mis the observed value of the angle ; let d' rep- 98 PRACTICAL ASTRONOMY resent it. Now, since an arc on the sextant is one-half the corresponding reading, we have 4(d - d 1 ) = e" sin (v+ 17) - e" sin 17 + \I, which reduces to d - d' = 4 e" cos (J v + 77) sin 4 v + I (175) = 4 e" cos j; sin J v cos 4 v — 4 e" sin 17 sin a i v + /. If we put 4 e" cos 17 = x, 4 e" sin rj = y, (176) we have sin 4 l/cos \vx —sin 2 \v y + I = d — d'. (177) This equation involves three unknown quantities, x, y, I. Three measured angles, each furnishing an equation of the form (177), are required for the solution of the problem. There are several ways in which to obtain the value of d — d' at any point of the arc. (a) For those who have access to a meridian circle, the most direct process known is the ingenious method proposed by Professor Schaeberle in der Astronomische Nachriohten, no. 2832. (5) When the latitude of the observer and the time are accurately known, make a series of measures of the double altitudes of a star just before and after its meridian pas- sage. The observed double altitude at the instant of tran- sit is obtained from these measures by the method of § 87. The apparent double altitude at the instant is obtained at once from the known declination, latitude and refraction. The latter minus the former is d — d'. (c) When the latitude and time are not accurately known, measure the distance between two stars and com- pare it with the known apparent distance. The apparent distance is found by the method of § 10, using a', B 1 and a", S" as affected by refraction, § 31. Example. The distance between Aldebaran and Aroturus was measured with the sextant at Ann Arbor, Thursday night, 1891 March 5, as below. It is required to form the CORRECTIONS TO SEXTANT READINGS 99 equation (177) for this pair of stars, correction A0 was + 15™ 7 s . The chronometer Chronometer Sextant Barom. 29 .400 inches 8* 37-» 25» 130° 14' 55" Att. Therm. 65°.0 F. 9 45 30 48 30 52 20 56 20 30 4 14 40 14 55 14 35 14 55 14 45 14 40 Ext. Therm. 18°.0 F. Amer. Ephem., pp. 322, 340 Aldebaran Arclurus a 4» 29" 39« .33 14» 10™ 41* .97 8 16° 17' 22" .7 19° 44' 47" .8 Means 8 52 A0 + 15 9 7 5 7 12 130 14 46.4 With these data we solve (41), (35), (36), (37), (32), (95), (100) and (101) as below. A Idebaran Arcturus sin t 9.97127 9.98667„ 9 s 7™ 12' 9* 7"> 12" cos 9.86915 9.86915 • a 4 29 39 14 10 42 cosec 2 0.04642 0.03729 t 4 37 33 18 56 30 cosec q 0.11316 0.10689 n *69°23' 15" 284° 7' 30" log 1 0.00000 6.00000 <£42 16 47 42 16 47 cot 0.04130 0.04130 True 2 63° 58' 41" 66° 35' 42" cos* 9.54660 9.38746 Mean ref r. 1 56 2 11 X21° 9' 54" 15° 1'24" App. 2 63 56 45 66° 33 31 816 17 23 19 44 48 log/i 1.75821 1.75766 tan* 0.42467 0.59921 n tan z 0.31078 0.36292 sin£ 9.55758 9.41366 ^llog.Br 9.99583 9.99583 sec(8 + L) 0.10027 0.08542 A logy 0.02746 0.02750 tan q 0.08252 0.09829„ logr 2.09228 2.14391 ?50°24' 39" 308° 34' 16" sin q 9.88684 9.8931 1„ cot(S + L) 0.11573 0.15849 sec 8 0.01780 0.02632 cosg 9.80433 9.79482 logrfa 1.99692 2.06334„ z 63° 58' 41" 66° 35' 42" cosy 9.80433 9.79482 d8 + r i8".8 + 1' 26".8 da + 1 39 .3 - 1 55 .7 Applying these refractions to the above star places we obtain the coordinates which are to be used in solving (53), (54), (55) and (50). 100 PRACTICAL ASTRONOMY a" - a' cot 8' COS (a" — a') G sin G tan (a" — a') sec (8" + G) tan B' 67° 26' 29" .2 16 18 41 .5 212 38 33 .8 19 46 14 .6 145 12 4 .6 0.533668 9.914429 n 289° 36' 52" .7 9.974038„ 9.841975„ 0.197546 0.013559 45° 53' 39".3 cot (8" + G) 9.914333, cosB' 9.842600 d 130° 17' 22" .4 sin (a" — a') 9.756404 cos 8' 9.982158 cosec B' 0.143841 cosec d 0.117597 l0gl 0.000000 d' 130° 14' 46".4 d-d' + 2 36 .0 d-d' + 156 .0 The angle v in (177) is not \d\ but one-half the read- ing corresponding to the line of the circle with which the vernier line coincides, and it is the eccentricity of this point which enters into d-d'. For the reading d' = 130° 14' 50" the 29th line of the vernier coincides with the circle line 135° 0', and therefore in this case \ v = 33° 45'. We now find sin \ v cos \v= 0.462, sin 2 £v = 0.309; and therefore 0.462 x - 0.309 y + I = 156.0. Similarly, from the meridian double altitude of a star, method (6), and from another pair of stars we find 0.259 x - 0.072 y + I= 165.0, 0.117 x- 0.014 y + 7=171.0. Solving these three equations we obtain log z = l. 61380„, logy = 0.43265, 7=175.8; whence, from (176), ^=176° 14', 4e" = 41".2. While the index correction varies from day to day, and its value should be determined by the methods of § 74 every time the sextant is used, the eccentricity is prac- tically constant. By neglecting the term I in (175) and DETERMINATION OF TIME 101 making 2 v successively 0°, 10°, etc., we obtain the follow- ing corrections for eccentricity to be applied to the circle readings. Circle Correction Circle Correction Circle Correction 0° 0".0 50° - 8".8 100° - 16" .2 10 -1 .8 60 -10 5 110 -17 .4 20 -3 .6 70 -12 .0 120 -18 .5 30 -5 .4 80 -13 .5 130 -19 .4 40 -7 .2 90 -14 .9 140 -20 .2 In order to determine the eccentricity very accurately, at least ten known angles distributed uniformly from 0° to 140° should be measured and the resulting equations solved by the method of least squares. The observations should be made in one night, so that I may be considered constant; but the observer should determine J several times during the night, to make sure that it does not change. DETERMINATION Ot TIME 76. Time is determined from observations on the heav- enly bodies by determining the corrections to the chro- nometer or other tit ; . '■,'<■■;. ^ n eo ■ + , observations are mad 77. By equal altitudes of a fixed star. When a star is from two to four hours east of the meridian and near the prime vertical, observe a series of its double altitudes [§ 71] with the sextant and sidereal chronometer, and let the mean of the chronometer times be 6'. When the star reaches the same altitude west of the meridian observe its double altitudes with the vernier of the sextant set at the same readings as before, in inverted order, and let the mean of the chronometer times be d". The chronometer time of the star's meridian passage is \(B' + 6"'). The 102 PRACTICAL ASTRONOMY sidereal time of the star's meridian passage equals its right ascension a. The chronometer correction Ad at this in- stant is given by A0 = a - i(6' + 6"). (178) If a mean time chronometer is employed the sidereal time a must be converted into mean time. The required chronometer correction is then given by (178) as before. Example. The following equal altitudes of Arcturus were observed with a sextant and sidereal chronometer at Ann Arbor, Saturday night, 1891 April 25. Required the chronometer correction. Chronometer Sextant Chronometer Star East reading Star West 10* 23" 21« 81° 30' 0" 17* 21™ 47' 24 14 81 50 20 52 25 9 82 10 19 56 26 4 82 30 19 2 27 82 50 18 7 & 10 25 9.6 0"17 19 56.8 Amer. Ephem., p. 340, a 14' 10™ 42«.8 \(fi' + 0") 13 52 33.2 A0 + 18 9 .6 78. By equal altitudes of the sun. Observe as described above [§§ 71, 77] the two series of equal double altitudes of the sun before and after noon, and let the chronometer times of the east and west observations be T' and T", a mean time-piece being used. The mean of the two times is not the chronometer time of the sun's meridian passage, since the sun's declination has changed during the interval, and a correction must be applied. To find its value let t — half the interval between the observations = £ (T" — T), 8 = the sun's declination at the observer's apparent noon, dh = the increment of the sun's declination in the interval t, dt = the increment of the sun's hour angle due to the increment of the declination. DETERMINATION OF TIME 103 Differentiating (15), regarding 6 and t as variables, and dividing by 15 to express dt in seconds of time, we have dt = /'tani_tanJ\rf8 \siat tan J / 15' <■ ' by which amount the east and west observed times are greater than they would be if the declination were con- stant and equal to 8. The chronometer time of the sun's meridian passage is therefore \ ( T' + T"~) — dt. The mean time of the sun's meridian passage is U, the equation of time at the observer's apparent noon; and therefore the chronometer correction at the mean of the two times is AT = E - \ {T + T") + dt. (180) If a sidereal chronometer is employed the sidereal inter- val t is converted into the equivalent mean interval [§ 16], dt is computed from (179) as before and subtracted from the mean of the two times, and the result is the chronome- ter time of the sun's meridian passage. The sidereal time at this instant is equal to the sun's apparent right ascen- sion a, and the chronometer correction is given by A0 = a - i(8' + 6")+ dt. (181) Example. The equal double altitudes of the sun were observed as below at Ann Arbor, Saturday, 1891 April 24- 25, a sidereal chronometer being used. Find the chro- nometer correction. Chronometer Sextant Chronometer Sun's Sun East reading Sun West limb 22» 5™ 4» 66° 46' 0" 5»42 m ■21* Upper 5 48 67 2 41 37 9.9587 Reduction -36.9 sin t 9.9192 Mean interval t 3 44 30.6 (1) 1.095 t 3\742 tan S 9.3725 t 56° 8' tan t 0.1732 <2> 42 17 (2) 0.158 Amer. Ephem., 8 + 13 16 log [(1) -(2)] 9.9717 48" .7 x 3.742 = dS + 182".2 \ogdS 2.2605 Sun's apparent a 2» 11™ 39'.3 log A 8.8239 W+6") 1 53 42.8 log dt 1.0561 dt + 11.4 A0 + 18 7 .9 79. It may be convenient to observe the equal altitudes in the afternoon of one day and the forenoon of the next day. In this case the mean of the two observed times minus the proper value of dt is the chronometer time of the sun's lower culmination. If t is half the mean time interval between the observations it must be replaced by 180 + t = t' when substituting in (179) ; and E in (180) and a in (181) must be increased by 12 A . The chronom- eter correction at midnight is then given by (180) and (181). Example. Find the (sidereal) chronometer correction from the following equal altitude observations of the sun. & 5" 33™ 56".3 Friday afternoon, 1891 April 24 6" 22 8 35 .3 Saturday forenoon, 1891 April 24 i 0" ~&)=t 8» 17" 19».5 tan 9.9587 Reduction -1 21.5 sin*' 9.9187„ Mean interval t 8 15 58.0 (1) - 1.096 t 8\266 tan 8 9.3668 t 123° 59' tani' 0.1713 n t' 303 59 (2) - 0.157 <*> 42 17 log [(1) -(2)] 9.9727„ 8 + 13 6 logrfS 2.6075 + 49".0 x 8.266 = dS + 405".0 !og^ 8.8239 12" + a 14» 9»46«.4 log dt 1.4041„ \ (6' + 6") 13 51 15.8 dt -25.4 A0 + 18 5 .2 DETERMINATION OP TIME 105 dt is a solar interval, and should be reduced to sidereal, but the correction is small, and for sextant work may be neglected. 80. The method of determining time by equal altitudes possesses the advantages that no corrections are applied for index error, eccentricity, refraction, parallax and semi- diameter; any undetermined errors are eliminated from the result ; and the latitude need not be accurately known. However, if the state of the atmosphere and the index correction are different at the two times of observation, the equal sextant readings do not correspond to equal true altitudes, and a correction must be applied. If the index correction is greater and the refraction less for the west observation than for the east, the true double altitude at the west observation is too great by the difference of the index corrections and twice the difference of the re- fractions, and the time of the observation must be increased by the interval required for the sextant reading to decrease that amount. This interval can be determined from the observations themselves. Thus in the example of § 78, the index correction and refraction for the east observation were I' = + 3' 8", r' = 1' 24" ; and for the west I" = + 3' 21", r" = V 22". The true double altitude at the west observation was too great by (J» - /') + 2 {r< - r") = + 17". From the observations it is seen that the sextant reading decreased 16' = 960" in about 44 s . If x is the correction to the time of the west observation, we have 960 : 17 = 44 : x, from which z=0 s .8. The correction to A0 is —\ x = -0 S .4, and therefore the true value of the chronometer correction is A0 = + 18™ V.b. 106 PRACTICAL ASTRONOMY 81. By a single altitude of a star. A series of double altitudes of a star having been observed in quick succes- sion, let R = the mean of the sextant readings, $' = the mean of the corresponding chronometer times ; and let / = the index correction, c = the correction for eccentricity, •■ h' = the apparent altitude of the star, z' = the apparent zenith distance of the star, r = the refraction, 2 = the true zenith distance of the star. Then and 2h' = R + I + e = 2(90°-z'), (182) V z = z< + r. (183) The latitude . (184) In case a mean time chronometer is used, the sidereal time must be converted into the mean time T and com- pared with the chronometer time T'. In determining the time from single altitudes of the stars and the sun, the observations should not be confined to one side of the meridian. It would be well to observe alter- nately east and west of the meridian, at about equal alti- tudes. A comparison of the results of such a series often leads to the detection of systematic errors whose presence would not be suspected from observations made wholly in one part of the sky. Hxample. The observations made on Areturus east of the meridian, recorded in § 77, give 9' = 10" 25» 9'.6, R = 82° 10' 0". DETERMINATION OF TIME 107 Find the chronometer correction. R 82° 10' 0" Barom. 29.100 inches 1 + 3 12 Att. Therm. 55° .0 F. £ ~ 14 Ext. Therm. 50 .0 F. 2V 82 12 58 h ' 41 6 29 sin £[>+(<£ -8)] 9.76646 z' 48 53 31 sin \\z - (<£ - S)] 9.35770 From (94), r 1 4 sec J [z ,.+ (<£ + 8)] 0.24652 z 48 54 35 sec'i [z - (0 + 8)] 0.00285 <£ 42 le 47 tan 2 J«, 9.37353 Ephem., 8 + 19 44 52 tan \ t 9.68676„ ^ - 8 22 31 55 % t 154° 4' 26" <*> + 8 '62 1 39 t 308 8 52 z + (<£ - S) 71 28 30 « 20» 32" 35'.5 z - W> - 8) 26 20 40 Ephemeris, a 14 10 42 .7 z + (<£ + 8) 110 56 14 io 43 18 .2 z-(4> + S) -13 7 4 6' 10 25 9.6 A0 + 18 8 .6 82. By a single altitude of the sun. If p = the parallax of the sun, S = the semidiameter of the sun, the true zenith distance of the center of the sun is given by z = z' + r-p±S; (185) S being + or — according as the upper or lower limb of the sun was observed. The value of t is given by (38) or (39) as before, t is the true time when the observation was made. The mean time T is given by applying the equation of time E. If T' is the chronometer time of observation, the chronometer correction is , AT=T~T. (186) If a sidereal time-piece is used the mean time T must be converted into the sidereal time and the resulting value compared with the chronometer time. Since the declination of the sun is changing, it is neces- sary to know the chronometer correction within 10 s ; other- wise the value of 8 taken from the Ephemeris may be 108 PRACTICAL ASTRONOMY slightly iu-e»©r, thus giving only an approximate value of the chronometer correction. With this value of the chronometer correction a more accurate value of 8 could be found, which substituted in (38) as before would give prac- tically exact values of t and the chronometer correction. Example. The observations made on the sun east of the meridian, recorded in § 78, give & = 22» 8» 35'.3, R = 67° 30' 0" The chronometer correction is assumed to be + 18 m 3 s ; required its value furnished by the observations. R I e 2h' h' z' r P z s <£-8 <£ + S z +(<£- 8) z-(4>-S) 2 +(<£ + 8) z -( + 8) sin i [* + (<£ -8)] sin J [a -(<£- 8)] sec l[z + (<£ + 8)] sec j [z - (if, + 8)] tan 2 J « tan | J True time 0" 67° 30' + 3 8 - 12 67 32 56 33 46 28 56 13 32 1 24 7 56 14 49 42 16 47 + 13 12 53 29 3 54 55 29 40 85 18 43 27 10 55 111 44 29 45 9 9.83097 9.37104 0.25099 0.00001 9.45301 9.72650„ 151° 57' 17" 303 54 34 20»15-»38".3 Barom. 29.036 inches Att. Therm. 50°.0 F. Ext. Therm. 47 .8 F. Amer. Ephem., p. 278, ir 8".8 log it 0.944 sin z 1 9.920 From (64), p 7" 22» 8™ 35* Approx. A0 + 18 3 Sid. time 22 26 38 Meantime April 24* 20 13 29 Longitude 5 34 55 Gr. mean time April 25 1 48 24 Ephem., 8 + 13° 12' 53" True time April 24*20* 15™ 38'.3 Longitude 5 34 55 .1 Gr. true time April 25 1 51 Eq. of time, E - 2 4.9 Mean time April 24 20 13 33 .4 Sid. time, 6 22 26 42 .4 Chron.time,0' 22 8 35.3 A0 + 18 7 .1 This value of A0 differs so little from the assumed value that another approximation to the value of 8 is unnecessary. GEOGEAPHICAL LATITUDE 109 Using + 3' 21" as the index correction, the value of A0 given by the afternoon solar observations, § 78, is + 18 m 8 s .l, which agrees well with the above, assuming the chronometer's daily rate to be + 3 S .6 [§ 64]. 83. The error in the hour angle — and therefore in the time — produced by a small error in the measured altitude or in the assumed latitude is readily found. Differentiat- ing (15), regarding z and t as variables, and reducing by (17), we obtain dt= . f- s (187) sin A cos that is, an error dz in the measured zenith distance pro- duces an error dt in the time, which is least when sin A is a maximum. Likewise, differentiating (15) with respect to and t and reducing by (16) and (17), we have dt = - . *+ . -; (188) tan A cos that is, an error d<\> in the latitude gives rise to an error dt in the time, which is small when tan A is large. For these reasons it appears that to obtain the best determination of time from observed altitudes, those stars should be selected which are as nearly as possible in the prime vertical. /\\ GEOGEAPHICAL LATITUDE 84. By a meridian altitude of a star or the sun. Observe the double altitude of the star or sun at the instant when it is on the meridian, and obtain the true zenith distance z as in §§ 81' and 82. The latitude is then found from 4> = 8±z, (189) the upper sign being used for a star south of the zenith, the lower sign for a star between the zenith and the pole. For a star below the pole we have $ = 180° -B-z. (190) 110 PRACTICAL ASTRONOMY Example. The double altitude of the sun's lower limb was observed at Ann Arbor at true noon, Friday, 1891 Feb. 6, as follows : Sextant 63° 49' 15". Barom. 28.98 inches, Ext. Therm. 38° F. Find the latitude. In this case r is computed from (96), and p may be taken from the table on page 27. R 63° 49' 15" z< 58° 3' 56" I + 35 r 1 28 e 12 P 8 2h' 63 52 8 S - 16 15 h> 31 56 4 z 57 49 1 z> 58 3 56 s - 15 32 11 42 16 50 85. By an altitude of a star, the time being known. Hav- ing determined the star's hour angle by (41), the latitude is given by (15), in which is the only unknown quantity. To determine it, assume / sin F = cos 8 cos /, (191) /cos F- sin 3, (192) and (15) becomes cos z =/sin($ + F) = sin SsecFsin ( + F). From these we obtain tan F = cot 8 cos t, (193) sin ( + F) = cos F cos z cosec 8, (194) which effect the solution. The quadrant of F is determined by (191) and (192). ($ + jF), being determined from its sine, may terminate in either of two quadrants, thus giving rise to two values of the latitude. That one is selected which agrees best with the known approximate value of the latitude. In case the sun is observed, t is the true solar time. GEOGRAPHICAL LATITUDE 111 Example. Find the latitude from the following double altitudes of Polaris observed Saturday, 1891 April 25 : Chronometer Sextant Barom. 29.17 inches 14* 55™ 5' 82° 18' 40" Ext. Therm. 39°.3 F. 56 35 19 20 58 20 19 35 Amer. Ephc 3m., p. 305 15 1 20 40 a 1*17"*48» + F 41° 10' 20" 42 16 52 86. Differentiating (15) with regard to z and <\> and reducing by (16) we obtain d = dz cos .4' (195) that is, an error dz in the measured zenith distance pro- duces the minimum error d$ in the latitude when the star is on the meridian. Differentiating (15) with regard to and t, and reduc- ing by (16) and (17), we obtain d = — tan A cos dt ; (196) that is, an error dt in the estimated time of making the observation gives rise to an error deft in the latitude, which will be small when the star is nearly on the meridian, and equal to zero when A is 0° or 180°. For these reasons it appears that to obtain the best de- termination of the latitude from observed altitudes, those stars should be selected which are as nearly as possible on the meridian. 112 PRACTICAL ASTRONOMY 87. By circummeridian altitudes. The method of § 84 is applicable to only one altitude observed when the star is on the meridian. If a series of altitudes be observed just before and after meridian passage, — called circum- meridian altitudes, — they can be reduced to the equivalent meridian altitudes and a quite accurate value of the lati- tude obtained by combining the results. Equation (15) may be written cos z = cos ( — 8) — cos cos 8 2 sin 2 \ t, (197) be- comes cos z = cos z — y. (198) Here 2 is a function of y, and we may write «=/(y)- Developing this in series by Maclaurin's formula, restoring the value of y, and dividing the abstract terms by sin 1" to express them in seconds of arc, we have cos ( 199 > which converges rapidly when t does not exceed 30™, and the star is more than 20° from the zenith, as it will be in sextant double altitudes. If we let cos<£cosS . .„ _ ,„... —^—— = A, A* cot z = B, (200) 2sin 2 i< 2 sin 4 4* ,. M . . if. = m, . * = n, (201) sm 1" sin 1" v ' and substitute the resulting value of z for z in (189), we have 4> = S±z^fAm±Bn, (202) the lower sign being employed for a star culminating between the zenith and the pole. When a star is observed near the meridian at lower cul- mination, it is convenient to reckon the hour angle from GEOGRAPHICAL LATITUDE 113 the lower transit, t in (15) must be replaced by 180° + t, and we obtain cos z = sin sin 8- cos cos 8 cos t = - cos ( + 8) + cos cos 8 2 sin 2 J «. Developing this in series as before and substituting the resulting value of s for z in (190), we have = 180° - 8 - z - Am - Bn. (203) The entire series of observations is conveniently reduced as a single observation by letting z, m and n in (202) and (203) represent the arithmetical means of the values of these quantities for the individual observations. The values of m and n are tabulated in the Appendix, Table III, with the argument t. An approximate value of is required in computing A. This may be obtained by the method of § 84, from the observation made nearest the meridian. If the sun is observed, the declination is taken from the Ephemeris for the instant of each observation in case the observations are reduced separately, and for the mean of the times in case they are reduced collectively. If a star is observed with a sidereal chronometer, the hour angles t are the intervals between each observed time and the chronometer time of the star's transit. If a star is observed with a mean time chronometer, the intervals must be reduced from mean to sidereal intervals before entering Table III for m and n. If the sun is observed with a mean time chronometer, the intervals should be reduced to apparent solar intervals by correcting for the change in the equation of time during the intervals. This, however, will never exceed S .5, and may be neglected in sextant observations. If the sun is observed with a sidereal chronometer, the intervals must be reduced to mean solar intervals and thence to apparent solar. If the rate of the chronometer is large it must be allowed for. 114 PRACTICAL ASTRONOMY Example. Wednesday, 1891 April 8, at a place in lati- tude about 42° 17' and longitude 5" 34 m 55 s the following- double altitudes of the sun were observed with a sextant and sidereal chronometer. Barom. 29.373 inches, Att. Therm. 66° F., Ext. Therm. 42°.o F. Required the lati- tude. [Each printed observation is the mean of three consecutive original observations.] Limb Sextant Chronometer Sidereal t Solar t m n Upper 109 c 58' 33".7 0»31" 28».0 — 19" 59».2 — 19» 55«.9 779".6 1".47 Lower 109 4 40 .0 34 51.0 -16 36.2 -16 33.5 538 .1 .70 K 109 14 6 .2 . 38 43.3 -12 43.9 — 12 41.8 316 .4 .24 Upper 110 25 16 .2 42 42.3 - 8 44.9 - 8 43.5 149 .5 .05 " 110 29 39 .0 45 44.7 - 5 42.5 - 5 41.6 63 .6 .01 Lower 109 27 27 .5 49 30.7 — 1 56.5 — 1 56.2 7 .4 .00 " 109 27 43 .7 53 9.7 + 1 42.5 + 1 42.2 5 .7 .00 Upper 110 29 .0 57 36.0 + 6 8.8 + 6 7.8 73 .8 .01 " 110 21 51 .2 1 2 56.0 + 11 28.8 + 11 26.9 257 .3 .16 Lower 109 11 33 .7 5 46.7 + 14 19.5 + 14 17.2 400 .6 .39 " 109 2 27 .7 9 12.3 + 17 45.1 + 17 42.2 615 .0 .92 Upper 109 56 20 .0 1 12 33.7 + 21 6.5 + 21 3.0 869 A 1 .83 109 45 43 .2 + 33.9 339 .7 .48 Apparent time of apparent noon 0* 0™ 8 .0 Equation of time + 1 52.1 Mean time of apparent noon 1 52.1 Sidereal time of apparent noon 1 8 37.2 Chronometer correction + 17 10.0 Chronometer time of apparent noon 51 27.2 The difference between this and the observed times gives the sidereal intervals t as above. The mean of the hour angles is + m 33 s .9, and there- fore the sun's declination is taken for the local mean time 0" l m 52M + m 33 s .9 = 0" 2™ 26 s , or Greenwich mean time 5" 37 m 21 s . An equal number of observations on the upper and lower limbs was made, hence there is no correction for semidiameter. The solution of (202) is made as follows : / + 2 51 .7 e - 18 .0 W 109 48 16 .9 h' 54 54 8 .4 z< 35 5 51 .6 T< 40 .5 P 5 .1 Z 35 6 27 .0 8 + 7 17 33 .4 Am 7 14 .7 Bn 1 .1 42 16 46 .8 GEOGRAPHICAL LONGITUDE 115 8 + 7° 17' 33»4 Sextant 109° 45' 43"'.2 42 17 z 34 59 26 .6 cos 9.86913 cos 8 9.99647 cosec z 0.24151 log^ 0.10711 logm 2.53110 Am 434".7 log^4 2 0.2142 cotz 0.1549 log n 9.6812 log fin 0.0503 Bn l".l A repetition of the computation with this value of does not change the result. [The latitude of the place is known to be about 42° 16' 47"!.] GEOGRAPHICAL LONGITUDE 88. By lunar distances. The moon's distance from a star nearly in the ecliptic is rapidly changing. Its geo- centric distances from the sun, Venus, Mars, Jupiter, Saturn and nine bright stars near its path are given in the Ameri- can Ephemeris [pp. XIII-XVIII of each month] at three- hour intervals of Greenwich mean time, from which the distances at any other instants may be found by interpola- tion. Conversely, if its distance from any of these objects is measured with the sextant and the apparent distance reduced to the corresponding geocentric distance, the Greenwich mean time at the instant of observation may be found. The Greenwich mean time minus the observer's mean time is the observer's longitude. This method of determining the longitude is occasionally of considerable importance to navigators and explorers. 