THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES SOUTHERN BRANCH, UNIVERSITY OF CALIFORNIA, LIBRARY, (LOS ANGELES, CALIF. - 3-3 ;&Uj0tt. MANUAL OP SPHERICAL AND PRACTICAL ASTRONOMY: EMBRACING THE GENERAL PROBLEMS OF SPHERICAL ASTRONOMY. THE SPECIAL APPLICATIONS TO NAUTICAL ASTRONOMY, AND THE THEORY AND USE OF FIXED AND PORTABLE ASTRO- NOMICAL INSTRUMENTS. WITH AN APPENDIX ON THE METHOD OF LEAST SQUARES BY WILLIAM CHAUVENET, PROFESSOR OF MATHEMATICS AND ASTRONOMY IX WASHINGTON ONIV1K8ITT, SAINT LOU VOL. 1. SPHERICAL ASTRONOMY. 43235 FIFTH EDITION, REVISED AND CORRECTED. PHILADELPHIA : J. B. LIPPINCOTT COMPANY. LONDON: lu JOHN STKEET, ADELPHI. EnUred, according to Act of Congress, in tbe year 1863, by J. B. LIPPINCOTT 4 CO. I the OJerk's Office of the District Court of the United States for the Earterc District of Pennsylvania. Copyright, 1891, by CATHERINE CHAUVRNET, WIDOW, AND CHILDREN OF WILLIAM CHAUVRNKT, DECKASRD. 00 PREFACE THE methods of investigation adopted in this work are ^ in accordance with what may be called the modern school <\ of practical astronomy, or more distinctively the Ger- man school, at the head of which stands the unrivalled BESSEL. In this school, the investigations both of the \ general problems of Spherical Astronomy and of the Theory of Astronomical Instruments are distinguished by the gene- ^ rality of their form and their mathematical rigor. When ( ^ approximative methods are employed for convenience in practice, their degree of accuracy is carefully determined by means of exact formulae previously investigated ; the latter being developed in converging series, and only such terms of these series being neglected as can be shown to be insen- ^ sible in the cases to which the formulae are to be applied. ^ And it is an essential condition of all the methods of com- < putation from data furnished by observation, that the errors 5th Method. By a single altitude at a given time SiO 6th Method. By the change of altitude near the prime vertical 311 7th Method. By the Pole Star 311 8th Method. By two altitudes with the elapsed time between them 313 CHAPTER VII. FINDING THE LONGITUDE BY ASTRONOMICAL OBSERVATIONS 317 1st Method. By portable chronometers 317 Chronometric expeditions 323 2d Method. By signals 337 Terrestrial signals 337 Celestial signals, (a) Bursting of a meteor. (6) Beginning or end- ing of an eclipse of the moon, (c) Eclipses of Jupiter's satellites, (rf) Occultations of Jupiter's satellites, (e) Transits of the satel- lites over Jupiter's disc. (/) Transits of the shadows of the satel- lites over Jupiter's disc, (g) Eclipses of the sun, Occultations of - stars and planets by the moon. [See Chapter X.] 339 8d Method. By the electric telegraph 341 Method of star signals v 342 4th Method. By moon culminations 350 PEIRCE'S method of correcting the ephemeris 358 Combination of moon culminations by weights , 363 5th Method. By azimuths of the moon, or transits of the moon and a star over the same vertical circle 371 6th Method. By altitudes of the moon 382 (A.) By the moon's absolute altitude 383 (B.) By equal altitudes of the moon and a star observed with the Zenith Telescope 386 7th Method. By lunar distances 393 (A.) Rigorous method 395 (B.) Approximative method 402 CONTENTS. MM FINDING THE LONGITUDE AT SEA 420 By chronometers 420 By lunar distances 422 By the eclipses of Jupiter's satellites 423 By the moon's altitude 423 By occultations of stars by the moon 424 CHAPTER VIII. FINDING A SHIP'S PLACE AT SEA BY CIRCLES OF POSITION SCMNER'S METHOD... 424 CHAPTER IX. THE MERIDIAN LINE AND VARIATION OF THE COMPASS 429 CHAPTER X. ECLIPSES 436 Bolar Eclipses. Prediction for the earth generally 436 Fundamental equations 439 Outline of the shadow 456 Rising and setting limits 466 Curve of maximum in horizon 475 Northern and southern limits 480 Curve of central eclipse 491 Limits of total or annular eclipse 498 Prediction for a given place 505 Correction for atmospheric refraction in eclipses 515 Correction for the height of the observer above the level of the sea 517 Application of observed solar eclipses to the determination of terrestrial longi- tudes and the correction of the elements of the computation 518 Lunar eclipses 542 Occultations of fixed stars by the moon 549 Terrestrial longitudes from occultations of stars 550 Prediction of occultations 557 Limiting parallels 561 Occultations of planets by the moon 565 Apparent form of a planet's disc 566 Terrestrial longitude from occultations of planets 7. 578 Transits of Venus and Mercury 591 Determination of the solar parallax 592 Prediction for the earth generally 593 Occultation of a fixed star by a planet 601 CHAPTER XL PRECESSION, NUTATION, ABERRATION, AND ANNUAL PARALLAX or THE FIXED STARS 602 Precession 604 Nutation 624 Aberration 628 Parallax % 643 Mean and apparent places of stars 645 CONTENTS. CHAPTER XII. MM DETERMINATION OF THE OBLIQUITY OF THE ECLIPTIC AND THE ABSOLUTE RIGHT ASCENSIONS AND DECLINATIONS OF STARS BY OBSERVATION 658 Obliquity of the ecliptic 669 Equinoctial points, and absolute right ascension and declination of the fixed stars 665 CHAPTER XIII. DETERMINATION OF ASTRONOMICAL CONSTANTS BY OBSERVATION 671 Constants of refraction 671 Constant of solar parallax 673 Constant of lunar parallax 680 Mean sernidiameters of the planets 687 Constant of aberration and heliocentric parallax of fixed stars 688 Constant of nutation 698 Constant of precession 701 Motion of the sun in space 703 SPHERICAL ASTRONOMY. CHAPTER I. THE CELESTIAL SPHERE SPHERICAL AND RECTANGULAR COORDINATES. 1. FROM whatever point of space an observer be supposed to view the heavenly bodies, they will appear to him as if situated upon the surface of a sphere of which his eye is the centre. If, without changing his position, he directs his eye successively to the several bodies, he may learn their relative directions, but cannot determine either their distances from himself or from each other. The position of an observer on the surface of the earth is, however, constantly changing, in consequence, 1st, of the diur* nal motion, or the rotation of the earth on its axis; 2d, of the annual motion, or the motion of the earth in its orbit around the sun. The changes produced by the diurnal motion, in the appa- rent relative positions or directions of the heavenly bodies, are different for observers on different parts of the earth's surface, and can be subjected to computation only by introducing the elements of the observer's position, such as his latitude and longitude. But the changes resulting from the annual motion of the earth, as well as from the proper motions of the celestial bodies themselves, may be separately considered, and the directions of all the known celestial bodies, as they would be seen from the centre of the earth at any given time, may be computed VOL. I. 2 17 18 THE CELESTIAL SPHERE. according to the laws which have been found to govern the motions of these bodies, from data furnished by long series of observations. The complete investigation of these changes and their laws belongs to Physical Astronomy, and requires the consi- deration of the distances and magnitudes as well as of the direc- tions of the bodies composing the system. Spherical Astronomy treats specially of the directions of the heavenly bodies; and in this branch, therefore, these bodies are at any given instant regarded as situated upon the surface of a sphere of an indefinite radius described about an assumed centre. It embraces, therefore, not only the problems which arise from the diurnal motion, but also such as arise from the annual motion so far as this affects the apparent positions of the hea- venly bodies upon the celestial sphere, or their directions from the assumed centre. SPHERICAL CO-ORDINATES. 2. The direction of a point may be expressed by the angles which a line drawn to it from the centre of the sphere, or point of observation, makes with certain fixed lines of reference. But, since such angles are directly measured by arcs on the surface of the sphere, the simplest method is to assign the position in which the point appears when projected upon the surface of the sphere. For this purpose, a great circle of the sphere, supposed to be given in position, is assumed as a primitive circle of refer- ence, and all points of the surface are referred to this circle by a system of secondaries or great circles perpendicular to the primi- tive and, consequently, passing through its poles. The position of a point on the surface will then be expressed by two spherical co-ordinates: namely, 1st, the distance of the point from the pri- mitive circle, measured on a secondary ; 2d, the distance inter- cepted on the primitive between this secondary and some given point of the primitive assumed as the origin of co-ordinates. We shall have different systems of co-ordinates, according to the circle adopted as a primitive circle and the point assumed as the origin. 3. First system of co-ordinates. Altitude and azimuth. In this system, the primitive circle is the horizon, which is that great circle of the sphere whose plane touches the surface of the SPHERICAL CO-ORDINATES. 19 earth at the observer.* The plane of the horizon may be con- ceived as that which sensibly coincides with the surface of a fluid at rest. The vertical line is a straight line perpendicular to the plane of the horizon at the observer. It coincides with the direction of the plumb line, or the simple pendulum at rest. The two points in which this line, infinitely produced, meets the sphere, are the zenith and nadir, the first above, the second below the horizon. The zenith and nadir are the poles of the horizon. Secondaries to the horizon are vertical circles. They all pass through the zenith and nadir, and their planes, which are called vertical planes, intersect in the vertical line. Small circles parallel to the horizon are called almucantars, or parallels of altitude. The celestial meridian is that vertical circle whose plane passes through the axis of the earth and, consequently, coincides with the plane of the terrestrial meridian. The intersection of this plane with the plane of the horizon is the meridian line, and the points in which this line meets the sphere are the north and south points of the horizon, being respectively north and south of the plane of the equator. The prime vertical is the vertical circle which is perpendicular to the meridian. The line in which its plane intersects the plane of the horizon is the east and west line, and the points in which this line meets the sphere are the east and tvest points of the horizon. The north and south points of the horizon are the poles of the prime vertical, and the east and west points are the poles of the meridian. * In this definition of the horizon we consider the plane tangent to the earth's surface as sensibly coinciding with a parallel plane passed through the centre ; that is, we consider the radius of the celestial sphere to be infinite, and the radius of the earth to be relatively zero. In general, any number of parallel planes at finite dis- tances must be regarded as marking out upon the infinite sphere the same great circle. Indeed, since in the celestial sphere we consider only direction, abstracted from dis- tance, all lines or planes having the same direction that is, all parallel lines or planes must be regarded as intersecting the surface of the sphere in the same point or the same great circle. The point of the surface of the sphere in which a number of parallel lines are conceived to meet is called the rani shiny point of those lines; and, in like manner, the great circle in which a number of parallel planes are conceived to meet may be called the vanishing circle of those planes. 20 THE CELESTIAL SPHERE. The altitude of a point of the celestial sphere is its distance from the horizon measured on a vertical circle, and its azimuth is the arc of the horizon intercepted between this vertical circle and any point of the horizon assumed as an origin. The origin from which azimuths are reckoned is arbitrary ; so also is the direction in which they are reckoned ; but astronomers usually take the south point of the horizon as the origin, and reckon towards the right hand, from to 360 ; that is, completely around the horizon in the direction expressed by writing the cardinal points of the horizon in the order S.W. N". E. We may, therefore, also define azimuth as the angle which the vertical plane makes with the plane of the meridian. Navigators, however, usually reckon the azimuth from the north or south points, according as they are in north or south latitude, and towards the east or west, according as the point of the sphere considered is east or west of the meridian : so that the azimuth never exceeds 180. Thus, an azimuth which is expressed according to the first method simply by 200 would be expressed by a navigator in north latitude by N. 20 E., and by a navigator in south latitude by S. 160 E., the letter prefixed denoting the origin, and the letter affixed denoting the direction in which the azimuth is reckoned, or whether the point consi- dered is east or west of the meridian. When the point considered is in the horizon, it is often referred to the east or west points, and its distance from the nearest of these points is called its amplitude. Thus, a point in the horizon whose azimuth is 110 is said to have an amplitude of W. 20 K Since by the diurnal motion the observer's horizon is made to change its position in the heavens, the co-ordinates, altitude and azimuth, are continually changing. Their values, therefore, will depend not only upon the observer's position on the earth, but upon the time reckoned at his meridian. Instead of the altitude of a point, we frequently employ itK zenith distance, which is the arc of the vertical circle between the point and the zenith. The altitude and zenith distance are, therefore, complements of each other. We shall hereafter denote altitude by A, zenith distance by '. azimuth by A. We shall have then C = 90 h h = 90 SPHERICAL CO-ORDINATES. 21 The value of for a point below the horizon will be greater than 90, and the corresponding value of h, found by the for- mula h = 90 , will be negative : so that a negative altitude will express the depression of a point below the horizon. Thus, a depression of 10 will be expressed by h 10, or = 100. 4. Se.cond system of co-ordinates. Declination and hour angle. Iu this system, the primitive circle is the celestial equator, or that great circle of the sphere whose plane is perpendicular to the axis of the earth and, consequently, coincides with the plane of the terrestrial equator. This circle is also sometimes called the equinoctial. The diurnal motion of the earth does not change the position of the plane of the equator. The axis of the earth produced to the celestial sphere is called the axis of the heavens : the points in which it meets the sphere are the north and south poles of the equator, or the poles of the heavens. Secondaries to the equator are called circles of declination, and also hour circles. Since the plane of the celestial meridian passes through the axis of the equator, it is also a secondary to the equator, and therefore also a circle of declination. Parallels of declination are small circles parallel to the equator. The declination of a point of the sphere is its distance from the equator measured on a circle of declination, and its hour angle is the angle at either pole between this circle of declination and the meridian. The hour angle is measured by the arc of the equator intercepted between the circle of declination and the meridian. As the meridian and equator intersect in two points, it is neces- sary to distinguish which of these points is taken as the origin of hour angles, and also to know in what direction the arc which measures the hour angle is reckoned. Astronomers reckon from that point of the equator which is on the meridian above the horizon, towards the west, that is, in the direction of the apparent diurnal motion of the celestial sphere, and from to 360, or from 0* to 24*, allowing 15 to each hour. Of these co-ordinates, the declination is not changed by the diurnal motion, while the hour angle depends only on the time at the meridian of the observer, or (which is the same thing) on the position of his meridian in the celestial sphere. All the observers on the same meridian at the same instant will, for the same star, reckon the same declination and hour angle. We have 22 THE CELESTIAL SPHERE. thus introduced co-ordinates of which one is wholly independent of the observer's position and the other is independent of his latitude. We shall denote decimation by ?, and north declination will be distinguished by prefixing to its numerical value the sign -4-, and south declination by the sign . We shall sometimes make use of the polar distance of a point, or its distance from one of the poles of the equator. If we denote it by P, the north polar distance will be found by the formula P = 90 _ 9 and the south polar distance by the formula P = 90 + d The hour angle will generally be denoted by t. It is to be observed that as the hour angle of a celestial body is continually increasing in consequence of the diurnal motion, it may be con- ceived as having values greater than 360, or 24* , or greater than any given multiple of 360. Such an hour angle may be re- garded as expressing the time elapsed since some given passage of the body over the meridian. But it is usual, when values greater than 360 result from any calculation, to deduct 360. Again, since hour angles reckoned towards the west are always positive, hour-angles reckoned towards the east must have the negative sign : so that an hour angle of 300, or 20 A , may also be expressed by 60, or 4 7 '. 5. Third system of co-ordinates. Declination and right ascension. In this system, the primitive plane is still the equator, and the first co-ordinate is the same as in the second system, namely, the declination. The second co-ordinate is also measured on the equator, but from an origin which is not affected by the diurnal motion. Any point of the celestial equator might be assumed as the origin; but that which is most naturally indicated is the vernal equinox, to define which some preliminaries are necessary. The ecliptic is the great circle of the celestial sphere in which the sun appears to move in consequence of the earth's motion in its orbit. The position of the ecliptic is not absolutely fixed in space ; but, according to the definition just given, its position at any instant coincides with that of the great circle in which the SPHERICAL CO-ORDINATES. 23 sun appears to be moving at that instant. Its annual change is, however, very small, and its daily change altogether insensible. The obliquity of the ecliptic is the angle which it makes with the equator. The points where the ecliptic and equator intersect are called the equinoctial points, or the equinoxes ; and that diameter of the sphere in which their planes intersect is the line of equinoxes. The vernal equinox is the point through which the sun ascends from the southern to the northern side of the equator ; and the autumnal equinox is that through which the sun descends from the northern to the southern side of the equator. The solstitial points, or solstices, are the points of the ecliptic 90 from the equinoxes. They are distinguished as the north- ern and southern, or the summer and winter solstices. The equinoctial colure is the circle of declination which passes through the equinoxes. The solstitial colure is the circle of decli- nation which passes through the solstices. The equinoxes are the poles of the solstitial colure. By the annual motion of the earth, its axis is carried very nearly parallel to itself, so that the plane of the equator, which is always at right angles to the axis, is very nearly a fixed plane of the celestial sphere. The axis is, however, subject to small changes of direction, the effect of which is to change the position of the intersection of the equator and the ecliptic, and hence, also, the position of the equinoxes. In expressing the positions of stars, referred to the vernal equinox, at any given instant, the actual position of the equinox at the instant is understood, unless otherwise stated. The right ascension of a point of the sphere is the arc of the equator intercepted between its circle of declination and the vernal equinox, and is reckoned from the vernal equinox east- ward from to 360, or, in time, from 0* to 24\ The point of observation being supposed at the centre &f the earth, neither the declination nor the right ascension will be affected by the diurnal motion: so that these co-ordinates are wholly independent of the observer's position on the surface of the earth. Their values, therefore, vary only with the time, and are given in the ephemerides as functions of the time reckoned at some assumed meridian. We shall generally denote right ascension by a. As its value reckoned towards the east is positive, a negative value resulting 24 THE CELESTIAL SPHERE. from any calculation would be interpreted as signifying an arc of the equator reckoned from the vernal equinox towards the west. Thus, a point whose right ascension is 300, or 20 A , may also be regarded as in right ascension 60, or 4 A ; but such negative values are generally avoided by adding 360, or 24 A . Again, in continuing to reckon eastward we may arrive at values of the right ascension greater than 24 A , or greater than 48 A , etc.; but in such cases we have only to reject 24* , 48 A , etc. to obtain values which express the same thing. 6. Fourth system of co-ordinates. Celestial latitude and longi- tude. In this system the ecliptic is taken as the primitive circle, and the secondaries by which points of the sphere are referred to it are called circles of latitude. Parallels of latitude are small circles parallel to the ecliptic. The latitude of a point of the sphere is its distance from the ecliptic measured on a circle of latitude, and its longitude is the arc of the ecliptic intercepted between this circle of latitude and the vernal equinox. The longitude is reckoned eastward from to 360. The longitude is sometimes expressed in signs, degrees, &c., a sign being equal to 30, or one-twelfth of the ecliptic. These co-ordinates are also independent of the diurnal motion. It is evident, however, that the system of declination and right ascension will be generally more convenient, since it is more directly related to our first and second systems, which involve the observer's position. We shall denote celestial latitude by /9 ; longitude by L Posi- tive values of /9 belong to points on the same side of the ecliptic as the north pole; negative values, to those on the opposite side. In connection with this system we may here define the nona- yesimal, which is that point of the ecliptic which is at the greatest altitude above the horizon at any given time. That vertical circle of the observer which is perpendicular to the ecliptic meets it in the nonagesimal ; and, being a secondary to the ecliptic, it is also a circle of latitude : it is the great circle which passes through the observer's zenith and the pole of the ecliptic. 7. Co-ordinates of the observer's position. "We have next to ex- press the position of the observer on the surface of the earth, according to the different systems of co-ordinates. This will be SPHERICAL CO-ORDINATES. 25 done by referring his zenith to the primitive circle in the same manner as in the case of any other point. In the first system, the primitive circle being the horizon, of which the zenith is the pole, the altitude of the zenith is always 90, and its azimuth is indeterminate. In the second system, the declination of the zenith is the same as the terrestrial latitude of the observer, and its hour angle is zero. The declination of the zenith of a place is called the geographical latitude, or simply the latitude, and will be hereafter denoted by (p. North latitudes will have the sign -f ; south latitudes, the sign . In the third system, the declination of the zenith is, as before, the latitude of the observer, and its right ascension is the same as the hour angle of the vernal equinox. In the fourth system, the celestial latitude of the zenith is the same as the zenith distance of the nonagesimal, arid its celestial longitude is the longitude of the nonagesimal. It is evident, from the definitions which have been given, that the problem of determining the latitude of a place by astro- nomical observation is the same as that of determining the declination of the zenith ; and the problem of finding the lon- gitude may be resolved into that of determining the right ascension of the meridian at a time when that of the prime meridian is also given, since the longitude is the arc of the equator intercepted between the two meridians, and is, conse- quently, the difference of their right ascensions. 8. The preceding definitions are exemplified in the following figures. Fig. 1 is a stereographic projection of the sphere upon the plane of the horizon, the projecting point being the nadir. Since the planes of the equator and horizon are both perpendicular to that of the meridian, their intersection is also perpendicular to it; and hence the equator WQE passes through the east and west points of the horizon. All vertical circles passing through the projecting point will be projected into straight lines, as the meridian NZS, the prime vertical WZE, and the vertical circle ZOffdr&wn through any point Oof the surface 26 THE CELESTIAL SPHERE. of the sphere. We have then, according to the notation adopted in the first system of co-ordinates, h = the altitude of the point = OH, C = the zenith distance " = OZ, A = the azimuth " = SJf, or = the angle SZH. If the declination circle POD be drawn, we have, in the second system of co-ordinates, S = the declination of OD, P = the polar distance " = PO, t = the hour angle " = ZPD, or QD. If T^is the vernal equinox, we have, in the third system of co-ordinates, d = the declination of = OD, a = the right ascension VD, or the angle VPD. In this figure is also drawn the six hour circle EPW, or the declination circle passing through the east and west points of the horizon. The angle ZPW, or the arc QW, being 90, the hour angle of a point on this circle is either + 6 /l or 6 A , that is, either 6 A or 18*. Fig. 2 is a repetition of the preceding figure, with the addi- Fig- 2. tion of the ecliptic and the circles related to it. C VT represents the ecliptic, P' its pole, P'OL a circle of latitude. Hence we have, in our third system of co-ordinates, /3 fa the celestial latitude of = OL, A " longitude " = VL, = the angle VP'L. We have also VPthe equinoctial colure, P'PAB the solstitial colure, P'ZGF the vertical circle passing through P', which is therefore perpendicular to the ecliptic at G. The point G is the nonagesimal ; ZG is its zenith distance, VG its longitude ; or ZG is the celestial latitude and VG the celestial longitude of the zenith. Finally, we have, in both Fig. 1 and Fig. 2, y> = the geographical latitude of the observer = ZQ = 90 PZ = PN SPHERICAL CO-ORDINATES. 27 Hence the latitude of the observer is always equal to the alti- hide of the north pole. For an observer in south latitude, the north pole is below the horizon, and its altitude is a negative quantity: so that the definition of latitude as the altitude of the north pole is perfectly general if we give south latitudes the negative sign. The south latitude of an observer considered independently of its sign is equal to the altitude of the south pole above his horizon, the elevation of one pole being always equal to the depression of the other. 9. Numerical expression of hour angles. The equator, upon which hour angles are measured, may be conceived to be divided into 24 equal parts, each of which is the measure of one hour, and is equivalent to & of 360, or to 15. The hour is divided sexagesimally into minutes and seconds of time, distinguished from minutes and seconds of arc by the letters " l and " instead of the accents ' and ". We shall have, then, 1* a* 15 l m = 15' I' = 15" To convert an angle expressed in time into its equivalent iir arc, multiply by 15 and change the denominations A m ' into ' "; and to convert arc into time, divide by 15 and change ' " into A "* '. The expert computer will readily find ways to abridge these operations in practice. It is well to observe, for this purpose, that from the above equalities we also have, 1 = 4 1' = 4' and that we may therefore convert degrees and minutes of arc into time by multiplying by 4 and changing ' into '" * ; and reciprocally. TRANSFORMATION OF SPHERICAL CO-ORDINATES. 10. Given the altitude (h) ami azimuth (A) of a star, or of any point of the sphere, and the latitude (?) of the observer, to find the declina- tion (d) and hour angle (t] of the star or the point. In other words, to transform the co-ordinates of the first system into those of the second. This problem is solved by a direct application of the formulae of Spherical Trigonometry to the triangle POZ, Fig. 1, in which, being the given star or point, we have three parts given, 28 THE CELESTIAL SPHERE. namely, 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 we can find PO the polar distance or complement of the required declination, and ZPO the required hour angle. But, to avoid the trouble of taking complements, and supplements, the formulae are adapted so as to express the declination and hour angle directly in terms of the altitude, azimuth, and latitude. Fio . 3 To show as clearly as possible how the formulae of Spherical Trigonometry are thus converted into formulae of Spherical Astronomy, let us first con- sider a spherical triangle ABC, Fig. 3, in which there are given the angle A, and the sides 6 and c, to find the angle B and the side a. The general rela- tions between these five quantities are [Sph. Trig. Art. 114]* cos a = cos c cos b -\- sin c sin b cos A "| sin a cos B = sin c cos b cos c sin b cos A > (01) sin a sin B sin b sin A j Now, comparing the triangle ABC with the triangle PZO of Fig. 1, we have A=PZO = ISO A a = P0 = 90 S b = ZO=Mh V=ZPO=t c= PZ= 90? Substituting these values in the above equations, we obtain sin S = sin