89. We shall suppose that the moon's distance from the sun has been observed. The formula? for a planet will be the same, save that the semidiameter of the planet may 116 PBACTICAL ASTRONOMY usually * be neglected. For a star the parallax and semi- diameter are zero. The sextant reading having been corrected for the index error and eccentricity, the result is the apparent distance between the nearest limbs of the sun and moon. It must be corrected for their semidiameters, refractions and parallaxes. To compute these corrections, the zenith distances of the two bodies must be known. When there are three observers, as frequently happens at sea, the altitudes of the sun and moon, and the distance between them, should be measured simultaneously. The observer's mean time can be obtained also from these observed altitudes of the sun [§ 82]. When it is not practicable to make these obser- vations at the same time, the observer may measure the altitudes immediately before and after measuring the lunar distance, and obtain the required altitudes at the instant of observation by interpolation. Again, the observer may assume an approximate value of the longitude (which he can usually do sufficiently accurately), and take from the Ephemeris the right ascensions and declinations of the sun and moon corresponding to the Greenwich time thus obtained. The hour angles, azimuths and zenith distances are then given by §§ 8 and 5. The parallax of the sun in azi- muth is negligible ; its parallax in zenith distance is given by (64). The parallax of the moon in azimuth is given by (71) and (72) ; and in zenith distance, by (80), (78) and (79). (95) gives the refractions, care being taken to use the apparent zenith distance. Fig. 18 The semidiameters of the sun and * In case the telescope is powerful enough to define the planet's disk, the moon's limb may be made to pass through the center of the disk. GEOGRAPHICAL LONGITUDE 117 moon are obtained by tbe methods of §§ 33-35. The solu- tion of (107) requires the values of q. In Fig. 18 let M be the moon's center, 8 the sun's center, and Z the zenith. For the sun, q = ZSM, and for the moon, q = ZMS. If we let Z' = the apparent zenith distance of the sun = ZS, z' = the apparent zenith distance of the moon = ZM, d' = the apparent distance between the centers = SM, we can write, for the sun, tan * a - J sin » (Z ' ~ Z ' + dl) shlHz ' + Z '~ d "> - f20« tan * q ~ X sin i (Z' + 2' + d') sin J (Z' - «' + rf') ' l J and for the moon, Vsin i (Z' - z' + d') sin J (Z' + z' - d') s: tan i n — * ° mf ^ ~ * T "- |; "" t ^ ^ * ~ " > - (Van) Adding the inclined semidiameters given by (107) to the corrected sextant reading, the sum is the distance d' between the centers as seen from the observer. The combined effect of the refraction and the parallax in zenith distance is to shift the bodies in their vertical circles without changing the angle SZM at the zenith, which we shall represent by V. If we let Z = the geocentric zenith distance of the sun, z = the geocentric zenith distance of the moon, d" = the corresponding distance between the centers, we can write cos d" = cos z cos Z + sin z sin Z cos V, (206) cos d' = cos z' cos Z' + sin z' sin Z' cos V. (207) Therefore cos d" — cos z cos Z _ cos d' — cos z' cos Z' . sin z sin Z sin z' sin Z' or cos d" — cos (z + Z) _ cos d' — cos (z' + Z f ) (208^ sin z sin Z sin z' sin Z' 118 PRACTICAL ASTRONOMY If we put z' + Z' + d' = 2 x, and substitute cos d' - cos (z' + Z') = 2 sin x sin (x — d'), cosd" = l-2sin a i<*", cos (2 + Z) = 2 cosH (z + Z) - 1, = l-2sin 2 £(z + Z), (208) reduces to sin* 4 d" = sin" \(z + Z)- sinz , sln f, sin * sin (* - *)■ ( 209 ) sin z' sin Z' Let an auxiliary angle M be denned by »in3 M - sin z sin .g . sin x sin (x - rf')_ f 21Q> ~ sin z' sin Z' sin" £ (z + Z) v ; Then (209) takes the form sin^rf" =sin4(z + Z)cosilf. (211) The parallax of the moon in azimuth produces a small change in V and therefore in d". From (206), by differ- entiation, Ad" = sin z sin Z sin V cosec d" A V, (21 2) in which AF"is the parallax in azimuth. The geocentric distance d between the centers is now given by d = d" + Ad". (213) In connection with the lunar distances, the Ephemeris gives a column "P. L. of Diff." (Proportional Logarithm of the Difference), which is the logarithm of 10800, the number of seconds in 3*, minus the logarithm of the change in the lunar distance, expressed in seconds of arc, in the next following three hours. That is, it is the logarithm of the reciprocal of the moon's average rate for the three hours, or the rate at the middle period of the three hours [see remarks on interpolation, § 15] . In order to interpolate for the Greenwich mean time corresponding to the given value of d, we have only to add the P. L. of Diff. for the middle period of the approximate interval to the logarithm of the number of seconds of arc by which d exceeds the GEOGRAPHICAL LONGITUDE 119 next smaller Ephemeris lunar distance. The sum is the logarithm of the number of seconds of time by which the Ephemeris time is to be increased. If the P. L. of Diff. given in the Ephemeris is used without change, a slight correction for the neglected second difference of the moon's rate can be taken from Table I, Appendix, American Ephemeris, and applied as there directed. If the resulting longitude differs considerably from the assumed longitude, a second approximation should be made by starting with the value of the longitude just obtained. A third approximation will not be necessary. Example. Tuesday, 1891 May 12, the distance between the bright limbs of the sun and moon was observed with a sextant and sidereal chronometer. The mean of ten observations gave 6' = 8 s 36» 10», R = 57° 28' 32".9. Chronometer correctipn, +19 m 15 s ; index correction, + 2'56".4; Barom. 29.25 inches, Att. Therm. 62° F., Ext. Therm. 57° F. ; latitude, + 42° 16' 47" ; longitude assumed, + 5 A 34™. Required a more exact value of the longitude. & 8*36»10' R 57°28'32".9 A0 + 19 15 / + 2 56 .4 8 55 25 e - 11 -3 Mean time 5 33 42 . Distance 57 31 18 .0 Longitude 5 34 Gr. mean time 11 7 42 Corresponding to this Greenwich mean time, we take from the American Ephemeris, pp. 74, 75, 77, 80 and 278, Sun Moon Right ascension, Declination, Semidiameter, Horizontal parallax, a 3* 17 m 54' 8 + 18° 15' 10" S 15 51.7 * 8.8 7» 29™ 31* + 25° 36' 55" 15 12.8 55 43.3 120 PEACTICAL ASTRONOMY By §§ 8 and 5 we find for the geocentric coordinates of the sun, t = 84° 22' 45", A = 100° 8' 50", Z = 73° 46' 5" ; and for the moon, t = 21° 28' 30", A = 53° 27' 21", z = 24° 15' 40". Computing the parallaxes we obtain, for the sun, .4' - 4 = 0, p = Z> - Z = 8".4. From (58) we find - 0' = 687".3 = 11' 27".3; and from (59) log p = 9.99935 ; therefore, for the moon, log m = 6.11807, A'-A= + 21".8, y = +6'49".2, logn = 8.20908, *' - z = 23' 6".2. The mean refraction of the sun, Table II, is about 3' 13", and therefore its apparent zenith distance is very nearly 73° 43' 0". The value of the refraction is now found from (95) to be 3' 8".6. Similarly, the refrac- tion for the moon is 25".6. The apparent zenith distances of the sun and moon are therefore Z' = 73° 43' 4".8, z' = 24° 38' 20".6. The apparent zenith distance of the upper limb of the sun is 73° 43' 4".8 - 15' 51".7 = 73° 27' 13".l. The cor- responding refraction is 3' 5".5. The apparent vertical semidiameter is therefore contracted 3".l [§ 35], and its value is 15' 48".6. ' The moon's apparent semidiameter is found from (106) to be 15' 26".3 ; and by refraetion its apparent vertical semidiameter is reduced to 15' 26" .0. The approximate distance between the centers of the sun and moon is d' = 57° 31' 18" + 15' 49" + 15' 26" = 58° 2' 33". Substituting these values of d', Z' and z' in (204) we obtain for the sun, q = 20° 58' ; and in (205) for the moon, q = 124° 34'. For the sun, a = 15' 51".7 = 951".7, I = 15'48".6 = 948".6 ; and by (107) the inclined semidiameter GEOGRAPHICAL LONGITUDE 121 is S" = 15' 49".l. Similarly, for the moon, S" = 15' 26".2. The apparent distance between the centers of the sun and moon is therefore d' = 57° 31' 18".0 + 15' 49".l + 15' 26".2 = 58° 2' 33".3. The solution of (210) gives M = 49° 48' 39".0 ; and thence, from (211), d" = 58° 18' 16".0. Substituting A' -A =AF= + 21".8 in (212), we obtain Ad" = + 7".4. The geocentric distance between the sun and moon is therefore, by (213), d = 58° 18' 16".0 + 7".4 = 58° 18' 23".4. From the American Ephemeris, pp. 86 and 87, at Greenwich mean time 9*, d = 57° 16' 23", P. L. of Diff. = 0.3169, Greenwich mean time 12 , d = 58 43 9 , P. L. of Diff. = 0.3184. We have to interpolate for the interval of time T after 9 A , corresponding to a change in d of 58° 18' 23".4- 57° 16' 23" = 3720".4. The value of T is approximately 2*. The value of P. L. of Diff. at the middle of the 2 h is 0.3167. Gr. mean time 11* 8 ra 34« Observer's mean time 5 33 42 Observer's longitude 5 34 52 The true value of the longitude is known to be 5* 34 m 55 s . The error of 3 s corresponds to an error of 2" in the meas- ured distance [or in the lunar tables], and is unusually small. The observations are difficult to make, and the measures of the best observers are easily liable to an error of 10". It is well, however, to carry the numerous correc- tions to tenths of a second to prevent the accumulated effect of neglected fractions. The above solution of this problem is essentially a rigorous one. Navigators are accustomed to employ abridged forms of solution, for which the reductions are much shorter. Likewise many of the functions are tabu- lated, which still further reduces the labor. P. L. of Diff. 0.3167 log 3720».4 3.5706 logT 3.8873 T 7714' T 2 s 8™ 34" CHAPTER VII THE TRANSIT INSTRUMENT 90. The transit instrument consists essentially of a telescope attached perpendicularly to a horizontal axis. The cylindrical extremities of this axis are the pivots. The straight line passing through their centers is the rotation axis. The supports for the two pivots are called the Vs. The straight line passing through the optical center of the object glass and the rotation axis and per- pendicular to the latter is the collimation axis. By revolv- ing the instrument about the rotation axis the collimation axis describes a plane called the collimation plane. In the common focus of the object glass and eye-piece is a system of wires called the r etfacle .(?) It consists either of spider threads attached to a frame, or of fine lines ruled on thin glass. An odd number of wires — usually five, seven or eleven — is placed parallel to the collimation plane and perpendicular to the collimation axis, over which the times of transit of a star's image are observed. The middle wire of the set is fixed as nearly as possible in the collimation plane. One or two wires are placed perpendicular to these to mark the center of the field of view. A micrometer wire parallel to the first set is arranged to move as nearly as possible in their plane. The axis of the instrument is hollow. A light is placed so that the rays from it enter the axis and fall on a small mirror in the center of the telescope, which reflects them to the eye-piece in such a way that the wires are seen as dark lines in a bright field. The illuminating apparatus in some instruments is arranged 122 THE TRANSIT INSTRUMENT 123 Fig. 19 so that the observer may change from dark lines in a bright field to bright lines in a dark field, — a necessary arrange- ment when the object to be observed is very faint. A very common distribution of the wires in the reticle is shown in Fig. 19. The instrument is so arranged that its rotation axis can be rotated 180° ; i.e., reversed, about a vertical line. The two posi- tions are defined conveniently by •stating the position of the clamp or graduated circle on the axis. Thus, clamp W or clamp E, circle W or circle E, denotes that position of the instrument in which the clamp or circle is west or east of the collimation plane. An excellent form of transit and zenith telescope com- bined is shown in Fig. 20. The circular base-plate of the instrument is supported on three screws, with which the instrument may be quickly leveled on its supporting pier. The two standards which support the pivots of the instru- ment are rigidly fixed to a circular plate, which may be rotated freely through 180° on the base-plate when the instrument is used as a zenith telescope [considered in the next chapter]. When the instrument is used as a transit, the two circular plates are clamped together, and remain clamped. To reverse the instrument, the observer turns the reversing crank (shown in the lower right corner of the cut), which raises the two small inner standards until the pivots are entirely free from the Vs ; the axis, supported on the two standards, is then turned gently through 180°, and carefully lowered by turning the crank in the reverse direction, until the pivots rest in the Vs. The illuminat- ing lanterns are in position for the rays to pass through the pivots. The level — in this form called the striding- level — is shown resting on the pivots. The micrometer i Fig. 20 THE TRANSIT INSTRUMENT 125 box can be rotated 90° to make the movable wire perpen- dicular to the rotation axis for the transit instrument, or parallel to it for the zenith instrument. The cut shows the micrometer in the latter position. A diagonal eye- piece enables the instrument to be used with zenith stars. The small circles attached to the sides of the telescope are used for setting the telescope at the zenith distance or altitude of the star to be observed. The vernier-arms bear both coarse and delicate levels. When one of the verniers is set at the proper reading for the star, the telescope is moved in altitude until the bubble of the coarse level "plays." The star will then pass through the approxi- mate center of the eyepiece. It is made to pass between the two horizontal wires by turning the slow-motion screw. Another common form of the transit instrument is that in which one end of the axis is made to take the place of the lower half of the telescope. A prism is placed at the intersection of the telescope and axis. This turns the rays of light through 90° to the eyepiece, which is in one end of the axis. This form is sometimes called the broken or prismatic transit. An excellent form of the prismatic transit is shown in Fig. 21. In this, the combined telescope and rotation axis is mounted east and west, and the totally reflecting prism is immediately in front of the object glass. It is provided with a reversing apparatus, with a micrometer and delicate zenith level, and can be used also as a zenith telescope. This instrument is very compact, and therefore well adapted for use in exploration or other cases where trans- portation is difficult. There are several considerations affecting all forms of the transit instrument; viz.: The instrument should be reversible without appreci- able jarring. The lamps used for illuminating the reticle, or for 126 PRACTICAL ASTRONOMY lighting the observing room, must be so placed that they will not heat the instrument appreciably. The supporting pier must be isolated from the floor of the observing room, and should extend down to a firm rock or soil foundation. The observing room should be constructed so that it may be thoroughly ventilated before observations-are begun. A sidereal chronometer or clock is a necessary com- panion of the transit instrument. Refined observations Fig. 21 should be made by the chronographic method, described in § 68. In a fixed observatory, the clock should not be mounted in the transit room, but in an interior room of more constant temperature, where it can be placed equally well in the electric circuit. 91. The transit instrument may be mounted so that its collimation plane is either in the prime vertical, or in the meridian. In the first case it may be used to determine the latitude ; but this method is practically superseded by that of the zenith telescope, to be described later. Mounted in the meridian, it is employed in connection with a sidereal clock or chronometer to determine the THE TRANSIT INSTRUMENT 127 time, the right ascensions of the stars or other celestial objects, and the longitude of the observer, when great accuracy is required; and we shall treat only this case. Let us suppose that the axis is mounted due east and west and that the middle wire is exactly in the collimation plane. If the image of a star whose apparent right ascen- sion is a is observed on the wire at the chronometer time 6', the chronometer correction A0 is given by (neglecting diurnal aberration) A0 = a - 6'. (214) The observer may adjust his instrument as accurately as he pleases, but the adjustments will not remain, owing to changes of temperature, strains, etc. It is customary to put the instrument very nearly in the meridian when it is first set up, and thereafter to vary the adjustments only at long intervals of time. In general, therefore, the star will be observed when it is slightly to one side of the meridian. A determination of the errors of adjustment of his instru- ment enables the observer to reduce the chronometer time of observation to the chronometer time of meridian pas- sage ; whence the chronometer correction is given by (214) as before. 92. Theoretically, the rotation axis should be in the prime vertical and in the horizon, and the middle wire should be in the collimation plane. The azimuth constant, a, is the angle which the rotation axis makes with the prime vertical. It is + when the west end of the axis is too far south. The level constant, b, is the angle which the rotation axis makes with the horizon. It is + when the west end of the axis is too high. The collimation constant, c, is the angle which a line through the middle wire and the optical center of the object glass — called the line of sight — makes with the 128 PRACTICAL ASTRONOMY collimation plane. It is + when the middle wire is west (in the eyepiece) of the collimation plane. It is required to correct the time of observation of a star for the small deviations a, b and c. Let SWNE in Fig. 22 represent the celestial sphere projected on the horizon, Z the observer's zenith, NS the meridian, WE the prime vertical, WQE the equator, and P the pole. Suppose the rotation axis of the instrument lies in the vertical circle AZB, and that the axis produced cuts the sphere in A and B ; that the great circle N'Z 1 S' lies in the collimation plane ; and that N"Z"S", parallel to JV'Z'S 1 , is described by the line through the middle wire and the center of the object glass. When the stars are observed on the middle wire they are on the circle N"Z"S", whereas we desire to know the chronometer time when they are on the meridian. Let be such a star. The time required for the star to pass from to the meridian is equal to the hour angle of measured from the meridian toward the east. Let t represent it. If we let 90° — m denote the hour angle and n th& declination of A, we have by definition, THE TRANSIT INSTRUMENT 129 ZPA = 90° - m, ZA = 90° - 6, PZA = 90° + a, PO = 90° - 8, PA = 90° - n, A = 90° + c, PZ = 90° - , OP4 = 90° - TO + T. i triangle ZPA we have sin n = sin 6 sir. .*- cos 6 cos sin a, (215) sin m cos n = sin b cos (ft + cos b sin sin a ; (216) and from OP A sin c = — sin n sin 8 + cos n cos 8 sin (t — m), or sin (t — m) = tan n tan 8 + sin c sec n sec 8. (217) These equations are true for any position of the instru- ment, and determine t when a, b and e are known. But for the instrument nearly in the meridian a, b, c, m and n are small, and the above equations become n = 6 sin — a cos tj>, (218) m = b cos <£ + a sin <£, • (219) t = m + n tan 8 + c sec 8. (220) (220) is Bessel's formula for computing the value of t. Eliminating m and n from the three equations we obtain Mayer's formula T = a . sin(4,-8) +6 . cos(«fr-8) +c 1 t (22 cos 8 cos 8 cos 8 in which the terms of the second member are the correc- tions, respectively, for errors of adjustment in azimuth, level and collimation. For convenience, let us put A== am(-8) i B = cos(-8) > c= _l_, ( 222) cos 8 cos 8 cos 8 and (221) becomes t =aA +bB + eC. (223) The effect of the diurnal aberration is to throw the star east of its true position. It is therefore observed too late, 130 PRACTICAL ASTRONOMY and the time of observation must be diminished by the quantity, (116), 0".31 cos sec 8 = 0-.021 cos <£ C. (224) For greater accuracy the star is observed over several wires. An odd number of wires is always used. They are generally placed very nearly equidistant, or very nearly symmetrical with respect to the middle wire. Were either of these arrangements exactly realized the mean of all the times of transit would be the most probable time of transit over the middle wire. This never happens, however, and it is necessary to determine the intervals between the wires. Let i denote the angular distance between a side wire and the middle wire ; * I the interval of time required by a star whose declination is 8 to pass through this distance. From Fig. 13, letting the two positions of the micrometer wire represent the side wire and the middle wire, we have in the triangle CS'P, CS' = i, S'P = 90° - 8, CPS' = I; and we can write sin / = sin i sec 8 = sin i C. (225) If the star is not within 10° of the pole, it is sufficiently accurate to use 7 = isec8 = iC. (226) Suppose there are five threads in the reticle, numbered I, II, III, IV, V, beginning on the side next to the clamp, and that the clamp is west. Let t v t 2 , t 3 , Z 4 , t b , be the observed times of transit of the star over the wires, and i v i 2 , i 4 , i 6 , the distances of the four side wires from the middle wire.f The five observed transits give for the time * That is, the angle subtended at the optical center of the object glass by lines drawn to the side wire and to the middle wire. It is also meas- ured by the interval of time required for a star in the equator to pass from the side wire to the middle wire. t For clamp west, i* and is are negative ; for clamp east, ii and i 2 are negative. THE TRANSIT INSTRUMENT 131 of crossing the middle wire either t x + i x 0, t 2 + i 2 0, t 3 , t 4 + i 4 0, or t s + i 5 0, which would all be equal if the observations were perfect. Taking their mean, the most probable time of crossing the middle wire is h + k + h + U + h , h + h + U + H n 5 + 5 C - If we let a _ k + h + 5 and ,• _ h + i 2 + i 4 + i a *"~ 5 ^ = <1 + <2 + t+<4 + ?S | ^^ (228) the most probable time of crossing the middle wire is n + i m C. (229) 6 m is the time of crossing a fictitious wire called the mean wire, and i m C is the reduction from the mean wire to the middle wire. The above method holds good also in the case of an incomplete transit ; that is, one in which the transits over some of the wires have been missed. Thus, suppose that the wires I and IV have been missed. The three remain- ing transits give for the times of crossing the middle wire t 2 + i 2 (7, t s , t h + i 5 C; and their mean is 'W, (230) and similarly in other cases. In accurate determinations of the time several stars will be observed, and if the chronometer has a sensible rate, the chronometer corrections at the several times' of observation will be different. To equalize them, let be some chro- nometer time near the middle of the series of observations, let a star be observed at the time m , and let the rate of the chronometer be 86. During the interval 6 m — 6 the chronomter loses (B M -e )W. (231) 132 PRACTICAL ASTRONOMY If this quantity be computed for all the stars observed and applied to the observed times, the resulting chronometer corrections furnished by the several stars will be the cor- rections at the instant O , and with perfect observations would all be equal. Collecting the expressions (223), (224), (229) and (231), we have for the observed time of crossing the meridian when the clamp is west, 6> = e m + aA+bB + cC- 0».021 cos C + i m C + (0„ - 6 ) 80 ; (232) and therefore, by (214), A6=a-[0 m +aA+bB+(c-O>.O21 cos + i m ) C+{6 K -6 a )W]. (233) For clamp east it is easily seen that e and i m change sign ; otherwise the formula remains the same. The formula has been deduced for a star observed at upper culmination. For a star observed at lower culmina- tion we have only to replace 8 by 180° — § in the factors A, B and C, and they become A _ sin ( + 8) ^ B _ cos ( + 8) ^ c _ 1_ . 234 . cos 8 cos 8 cos 8 The factors A, B and are readily computed with four- place tables. But when an instrument is set up perma- nently, as in an observatory, their values should be- computed for every degree of declination, and tabulated. For polar distances less than 15° it is convenient to have them tabulated for every ten minutes of declination. DETERMINATION OF THE "WIRE INTERVALS 93. (a) If the instrument is provided with a micrometer in right ascension, set the micrometer wire in succession on each of the fixed wires.* The differences of the microm- * More accurate readings will be obtained, as in many other cases, by setting the micrometer wire on each side of the fixed wire and just in apparent contact with it. The mean of the readings in the two positions is the reading for the coincidence of the two wires. THE TRANSIT INSTRUMENT 133 eter readings on the side wires and the middle wire give the intervals in terms of one revolution of the screw, which will have been obtained by the methods of § 61. Example. Friday, 1891 Feb. 20. The transit instrument of the Detroit Observatory. Four sets of micrometer read- ings were made when the micrometer wire was in contact with each side of the fixed wires, to find the wire intervals and i m . The numbers in the last line are the means of all the readings on the corresponding wires. The value of one revolution of the screw is, § 61, B = 45" .042 = 3 S .003. I n m rv V 29.966 27.443 24.891 22.361 19.797 30.130 27.620 25.054 22.523 19.965 29.969 27.445 24.890 22.357 19.799 30.131 27.621 25.058 22.527 19.970 29.968 27.448 24.895 22.363 19.802 30.135 27.618 25.060 22.528 19.969 29.968 27.445 24.898 22.356 19.797 30.135 27.623 25.062 22.526 19.969 30.050 27.533 24.976 22.443 19.883 ?! = (30.050 - 24.976) R = + 15 8 .237, ~i 2 = (27.533 - 24.976) R = + 7 .679, i 4 = (22.443 - 24.976) R = - 7 .607, t 5 = (19.883 - 24.976) R = - 15 .294. i m = + 0".003 for clamp west, i m = — .003 for clamp east. (5) Observe the transits of a close circumpolar star over the several wires, and solve (225) for the intervals i. It is convenient and sufficiently accurate to use (225) in the form i=sin/-^L8 . (230 15 sin 1" 134 PRACTICAL ASTRONOMY Solving as in the case of (159), the resulting values of i will be expressed in seconds of time. Example. Monday, 1891 March 16, \ Ursce Minoris was observed at lower culmination with the transit instru- ment of the Detroit Observatory, clamp east, as below. Required the wire intervals. From the Amer. Ephem., p. 804, 8 = 88° 57' 47".3. Wires Chronom. / I sin I i I 7* 2™ 37' - 14"> 3' - 3° 30' 45" 8.787222,, - 15-.245 II 9 35 - 7 5 - 1 46 15 8.489986,, - 7.689 III 16 40 IV 23 40 + 70 + 1 45 8.484848 + 7.599 V 30 46 + 14 6 + 3 3,1 30 8.788762 + 15 .299 A number of stars should be observed in this way, and the mean of all the results adopted as the wire intervals. DETERMINATION OF THE LEVEL CONSTANT 94. The level constant b is generally found by means- of a spirit level, as explained in § 62. However, the level is applied to the outer surface of the cylindrical pivots and does not give the inclination of the axis, which passes through their centers, unless their radii are equal. To determine the inequality of the pivots and the method of eliminating it, let A and B, Fig. 23, be the Fig. 23 centers of the west and east pivots, for clamp west; M and M' the vertices of the V's in which the pivots rest; L « DETERMINATION OP THE LEVEL CONSTANT 135 and L' the vertices of the V's of the level ; and HM' a horizontal line. Then BAG = BAB is the inequality of the pivots, which we shall represent by p. If we let B' = the inclination given by the level for clamp west, B" = the inclination given by the level for clamp east, V = the true inclination for clamp west, b" = the true inclination for clamp east, /3 = the constant angle HM'M, we can write V = B' +p = |8 — p, for clamp west, 6" = B" — p = ($ + p, for clamp east ; and therefore P=- B" - B< (236) (237) (238) When p has been determined, the value of the level constant is given by (236) or (237). The value of p should be determined a large number of times, and the mean of all the individual results adopted as its final value. In making the observations the telescope should be set at different zenith distances, to detect any variations of the pivots from a cylindrical form. Example. 1891 Feb. 17. The following observation was made on the pivots of the Detroit Observatory transit instrument. Required the inequality of the pivots and the inclinations of the rotation axis. The value of one division of the striding level is d = 1".878 = 0U25. [See § 63.] Clamp Zenith Distance Level Siiect Level Reversed w e w 1 e' w E N. 30° S. 30 7.2 5.2 9.6 11.7 15.6 13.5 1.3 3.4 136 PRACTICAL ASTRONOMY From (166), b = B< = + 2.975 d = + 0».372, for clamp west ; b = B" = + 0.900 rf = + 8 .112, for clamp east. Substituting these values in (238), we find p = - s .O65 ; and therefore, from (236) and (237), b< = + C.372 - 0-.065 = + 0«.307, 6" = + C.112 + 0-.065 = + 0M77. The mean of twenty-two determinations of p for this in- strument gave p —— S .066 ± O'.OOl. Another method of determining the level constant is given in § 97, ( C;* and for clamp east A0 = a - 2 - aA - b"B + cC + 8 .021 cos C. Subtracting and solving for c we. obtain c = i (0 2 - 6J cos 8 + J (&" - b') cos (-8). (242) For lower culmination, S being replaced by 180° — S, c = - K#2 - #i) cos S - J (b" - V) cos (0 + 8) . (243) An example is given in § 103. (c?) By the nadir. If the telescope be directed verti- cally downward to a basin of mercury, and a piece of glass be placed diagonally over and close to the eyepiece in such a way that light from a lamp at one side will be reflected into the telescope, the middle wire and its image reflected from the mercury may be seen near together. Measure with the micrometer the distance between the middle wire and its reflected image. Let M be this dis- tance, and consider it positive when the wire is west (in * The correction for rate will be small compared with the probable error of a transit of a slowly-moving northern star, and may be neglected. 140 PRACTICAL ASTRONOMY the eyepiece) of its image. If the rotation axis is hori- zontal we have M= 2c; but if there is a level constant b, the distance is diminished by 2 b, so that M= 2c — 2b; or c = iM+b. (244) With well-constructed instruments, the collimation con- stant usually remains practically unchanged during a series of observations. The level constant, on the contrary, sometimes varies rapidly. Further, the spirit level is not always trustworthy. Many excellent observers do not use the striding level, but determine the level constant by the method of the nadir, described above. The collimation con- stant having been determined by one of the many available methods — usually by the aid of two collimations in the case of large instruments — the value of the level constant is given by (244), thus : b = c-\M. (245) If we wish to determine both the level and collimation constants by the method of the nadir, we measure the dis- tances of the middle wire from its reflected image in the two positions of the instrument; calling this distance -f- or — according as the middle wire is west or east of its image. Let M 1 = the distance for clamp west, M"= the distance for clamp east, V = the level constant for clamp west, b" = the level constant for clamp east. We have, for clamp west, c = \M'+V, and for clamp east, -c= M'i+Vi. Therefore c = \(M'-M") + \(V- b"), b'+b"=-l(Mi+M"). DETERMINATION OF THE COLLIMATION CONSTANT 141 From (236) and (237), V - b" = - 2p. Therefore c = \(M'-M") -p, clamp west, .(246) c = - i (Mi - M") + p, clamp east, (247) V =-\ (M' + M") - p, clamp west, (248) b" = - $ (M ' + M ") + p, clamp east. (249) Example. 1891 July 24. The following nadir observa- tions were made with the transit instrument of the Lick Observatory. Required the values of c, b' and b". Clamp Micrometer on middle wire Micrometer on W 11.025 10.847 .125 .852 Mean 11.075 Mean 10.849 E 11.020 11.164 .135 .166 Mean 11.077 Mean 11.165 The middle wire was east of its image in both cases. For this instrument, p=- S .021, R = 2 S .931. We have M =- (11.075 - 10.849) R = - 0«.662, M"= - (11.165 - 11.077) R = - .258. Therefore c = - 0M01 + 8 .021 = - ! .080, clamp west, V = + .230 + .021 = + .251, b" = + .230 - .021 = + .209. (e) By two collimators. When the observing telescope is large, it is inconvenient and very undesirable to determine the collimation constant by any method which involves reversing. This is avoided by using two collimators, one north and the other south of the instrument, the object glasses of the two collimators being turned toward each other and toward the center of the transit instrument. The view of one collimator from the other collimator is obstructed by the intervening transit instrument ; but in large instruments apertures are provided on opposite sides of the enlarged central section of the transit telescope, so 142 PRACTICAL ASTRONOMY that when the telescope is directed to the nadir, and the coverings of the apertures removed, the view is unob- structed. The vertical thread in one collimator and the horizontal thread in the other collimator are usually movable by micrometer screws. Let the vertical micrometer thread in one collimator be brought into exact coincidence with the fixed vertical thread in the other. The lines of sight of the two colli- mators will then be exactly parallel, and the two vertical threads, viewed by the transit telescope, will represent objects virtually at an infinite distance and having azimuths differing exactly 180°. Measure the distance B' from the middle wire of the transit reticle to the image of the north collimator thread, and the distance D" from the middle wire to the image of the south collimator thread, calling these distances + or — according as the middle wire is west or east (in the eyepiece) of the collimator images. Then we shall have c = l(D + D"). (250) DETERMINATION OF THE AZIMUTH CONSTANT 98. The azimuth constant a can be determined only from observations of stars. Let two stars (a v S x ) and (a 2 , S 2 ) be observed. When all the constants except a have been determined, the times of observation of the two stars can be corrected for all errors save the azimuth. Let 6 X and 2 be the times so corrected. Then (233) reduces for the first star, to A0 = a L — 6 l — aA v and for the second star, to A0 = a 2 - 6 2 - aA 2 , A x and A 2 being the values of A corresponding to 8j and S 2 . Combining these equations, we obtain a = (q, - y - (a, - g,\ (251) A 1 — A 2 MERIDIAN MAEK, OR MIKE 143 It will be seen that to determine a accurately, all the other constants of the instrument must be well determined, since errors in any one or more of them affect the values of 6 X and r If the instrument is not mounted in a very stable manner, the right ascensions a x and a 2 should differ as little as possible. The value of a will be determined best when the denominator A l — J. 2 is as large as possible. If both stars are observed at upper culmination, one should be as far south as possible and the other as near the pole as possible, in which case A x and A 2 will be large and opposite in sign. This condition will be fulfilled still better by observing one star (a v S x ) at lower culmination and the other (o 2 , 8 2 ) at upper culmination, both as near the pole as possible and differing nearly 12* in right ascen- sion. In this case a t must be replaced by 12* + a t and B x by 180° — 8 1 in the various formulae. Stars observed at lower culmination are marked S. P. (sub polo). MERIDIAN MARK, OR MIRE 99. If a transit instrument is to be used for making long series of observations, as at a fixed observatory, it is well to have a permanent meridian mark, or mire, to assist in determining the azimuth constant. The mark consists usually of a minute circular hole in a metal plate mounted on a firm pier at a considerable distance to the north or south of the instrument. An isolated pier in the transit room carries a lens whose center is in the line joining the mark and the center of the transit instrument, the focal length of this lens being equal to the distance of the mark from the lens. When the mark is illuminated by a lamp or electric light [controlled by a switch in the transit room] placed behind the metal plate, the rays which fall on the mire lens will be transmitted as parallel rays to the observing telescope, and the observer will see a well- defined image of the mark in the focus of his instrument. 144 PRACTICAL ASTRONOMY The focal length of the mire lens should be great, in order that the mark may be at a considerable distance, thereby- reducing the angular value of any possible motion of the mark. The mire lens for one of the - instruments at Pul- kowa has a focal length of 556 feet. One at the Lick Observatory has a focal length of 80 feet. Well mounted mires have been found to be almost constant in azimuth for months at a time. The azimuth of the transit instrument having been de- termined from observations of a pair of azimuth stars, by the methods of the preceding section, the azimuth of the mire may be determined by measuring the angle between the mire and the middle wire with the micrometer, and combining the result with the known collimation and azi- muth constants. The mean of a long series of such deter- minations may be adopted as the azimuth of the mire ; and thereafter a measure of the angle between the mire and middle wire, combined with the known collimation con- stant, will determine the azimuth constant. Nevertheless, the observation of star pairs for azimuth should be made as usual, and the results thus obtained combined with those obtained from the mire. The relative weights to be assigned to the results from the two methods will be evi- dent after a short experience with them. If the mire is mounted at a small angle I above or below the horizon of the instrument, the measured angle between the mire and meridian should, in reality, be multiplied by sec I, but that is a constant factor, and with most mires need not be taken into account. ADJUSTMENTS 100. To set up the instrument, it should first be placed by estimation as nearly as possible in the meridian, and the following adjustments' made in the order indicated. 1st. To bring the wires in the common focus of the eye- piece and objective, slide the eyepiece in or out until the ADJUSTMENTS 145 wires are perfectly well defined. Then direct the telescope to a very distant terrestrial object, or to a star, and move the tube carrying the wires and eyepiece until an image of the object seen on one of the wires will remain on the wire when the position of the eye is changed. Polaris is a good star for this purpose, since its image will move very slowly. When the wires are placed satisfactorily in the focus of the objective, the tube carrying them should be clamped firmly, and remain unmolested indefinitely. Different observers will require only to alter the distance of the eyepiece from the wires in order to bring both star and reticle into focus. This adjustment should be made when the atmosphere is steady. 2d. Make the level constant very nearly zero, testing it by the method of § 94. 3d. To make the wires perpendicular to the axis, direct the telescope to a well-defined mark and bisect it with the middle wire. Adjust the reticle so that the object remains on the wire when the telescope is rotated on its axis. The intersection of the two wires of a collimator furnishes an excellent mark for this purpose. 4th. Test the collimation by the methods of § 97, (a), (6), (c?) or (e), and move the reticle sidewise until c is made very small. 5th. To set the finding circle, direct the telescope to a bright star near the zenith, whose declination is S. When the star enters the field of view move the telescope so that the star describes a diameter of the field, and clamp the instrument. If the circle is designed to give the zenith distances, set it at the reading z = r Rotate the whole instrument horizontally so that the star is on the middle wire at the instant when the chronometer indicates the time 6 r 7th. Repeat the 2d adjustment. 8th. Repeat the 6th adjustment. 9th. The final adjustment in azimuth should be tested by the method of § 98. DETERMINATION OP TIME 101. When the chronometer correction is required to be known very accurately, it is customary to observe the transits of ten or twelve stars. The observing list should be made out very carefully, in advance. Half the stars should be observed with clamp west, the other half with clamp east, since any errors in the adopted values of i m7 p and e will be practically eliminated by reversing the instrument. To determine a well, a pair of azimuth stars should be observed before reversing, and another pair after reversing. The remaining stars on the list should be those which culminate near the zenith, or between the zenith and equator ; since the zenith stars are affected least by an error in the adopted value of a, and the time of transit can be estimated most accurately for the rapidly moving equa- torial stars. There is no method of eliminating an error in b, and it must be very carefully determined. A good DETERMINATION OP TIME 147 program to follow, with small or medium-sized instru- ments, is Take the level readings Observe half the stars Take the level readings Reverse the instrument Take the level readings Observe half the stars Take the level readings If there is time between the stars for making further level readings, they should be made. In reversing, the instru- ment should be handled very carefully to avoid changing the constants. 102. Example. Wednesday, 1891 Feb. 25. The follow- ing observing list was prepared and the stars observed by No. Object Mag. a S Setting 0) Level (2) it Cephei, S. P. 4.6 23» 4" 20« 74° 47'.9 N. 62° 55' (3) 8 Leonis 2.3 11 8 20 21 7 S. 21 10 (4) v Ursce Majoris 3.3 12 37 33 41 S. 8 36 (5) cr Leonis 4.1 15 32 6 38 S. 35 39 (6) \ Draconis 3.3 25 69 56 N. 27 39 (7) Level Reverse (8) Level (9) X Ursce Majoris 3.8 40 19 48 23 N. 6 6 (10) fi Leonis 2.0 43 31 15 11 S. 27 6 (11) /3 Virginia 3.3 45 2 2 23 S. 39 54 (12) y Ursce Majoris 2.3 48 8 54 18 N. 12 1 (13) Level (14) e Corvi 3.0 12 4 32 -22 1 S. 64 18 (15) 4 H. Draconis 4.6 7 12 78 13.2 N. 35 56 (16) Level the eye and ear method with the transit instrument of the Detroit Observatory, to determine the correction to side- real chronometer Negus no. 721. The stars were selected from the list in the Berliner Astronomisches Jahrbuch, pp. 148 PRACTICAL ASTRONOMY 190-327. For convenience in referring to them in the re- ductions they are numbered, together with the level obser- vations, in the first column. Their magnitudes are given in the third column. The "Setting" is the reading at which the circle is to be set for observing each star. The circle of this instrument reads zero when the telescope points to the zenith and the degrees are numbered in both directions from the zero. The setting is therefore the zenith distance. The level observations and their reductions are (1) (7) (8) (13) (16) W. E. W. E. W. E. W. E. W. E. 15.1 9.1 15.1 9.3 15.3 9.0 14.3 10.3 14.6 10.1 12.0 12.1 13.7 10.8 8.6 15.9 11.0 13.5 11.0 13.5 12.1 12.1 13.6 10.8 9.0 15.4 10.7 13.9 11.1 13.5 15.2 8.9 14.8 9.5 15.1 9.4 14.0 10.5 14.4 10.1 B' + 1 .525 42 16.8 B -1.736 4> 42 17 B +1.000 sin (<£ + 8) 9.9496 C -3.813 sin (<£ — 8) 9.5576 C +1.072 cos (<£ + 8) 9.6582„ cos (<£ — 8) 9.9697 sec 8 0.5813 sec 8 0.0302 The apparent right ascensions are taken as accurately as possible from the Jahrbuch. We are now prepared to 150 PRACTICAL ASTRONOMY fill in the columns A, B, C, Mate, c'C, bB and a; after which we can determine a, and thence aA and 0'. Star A B C Bate c'C bB aA «' a A9 Wt, s s s 8 h m 8 h m a m 8 (2) + 3.895 -1.786 -3.813 -.OS -0.37 -.24 -1.86 10 49 48.39 11 4 20.01 + 14 36.62 (3) + .887 + 1.000 + 1.072 -.07 + .11 + .15 - .15 58 43.08 8 19.66 36.58 2 (4) + .179 + 1.187 + 1.200 -.06 + .12 + .19 - .07 58 0.22 12 36.77 86.55 2 (5) + .587 + .818 + 1.007 -.05 + .10 + .18 - .23 11 55.29 15 81.78 86.49 2 (6) -1.353 + 2.582 + 2.915 -.02 + .29 + .49 + .54 10 22.90 24 59.52 86.62 1 (9) - .160 + 1.497 + 1.506 + .01 - .19 + .09 + .06 25 42.69 40 19.26 86.57 2 (10) + .472 + .922 + 1.036 + .02 - .18 + .06 - .16 28 54.28 43 80.82 86.59 2 (11) + .642 + .768 + 1.001 + .03 -8.95 + .06 - .22 SO 25.27 45 1.77 86.50 2 (12) - .857 + 1.676 + 1.714 + .08 - .22 + .13 + .12 88 81.12 48 7.67 86.55 1 (14) + .972 + .468 + 1.078 + .07 - .14 + .05 - .84 49 55.30 12 4 81.81 86.51 1 (15) -2.875 + 8.966 + 4.898 + .08 - .68 + .48 + 1.00 52 85.07 7 11.58 + 14 86.51 Using stars (2) and (6) to determine a we have a, = ll» 4-»20 , .01, 0, = 10 49 44 .75, ^-0, = + 14 35.26, A 1 = + 3.395, a 2 = ll»24™59'.52, 2 = 11 10 22 .36, a 3 -0 2 = + 14 37.16, A 2 = - 1.353; and therefore, from (251), a — — S .400. Similarly, from (14) and (15) we obtain a = — S .348. Using these as the values of a for clamp west and east respectively, we form the column aA. All the corrections have now been computed. Substituting them in (233) for each star, we obtain the values A#. Stars (2) and (15) were observed solely to determine a, and the values of A# furnished by them will be given a weight 0, in the last column. Assigning a weight 2 to the stars which culminate near the zenith and between the zenith and equator, and a weight 1 to those outside these limits, for the reasons given in § 99, we obtain for the weighted mean of the chronometer corrections, A0 = + 14" 36'.55 ± 0«.009, which we shall adopt as the chronometer correction at the time 6 = 11" 20 m . DETERMINATION OF TIME 151 103. To illustrate the determination of e by the method of § 97, (c), Polaris was observed at lower culmination the -same night, as below. Polaris Level Clamp W V 12*52»> 6' Clamp W Clamp E " IV 12 57 51 WE WE Reversed 13.9 10.2 9.9 14.0 Clamp E in 13 3 22 12.0 12.0 11.4 12.5 IV 13 9 3 12.1 12.0 11.4 12.4 V 13 14 54 13.8 10.2 9.8 14.0 The intervals of time required for Polaris to pass from V to III and from IV to III are given by (225) first put- ting it in the form sin / = 15 sin 1" sec 8. i. The value of S was + 88° 43' 49". Substituting i 4 = 7 S .607 and i 5 = 15".294 successively for i in the formula, we find I t = 1° 25' 50" = 5™ 43«.3, I 6 = 2° 52' 37" = ll-» 30».5 ; and therefore the equivalent times of transit over III are Clamp W 12* 52"' 6« + 11" 30'.5 = 13* 3»* 36'.5 12 57 51 + 5 43 .3 = 13 3 34 .3 Clamp E 13* 3™ 22" = 13* 3 m 22». 13 9 3 - 5»43«.3 = 13 3 19.7 13 14 54 - 11 30 .5 = 13 3 23 .5 Taking the means for clamp west and clamp east, we obtain 0j = 13* 3-» 35«.4, 6 2 = 13* 3™ 2P.7. The level constants given by the above observations are V = + 0'.050, b" = -0'.096. Substituting these in (243) we obtain c = + 0".091, clamp west. 152 PRACTICAL ASTRONOMY REDUCTION BY THE METHOD OF LEAST SQUARES 104. In case the chronometer correction is required with all possible accuracy, the series of transit observations should be reduced by t^he method of least squares. Let us assume that the level constant, the rate and i m are accu- rately determined, and that the chronometer correction, the azimuth constant and the collimation constant are to be obtained from the observations. To avoid dealing with large quantities, lei_A0<; be an approximate value of A0, and x a small correction to A0 Q , such that A0 O + x = A0. (252) Further, let A0 O + 6 m + bB - O.021 cos C + i m C + (0 m - 6 B ) 80 - a = d. (253) The.n (233) takes the form aA ± cC + x + d = 0, (254) the lower sign being for clamp east. A value for A# having been assumed, all the terms in (253) are known for each star. Therefore, a, c and x are the only unknown quantities in (254). Each star furnishes an equation of this form, and their solution by the method of least squares gives the most probable values of a, c and x; and therefore, by (252), the most probable value of A0. 105. The accuracy with which the time of transit of a star over a wire can be estimated depends upon the power of the instrument and the declination of the star. As- sistant Schott of the Coast Survey* discussed a large number of observations, and found that the probable error of the observed time of transit over one wire is best repre- sented by £ = V (0.063) 2 + (0.036) 2 tan 2 S, for large instruments, £ = V (0.080) 2 + (0.063) 2 tan 2 8, for small instruments. * See U. S. Coast and Geodetic Survey Report for 1880. REDUCTION BY THE METHOD OF LEAST SQUARES 153 The values of e given in the table below are computed from these for the different values of 8. If 1 be the weight of an observation of an equatorial star, e its probable error, and p the weight of an observation of any other star we have, from theory, ■n — • For large instruments, e = (K063, and for small ones, e = (K080. Substituting the values of e in this equation we find the following values of p. s Large Instruments Small Instruments e P Vp e P Vp 0° ± 0«.G6 1.00 1.00 ± 0'.08 1.00 1.00 10 .06 1.00 1.00 .08 .98 1.00 20 .06 .98 1.00 .08 .92 .96 30 .07 .91 .95 .09 .83 .91 40 .07 .82 .90 .10 .70 .83 50 .08 .69 .83 .11 .53 .73 55 .08 .61 .78 .12 .44 ' .66 60 .09 .51 .71 .14 .34 .59 65 .10 .40 .63 .16 .26 .51 70 .12 .29 .54 .19 .18 .42 75 .15 .18 .43 .25 .10 .32 80 .21 .09 .30 .37 .05 .22 85 .42 .02 .15 .72 .01 .11 86 .52 .015 .122 .90 .008 .088 87 .69 .008 .091 1.21 .004 .066 88 1.03 .004 .061 1.82 .002 .044 89 2.06 .001 .031 3.70 .000 .022 90 co .000 .000 CO .000 .000 The observation equations (254) should be multiplied through by the square roots of their respective weights before forming the normal equations. (254) becomes Vp(aA ± cC + x + d) = 0. (255) 154 PRACTICAL ASTRONOMY In case some of the wires have heen missed, the weight is diminished. If we let N = the whole number of wires, n = the number of wires observed, 1 = the factor for an observation over the N wires, P = the factor for an observation over n wires, then the weight for an incomplete transit is pP. Assistant Schott found that we should use P = 5-3, for large instruments, 1+: 1 + P = n N 1+2 _o , for small instruments. (256) (257) The following table gives the value of P for reticles con- taining seven and five wires, for the different values of n. Large Instruments Small Instruments n P n P n P n P 7 1.00 5 1.00 7 1.00 5 1.00 6 .97 4 .94 6 .96 4 .93 5 .93 3 .86 5 .92 3 .84 4 .88 2 .73 4 .86 2 .70 3 .80 1 .51 3 .77 1 .47 2 .68 2 .64 1 .47 1 .43 106. We shall now apply these methods to the reduction of the transit observations in § 102. We shall assume A0 O = + 14™ 36 s .5. The values of m , bB, (6 m — O ) Bd, and a are obtained as before, and we shall use their values tabulated in § 102. To compute the terms Then S .021 cos + i m = = /.". SEDUCTION BY THE METHOD OP LEAST SQUARES 155 c" = - 0«.015 + 0'.003 = - 0'.012, for clamp west, c" = - .015 - .003 = - .018, for clamp east, c" = - .015 - 3 .827 = - 3 .842, for star (11). The products c"C are given in the table below. The value of d is found for each star by (253). The column Vp is taken from the table for the large instruments ; but for star (11), which is incomplete, the square root of the weight is found from pP. Star c"C d Vp (2) + 0».05 + 1'.66 0.43 (3) - .01 - .05 .99 (4) - .01 - .11 .93 (5) - .01 + .13 1.00 (6) - .03 - .98 .54 (9) - .03 + .03 .84 (10) - .02 + .18 1.00 (U) -3.84 + .33 .97 (12) - .03 + .02 .79 (14) - .02 + .45 .99 (15) - .09 - .47 .34 Substituting the values of A, O, d and Vp in (255), N being careful to change the sign of the c term for clamp east, we have the weighted observation equations + 1.462 a - 1.640 c + 0.43 x + 0.714 = 0, + .383 + 1.061 + .99 - .049 = 0, + .166 + 1.116 + .93 - .102 = 0, + .587 + 1.007 + 1.00 + .130 = 0, - .731 + 1.574 + .54 - .529 = 0, - .134 - 1.267 + .84 + .025 = 0, + .472 - 1.036 + 1.00 + .180 = 0, + .623 - .971 + .97 + .320 = 0, - .282 -1.354 + .79 + .016 = 0, + .962 - 1.067 + .99 + .445 = 0, — .977 -1.696 + .34 — .160 = 0. (258) 156 PRACTICAL ASTRONOMY The normal equations formed from these are + 5.780 a - 2.278 c + 2.714 x + 2.331 = 0, -2.278 +18.020 -2.505 -2.807 = 0, (259) + 2.714 - 2.505 + 7.689 + 0.918 = 0. . Their solution gives a = - 8 .383, c = + 0M15, x = + 0».053 ; and therefore A0 = A0 O + x = + 14-» 36».5 + 0'.053 = + 14"* 36«.553. The weights of the quantities just determined are p a = 4.71, p c = 16.80, p x = 6.29. Substituting the values of a, c and x in (258), we obtain the residuals Vpv, - 0.012, - .021, + .011, + .074, - .089, - .024, - .067, + .021, + .010, + .007, + .001. The sum of the squares of these is 2pvv = 0.0134. The probable error r x of an observation of weight unity is given by where m is the number of observation equations, and q is the number of unknown quantities. In this ease m = 11 and q = 3. Therefore r t = ± S .028. The probable errors of the unknowns are given by = -£=• (261) r« = ± O'.Oll, Vp a r c — , — i r x Therefore fa and = ± 0-.013, r c = ± 8 .007, a - = - 0'.383 ± 0'.013, c - = + 0M15 ± 0'.007, A6-- = + 14™ 36'.553 ± O'.Oll. PERSONAL EQUATION 157 CORRECTION FOR FLEXURE 107. In the broken or prismatic transit instrument (§ 90), a correction for flexure due to the bending o£ the axis must be applied. The effect of the flexure is to change unequally the positions of the eyepiece and objec- tive, which is the same as changing the inclination of the axis. It can therefore be allowed for by changing the measured inclination b, using b +f for clamp west, b — f for clamp east, / being the coefficient of flexure, and the eyepiece being on the clamp end of the axis. It requires special apparatus to determine / directly, so that unless its value for a particular instrument has been well determined, it is best to reduce all the transit obser- vations by the method of least squares, inserting another unknown quantity/, thus : Vp(aA ±/B±cC + x + d) = 0. (262) PERSONAL EQUATION 108. It generally occurs that two observers differ appre- ciably in their estimates of the time of transit of a star over a wire. Some observers acquire the habit of noting a transit too early, while others note it too late. To illus- trate, if a star actually transits at 9 S .5, one observer may note it systematically at 9\7, whereas another may note it systematically at 9 s . 