180 we have I > 180, conditions which also follow directly from the nature of our problem, since the star is west or east of the meridian according as A < 180 or A > 180. The formula (1) or (4) fully determines 3, which will always be taken less than 90, positive or negative according to the sign of its sine.* To adapt the equations (4), (5), and (6) for logarithmic compu- tation, let m and M be assumed to satisfy the conditions [PI. Trig. Art. 174], m sin M =sin C cos A 1 m cos M = cos C / the three equations may then be written as follows : sin <5 = m sin (

M ) v (8) cos d sin t = sin C sin A ) If we eliminate m from these equations, the solution takes the following convenient form : * There are, however, special problems in which it is convenient to depart from (his general method, and to admit declinations greater than 90, as will be seen hereafter. 30 THE CELESTIAL SPHERE. tan M= tan C cos A tan A sin M tan t = cos ( M ) tan 5 = tan (^ Jf ) cos t In the use of which, we must observe to take t greater or less than 180 according as A is greater or less than 180, since the Vour angle and the azimuth must fall on the same side of the meridian. EXAMPLE. In the latitude if 38 58' 53", there are given for a certain star == 69 d 42' 30", A = 300 10' 30" ; required 3 and /. The computation by (9) may be arranged as follows :* log tan C 0.4320966 = 38 58 / 53" log cos A 9.7012595 log tan A nO. 2355026 M= 53 39 41.98 log tan M 0.1333561 log sin M 9.9060828 cos A In cos 8 sin q = cos

sin A or in the following : (11) cos 5 cos sin or again in the following : g sin G = sin y g cos G = cos

cos 8 cos t sin C sin A = cos 8 sin t which express and A directly in terms of the data. Adapting these for logarithmic computation, we have m sin M = a\n 8 m cos M = cos 8 cos t cos C = m cos (

M) 0.0939172 A = 300 10 30 log cos A 9.7012595 log tan A nO. 2355026 C = 69 42 30 log tan C 0. 4320966 SPHERICAL CO-ORDINATES. 32 For verification we can use the equation sin C sin A = cos 3 sin t log sin : 9.9721748 log cos 3 9.9951697 log sin A 9.9367621 log sin t 9.9137672 9.9089369 9.9089369 EXAMPLE 2. In latitude tp = 48 32', there are given for a star, <5 = 44 6' 0", * = 17 A 25" 1 4; required A and . We find A = 241 53' 33".2, = 126 25' 6".6 ; the star is below the horizon, and its negative altitude, or depression, is h = -36 25'6".6. If the zenith distance of the same star is to be frequently com- puted on the same night at a given place, it will be most readily done by the following method. In the first equation of (14) substitute cos t = I 2 sin* i t then we have cos = cos O -r 8) 2 cos

sin 8 cos sin C sin q = cos ^ sin t which are adapted for logarithms as follows : n sin N= cos

sin or, eliminating n, thus : tan -ZV=cot

d) (21) 16. When the altitude, azimuth, and parallactic angle of known stars are to be frequently computed at the Fi - 4 - same place, the labor of computation is much diminished by an auxiliary table pre- pared for the latitude of the place accord- ing to formulae proposed by Gauss. A specimen of such a table computed for the latitude of the Altona Observatory will be found in "Schumacher's Hulfstafeln, neu herausg. v. Warnstorif." The requisite formulae are readily deduced as follows : Let the declination circle through the object 0, Fig. 4, be produced to intersect the horizon in F. By the diurnal motion SPHERICAL CO-ORDINATES. 35 the point F changes its position on the horizon with the time , but its position depends only on the time or the hour angle ZPO, and not upon the declination of 0. The elements of the position of F may therefore be previously computed for succes- sive values of t. We have in the triangle PFS, right-angled at S, FPS=t, PS = 180

tan , tan B = cot

sin 2 8 sin 2 ^> COS 2 sin (8 -(- y>} sin (8 19. To find the hour angle, zenith distance, and parallactic angle of a given star on the prime vertical of a given place. In this case, the point in Fig. 1 being in the circle WZE, the angle PZO is 90, and the right triangle PZO gives tan 8 cos t = cos C = -3 q = tan cos 8 We may also deduce the following convenient and accurate formulae for the case where the star's declination is nearly equal to the latitude [see Sph. Trig. Arts. 60, 61, 62] : tan*C= //tapfr-*)\ \ Vtan * & + 9)) tan (45 J 0) = ,/[tan J O -f- 5) tan } O (27) If d > ^>, these values become imaginary; that is, the star can- not cross the prime vertical. EXAMPLE. Required the hour angle and zenith distance of the star 12 Canum Venaticorum (3 = + 39 5' 20") when on the prime vertical of Cincinnati (

0 = 85n[-l5 K ~ <*)] si n (45 -fi ) cos (45 J /?) cos i (E + A) = cos [45 * (e + <*)] cos (45 + * a) sin (45 10) sin * (E A) = sin [45 1 (e + S)] cos (45 + Ja) sin (45 J0)cosJ(^ A) = cos [45 J(e <*)]sin (45 + *a) 25. If the angle at the star is required when the Gaussian Equations have not been employed, we have from the triangle POP', Fig. 0, putting P'OP se.f = 90 JS, 42 THE CELESTIAL SPHERE. cos ft cos TJ = cos e cos 8 -j- sin e sin 8 sin a cos ft sin iy sin e cos a or, adapted for logarithms, n sin N = sin e sin a ;\ COS jV = COS e cos ft cos 7 n cos (.ZV <5) cos # sin 73 = sin e cos a 26. Given the latitude (ft) and longitude (/) of a star, and the obliquity of the ecliptic (e), to Jind the declination and right ascension of the star. By the process already employed, we derive from the triangle PP'O, Fig. 6, for this case, sin 8 = cose sin ft -f sin e cos ft sin A ^ cos 8 sin = sin e sin ft -\- cos e cos ft sin A V (34) cos 6 cos a = cos ft cos A J which, it will be observed, may be obtained from (29) by inter- changing a with ^, and 3 with /9, and at the same time changing the sign of e, that is, putting e for e, and, consequently, sin e for sin e. For logarithmic computation, we have m sin M = sin /5 m cos J.f cos /3 sin Jt sin 5 = m sin (Jlf -{- e) cos 8 sin a = m cos (Jf -f e) COS 8 COS a = COS /9 COS >l or the following, analogous to (31) : A- sin M = tan ft k cos M= sin /I A' sin a = cos (M -\- e) k' cos a cos M cot A tan 5 sin a tan (Jf + e) cos 5 sin a cos (Jf -f e) For verification : ^. r-= & cos ft sin A cos M .(35) (36) 27. The angle at the star, POP', Icing denoted, as in Art, 24, RECTANGULAR CO-ORDINATES. 43 by 90 E, the solution of this problem by the Gaussian Equations is sin (45 J (J) sin * (E + a) = sin [45 \ (e + /?)] sin (45+ * A) sin (45 J 5) cos J (JJ-fr a) == cos [ 45 * ( &f\ cos ( 45 + * A ) cos (45 J 3) sin J (J? a) = sin [45 } (e ,9)] cos (45+ * A) cos (45 | J) cos * (^ a) = cos [45 J (e + /?)] sin (45+ J A) (37) 28. But if the angle y = 90 - E is required when the Gaussian Equations have not been employed, we have directly cos 8 cos t) = cos e cos /3 sin e sin /? sin A cos 5 sin r t = sin e cos X or, adapted for logarithms, N= sin e sin A N COS e (38) w sin 2F= sin e sin A n cos JV= cos e cos d cos 5j n cos (JV -j- cos 5 sin i = sin e cos A 29. Jf^or ^Ae sun, we may, except when extreme precision is desired, put ft = 0, and the preceding formulae then assume very simple forms. Thus, if in (34) we put sin ft -= 0, cos 8 =. 1, we lind sin 3 = sin e sin A cos 3 sin a = cos e sin A cos 3 cos a = cos A whence if any two of the four quantities 3, a, ^, e are given, we can deduce the other two. RECTANGULAR CO-ORDINATES. 30. By means of spherical co-ordinates we have expressed only a star's direction. To define its position in space com- pletely, another element is necessary, namely, its distance. In Spherical Astronomy we consider this element of distance only so far as may be necessary in determining the changes of apparent direction of a star resulting from a change in the point from which it is viewed. For this purpose the rectangular co- ordinates of analytical geometry may be employed. Three planes of reference are taken at right angles to each other, their common intersection, or origin, being the point of 44 THE CELESTIAL SPHERE. observation; and the star's distances from these planes are denoted by x, #, and z respectively. These co-ordinates are respectively parallel to the three axes (or mutual intersections of the planes, taken two and two), and hence these axes are called, respectively, the axis of x, the axis of ?/, and the axis of z. The planes are distinguished by the axes they contain, as "the plane of xy" the "plane of xz" the "plane of yz" The co- ordinates may be conceived to be measured on the axes to which they belong, from the origin, in two opposite directions, distinguished by the algebraic signs of plus and minus, so that the numerical values of the co-ordinates of a star, together with their algebraic signs, fully determine the position of the star in space without ambiguity. Of the eight solid angles formed by the planes of reference, that in which a star is placed will always be known by the signs of the three co-ordinates, and in one only of these angles will the three signs all be plus. This angle is usually called the first angle. To simplify the investigations of a problem, we may, if we choose, assume all the points considered to lie in the first angle, and then treat the equations obtained for this simplest case as entirely general; for, by the principles of analytical geometry, negative values of the co-ordinates which satisfy such equations also satisfy a geometrical construction in which these co-ordinates are drawn in the negative direction. The polar co-ordinates of analytical geometry (of three dimen- sions) when applied to astronomy are nothing more than the spherical co-ordinates we have already treated of, combined with the element distance ; and the formulae of transformation fi 3m rectangular to polar co ordinates are nothing more than the values of the rectangular co-ordinates in terms of the dis- tance and the spherical co-ordinates. For the convenience of reference, we shall here recapitulate these formulae, with special reference to our several systems of spherical co-ordinates. 31. We shall find it useful to premise the following LEMMA. The distance of a point in space from the plane of any great circle of the celestial sphere is equal to its distance from the centre of the sphere multiplied by the cosine of its angular distance from the pole of that circle; and its distance from the axis of the circle is equal to its distance from the centre of the sphere multiplied by the sine of its angular distance from the pole. RECTANGULAR CO-ORDINATES. 45 For, let Alt, Fig. 7, be the given great circle orthographi- cally projected upon a plane passing through its axis OP and the given point C; P its pole. The dis- tance of the point C from the plane of the great circle is the perpendicular CD ; CE is its distance from the axis ; CO its dis- tance from the centre of the sphere ; and the angle COP the angular distance from the pole. The truth of the Lemma is, therefore, obvious from the figure. 32. The values of the rectangular co-ordinates in our several systems may be found as follows : First system. Altitude and azimuth. Let the primitive plane, or that of the horizon, be the plane of xy ; that of the meridian, the plane of xz; that of the prime vertical, the plane of yz. The meridian line is then the axis of x; the east and west line, the axis of y; and the vertical line, the axis of z. Positive x will be reckoned from the origin towards the south, positive y towards the west, and positive z towards the zenith. The fast angle, or angle of positive values, is therefore the southwest quarter of the hemisphere above the plane of the horizon. Let Z, Fig. 8, be the zenith, S the south point, W the Fig. s. west point of the horizon. These points are respectively the poles of the three great circles of reference ; if, then, A is the position of a star on the surface of the sphere as seen from the centre of the earth, and if we put h = altitude of the star = AN, A = azimuth " = SH, J its distance from the centre of the sphere we have immediately, by the preceding Lemma, x = A cos AS, y = J cos A \V, z = J cos AZ, which, by considering the right triangles AHS, AHW, become x = J cos h cos A "| y J cos A sin A > (39) z = A sin h ) These equations determine the rectangular co-ordinates x,y,z, 46 THE CELESTIAL SPHERE. when the polar co-ordinates J, h, A are given. Conversely, if x, y, and z are given, we may find J, A, and A ; for the first two equations give tan A = x and then we have A sin h = z A cos h = cos J. sin A whence J and A. Or, by adding the squares of the first two equations, we have A cos h = \/x* + y' whence tan h = and the sum of the squares of the three equations gives Second system. Decimation and hour angle. Let the plane of the equator be the plane of xy ; that of the meridian, the plane of xz; that of the six hour circle, the plane of yz. In the pre- ceding figure, let Zi now denote the north pole, S that point of the equator which is on the meridian above the horizon and from which hour angles are reckoned, Wthe west point. Posi- tive x will be reckoned towards S, positive y towards the west, positive z towards the north. If then A is the place of a star on the sphere as seen from the centre, and we put 3 = the star's declination = AH, t = " hour angle SH, A = " distance from the centre, and denote the rectangular co-ordinates in this case by x',y',z', we have x' = A cos S cos t ^ y' = A cos S sin t > (40) z' = J sin S J Third system. Declination and right ascension. Let the plane of the equator be the plane of xy ; that of the equinoctial coin re, the plane of xz; that of the solstitial colure, the plane of yz. RECTANGULAR CO-ORDINATES. 47 The axis of x is the intersection of the planes of the equator and equinoctial colure, positive towards the vernal equinox ; the axis of y is the intersection of the planes of the equator and sol- stitial colure, positive towards that point whose right ascension j s -f 90 ; and the axis of z is the axis of the equator, positive towards the north. If then, in Fig. 8, Z is the north pole, W the vernal equinox, A a star in the first angle, projected upon the celestial sphere, and we put 8 = declination of the sta\ = AH, a = right ascension u . = WH, A = distance from the centre, , , while x" ', y", z" denote the rectangular co-ordinates, we have x" = J cos A W, y" =. J cos AS, -z"= J cos AZ t which become x" J cos d cos a ^ y" = J cos d sin a \ (41) z" = J sin d ) Fourth system. Celestial latitude and longitude. Let the plane of the ecliptic be the plane of xy ; the plane of the circle of latitude passing through the equinoctial points, the plane of xz ; the plane of the circle of latitude passing through the solstitial points, the plane of yz. The positive axis of x is here also the straight line from the centre towards the vernal equinox ; the positive axis ofy is the straight line from the centre towards the north solstitial point, or that whose longitude is +90;. and the positive axis of z is the straight line from the centre towards the north pole of the ecliptic. If then, in Fig. 8, Z now denotes the north pole of the ecliptic, W the vernal equinox, A the star's place on the sphere, and we put B = latitude of the star AH, A = longitude of the star = WH, A distance of the rrtar from the centre, and x'", y'", z'", denote the rectangular co-ordinates for this system, we have x'" =. J cos /9 cos X y'" = J cos ,5 sin A }. (42) ' z'" = J sin a 48 THE CELESTIAL SPHERE. TRANSFORMATION OF RECTANGULAR CO-ORDINATES. 33. For the purposes of Spherical Astronomy, only the most simple cases of the general transformations treated of in analy- tical geometry are necessary. We mostly consider but two cases : First Transformation of rectangular co-ordinates to a new origin, without changing the system of spherical co-ordinates. The general planes of reference which have been used in this chapter may be supposed to be drawn through any point in space without changing their directions, and therefore without changing the great circles of the infinite celestial sphere which repre- sent them. We thus repeat the same system of spherical co-ordi- nates with various origins and different systems of rectangular co-ordinates, the planes of reference, however, remaining always parallel to the planes of the primitive system. The transformation from one system of rectangular co-orrli- nates to a parallel system is evidently effected by the formulae in which x v y z l are the co-ordinates of a point in the primitive system ; x. 2 , y z , z 2 the co-ordinates of the same point in the new system ; and a, b, c are the co-ordinates of the new origin taken in the first system. As we have shown how to express the values of x v y v z t and of x v y v z 2 in terms of the spherical co-ordinates, we have only to substitute these values in the preceding formulae to obtain the general relations between the spherical co-ordinates correspond- ing to the two origins. This is, indeed, the most general method of determining the effect of parallax, as will appear hereafter. Second. Transformation of rectangular co- ordinates when the system of spherical co-ordi- nates is changed but the origin is unchanged. This amounts to changing the directions of the axes. The cases which occur in practice are chiefly those in which the two systems have one plane in common. Suppose this plane is that of xz, and let OX, OZ, Fig. 9, be the axes of x and z in the first system; OX RECTANGULAR CO-ORDINATES. 49 OZ { , the axes of x l and Zj in the new system. Let A he the projection of a point in space upon the common plane ; and let x = AB, z = OB, x l = AB V z, =OB r The distance of thb point from the common plane heing unchanged, we have The axis of y may be regarded as an axis of revolution about which the planes of yx and yz revolve in passing from the first to the second system ; and if u denotes the angular measure of this revolution, or u = XOX^ = ZOZ l BAB^ we readily derive from the figure the equation x sec u=x t z l tan u or, multiplying by cos w, x = x t cos w z t sin u and 2 = x tan u-\- z l sec u or, substituting in this the preceding value of x, z = Xi sin u -\- z l cos u Thus, to pass from the first to the second system, we have the formulae x = x v cos u z 1 sin u ~| y = ?i > (44) z = x t sin u -f- 2 cos u ) And to pass from the second to the first, we obtain with the same ease, x l = x cos u -f 2 sin u ^| y.= y } (45) z l = x sin u -f 2CO8 u ) As an example, let us apply these to transforming from our second system of spherical co-ordinates to the first ; that is, from declination and hour angle to altitude and azimuth. The 'meri- dian is the common plane ; the axis of z in the system of declina- tion and hour angle is the axis of the equator, and the axis of ?, in the system of altitude and azimuth is the vertical line ; the angle between these axes is the complement of the latitude, or u = 90 (p. Substituting this value of u in (44), and also tne values of x, #, 2, ar 15 y v z v given by (39) and (40), we have, after omitting the common factor d, VOL. I. 4 50 THE CELESTIAL SPHERE. cos h cos A = sin cos S cos t cos if sin 8 cos h sin A = cos d sin sin h = cos ^ cos 3 cos < -j- sin