2. An observer's absolute personal equation is the quantity which must be applied to his observed time of transit to produce the actual time of transit. The relative personal equation of two observers is the quantity which must be applied to the time of transit noted by one observer to produce the time noted by the other. The personal equation arises from the observers' habits of 158 PRACTICAL ASTRONOMY observation, and under uniform conditions may be regarded as sensibly constant for short periods of time. The relative personal equation of two most skilful observers, Bessel and Struve, was zero in 1814, but in 1821 it had increased to S .8 and in 1823 to l'.O. An observer's absolute per- sonal equation will depend very considerably upon the circumstances under which he observes. It will in general be different for observations made with a chronograph and for those made by the eye and ear method ; for those made with a clock beating seconds and with a chronome- ter beating half -seconds ; for large and for small instru- ments ; for equatorial and for circumpolar stars ; for bright and for faint stars ; for stars and for the moon's edge ; for different positions of the observer's body ; for the observer's different degrees of fatigue; and for other variable cir- cumstances. It is seldom that an observer's absolute personal equa- tion exerts an injurious effect upon results obtained in completed form from his own observations. But when results obtained by two observers are to be compared or combined, it is often essential that their personal equation be eliminated. The relative personal equation of two observers A and B may be determined by one of many methods. (a) Let A observe the transit of a star over the first three or four threads of a transit instrument, and B its transit over the remaining threads. For a second star let the observers alternate, B observing the transits over the first threads, and A over the last threads. When twenty-five or more stars have been observed, let the observations of each be reduced to the corresponding times of transit over the middle wire, by equation (226). The difference of times thus obtained for the two observers will be their relative personal equation. The objection to this method is that the observers are liable to be unduly hurried in exchanging positions at the eyepiece. DETERMINATION OF LONGITUDE 159 (J) Let A observe a star's transit over all the threads as usual. Let a second star's transit be observed by B as usual. In this manner let the observers alternate until each has observed a long and well selected list of stars for determining the clock correction. Let each reduce his observations as usual. The difference of their clock cor- rections will be their relative personal equation. (c) Various personal equation machines have been de- vised for measuring personal equation. In these an artifi- cial star is made to cross a field of view arranged with a reticle just as in the transit instrument, and the observer notes the times of transit in the usual manner. The actual times of transit are recorded automatically by an electrical device. The difference of the times determined in the two ways is the observer's absolute equation, pro- vided the machine has no personal equation in making its automatic record. At airy rate, the difference of the results thus obtained for two observers is their relative personal equation. The original programs of observation should always be arranged, if possible, with reference to the direct elimi- nation of the personal equation. For example, in the case of longitude determinations, the personal equation of the observers is eliminated by their exchanging places when the program of observations is half completed; and similarly in other cases. DETERMINATION OP GEOGRAPHICAL LONGITUDE 109. The accurate determination of the difference of longitude of two places requires the accurate determination of the time at each place and a method of comparing these times. One of the following methods of comparison is generally employed. (a) By Transportation of Chronometers. Let the eastern place be E, the western place W, and the difference of their 160 PRACTICAL ASTRONOMY longitude, L. Determine the correction &6 e and the rate 80 of a chronometer at E, at the chronometer time 6 e . Carry the chronometer to W, and there determine its cor- rection A0 W at the chronometer time 6 W . Then 6 W + A0„ = correct time at W at chronometer time 8„; w + A0 e + 80 (0«, - e ) = correct time at E at chronometer time 0„. Their difference is L = A0„ + 80 (6„ - 0.) - A0„. (263) The rate of the chronometer during transportation gen- erally differs from its rate when at rest. The change may be eliminated largely by transporting it in both directions between E and W. The rate is also a function of the tem- perature and the lubrication of the pivots. It has been found that the rate m at any temperature $ can be repre- sented by the formula m = m + k (i? - # ) - k't, (264) in which i? is the temperature of best compensation, m the rate at that temperature with t = 0, t the time meas- ured from that instant, k the temperature coefficient and k' the lubrication coefficient. By determining m , k, $ and k' for each chronometer, keeping a record of the tem- perature during transportation, and transporting several chronometers in both directions, the method yields good results. It should never be employed, however, except when the telegraphic method is impracticable. _ (6) By the Electric Telegraph. To illustrate the sim- plest application of the method first, let the observers at E and W determine their chronometer corrections. Next, let the observer at E tap the signal key of the telegraph line joining E and W simultaneously with the beats of his chronometer, and let the observer at W note on his chronometer the times of receiving these signals. In the DETERMINATION OP LONGITUDE 161 same way let the observer at W send return signals to the observer at E. 'Let e = correct time at E of sending signal, 0„ = correct time at Wot receiving signal, OJ = correct time at Woi sending return signal, 9 e ' = correct time at E of receiving return signal, jh = the transmission time. Then e + P — L = W , 6,'- l i.-L = Oj. Therefore L = i (0 e + OJ) -\{0 a + OJ), (265) P = i(0 w - OJ) -i(0 e - e >). (266) There are several small errors affecting the value of L obtained by this method of comparison, viz. : 1st. The personal equation of the observers in sending and receiving the signals ; 2d. The time required to close the circuit after the finger touches the key, and to move the armature of the receiving magnet through the space in which it plays — called the armature time. 3d. The personal equation of the observers in determin- ing the chronometer corrections, and errors in the right ascensions of the stars employed. These must be eliminated as far as possible in refined determinations. This is best done by a modification of the above method, called the method of star signals. One clock or chronometer, provided with a break-circuit, is placed in the circuit of the telegraph line, and at each station a chronograph and the signal key of a transit instrument are placed in the same circuit. The same list of stars is observed at both places, thereby eliminating errors in the right ascensions. When the first star crosses the wires of the transit instrument at E, the observer makes the records on both chronographs by tapping his 162 PRACTICAL ASTRONOMY key. When the same star reaches the meridian of W the observer there makes a similar record on both chronographs, and similarly for the other stars. The observers must also make suitable observations for determining the constants of their instruments and the rate of the clock. Let $ e = the clock time when a star is on the meridian of E, from the chronograph at E, OJ = the same, taken from the chronograph at W, d w = the clock time when the same star is ore the meridian of W, taken from the chronograph at E, QJ = the same, taken from the chronograph at W, e = the absolute personal equation of the observer at E, w — the absolute personal equation of the observer at W, 86 = the correction for rate in the interval 6 a — e . Then 0,„ + 80 + w - ju, - L = 6. + e, 6J + 8(9 + w + ^ - L = 6J + e. Therefore L = I (d„ + 6J) -$(&„ + e ') +W + w-e, which we may write L = L 1 + w-e. (267) If now the observers exchange places and repeat the observations we shall obtain L = i 2 + e - w, (268) provided their relative personal equation has not changed. Therefore, L = i (£, + £,)• (269) Great care must be taken in arranging the circuits to insure that the electric constants are the same at both stations. This condition can be secured by means of a rheostat and galvanometer placed in the circuit at each station. If there is any doubt as to the equality of the constants, any difference in the armature times at the two stations may be eliminated by exchanging the electrical DETERMINATION OF LONGITUDE 163 apparatus, along with the observers, at the middle of the series. If the above conditions are realized, the resulting longi- tude will be free from all errors except the accidental errors of observation. The method of star signals requires the exclusive use of the connecting telegraph line for several hours on each observing night. If such an arrangement is impossible, the observers must adopt some practicable method. Thus, if the telegraph line can be used only a few minutes each night, a set of adopted signals can be sent back and forth in such a way as to be recorded on both chronographs. The time at the two stations having been accurately deter- mined, from a carefully selected list of stars, the results obtained by this method are nearly as accurate as those obtained by star signals. The clock or chronometer should never be placed directly in the circuit joining the two stations, as the cur- rent would generally be strong enough either to change its rate or to injure its mechanism. It should be placed in a local circuit of its own, with a current just sufficient to work a relay connecting it with the main circuit. If so desired, a clock or chronometer may be connected with the circuit at each station, so that the beats of both will be recorded on both chronographs. It is the custom of the Coast Survey to determine longi- tudes from observations and signals on ten nights, the observers exchanging places at the middle of the series. (c) By the Heliotrope. In mountainous regions the telegraphic method of determining longitudes is usually unavailable. The difference of longitude of two points in sight of each other can be determined from heliographic signals. The necessity for sending signals in both direc- tions and for the observers exchanging stations will be obvious. The equations involved will be similar to those in method (J). 164 PRACTICAL ASTRONOMY ( i s T ™ : — The sidereal interval is therefore 60 — Aa - 60 " 164 Let iff = 60 - 164 60.164- Aa 60.164 - Aa DETERMINATION OP LONGITUDE 165 The values of log M can be taken from the following table : Ac. log M Ao log M An log M Ac log M 1«.65 0.0121 l s -95 0.0143 2«.25 0.0166 2».55 0.0188 1.70 .0124 2.00 .0147 2.30 .0169 2.60 .0192 1.75 .0128 2.05 .0151 2.35 .0173 2.65 .0196 1.80 .0132 2.10 .0154 2.40 .0177 2.70 .0199 1.85 .0136 2.15 .0158 2.45 .0181 2.75 .0203 1.90 .0139 2.20 .0162 2.50 .0184 2.80 .0207 The "sidereal time of semidiameter passing meridian" is tabulated in the American Ephemeris, pp. 385-392. Let 8 represent it. The right ascension of the moon's center when on the meridian is equal to the observer's sidereal time 0, and is given by a = = 6 m + A0 + tM±S, (271) the upper or lower sign being used according as the west or east limb is observed. Example. The moon's east limb and seven stars were observed with the transit instrument of the Detroit Observatory, Saturday, 1891 May 23, to determine the longitude. The star transits gave A0 = + 15™34».93 at chronometer time 16»13™, a = - 0».360, 6 = + 0.674, c' = + 0.100. The mean of the observed times of transit of the moon's second limb over the five wires was 6 m = 16* 12™ 45'.60. The moon's geocentric declination = — 22° 3', Parallax = 51 , The moon's apparent declination = — 22 54 . 166 PRACTICAL ASTRONOMY Therefore, A=+ 0.985, B = + 0.455, (7= + 1.085; and r = S .04 = tM. From the Ephemeris, p. 388, S= l m 9 S .90. Therefore a = 6 = 16 s 12™ 45 8 .60 + 15™ 34 8 .93 + 8 .04 - 1™ 9 8 .90 = 16* 27™ 10-.67. From the Ephemeris, p. 83, the right ascension at Green- wich mean time 18" was 16* 27 m 20 s .32. The difference, 9 s . 65, corresponds to a difference in time of ahout 4 m . The average increase of right ascension per minute during this interval was 2 S .3058. The exact value of the interval before 18" is '9.65 -h 2.3058 = 4 m .185 = 4™ 11M0. The Greenwich mean time corresponding to the observed value of a was therefore 17* 55™ 48 s .90. The equivalent sidereal time was 22* 2 m S .38, and the longitude of the observer was L = 22* 2™ 0\38 - 16* 27™ 10«.67 = 5» 34™ 49'.71. Longitudes obtained by this method can be regarded only as approximately correct, for two reasons : 1st. An error in the observed right ascension introduces an error — times as great in the resulting longitude ; 2d. The tables of the moon's motion are imperfect, and the tabulated right ascensions may be slightly in error. This would introduce an error about — as great in a resulting longitude. The above example should be reduced anew when the corrections to the moon's right ascensions for 1891 are published. CHAPTER VIII THE ZENITH TELESCOPE 110. When a sensitive spirit level at right angles to the rotation axis, and a micrometer with wire moving parallel to the axis are added to the transit instrument, it becomes a zenith telescope. The level is called a zenith level. The transit instrument and the zenith telescope are fre- quently combined in this way, as shown in Fig. 20. DETERMINATION OP GEOGRAPHICAL LATITUDE 111. The zenith telescope is specially adapted to deter- mining the latitude when great accuracy is required. The method employed is known as Talcott's method. It con- sists in measuring the difference of the zenith distances of two stars, one of which culminates south of the zenith and the other north of the zenith. The difference of their zenith distances should not exceed half the diameter of the field of view, to avoid observing near the edge of the field. The difference of their right ascensions should not exceed 15 m or 20 ro , to avoid any change in the constants of the instrument between the two halves of the observation ; nor should the difference be less than 2 m or 3™ to avoid undue haste. The zenith distances should never exceed 35°, to avoid uncertainty in the refractions. To prepare the observing list, an approximate value of the latitude must be known. This can be found from a map, or from a sextant meridian double altitude (§§ 84-87). 167 168 PEACTICAL ASTRONOMY Letting the primes refer to the southern star and the seconds to the northern star, we have 8' 8": *. Therefore + 2". 8' + 8" = 2 + (z" - z'), (272) (273) (274) which is the condition that the two stars of the pair must fulfill. Thus, in latitude 42° 17', and with an instrument whose field of view is 40' in diameter, we must have two stars such that 8' + 8" is greater than 84° 14' and less than 84° 54'. A pair is given below which meets these require- ments. The " Setting " is the mean of the zenith distances. The assumed latitude is 42° 17'. Star Mag. Apparent a 8 z Setting k Ursce Majoris 38 Lyncis 3.3 4.1 8»56™12 s 9 12 5 + 47° 35' 37 16 N. 5° 18' S. 5 1 N. 5° 9' S. 5 9 Care must be taken, in forming the observing list, to employ only those stars whose declinations are well deter- mined. To observe the first star, the circle to which the zenith level is usually attached is made to read the " Setting," the telescope is rotated until the bubble moves to the middle of the tube, and the micrometer wire is moved to the part of the eyepiece where it is known the star will pass. Thus, in the pair above, it is known that the first star will cross 9' [ = 5° 18' — 5° 9'] above the center. When the first star culminates, or within a few seconds of culmi- nation, bisect the star by the micrometer wire, and read the zenith level and the micrometer. Reverse the instrument without jarring it, bring the bubble to the center of the level again, and observe the second star in the same way as the first. It is sometimes preferable not to clamp the DETERMINATION OF LATITUDE 169 instrument during the observations. Care must be taken not to change the position of the level with respect to the line of sight during the progress of an observation ; the angle between the two must be preserved. Let m be the micrometer reading on any point of the field assumed as the micrometer zero ; z the apparent zenith distance corresponding to m when the level bubble is at the center of the tube ; m', m" the micrometer read- ings on the two stars, the readings being supposed to increase with the zenith distance ; _R the value of a revolu- tion of the micrometer screw; V, b" the level constants for the two stars, plus when the north end is high ; r', r" the refractions for the two stars. Then the true zenith distance of the southern star is given by z'=z + (m'-m^R + b' + r'; and of the northern star z"= z + (to"- m )R - b"+ )-". Substituting these in (274) and solving for , we obtain 4, = 4 (8' + 8") + J (to' - m")R + 4(6' + 6") + 4 (W - r"). (275) If the micrometer readings decrease for increasing zenith distanced the sign of the second term is minus. In case the zero of the level scale is at the center of the tube, 4 (V+ 6") = \ [(n'+ »") - (sj+ si-)-] d, (276) in which n', n", «', s" are the level readings for the two stars, and d is the value of a division of the level. In case the zero of the level scale is at one end of the tube, 4 (6'+ 6") = \ [± (n'+ *') T (»"+ «")] d, (277) the upper sign being used when n' is greater than s', the lower when n' is less than s'. 170 PRACTICAL ASTRONOMY The refraction correction is small, and can be computed differentially by the formula £ If for any reason the star cannot be observed at the instant of culmination, the bisection may be made when the star is at some distance from the center of the field, the time of observation being noted. The polar distance of every star observed in this way will be too small." A slight correction, called the reduction to the meridian, must be applied. Let x represent it; and let t be the distance of the star from the meridian when it was ob- served, in seconds of time. In the right triangle formed by the meridian, the star's declination circle, and the micrometer wire projected on the sphere, we have the side 90° — S and the angle t at the pole, to find the side 90° — (6" ± x). We can write cot (S ± x) = cos t cot 8. Expanding and solving for tan x, , (1 — cos t) sin 8 cos 8 sin 2 8 4- cos t cos 2 8 We can put the denominator equal to unity without sensi- ble error, since t is always small. Therefore tana; = ± 2 sin 2 £« sin 8 cos 8 = ±$sin2S ■ 2sin 2 £<; or, * = ±sin2 8 s -i^; (281) sin 1" the lower sign being used for stars observed near lower culmination. 172 PRACTICAL ASTRONOMY The correction to the observed latitude will always he^x. If both stars of the pair are observed off the meridian, there will be two such terms to apply. The values of x are tabulated below with the arguments B and t. Values op x \t o\ 5» 10» ''.00 15« ".00 20 s ".00 25" ".00 30" ".00 35« ".00 40« ".00 45 ".00 50 s ".00 55« ".00 60« ".00 t/ 0° ".00 90° 5 .00 .00 .01 .02 .03 .04 .06 .08 .10 .12 .14 .17 85 10 .00 .01 .02 .04 .06 .08 .11 .15 .19 .23 .28 .34 80 15 .00 .01 .03 .05 .09 .12 .17 .22 .28 .34 .41 .49 75 20 .00 .02 .04 .07 .11 .16 .22 .28 .36 .44 .53 .63 70 v25 .01 .02 .05 .08 .13 .19 -.26 .34 .42 .52 .63 .75 65 30 .01 .02 .05 .09 .15 .21 .29 .38 .48 .59 .71 .85 60 35 .01 .03 .06 .10 .16 .23 .31 .41 .52 .64 .77 .92 55 40 .01 .03 .06 .Ji- -07- ~£4 .33 .43 .54 .67 .81 .97 50 45 .or .03 .06 ll .17 .25 .33 .44 .55 .68 .82 .98 45 112. The adjustments for the transit instrument, § 100, apply equally well, for the most part, to the zenith tele- scope. Special forms of the instrument, however, will call for special methods, which the intelligent observer will easily devise. The micrometer wire must be made perpendicular to the meridian. If this adjustment is perfect, an equatorial star will travel on the wire throughout its entire length. 113. Example. The following observations were made with the zenith telescope of the Detroit Observatory, Mon- day, 1891 March 16. Star Chronometer Micrometer Level n s k UrscB Majoris 38 Lyncis 8* 40" 36« 8 56 10 13.647 37.359 8.9 39.6 35.7 12.4 Required the latitude. -') DETERMINATION OF IaTITUDE 173 The chronometer correction was + 15™ 47 s . The value of one revolution of the micrometer screw is i2=45".042. The value of one division of the level is d = 2".74. The mean places of the stars are given in the Jahrbuch, p. 180. Their apparent places are found by the methods of § 55 to be a" = 8" 56™ 12», 8" = + 47° 35' 21".33, a' = 9 12 5, 8' = + 37 15 53 .05. Therefore J (8' + 8") = 42° 25' 37".19. The micrometer readings decreased with increasing zenith distances. Therefore - £ O' - to") R = - 11.856 R = - 8' 54".02. The zero of the level was at one end ; therefore, by (277), I (V + b") = \ (52.0 - 44.6) d = + 5".07. The half difference of the zenith distances is 8' 54", and the mean zenith distance is z = 5° 9'. Therefore, from the table for differential refraction, i (r- - r") = - 0".16. The first star was observed at an hour angle t = + 11 s ; therefore, from the table, the value of J x for the northern star is £z"= + 0''.02. The second star was observed at the hour angle t = — 8 s ; therefore the value oi\x for the southern star is £x'= + 0".01. Combining the terms of (275) and the reductions to the meridian, we obtain 4> = 42° 16' 48".ll. [The known value of the latitude is about 42° 16' 47".3.] 114. In very accurate determinations of the latitude, a number of pairs of stars should be observed several times 174 PRACTICAL ASTRONOMY in this way, and the results combined by the method of least squares. If we let <£ , R and d be veiy nearly the true values of <£, B and d, and let A, AB and Ac? be slight corrections to <£ , B and d , each observation fur- nishes an equation of the form <£ + A<£ = l(8> + 3") + i(m' - «") («o + Ai? ) + i[±( n ' + s')T(n" + *")]( d o + A <9 + K»" - r ") + **' + i x "- Let £ = <£ - i(8' + 8") - i(m' - m")R - il±(n',+ s')T{n" + 8»)-]d„- l(r' - r")- lx -\x". (282) Then A<£ - \ (m> - m") Ai? - \ [ ± (n' + «') =F (n" + *")] Ad + A = 0, (283) is an observation equation for determining A, AB and Ad. Thus, in the example above, if we assume O =42° 16' 47".0, i£ = 45".040, d = 2".70, we find k = - 1".06, and (283) DGCOU16S A<£ + 11.856 AiJ - 1.85 Arf - 1.06 = 0. (284) Forming the corresponding equations for the other pairs observed and solving by the method of least squares, the most probable values of A<£, AB and Ad, and therefore of $, R and d, are obtained. However, it has recently been shown that the latitude of a place varies appreciably, sometimes in the course of a few weeks ; and latitude observations, to be combined by any direct method, must be made inside of a few days and the result be taken as the latitude at the mean of the observation times. CHAPTER IX THE MERIDIAN CIRCLE 115. The Meridian Circle consists essentially of a transit instrument with a graduated circle attached at right angles to, and concentric with, the rotation axis. The graduated circle rotates in common with the telescope, and is read by- reading microscopes firmly attached to one of the support- ing piers of the instrument. The best instruments are provided with two graduated circles. One of these circles usually remains fixed on the axis for an indefinite time; whereas the other is movable, and many observers are accustomed to rotate it through any desired angle [and clamp it firmly to the axis] from time to time. An excellent form of the meridian circle is illustrated, with many details omitted, in Fig. 25. Two massive supporting piers extend down to solid earth or rock foun- dation ; and, as in the case of all telescope piers, are com- pletely isolated from the floor and building. The circular drum on each pier carries the four long slender reading microscopes for reading the graduated circle, on which they are focused. The pivots rest in V's attached to the inner head plates of the drums. The counterbalance levers are shown on the tops of the drums. A lifting arm de- scends from the inner end of each lever and rests, with roller bearings, on the under side of the axis. The chain descending from the outer end of the lever, through a hole in the pier, carries a counterweight. Nearly the whole weight of the instrument is supported in this manner, leav- ing only a small residual weight to be borne by the V's. 175 Fig. 25 THE MERIDIAN CIRCLE 177 The level is in position on the instrument, suspended from the pivots. It is visible immediately under the circles. The eyepiece is supplied with the usual number of verti- cal threads, and with vertical and horizontal micrometer wires. A basin of mercury, not visible in the cut, is mounted below the level of the floor, immediately under the center of the instrument, on a pier isolated from the floor. The small telescope showing just below the ob- jective of the lower left reading microscope is the " setting telescope," used for setting the instrument at any desired circle reading. The auxiliary apparatus shown is, the observing chair in the foreground, the adjustable mercury basin for reflection observations, and the reversing carriage in the background. The field of view and the wires are illuminated by light from a lamp at the side of the room, shining through the hollow axis, as in the case of the transit instrument. A system of small mirrors receives light from.the same source and reflects it where it is needed for reading the circles and setting the telescope. The graduated circles are about two feet in diameter, and the graduations are two minutes of arc apart. The micrometer head of each microscope is divided into 60 parts, each division corresponding to 1". The divisions may be sub- divided, by estimation, into 10 parts, each part being 0".l. The meridian circle is used principally to determine the accurate positions of the heavenly bodies on the celestial sphere; i.e., their right ascensions and declinations. It is further adapted to determining the time and the geo- graphical longitude by the methods of Chapter VII, and to determining the geographical latitude. 116. The determination of right ascensions. The princi- ples involved in this problem have been treated in Chap- ter VII. The stars whose right ascensions are to be determined (which we shall call undetermined stars), are placed in an observing program which includes a 178 PRACTICAL ASTRONOMY considerable number of stars whose positions are accurately known (which we shall call standard stars), and which are suitable for determining the azimuth of the instrument, and the time. The transits of all the stars, both unde- termined and standard, are observed, the constants of the instrument are determined, and all the observations are reduced in the usual manner. The clock correction is determined from the standard stars, as usual. The right ascensions of the undetermined stars are found by means of equation (233), which may be written a = 6 m + A0 + aA + bB + c'C + (6 m - ) 89. (285) In forming the observing program, the undetermined stars should be preceded, accompanied and followed by standard stars. Likewise, the declinations of the standard stars should be nearly the same as, or at least should in- clude, the declinations of the undetermined stars, thereby eliminating largely the uncertainties or progressive changes in the instrumental constants. Example. In the example of § 102, let it be assumed that the star (3) 8 Leonis and star (10) /3 Leonis are undeter- mined stars, and that the remaining nine stars of the list are standard stars. Required the right ascensions of stars (3) and (10). The value of the chronometer correction from the nine standard stars is A0 = + 14™ 36 8 .54. The right ascension computations may i now be tabulated, as below. Star (3) 8 Leonis Star (10) B Leonis 8m 10'>53™43 8 .04 : LP2S™54».44 Ad + 14 36.54 + 14 36.54 aA 0.15 — 0.16 LB + 0.15 + 0.06 c'C + . 0.11 — 0.13 ce m - O ) 80 - .07 + 0.02 U 11 8 19.62 11 43 30.77 THE MERIDIAN CIRCLE 179 117. Declinations and the latitude are determined from observations which involve readings of the graduated circle. The method of reading the circle by reading microscopes, and correcting for error of runs, is given in § 58. In modern instruments, provided with eyepiece micrometers moving in declination, the error of runs may and should be practically eliminated by a suitable method of observing. To illustrate, if it is the observer's custom to read each microscope on only one graduation, the telescope should be directed, by means of the setting telescope, so that the graduation to be set on will always fall in the same posi- tion with reference to the zero of the microscope for all the observations of a series. Again, if it is the observer's custom to read each microscope on two adjacent gradua- tions, these should always fall in the same positions with reference to the micrometer zero, the two graduations being, preferably, on opposite sides of the zero. Knowing the approximate position of the star to be observed, its " setting " — usually zenith distance — can be computed to the nearest even minute, and the instrument set for that reading. As the star crosses the middle thread in the eye- piece, its distance from the zero position of the micrometer wire is measured with the declination micrometer. The microscope readings on the graduations are then secured, and later the declination micrometer is read. In reading the circle it is customary to take the degrees and even minutes from the circle as seen in one of the microscopes, and the seconds and fractions from the mean of the four microscope readings. The reading thus obtained must be corrected for runs, for flexure, for the distance of the declination micrometer wire from its zero position, and possibly for errors of the graduations. 118. The zero reading of the micrometer may be obtained from nadir observations [§97, (i)]. Let the observing telescope be directed vertically downward to the mercurial 180 PRACTICAL ASTRONOMY basin. Obtain the micrometer readings when the wire is on each side of its reflected image, at minute and equal distances. The mean of the two readings is the reading for coincidence of the wire and its image, and is the zero reading of the micrometer. Let the microscopes be read for this position of the instrument, and corrected for runs and graduation error. The result is the nadir reading of the circle. The nadir reading plus 180° is the zenith reading. Many of the older forms of meridian circles are not pro- vided with declination micrometers, but have two hori- zontal fixed wires marking the center of the field, as shown in Fig. 19. In this case, when a star is crossing the middle transit thread, the entire instrument is moved by a slow motion screw until the star travels midway between the two horizontal wires. The microscope readings may thus have any value up to 2', and the correction for runs must be carefully determined. Again, some instruments have a single horizontal fixed wire. 119. Determination of the value of one revolution of the micrometer screw. The method of § 61, (a), is applicable, but a better method is the following : Direct the observ- ing telescope to one of the collimators (described in § 97), so that the image of the horizontal wire -of the collimator falls about half a radius above the center of the field. Determine the micrometer reading when the micrometer wire is coincident with the image of the collimator wire, and read the circle. Rotate the instrument so that the collimator image moves to the opposite side of the field of view, and again determine the corresponding micrometer and circle readings. The difference of the circle readings divided by the difference of the micrometer readings is the value of a revolution of the screw. If a movable circle is available, several different arcs may be used for this purpose, thereby eliminating very largely the effect of graduation errors. THE MERIDIAN CIRCLE 181 120. Eccentricity of the circle. As explained in § 59, the effect of eccentric mounting of the circle is eliminated by the use of two or more equidistant verniers or reading microscopes. 121. Flexure. When the instrument is rotated from one position to another, the form of the observing telescope (and sometimes also the form of the graduated circle), is appreciably changed under the action of gravity. The bending of the telescope tube will do no harm provided the objective and the eyepiece are displaced the same amount, but a difference in their displacements changes the direction of the line of sight with reference to the circle graduations. This effect is called the flexure. In most modern instruments the flexure is very small, since the observing telescope is symmetrical with refer- ence to the rotation axis, and the mechanism at the eye end is made of the same weight as the objective and its cell. They are further designed so that the objective and eyepiece mechanism may be interchanged on the telescope tube. If we combine two observations of the same body, one made before interchanging the objective and ocular, and the other after interchanging them (the interchange involving at the same time a rotation of the telescope tube through 180°), the result will be free from flexure, theo- retically at least. The two collimators furnish a simple method of measur- ing the horizontal flexure, i.e., the flexure when the tele- scope is in a horizontal position. Let the horizontal threads of the collimators be brought into coincidence, as explained in § 97, (e). The two threads then represent two infi- nitely distant lines whose angular distance apart, measured through the zenith, is exactly 180°. Measure this distance in the usual manner. If there is no flexure, the difference of the circle readings should be exactly 180°. If any excess or deficiency exists, that excess or deficiency is 182 PRACTICAL ASTRONOMY twice the horizontal flexure, plus the accidental and un- avoidable errors of the observation. Example. Repsold Meridian Circle, Lick Observatory, Saturday, 1898 June 11, the following observations were made by R. H. Tucker for determining the horizontal flexure.. Circle east. Circle Reading on South Collimator .... 224° 56' 49".07 North Collimator set on South Collimator Circle Reading on North Collimator .... 44 56 48 .64 North Collimator set on South Collimator Circle Reading on North Collimator .... 44 56 48 .58 Circle Reading on South Collimator .... 224 56 48 .75 Mean Circle Reading, North 44 56 48 .61 Mean Circle Reading, South 224 56 48 .91 Difference, North-South 179 59 59 .70 The deficiency is 0".30 and the horizontal flexure, /, is 0".15. The sign of the flexure correction to the circle readings is readily found. As the telescope was turned from the south collimator through the zenith to the north collimator, the readings increased from 224° through 360° to 44°, and the measure of the angle is 0".30 too small. The correction to the circle reading for a star south of the zenith is minus, and for a star north of the zenith is plus. If the instrument were reversed, circle west, the signs of the corrections would be reversed. The mean value of /, resulting from 23 determinations by the same observer, extending through two years, is 0".065, but the value 0".l has been adopted, provisionally. Since the gravitational moment of any given mass in the telescope, with reference to the rotation axis, varies with the sine of the zenith distance of the line of sight, the general expression for the flexure is assumed to be Flexure = /sin z, (286) though it is not probable that the flexures in all instru- ments can be represented by this law. THE MERIDIAN CIECLE 183 The order of observation followed in the above example illustrates a general principle which should be taken into account, whenever possible, in forming programs of ob- servation with any instrument. The observations were made in one order, and repeated in reverse order, thereby eliminating largely any possible progressive changes in the apparatus. 122. Errors of graduation exist in all circles and affect the angles measured by them. Whether these errors are negligible, or must be taken into account, depends largely upon the degree of refinement exacted by the problem in hand. In the case of small instruments constructed by first-class makers, the errors of graduation will generally be smaller than the least reading of the instrument, and may be neglected. The circles provided by the best makers for modern meridian instruments are nearly perfect. It is seldom that one of the graduations is displaced as much as 1" from its theoretical position, or that the mean of four graduations 90° apart is in error by as much as 0".5. Nevertheless, it is necessary to investigate every such circle to determine the degree of refinement which it will impart to observations depending upon its readings, and to secure data for eliminating errors arising from its imper- fect graduation. The investigation of 10,800 graduations on a circle taxes the resources of most long established observatories so prohibitively that it is seldom or never carried to a finish. After the investigation has extended to all the graduations marking the degrees, or at the most to those marking the 20' divisions, the nature and magnitude of the systematic errors and the magnitude of the accidental errors will have been revealed, and further determinations may generally be confined to the gradua- tions which are used with special frequency, e.g., the graduations used in determining the nadir reading, or those used with particular stars, or zones of stars. 184 PRACTICAL ASTRONOMY It will be seen from the above that the problem is one for the professional astronomer and his assistants, rather than for the student. A complete solution is therefore not called for in this place, but an outline of one of the best methods may be given. Let us suppose that the instrument has a fixed and a movable circle, each read by four microscopes 90° apart. As an origin for the entire system of measures, let it be assumed that the mean of the readings of the four micro- scopes on the 0°, 90°, 180° and 270° lines is free from graduation error, in both circles. If now the axis of the instrument be rotated through a given angle, 30° for example, and the circle reading be taken, the observed angle will differ slightly from 30°, from several causes : first, the unavoidable errors of observation, which may be reduced materially by increasing the number of indepen- dent observations ; second, progressive changes in the ap- paratus, largely due to temperature variations, which may be reduced materially by repeating the observations back- wards ; third, differential flexure of the circle, which may be eliminated, it is assumed, by rotating the instrument on its axis through 180° and repeating the observations on the same lines ; and fourth, the graduation errors of the divisions used. These considerations suggest the principal features of the program of observations. Let it be required, first, to determine the division errors of the 45° points of both circles; i.e., the error for each circle affecting the mean of the readings obtained from the four microscopes on the points 45°, 135°, 225° and 315°. Place the 0° of the two circles in coincidence and read the microscopes for both circles. To increase the accuracy of the determination by increasing the number of observations, and at the same time eliminate the circle flexure, let these observations be repeated with the instru- ment rotated through one, two, three and four quadrants. Now let the 45° line of the movable circle be made coinci- THE MERIDIAN CIRCLE 185 dent with the 0° line of the fixed circle, and a series of readings similar to the above be secured. Next let the 90° division of the movable circle be made coincident with the 0° line of the fixed circle, and so on until a series has been secured with each 45° division of the movable circle in coincidence with the 0° of the fixed circle. In order to eliminate progressive changes in the instrument, as far as possible, let the above program of observations be re- peated in reverse order. The data will then be at hand for a thorough determination of the errors of the 45° divi- sions of both circles. Each arc of 45° on one circle has been compared with each such arc on the other circle. For example, the first 45° arc of the fixed circle has been used to measure each of the eight 45° arcs on the movable circle. The true sum of these eight arcs is 360°. If their sum measured by the fixed-circle arc differs from 360° by any quantity n, the relative division error of the mean reading of the fixed-circle microscopes on the 45° lines is \ n. Similarly, the 45° division error for the movable circle may be computed. The division errors of the 15° readings of the two circles may be obtained by subdividing and comparing the 45° arcs just determined, or from the complete circles, as before ; and so on for the 5°, 1° and other readings. When a circle has been investigated, the zero of the system may be changed arbitrarily by applying a constant to all the division errors secured, either to make them all of the same sign, or to make their algebraic sum zero. The 1° readings of the fixed circle, and the 3° readings of the movable circle of the Repsold instrument of the Lick Observatory, were investigated by the above methods by It. H. Tucker.* The average errors of the fixed and * For further and fuller details, see articles by Professor Tucker in Publications Astronomical Society of the Pacific, 1895, pp. 330-338, and 1896, pp. 270-272. Also an article by Professor Boss in The Astronomi- cal Journal, 1896, Nos. 382, 383. 186 PRACTICAL ASTRONOMY movable circle readings were ± 0".18 and ± 0".15, re- spectively. The following table contains the errors for the 9° readings, by way of illustration. Beading Fized Circle Movable Circle 0° + 0".18 + 0'UO 9 + .12 + .14 18 -0 .23 + .04 27 -0 .04 + .37 36 -0 .52 -0 .34 45 -0 .34 + .03 54 + .02 -0 .18 63 + .11 -0 .07 72 -0 .22 -0 .14 81 + .16 + .08 90 + .18 + .10 In case the instrument has only one circle, its errors may be determined by means of two extra microscopes, placed 180° apart, in connection with the four regular microscopes.* It should be explained that many observers shift the movable circle from time to time, so that the several observations of any star will depend upon many different graduations of the circle, thereby reducing the magnitude of the division error affecting the mean result. The flexure of the circles in modern instruments is so small that to apply a correction for it is generally more objectionable than to omit it. This, and similar small and uncertain corrections are not ignored, however. Good practice requires that a star's position be determined from an equal number of observations with circle west and circle east, with the result that several slight errors are largely eliminated from the mean of all the observations. * For an exposition of this method, see Annalen der Sternwarte in Leiden, Band II, seite [50-92]. THE MERIDIAN CIRCLE 187 123. Reduction to the meridian. Theoretically, the ob- server is supposed to bisect the image of a star with the declination micrometer at the instant of meridian passage. If for any reason the bisection is made at t seconds before or after meridian passage, the necessary correction x to reduce to the meridian may be found from equation (281) and the corresponding table in § 111. The correction x must be applied to the circle reading with the proper sign to increase the observed polar distance. 124. Refraction. The refractions given by (95) must be applied to the circle readings in such a way as to increase the zenith distances. 125. Parallax. Observations on bodies within the solar system will require correction for parallax, by the methods of §§ 25-27. 126. The meridian circle is applied to the determination of declinations by two general methods. 1st. Fundamental Determinations. The latitude of the observer being known, the equator reading of the circle is given by Equator reading = Zenith reading ± , (287) the lower sign being for circle east. The difference between the circle reading for a star (corrected for refrac- tion, etc.), and the equator reading, is the declination of the star, determined fundamentally. 2d. Differential Determinations. If a standard star be observed in the usual manner, its circle reading (corrected for refraction, etc.), plus or minus its known declination, will be the equator reading of the circle. The corrected circle reading for an undetermined star will differ from the equator reading by the declination of the star. Fur- ther, the equator reading obtained from the standard star, combined with the zenith reading, will furnish a new deter- mination of the latitude. 188 PRACTICAL ASTRONOMY A circumpolar star observed for latitude at both upper and lower culmination has the advantage that any error of declination is eliminated from the mean result ; but the disadvantage, for observers situated well toward the equa- tor, that the refractions are large. Since the latitude of a place varies appreciably, funda- mental determinations of declination require a knowledge of the current value of the latitude. Programs for funda- mental work should contain a few standard stars, as checks on each night's results. In programs for differential work, the undetermined stars should be preceded, accompanied and followed by standard stars ; the range of declinations for the two kinds being about equal, to assist in eliminating uncertainties in refractions. The equator reading should be obtained from all the standard stars. A program covering four or five hours should contain eight or ten standard stars. If the value of the latitude is known, a long series of such obser- vations of the standard stars will furnish corrections to the standard declinations themselves. The following abridged program of observations, made with the Repsold meridian circle of the Lick Observatory, illustrates many of the important principles treated in this chapter. The mean of the nadir and micrometer readings taken just before and after the observations of stars furnished the values Nadir reading = 134° 56' 29".82, Micrometer (zero reading) = 17 r .OOO. The meteorological observations required for computing the refractions were made at 15.2 hours sidereal time, thus : Barom. 25.70 inches, Att. Therm. + 64° F., Ext. Therm. + 63°.0 F. Other meteorological observations were taken throughout the program. THE MERIDIAN CIRCLE 189 a K o D h a pi - o Ed ►J « S5 «! a co M a H CO cm 3 rH CO CO ■* t- to © 00 O O ^ (M iN lO ob O © OS o o Ttl ia CO CD ** CN IN + • 1 rH r-l CM <3 C5 *— I CO >o rH -* CO © CQ iO CO o lO | CO CM rfS o CQ U3 o o d CO t-^ co' co' 6 „, CM IN I r-i ■* rH & & w rH CO uo rH CO CO CM H "5 iO rH o ■* 1 ■* CO *o ca * o 1 -tJ lO o 00 ■* CO CO t^ © 03 r-l 1 lO T— 1 CO (N CM CM CM 1 IN © O CO r- ■* CO -* -* CO rH o o CO CM lO CO IN lO CO 00 d „ CM IN I r-i lO & & St -* CD IN . i r-l o CO >o n ■* IN >o 00 1 CO CO 03 « 1 £ ° 03 CO -* CO CO t~ on CO rH 1—1 1 IQ T— 1 CO U3 o + 1 r-l O CN ^ IN ?- CO CO © 1 CO CO © o -* 1—1 ■* CO CO t^ t- r-l i— i 1 u- i-H CO CO IN CO CM CM CO CO IN CO rH CO >o CO © IN CD CO CO o o t- r-l lO CO a -." oi O CO co a> CO o o co © CO © ^ o IN CO 1 + CO + id CO CM O j> <" 1—1 o IQ ■* H* CO CM s * o ^ "O t— r- ?■■ r-l + CO CO CO CO CM CO SO M a -3 •3 c3 '•3 o fH U 03 a CD a M £ O -t-= CO T3" • a o a co u 03 Oh g (S 0) ■3 3 co 60 .a ■a o3 CO u u 1 ■§ el _o o cS CO o 'o ^1 CO CO o 03 P CD o CO TJ OJ t CO CO CS CO > u . (289) The value of b is found by (166) or (167), or by the methods of § 94. If the illuminated mark is not in the horizon, the circle readings on the mark must be corrected by (288), using its zenith distance z. ^ DETEBMINATION OP AZIMUTH 193 129. Correction for diurnal aberration. Owing to diur- nal aberration the star will be observed too far east. In the most refined observations this must be allowed for. The correction to the circle reading is given by (118); which, for circumpolar stars, is approximately dA = + 0».31cos4. (290) If the circles cannot be read to less than 1", this correc- tion is negligible. 130. Correction for error of runs. If reading microscopes are used [§ 58], the circle readings must be further cor- rected for error of runs. AZIMUTH BY A CIBCUMPOLAE STAE NEAE ELONGATION 131. A star is at western or eastern elongation when its azimuth is the least or greatest possible. It is then moving in a vertical circle, and is in the most favorable position for azimuth observations. Only one observation can be made at the instant of elongation, but it is customary to make several observations just before and after elongation, and allow for the change in azimuth during the intervals. At the instant of elongation, the triangle formed by the pole, star and zenith, which we shall denote by PSZ, is right-angled at the star. If we let 6 Q be the sidereal time, and A , t and z the azimuth, hour angle and zenith dis- tance of the star at elongation, a and 8 its right ascension and declination, and the observer's latitude, _ we shall have, for western elongation, PZ = 90°- <£, PS = 90°- 8, ZS = z , PZS = 180° - A , ZPS = t„, PSZ = 90° ; and for eastern elongation, PZS = A„ - 180°, ZPS = 360° - t . 194 PRACTICAL ASTRONOMY We can write tan A sin <4 . cos 8 /of)1 . tan 8 ° sin 8 cos

sin ± cos A = — sin 8 sin t a , (294) in which the upper signs are for western elongation, the lower for eastern. If the star is observed at any other hour angle t, its azimuth A is given by (16) and (17). Multiplying (16) by (293), (17) by (294), and subtracting one product from the other, we obtain sin z sin (A - A) = =p sin 8 cos 8 2 sin 2 J (« - t). (295) If the observations are made near elongation, t will not differ much from t , A — A will be small, and for the circumpolar stars z will not differ much from z . There- fore we can write, without sensible error, A _ A _ sin 8 cos 8 2sin 2 ^(g -Q , 2g6) sinz sinl" in which the lower sign is for eastern elongation, as before. A — A is the correction to be applied to the circle reading for an observation made at hour angle t, to reduce to the corresponding circle reading for an obser- vation made at hour angle t . For convenience, let and (296) becomes 2 gin' jQ, -Q m ~ shTT* ' Tm sin8cos8 (297) DETERMINATION OP AZIMUTH ' 195 The values of m can be taken from Table III, Appendix, for the different values of t — t. If we let m be the mean of the several values of m, the corrections can be applied collectively to the mean of the circle readings on the star, and the equation (297) becomes A -A=^m sinScosS , (298) smz in which A — A is the correction to the mean of the circle readings. Further, if the level readings have been taken symmetrically with reference to the program, which can always be done, the mean value of y, equation (289), can be applied to the mean of the circle readings. 132. The values of t and O having been computed for the star to be observed, the instrument is carefully adjusted, and a program similar to this is followed : Make two readings on the mark Read the level Make four readings on the star Read the level Make two readings on the mark Reverse Make two readings on the mark Read the level Make four readings on the star Read the level Make two readings on the mark The times of observation are noted on a time-piece, pref- erably a sidereal chronometer. Its correction must be known within one or two seconds if the most refined form of instrument is employed, or to the nearest minute if an ordinary surveyor's transit is used. This correction can be obtained by any of the methods described in the pre- ceding chapters, or by a comparison with the time signals at the nearest telegraph station. The chronometer time of elongation is now known. Subtracting from it the several times of observation, the values of t — t are found, and 196 PRACTICAL ASTRONOMY the values of m corresponding to them taken from Table III. Forming the mean m and computing z from (291), the value of A — A is found and applied to the mean of the circle readings. The mean of the corrections for level errors and the correction for diurnal aberration are now applied. The corrected mean circle reading, which we shall call s, corresponds to the azimuth A of the star at elongation, which is computed by (291). If k is the mean of the circle readings on the mark, and M the azimuth of the mark, then M=k-(s- A ). (299) The circle reading when the telescope is directed to the south point of the horizon will be equal to s — A . 133. Example. Detroit Observatory, Wednesday, 1891 May 6. Find the azimuth of a given point (nearly in the horizon) from observations on 8 Ursce Minoris near its eastern elongation. Observer's latitude, 42° 16' 48". The apparent place of the star was a= 18*7" 44 s , 3 = + 86° 36' 25".0. Equations (291) and (292) are solved as below. tan<£ 9.958704 sin<£ 9.827856 cos S 8.772214 tan 8 1.227024 sin 8 9.999236 cos<£ 9.869153 t 273° 5' 25" z 47° 37' 42" A 184° 35' 17".8 t„ 18 6 12»22» a 18 7 44 e 12 20 6 The chronometer correction was + 18" 1 52 s , and, there- fore, the chronometer time of elongation was 12 A l m 14 s . A good surveyor's transit, whose horizontal circle was read to 10" by two verniers, and which was provided with plate levels and a delicate striding level, was placed over the point of observation and carefully leveled a short time before elongation. The following observations were made : DETERMINATION OP AZIMUTH 197 No. Object Telescope Chronometer Vernier A Vernier B (1) Mark Reversed 96° 16' 40" 276° 16' 30" (2) a " 96 16 35 276 16 25 (3) Level (4) Star « 11»44" 52" 243 39 20 63 39 20 (5) a tt 48 40 243 39 50 63 39 50 (6) ti tt 51 6 243 39 50 63 39 50 (7) a tt 53 11 243 40 63 40 5 (8) Level (9) Mark tt 96 16 45 276 16 40 (10) IS. •• 96 16 50 276 16 40 (11) it Direct 276 17 96 16 45 (12) " £( 276 16 55 96 16 45 (13) Level • (14) Star ti 12 5 50 63 40 10 243 40 10 (15) it it 7 54 63 40 243 40 (16) a t* 9 44 63 39 50 243 39 50 (17) u (( 11 27 63 39 45 243 39 55 (18) Level (19) Mark it 276 16 45 96 16 30 (20) a a 276 16 45 96 16 35 The level readings given by the striding level were (3) (8) (13) (18) W E W E W E W E 4.4 4.1 4.2 4.0 4.0 4.4 4.0 4.2 4.3 4.0 4.3 3.8 4.0 4.2 4.1 4.2 4.4 4.0 4.2 3.9 4.0 4.2 4.0 4.2 4.3 3.8 4.4 3.8 4.1 4.4 4.2 4.2 The value of one division of the level was 10".7, and therefore, from (166), the inequality of the pivots being negligible, (3) (8) (13) (18) b= + 2".0, +2ii.l, -VIA, -0".6, and by (289), (3) (8) (13) (18) y = + 1".8, + 1".9, - 1".3, - 0».6. 