(46) cos B da cos A db -f dc = sin a sin B dC ) From these we obtain the following by eliminating da: sin C db cos a sin B dc sin b cos C dA -f- sin a then the first equations of (46) and (47) give dS = cos q d^ -\- sin q sin C dA -\- cos t d

) which determine the errors dd and dt in the values of d and t computed according to the formulae (4), (5), and (6), when , A, and (p are affected by the small errors rf, cL4, and efy? respectively. In a similar manner we obtain dZ = cos q dd -|- sin q cos 8 dt -\- cos vl rfy ") r .-, sin C dA = sin ; ^/,? -f- S1 ' n ^ co9 ^ dX -)- sin a eg cos 5 t?a = sin ^ rf/3 -j- cos TJ cos p dA sin 3 cos ade 52 TIME. CHAPTER II. TIME USE OF THE EPHEMERIS INTERPOLATION STAR CATALOGUES. 37. TRANSIT. The instant when any point of the celestial sphere is on the meridian of an observer is designated as the transit of that point over the meridian ; also the meridian passage, and culmination. In one complete revolution of the sphere about its axis, every point of it is twice on the meridian, at points which are 180 distant in right ascension. It is therefore necessary to distinguish between the two transits. The meri- dian is bisected at the poles of the equator: the transit over that half of the meridian which contains the observer's zenith is the upper transit, or culmination ; that over the half of the meri- dian which contains the nadir is the lower transit, or culmina- tion. At the upper transit of a point its hour angle is zero, or 0* ; at the lower transit, its hour angle is 12*. 38. The motion of the earth about its axis is perfectly uni- form. If, then, the axis of the earth preserved precisely the same direction in space, the apparent diurnal motion of the celestial sphere would also be perfectly uniform, and the inter- vals between the successive transits of any assumed point of the sphere would be perfectly equal. The effect of changes in the position of the earth's axis upon the transit of stars is most per- ceptible in the case of stars near the vanishing points of the axis, that is, near the poles of the heavens. We obtain a measure of time sensibly uniform by employing the successive transits of a point of the equator. The point most naturally indicated is the vernal equinox (also called the First point of Aries, and de- noted by the symbol for Aries, T). 39. A sidereal day is the interval of time between two succes- sive (upper) transits of the true vernal equinox over the same meridian. The effect of precession and nutation upon the time of transit TIME. 53 of the vernal equinox is so nearly the same at two successive transits, that sidereal days thus denned are sensibly equal. (See Chapter XI. Art. 411.) The sidereal time at any instant is the hour angle of the vernai equinox at that instant, reckoned from the meridian westward from 0* to 24*. When HP is on the meridian, the sidereal time is 0* O m 0* ; and this instant is sometimes called sidereal noon. 40. A solar day is the interval of time between two successive upper transits of the sun over the same meridian. The sofar time at any instant is the hour angle of the sun at that instant. In consequence of the earth's motion about the sun from west to east, the sun appears to have a like motion among the stars, or to be constantly increasing its right ascension ; and hence a solar day is longer than a sidereal day. 41. Apparent and mean solar time. If the sun changed its right ascension uniformly, solar days, though not equal to sidereal days, would still be equal to each other. But the sun's motion in right ascension is not uniform, and this for two reasons : 1st. The sun does not move in the equator, but in the ecliptic, so that, even were the sun's motion in the ecliptic uniform, its equal changes of longitude would not produce equal changes of right ascension ; 2d. The sun's motion in the ecliptic is not uni- form. To obtain a uniform measure of time depending on the sun's motion, the following method is adopted. A fictitious sun, which we shall call the first mean sun, is supposed to move uniformly at such a rate as to return to the perigee at the same time with the true sun. Another fictitious sun, which we shall call the second mean sun (and which is often called simply the mean sun), is sup- posed to move uniformly in the equator at the same rate as the first mean sun in the ecliptic, and to return to the vernal equinox at the same time with it. Then the time denoted by this second mean sun is perfectly uniform in its increase, and is called mean time. The time which is denoted by the true sun is called the true or, more commonly, the apparent time. The instant of transit of the true sun is called apparent noon, and the instant of transit of the second mean sun is called mean noon. 54 TIME. The equation of time is the difference between apparent and mean time ; or, in other words, it is the difference between the hour angles of the true sun and the second mean sun. The greatest difference is about 16"' The equation of time is also the difference between the right ascensions of the true sun and the second mean sun. The right ascension of the second mean sun is, according to the preceding definitions, equal to the longitude of the first mean sun, or, as it is commonly called, the sun's mean longitude. To compute the equation of time, therefore, we must know how to find the longi- tude of the first mean sun ; and this is deduced from a knowledge of the true sun's apparent motion in the ecliptic, which belongs to Physical Astronomy. Here it suffices us that its value is given for each day of the year in the Ephemeris, or Nautical Almanac. 42. Astronomical time. The solar day (apparent or mean) is conceived by astronomers to commence at noon (apparent or mean), arid is divided into twenty-four hours, numbered succes- sively from to 24. Astronomical time (apparent or mean) is, then, the hour angle of the sun (apparent or mean), reckoned on the equator west- ward throughout its entire circumference from 0'' to 24*. 43. Civil time. For the common purposes of life, it is more convenient to begin the day at midnight, that is, when the sun is on the meridian at its lower transit The civil day is divided into two periods of twelve hours each, namely, from midnight to noon, marked A.M. (Ante Meridiem), and from noon to midnight, marked P.M. (Post Meridiem) 44. To convert civil into astronomical time. The civil day begins 12* before the astronomical day of the same date. This remark is the only precept that need be given for the conversion of one of these kinds of time into the other. EXAMPLES. Ast. T. May 10 ? 15*= Civ. T. May 11, 3' A.M. Ast. T. Jan. 3, 7*= Civ. T. Jan. 3, 7* P.M. Ast. T. Aug. 31, 20* = Civ T. Sept. 1, 8* A.M. TIME. 55 45. Time at different meridians. The hour angle of the sun at any meridian is called the local (solar) time at that meridian. The hour angle of the sun at the Greenwich meridian at the same instant is the corresponding Greenwich time. This time we shall have constant occasion to use, both because longitudes in this country and England are reckoned from the Greenwich meridian, and because the American and British Nautical Almanacs are computed for Greemvich time.* The difference between the local time at any meridian and the Greenwich time is equal to the longitude of that meridian from Greenwich, expressed in time, observing that 1 A = 15. The difference between the local times of any two meridians is equal to the difference of longitude of those meridians. In comparing the corresponding times at two dif- ferent meridians, the most easterly meridian may be distinguished as that at which the time is greatest ; that is, latest. If then PM, Fig. 10, is any meridian (referred to the celestial sphere), PG the Greenwich meridian, PS the declination circle through the sun, and if we put T n = the Greenwich lime GPS, T = the local time = MPS, L = the west longitude of the meridian PM = GPM, we have If the given meridian were east of Greenwich, as PM' , we should have its east longitude = T 7^; but we prefer to use the general formula L = T (} T in all cases, observing that east longitudes are to be regarded as negative. In the formula (54), T and T are supposed to be reckoned always westward from their respective meridians, and from 0* to 24 A ; that is, T and T are the astronomical times, which should, of course, be used in all astronomical computations. As in almost every computation of practical astronomy we are dependent for some of the data upon the ephemeris, and these * What we have to say respecting the Greenwich time is, however, equally appli- cable to the time at any other meridian for which the epheineris may be computed. 56 TIME. are commonly given for Greenwich, it is generally the first step in such a computation to deduce an exact or, at least, an ap- proximate value of the Greenwich astronomical time. It need hardly he added that the Greenwich time should never be other- wise expressed than astronomically.* EXAMPLES. 1. In Longitude 76 32' W. the local time is 1856 April 1, 9* 3 m 10* A.M. ; what is the Greenwich time ? Local Ast, T. March 31, 21* 3" 10' Longitude -f 5 6 8 Greenwich T. ApriFT^ 2 9 18 2. In Long. 105 15' E. the local time is August 21, 4 7 ' 3'" P.M ; what is the Greenwich time ? Local Ast. T. Aug. 21, 4* 3" Longitude 7 1 Greenwich T. Aug. 20, 21 2 3. Long. 175 30' W. Loc. T. Sept. 30, 8" 10'" A.M.^G. T. Sept. 30, l h 52'". 4. Long. 165 0' E. Loc. T. Feb. 1, 7" 11* P.M. = G. T. Jan. 31, 20" 11'". 5. Long. 64 30' E. Loc. T. June 1, 0* M. (Noon) = G. M. T. May 31, 19* 42. 46. In nautical practice the observer is provided with a chro- nometer which is regulated to Greenwich time, before sailing, at a place whose longitude is well known. Its error on Green- wich time is carefully determined, as well as its daily gain or loss, that is, its rate, so that at any subsequent time the Green- wich time may be known from the indication of the chronometer corrected for its error and the accumulated rate since the date of sailing. As, however, the chronometer has usually only 12* marked on the dial, it is necessary to distinguish whether 'it indicates A.M. or P.M. at Greenwich. This is always readily done by means of the observer's approximate longitude and local * On this account, chronometers intended for nautical and astronomical purposes should always be marked from C* to 24*, instead of from 0* to 12* as is now usual. It is surprising that nav : gators have not insisted upon this point. TIME. 57 time. As this is a daily operation at sea, it may be well to illus- trate it by a few examples. 1. In the approximate longitude 5* W. about 3* P.M. on Au- gust 3, the Greenwich Chronometer marks 8 A 11'" 7", and is fast on G. T. 6"' 10* ; what is the Greenwich astronomical time ? Approx. Local T. Aug. 3, 3* Gr. Chronom. 8* 11" 7* " Longitude, -f- 5 Correction, - 6 10 Approx. G. T. Aug. 3, 8 Gr. Ast. T. Aug. 3, 8 4 57 2. In Long. 10* E. about l h A.M. on Dec. 7, the Greenwich Chronometer marks 3* 14 W 13'.5, and is fast 25 m 18'.7 ; what is theG. T.? Approx. Local T. Dec. 6, 13* Gr. Chronom. 3* 14" 13-.5 " Long. 10 Correction, 25 18 .7 Approx. G. T. Dec. 6, 3~ G. A. T. Dec. 6, 2 48 54.8 3. In Long. 9* 12" W. about 2* A.M. on Feb. 13, the Gr. Chron. marks 10'' 27" 13*.3, and is slow 30"' 30*.3; what is the G. T.? Approx. Local T. Feb. 12, 14* Gr. Chronom. 10* 37" 13'.3 " Long. +9 Correction, -f 30 30 .3 Approx. G. T. Feb. 12, 23 G. A. T. Feb. 12, 23 7 4& -6 The computation of the approximate Greenwich time may, of course, be performed mentally. 47. The formula (54), L= T 9 T, is true not only when T 9 and T are solar times, but also when they are any kinds of time whatever, or, in general, when T and T 7 express the hour angles of any point whatever of the sphere at the two meridians whose difference of longitude is L. This is evident from Fig. 10, where S' may be any point of the sphere. 48. To 'convert the apparent time at a given meridian into the mean time, or the mean into the apparent time. If M = the mean time, A = the corresponding apparent time, E = the equation of time, we have M =A + E or A =ME 58 TIME. in which E is to be regarded as a positive quantity when it is additive to apparent time. The value of E is to be taken from the Nautical Almanac for the Greenwich instant corresponding to the given local time. If apparent time is given, find the Gr. apparent time and take E from page I of the month in the Nautical Almanac; if mean time is given, find the Gr. mean time and take E from page II of the month. EXAMPLE 1. In longitude 60 W., 1856 May 24, 3* 12 m 10' P.M., apparent time ; what is the mean time ? We have first Local time May 21, 3* 12" 10' Longitude, 400 Gr. app. time May 24, 7 12 10 We must, therefore, find E for the Gr. time, May 24, 7* 12" 1 10", or 7*.21. By the Nautical Almanac for 1856, we have .E'at apparent Greenwich noon May 24 = 3"' 25".43, and the hourly difference + 0".224. Hence at the given time E = 3" 25'.43 + 0-.224 X 7.21 = 3 m 23.81 and the required mean time is M = 3* 12 10' 3- 23-.81 = 3 8 46M9. EXAMPLE 2. In longitude 60 W., 1856 May 24, 3* 8 m 46M9 mean time ; what is the apparent time ? Gr. mean time, May 24, 7* 8 m 46-.19 (= 7M5) E at mean noon May 24 = 3 m 25'.41 Hourly diff. = 0'.224 Correction for 7M5 + 1-60 7.15 E= 3 23.81 1.60 and hence M 3* 8" 46'.19 E = + 3 23.81 A = 3 12 10 .00 As the equation of time is not a uniformly varying quantity, it is not quite accurate to compute its correction as above, by mul- tiplying the given hourly difference by the number of hours in the Greenwich time, for that process assumes that this hourly difference is the same for each hour. The variations in the hourly difference are, however, so small that it is only when TIME. 59 extreme precision is required that recourse must be had to the more exact method of interpolation which will be given here- after. 49. To determine the relative length of the solar and sidereal units of time. According to BESSEL, the length of the tropical year (which is the interval between two successive passages of the sun through the mean vernal equinox) is 365.24222 mean solar days;* and since in this time the mean sun.has described the whole arc of the equator included between the two positions of the equinox, it has made one transit less over any given meridian than the vernal equinox ; so that we have 366.24222 sidereal days = 365.24222 mean solar days whence we deduce 365 242 9 2 1 sid. day = ^ ^ sol. day = 0.99726957 sol. day or 24* sid. time = 23* 56 m 4'.091 solar time Also, 366 ? -499? 1 sol. day = - ^ ^ sid. day = 1.00273791 sid. day or 24* sol. time = 24* 3 ra 56'.555 sid. time If we put 366 24222 "=Hii= 1 - 00273791 and denote by 7 an interval of mean solar time, by I' the equiva- lent interval of sidereal time, we always have 7'=A*7 = 7+0* 1) / =/+ 00273791 I \ I = - = I' - (1 1)7' = 7' _ .00273043 I' f ( 55 ) n ^ t* J ) Tables are given in the Nautical Almanacs to save the labor of computing these equations. In some of these tables, for each solar interval 7 there is given the equivalent sidereal interval /' = ///, and reciprocally : in others there are given the correc- tion to be added to 7 to find P (i.e. the correction .00273791 7), * The length of the tropical year is not absolutely constant. The value given in the text is for the year 180C Its decrease in 100 years is about 0'.6 (Art. 407). 60 TIME. and the correction to he Subtracted from /' to find I (i.e. the correction .00273043 1'). The latter form is the most conve- nient, and is adopted in the American Ephemeris. The correction ([j. 1) / is frequently called the acceleration of the fixed stars (re- latively to the sun). The daily acceleration is 3'" 56'. 555. 50. To convert the mean solar time at a giccn meridian into the corresponding sidereal time. In Fig. 1, page 25, if PQ is the given meridian, VQ the equator, D the mean sun, Vthe vernal equinox, and if we put T '= DQ = the mean solar time, 0= VQ=t\\G sidereal time, = the right ascension of the meridian, V= the right ascension of the mean sun, we have e = r+ V (56) The right ascension of the mean sun, or V, is given in the American Ephemeris, on page II of the month, for each Green- wich mean noon. It is, however, there called the "Sidereal Time," because at mean noon the second mean sun is on the meridian, and its right ascension is also the right ascension of the meridian, or the sidereal time. But this quantity V is uni- formly increasing* at the rate of 3"' 56'.555 in 24 mean solar hours, or of 9*. 8565 in one mean hour. To find its value at the given time 7 1 , we may first find the Greenwich mean time 7 T by applying the longitude ; then, if we put F = the value of Fat Gr. mean noon, = the " sidereal time" in the ephemeris for the given date, we have V= F H- 9-.S565 X T, in which T must be expressed in hours and decimal parts. It is easily seen that 9'.8565 is the acceleration of sidereal time on solar time in one solar hour, and therefore the term 9'.8565 X T is the correction to add to T to reduce it from a solar to a side- real interval. This term is identical with (p 1)3^ as given by * The sidereal time at mean noon is equal to the true R.A. of the mean sun, or it is the R.A. of the mean sun referred to the true equinox, and therefore involves the nutation, so that its rate of increase is not, strictly, uniform. But it is sufficiently so for 24 hours to be so regarded in all practical computations. See Chapter XI. TIME. 61 the preceding article, if T in tlie latter expression is expressed in seconds, since we have .- 0.00273791 =/*-! 3600- We may then write (56) in the following form, putting L = the west longitude of the given meridian, and T = T + L: Q = T + F + (/, - 1) (T + L) (57) The term (p. 1) (T -f .L) is given in the tables of the Amer- ican Ephemeris for converting "Mean into Sidereal Time," and may be found by entering the table with the argument T + Z, or by entering successively with the arguments T and L and adding the corrections found, observing to give the correction for the longitude the negative sign when the longitude is east. If no tables are at hand, the direct computation of this term will be more convenient under the form 9*.8565 X T v EXAMPLE 1. In Longitude 165 W. 1856 May 17, 4* A.M.: what is the sidereal time ? The Greenwich time is May 17, 3' 1 ; and the computation may be arranged as follows : Local Ast. Time T = 16* 0" -5 ~ * * = 0.28 j and from the Ephemeris : March 30, 12 (d) 36 17' 53" Q, .2993 A#, + .0041 j' Q 40 28 .0011 .28 At 13* 20" 24 d = 35 37 25 Q', .2982 + .0011 log*, 3.6834 log J', 3.3852 66. To find the Greenwich time corresponding to a given lunar dis tance on a given day. We find in the Ephemeris for the given day the two distances between which the given one falls; and if A' = difference be- tween the first of these and the given one, J = difference of the distances in the Ephemeris, we find the interval t, to be added to the preceding Greenwich time, by simple interpolation, from tho formula f = 3x- A or log t = log J' -j- P. L. J = log J' 4- Q (65) and, with regard to second differences, the true interval; t 1 ', by the formula log' = k>g J'+ Q' (66) where Q' has the value given in the preceding article. But to find Q' by (64) we must first find an approximate value of t. To avoid this double computation, it is usual to find t by (65), and to give a correction to reduce it to I' in a small table which is computed as follows. We have from (64), (65), and (66) 78 EPHEMERIS. By the theory of logarithms, we have, M being the modulus of the commpn system, log x = M[(x 1) \(x I) 2 + &c.] so that or, neglecting the square and higher powers of the small fraction t' - t t ' log t' log t = Ml ) \ t I This, substituted above, gives MX 3* 2Jlfx3 by which a table is readily computed giving the value of t' t [or the correction of found by (65)], with the arguments A^and t. In this formula t and /' t are supposed to be expressed in hours; and to obtain t' t in seconds we must multiply the second member by 3600; this will be effected if we multiply each of the factors t and 3 A t by 60, that is, reduce them each to minutes, so that if we substitute the value of M = .434294 the formula becomes *< in which t is expressed in minutes, and t' t in seconds. EXAMPLE. 1856 March 30, the distance of the moon and Fomalhaut is 35 37' 25" ; what is the Greenwich time ? March 30, 12* 0" 0* (d) = 36 17' 53" Q= .2993 A# = + 41 t= 1 20 36 d =35 37 25 log J' = 3.3852 A p. Gr. time =13 20 36 J' 4028 log* =3.6845 By(67)V * = -12 True Gr. time = 13 20 24 * Or from the " Table showing (he correction required on account of the second differences of the moon's motion in finding the Greenwich time corresponding to a corrected lunar distance," which is given in the American Ephemeris, and is also included in the Table? for Correcting Lunar Distances given in Vol. II. of this work. INTERPOLATION IN GENERAL. 79 INTERPOLATION BY DIFFERENCES OF ANY ORDER. 07. When the exact value of any quantity is required from the Ephemeris, recourse must be had to the general interpolation formulae which are demonstrated in analytical works. These enable us to determine intermediate values of a function from tabulated values corresponding to equidistant values of the variable on which they depend. In the Ephemeris the data are in most cases to be regarded as functions of the time considered as the variable or argument. Let T, T+ w, T-\- 2w, T+ 3w, &c., express equidistant values of the variable; F, F', F", F'", &c., corresponding values of the given function ; and let the differences of the first, second, and following orders be formed, as expressed in the following table : Argument. Function. 1st Diff. 2d Diff. 3d Diff. 4th Diff. 5th Diff. 6th Diff. T F a T+ w F' b a' C T+2w F" V d a" 4 e T+3w .F'" b" d f / a'" c" e' T-f-4u; F" b'" d" a C"' T+6w F" 6" a" T+ Qw F The differences are to be found by subtracting downwards, that is, each number is subtracted from the number below it, and the proper algebraic sign must be prefixed. The differences of any order are formed from those of the preceding order in the same manner as the first differences are formed from the given func- tions. The even differences (2d, 4th, &c.) fall in the same lines with the argument and function ; the odd differences (1st, 3d, &c.) between the lines. Now, denoting the value of the function corresponding to a value of the argument T -\- nw by F ( "\ we have, from algebra, . (68) 1.2 1.2.3 1.2.3.4 in which the coefficients are those of the n th power of a binomial. 80 INTERPOLATION IN GENERAL. Iii this formula the interpolation sets out from the first of the given functions, and the differences used are the first of their respective orders. If n be taken successively equal to 0, 1, 2, 3, &c., we shall obtain the functions F, F', F", F'", &c., and in- termediate values are found by using fractional values of??. We usually apply the formula only to interpolating between the function from which we set out and the next following one, in which case n is less than unity. To find the proper value of n in each case, let T -\- t denote the value of the argument for which we wish to interpolate a value of the function : then t n = w nw = t that is, n is the value of t reduced to a fraction of the interval w. EXAMPLE. Suppose the moon's right ascension had been given in the Ephemeris for every twelfth hour as follows : D'sR. A. 1st. Diff. 2d Diff. 3d Diff. 4th Diff. 1856 March 5, A 21* 58 m 28'. 39 + 28 m 47'. 04 5, 12 22 27 15.43 36'. 97 28 10.07 + 4'. 79 " 6, 22 55 25.50 32.18 + K74 27 37.89 6.53 " 6, 12 23 23 3. 39 25.65 1.08 27 12.24 7.61 " 7, 23 50 15.03 18.04 26 54.20 " 7, 12 17 9.83 (J'.66 Required the moon's right ascension for March 5, 6*. Here T= March 5, 0*, * = 6*, w = 12\ n:==y^=l; and if we denote the coefficients of a, 6, e, rf, e in (68) by A, B, (7, D, JS, we have .F = 2158'"28'.39 a = + 28 47'.04, A = n == i, 36 .97, B = A . = 4, Aa = + 14 23 .52 Bb = + 4 .62 4.79, 0=B. 1.74, D = C. 0.66, E = D. D's E. A. 1856 March 5, 6* Cc=+ = , Ee = 0.30 0.07 0.02 = 22 12 56 .74 INTERPOLATION. which agrees precisely with the value given in the American Epheraeris. 68. The formula (68) may also be written as follows : Thus, in the preceding example, we should have n 4 5 n 3 4 n 2 3 n 1 - T 7 o X 0-.66 - I (+ l'J4 + 0*.46) - (+ 4 '- 79 1 '- 38 ) _ i (_ 3G-.97 I'.Tl) + 0'.46 - 1 .38 - 1 -71 + 9 .67 (+ 2847'.04 + 9'.67) = + 14" 28'.35 and adding this last quantity, 14" 1 28'.35, to 21* 58"' 28'.39, we obtain the same value as before, or 22 A 12'" 56'.74. 69. A more convenient formula, for most purposes, may be deduced from (68), if we use not only values of the functions following that from which we set out, but also preceding values; "hat is, also values corresponding to the arguments T w, T 2w, &c. We then form a table according to the following schedule : rgument. Function. 1st Diff. 2d Diff. 3d Diff. 4th Diff. 5th Diff. 6th Diff. T 3u? F nl a ,n T 2w F u U T w p t " b t " d , * a i C e t T F b d f a' c' e' T + w F' b' d' a" c" T -\- 2w F" b" a'" T+3w put VOL. I.- &S INTERPOLATION. According to the formula (68), if we set out from the function Fj we employ the differences denoted in this table by ', &', c", &c., and hence for the argument T + nw we find the value of jp<> \) the formula 1.2 1.2.3 1.2.3.4 But we have V = b +d c" = c f + d' = c' + d + e' d" = d' + e" = d + ef + e' + /' = d + 2e' + /' &c. &c. in which 6', c",- &c. are expressed in terms of the differences that lie on each side of a horizontal line drawn in the table immediately under the function from which we set out. These values substituted in the formula give l)(*)(n-l)('.-2) + ke. (69) in which the law of the coefficients is that one new factor is introduced into the numerator alternately after and be/ore the other factors, observing always that the factors decrease by unity from left to right. The new factor in the denominator, as in the original formula (68), denotes the order of difference. The interpolation by this formula is rendered somewhat more accurate by using, instead of the last difference, the mean of the two values that lie nearest the horizontal line drawn under the middle function : thus, if we stop at the fourth difference, we use a mean between d and d' instead of d. We thus take into account a part of the term involving the fifth difference. EXAMPLE. Find the moon's right ascension for 1856 March 5, 6 fc , employing the values given in the Ephemeris for every twelfth hour. This is the same as the example under Art. 67, where it is worked by the primitive formula (68). But here we take from the Ephemeris three values preceding that for March 5, 0*, and three vsdues following it, and form our table as follows: INTERPOLATION. 1856 March 3, 12* " 4, " 4, 12 ' 5, D'sR.A. 1st Diff. 2dDiff. 3d Diff. 4th Diff. Sth Diff. 20* 28 m 17'. 88 20 58 57.08 21 29 2.01 21 58 28.39 + 30" 39.20 30 4.93 29 26.38 34'. 27 38.55 39.34 4'. 28 0.79 + 3'.49 3.16 0.33 5, 12 " 6, 6, 12 22 27 15.43 22 55 25.50 23 23 3.39 28 47.04 28 10.07 27 37.89 36.97 32.18 + 2.37 + 4.79 2.42 0.74 Drawing a horizontal line under the function from which we set out, the differences required in the formula (69) stand next to this line, alternately below and above it. F = 21* 58" 28'.39 a' = - f- 28" 47' QQ .04, .34, .37, .16, .74, A = C' A . B . C. D . n n 1 ll I T! S > 3l> ^4a' = + fih - -I- 14 23.52 4.92 0.15 0.07 0.01 t? -I h 2 h 3 o 2 n + 1 fV 3 n 2 E 4 9 | 7/V 5 T" D's R. A. 1856 March 5, 6 = f 22 12 56 .74 69*. If in (69) we substitute the values a' = a, + b &c. we find 1.2 b 1.2.3.4 1.2.3 (70) in which the law of the coefficients is that one new factor is introduced into the numerator alternately before and after the other factors, observing still that the factors decrease by unity from left to right. The differences employed are those which lie on each side of the horizontal line drawn immediately above the function from which we set out. 84 INTERPOLATION. If in the preceding formulae we employ a negative value of n less than unity, we shall obtain a value of the function between F and F and in that case (70) is more convergent than (69). In general, if we set out from that function which is nearest to the required one, we shall always have values of n numerically less than |, and we should prefer (69) for values of n between and -f- J, and (70) for values of n between and -|. 70. If we take the mean of the two formulae (69) and (70), and denote the means of the odd differences that lie above and below the horizontal lines of the table, by letters without ac- cents, that is, if we put a = } (a, -f a'), c = * (c, + c') &c. we have The quantities a, c, &c. may be inserted in the table, and will thus complete the row of differences standing in the same line with the function from which we set out. The law of the coefficients in (71) is that the coefficient of any odd difference is obtained from that of the preceding odd dif- ference by introducing two factors, one at the beginning and the other at the end of the line of factors, observing as before that these factors are respectively greater and less by unity than those next to which they are placed; and the coefficients of the even differences are obtained from the next preceding even differences in the same manner. The factors in the denominator follow the same law as in the other formulae. EXAMPLE. Find the moon's right ascension for 1856 March 5, 6*, from the -values given in the Ephemeris for noon and mid- night The table will be as below: INTERPOLATION. 85 Mar. 3, 12 4, 4, 12 >s R. A. 1st Diff. 2d Diff. 3d Diff. 4th Diff. 5th Diff. 20*28'" 17'.88 20 58 57 .08 21 29 2 .01 + 30 m 29.20 30 4 .93 29 26.38 34'.27 38.55 4'.28 0.79 +3'.49 C-.33 5, 21 58 28.39 [+29 6 .71] 39.34 [+0 .79] + 3.16 [-0.54] " 5, 12 6, " 6, 12 22 27 15.43 22 55 25.50 23 23 3 .39 28 47.04 28 10.07 27 37.89 36.97 32.18 + 2.37 + 4.79 2 42 0.74 Drawing two lines, one above and the other below the func- tion from which we set out, and then tilling the blanks by the means of the odd differences above and below these lines (which means are here inserted in brackets), we have presented in the same line all the differences required in the formula (71) ; and we then have F= 21* 58"28'.39 a = + 29- 6-.71, .4 = n = J, Aa = + 14 33 .36 b=~ 39.34, = =+l, b= 4.92 .79, C^ 3.16, J>= 0.54, E=C. p Ee = 0.05 0.02 .01 poo = 22 12 56 .75 agreeing within O'.Ol with the value found in the preceding article. HANSEN has given a table for facilitating the use of this formula. (See his Tables de la Lune). . 71. Another form, considered by Bessel as more accurate than any of the preceding, is found by employing the odd differences that fall next below the horizontal line drawn below the function from which we set out, and the means of the even differences that fall next above and next below this line. Thus, if we put *. = i (* -f V\ <*. = t(* + 's2dlimb. 1st Diff. 2d Diff. 3d Diff. 4th Diff. 5th Diff. May 14, U. C. 15, L. C. " 15, U. C. 15* 12 m 30'. 04 15 41 3.41 16 9 39.89 -f 28 24'.37 28 36.48 + 12M1 -f 9.49 2.62 K68 28 45.97 [+7.39] 4.^0 f-1.42] 4-0-.33 " 16, L. C. " 16, U. C. ' 17. L. C. 16 38 25.86 17 7 17.12 17 36 8.22 28 51 .26 28 51.10 + 5.29 0.16 5.45 1.25 INTERPOLATION. 87 For interpolation by formula (72) we draw a horizontal lino below the function from which we set out, and one above the next following function. These lines enclose the odd differences regularly occurring in the table. Inserting in the blanks in the columns of even difterences the means of the numbers above and below, all the differences to be employed in the formula stand in the same line, namely: a' = -f 1725-.97, b = + 7'.39, c' = 4.20, d = 1-.42, ef = + (K33 As n is here not a simple fraction, the computation will be most conveniently performed by logarithms, as follows : 4* 42" 19* = 16939* log 4.2288878 12 ^43200 log 4.6354837 log A = log n = 9.5934041 n = 0.39210651 9.59340 9.5934 9.5934 9.5934 n 1 = 0.60789 719.78383 n9.7838 7i9.7838 719.7838 n i = 0.10789 n9.0330 n9.0330 n -2 = 1.6079 nO.2063 nO.2063 n -|- 1 = + 1.3921 (A) 9.5934041 (a') 3.2370332 Q) 9.698-97 (g) 9-2218 0.1437 ( 5 L) 8.6198 0.1437 (5)7i9.07620 (&) 0.86864- (C) 7.6320 (c')n0.6232 (Z>) 8.3470 (e') 9.5185 2.8304373 n9.94484 n8.2552 nS.4993 n6.199S Increase of R. A. Aa' = 11" 16'.764 Bb Q = .879 (y . _ .018 Z>^ = .032 Eg = 0.000 = 11 15.835 K. A. Greenwich Culm. = 16* 9" 39*. 890 E. A. on given meridian = 16* 20 W 55'.725 The use of BESSEL'S formula of interpolation is facilitated bj a table in which the values of the coefficients above denoted 1 y A, B, (7, Z), &c., and also their logarithms, are given with the argument n. 72. Interpolcttion into the middle. When a value of the functkn is sought corresponding to a value of the argument which is a 88 INTERPOLATION. mean between two values for which the function is given, that is, when n J, we have hy (72), since n | = 0, or, since F+ J a' = which is known as the formula for interpolating into the middle. When the third differences are constant, d w / , &c. are zero, and the rule for, interpolating into the middle between two func- tions is simply : From the mean of the two functions subtract one- eighth the mean of the second differences which stand against the func- tions. Interpolation by this rule is correct to third differences inclusive. The formula (73) is especially convenient in computing tables. Values of the function to be tabulated are directly computed for values of the argument differing by 2 m w ; then interpolating a value into the middle between each two of these, the arguments now differ by 2 m ~ l w ; again interpolating into the middle between each two of the resulting series, we obtain a series with argu- ments differing by 2 m ~ 2 iv ; and so on, until the interval of the argument is reduced to 2 m ~ in w or w. EXAMPLE. Find the moon's right ascension for 1856 March 5, 6 A , from the values of the Ephemeris for noon and midnight. This is the same as the example of Art. 69 ; but, as 6 A is the middle instant between noon and midnight, the result will be obtained by the formula (73) in the following simple manner. We have from the table in Art. 69 b = 38'.16 =2'2 12 56.74 73. In case we have to interpolate between the last two values of a given series, we may consider the series in inverse order, the arguments being T, Tw, TZw, &c., T being the last argument. The signs of the odd differences will then be changed, and, taking the last differences in the several columns as a, 6, c, d, &c., the interpolation will be effected by (68). INTERPOLATION. 89 74. The interpolation formulas, arranged according to the powers of the fractional part of the argument. When several values of the function are to be inserted between two of the given series, it is often convenient to employ the formula arranged according to the powers of n. Performing the multiplications of the factors indicated in (68), and arranging the terms, we obtain F* = F + n (a - \ b -f \ c - \ d + I e - &c.) 3 (c I d -f I e &c.) 1.2.3 + &c ......... (74) where the differences are obtained according to the schedule in Art. 67. Transforming (71) in the same manner, we have J*"> = F + n (a - ],c + 3 ' e - &c.) + &c ......... (75) where the differences a, e, ), f'"(T + nw}, &c., and f(T\ f'(T), f"(T], Ac., will denote the values of the function and its derivatives corresponding to the argument T, or when n = 0. Hence, if we regard nw as the variable, we shall have, by Mac- laurin's Theorem, /( T + nw) = /( 3T) + f( T) nw + /"( T) ^ + &c. Comparing the coefficients of the several powers of n in this formula with those in (74), we have f'(T) = (a - \ b -f | c - \d + i e - fec.) w ==~ (6 - c + \$d - | e + &c.) ^r) =-(4- 2e + Ac.) r(r)=~-(e-M) &c. &c ....... (76) the differences being taken as in Art. 67. Still more convenient expressions are found by comparing Maclaurin's Theorem with (75); namely: /'(T) =-i (a - b i c + J e - Ac.) w /"(T) = -L(&- T ' 2 d + &c.) /'"(!T)=i (c-tc + Ac.) &c. Ac. (77) the differences being found according to the schedule in Art. 69, and the odd differences, , c, e, &c., being interpolated means. STAR CATALOGUES. 91 The preceding formulae determine the derivatives for the value T of the argument. To find them for any other value, we have, by differentiating Maelaurin's Formula with reference to nw, f'(T + nw) =f'(T} + f"(T) . nw + J/"'(!T) . nW + Ac. (78) in which we may substitute the values of f(T),f f (T), &c. from (76) or (77). In like manner, by successive differentiations of (78) we ob- tain /" (T+nw) =f" (T} +/'" (T). nw + */* (T). nw + &c- /'" (T+ nw}=f"(T) + /" (T).nw + &c. &c. &c. 76. An immediate application of (76) or (77) is the compu- tation of the differences in a unit of time of the functions in the Ephemeris ; for this difference is nothing more than the first derivative, denoted above by the symbol /'. EXAMPLE. Find the difference of the moon's right ascension in one minute for 1856 March 5, 0*. We have in Art. 70, for T = March 5, A , a = 29 m 6'.71, c = + 0'.79, e= - s . 54, and w = 12 A = 720" 1 . Hence, by the first equation of (77), f'(T) = , (29- 6'.71 0-.13 0-.02) == 2'.4258 On interpolation, consult also ENCKE in the Jahrbuch for 1830 and 1837. STAR CATALOGUES. 77. The Nautical Almanac gives the position of only a small number of stars. The positions of others are to be found in the Catalogues of stars. These are lists of stars arranged in the order of their right ascensions, with the data from which their apparent right ascensions and declinations may be ob- tained for any given date. The right ascension and declination of the so-called fixed stars are, in fact, ever changing: 1st, by precession, nutation, and aberration (hereafter to be specially treated of), which are not changes in the absolute position of the stars, but are either changes in the circles to which the stars are referred by sphe- rical co-ordinates (precession and nutation), or apparent changes arising from the observer's motion (aberration^; 2d, by the 92 STAR CATALOGUES. proper motion of the stars themselves, which is a real change of the star's absolute position. In the catalogues, the stars are referred to a mean equator and a mean equinox at some assumed epoch. The place of a star so referred at any time is called its mean place at that time ; that of a star referred to the true equator and true equinox, its true place ; that in which the star appears to the observer in motion, its apparent place. The mean place at any time w r ill be found from that of the catalogue simply by applying the preces- sion and the proper motion for the interval of time from the epoch of the catalogue. The true place will then be found by correcting the mean place for nutation ; and finally the appa- rent place will be found by correcting the true place for aber- ration. To facilitate the application of these corrections, BESSEL pro- posed the following very simple arrangement. He showed that if a , d + E I d = 3 U + T M '+ Aa' + Bb' + Cc' + Dd' / in which a, 6, c, d, a', &', c', d' are functions of the star's right ascension and declination, and may, therefore, be computed for each star and given with it in the catalogue ; A, _B, (7, D, E are functions of the sun's longitude, the moon's longitude, the longitude of the moon's ascending node, and the obliquity of the ecliptic, all of which depend on the time, so that -A, B, C, D, E may be regarded simply as functions of the time, and given in the Nautical Almanac for the given year and day; E is a very small correction, usually neglected, as it can never ex- ceed 0".05. If the catalogue does not give the constants a, 6, c, d, a', &', c r , d f , they may be computed, for the year 1850, by the following formulae (see Chap. XI. p. 648): STAR CATALOGUES. 93 a = 46".077 -f 20".05G sin a tan d a' = 20".056 cos a b = cos a tan 3 b' = sin a c = cos a sec 8 c' = tan s cos d sin a sin S d sin a sec d d' = cos a sin 3 hi which e = obliquity of the ecliptic. Or we may resort to what are usually called the independent constants, and dispense with the , 6, c, d, ', b', c', d' altogether, proceeding then by the formula = -f T ^ +/ + g sin (G + ) tan for June 15, 1865, f from the Catal. logs, a 0.5352 logs. A 9.7877 logs, a' 0.8934 b 7.8794 B 0.9437 V n9.9607 c 8.4329 CnO.2125 c' 9.2019 d 8.8058 /> n 1.3089 rf wO.1147 />rf' nO.3467 Corr. of o , Aa = -f 2M03, /?A = -f C'.067, (7c = - O.044, Dd = 1'.302 Corr. of (V 4a' = + 4".80, Bb' = 8".02, C'c' = - 0".2<5, O = + 16 14 .77 78. When the greatest precision is required, we should con- sider the change in the star's place even in a fraction of a day, and therefore also the change while the star is passing from one meridian to another; also the secular variation and the changes PIAZZI, SANTINI ; and the published observations of the principal observatories. See also a list of catalogues in the introduction to the B. A. C. THE EARTH. 95 in the precession and in the logarithms of the constants. Fur- ther, it is to be observed that the annual precession of the cata- logues is for a mean year of 365 d 5*.8. But for a fuller consider- ation of this subject see Chapter XI. CHAPTER III. FIGURE AND DIMENSIONS OF THE EARTH. 79. THE apparent positions of those heavenly bodies which are vvithin measurable distances from the earth are different for ob- servers on different parts of the earth's surface, and, therefore, before we can compare observations taken in different places we must have some knowledge of the form and dimensions of the earth. I must refer the reader to geodetical works for the methods by which the exact dimensions of the earth have been obtained, and shall here assume such of the results as I shall have occasion hereafter to apply. The figure of the earth is very nearly that of an oblate spheroid, that is, an ellipsoid generated by the revolution of an ellipse about its minor axis. The section made by a plane through the earth's axis is nearly an ellipse, of which the major axis is the equatorial and the minor axis the polar diameter of the earth. Accurate geodetical measurements have shown that there are small deviations from the regular ellipsoid ; but it is sufficient for the purposes of astronomy to assume all the meridians to be ellipses with tbe mean dimensions deduced from all the measures made in various parts of the earth. 80. Let EPQP', Fig. 11, be one of the elliptical meridians of the earth, EQ the diameter of the equator, PP' the polar diameter, or axis of the earth, C the centre, jf a focus of the ellipse. Let a the semi-major axis, or equatorial radius, = CE, b = the semi-minor axis, or polar radius, = CP, c = the compression of the earth, e =the eccentricity of the meridian. 96 REDUCTION OF LATITUDE. By the compression is meant the difference of the equatorial and polar radii expressed in parts F i g . 11. of the equatorial radius as unity, or The eccentricity of the meridian is E ' the distance of either focus from the centre, also expressed in parts of the equatorial radius, or, in Fig. 11, CF CE But, since PF= CE, we have, that is, CF 2 _PF 2 -PC 2 _ l PC* CE 2 ~ CE 2 CE 2 e= v /2c c* (81) By a combination of all the most reliable measures, BESSEL deduced the most probable form of the spheroid, or that which most nearly represents all the observations that have been made in different parts of the world. He found* 6 _ = 298.1528 a ~ ~ 299.1528 or whence, by (81), 299.1528 e = .0816967 log e = 8.912205 log i/(l ee) = 9.9985458 * Attronomitche Nachrichten, No. 438. See also Encke's Tables of the dimensions of the terrestrial spheroid in the Jahrbuch for 1852. REDUCTION OF LATITUDE. 97 The absolute lengths of the semi-axes, according to BESSEL, are, a = 6377397.15 metres = 6974532.34 yds. = 3962.802 miles b = 6356078.96 " = 6951218.06 = 3949.555 " 81. To find the redaction of the latitude for the compression of the earth. Let A, Fig. 11, be a point on the surface of the earth; AT the tangent to the meridian at that point ; AO, perpendicular to A T, the normal to the earth's surface at A. A plane touching the earth's surface at A is the plane of the horizon at that point (Art. 3), and therefore A 0, which is perpendicular to that plane, represents the vertical line of the observer at A. This vertical line does not coincide with the radius, except at the equator and the poles. If we produce CE, OA, and CA to meet the celestial sphere in E', Z, and Z' respectively, the angle ZO'E' is the declination of the zenith, or (Art. 7) the geographical latitude, and Z is the geographical zenith ; the angle Z'CE' is the declination of the geocentric zenith Z', and is called the geocentric or reduced latitude ; and ZAZ' = CAO is called the reduction of the latitude. It is evident that the geocentric is always less than the geogra- phical latitude. Now, if we take the axes of the ellipse as the axes of co-ordi- nates, the centre being the origin, and denote by x the abscissa, and by y the ordinate of any point of the curve, by a and 6 the semi-major and semi-minor axes respectively, the equation of the ellipse is *+* = ! a*^ b* If we put tp = the geographical latitude, . 6in 4 " - &c - (88) in which p + 1 1 e* + 1 2 Employing BESSEL'S value of , we find = 690".65 . sin 1" 2 sin 1" and, the subsequent terms being insensible, v p' = 690".65 sin 2 ^ 1".16 sin 4^ (83*) by which