198 PRACTICAL ASTRONOMY The solution of (298) for the eight readings on the star is given below. The column " Circle Readings " is formed by taking the means of Vernier A and Vernier B. No, Circle Readings t -t m (4) 243° 39' 20" + 16"* 22* 525".7 (5) 243 39 50 + 12 34 310 .0 (6) 243 39 50 + 10 8 201 .6 (7) 243 40 2 + 83 127 .2 (14) 63 40 10 - 4 36 41 .5 (15) 63 40 - 6 40 87 .3 (16) 63 39 50 - 8 30 141 .8 (17) 63 39 50 -10 13 204 .9 Means 63 39 51.5 m = 205 .0 logro 2.31175 sin 8 9.99924 cos S 8.77221 cosec z 0.13148 log(il -^) 1.21468 A -A + 16".4 The mean of the four values of y is + 0".4. The value of dA, from (290), is — 0".3. The corrected circle read- ing on the star at elongation is therefore s = 63° 39' 51".5 + 16".4 + 0".4 - 0".3 = 63° 40' 8".0. The mean of all the readings on the mark is k = 276° 16' 41".6 ; and, therefore, by (299), ilf=37 ll'51".4. Since the verniers on this instrument read to only 10", the diurnal aberration could have been neglected, and the other corrections computed to the nearest second only. But all the corrections have been applied here, to illustrate the method. DETERMINATION OF AZIMUTH 199 AZIMUTH BY POLAEIS OBSERVED AT ANY HOUR ANGLE 134. When the azimuth is required with the greatest possible accuracy, the observations should always be made at or near elongation, and reduced as in the preceding sec- tion. However, good results can be obtained by observing Polaris in any position, if the time is accurately known. The time should be known within S .5 when using the finest instruments, and within 5 s or 10* when using a good surveyor's transit whose least reading is 10". As in the preceding method, the observations should be made on the mark and star in both positions of the tele- scope. If the observations are made in quick succession, the mean of two or three observations made before revers- ing may be treated as a single observation, and similarly for those made after reversing. But if the separate obser- vations are several minutes apart, each observation should be reduced separately. The sidereal time 6 of observation having been noted with great care, the hour angle t of Polaris is given by (41). If we let the azimuth A of the star be measured from the north point, + if the star is west of the meridian and — if east, and let q be the star's parallactic angle [§ 6], we may write [Chauvenefs Sph. Irig., § 24] COS (8-*). tanKg+^) = ^g + g ootit=/cotil, (300) tanK?-^) = ^|^goot^=/'cot^, (301) A = $(q + A)-l(q-A). . (302) The auxiliary quantities,/ and/', depending on 8 and , are constant for a night's observations, and with surveyors' instruments may be considered constant for several weeks. When they have been computed, once for the whole series of observations, they may be combined rapidly with the 200 PRACTICAL ASTRONOMY values of cot \ t for the individual observations, to deter- mine |( + 131 17 1(8-*) + 23 13 20 1(8+*) + 65 30 8 cos J (8 — ) 9.963307 sin I (8 + 4>) 9.959031 !og/ 0.004276 0.004276 cot \t 8.767417„ 8.893338 n 1(9 + 4) - 3° 22' 59" -4° 31' 1" sin£(S--*) 9.595825 cos \ (8 + *) 9.617690 , ^g/' 9.978135 9.978135 cot£t 8.767417„ 8.893338„ 202 PRACTICAL ASTRONOMY A Circle on star y dA S S + A K N Mean N The corrections should be carried to tenths of seconds when refined instruments are used. The azimuth of the same mark was measured on the same night with the same instrument, by means of obser- vations of 8 Ursoe Minoris taken near eastern elongation [see example of the preceding section]. The azimuth obtained, measured from the south point, was M = 37° 11' 51''.4. -3° 11' 9" -0 11 50 - 4° 15' 14" -0 15 47 59° 16' 42" -3 59 16 39 239° 20' 31" + 2 239 20 33 59 4 49 276 16 36 239 4 46 96 16 36 142 48 13 142 48 10 142° 48' 12" CHAPTER XI THE SURVEYOR'S TRANSIT 135. The surveyor's transit is adapted to the determina- tion of the time, latitude and azimuth by many of the preceding methods. These elements can easily be deter- mined to an accuracy within the least readings of the cir- cles, if the instrument is of reliable make, and is provided with spirit levels. We shall assume that the observer uses a mean time-piece, which we shall call a watch, and that he has a thorough knowledge of the subject of Time, Chapter II, without which the Ephemeris cannot be used intelligently. We shall assume, also, that the vertical circle of his instrument is complete, and that the degrees are numbered consecutively from to 360. In case they are not, the observer can readily reduce his readings to that system. The instrument is supposed to be carefully adjusted. A method of illuminating the wires at night is given in § 127. Figure 26 illustrates a form of instrument well adapted to the solution of the problems described in this chapter ; but the methods can be used, within limits, with nearly all forms of the surveyor's transit instrument. DETERMINATION OF TIME 136. By equal altitudes of a star. Set the instrument up firmly, level it, and direct the telescope to a known bright star east of the meridian. Pointing the telescope slightly above the star, clamp the vertical circle and note the time T' when the star crosses the horizontal wire. 203 IP ': 1 Fig. 26 DETERMINATION OF TIME 205 The vertical circle must not be undamped. , A short time before the star reaches the same altitude west of the meridian, level the instrument, move it in azimuth until the telescope is directed to a point just below the star, wait for the star to enter the field, and note the time T" when it crosses the horizontal wire. The sidereal time 9 when the star was on the observer's meridian equals its right ascension a, and this corresponds to the mean of the two watch times. Converting the sidereal time 6 = a into the corresponding mean time T, the watch correction AT is given by AT = T - i (7" + Til). (304) Example. Thursday, 1891 March 5. In longitude 5'' 34™ 55 s , Regulus was observed at equal altitudes east and west of the meridian, at the watch times V = 8 s 7™ 34 s , T" = 14* 10™ 20". Required the watch correction. From the American Ephemeris, p. 332, a=0=lO* 2 m 35 s . Converting this into mean time, § 18, we find T=ll h 8">&; and, therefore, by (304), the watch correction was Ar = -54»; or the watch was 54 s fast. 137. By a single altitude of a star. Level the transit instrument. Direct the telescope very slightly above a known star in the east or below a known star in the west, and clamp the telescope. Note the watch time when the star crosses the horizontal wire and read the vertical circle. Unclamp the telescope and repeat the observation once or twice, as quickly as possible. Double reverse the instrument, and make the same number of observations as before. Form the means of the circle readings made be- 206 PRACTICAL ASTRONOMY fore reversal and of those made after. Subtract one from the other in that order which makes their difference less than 180°. One half this difference is the apparent zenith distance of the star at T', the mean of the several watch times of observation. Adding the refraction given by (97), r = 58" tan z, (305) the result is the true zenith distance z. Substituting the values of z, + 8)] 0.28114 <£ - 8 22 31 55 log sec £ [z - (<£ + 8)] 0.00085 + 8 62 139 log tan 2 \t 9.52243 z + (<£_8) 77 22 47 log tan \ t 9.76121„ z _(- 8) 32 18 57 J* 150° 0' 48" z + (4, + g) 116 52 31 < 300 1 36 z _(0 + 8) - 7 10 47 t 20 ft 0'» 6« Solving (306), 6 = 10" 10 m 49 s . The equivalent mean time is T=l h 55 m 45 s ; and the mean of the six times of observation is 2" = T 55™ 2 s . Therefore, AT = + 43«. 138. i?y « single altitude of the sun* Observe the transits of the sun's upper and lower limbs over the hori- zontal wire by the method used for a star, § 137. Double reverse, and repeat the observations.! Form half the difference of the means of the circle readings, and add the refraction given by (305), as before. Further, subtract the parallax given by (64) p = 9"sinz, (308) and the result is the sun's true zenith distance z at the mean of the times, T'. The correct mean time is probably known within 5 ro or 10™. Increase it by the longitude, and the result is an approximate value of the Greenwich mean time. Take from the Ephemeris, p. II of the month, the value of the sun's declination 8 at that time. The * The observer must cover the eyepiece with a small piece of very dense neutral-tint glass before looking through at the sun. The observa- tions can be made, also, by holding a piece of paper a short distance back from the eyepiece, and focusing the eyepiece so that the images of the sun and wire are seen on the paper. t While waiting for the second limb to approach the wire, the time may well be spent in reading the vertical circle. 208 PRACTICAL ASTRONOMY Ephemeris contains the apparent declination for Greenwich mean noon, and the " difference for one hour," whence the declination at any instant can be found. Solve (38) or (39) for these values of z, 8 and <£. The resulting hour angle t is the observer's apparent solar time. Convert this into the equivalent mean time T, by § 15. The watch correction is given by (307), as before. Example. Thursday morning, 1891 May 6. In latitude + 42° 16' 50" and longitude 5 h 35 m , the following observa- tions of the sun were made with Buff & Berger transit No. 1554. Required the watch correction. Telescope Limb Watch Circle reading Direct Upper 20* 38» 59» 41° 48' 30" a Lower 41 59 41 48 30 Reversed Upper 46 49 137 33 u Lower 49 50 137 33 One half the difference of the circle readings is 47° 52' 15". The refraction, by (305), is 64", and the parallax, by (308), is 7". Therefore the true zenith dis- tance z of the sun's center is 47° 53' 12". The mean of the four watch times is 20* 44'" 24 s . We have T' 1891 May 6" 20* 44" 24' Longitude 5 35 Gr. mean time 7 2 19 " " " 7 2".32 From the American Ephemeris, p. 75, the sun's declina- tion at Greenwich mean noon, May 7, was + 16° 48' 53", and the difference for one hour, + 41". The change for 2\32 was therefore 96", and the required value of the declination was 8 = + 16° 50' 29". Substituting the values of z, cf> and S in (38), and solv- ing as was done in § 137, we obtain the hour angle t = 312° 12' 10" = 20" 48 m 49 s . The observer's true time is therefore May 6" 20" 48™ 49 s . Converting this into the DETERMINATION OP LATITUDE 209 mean time T, by § 15, we find T = 1891 May 6<* 20" 45 m 15'. The watch correction is AT = 20 45™ 15' - 20* 44 m 24' = + 51'. DETERMINATION OF GEOGRAPHICAL LATITUDE 139. By a meridian altitude of a star. A star is on the observer's meridian when the sidereal time 6 is equal to its right ascension a. Convert this into the corresponding mean time, subtract the watch correction obtained by any of the above methods from it, and the result is the watch time of the star's meridian passage. A few seconds before this watch time direct the telescope to the star, bring the star's image on the horizontal wire, and read the circle. Double reverse quickly, and make another observation. As before, form one-half the difference of the circle read- ings, add the refraction given by (305), and the sum is the star's true zenith distance z. Take the value of 8 from the Ephemeris. For a star observed south of the zenith, <£ = S + z; (309) and for a star observed between the zenith and pole, = 8 - z. (310) For a star below the pole the sidereal time of meridian passage is 12" + a. Obtaining the value of z as before, the latitude is given by <£ = 180° -B-z. (311) Example. Ann Arbor, Friday, 1891 April 24. a Sydrae was observed on the meridian with a surveyor's transit, as below. Required the latitude. Telescope Circle reading Reversed 140° 26' 30" Direct 39 33 One half the difference of the circle readings is 50° 26' 45". The refraction is 70". Therefore, z = 50° 27' 55". 210 PRACTICAL ASTRONOMY From the Ephemeris, p. 331, S = - 8° 11' 18". Therefore, from (309), = 42° 16' 37". To find the watch time when the star is on the meridian, we have, from the Ephemeris, a = = 9" 22 m 14'. The corresponding mean time, by § 18, is 7 A 11"' 13 s . The watch correction is + 43 s , whence the required watch time is 7" 10"* 30 s . 140. By a meridian altitude of the sun. The sun is on the meridian at the apparent time h m s . Apply the equation of time to this, by § 15, and subtract the known watch correction. The result is the watch time of the sun's meridian passage. One. or two minutes before this watch time, direct the horizontal wire of the telescope to the upper limb of the sun, and read the vertical circle. Observe the lower limb in the same way. Double reverse and observe both limbs again. Form half the difference of the means of the readings in the two positions. Add the refraction given by (305), and subtract the parallax given by (308). The result is the value of z. Take from the Ephemeris the value of h for the time of meridian passage. The latitude is now given by (309), as in the case of a star. Example. Wednesday, 1891 March 25. In longitude b h 35 m the following meridian altitude observations of the sun were made with a surveyor's transit. Required the latitude. Telescope Limb Circle reading Direct Upper 49° 54' 30" a Lower 49 22 30 Keversed a 130 5 30 (( Upper 130 38 30 One half the difference of the means of the circle read- ings is 40° 21' 45". The refraction is 49". The parallax is 6". Therefore, z = 40° 22' 28". DETERMINATION OP AZIMUTH 211 The Greenwich apparent time of observation was March 25" 5" 35 m . The value of 8 at that instant was + 1° 54' 32", Ephemeris, p. 38. Therefore, by (309), = 42° 17' 0". To find the watch time of meridian passage, we have, Apparent time 0* 0™ 0« Equation of time +6 3 Mean time 6 3 Watch correction — 15 Watch time 6 18 DETERMINATION OF AZIMUTH 141. The two methods of " determining azimuth which are described in the preceding chapter are adapted to the surveyor's transit, and need no further explanation. With this instrument the diurnal aberration can be neglected. If the transit is provided with plate levels only, they should be kept in perfect adjustment. If the bubble of the level which is parallel to the rotation axis of the tele- scope remains constantly in the center, no correction for level is required. But if the bubble is n divisions of the level from the center when an observation on a star is made, and d is the value of one division of the level, the circle reading must be corrected by y = ± nd cot z ; (312) + if the bubble is too far west, — if too far east. CHAPTER XII THE EQUATORIAL 142. This instrument consists essentially of the follow- ing parts : A supporting pier ; a polar axis parallel to the earth's axis, supported at two or more points by the pier in such a way that it can rotate ; a declination axis attached to the upper end of the polar axis, and at right angles to it, in such a way that it can rotate ; a telescope firmly attached to one end of the declination axis, and at right angles to it ; a graduated declination circle attached to the other end of the declination axis ; a graduated hour circle attached to the polar axis, and at right angles to it; a finding telescope or finder, to assist in pointing the principal telescope, and attached to it ; a driving clock and train of wheels for rotating the instrument about its polar axis at a uniform rate. The various moving parts are so counter- poised that the telescope will be in equilibrium in all posi- tions. The principal features of the equatorial are well illustrated in Fig. 27. A sidereal chronometer is an almost indispensable com- panion of the equatorial. The equatorial serves two purposes : 1st. As an instrument of direct observation and dis- covery, by assisting the vision. 2d. As an instrument for determining very accurately the relative positions of two objects comparatively near each other, by means of a micrometer eyepiece [§ 60]. If the position of one of the objects is known, the position of 212 THE EQUATORIAL 213 the other is known as soon as their relative positions are determined. Fig. 27 143. By the above system of mounting it is evident that the telescope can be directed to any part of the sky ; and 214 PRACTICAL ASTRONOMY that it will follow a star in its diurnal motion, by revolv- ing the instrument about the polar axis alone; for in that case the line of sight maintains a constant angle with the celestial equator, and therefore describes a circle which is identical with the star's diurnal circle. Since the star's angular motion is uniform, the telescope is made to follow it by means of the sidereal driving clock. In some obser- vations the driving clock is not used; in others it is indispensable. When the telescope is revolved upon the declination axis, its line of sight describes an hour circle on the celestial sphere. The position of this hour circle is indicated by the reading of the graduated hour circle of the instrument. The position of the telescope in this hour circle is indicated by the reading of the graduated declination circle. When the telescope is directed exactly to the south point of the equator, the hour circle reading should be h Q™ s , and the declination circle reading should be 0° 0' 0". Then, if the other parts of the instrument are in adjustment, and the telescope is directed to a star, the hour angle and declina- tion of the star will be indicated (neglecting the refraction and parallax), by the hour circle and declination circle readings.* ADJUSTMENTS 144. It is not essential that the errors of adjustment of an equatorial be entirely eliminated, or that their values be accurately known ; but it is a practical convenience to have the errors small, particularly so for observations on objects near the poles of the equator. It is expected that the maker of the instrument will * The hour circle should read time. It should he graduated from 0» to 24* toward the west, or from 0* to 12* in both directions from 0*. The declination circle will read arc. It may be graduated from 0° to 360°, or from 0° to 180° in both directions from one of its equator points, or from 0° to 90° in both directions from its two equator points. We shall suppose it to read from 0° to 360°. ADJUSTMENTS 215 adjust the various parts of it as perfectly as possible with reference to each other. It remains for the observer to place the instrument as a whole in correct position. The polar axis should be in the plane of the meridian ; the elevation of the polar axis should equal the latitude of the place ; the hour circle should read zero when the telescope is in the meridian ; the declination circle should read zero when the telescope is in the equator; and the lines of sight of the finder and telescope should be parallel. The instrument should first be placed as nearly as possible in position, by estimation. Then direct the telescope to any known star near the southern horizon whose right ascension is a. The star will be on the meridian at the sidereal time = a. Move the whole instrument in azimuth so that the star is in the center of the telescope when the chronometer time plus the chronometer correction is equal to 6 = a. The order of making the final adjustments is as below. 145. To adjust the finder. Using the lowest power eye- piece, direct the telescope to a bright star. Replace the low-power eyepiece by a high-power. Keeping the star in the center of the field of view, turn the adjusting screws of " the finder so that the star is on the intersection of the cross wires in the finder. The two telescopes will then be sufficiently near parallelism. 146. To determine the angle of elevation of the polar axis, and the index correction of the hour circle.* Across the object end of the telescope firmly tie a piece of wood which projects several inches from the telescope tube on the side opposite the pier. Pass a fine thread through a very small hole in the projecting end, and fasten it. Direct the tele- scope to the zenith. Near the eye end and on the same * This very simple and satisfactory method was proposed by Professor Schaeberle, in der Astronomische Nachrichten, No. 2374. It has the advantage that the errors can he determined, and corrected, in the day- time. 216 PRACTICAL ASTRONOMY side as the projecting arm, fasten a block of wood. To this screw a metal plate so that it will be perpendicular to the axis of the tube, and in which is a very small circu- lar hole as nearly as possible (by estimation) under the hole above. Pass the thread through it, tie a plumb-bob to the end of the thread near the floor, and let it swing in a vessel of water. Move the telescope by the slow-motion screws until the plumb-line passes through the center of the lower hole. Read both verniers of the hour and decli- nation circles. Unclamp, hold the plumb-bob in the hand to avoid displacing the metal plate, reverse the telescope to the other side of the pier, and set it so that the plumb- line again passes centrally through the hole. Read both circles as before. Let h equal the angle of elevation of the polar axis. The difference of the readings of the declination circle in the two positions is 180° — 2 h. The elevation should equal the known latitude . The error is h — (j>. Change the last circle reading by this amount, by moving the telescope in declination in the proper direction. Adjust the angle of elevation by the proper screws until the plumb-line again passes through the center of the hole. If the declination circle is graduated so as to read from 0° to 90° in both directions from its two equator points, then the mean of the circle readings for the two positions of the telescope is at once the inclination of the polar axis to the horizon. The mean of the hour circle readings in the two posi- tions is the reading of the circle when the telescope is in the meridian. This should be 0'' m s . The index error of the hour circle is the mean of the readings minus 0* m s (or minus 24 A 0'" s ). To correct for it, set the circle at this mean reading ; then move the vernier screws until the reading is 0" 0'" s . The index correction of the hour circle is equal to the index error with its sign changed. If the error is not ADJUSTMENTS 217 removed by adjusting the verniers, the index correction must be applied to every reading made with the hour circle, in order to obtain the true reading. If the errors are large, these adjustments should be repeated once or twice. Example. 1891 Feb. 20. The following plumb-line observations were made on the 6-inch equatorial of the Detroit Observatory. Determine the errors. The last column gives the position of the telescope with reference to the pier: Hour Circle Declination Circle Telescope Vernier A Vernier B Vernier A Vernier B 24" 2*» 53* 11 56 56 12* 2™ 58« 23 57 6 135° 45' 30" 40 16 45 315° 45' 00" 220 16 30 E W The means of the declination circle readings were 135° 45' 15" and 40° 16' 38", and therefore h equaled 42° 15' 42". The value of is 42° 16' 47". The axis was therefore 1' 5" too low. The telescope was moved in declination until the verniers read 40° 17' 45" and 220° 17' 30", and the axis adjusted until the thread was again central in the hole. The hour circle readings were 24 A 2 m 55 s .5 and 23 A 57 m I s , and their mean was 23 A 59 m 58".2. The error was there- fore — l s .8. The verniers were moved to the west 2 s . A repetition of the observations gave h = 42° 16' 49", and the mean of the hour circle readings, 24* m S .5. Further adjustment was not required. The index error of the hour circle was + s . 5, and the index correction to be applied to future readings was — 0'.5. 147. To determine the azimuth correction of the vertical plane containing the polar axis. This is best determined 218 PRACTICAL ASTRONOMY by observations on one of the four Ephemeris circumpolar stars near its culmination. Using the micrometer eyepiece (§ 60), direct the tele- scope to the star a few minutes before its culmination, note the chronometer time 1 when the star is on the point of intersection of the wires (or any well defined point in the eyepiece), and read the hour circle. Reverse the tele- scope to the other side of the pier, note the time 8 2 when the star is at the same point of the eyepiece, and read the hour circle. Let t x and £ 2 be the hour circle readings in the two positions, corrected for index error, if any ; let a be the required azimuth correction ; and let Ad be the known chronometer correction [see §152]. Neglecting the quantities which are eliminated by the reversal, we have for the sidereal times when the star is in the vertical plane of the polar axis, 6 l + A6 - t v 2 + A0 - t 2 . Therefore, as with the transit instrument, § 98, aA =a -(0, + A0- t{), aA = a - (0 2 + A0 - t 2 ), in which A is given by, (222) and (234), sin (<£ t 8) A. — s ' cos 6 the lower sign being for lower culmination. Solving for a we find a = [a - !(*! + 2 ) - A0 + I (t, + g] . c ° s8 • (313) sin (<£ T o) a is expressed in time: in arc, the azimuth correction is 15 a. If a is + , the south end of the axis requires to be moved to the west; if — , to the east. This is readily done. Direct the telescope to a distant terrestrial object nearly in the horizon, make the movable micrometer wire vertical, ADJUSTMENTS 219 and set it on the object. Next move the wire through the distance a in the proper direction. This can be done when the value of one revolution of the screw is known [§ 61]. Shift the whole instrument in azimuth by the proper screws until the micrometer wire is again on the object. The vertical plane of the polar axis should now coincide with the meridian. If the value of a is large the observations should be repeated. If a is less than 3 s , it will cause no inconven- ience and scarcely need be corrected. Example. Wednesday, 1891 February 25. 51 Oephei was observed at upper culmination with the 6-inch equa- torial of the Detroit Observatoiy, as below. Determine the azimuth correction. The value of Ad was + 14 m 36 s .O. Telescope Hour circle Chronometer West 23» 56" 31' 6" 31" 42" East 24 42 6 35 29 The index correction of the hour circle was — S .5. Amer. Ephem., p. 303, v. 6» 49" 27'.5 8 + 87° 13' £(0! + 2 ) 6 33 35.5 $ +42 17 A0 + 14 36.0 cos 8 8.6863 K'i + 's) 2358 36.0 sin(0-S) 9.8490„ _ 8 .0 £2i§ - 0.069 sin(<£ - 8) The value of a was - 0.069 x - 8 s .O = + S .6 = + 9" ; that is, the south end of the axis should be moved 9" west. This was too small to require correction. 148. To adjust the declination verniers. Direct the tele- scope to a star, nearly in the zenith, whose declination is known. Bring the star to the center of the eyepiece, using a high power, and clamp the instrument in declina- tion. Set the verniers so that they read the star's declina- tion. They will then be in adjustment. 220 PRACTICAL ASTRONOMY 149. To center the object glass. Imperfect images are often due to the fact that the object glass is not properly centered. To test this adjustment, remove the eyepiece and hold a candle flame in such a position that the images of the flame reflected from the surfaces of the object glass can be seen through the flame. If the object glass is per- fectly centered all the images should coincide when the observer's eye and the center of the flame are in the axis of the telescope. If they do not coincide, raise one side of the object glass cell by the set screws until the coincidence is perfect. 150. The magnifying power of a telescope is equal to the focal length of the objective divided by the focal length of the eyepiece. It is therefore different for different eye- pieces on the same telescope, or for the same eyepiece on different telescopes. The following method of determin- ing it is simple, and abundantly accurate. Focus the telescope on a distant object, and direct it in the daytime to the bright sky. Hold a piece of thin, un- glazed paper in front of the eyepiece at such a distance that the bright disk formed on it is clearly defined. This disk is the minified image of the object glass. Measure its diameter by a finely divided scale held against the paper, and measure the diameter of the object glass. It can be shown that these diameters are to each other as tl:e focal lengths of the eyepiece and object glass. Their quotient is therefore the magnifying power. Thus, for the equa- torial mentioned above, the diameter of the object glass is 6.05 inches, and the diameter of the bright disk for a cer- tain eyepiece is 0.08 inch. The magnifying power is, therefore, 6.05 -r- 0.08 = 76. A definite statement of the magnifying power to be used in observing an object cannot be made. A higher power can be used when the seeing is good, i.e., whea the images in the telescope are steady and well defined, than when CHRONOMETER CORRECTION 221 the seeing is poor. Lower powers must in general be used with nebulae and comets. The very highest powers can be used with stars and some of the planetary nebulas, if the seeing is good. Further than this, the observer must select that eyepiece which on trial gives the best results. 