, we have from the equation of the ellipse and its differential equation, after substituting 1 & - == (1 e 2 ) tan ? from which by a simple elimination we find a cos

, putting = 1, is given in our Table III. Vol. II. But the logarithm of p may be more conveniently found by a series. If in (84) we substitute sinV = J (1 cos we find, putting a = 1, /n+/ 4 + (l-/ 4 ) \ Ll +/' + (!-/') f Now (PI. Trig. Art 260) if we have an expression of the form X =|/(1+ m 2 2m cos C) (A) 100 RADIUS OF THE EARTH. we have, if M the modulus of the common system of loga- rithms, ,,/ m? cos 2(7 m 3 cos 3(7 \ log X = M I m cos C -f - - -f - - + &c. ) (B) \ O r by which we may develop the logarithms of the numerator and denominator of the above radical. Hence we find 1 _|_/2 / m 2 __. m 'l log p = log l -l- + M( (m m') cos 2y> cos 4p , m 3 m' 3 \ -| cos 6^ &c. I 3 / in which we have put for brevity I/ 2 I/ m = m' = - 1+/ 2 !+/ Restoring the value of / = j/(l e 2 ) and computing the numerical values of the coefficients, we find log p = 9.9992747 + 0.0007271 cos 2 ^ 0.0000018 cos 4 ? (85) as given by ENCKE in the Jahrbuch for 1852. The values of p and (p' may also be determined under another form which will hereafter be found useful. We have in Fig. 11, p sin ' = a 1 * a cos

so that sin 4- e sin y p sin tp' = a (1 e s ) sin

(oo) p COS (y y) = 0- COS 4 NORMAL. 101 Hence, also, the following: 83. To ft/id ihe length of (he normal terminating in the axis, for a g'tcen latitude. Putting N^= the normal = AO (Fig. 11), \ve have evidently N= ? C 9 y/ =, - a - - (901 cos if y'(\ e* sin 2 ^) or, emf cfiyittg the auxiliary $, of the preceding article, N = a sec $ 84. To find the distance from the centre to the intersection of the normal with the axis. Denoting this distance by ai (so that i denotes the distance when a = 1), Ave have in Pig. 11, ai *= CO and, from the triangle ACO, ai = p sin ^ ~ ^ COS

(ii = 77^ .. . , - N ^= ae* sin e> sec 4. (91) j/(l e l sin 2 64 43> ? anc i the horizontal parallax ic = 20'' .0; find the geocentric zenith distance. log* 1.3010 C' = 64 43' 0"0 log sin C' 9.9563 p= 18.1 log/) 1.2573 C = 64 42 41.9 When the true zenith distance is given, to compute the paral- lax, we may first use this true zenith distance as the apparent, and find an approximate value of p by the formula p = TT sin ; then, taking the approximate value of ' = -f p, we compute a more exact value of p by the formula (94) or (96). This second approximation is unnecessary in all cases except that of the moon, and the parallax of the moon is so great that it becomes necessary to take into account the true figure of the earth, as in the following more general investigation of the subject. 91. In consequence of the spheroidal figure of the earth, the vertical line of the observer does not pass through the centre, and therefore the geocentric zenith distance cannot be directly PARALLAX. 101 referred to this line. If, however, we refer it to the radius drawn from the place of observation (or CAZ', Fig. 11), the zenith dis- tance is that measured from the geocentric zenith of the place; whereas it is desirable to use the geographical zenith. Hence we shall here consider the geocentric zenith distance to be the angle which the straight line drawn from the centre of the earth to the star makes with the straight line drawn through the centre of the earth parallel to the vertical line of the observer. These two vertical lines are conceived to meet the celestial sphere in the same point, namely, the geographical zenith, which is the common vanishing point of all lines perpendicular to the plane of the horizon. Thus both the true and the apparent zenith distances will be measured upon the celestial sphere from the pole of the horizon. The azimuth of a star is, in general, the angle which a vertical plane passing through the star makes with the plane of the meri- dian. When such a vertical plane is drawn through the centre of the earth, it does not coincide with that drawn at the place of observation, since, by definition (Art. 3), the vertical plane passes through the vertical line, and the vertical lines are not coincident. Hence we shall have to consider a parallax in azimuth as well as in zenith distance. 92. To find the parallax of a star in zenith distance and azimuth when the geocentric zenith distance and azimuth are gicen, and the earth is regarded as a spheroid.* Let the star be referred to three co-ordinate planes at right angles to each other : the first, the plane of the horizon of the observer; the second, the plane of the meridian; the third, the plane of the prime vertical. Let the axis of x be the meridian line, or intersection of the plane of the meridian and the plane of the horizon ; the axis of y, the east and west line ; the axis of z, the vertical line. Let the positive axis of x be towards the south; the positive axis of y, towards the west; the positive axis of z, towards the zenith. Let J' = the distance of the star from the origin, which is the place of observation, C' = the apparent zenith distance of the star, A' = the apparent azimuth " " " * The investigation which follows is nearly the same as that of OLBERS, to whom the method itself is due. 108 PARALLAX. then, x' y', z' denoting the co-ordinates of the star in this system, we have, by (39), x 1 = A' sin C' cos A 1 y 1 = A' sin C' sin A' z 1 = A' cos ' Again, let the star be referred by rectangular co-ordinates to another system of .planes parallel to the former, the origin now being the centre of the earth. In the celestial sphere these planes still represent the horizon, the meridian, and the prime vertical. If then in this system we put A = the distance of the star from the origin, = the true zenith distance of the star, A = the true azimuth " " and denote the co-ordinates of the star in this system by x, y, and z, we have, as before, x == A sin y = A sin z = A cos COS ^ sin A Now, the co-ordinates of the place of observation in this last system, being denoted by , 6, c, we have a = p sin (y f ') b = c = p cos (

(97) A' cos C' = A cos C p cos ( ?'} ) which are the general relations between the true and apparent zenith distances and azimuths. All the quantities in the second members being given, the first two equations determine J'sin ', and A ' ; and then from this value of J'sin ', and that of J'cos ' given by the third equation, A' and ' are determined. PARALLAX. 109 But it is convenient to introduce the horizontal parallax instead of J. For, if we put the equatorial radius of the earth 1, we have sin T: = J and hence, if we divide the equations (97) by J, and put we have /sin C' cos A' = sin / sin C' sin A' = sin / cos C' = cos cos A f> sin TT sin (^> y/) ^ sin A I (98) p sin TT cos (y> ?>') sin A \ f sin C' cos (A' A) = sin C p sin rr sin (') cos A J Multiplying the first of these by sin J (A' A), the second by cos J (A f A), and adding the products, we find, after dividing the sum by cos J (A 1 A), ,. cos i (A' + ^) /sm C = sin r-siil * sin ( - y ) which with the third equation of (98) will determine '. If we assume such that (100) we have the following equations for determining ' : / sin ' = sin p sin - cos (

') J ^ which, by the process employed in deducing (99), give sin (C (102) / sin (' C) = p sin TT cos (y> - f cos (C' C) = 1 / sin JT cos (y> ?>') 110 PARALLAX. By multiplying the first of these by sin \ (' ), the second by cos \ (' ), and adding the products, we find, after dividing by cos J(C'-C). f __ i _ P sin K cos (? ~ ?') C08 [* ( g> + c) r] cos r cos } (C' or multiplying by J, cos r cos * (C' C) The equations (99) determine rigorously the parallax in azimuth ; then (100) and (102) determine the parallax in zenith distance, and (103) the distance of the star from the observer. The relation between J and J' may be expressed under a more simple form. By multiplying the first of the equations (101) by cos j-, the second by sin 7% the difference of the products gives 93. The preceding formulae may be developed in series. Put P sin TT sin (

and therefore [PI. Trig. Art. 258], A' A being in seconds, A- - A = - + - + - + &o. (106) sin 1" 2 sin 1" 3 sin 1" To develop f in series, we take ,. cos [A + J (.4' 4)] tan r == tan (

') [cos ^1 sin ^1 tan J (J.' ^4)] whence, by interchanging arcs and tangents according to the PARALLAX. 1H formulae tan" 1 y = y \ y* + &c., tan x - - x + rr 3 + &c. fPl. Trig. Arts. 209, 213], (' COS f we find from (102) 1 _ w cos (C whence, ' being in seconds, (1081 _ r = '1^__- + p'i"2C-r) n'sin8(C-r) &c 1Q sin 1" 2 sin 1" 3 sin 1" Adding the squares of the equations (102), we have / 2 ( ) =1 2 n cos ( r) ~\~ w * \J / whence [equations (A) and (J5), Art. 82] log J' = log J M (n cos (C Y) -\- ~\~ & C -J (HO) where Jf=the modulus of common logarithms. 94. The second term of the series (107) is of wholly inappre- ciable effect ; so that we may consider as exact the formula ? = (') cos A sin & m cos A = then 1 sin . tan tan (45 -f i *) tan A (112) 112 PARALLAX. Put sin *' = n cos (: - ,) = cos Y then tan r _ = 1 sin *' = tan A' tan (45 + } *') tan (C (US) EXAMPLE. In latitude ^ = 38 59', given for the moon, A = 320 18', C = 29 30', and n = 58' 37".2, to find the parallax in azimuth and zenith distance. We have (Table III.) for

0') 9.999998 log cosec C 0.30766 log sec y 0.000001 log cos A 9.88615 log cos (C 7) 9.940313 & = 18", log sin # 5.93987 #'= 61' 1".5, log sin #' 8.171491 log tan i? 5.93987 log tan #' 8.171539 log tan (45 + J tf) 0.00004 log tan (45 -f J #') 0.006446 log tan A 7i9.91919 log tan (C 7) 9.750087 log tan (A 1 A) n5.85910 log tan (C' C) 7.928072 A' A = 14".91 C' C = 29' 7". 79 A' = 320 17' 45".09 f = 29 59' 7". 79 It is evident that we may, without a sacrifice of accuracy, omit the factors cos (

(115) Again, if we multiply the first of the equations (98) by sin A' and the second by cos A', the difference of the products gives sin (A' -A} = to compute which, must first be found by subtracting the value of the parallax ' , found by (114), from the given value of ". EXAMPLE. In latitude tp = 38 59', given for the moon A' =j 320 17' 45".09, == 29 59' 7".79, TT = 58' 37".2, to find the parallax in zenith distance and azimuth. We have, as in the example Art. 94, ^> ?>') 7.51488 log sin (:' C) 7.928058 log sin A' n9.80538 ? c = 29' 7".79 log cosec C 0.30766 C = 29 30' 0" log sin (A' A} n5.85910 A' A = 14".91 ^1 = 320 18' 0" agreeing with the given values of Art. 94. 96. For the planets or the sun, the following formulae are always sufficiently precise : C' C = pi: sin (C' f) \ A' A = pi: sin ( tp') sin A' cosec C' j ( A ' ) and in most cases we may take ' TT sin ', and A' A = 0. The quantity />TT is frequently called the reduced parallax, and TT />TT = (1 />)/r the reduction of the equatorial parallax for the given latitude ; and a table for this reduction is given in some collections. This reduction is, indeed, sensibly the same as the correction given in our Table XIII , which will be explained more particularly hereafter. Calling the tabular correction ATT, we shall have, with sufficient accuracy for most purposes, pit = TT ATT VOL. I. 8 114 PARALLAX. 97. The preceding methods of computing the parallax enable us to pass directly from the geocentric to the apparent azimuth and zenith distance. There is, however, an indirect method which is sometimes more convenient. This consists in reducing both the geocentric and the apparent co-ordinates to the point in which the vertical line of the observer intersects the axis of the earth. I shall briefly designate this point as the point (Fig. 11). "We may suppose the point to be assumed as the centre of the celestial sphere and at the same time as the centre of an imaginary terrestrial sphere described with a radius equal to the normal OA (Fig. 11). Since the point is in the vertical line of the observer, the azimuth at this point is the same as the appa- rent azimuth. If, therefore, the geocentric co-ordinates are first reduced to the point 0, we shall then avoid the parallax in azimuth, and the parallax in zenith distance will be found by the simple formula for the earth regarded as a sphere, taking the normal as radius. Since the point is in the axis of the celestial sphere, the straight line drawn from it to the star lies in the plane of the declination circle of the star; the place of the star, therefore, as seen from the point 0, differs from its geocentric place only in declination, and not in right ascension. We have then only to find the reduction of the declination and of the zenith distance to the point 0. 1st. To reduce the declination to the point 0. Let PP', Fig. 13, be the earth's axis ; C the centre ; the point in which the vertical line or normal of an observer in the given latitude

= the geographical latitude 9 log A 7.8244 10 7.8245 20 7.8246 30 7.8248 40 7.8250 50 7.8253 60 7.8255 70 7.8257 80 7.8258 90 7.8259 We shall then compute d l forms : sin cos <5 t = J (1 -f A sin T: sin

) and this value is sufficiently accurate for the compu- tation of the parallax in all cases. If then we put a = 1, we have and in series, Or, rigorously, sin * = sin K, cos C, tan (:' c t ) = tan * tan (45 -f J *) tan To find TTj we have 1 sin ffj = joj or sin /> J (1 -)- A sin TT sin y> sin PARALLAX. LIT AO = N=- P If now in the vertical plane passing through the line ZO and the star S we draw SB perpendicular to OZ, and put d the zenith distance at = SOZ ? = the apparent zenith dist. = SAZ the triangles OSB, ASB give ^' cos :< = J, cose,-! A' sin ' d t sin r t j Dividing these equations by J p and putting J' 1 =/ sin 7T, = - 4 M they become /! cos C' = cos d sin ir t /! sin f = sin C t from which we deduce /! sin (C' CJ = sin 7r t sin ^^ / t cos (' d) 1 sin -, cos C, tan (:' - r i} = 1 "^tBn t (124) 1 sin jfj cos C t />(! -f- A sin JT sin tp sin 5) (127) 118 PARALLAX. But this very- precise expression of ^ will seldom be required : it will generally suffice to take sin TT ' ; P P which will be found to give the correct value of TT,, even for the moon, within 0".2 in every case. Where this degree of accu- racy suffices, we may employ a table containing the correction for reducing TT to n v computed by the formula Table XIII., Vol. II., gives this correction with the arguments it and the geographical latitude ' cos B 'v / cos 3' sin oi' = cos S sin a p sin cos