151. The field of view is the circular portion of the sky which can be seen through the telescope at one time. Its diameter is equal to the angle contained by two rays drawn from the center of the object glass to the two extremities of a diameter of the eyepiece. The diameter, expressed in arc, is equal to 15 times the interval of time required by an equatorial star to traverse it. This can be directly observed. 152. The chronometer correction is quickly obtained, with an accuracy sufficient for all ordinary uses of the equa- torial, by the following method : Direct the telescope to a known star nearly in the zenith, note the chronometer time X when the star is on the point of intersection of the wires, and read the hour circle. Carry the telescope to the other side of the pier, observe the star as before at the time 2 , and read the hour circle. Let the hour circle readings corrected for index error be t t and t 2 . We have, by (40), a = 8 1 + A0 - t v a = $ 2 + A0 - « 2 , neglecting only very small quantities and those eliminated by reversal. Therefore A6 = a-l(6 1 + 6 2 )+l ft + g. (314) For many purposes an observation on one side of the pier will suffice, and we have A0 = a + k - e v , (315) Example* Ann Arbor, Wednesday, 1891 Feb. 25. The following observation of Castor was made with the 6-inch 222 PRACTICAL ASTRONOMY equatorial, to determine an approximate value of the chro- nometer correction. Chronometer time, 6 1 7 5 m 9* Hour circle, t x 23 52 3 Amer. Ephem., p. 327, a 7 27 39 Therefore, by (315), A0 = + 14 m 33 s . 153. To direct the telescope to an object whose right ascension (a) and declination (8) are known, first deter- mine whether the object is east or west of the meridian. If the right ascension is greater than the sidereal time, it is east ; if less, it is west. Generally, if the object is east, the telescope should be west of the pier ; if the object is west, the telescope should be east of the pier. Move the telescope in declination till the declination circle reads 8. To the reading of the chronometer add the chronometer correction, and one or two minutes more for the time con- sumed in setting. Subtract a from this sum and set off the difference (which is the hour angle) on the hour circle. When the chronometer indicates the time for which the hour angle was computed, the object should be seen in the finder. Move the telescope until the star is on the inter- section of the finder cross-wires. . The star should then be visible near the center of the field of view of the (prin- cipal) telescope. Conversely, if an unknown star is seen in the telescope, the chronometer time noted and the circle readings taken : then the declination circle reading is the star's declination ; and the chronometer time of observation, plus the chro- nometer correction, minus the hour circle reading, is its right ascension. These results are only approximate, of course, since the instrument will never be in perfect adjustment, and the star will not be seen in its true place, owing to the refrac- tion, etc. DETERMINATION' OP APPARENT PLACE 223 DETERMINATION OF APPARENT PLACE OF AN OBJECT 154. By the method of micrometer transits. Select a known * star, called a comparison star, whose right ascen- sion and especially whose declination differ as little as possible from that of the object. Revolve the filar microm- eter [§ 60] until the star in its diurnal motion follows along the micrometer wire. The wire in this position is exactly east and west, or parallel to the equator, and the reading of the position circle for this direction of the wires is called the equator reading. If the object and comparison star are in the vicinity of the pole, their diurnal circles will be sharply curved, and in this case the equator read- ing of the circle should, first of all, be determined from an equatorial star. Direct the telescope just in advance of the two objects. The diurnal motion will carry them across the field. Note the chronometer times when they cross the transverse wire or wires. The difference of these times for the two objects is the difference of their right ascen- sions. Also, when the first or preceding object approaches the center of the field, move the whole system of wires until the object follows along the fixed wire. When the second or following object approaches the center of the field, bisect it with the micrometer wire. Read the microm- eter in this position, and also when the micrometer wire is in coincidence with the fixed wire. The difference of the two readings, multiplied by the value of a revolution of the screw [§ 61], is the difference of the declinations of the two objects. Care should be taken to have the micrometer and fixed wires exactly parallel,! and the transverse wire (or wires) exactly perpendicular to the micrometer wire. To test the * That is, a star whose accurate position is given in one or more of the star catalogues. t In good forms of the micrometer, an adjusting screw is provided for bringing them into parallelism. 224 PRACTICAL ASTRONOMY relative positions of the two sets of wires, direct the tele- scope to an equatorial star, adjust the micrometer wire so that the star's diurnal motion will carry it across the field in coincidence with the wire, and read both verniers of the position circle. Rotate the system of wires, adjust the transverse wire so that the star will cross the field in coin- cidence with the wire, and read both verniers. The dif- ference of the two position circle readings should be exactly 90°. Many observers prefer to observe only one coordinate at a time. A good program to follow is, measure the differ- ence of declination, revolve the micrometer 90° and ob- serve the difference of right ascension, then revolve 90° more and measure the difference of declination again. In any case, the observations should be repeated several times, and the mean of all the observations adopted. If the object has a proper motion, the differences in right ascension and declination are those corresponding to the instant ivhen that object was observed: that is, the mean of the chronometer times for the object, plus the chronometer correction. The mean place of the comparison star* will be given for the epoch of the catalogue which contains it. Reduc- ing this to the mean place for the beginning of the year of observation by § 46, 47 or 52, thence to the apparent place for the instant of observation by § 55, and applying the micrometer differences to the apparent place, we obtain the observed place of the object. This must be corrected for refraction and parallax. The refraction correction will be small, since the star * If the star is a very bright one, it may be identified satisfactorily, both in the sky and in the catalogue, by the methods of § 153. But, in case it is faint, the observer should always compare a chart of the neigh- boring stars, prepared from the catalogue, with that region of the sky, making sure that the configurations of the stars on the chart and in the sky agree. DETERMINATION OF APPARENT PLACE 225 and object will be refracted nearly tbe same amount in nearly tbe same direction. An equatorial is a fixed part of an observatory, and tables of differential refractions in right ascension and declination in every part of the sky should be computed for tbe latitude of the observatory. The corrections can then be taken from the tables very quickly. Until such tables are constructed, the corrections can be computed by the following method: Let t be the mean of the hour angles of the star and object, B the mean of their declinations, and z the mean of their zenith dis- tances. Substitute these values of t and 8 in (35), (36) and (37), to determine L and z. The corrections to the observed place will be given by * 8' — 8 tan t sin L cos (2 8 + L) Aa: 15 sin 2 (8 + Z) cos 2 8„ A8=K. 8' -8 sin 2 (S + Z)' (316) (317) in which S' — 8 is the declination of the object minus the declination of the star expressed in seconds of arc, and k is defined by K = fi."BTy"i. (318) B, T and 7 have the same significance as in § 30, and their values are given in Table I of the Appendix. The values of log fi" and X" are tabulated below with the argument z. z log /i" V z log m" X" 0° 6.446 1.00 80° 6.395 1.10 45 6.444 1.00 82 6.370 1.15 60 6.440 1.01 83 6.351 1.18 70 6.433 1.03 84 6.323 1.21 75 6.422 1.05 85 6.285 1.24 * These equations are derived in Chauvenet's Spherical and Practical Astronomy, Vol. II, pp. 450-460. Q 226 PRACTICAL ASTRONOMY It is only in the most refined measurements and in extreme states of the weather that the barometer and thermometer readings need be taken into account. With comets it will scarcely ever be necessary, except when they are very near the horizon. If one or both of the bodies is in the solar system, and at different distances from the observer, the observations will require correction for parallax, by the methods ex- plained in full in § 28. Four-place tables are sufficient for computing the refrac- tion and parallax corrections. Example. Wednesday, 1890 July 23, the author made the following observation of Coggia's comet with the 12-inch equatorial of the Lick Observatory. Required its apparent place. The comet was south of and preceding the 6th magni- tude star No. 1518 Pulkowa Catalogue for 1855.0. The reading of the position circle when the star followed along the micrometer wire was 201°. 35. The micrometer readings when the micrometer and fixed wires coincided were 19.947 .947 .947 Mean 19.947 When the comet was in the center of the field the fixed wire was made to bisect it, and the chronometer time was noted. When the star reached the center of the field (nearly four minutes later) the micrometer wire was made to bisect it, and the micrometer reading noted. In this way the difference of the declinations was observed, as below. Chronometer Micrometer Remarks 16» 43™ 11- 22.954 Very windy 48 59 23.822 Very windy 54 2 24.550 Very windy Means 16 48 44 23.775 DETERMINATION OP APPARENT PLACE 227 The micrometer was rotated 90° until the circle read 291°. 35, and the chronometer times of transit over the two wires noted, as below. 17* Comet Star Difference 0™ 51\7 0™ 59'.2 17» 4 m 44'.6 4"* 51'.8 - 3™ 52».75 5 42.3 5 49.5 9 34.0 9 41.3 3 51.75 10 37.8* 10 46.1 14 28.0 14 36.3 3 50.20 23 3.5 23 11.5 26 49.8 26 58.1 3 46.45 27 39.5* 27 52.2 31 24.6 31 37.2 -3 45.05 ans 17* 1 3-»39« -3 49.24 The micrometer was rotated 90° further until it read 21°.35, and the difference of the declinations measured again, as below, Chronometer Micrometer Remarks 17» 34"* 22 s 9.751 Very windy 39 44 8.867 Very windy 48 40 7.545 Very windy Means 17 40 55 8.721 Readings for coincidence of wires, 19.944 .946 .943 Mean 19.944 The value of one revolution of the screw is i2 = 14".058. We shall combine the two differences of declination, thus : Chronometer 16* 48" 44« 17 40 55 Means 17 14 49 Diff . of decl. - 3.828 R - 11.223 R - 7.525 R = 1' 45".8. The chronometer time for the declinations is l m 10 s greater than that for the right ascensions. From the two measured declinations it is found that the declination changed 2".3 in l m 10 s . Therefore, at 11" 13 m 39 s the difference of decli- nation was — 1' 43". 5. * The distance between the wires was changed intentionally. 228 PRACTICAL ASTRONOMY The mean place and proper motion of the comparison star for 1855.0 given by the catalogue were a = 9" 29™ 17-.57, 8 = + 40° 53' 16 '.1, m = - 0-.0022, fi'= + 0".008. The mean place for 1890.0 was, by § 47, a = 9" 31™ 29".56, 8 = + 40° 43' 58".8 ; and the apparent place for sidereal time 1890 July 23 d 17'', by § 55, a'=9»31»28».90, 8'= + 40° 44' 4".2. Therefore the observed place of the comet at 17* 13 m 39 s was a = 9» 27™ 39 8 .66, 8 = + 40° 42' 20".7. The chronometer correction was — 1™ 19 s . We have Chronometer time, & 17" 13" 39« Chronometer corr., A0 — 1 19 Sidereal time, 17 12 20 Right ascension, a 9 27 40 Hour angle, t 7 44 40 The corrections for differential refraction corresponding to this hour angle and declination, taken from the tables constructed for the Lick Observatory, or computed from (316) and (317), are Aa = - 0».04, A8 = - 0".6. The corrections for parallax at the unit distance, i.e. the parallax factors, taken from tables constructed for the Lick Observatory, or computed from (92), (90) and (93), are Aa = + 0'.56, AS = + 6' .1. The comet's distance was 1.57, and therefore the required corrections for parallax are Aa = + S .36, A8 = + 3 .9. DETERMINATION OF APPARENT PLACE 229 Applying these corrections to the observed place we ob- tain the following apparent place of the comet : Mt. Hamilton sid. time Apparent a Apparent 8 1890 July 23, 17* 13™ 39» 9* 27™ 39".98 + 40° 42' 24".0 155. By the method of direct micrometer measurement. When the object whose position, (a", <$"), is to be deter- mined is comparatively near the comparison star, (a 1 , 8'), so that both are well within the field of view, their differ- ences of right ascension and declination are conveniently determined by direct micrometer measurement. Let the micrometer wire be placed parallel to the equa- tor, as before. By means of the driving clock keep the telescope directed to the star and object so that the point midway between them will be as nearly as possible in the center of the field. Bisect the star's image with the fixed wire and the object's image with the micrometer wire, and note the chronometer time and the reading m of the microm- eter. If m is the reading for coincidence of the wires, and M the value of a revolution of the screw, then the apparent difference AS of the declinations is given by AS = (m - m )R. (319) Rotate the wires through 90°, bisect the two images as before and note the time and the micrometer reading m'. The apparent difference of the right ascensions is given by Aa = (m! -m B )R sec J (8" + 8') . (320) This method cannot be used with safety near the pole unless the instrument is in good adjustment and the differ- ence of right ascension is small. The apparent differences of right ascension and declina- tion will require correction for differential refraction. The corrections could be computed by differential formulae, but an equally satisfactory method consists in computing the absolute refractions in right ascension and declination for the star and object, and taking their differences as the cor- 230 PRACTICAL ASTKONOMY rections to the observed intervals Aa and AS. The values of the parallactic angles and zenith distances, q', z' for the star and q", z" for the object, may be taken from general tables constructed for the point of observation, or computed by (35), (36) and (37). These should be par- tially checked by the formula z i, _ z i _ a 2 = _ cos 8' sin q< Aa — cos q' A8, (321) formed by differentiating (30) and (41) and combining the results with (31). The refraction r' for the star may be computed by means of (95), (96) or (97), remembering that z in these formulae is the apparent zenith distance. Formula (96) will be sufficiently accurate, except in case of observations made very near the horizon. The refrac- tion r" for the object should now be found differentially. The change Ar in refraction due to a change Az in zenith distance is obtained from (96), thus: Ar= 9836_ 2 ;nl „ A „ 22 v 460 + ( *■ ' in which Ar and Az are expressed in terms of the same unit. The refraction for the object will be given by r" = r> + Ar. (323) The corrections to the apparent places may be obtained from (100) and (101), thus : da 1 =-r> sin q< sec 8', d$' =-r l cos q', (324) da" = - r" sin q" sec 8", d8" = - r" cos q". (325) Therefore, we shall have the true differences of right ascen- sion and declination, as seen from the point of observation, a" - a' = Aa + (da" - da'), (326) 8" - 8' = AS + (rf8" - dS'), (327) from which the values of a" and B" may be obtained. If the object is in the solar system, it will further require correction for parallax. DETERMINATION OP APPARENT PLACE 231 Example. Lick Observatory, Friday, 1898 Nov. 11. 12-inch equatorial. Observer, W. J. Hussey. The differ- ences of right ascension and declination of the minor planet Eros and the star DM. — 4°.5413 were measured directly with the micrometer, to determine the apparent place of the planet. The mean of five measures of difference of declination was, (the planet being north of the star), m = 55 r .007 at (sidereal) chronometer time 22* 48™ 21 s .O. The mean of ten measures of difference of right ascension was, (the planet being west of the star), m' = 67--.180 at chronometer time 22 s 52™ 22 8 .7. The mean of five additional measures of difference of declination was to = 55 r .228 at chronometer time 22 s 56*»27«.2. The coincidence of the wires was at 46 r .650 ; the value of one revolution of the screw is 14". 058 ; the chronometer correction was -f 2 m 53 s .7 ; and the apparent place of the star for the instant of observation was [from Karlsruhe Beobaehtungen\ , a' = 21" 13" 10«.63, 8' = - 4° 6' 10".9. The observations for determining AS will be combined as follows, the reductions — 1".4 and — C.001 being applied so that the declination observations will refer to the same instant as the right ascension observations. Chronometer, 22» 48" ' 21-.0 m, 55'.007 22 56 27.2 55 .228 Means, 22 52 24.1 55 .118 Reductions, -1.4 -.001 22 52 22.7 M , AS, 55 .117 46 .650 + 8 .467 R .= + 119".04 Chronometer, 22 52 22.7 m', 67 .180 Chron. corr., + 2 53.7 "io> 46 .650 Sid. time, 6, 22 55 16.4 Aa, Aa, -20.530 R sec (-4° 5. 2) - 289".36 232 PRACTICAL ASTRONOMY Applying these values of Aa and AS to the star's place we have the approximate position of the planet, a" = 21* 12™ 51".3, 8" = - 4° i> 12". The data for solving (35), (36) and (37) are <£ = 37° 20' 26" 6= 22» 55">16«.4 t> = + 1*42™ 6" t" = + 1 42 25 I' = 4- 25° 31' 30" t" = + 25° 36' 15" 8' = - 4 6 11 8" = - 4 4 12 The quantities obtained from the solution are qi = 27° 33' 50" q" = 27° 38' 46" z< = 47 45 42 2" = 47 46 10 The value of z" — z 1 = As = + 28" agrees exactly with that obtained from (321). The mean value of the refraction at true zenith distance z' = 47° 46' is about 1'. Solving (96), (322) and (323) for 2 = 47° 45' and As = + 28", assuming b = 25.8 inches and t = + 55° as the average for observing weather at that season of the year, we obtain »•' = 54".2, r" = 54 .2 + 0".015 = 54".215. Substituting these in (324) and (325) we obtain da' = - 25".14, tfS' = - 48".05, da" = - 25 .22, d&" = - 48 .03 ; and, therefore, from (326) and (327), a" - a' = - 289".36 - 0".08 = - 289".44 = - 19«.30, 8" - 8' = + 119 .04 + .02 = + 119 .06 = + 1' 59".l ; whence a" = 21" 12" 1 50».33, 8 = - 4° 4' 11".8. The distance A of the planet from the earth being 1.225 units, the parallax corrections taken from general tables or computed from (89), (90) and (91), are + S .17 and POSITION ANGLE AND DISTANCE 233 + 4" .7. The apparent place of the planet referred to the center of the earth was therefore Mt. Hamilton sidereal time Apparent a Apparent 8 1898 Nov. 11, 22* 55" 16-.4 21» 12" 50-.50 - 4° M 7".l DETERMINATION OP POSITION ANGLE AND DISTANCE 156. The relative positions of two objects close together are conveniently expressed in terms of position angle and distance. The position angle, p, of one star, B, with refer- ence to another star, A, is the angle which the great circle passing through the two objects makes with the hour circle passing through A, reckoned from the north toward the east through 360°. Their distance, s, is the length of the arc of the great circle joining them. To determine the position angle, revolve the micrometer until one of the stars, by its diurnal motion, follows along the micrometer wire, and note the reading P of the posi- tion circle. Keeping the telescope directed upon the stars by means of the driving clock, revolve the micrometer until the micrometer wire passes through the two stars, and note the circle reading P The position angle re- quired is P = P-(P ± 90°). (328) To determine the distance, revolve the micrometer to the circle reading P ± 90°. Bisect one of the stars with the fixed wire by moving the whole system of wires, then bisect the other star with the micrometer wire, and note the reading m of the micrometer. If m is the reading of the micrometer when the two wires coincide, and It the value of a revolution of the screw, the required distance is s=(m-m )R. (329) In very accurate measurements bisect the two stars as above, and take the reading m. Move the micrometer wire to the other side of the fixed wire, bisect the stars 234 PRACTICAL ASTRONOMY with the wires in that order, and take the reading m'. The distance is now given by s = J (m' - m) R. (330) In this way several systematic and personal errors are eliminated, partially at least. This method is called the method of double distances. The values of s and p will be affected by refractien ; but in the case of double stars, to which the method is especially applied, the correction for refraction may usually be neglected. If the distance between the two stars is large, the tele- scope should be directed so that the two stars will fall on opposite sides of the center of the field, and at equal dis- tances from the center. In this case the measured position angle is the angle between the arc joining the two stars, and the hour circle passing through the middle point of that arc. Let p' and s' be the observed angle and dis- tance. Let their values corrected for refraction be p and s. Let z and q be determined for the point midway be- tween the two stars from (35), (36) and (37), and let k be denned as in (318). It can be shown * that p =p' — k cosec 1" [tan 2 z cos ( p' — q) sin (p 1 — q) — tan z sin q tan J (8 + S')], (331) s = s' + sk [tan 2 z cos 2 (p 1 - q) + 1] . (332) This value of p, referring to the point midway between the two stars, may readily be converted into the position angle of each star with reference to the other star. Let *S", in the position a', 8', represent the western star ; S", in the position a", S", the eastern star ; M the point mid- way between them ; and P the pole of the equator. Let p' be the position angle of the eastern star with reference to the western, and 180° +p" the position angle of the * See Chauvenet's Sph. and Prac. Astronomy, Vol. II, pp. 450-459. POSITION ANGLE AND DISTANCE 235 western star with reference to the eastern. In practice, the declination of one star will always be known ; and if the declination of the other is unknown, its value may be found with sufficient accuracy from equation (338) below. Let S = ^ (S" + 8'). Without sensible error, we may assume S as the declination of the point M defined above. Then, from the triangles PS'M and PS"M we can write [Ohauvenefs Sph. Trig., § 70, (N)], sixip tan/ = — -j S ^ F . , . . , (333) r cos iscosp + sin Jstan \ i, _ sinp ™ — cos %scosp — sin Js tan S ' a--- _ (334) which determine p' and p". Their values will differ very little from p, unless the stars are very near the pole. It is frequently required to convert position angle and distance into the corresponding differences of right ascen- sion and declination. From the triangle PS'S" defined above, we may write [ Chauvenefs Sph. Trig., (44)], sin \ (a" - a') = sin \ s sin \ (p" + p') sec 8 , (335) sin i (8" - 8') = sin \ s cos \ (p" + p<) sec \ (a" - a'), (336) which solve the problem. If the stars are at some distance from the pole, we may safely substitute p for \(p" +p'} in these equations. If the stars are not far from the equator, or if s is rela- tively small, or if only moderate accuracy is required, these equations may be written, a" — a' = s sinp sec 8 , (337) 8" - 8' = s cos .p. (338) Example. 36-inch equatorial, Lick Observatory, Thurs- day night, 1898 November 17. Observer, W. J. Hussey. The position angle and distance of the faint companion of Sirius with reference to the principal component were measured as below. The distance was determined by the 236 PRACTICAL ASTRONOMY method of double distances. The equator reading of the position circle was 111°. 7, and the value of a revolution of the micrometer screw is 9".907. P, 185°.6 to', 56' .199 m, 55'.344 184.7 .182 .350 184.3 .180 .336 184.7 .178 .342 182 .2 .193 .336 185.0 Mean m', 56 .186 __ Mean m, 55 .342 184 .3 m, 55 .342 * MeanP, 184.4 | (m' - to), 0' .432 P , 111 .7 R, 9''.907 72.7 s, 4". 18 + 90 .0 p, 162°.7 The correction for refraction is not appreciable. THE RING MICROMETER 157. This consists of a narrow metal ring, one or both of its edges turned exactly circular, attached to a thin piece of glass in the focal plane of an eyepiece. When the eyepiece is put on the telescope and focused, the ring is also in the focal plane of the object glass. If the times qf transit of two stars over the edges of the ring are observed, the differences of their right ascensions and declinations can be found. But results obtained in this way can be regarded as only approximately correct, and the ring micrometer should never be used with an equatorial telescope unless, in case of great haste, there is not time to attach the filar micrometer and adjust its wires by the diurnal motion. The principal advantage of the ring micrometer is that it can be used with an instrument mounted in altitude and azimuth as well as with an equa- torial, whereas a filar micrometer cannot. 158. To find the radius of the ring. Select two stars whose declinations are accurately known, the difference RING MICROMETER 237 of whose declinations is a little less than the diameter of the ring, and whose right ascensions do not differ more than three or four minutes. Two stars to fulfil these con- ditions can always he found in the Pleiades. When these stars are nearly on the observer's meridian, observe their transits over the edge of the ring. In Fig. 28 let CDD' C represent the ring ; CD the path of one star (a, 8), and t x and £ 2 the observed sidereal times of its transit over C and D ; CD' the path of the other star (a', 8'), and t-[ and f 2 ' the times of its transit over C and D'. Draw MM 1 perpendicu- lar to CD and CD'. Draw the radii CO and C 0, and let r represent their value in seconds of arc. If we put COM = y, COM' = y', we can write OM = r cos y, CM = r sin y, OM' = r cos y', CM' = r sin y' ; and therefore MM 1 = r (cos y ' + cos y) = 2 r cos \ (y' + y) cos \ (y ' - y) , (339) C'.W + CM = r (sin y< + sin y) = 2 r sin J (y' + y) cos i (y' - y), (340) CM' - CM = r (sin y' - sin y) = 2 r cos J (y' + y) sin J (y' - y) . (341) We have MM' = 8' - 8, CM = ^- (t 2 - «0 cos 8, CikP = ^ (*„' - V) cos 8 ' i and if we put i(y' + y) = ^> Hy'-y) = £ > we can write _ C'Jf + CM _ ^ (<„' - «,') cos 8' + J# (<„ - «0 cos 8 tanjl - ]^p - 8' -8 ' v „ CM' -CM ^ (t 2 ' - t^) cos 8' - ^(t^-t^) cos S 4 _ tan£= MW — = 8' -8 ~ ' V ' MM' ' 2 cos A cos £ 2 cos ^4 cos £ (344) 238 PRACTICAL ASTRONOMY The apparent distance between the stars is affected by- refraction. Since the observations are made near the meridian, the refraction in right ascension can be neg- lected, and it will be sufficiently accurate to consider that its effect upon the difference of the declinations is equal to the difference of refraction in zenith distance of the stars when they are on the meridian. The difference of the declinations furnished by the star catalogues requires to be decreased numerically by the difference of the re- fractions before substituting in the above equations. The difference of the refractions may be readily obtained with the assistance of the table in § 111, based upon equation (280), or from the table of mean refractions, Appendix, Table II. Example. Friday, 1889 Jan. 25. The following times of transit of 23 Tauri and 27 Tauri over the outer edge of the ring micrometer of the 12^-inch equatorial of the Detroit Observatory were noted, to determine the radius of the ring. 23 Tauri 27 Tauri t L = 3 s 30 m 41«.5, t{ = 3* 33™ 33 8 .5, t 2 = 3 30 53 .0, q = 3 33 41 .0. The mean places of these stars for 1850.0 are given in NewcomFs Standard Stars. Reducing them to the mean place for 1889.0 by § 52, and thence to the apparent place at the instant of observation by § 55, (5), we obtain 8 = 23° 36' 4".13, 8' = 23° 42' 4o".40. The zenith distance of the stars is about 18° and the difference of their zenith distances is about 7'. Entering the table in § 111 with l (a' - a") = 3'.5 and z = 18°, one- half the difference of the refractions is 0".07, or the whole difference is 0".14. Therefore the apparent difference of declination of the stars is 8' - 8 = 401".13. RING MICROMETER 239 From (342) and (343) we find A = 18° 1' 34", B = 3° 55' 37" ; and then, from (344) r = 211".4. The mean value of r from nine complete sets of transits is r = 210".7 ± 0".18. 159. To determine the difference of the right ascensions and declinations of two stars. Observe the transits over the edge of the ring, as in § 158. Using the notation of § 158, the difference of the right ascensions is given by a' - a = | ( V + t 2 ') - i ('i + Q . (345) Letting OM= d, and OM! = d' we can write Sin y = 3^ (t 2 - tj), sin y' = IE (« 2 ' - , z and A are inserted. From the tables tan z and sin z are found and written oppo- site their symbols ; and likewise cos A, tan A and sin A. The sum of tan z and cos A is tan M. Add them mentally, enter the tables and take out M. Take out sinM at once. Subtract M from cj>, mentally, find sec ($ — M ) and tan ($-M). The value of sec (-M) is 10 -cos ($-M). Add the three logarithms to find tan t. Determine t from the tables and take cos t and cosec t out. The sum of tan ($ — M~) and cos t is tan 8. Add them mentally and take 8 and sec 8 from the tables. The sum of the last four logarithms is log 1. It should not differ more than one or two units of the last place from zero or ten. The printed solution contains every figure that need be written down. But possibly the beginner should write down tan M, — M&nd tan 8. B. Interpolation Formulae The Ephemeris tabulates the values of any required function corresponding to equidistant values of the time. If the value of the function for any intermediate date is desired, it is sometimes determined most conveniently by means of a general interpolation formula. Let The the date in the Ephemeris nearest the instant for which the value of the function is required, and let a> be the tabular interval of time. Then the adjacent dates APPEHDICES 245 in the Ephemeris may be represented as in the first column of the following table, and the corresponding values of the function as in the second column. Subtracting each value Argument Funotion 1st Diff, 2d Diff, 3d Diff, 4th Diff, 5th Diff. 6th Diff. T-3a> f(T - 3 f{T- <-) Oil a, »/ rf, «/ T f(T) [a] 6 W d H / a' c> e' T + o> f{T+ «.) a" 6' c" d> r + 2<» /(r + 2oi) a'" J" r + 3« /(r + 8«) of the function from the next following value, we obtain the " 1st differences " in column three. Subtracting each 1st difference from the next following, we obtain the 2d differences in column four ; and so on, for the differences of higher orders. Lastly, the quantities [a] = \ (a, + a'), [c] =!( ; i.e., n = t/(o, or t = wo). The value of the function required is 2-3 + „(„»-!)(„»_- 4) [8] + S-3-4-5 Example. Determine from the American Ephemeris, page 219, the apparent declination of Mercury at Green- wich mean time 1899 April 2 d 16" 0™ 246 PRACTICAL ASTRONOMY The epoch T is April 2> d . 0, the tabular interval eo is 24\ and t is — 8\ Therefore, n = — J. The functions and differences are as below: Argument Junction a 6 c d e March 3K0 April 1 .0 2.0 + 13°16'34".4 13 22 29 .6 13 24 29 .6 +5'55".2 +2 .0 -1 54 .4 -3'55''.2 -3 54 .4 +0".8 + 3 .2 +2".4 + 0".3 3.0 13 22 35 .2 [-3 50 .0] -3 51 .2 [+4 .6] + 2 .7 [+0 .1] 4.0 6.0 6.0 13 16 49 .6 13 7 18 .7 + 12 54 11 .0 -6 45 .6 -9 30 .9 -13 7 .7 -3 45 .3 -3 36 .8 +5 .9 + 8 .5 +2 .6 -0 .1 The quantities to be taken from this table are all in- cluded, with April 3 d .O, between the two horizontal lines. Substituting these values and n = — i in the above equa- tion we obtain the following values of the individual terms, and their sum, respectively : + 13° 22 35''.2 + 1 16 .7 — 12 .8 + .2 — .1 .0 + 13° 23' 39' .2 f(T+t) It is often required to determine by interpolation the value of a function for a date midway between two tabu- lar dates. The required value is determined as follows : From the arithmetical mean of the two values of the function corresponding to the two tabular dates, subtract one-eighth of the arithmetical mean of the second differ- APPENDICES 247 ences found on the same horizontal lines as the two dates, and add three one hundred and twenty-eighths of the arithmetical mean of the fourth differences found on the same horizontal lines. Example. Determine the apparent declination of Mer- cury at Greenwich mean time 1899 April 3 d .5. From the above data i (13° 22' 35".2 + 13° 1© 49".6) = + 13° 19' 42''.4 - i -U- 3 51.2- 3 45.3) = + 28.5 + Tf*-U+ 2.7+ 2.6)= (KO App. S, 1899 April 3<*.5 = + 13° 20' 10".9 C. Combination and Comparison of Observations Formula, resulting from the Method of Least Squares 1. Direct observations of a quantity : n separate results, m v m v ••• m n of equal weight. Most probable value of quantity, z = i^±. Kesiduals, z — m 1 = v v z - m 2 = v 2 , ■■■ z - m„ = v n . Probable error of z, ,- = ± 0.6745 * / W Yn(ra-l) Probable error of a single observation, r = ± 0.6745 A /JZ!!]_. \n — 1 2. Direct observations of a quantity : n separate results, m v m 2 , ••• m n of unequal weights, p v p 2 , •••p n . Most probable value of quantity, z = <-P™l . \_pvv~] Probable error of z, r n = ± 0.6745 \/^^r V[>](n-1) Probable error of an obs'n of weight unity, r = ± 0.6745 yj LEO^A. * The symbols [ ] signify the sum of all similar quantities. Thus [m]=mi + wi 2 + ••• + m n . [pvv] =x>\, i = 5? + *!+...*!. ^ ft ft ft 4. In general, if Z =f(z v z 2 , ■■• z n ), the probable error of Z is '-±V(£)V + ($ , * + - + (£)V 5. Direct observations of a function of a quantity z: the separate results, m v m 2 , •■• m n of equal weight, and the form of the function, az. The observation equations are OjZ + mj = 0, 022 + m 2 = 0, a„z + m n = 0. The most probable value of z and its probable error are z = - t_J r = ± 0.6745-y L J • [_aaj \ [aaj (re — 1) If the observations are of unequal weights, multiply the observation equations through by the square roots of their respective weights, and proceed as before. 6. Direct observations of a function of two quantities, w and z: the separate results, m v m 2 , . . . m n of equal weights, and the form of the function, aw + bz. The ob- servation equations are a^ + ftjZ + m l = 0, a 2 w + b^z + m, = 0, a n w + J„2 + m„ = 0. APPENDICES 249 The normal equations are [aa\ w + [a6] z + [am\ = 0, [aft] w + [66] z + [6m] = 0. Let [W] - [S] ^ = t^ 1 ]' [*»] - ^ [«"•] = [6m.l]. Then the most probable values of w and z are given by z [frm.l] - [66.1]' [a6] [ami [aa] [aa] The weights of w and z are A = [«■!]. *. = -^[aa]. The probable error of a single observation (of weight unity) is ■2' and the probable errors of w and z are r r«, = > r, = ■ If the observations are of unequal weights, multiply the observation equations through by the square roots of their respective weights, and proceed as before. 7. Direct observations of a function of three quantities, x, y and z : the separate results, rn v m 2 , . . . m n of equal weight, and the form of the function, ax + by + ce. The observation equations are a r x + b-tf) + CjZ + m l = 0, a%c + b^y + c%n + m 2 = 0, a„x + 6„y + c n z + m„ = 0. The normal equations are [aa] x + [ab] y + [ac] z + [am] = 0, [a6] a + [66] y + [6c] z + [6m] = 0, [ac] £ + [6c] y + [cc] z + [cm] = 0. 250 P3 JACTICAI Let [»] [aa] = [66.1], [6m] [aa] L J = [6m.l], M [ ac ] r n [aa] L = [cc.l], M -^[«]-Pcil Tcm] — i — r[ am 3 = [cm.l], L [aa] [cc.l] - j^}[fc.l] = [w-2], [c«.l] -gg [i«.l] = [<™.2]. Then the most probable values of a;, «/ and z are given by [cm.2] Z ~ [cc.2]' [&cl] [ftm.1] [66.1] [66.1]' [a6] [ac] [am] [aa] [aa] [aa] The weights of x, y and 2 are given by p.= [pel], [cc.2] a=M cm-1] ' [cc.2] [66.1] r n * = feit [66] M ' in which [cc.l] a = [cc]-gl[6c]. The probable error of a single observation (of weight unity) is r = ± 0.6745 JjS, \n — 3 and the probable errors of x, y and z are VS ^ Vi^ If the observations are of unequal weights multiply the observation equations through by the square roots of their respective weights, and proceed as before. APPENDICES 251 D. Objects for the Telescope Besides the moon, the planets and the Milky Way, the objects in the following list will be of interest to the stu- dent. Fuller descriptions of them, with many valuable hints on the use of the telescope, can be found in Webb's Celestial Objects for Common Telescopes, which is an excel- lent guide for the observer. Every student should provide himself with a good star atlas. Klein's Star Atlas, or Heiss Atlas Coelestis is recommended. a 1 1900.0 5, 1900.0 Object: description : remarks 0" 37"* .3 + 40° 43' The Great Nebula in Andromeda. One of the most interesting in the sky, large, 2J° by 4°, easily visible to the naked eye. A small companion nebula lies 22' south. 53 .4 + 81 20 U Oephei, variable, 7 m .l to 9<".2, period 2<*.5. 1 18 .9 + 67 36 \f> Cassiopeice, triple, Ai m .5, B9 m , C10™. AB = 30", BC=Z". 1 22 .6 + 88 46 a Ursce Minoris or Polaris, the standard 2 m star ; a 9™ companion at s = 18". 5. 1 48 .0 + 18 48 1 Arietis, double, 4 m .5 and 5™, p = 179°, s = 8". y Andromeda,, double, 3™.5 and 5™.5, p = 1 57 .7 + 41 51 63°, s = 10". The 5™.5 is also double, but close and difficult even for the largest telescopes. 2 12 .0 + 56 41 Cluster in Perseus. A magnificent object with a low power. Another fine cluster 3 minutes east. 2 14 .3 - 3 26 o Ceti, interesting variable, irregular, 1™.7 to 9™. 5, period about SSI*. 3 1 .7 + 40 34 p Persei (Algol), interesting variable, 2™. 3 to 3™. 5, period 2<* 20» 48™ 55". 3 40 .2 + 23 27 Nebula in the Pleiades, very faint and diffi- cult, Merope in its north extremity. 4 7 .6 + 50 59 Cluster in Perseus, good with low power. 4 9 .6 -13 Planetary nebula in Eridanus, circular, 12™ star in center. 4 30 .2 + 16 19 a Tauri (Aldebaran) ,- 1™ star, red. 5 9 .3 + 45 54 a Auriga, (Capella), 1™ star. 5 9 .7 - 8 19 /3 Orionis (Bigel), double, l" 1 and 9™, s = 9". 5. The 9™ is a close double, very dif- ficult even with the largest instruments. 5 28 .5 + 21 57 Nebula in Taurus large, faint, oblong. 252 PRACTICAL ASTRONOMY <*! 1900.0 S, 1900.0 Object; description! remarks 6» 30" '.4 - 5° 27' The Great Nebula in Orion, one of the most interesting nebulae in the sky, about 3° by 5° in size. Near its densest part is the multiple star 6 Orionis, called the Trapezium. The spectrum of the nebula indicates a gaseous composition. 5 35 .7 - 2 i Orionis, triple, A 3™, B6 m .b, CIO™, AB = 2".5, ^tC = 57". 6 2 .7 + 24 21 Cluster in Gemini, fine field with low power. 6 37 .4 + 59 33 12 Lyncis, triple, A 6", S6™.5, 07™ 5, AB = 1".5, ^4C = 8".6. 6 40 .7 - 16 34 a Canis Majoris (Sirius), the brightest star in the sky. A close 10 mag. companion is now (1899) difficult in powerful tele- scopes. 7 14 .1 + 22 10 5 Geminorum, double, one yellow 3™.5, the other red S m , p = 205°, s = 7". 7 28 .2 + 32 6 a Geminorum (Castor), fine double, 3™ and 3™.5, p = 226°, s = 5".7. 7 34 .1 + 5 29 a Canis Minoris (Procyori) 1™, with 13™ companion discovered in 1896, difficult with large instruments. At discovery, i> = 320°, s = 4".7. 8 34 .5 + 20 17 Cluster in Cancer (Prcesepe), fine field with low power. 8 45 .7 + 12 10 Cluster in Cancer, about 200 stars, 9™ to 15™. 9 47 .2 + 69 36 Nebulae in Ursa Major, two nebulas, 30' apart, preceding one brighter with bright nucleus. 10 14 .4 + 20 21 y Leonis, fine double, 2™ and 3™. 5. In 1897, p = 115°, s = 3".8. 10 19 .9 - 17 39 Planetary Nebula in Hydra, fairly bright. 11 12 .5 + 59 19 Nebula in Ursa Major, small, bright, with nucleus. 11 47 .7 + 37 33 Nebula in Ursa Major, bright, 3' to 4' in diameter. 12 5 .0 + 19 6 Cluster in Coma Berenices, globular, bright, well resolved in large telescopes. 12 34 .8 - 11 4 Nebula in Virgo, elliptical, 30" by 5', fine field with low power. 12 36 .6 - 54 y Virginis, double, 4™ and 4™. In 1898, p = 330°, s = 6". 13 19 .9 + 55 27 f Urs(B Majoris, fine double, 3™ and 5™, s = 14". 13 37 .5 + 28 52 Cluster in Canes Venatici, bright, globular, probably more than 1,000 stars. 14 11 .1 + 19 43 a Bootis (Arcturus), l m star, yellow. APPENDICES 253 1900.0 S, 1900.0 Object: description: remarks 14" 40 m .6 15 14 .1 16 23 .3 16 37 .5 16 38 .1 16 40 .3 17 10 .1 17 11 .5 17 61 .1 17 58 .6 18 7 .3 18 33 .6 18 41 .0 18 46 .4 18 49 .8 19 26 .7 19 48 .5 19 55 .2 20 42 .0 20 58 .7 21 2 .4 21 8 .2 21 28 .2 22 23 .7 23 21 .1 + 27° 30' + 32 1 -26 13 + 31 47 + 36 39 + 23 59 + 14 30 + 1 19 - 18 59 + 66 38 + 6 50 + 38 41 + 39 34 + 33 15 + 32 54 + 27 45 + 70 1 + 22 27 + 15 46 -11 45 + 38 15 + 68 5 - 1 16 - 32 + 41 59 e Bo'dtis, beautiful double, 3™ yellow and 7™ blue, s = 3", p = 328°. U Coronas, variable, 7 m .5 to 8™.9, period 3* 10» 51». o Scorpii, double, 1"» and 7«", s = 3". f Herculis, double, 3™ and 6"*, s = 0".6 in 1899, period about 35 years, now (1899) very difficult in large instruments. The Cluster in Hercules, globular, one of the finest of its kind. [eter. Nebula in Hercules, planetary, 8" in diam- a Herculis, variable, 3"'.l to 3™. 9, irregular period ; companion 6™.6 at p = 116°, s = 4".7. V Ophiuchi, 6">.0 to 6™. 7, period 20» 8». Cluster in Ophiuchus, good field with low power. Nebula in Draco, planetary, bright, diam- eter 35", very near pole of ecliptic, very interesting. Nebula in Ophiuchus, planetary, bright, diameter 5". a Lyras (Vega), brightest star in northern hemisphere. e Lyra,, a multiple star, A 5"*, B 6™. 5, C 5™, D 5».5, AB 3", CD 2".3, AC 207". Nu- merous small stars between AB and CD. (3 Lyrce, variable, 3 m .4 to 4 m .5, period 12* 21» 47". Bing Nebula in Lyra, annular, gaseous, most interesting of its kind. |3 Cygni, fine double, 3™ yellow and 7™ blue, p = 56°, s = 35". e Draconis, double, 5 m .5 and 7™.5, s = 3". Nebula in Vulpecula, the "Dumb Bell Neb- ula," double, large. 7 Delphini, double, 4™ and 6», s = 11". Nebula in Aquarius, planetary, bright, very interesting in a large telescope. 61 Cygni, double, 5»5 and 6™, s = 21", one of the nearest stars to us. T Cephei, variable, 5»6 to 9».9, period 383<*. Cluster in Aquarius, large, globular. i Aquarii, double, 4» and 4™.5, s = 3".3 in 1897. Nebula in Andromeda, planetary, small, very bright, round. 254 PRACTICAL ASTRONOMY Table I. Pulkowa Refraction Tables App't z log^ K App't z logM \ App't z logM \ A o 1 1.76032 a 1 71 1.75614 1.0115 o ; 77 1.75131 1.0253 1.0029 5 1.76032 10 1.75606 1.0118 10 1.75107 1.0259 1.0029 10 1.76030 2Q 1.75598 1.0120 20 1.75083 1.0264 1.0030 15 1.76028 30 1.75590 1.0123 30 1.75058 1.0271 1.0030 20 1.76025 40 1.75582 1.0125 40 1.75032 1.0278 1.0031 25 1.76021 50 1.75573 1.0128 50 1.75005 1.0285 1.0032 30 1.76015 72 1.75564 1.0130 78 1.74976 1.0293 1.0033 35 1.76006 10 1.75555 1.0133 10 1.74947 1.0300 1.0033 40 1.75995 20 1.75546 1.0136 20 1.74917 1.0309 1.0034 45 1.75980 1.0018 30 1.75536 1.0138 30 1.74886 1.0318 1.0035 50 1.75960 1.0022 40 1.75526 1.0141 40 1.74853 1.0327 1.0036 51 1.75955 1.0024 50 1.75516 1.0144 50 1.74819 1.0335 1.0037 52 1.75949 1.0025 73 1.75506 1.0147 79 1.74783 1.0344 1.0038 53 1.75943 1.0026 10 1.75496 1.0150 10 1.74746 1.0354 1.0039 54 1.75936 1.0027 20 1.75485 1.0153 20 1.74707 1.0364 1.0040 55 1.75928 1.0029 30 1.75474 1.0157 30 1.74665 1.0374 1.0041 56 1.75920 1.0032 40 1.75462 1.0160 40 1.74623 1.0.385 1.0042 57 1.75912 1.0035 50 1.75450 1.0163 50 1.74579 1.0397 1.0043 58 1.75902 1.0038 74 1.75438 1.0166 80 1.74533 1.0409 1.0044 59 1.75892 1.0041 10 1.75425 1.0170 10 1.74484 1.0421 1.0045 60 1.75881 1.0044 20 1.75412 1.0173 20 1.74433 1.0433 1.0046 61 1.75868 1.0047 30 1.75398 1.0177 30 1.74380 1.0447 1.0048 62 1.75853 1.0051 40 1.75384 1.0181 40 1.74325 1.0461 1.0049 63 1.75837 1.0055 50 1.75369 1.0185 50 1.74266 1.0475 1.0050 64 1.75820 1.0059 75 1.75354 1.0188 81 1.74204 1.0491 1.0052 65 1.75801 1.0064 10 1.75338 1.0191 10 1.74139 1.0508 1.0053 66 1.75780 1.0070 20 1.75322 1.0195 20 1.74071 1.0525 1.0055 67 1.75755 1.0077 30 1.75306 1.0200 30 1.73999 1.0542 1.0057 68 1.75727 1.0085 40 1.75289 1.0205 40 1.73924 1.0561 1.0059 69 1.75694 1.0093 50 1.75271 1.0211 50 1.73844 1.0580 1.0061 70 1.75657 1.0103 76 1.75253 1.0216 82 1.73760 1.0600 1.0063 10 1.75650 1.0105 10 1.75235 1.0223 10 1.73671 1.0622 1.0065 20 1.75643 1.0107 20 1.75216 1.0229 20 1.73577 1.0645 1.0068 30 1.75636 1.0109 30 1.75196 1.0235 30 1.73478 1.0669 1.0070 40 1.75629 1.0111 40 1.75175 1.0241 40 1.73373 1.0694 1.0073 50 1.75622 1.0113 50 1.75153 1.0246 50 1.73260 1.0720 1.0076 71 1.75614 1.0115 77 1.75131 1 1.0253 83 1.73143 1.0747 1.0078 Supplement App't z log Mtanz X A App't . z log Mtan .r X A o 1 82 30 83 83 30 84 84 30 85 85 30 86 2.61534 2.64226 2.67076 2.70088 2.73294 2.76717 2.80376 2.84304 1.0669 1.0747 1.0839 1.0949 1.1080 1.1235 1.1424 1.1652 1.0070 1.0078 1.0087 1.0098 1.0112 1.0127 1.0148 1.0172 o 1 86 30 87 87 30 88 88 30 89 89 30 90 2.88535 2.93113 2.98087 3.03519 3.09458 3.15994 B.23206 3.31186 1.1934 1.2277 1.2708 1.3241 1.3902 1.4729 1.5762 1.7046 1.0203 1.0241 1.0294 1.0357 1.0437 1.0541 1.0680 1.0859 APPENDICES Table I. Pcjlkowa Refraction Tables B. Factor depending on the Barometer 255 English inohea log B English inches log B French metres log B 25.0 - 0.07330 28.0 - 0.02409 0.724 - 0.01621 25.1 - 0.07157 28.1 - 0.02254 0.726 - 0.01500 25.2 - 0.06984 28.2 - 0.02099 0.728 - 0.01380 25.3 - 0.06812 28.3 - 0.01946 0.730 -0.01261 25.4 - 0.06641 28.4 - 0.01793 0.732 -0.01142 25.5 - 0.06470 ,28.5 - 0.01640 0.734 - 0.01024 25.6 - 0.06300 28.6 - 0.01488 0.736 - 0.00906 25.7 - 0.06131 28.7 - 0.01336 0.738 - 0.00788 25.8 - 0.05962 28.8 -0.01185 0.740 - 0.00670 25.9 - 0.05794 28.9 - 0.01036* 0.742 - 0.00553 26.0 - 0.05627 29.0 - 0.00885 0.744 - 0.00436 26.1 - 0.05640 29.1 - 0.00735 0.746 - 0.00319 26.2 - 0.05294 29.2 - 0.00586 0.748 - 0.00203 26.3 - 0.05129 29.3 - 0.00438 0.750 - 0.00087 26.4 - 0.04964 29.4 - 0.00290 0.752 + 0.00028 26.5 - 0.04800 29.5 - 0.00142 0.754 + 0.00144 26.6 - 0.04636 29.6 + 0.00005 0.756 + 0.00259 26.7 - 0.04473 29.7 0.00151 0.758 + 0.00374 26.8 - 0.04311 29.8 0.00297 0.760 + 0.00488 26.9 - 0.04149 29.9 0.00443 0.762 + 0.00602 27.0 " - 0.03988 30.0 0.00588 0.764 + 0.00716 27.1 - 0.03827 30.1 0.00732 0.766 + 0.00830 27.2 -0.03667 30.2 0.00876 0.768 + 0.00943 27.3 - 0.03508 30.3 0.01020 0.770 + 0.01056 27.4 - 0.03349 30.4 0.01163 0.772 + 0.01168 27.5 - 0.03191 30.5 0.01306 0.774 + 0.01281 27.6 - 0.03033 30.6 0.01448 0.776 + 0.01393 27.7 - 0.02876 30.7 0.01589 0.778 + 0.01505 27.8 - 0.02720 30.8 0.01731 0.780 + 0.01616 27.9 - 0.02564 30.9 0.01871 0.782 + 0.01727 28.0 - 0.02409 31.0 + 0.02012 0.784 + 0.01837 T. Factor depending on Attached Thermometer Fahr, log T Gent. log T -20° + 0.00201 -30° + 0.00209 -10 0.00162 -25 0.00174 0.00123 -20 0.00139 + 10 0.00085 -15 0.00104 20 0.00047 -10 0.00069 30 + 0.00008 - 5 + 0.00035 40 - 0.00030 0.00000 50 - 0.00069 , „ + 5 - 0.00035 60 -0.00108 10 - 0.00069 70 - 0.00146 15 - 0.00104 80 - 0.00184 20 - 0.00138 90 - 0.00222 25 - 0.00173 + 100 - 0.00262 + 30 - 0.00207 256 PRACTICAL ASTRONOMY Table I. Pulkowa Refraction Tables 7. Factor depending on External Thermometer Fair. log 7 Fahr. log 7 Gent. log 7 -22° + 0.06560 + 35° + 0.01200 -30° + 0.06560 -21 ■ 0.06461 36 0.01112 -29 0.06381 -20 0.06361 37 0.01023 -28 0.06202 -19 0.06262 38 0.00935 -27 0.06023 -18 0.06162 39 0.00848 -26 0.05846 -17 0.06063 40 0.00760 -25 0.05669 -16 0.05964 41 0.00672 -24 0.05493 -15 0.05866 42 0.00585 -23 0.05317 -14 0.05767 43 0.00498 -22 0.05142 -13 0.05669 44 0.00411 — 21 0.04968 -12 0.05571 45 0.00324 -20 0.04795 • -11 0.05473 46 0.00238 — 19 0.04622 -10 0.05376 47 0.00151 -18 0.04451 - 9 0.05279 48 + 0.00064 — 17 0.04279 - 8 0.05182 49 - 0.00022 -16 0.04108 - 7 0.05085 50 — 0.00107 -15 0.03938 - 6 0.04988 51 - 0.00193 -14 0.03769 - 5 0.04891 52 — 0.00279 -13 0.03601 — 4 0.04795 53 - 0.00364 -12 0.03433 - 3 0.04699 54 — 0.00449 -11 0.03265 - 2 0.04603 55 — 0.00535 -10 0.03099 - 1 0.04508 56 - 0.00620 - 9 0.02933 0.04413 57 - 0.00704 — 8 0.02767 + 1 0.04318 58 — 0.00789 — 7 0.02602 2 0.04223 59 - 0.00873 - 6 0.02438 3 0.04128 60 — 0.00957 — 5 0.02274 4 0.04033 61 — 0.01041 — 4 0.02112 5 0.03938 62 — 0.01125 — 3 0.01950 6 0.03844 63 - 0.01209 — 2 0.01788 7 0.03750 64 — 0.01293 — 1 0.01027 8 0.03657 65 - 0.01376 0.01466 9 0.03563 66 — 0.01459 + 1 0.01306 10 0.03470 67 — 0.01543 2 0.01147 11 0.03377 68 — 0.01626 3 0.00988 12 0.03284 69 — 0.01709 4 0.00830 13 0.03191 70 - 0.01792 5 0.00672 14 0.03099 71 — 0.01874 6 0.00515 15 0.03007 72 - 0.01956 7 0.00359 16 0.02915 73 - 0.01838 8 0.00203 17 0.02822 74 — 0.01920 9 + 0.00047 18 0.02730 75 - 0.02202 10 — 0.00107 19 0.02639 76 — 0.02284 11 — 0.00261 20 0.02548 77 - 0.02366 12 — 0.00415 21 0.02456 78 — 0.02447 13 - 0.00569 22 0.02364 79 — 0.02528 14 - 0.00721 23 0.02273 80 - 0.02609 15 — 0.00873 24 0.02183 81 - 0.02690 16 — 0.01025 25 0.02094 82 — 0.02771 17 — 0.01176 26 0.02004 83 — 0.02851 18 — 0.01326 27 0.01914 84 - 0.02932 19 - 0.01476 28 0.01824 85 - 0.03012 20 — 0.01626 29 0.01734 86 - 0.03093 21 — 0.01775 30 0.01645 87 — 0.03173 22 - 0.01923 31 0.01555 88 — 0.03253 23 - 0.02071 32 0.01466 89 - 0.03333 24 - 0.02219 33 0.01377 90 - 0.03413 25 — 0.02366 34 0.01288 91 — 0.03492 30 — 0.03093 + 35 0.01200 | 92 - 0.03572 + 35 — 0.03810 APPENDICES 257 Table I. Pulkowa Refraction Tables logo- App't z logo App't z logo o 1 80 0.00019 ° i 85 0.00146 80 30 0.00022 85 30 0.00185 81 0.00025 86 0.00241 81 30 0.00029 86 30 0.00320 82 0.00035 87 0.00421 82 30 0.00045 87 30 0.00561 83 0.00057 88 0.00749 83 30 0.00073 88 30 0.01006 84 0.00091 89 0.01352 84 30 0.00116 89 30 0.01813 85 0.00146 90 0.02424 Date i Jan. 15 + 0.34 i'eb. 15 + 0.27 Mar. 15 + 0.05 April 15 -0.08 May- 15 -0.20 June 15 -0.26 July 15 -0.33 Aug. 15 -0.30 Sept. 15 -0.19 Oct. 15 + 0.16 Nov. 15 + 0.33 Dec. 15 + 0.37 logr = log/i + log tan z + A (\o%B + log T) + Mog7 + ilogo Table IT. Pulkowa Mean Refractions Bafom. 29.5 inches, Att. Therm. 50° F., Ext. Therm. 50° F. App't Mean App't Mean App't Mean App't Mean z refr'n z refr'n z refr'n z refr'n o / i ii o ) / ii o 1 / // ° l / ii 0.0 58 1 31.2 73 3 4.7 80 40 5 34 5 5.0 59 1 34.8 73 20 3 8.5 81 5 46 10 10.1 60 1 38.7 73 40 3 12.5 81 20 5 58 15 15.3 61 1 42.8 74 3 16.6 81 40 6 12 20 20.8 62 1 47.1 74 20 3 20.9 82 6 26 25 26.7 63 1 51.7 74 40 3 25.4 82 20 6 41 30 33.0 64 1 56.6 75 3 30.0 82 40 6 58 32 35.7 65 2 1.9 75 20 3 34.8 83 7 15 34 38.5 65 30 2 4.7 75 40 3 39.9 83 20 7 35 36 41.5' 66 2 7.6 76 3 45.2 83 40 7 56 38 44.6 66 30 2 10.6 76 20 3 50.7 84 8 19 40 47.9 67 2 13.8 76 40 3 56.5 84 20 8 43 42 51.4 67 30 2 17.1 77 4 2.5 84 40 9 10 44 55-1 68 2 20.5 77 20 4 8.8 85 9 40 46 59.1 68 30 2 24.1 77 40 4 15.5 85 30 10 32 48 1 3.4 69 2 27.8 78 4 22.5 86 11 31 50 1 8.0 69 30 2 31.7 78 20 4 29.8 86 30 12 42 52 1 13.0 70 2 35.7 78 40 4 37.6 87 14 7 53 1 15.7 70 30 . 2 39.9 79 4 45.7 87 30 15 49 54 1 18.5 71 2 44.4 79 20 4 54.4 88 17 55 55 1 21.4 71 30 2 49.1 79 40 5 3.5 88 30 20 33 56 1 24.5 72 2 54.0 80 5 13.1 89 23 53 57 1 27.8 72 30 2 59.2 80 20 5 23.4 89 30 28 11 58 1 31.2 73 3 '4.7 80 40 5 34.3 90 33 51 258 PRACTICAL ASTRONOMY 2 sin 2 it 2 sin 2 A (to - «1 Table III. m sinl" ' sinl" t, or h-t m t, or m t, or h-t m t, or h-t m t, or h-t m 771 8 « 1)1 s » m s " ni 8 " m 8 " 0.00 4 31.42 8 125.65 12 282.68 16 502.5 5 0.01 4 5 32.74 8 5 128.28 12 5 286.62 16 5 507.7 10 0.05 4 10 34.09 8 10 130.94 12 10 290.58 16 10 513.0 15 0.12 4 15 35.46 8 15 133.63 12 15 294.58 16 15 518.3 20 0.22 4 20 36.87 8 20 136.34 12 20 298.60 16 20 523.6 25 0.34 4 25 38.30 8 25 139.08 12 25 302.64 16 25 529.0 30 0.49 4 30 39.76 8 30 141.85 12 30 306.72 16 30 534.3 35 0.67 4 35 41.25 8 35 144.64 12 35 310.82 16 35 539.7 40 0.87 4 40 42.76 8 40 147.46 12 40 314.95 16 40 545.2 45 1.10 4 45 44.30 8 45 150.31 12 45 319.10 16 45 550.6 50 1.36 4 50 45.87 8 50 153.19 12 50 323.29 16 50 556.1 55 1.65 4 55 47.46 8 65 156.09 12 55 327.50 16 65 561.6 1 1.96 5 49.09 9 159.02 13 331.74 17 567.2 1 5 2.31 5 5 50.73 9 5 161.98 13 5 336.00 17 5 572.8 1 10 2.67 5 10 52.41 9 10 164.97 13 10 340.30 17 10 578.4 1 15 3.07 5 15 54.11 9 15 167.97 13 15 344.62 17 15 584.0 1 20 3.49 5 20 55.84 9 20 171.02 13 20 348.97 17 20 589.6 1 25 3.94 5 25 57.60 9 25 174.08 13 25 353.34 17 25 595.3 1 30 4.42 6 30 59.40 9 30 177.18 13 30 357.74 17 30 601.0 1 35 4.92 5 35 61.20 9 35 180.30 13 35 362.17 17 35 606.8 1 40 5.45 5 40 63.05 9 40 183.46 13 40 366.64 17 40 612.5 1 45 6.01 6 45 64.91 9 45 186.63 13 45 371.11 17 45 618.3 1 50 6.60 5 50 66.81 9 50 189.83 13 50 375.12 17 50 624.1 1 55 7.21 5 55 68.73 9 55 193.06 13 55 380.17 17 55 630.0 2 7.85 6 70.68 10 196.32 14 384.74 18 635.9 2 5 8.52 6 5 72.66 10 5 199.60 14 5 389.32 18 5 641.7 2 10 9.22 6 10 74.66 10 10 202.92 14 10 393.94 18 10 647.7 2 15 9.94 6 15 76.69 10 15 206.26 14 15 398.58 18 15 653.6 2 20 10.69 6 20 78.76 10 20 209.62 14 20 403.26 18 20 659.6 2 25 11.47 6 25 80.84 10 25 213.02 14 25 407.96 18 25 665.6 2 30 12.27 6 30 82.95 10 30 216.44 14 30 412.68 18 30 671.6 2 35 13.10 6 35 85.09 10 35 219.88 14 35 417.44 18 36 677.7 2 40 13.96 6 40 87.26 10 40 223.36 14 40 422.23 18 40 683.8 2 45 14.85 6 45 89.45 10 45 226.86 14 45 427.04 18 45 689.9 2 50 15.76 6 50 91.68 10 50 230.39 14 50 431.87 18 50 696.0 2 55 16.70 6 55 93.92 10 55 233.95 14 55 436.73 18 55 702.2 3 17.67 7 96.20 11 237.54 15 441.63 19 708.4 3 5 18.67 7 5 98.50 11 5 241.14 15 5 446.55 19 5 714.6 3 10 19.69 7 10 100.84 11 10 244.79 15 10 451.50 19 10 720.9 3 15 20.74 7 15 103.20 11 15 248.45 15 15 456.47 19 15 727.2 3 20 21.82 7 20 105.68 11 20 252.15 15 20 461.47 19 20 733.5 3 25 22.92 7 25 107.99 11 25 255.87 15 25 466.50 19 25 739.8 3 30 24.05 7 30 110.44 11 30 259.62 15 30 471.55 19 30 746.2 3 35 25.21 7 35 112.90 11 35 263.39 15 35 476.64 19 35 752.6 3 40 26.40 7 40 115.40 11 40 267.20 15 40 481.74 19 40 759.0 3 45 27.61 7 45 117.92 11 45 271.02 15 45 486.88 19 45 765.4 3 50 28.85 7 50 120.47 11 50 274.88 15 50 492.05 19 50 771.9 3 55 30.12 7 55 123.05 11 55 278.76 15 55 497.23 19 55 778.4 4 31.42 8 125.65 12 282.68 16 502.46 20 784.9 APPENDICES 259 Table III. m= 2si ° 2 !', or m= l^*«o-Q sin 1" sinl" 2sin