/ cos ff = cos 5 p sin it sin ?' cot Y ) whence sin Y (136) / cos (9 8) = 1 p sin K sin p'^4? '-4-Eg r w The equations (133) determine, rigorously, the parallax in right 122 PARALLAX. ascension, or a' a ; (136) the parallax in declination, or d f d; and (137) determines A'. 99. To obtain the developments in series, put P sin TT cos of m = cos 8 then from (133) we have . , m sin (o ^/) tan (a a) = q - -. 4^\ ' 1 m cos (a 6) whence Putting sin f we have from (136) whence 100. The quantity a is the hour angle of the star east of the meridian. According to the usual practice, we shall reckon the hour angle towards the west, and denote it by t, or put *= 6 -a and then we shall write (138) and (140) as follows : m sin t tan (a a') = tan (t 8) = (U2) sin Y tan ( &) = tan 0' tan (45 -f i 0') tan (y We have here a' equal to the apparent or observed hour angle; and hence, putting t' o' the computation may be made under the following form : p sin T: cos ' sin t' sin (o a') = cos 3 COS p' - i (a - a')] 8in rt~ O = ? Bip ' ip y' Bip fr- (1 III the first computation of a a' we employ <5' for 3. The vulue of a a' thus found is sufficiently exact for the compu- tation of f and d 3'. With the computed value of d d' we then find d and correct the computation of a a'. EXAMPLE. Suppose that on a certain day at the Greenwich Observatory the right ascension and declination of the moon were observed to be ' = 7* 41" 20.436 ^ = 15 50' 27".66 when the sidereal time was _ \ ( a a') = 53 39 58 log sec \i' $ (a a')] 0.227319 log cos HO o') 9.999996 log tan 0' 0.096133 7.922008 9.983186 7.938823 29' 51". 6 7.922008 9.981835 log tan y 0.323448 64 35' 58" 48 45 30 7.940173 29' 57".23 log /> sin TT log sin <>' log sin (y log cosec y log sin (<5 rf') rf ') 8.218377 9.892275 9.876181 0.044153 a =115 = 7*4 50' 3". 77 3 m 20-.251 8.030986 6 sin y>' = 9.7955. Converting the mean into sidereal time (Art. 50), we find = 19* 55 m 16'. 98. Hence, by (145) and (146), = 298 49'.2 log tan ?' 9.9038 a' = 236 48.0 log cos t' 9.6713 f = 62 1 .2 log tan r 0.2325 log TT O 0.933C r = 59 39'.2 log J 9.7444 r 8' = 67 16 .1 log* 1.1886 log pit cos = ' and p = 1. REFRACTION. 105. General laws of refraction. The path of a ray of light is a straight line so long as the ray is passing through a medium of uniform density, or through a vacuum. But when a ray passes obliquely from one medium into another of different density, it is bent or refracted. The ray before it enters the second medium is called the incident ray ; after it enters the second medium it is called the refracted ray; and the difference between the directions of the incident and refracted rays is called the refraction. If a normal is drawn to the surface of the refracting medium at the point where the incident ray meets it, the angle which the incident ray makes with this normal is called the angle of inci- dence, the angle which the refracted ray makes with the normal is the angle of refraction, and the refraction is the difference of these two angles. 128 REFRACTION. Thus, if 4, Fig. 15, is an incident ray upon the surface BE' of a refracting medium, AC the refracted ray, MN the normal to the surface at A, SAM is the angle of incidence, CAN\s the angle of refraction ; and if CA be produced backwards in the direction AS', SAS' is the refraction. An observer whose eye is at any point of the line AC will receive the ray as if it had come directly to his eye without refraction in the direction S'AC, which is therefore called the apparent direction of the ray. Now, it is shown in Optics that this refraction takes place according to the following general laws : 1st. When a ray of light falls upon a surface (of any form) which separates two media of different densities, the plane which contains the incident ray and the normal drawn to the surface at the point of incidence contains the refracted ray also. 2d. When the ray passes from a rarer to a denser medium, it is in general refracted towards the normal, so that the angle of refraction is less than the angle of incidence ; and when the ray passes from a denser to a rarer medium, it is refracted from the normal, so that the angle of refraction is greater than the angle of incidence. 3d. Whatever may be the angle of incidence, the sine of this angle bears a constant ratio to the sine of the corresponding angle of refraction, so long as the densities of the two media are constant. If a ray passes out of a vacuum into a given medium, the number expressing this constant ratio is called the index of refraction for that medium. This index is always an improper fraction, being equal to the sine of the angle of incidence divided by the sine of the angle of refraction. 4th. When the ray passes from one medium into another, the sines of the angles of incidence and refraction are reciprocally proportional to the indices of refraction of the two media. 106. Astronomical refraction. The rays of light from a star in coming to the observer must pass through the atmosphere which surrounds the earth. If the space between the star and the upper limit of the atmosphere be regarded as a vacuum, or as filled with a medium which exerts no sensible effect upon the REFRACTION. 129 direction of a ray of light, the path of the ray will be at first a straight line; but upon entering the atmosphere its direction will be changed. According to the second law above stated, the new medium being the denser, the ray will be bent towards the normal, which in this case is a line drawn from the centre of the earth to the surface of the atmosphere at the point of incidence. The atmosphere, however, is not of uniform density, but is most dense near the surface of the earth, and gradually decreases in density to its upper limit, where it is supposed to be of such extreme tenuity that its first effect upon a ray of light may be considered as infinitesimal. The ray is therefore continually pass- ing from a rarer into a denser medium, and hence its direction is continually changed, so that its path becomes a curve which is concave towards the earth. The last direction of the ray, or that which it has when it reaches the eye, is that of a tangent to its curved path at this point; and the difference of the direction of the ray before en- tering the atmosphere and this last direction is called the astro- nomical refraction, or simply the refraction. Thus, Fig. 16, the ray Se from a star, entering the atmosphere at e, is bent into the curve ecA which reaches the observer at A in the direction of the tangent S'A drawn to the curve at A. If CAZ is the vertical line of the observer, or normal at A, by the first law of the preceding article, the vertical plane of tho observer which con- tains the tangent AS' must also contain the whole curve Ae and the incident ray Se. Hence refrac- tion increases the apparent altitude of a star, but does not affect its azU muth. The angle S'AZ is the apparent ze- nith distance of the star. The true zenith distance* is strictly the angle which a straight line drawn from the star to the point A Fig. 16. * By true zenith distance we here (and so long as we are considering only the effect of refraction) mean that which differs from the apparent zenith distance only by the refraction. VOL. I. 9 130 REFRACTION. makes with the vertical line. Such a line would not coincide with the ray Se; but in consequence of the small amount of the refraction, if the line Se be produced it will meet the vertical line AZ at a point so little elevated above A that the angle which this produced line will make with the vertical will differ very little from the true zenith distance. Thus, if the produced line Se be supposed to meet the vertical in 6', the difference between the zenith distances measured at b' and at A is the parallax of the star for the height Ab', and this difference can be appreciable only in the case of the moon. It is therefore usual to assume Se as identical with the ray that would come to the observer directly from the star if there were no atmosphere. The only case in which the error of this assumption is appre- ciable will be considered in the Chapter on Eclipses. 107. Tables of Refraction. For the convenience of the reader who may wish to avail himself of the refraction tables without regard to the theory by which they are computed, I shall first explain the arrangement and use of those which are given at the end of this work. Since the amount of the refraction depends upon the density of the atmosphere, and this density varies with the pressure and the temperature, which are indicated by the barometer and the thermometer, the tables give the refraction for a mean state of the atmosphere; and when the true refraction is required, supple- mentary tables are employed which give the correction of the mean refraction depending upon the observed height of the barometer and thermometer. TABLE I. gives the refraction when the barometer stands at 80 inches and the thermometer (Fahrenheit's) at 50. If we put r = the refraction, z = the apparent zenith distance, C = the true zenith distance, then C = 2 + r "WTiere great accuracy is not required, it suffices to take r directly from TABLE I. and to add it to z. (The resulting is that zenith distance which we have heretofore denoted by ' in the discussion of parallax.) The argument of this table is the apparent zenith distance z. REFRACTION. 131 TABLE II. is BESSEL'S Refraction Table,* which is generally regarded as the most reliable of all the tables heretofore con- structed. In Column A of this table the refraction is regarded as a function of the apparent zenith distance 2, and the adopted form of this function is r= *p A r* tan z in which a varies slowly with the zenith distance, and its loga- rithm is therefore readily taken from the table with the argu- ment z. The exponents A and A differ sensibly from unity only for great zenith distances, and also vary slowly ; their values are therefore readily found from the table. The factor ft depends upon the barometer. The actual pres- sure indicated by the barometer depends not only upon the height of the column, but also upon its temperature. It is, therefore, put under the form ,3 BT and log B and log J"are given in the supplementary tables with the arguments "height of the barometer," and "height of the attached thermometer," respectively ; so that we have log /3 = log B -f log T Finally, log ? is given directly in the supplementary table with the argument "external thermometer." This thermometer must be so exposed as to indicate truly the temperature of the atmo- sphere at the place of observation. In Column B of the table the refraction is regarded as a function of the true zenith distance expressed under the form and log a', A 1 ', and /' are given in the table with the argument ; ft and f being found as before. Column A will be used when z is given to find ; and Column B, when is given to find z. Column C is intended for the computation of differential re- fraction, or the difference of refraction corresponding to small * From his Astronomische Untersuchungen, Vol. I. 132 REFRACTION. differences of zenith distance, and will be explained hereafter (Miorometric Observations, Vol. II.). These tables extend only to 85 of zenith distance, beyond which no refraction table can be relied upon. There occur at times anomalous deviations of the refraction from the tabular value at all zenith distances; and these are most sensible at great zenith distances. Fortunately, almost all valuable astrono- mical observations can be made at zenith distances less than 85, and indeed less than 80 ; and within this last limit we are justified by experience in placing the greatest reliance in BESSEL'S Table. . In an extreme case, where an observation is made within 5 of the horizon, we can compute an approximate value of the refraction by the aid of the following supplement- ary table, which is based upon actual observations made by ARQELANDER.* App. zen. distance. log Refract. A A 85 0' 2.76687 1.0127 1.1229 30 2.80590 1.0147 1.1408 86 2.84444 1.0172 1.1624 30 2.88555 1.0204 1.1888 87 2.93174 1.0244 1.2215 30 2.98269 1.0298 1.2624 88 3.03686 1.0368 1.3141 30 3.09723 1.0465 1.3797 89 3.16572 1.0593 1.4653 30 3.24142 1.0780 1.578S If we call H the refraction whose logarithm is given in this table, the refraction for a given state of the air will be found by the formula r = EXAMPLE 1. Given the apparent zenith distance z = 78 30' 0", Barom. 29.770 inches, Attached Therm. 0.4 F., Ex- ternal Therm. 2.0 F. We find from Table II., Col. A, for 78 30', log a = 1.74981 A = 1.0032 /I = 1.0328 and from the tables for barometer and thermometer, * Tabulx Regiomontanx, p. 639. REFRACTION. 133 /og B = + 0.00253 log r = + 0.04545 log T= + 0.00127 log = + 0.00380 Hence the refraction is computed as follows : log a = 1.74981 A log = log $ A = + 0.00381 A log r = log r x = + 0.04694 log tan z = 0.69154 r = 310".53 = 5' 10".53 log r = 2.49210 The true zenith distance is, therefore, 78 30' 0" + 5' 10".53 => 78 35' 10".53. EXAMPLE 2. Given the true zenith distance = 78 35' 10".53, Barom. 29.770 inches, Attached Therm. 0.4 F., External Therm. 2.0 F. We find from Table II., Col. B, for 78 35' 10", log a! = 1.74680 A' = 0.9967 A' = 1.0261 and from the tables for barometer and thermometer, as before, log B == + 0.00253 log r = + 0.04545 log T= + 0.00127 log ft = + 0.00380 The refraction is then computed as follows : log of = 1.74680 A' log/9 = log ft*'= -f 0.00379 A' log r = log r x = + 0.04663 log tan C = 0.69489 r = 310".53" == 5' 10".53 log r = 2.49211 and the apparent zenith distance is therefore 78 30'. EXAMPLE 3. Given z = 87 30', barometer and thermometer as in the preceding examples. By the supplementary table above given, log R = 2.98269 A = 1.0298 log ft = + 0.00380 log $ A = + 0.00391 \ =.- 1.2624 log r = + 0.04545 log r A = + 0.05738 r = 18' 26".6 log r = 3.04398 134 REFRACTION. It is important in all cases where great precision is required that the barometer and thermometer be carefully verified, to see that they give true indications. The zero points of thermo- meters are liable to change after a certain time, and inequalities in the bore of the tube are not uncommon. A special investi- gation of every thermometer is, therefore, necessary before it is applied in any delicate research. If the capillarity of the baro- meter has not been allowed for in adjusting the scale, it must be taken into account by the observer in each reading. We may obtain the true refraction for any state of the air within 1" or 2", very expeditiously, by taking the mean refrac- tion from Table I. and correcting it by Table XIV. A, and Table XIV. B. The mode of using this table is obvious from its arrangement. Thus, in Example 1 we find from Table I., Mean refr. = 4' 38".9 " XIV. A, for Barom. 29.77, Corr. = 2 . " XIV. B, " Therm. 2. " = -f 32 . True refr. = 5' 9". which agrees with BESSEL'S value within 1".5. For greater accuracy, the height of the barometer should be reduced to the temperature 32 F., which is the standard assumed in these tables. The corrected height of the barometer in this example is 29.85, and the corresponding correction of the refraction would then be 1"; consequently the true refraction would be 5' 10", which is only 0".5 in error. These tables furnish good approximations even at great zenith distances. Thus, we find by them, in Example 3, r = 18' 24". 108. INVESTIGATION OF THE REFRACTION FORMULA. In this investigation we may, without sensible error, consider the earth as a sphere, and the atmosphere as composed of an infinite number of concentric spherical strata, whose common centre is the centre of the earth, each of which is of uniform density, and within which the path of a ray of light is a straight line. Let C, Fig. 16, be the centre of the earth, A a point of observation on the surface; CAZ ihe vertical line ; Aa 1 ', a'b', b'c', &c. the vertical thicknesses of the concentric strata; Se a ray of light from a star S, meeting the atmosphere at the point e, and successively re- REFRACTION. 135 fracted in the directions ed, dc, &c. to the point A. The last direction of the ray is a.A, which, when the number of strata is supposed to be infinite, becomes a tangent to the curve ecA at A, and consequently AaS' is the apparent direction of the star. Let the normals Ce, G/, &c. be drawn to the successive strata. The angle Sef is the first angle of incidence, the angle Ced the first angle of refraction. At any intermediate point between e and A, as c, we have Ccd, the supplement of the angle of incidence, and Ccb, the angle of refraction. If now for any point, as - r j (159) If we denote the horizontal refraction, or that for z = 90, by r , this formula gives tan I r = nto, cot | r _ or tan r j/A - -8 or, by (165), (168) which shows that equal increments of x correspond to equal decrements of 8. This last equation also gives for the upper limit of the atmo- sphere, where 3 = 0, x = 2 I ; that is, in this hypothesis the height of the atmosphere is double that of a homogeneous atmosphere of the same pressure. Again, we have, by (164), (165), and (168), = =l- (169) Po 3 S 21 The function *= expresses the law of heat of the strata of the PJ atmosphere. For let r be the temperature at the surface of the earth, r the temperature at the height x. If the temperature were r in both cases, we should have P -= 8 - (HO) Po 8 o but when the temperature is changed from r to r the density ia diminished in the ratio 1-f (r r ) : 1, e being a constant which SECOND HYPOTHESIS. 143 is known trom experiment; so that the true relation between the pressures and densities at different temperatures is expressed by the known formula whence which combined with (169) gives and hence equal increments of x correspond to equal decrements of r. Hence, in this hypothesis, the heat of the strata of the atmo- sphere decreases as their density in arithmetical progression. The value of e, according to RUDBERG and REGNAULT, is very nearly 1 2 1 . Hence we must ascend to a height = 58.6 metres, in order to experience a decrease of temperature of 1 C. But, according to the observations of GAY LUSSAC in his celebrated balloon ascension at Paris (in the year 1804), the decrease of temperature was 40.25 C. for a height of 6980 metres, or 1 C. for 173 metres, so that in the hypothesis under consideration the height is altogether too small, or the decrease of temperature is too rapid. This hypothesis, therefore, is not sustained either by the observed refraction or by the observed law of the decrease of temperature. 112. Second hypothesis. Before proposing a new hypothesis, let us determine the relation between the height and the density of the air at that height, when the atmosphere is assumed to be throughout of the same temperature, in which case we should have the condition (170). Resuming the differential equation (161), put a . ~ 144 REFRACTION. in which 5 is a new variable very nearly proportional to x. We then have dp = g adds which with the supposition (170) gives Integrating, dp __ _ g 3 ads J~ Po _ a5-f C in which the logarithm is Napierian. The constant being determined so that p becomes p when s = 0, we have and therefore , p q 8 as log i- = ^ as = -- r to Po Po I where I has the value (163). Hence, e being the Napierian base, r> i^ a * 2- = 1 = eT (172) Po 8 which is the expression of the law of decreasing densities upon the supposition of a uniform temperature. In our first Hypo- thesis the temperatures decrease, but nevertheless too rapidly. We must, then, frame an hypothesis between that and the hypothesis of a uniform temperature. Now, in our first hypothesis we have by (169), within terms involving the second and higher powers of 5, and in the hypothesis of a uniform temperature, ^ = 1' Po$ The arithmetical mean between these would be ' ?*> - i __ - ~ SECOND HYPOTHESIS. 145 but, as we have no reason for assuming exactly the arithmetical mean, BESSEL proposes to take (173) p d h being a new constant to be determined so as to satisfy the observed refractions. This equation, which we shall adopt as our second hypothesis, expresses the assumed law of decreasing tempe- ratures, since, by (171), it amounts to assuming 1 + ( r _ Tfl ) = e ~ a h (174) and it follows that in this hypothesis the temperatures will not decrease in arithmetical progression with increasing heights, though they will do so very nearly for the smaller values of s. that is, near the earth's surface. Now, combining the supposition (173) with the equation dp = we have dp < JL _ _^_ P Po I Integrating and determining the constant so that for s = 0, p becomes p^ we have which with (173) gives Inasmuch as the law of the densities thus expressed is still hypothetical, we may simplify the exponent of e. For if A is much greater than I (as is afterwards shown), we may in this ex- ponent put e h 1 y and we shall have as the expression of our hypothesis a = Je~7' + ? = de~~^~'^ ( 175 ) * BESSEL. Fundament a Astronomiae, p. 28. VOL. I. 10 l46 REFRACTION. By comparing this with (172), we see that this new hypothesis differs from that of a uniform temperature by the correction applied to the exponent of e. Putting, for brevity, (M-^i-l (") we have 8 = d e~ fts (177) in which ft is constant. This expression of the density is to be introduced into the differential equation of the refraction (150j. Now, by (149), in which q = a -}- x, we have whence . . _ aju sin 2 _ (1 ~ (a + x)n ~ sin i (1 s) sin z tan i =_ = 2 '- .. -*o /r/i;_ (1 _ s) , 8 (1 s) sin z ^1 [cos* z 1 1 ' j + (2s s') sin 1 z\ From the equation // 1 -f- 4 kd we deduce y and the functions /and being given. If from this equation y could be found as an explicit function of x and substituted in the equation u = fy, the development could be effected at once by Maclaurin's Theorem, according to which we should have + &c - where M O , DJ.I W &c. denote the values of u and its successive derivatives when x = 0. It is proposed to find the values of the derivatives without recourse to the elimination of y, as this elimination is often impracticable. For brevity, put Y S we have /* =f*+i- [>'. D/S' ] + ~ In which/ and ^ denote any functions whatever, and Z), Z) 2 , &c. the successive derivatives of the functions to which they are prefixed. Hence, by putting this theorem gives Z? z y = F + xD 9 YD x y D t y = l +x D, whence, eliminating z, D.y = YD t y Multiplying this by D y u, it gives D x u = KZ>, (a) The derivative of this equation relatively to t is This is a general theorem, whatever function u is of y, and consequently, also, what- ever function D t u is of y. We may then substitute in it the function Y n D t u for D ( u, and we shall have D,[ Y*D t it] =D t [Y n + i/? t w] (6) Nw, the successive derivatives of (a) relatively to z are, by the successive appli- cation of (6), making n = 1, 2, 3, &c., D* u = D z [ YD t u] = D t [ I^a] D x s u = />,*,[ r/? ( i] = Z>,* But when z = 0, we have y = t, Y = ^/, and hence "o=A D x u = ft . Dft, ... Z? x M = Z)"-i[(^ n #/0 where the subscript letter of the D is omitted in the second members as unnecessary, since t is now the only variable. These values substituted in Maclaurin's Theorem give Lagrange's Theorem : 150 REFRACTION. e-*=e-/"'_^_ fl - e - siri*z L &c. (182) But we have in the numerator of (181) and hence, differentiating (182) and substituting the result in (181), we find dr= Bin* 1 1.8.8... ti&* LV J + &c. 1 (183; To effect the differentiations expressed in the several terms of this series, we take the general expression where the upper sign is to be used when n is even, and the lower sign when n is odd. * Differentiating this n times successively, we have SECOND HYPOTHESIS. 151 by means of which, making n = 1 . 2 . 3 . . . successively, we reduce (183) to the following form : dr = (1 a) [cos 2 z + 2 s' sin 2 2]* I sin 1 z 1 . 2 sin* 2 + &c. (184) We have now to integrate the terms of this series, after having multiplied each by the factor without the brackets. The inte- grals are to be taken from the surface of the earth, where s = 0, to the upper limit of the atmosphere ; that is, q being the nor- mal to any stratum (Art. 108), they are to be taken between the limits q = a and q a -f H, H being the height of the atmo- sphere. Now, this height is not known ; but since at the upper limit the density is zero and beyond this limit the ray suffers no refraction to infinity, we can without error take the integrals between the limits q = a and q = oo , i.e. between s = and 5 = 1. But we may make the upper limit of s also equal to in- finity. For, by (176), /9 will not differ greatly from -, and conse- quently will be a very large number, nearly equal to 800, as we find from (167) ; hence for s = 1 we have in (172) d = - * - (2.718..)'" which will be sensibly equal to zero, and consequently the same as we should find by taking s = oo . Hence the integrals may be taken between the limits 5 = and s = oo ; consequently, also, according to (180), between the limits 5' = and s' = oo . Now, as every term of the series will be of the form ft sin z ds'e-*P>' pds'e-*?*' [cos 1 2-J-25' sin 2 2]* ' [cot 2 2 + 2s']* multiplied by constants, we have only to integrate this general form. Let t be a new variable, such that 2/ cot'2-f ^s' =~ (186) 152 REFBACTION. then (185) becomes the integral of which is to be taken from t = r (187) to t = oo , which are the limits given by (186) for s' = and *' = oo . If, therefore, we denote by ^ (n) a function such that or j^(n') = e TT f t/> dte- tt (188) the integral of (185) will become r *"' 'o 2 ' t=1 /g?. iff [cos 2 z + 2s' sin 2 z\* V* Substituting this value in (184), making successively n = \, 2, 3, &c., we find the following expression of the refraction: + &c. (190) since we have in general !_ + ^L. _^+. .. = ,-. 1 T 1.2 1.2.3 T SECOND HYPOTHESIS. can also be written as follows :* 353 f :- e ^rr^(L N ; + 2* 0/9 '*, sin 2 2 1/27 3 3 1 Sa/J 1-a ' f> .inlz J./^ + 1.2 Bin*2 6 5 _S 03 in? r .i n aT f.\\ + 1.2.3 sin 6 2 6 + &C. (191) 113. The only remaining difficulty is to determine the func- tion ^(n),(l88). In the case of the horizontal refraction, where cot z = and therefore also T= 0, this function becomes independent of (w), and reduces to the well-known integralf (192) * LAPLACE, Mecanique Celeste, Vol. IV. p. 186 (BOWDITCH'S Translation); where, however,- stands in the place of the more general symbol (3 here employed. This form of the refraction is due to KRAMP, Analyse des refracliom astronomiques ft ter- restres, Strasbourg, 1799. f This useful definite integral . may be readily obtained as follows. Put k = J dt e tt. Then, since the definite integral is independent of the variable, we also have k ( doe~ vv , and, multiplying these expressions together, P= r J Q J Q J o J Q the order of integration being arbitrary Let v = tu ; whence dv = t du (for in integrating, regarding v as variable, t is regarded as constant) : then we have o *^o ^o *^o whance ' J Q 154 REFRACTION. where it 3.1415926 .... The expression for the horizontal refraction is therefore found at once by putting $i/n for 4/ (n) in every terra of (191), and sin 2 = 1, namely: ,f 1.2.3 o'' (193) For small values of T, that is, for great zenith distances, we may obtain the value of the integral in (188) by a series of ascending powers of T. We have /Q I J dte-"= dte~ n - dte (194) The first integral of the second member is given by (192). The second is 1.2 5 1.2.3 7 (195) Another development for the same case is obtained by the suc- cessive application of the method of integration by parts, as follows:* -" = te~ lt + 2 f t'dte- * By the formul&fxdy = xy fydx, making always x = e~ lt , and dy SUCCCB wvely = dt, t*dt, t*dt, &c. SECOND HYPOTHESIS. 155 33.5 whence, by introducing the limits, As the denominators increase, these series finally become con- vergent for all values of T; but they are convenient only for small values. For the greater values of T, a development according to the descending powers may be obtained, also by the method of integration by parts, as follows :* We have fcft <>- = - J_ e -_f^ e - J 2t * J i* J_ e - + _!_,-+ 2t 2'* Hence ., ] 2 T 2 T 2 (2 T')* (2 T*)' 1.8.6...(2n-l)\_1.8.5...(2n + l)f a dt t (2T)- I - 2+> JT + The sum of a number of consecutive terms of this series is alternately greater and less than the value of the integral. But since the factors of the numerators increase, the series will at last become divergent for any value of T. Nevertheless, if we stop at any term, the sum of all the remaining terms will be less than this term; for if we take the sum of all the terms in the brackets, the sum of the remaining terms is dt * By the formulay*x dy = xy fy dx, making always dy = t dt e~~ lt , and x ,, , &c. 156 REFRACTION. The integral in this expression is evidently less than the product of the integral f dt - 1 J T fi. + (2 n + i)T* n - multiplied by the greatest value of e~ n between the limits 7 and QO , and this greatest value is e~ TT . Hence the above remainder is always numerically less than which expression is nothing more than the last term of the series (when multiplied by the factor without the brackets), taken with a contrary sign. Hence, if we do not continue the summation until the terms begin to increase, but stop at some sufficiently small term, the error of the result will always be less than this term. Finally, the integral may be developed in the form of a con- tinued fraction, as was shown by LAPLACE. Putting for brevity +(n) = 1 , 1 = JL(l_J_ + J_JL_L^ \ (198) 2T\ 2T* (27 12 ) 2 (27V / and denoting the successive derivatives of u relatively to !Tby MJ, u v &c., we have first w, = - -j ' + &c. (199) or 1 (200) Differentiating this equation, successively, we have &c. or, in general, n having any value in the series 1.2.3.4... &c. SECOND HYPOTHESIS. 157 Hence we derive u n 2n 22 1 n + 1 or, putting * = ^5 (20D -S_ = _- (202) M-I i /*vt+-i v ' By (200) we have or 2rM o = Lnr ( 203 ) But from (202), by making n successively 1, 2, 3, &c., we have /*\* (2) 2 W. \-/ a x~/ JL-, 1 ~ which successively substituted in (203) give 1 1 4- *" F+ &c. (204) This can be employed for all values of T, but when k exceeds it will be more convenient to employ (195) or (196). The successive approximating fractions of (204) are 1 1 1 + 2A- 1 + 5* 1 + 9A- + 8A-' 1 I ifrrt -\-k 1+3* 1 + 6* + 3A* 1 + 10A- and, in general, denoting the n th approximating fraction by , 158 REFRACTION. n _ a n -\ -f (n By the preceding methods, then, the function ^(ri) can be computed for any value of T. A table containing the logarithm of this function for all values of Tfrom to 10, is given by BESSEL (Fundamenta Astronomice, pp. 36, 37), being an extension of that first constructed by KRAMP. By the aid of this table the computation of the refraction is greatly facilitated. 114. Let us now examine the second term of (179.) This term will have its greatest value in the horizontal refraction, w r hen z = 90, in which case it reduces to Moreover, the most sensible part of the integral corresponds to small values of s, and therefore, since a is also very small, we may put 2a(l e~^ s ) = 2aps. The integral thus becomes Now we have, by integrating by parts, and hence. Jo * ^i/S^o Putting /9s = z 2 , this becomes, by (192), Hence the term becomes a (3 4a/3) l^T ~ 8 (1 - a) (1 - a/?)l \ 2/9 SECOND HYPOTHESIS. lf>9 Taking BESSEL'S value of h = 116865.8 toises* = 227775.7 metres, and the value of 1 = 7993.15 metres (p. 141), we find by (176) /? 768.57. Substituting this and a = 0.000294211 (p. 146), the value of the above expression, reduced to seconds of arc by dividing by sin 1", is found to be only 0".72, which in the hori- zontal refraction is insignificant This term, therefore, can be neglected (and consequently also all the subsequent terms), and the formula (191) may be regarded as the rigorous expression of the refraction. 115. In order to compute the refraction by (191), it only re- mains to determine the constants a and /9. The constant a might be found from (178) by employing the value of k deter- mined by BIOT by direct experiment upon the refractive power of atmospheric air, but in order that the formula may represent as nearly as possible the observed refractions, BESSEL preferred to determine both a and /? from observations, f Now, a depends upon the density of the air at the place of observation, and is, therefore, a function of the pressure and temperature; and /9, which involves I, also depends upon the ther- mometer, since by the definition of I it must vary with the tem- perature. The constants must, then, be determined for some assumed normal state of the air, and we must have the means of deducing their values for any other given state. Let p = the assumed normal pressure, T O = " " temperature, p = the observed pressure, T = " " temperature, = the normal density corresponding to p and r f , S = the density corresponding to^> and T- * Fundamenta Astronomise, p. 40. f It should be observed that the assumed expression of the density (177) may represent various hypotheses, according to the form given to ft. Thus, if we put 3 = -, we have the form (172) which expresses the hypothesis of a uniform tem- perature. We may therefore readily examine how far that hypothesis is in error in the horizontal refraction: for by taking the reciprocal of (167) we have in this cast 8 = 796.53, and hence with o = 0.000294211 we find, by taking fifteen terms of the series (193), r = 39' 54".5, which corresponds to Barom. 0. 76, and Therm. C. This is 2' 23".5 greater than the value given by AROELANUER'S Observations (p. 141). Our first hypothesis gave a result too small by more than 7', and hence a true hypo- thesis must be intermediate between these, as we have already shown from a con 160 REFRACTION. then we have by (171) in which e is the coefficient of expansion of atmospheric air, or the expansion for 1 of the thermometer. If the thermometer is Centigrade, we have, according to BESSEL,* e = 0.0036438 From (178) it follows that a is sensibly proportional to the density, and hence if we put a a = the value of a for the normal density S , we have, for any given state of the air, (205) in which for p and p we may use the heights of the barometric column, provided these heights are reduced to the same tem- perature of the mercury and of the scales. Again, if 1 = the height of a homogeneous atmosphere of the temperature T 0> and any given pressure, then the height I for the same pressure, when the temperature is r, is Z = / [l + e(r-T )] (206) The normal state of the air adopted by BESSEL in the determi- nation of the constants, so as to represent BRADLEY' s observa- tions, made at the Greenwich Observatory in the years 1750- 1762, was a mean state corresponding to the barometer 29.6 inches, and thermometer 50 Fahrenheit 10 Centigrade; and for this state he found a = 0.000278953 gideration of the law of temperatures. At the same time, we see that the hypothesis of a uniform temperature is nearer to the truth than the first hypothesis, and we are so far justified in adhering to the form <5 = A t-fi' with the modification of substi- tuting a corrected value of 8. * This value, determined by BESSEL, from the observations of stars, differs slightly from the value ?fa more recently determined by RUDBERO and REONAULT by direct experiments upon the refractive power of the air. SECOND HYPOTHESIS. 161 or, dividing by sin V, Oo = 57".538 and h = 116865.8 toises = 227775.7 metres. For the constant / at the normal temperature 50 F., BESSEL employed 1 = 4226.05 toises = 8236.73 metres.* Since the strata of the atmosphere are supposed to be parallel tc the earth's surface, BESSEL employed for a the radius of curva- ture of the meridian for the latitude of Greenwich (the observa- tions of Bradley being taken in the meridian), and, in accordance with the compression of the earth assumed at the time when this investigation was made, he took a = 6372970 metres. Hence we have = 745.747 These values of a and /9 being substituted for a and ft in (193), the horizontal refraction is found to be only about 1' too great, which is hardly greater than the probable error of the observed horizontal refraction. At zenith distances less than 85, however, BESSEL afterwards found that the refraction com- puted with these values of the constants required to be multi- plied by the factor 1.003282 in order to represent the Konigsberg observations. 116. By the preceding formulae, then, the values of the con- stants a and p can be found for any state of the air, as given by the barometer and thermometer at the place of observation, and then the true refraction might be directly computed by (191). But, as this computation would be too troublesome in practice, the mean refraction is computed for the assumed normal values of a and /9, and given in the refraction tables. From this mean * According to the later determination of REONAULT which we have used on p. 143, we should have 1 = 8286.1 metres. The difference does not affect the value of BESSEL'S tables, which are constructed to represent actual observations. VOL. I. 11 162 REFRACTION. refraction we must deduce the true refraction in any case by applying proper corrections depending upon the observed state of the barometer and thermometer. For facility of logarithmic computation, BESSEL adopted the form in which r is the tabular refraction corresponding to p and r , and r is. the refraction corresponding to the observed p and r. Let us see what interpretation must be given to the exponents A and L If the pressure remained p , the refraction correspond- ing to the temperature r would be or, with sufficient precision, In like manner, if the temperature were constant, and the pres- sure is increased by the quantity p p , the refraction would become nearly Hence, when both pressure and temperature vary, we shall have, very nearly, Now, putting in (207) under the form 1 + - 9 , and develop- ing by the binomial theorem, we have r = r.{l + -(j> -j.) + Ac. } X {l-Je(T-T )-f &c.} Therefore, neglecting the smaller terms, we must have JL=*. (1), ft = 2 H( 2 )> ft = 3 *^(3), &c., or in general ^ = n^4(n) (211) then, if we also put Q = x e~ x q t -f ^fe~ Zx q a -\ e~ M a -f &c. (212) 1-2 '1.2 n the formula (191) becomes ?.G (213) in which, since the variations of j " - in (191) are sensibly the same as those of a, we may regard 1 a as constant. Differen- tiating this, observing that Q varies with both p and T, while ft varies only with r, we have (214) In differentiating Q, it will be convenient to regard it as a func- tion of the two variables x and ft, the quantities q v q v &c. vary- ing only with ft. We have, since ft does not vary with p, ~dp~dx ' ~dp and since both x and d vary with r, - } dp \ft dp a -.) = * +v (216) dr ^ d{3 dr ^ J From (212) we find 164 REFRACTION. in which Q' = xe' x q t + ^ r* 20, + p^ *- 3 * 3& + &o. (218) Also, ^ = *T ^ + *- e ~ ** ** + &c. (219) rf/3 dp, 1 .3 ^5 in which we have generally, by (211), ^ = n ! T ll li(l) ^ r//5 dT ' dfi But by (200), in which M O = ^(?i), we have and by (187) whence dq T* T !5=1 _ i i = _ -- n * dp ft * n 2/3 cot 2 z cot 2 = - nq - n n 2 1/2/8 Substituting the values of this expression for n 1, 2, 3, &c. in (219), we have The first series in this expression = Q'. The second, wher -*, e- 2 *, &c. are developed in series, becomes &c. = 1- r SECOND HYPOTHESIS. 165 and hence dQ = cot*z Q, cot x 22(r We have, further, from (210) and the values of a, I, and ft in the preceding article, dx X da X a X dp a. dp a p p dp dp dl a h dr dl dr~ I 2 i__l dx x d* x dp 2 h l _ . _ . __ __ . _ L _ e /p . _ dr a dr ~ p dr h l Substituting these values in (215) and (216), and then substituting in (214), we find* _ )^L =sin . i? /1.0'.ln^ ; dp \ p s p (221) These formulae are to be computed with the normal values of a, /9, r, ^, and p, and for the different zenith distances, after which A and X are computed by (209). The values of A and ^ thus found are given in Table II. 117. Finally, in tabulating the formula (207), BESSEL puts r = a tan z (222) + K 7 - *) (where a and ^9 no longer \iave the same signification as in the preceding articles). * BESSEL, Fundamenla Astronomix, p. ?A. 166 REFRACTION. The true refraction then takes the form r = a/3-V tan z (223) The quantity here denoted by /? is the ratio of the observed and normal heights of the barometer, both being reduced to the same temperature of the mercury and of their scales. First, to correct for the temperature of the scale, let b (l \ b (e \ or b (m) denote the ob- served reading of the barometer scale according as it is graduated in Paris lines, English inches, or French metres. The standard temperatures of the Paris line is 13 Reaumur, of the English inch 62 Fahrenheit, and of the French metre Centigrade ; that is, the graduations of the several scales indicate true heights only when the attached thermometers indicate these temperatures respectively. The expansion of brass from the freezing point to the boiling point is .0018782 of its length at the freezing point. If then the reading of the attached thermometer is denoted either by r',/', or c', according as it is Reaumur's, Fahrenheit's, or the Centigrade, the true height observed will be (putting 5 0.0018782) 1 4- - 13 1 + 30 T 80 ^180 or b ^ * + r ' s ^ 180 + (/' -32)s 6 ,.> 100 + o' 224 SO + 13s' 180 + 30s 100 where the multipliers 1 -f ^r', &c. evidently reduce the reading to what it would have been if the observed temperature had been that of freezing, and the divisors 1 + -jj- 13, &c. further reduce these to the respective temperatures of graduation, and conse- quently give the true heights. This true height of the mercury will be proportional to the pressure only when the temperature of the mercury is constant. We must, therefore, reduce the height to what it would be if the temperature were equal to the adopted normal temperature, which is in our table 8 Reaumur = 50 F. = 10 C. Now, mercury expands of its volume at the freezing point of water, when 55.5 SECOND HYPOTHESIS. 167 its temperature ir; raised from that point to the boiling point of water. Hence, putting q = , the above heights will be reduced to the normal temperature by multiplying them respectively by the factors 80 +Sq 180 -f 18g 100 -f lOq 80 -|- r'q 180 + (/' 32)?' 100 -f- c'q The normal height of the barometer adopted by BESSEL was 29.6 inches of Bradley's instrument, or 333.28 Paris lines ; but it after^ wards appeared that this instrument gave the heights too small by | a Paris line, so that the normal height in the tables is 333.78 Paris lines, at the adopted normal temperature of 8 R. Reducing this to the standard temperature of the Paris line = 13 R., we have o o = 333.78 8 + 8s (226) 80 -f 135 In comparing this with the observed heights, the 6 (e) and 6 (m) must be reduced to lines by observing that one English inch = 11.2595 Paris lines, and one metre = 443.296 Paris lines. Making this reduction, the value of f==- is found by dividing the product of (224) and (225) by (226). The result may then be separated into two factors, one of which depends upon the observed height of the barometric column, and the other upon the attached ther- mometer ; so that if we put 333.78 80 -f 8s -& 11 - 2595 80 + 13s 180 + 18g ' 333.78 ' 80 -f- 8s ' 180 + 30s (w) 443.296 80 + 13s 100 + 10 g (227} ' 333.78 ' 80 -f 8s " 100 and T _80 -f-r's J_ 180 -f- (f 32) s _ 100 + c's '" SO -;- r'? . 180 + (/' 32) q ~ 100 + c'q we shall have fi = BT, or log ft = log B + log T (228) 168 REFRACTION. The quantity f would be computed directly under the form r = if r were at once the freezing point and the normal temperature of the tables ; for e is properly the expansion of the air for each degree of the thermometer above the freezing point, the density of the air at this point being taken as the unit of density. But if the normal temperature is denoted by r , that of the freezing point by r t , the observed by r, we shall have Y = !_ e (vulil l+c(r -r,) an expression which, if we neglect the square of e, will be reduced to the above more simple one by dividing the numerator and denominator by 1 + e(r r,). BESSEL adopted for r the value 50 F. by BRADLEY'S thermometer; but as this thermometer was found to give 1.25 too much, the normal value of the tables is r = 48.75F. Hence, if r, /, or c denote the temperature indi- cated by the external thermometer, according as it is Reaumur, b'ahr., or Cent., we have* 180 + 16.75 X 0.36438 r ~ 180 + r X 0.36438 180 -f 16.75 X 0.36438 ~ 180 + (/ 32) X 0.36438 ( 229 ) 180 -f- 16.75 X 0.36438 180 + f c X 0.36438 The tables constructed according to these formulae give the values of log B, log 7 1 , and log ft with the arguments barometer, attached thermometer, and external thermometer respectively, and the computation of the true refraction is rendered extremely simple. An example has already been given in Art. 107. 118. In the preceding discussion we have omitted any con- sideration of the hygrometric state of the atmosphere. The * Tabulte Regiomontante, p. LXII. REFRACTION. 169 refractive power of aqueous vapor is greater than that of at- mospheric air of the same density, but under the same pressure its density is less than that of air ; and LAPLACE has shown that " the greater refractive power of vapor is in a great degree com- pensated by its diminished density."* 119. Refraction table with the argument true zenith distance. When the true zenith distance is given, we may still find the refrac- tion from the usual tables, or Col. A of Table II., where the apparent zenith distance ^ is the argument, by successive ap- proximations. For, entering the table with instead of z, we shall obtain an approximate value of r, which, subtracted from , will give an approximate value of z; with this a more exact value of r can be found, and a second value of z, and so on, until the computed values of r and z exactly satisfy the equation z = r. But it is more convenient to obtain the refraction directly with the argument . For this purpose Col. B of Table II. gives the quantities a', A 1 ', A', which are entirely analogous to the a, A; and /I, so that the refraction is computed under the form r = a'/9'4>*'tan C (230) where /9 and f have the same values as before. The values of a', A', and X' are deduced from those of a, A, and /I after the latter have been tabulated. They are to be so determined as to satisfy the equations a,9^V tan z = a! P A 'Y* tan : (231) 3 = C a'/?^Y A ' tan : (232) and this for any values of /9 and f. Let (z) denote the value of z which corresponds to when /3 = 1, f = 1 ; that is, when the refraction is at its mean tabular value. The value of (z) may be found by successive approximations from Col. A., as above ex- plained. Let (a), (A), (/I), and (r) denote the corresponding values of a, A, ^, r. We have (r) = () tan (2) = a' tan C (2) = C o' tan C whence, by (232), * Mf'c. Cel. Book X. 170 REFRACTION. * = () a' tan C (ft A 'r* 1) But, taking Napierian logarithms, we have t(fi*jrfj**A'ij + xir and hence, e being the Napierian base, fi* r * = e W + *'Y = 1 + (A' Z/3 4- r /r) -f &c. where, as ^3 and p differ but little from unity, the higher poweir of A'lft H- X'l f may be omitted. Hence jr^JMr-ibCi'M-'+^Iri Now, taking the logarithm of (231), we have I (a, tan 2) 4- A/,9 + /1/r = l(o' tanC) -f A' I ft + Wy The first member is a function of 2, which we may develop as a function of (z) ; for, denoting this first member by/?, and putting y = - (r) [A'Z/9 + ^ r ] we have z = (z}+ y> and hence y] =/(*) + y -f &c., where we may also neglect the higher powers of y. But since f(z) is what/2 becomes when z = (2), and consequently A = (A), ; = (yfy we have /(*) = ^ CO) tan (2)] + (^) //? + (A) l r df(z) = ^[(q) tan (2)] = d[()tan(z)] = 1 d(z) d(z} (a) tan (z)d (2) (r) Hence we have /z = / [(a) tan (2)] + (A) If + 00 r - [^' ^ -f X l r ] = I [' tan C] + A' I ft -f A' J r or, since (a) tan (2) = a' tan , REFRACTION. 171 Since this is to be satisfied for indeterminate values of /9 and 7-, the coefficients of I ft and If in the two members must be equal; and therefore (233) * T *55 and also All the quantities in the second members of these formulae may be found from Column A of Table II., and thus Column B may be formed.* If we put we shall now find the refraction under the form r = k' tan C 120. To find the refraction of a star in right ascension and decli- nation. The declination 3 and hour angle t of the star being given, together with the latitude tp of the place of observation, we first compute the true zenith distance and the parallactic angle g by (20). The refraction will be expressed under the form r = k' tan C in which ft' = a'/HV The latitude and azimuth being here constant (since refrac- tion acts only in the vertical circle), we have from (50), by put- * See also BESSEL, Astronomische Untersuchim-ien. Vol I p. 159 172 REFRACTION. ting d(p = 0, dA = 0, d = r = k' tan , dt = da, (a = star's right ascension), dd = k' tan f cos ' = J*5J l_ a ( T _ T where D is the dip, computed by (235), when the refraction is neglected, the sine of so small an angle being put for its tan- gent. If we substitute the values a = 0.00027895, sin D = D sin 1", and s = 0.002024, this formula becomes 176 DIP OF THE HORIZON. D in which D is in seconds. If D is expressed in minutes in the last term, it will be sufficiently accurate to take D' = D 400 X ^=p (238) This w r ill give D' = D when r = r c , as it should do, since in that case the atmosphere is supposed to be of uniform density from the level of the sea to the height of the observer. If r -~C ~ > we l iave -D' > D' I' 1 extreme cases, where r is much greater than r u , we may have D' < 0, or negative, and the visible horizon will appear above the level of the eye, a phenomenon occasionally observed. I know of no observations sufficiently precise to determine whether this simple formula, deduced from theoretical considerations, accurately represents the observed dip in every case. 124. If, however, we wish to compute the value of D f for a mean state of the atmosphere without reference to the actually observed temperatures, we may proceed as follows : In the equa- tion above found, IJL a -\-X we may substitute the value a + x which is our first hypothesis as to the law of decrease of density of the strata of the atmosphere, Art. 109. This hypothesis will serve our present purpose, provided n is so determined as to represent the actually observed mean horizontal refraction. "We have, then, COB IX = and developing, neglecting the higher powers of, DIP OF THE HORIZON. 177 n 4- 1 a or To determine n, \ve have by (160), reducing r to seconds. -(r 8inl")' where, for Barom. 0'".76, Therm. 10C., which nearly represent the mean state of the atmosphere at the surface of the earth, we have 4k 3 = 0.00056795, and r = 34' 30" (which is about the mean of the determinations of the horizontal refraction by dif- ferent astronomers) ; and hence we find 0.0784 n = 5.639, J n = 0.9216 =10. \n-j-l D' = D .0784 D (239) The coefficient .0784 agrees very nearly with DELAMBRE'S value .07876, which was derived from a large number of observations upon the terrestrial refraction at different seasons of the year To compute D' directly, we have sin 1" If x is in feet, we must take a in feet. Taking the mean value a = 20888625 feet, and reducing the constant coefficient of j/:r, we have D' = 58".82 -\/x in feet. (240) Table XI., Vol- II., is computed by this formula. VOL. I. 11 178 DISTANCE OF THE HORIZON. 125. To find the distance of the sea horizon, and the distance of an object of known height just visible in the horizon. The small portion TA, Fig. 19, of the curved path of a ray of Flg ' 19 ' light, may be regarded as the arc of a circle; and then the refraction elevates A as seen from T as much as it elevates T as seen from A. Drawing the tangent TP, the ob- server at T would see the point A at P ; and if the chord TA were drawn, the angle PTA would be the refraction of A. This refraction, being the same as that of T as seen from -4, is, by (239), equal to .0784 Z). In the triangle TPA, TAP is so nearly a right angle (with the small elevations of the eye here considered) that if we put x l = AP we may take as a sufficient approximation x, = TA X tan PTA = a tan D X -0784 tauZ> But we have a tan 2 D = 2x, and hence x l = .1568* Putting d the distance of the sea horizon, we have PT = )/(2CB + P) X PB or, nearly, d = i/2a (x + *,) = ]/23lMax If x is given in feet, we shall find d in statute miles by dividing this value by 5280. Taking a as in the preceding article, we find 5280 and, therefore, d (in statute miles) = 1.317 -\/x in feet (241) If an observer at A' at the height A'B' x' sees the object .4, whose height is x, in the horizon, he must be in the curve de- DIP OF THE SEA. 179 scribed by the ray from A which touches the earth's surface at T. The distance of A' from Twill be = 1.317 i/P, and hence the whole distance from A to A' will be = 1.317 (Vx + Vx'). The above is a rather rough approximation, but yet quite as accurate as the nature of the problem requires ; for the anoma- lous variations of the horizontal refraction produce greater errors than those resulting from the formula. By means of this formula the navigator approaching the land may take advantage of the first appearance of a mountain of known height, to deter- mine the position of the ship. For this purpose the formula (241) is tabulated with the argument "height of the object or eye ;" and the sum of the two distances given in the table, cor- responding to the height of the object and of the eye respect- ively, is the required distance of the object from the observer. 126. To find the dip of the sea at a given distance from the observer. By the dip of the sea is here understood the apparent depres- sion of any point of the surface of the water nearer than the visible horizon. Let T 7 , Fig. 20, be such a point, and A the position of the observer. Let TA' be a ray of light from 7} tangent to the earth's surface at T 7 , meeting the ver- tical line of the observer in A'. Put D" the dip of T as seen from A, d = the distance of T in statute miles, x = the height of the observer's eye in feet = AB, x' = A'B. We have, by (241), and the dip of T, as seen from A', is, therefore, by (240), = 58".82 i/* 7 = 44".66 d. Now, supposing the chords TA, TA' to be drawn, the dip of T at A exceeds that at A' by the angle A TA' , very nearly ; and we have nearl angle ATA' = - X -- = TA' sin 1" 5 5280 d sin 1" 180 SEMIDIAMETERS. whence 5280 d sin 1" Substituting the value of x' in terms of d, D" = 22". 14 d -f 39".07 - (x being in feet and d in statute miles). (242) If d is given in sea miles, we find, by exchanging d for -d, D" = 25".65 d + 33".73 (.r being in feet and d in sea d miles). (243\ The value of D" is given in nautical works in a small table with the arguments x and d. The formula (243) is very nearly the same as that adopted by BOWDITCH in the Practical Navigator. 127. At sea the altitude of a star is obtained by measuring its angular distance above the visible horizon, which generally appears as a well -defined line. The observed altitude then exceeds the apparent altitude by the dip, remembering that by apparent altitude we mean the altitude referred to the true horizon, or the complement of the apparent zenith distance. Thus, h' being the observed altitude, h the apparent altitude, or, when the star has been referred to a point nearer than the visible horizon, h = h' - D" SEMIDIAMETERS OF CELESTIAL BODIES, 128. In order to obtain by observation the position of the centre of a celestial body which has a well-deftned disc, we observe the position of some point of the limb and deduce that of the centre by a suitable application of the angular semi- diameter of the body. I shall here consider only the case of a spherical body. The apparent outline of a planet, whether spherical or spheroidal, and whether fully or partially illuminated by the sun, will be SEMIDIAMETERS. 181 Fig. 21. discussed in connection with the theory of occultations in Chapter X. The angular semidiameter of a spherical body is the angle subtended at the place of observation by the radius of the disc. I shall here call it simply the semidiameter, and distinguish the linear semidiameter as the radius. Let 0, Fig. 21, be the centre of the earth, A the position of an ob- server on its surface, M the centre of the observed body; OB, AB', tangents to its surface, drawn from and A. The triangle OB M re- volved about OM as an axis will de- scribe a cone touching the spherical body in the small circle described by the point B, and this circle is the disc whose angular semidiameter at is MOB. Put S = the geocentric semidiameter, MOB, S' = the apparent semidiameter, MAB' , J, J' = the distances of the centre of the body from the centre of the earth and the place of observation respectively, a = the equatorial radius of the earth, a' = the radius of the body, then the right triangles OMB, AMB' give sin 8 = - J &t lt sin S = J' (244:) But if T: = the equatorial horizontal parallax of the body, we have, Art. 89, a sin TT = J and hence c, ' sin >S = sin a sin S' = sin S J' (245) or, with sufficient precision in most cases, (246) 182 SEMIDIAMETERS. The geocentric semidiameter and the horizontal parallax have therefore a constant ratio = . For the moon, \ve have a = 0.272956 (247) a as derived from the Greenwich observations and adopted by HANSEN (Talks de la Lune, p. 39). If the body is in the horizon of the observer, its distance from him is nearly the same as from the centre of the earth, and hence the geocentric is frequently called the horizontal semidiameter ; but this designation is not exact, as the latter is somewhat greater than the former. In the case of the moon the difference is between 0".l and 0".2. See Table XII. If the body is in the zenith, its distance from the observer is less than its geocentric distance by a radius of the earth, and the apparent semidiameter has then its greatest value. The apparent semidiameter at a given place on the earth's surface is computed by the second equation of (245) or (246), in which the value of is that found by (104) ; so that, putting z = the true (geocentric) zenith distance of the body, = the appa- rent zenith distance (affected by parallax), A = its azimuth,

' _ a ' sin rr ~ a ~~a sin 1" * The greatest declination of the moon being less than 30, it can reach great altitudes only in low latitudes, where the compression is less sensible. A rigorou* investigation of the error produced by neglecting the compression shows that the maximum error is less than 0".06. 184 SEMIDIAMETERS. arid if we put h =* ^L sin 1", log h = 5.2495 we have sin it = hS, which substituted above gives the follow- ing formula for computing the augmentation of the moon's semidiameter : S' S = h S* cos :' -f \ h 2 S 3 + i A 2 S 3 cos 2 C (251) EXAMPLE. Find the augmentation for ' = 40, S = 16' 0" == 960". log /S' 2 5.9645 log h 5.2495 log cos C' 9.8843 log S 3 8.947 log J A 2 0.198 1st term 2d 3d " S' - S = 12".54 = .14 = 0. 08 log 2d term 9.145 log cos 2 :' 9.769 log 1st term 1.0983 = 12 .76 log 3d term 8.914 The value of S f S may be taken directly from Table XII. with the argument apparent altitude = 90 '. 131. If the geocentric hour angle (t) and declination (3) are given, we have, by substituting (137) in (245), (252) sin (5 for which f and 8' are to be determined by (134) and (136), or with sufficient accuracy for the present purpose by the formulae tan S' q= 2 sin # sin 2 J p where the last two terms never amount to 0".2, and therefore the formula may be considered exact under the form sin (C' C) sin (p =p S~) =p i (/> T $) sin 1" sin p sin # Since ^' and p^+ S differ by so small a quantity, there will J92 REDUCTION OF ZENITH DISTANCES. be no appreciable error in regarding them as proportional to their sines ; and hence we have C' C = p + S qr sn p sin (257) the upper signs being used for the upper limb and the lower signs for the lower limb. In this formula, p is the parallax computed for the zenith distance of the limb, and the small term %(p + S)sin p sin S may be regarded as the correction for the error of assuming the parallax of the limb to be the same as that of the centre. EXAMPLE. In latitude

y the clock. The star's hour angle at that instant is == 0*, whence the local sidereal time T' is (Art. 55) T' = a = the star's right ascension. If the clock is regulated to the local sidereal time, we have, therefore, AT=a T But if the clock is regulated to the local mean time, we first con- vert the sidereal time a into the corresponding mean time T' (Art. 52), and then we have A T = T' T This, then, is in theory the simplest and most direct method possible. It is also practically the most precise when properly carried out with the transit instrument. But, as the transit in- strument is seldom, if ever, precisely adjusted in the meridian, the clock time T of the true meridian transit of a star is itself deduced from the observed time of the transit over the instru- ment by applying proper corrections, the theory of which will be fully discussed in Vol. II. It will there be seen, also, that the time may be found from transits over any vertical circle. SECOND METHOD. BY EQUAL ALTITUDES. 139. (A.) Equal altitudes of a fixed star. The time of the meri- dian transit of a fixed star is the mean between the two times when it is at the same altitude east and west of the meridian; so that the observation of these two times is a convenient substi- tute for that of the meridian passage when a transit instrument is not available. The observation is most frequently made with the sextant and artificial horizon ; but any instrument adapted to the measurement of altitudes may be employed. It is, however, not required that the instrument should indicate the true alti- tude ; it is sufficient if the altitude is the same at both observa- BY EQUAL ALTITUDES. 197 tions. If we use the same instrument, and take care not to change any of its adjustments between the two observations, we may generally assume that the same readings of its graduated arc represent the same altitude. Small inequalities, however, may still exist, which will be considered hereafter.* The clock correction will be found directly by subtracting the mean of the two clock times of observation from the com- puted time of the star's transit. EXAMPLE 1. March 19, 1856; an altitude of Arcturus east of the meridian was noted at 11* 4" 1 51*.5 by a sidereal clock, and the same altitude west of the meridian at 17* 21'" 30*.0; find the clock correction. East 11 4"51'.5 West 17 21 30.0 Merid. transit by clock = T = 14 13 10 .75 March 19, Arcturus R. A . = a = 14 9 7 .11 Clock correction = A T = 4 3 .64 This is the clock correction at the sidereal time 14* 9 m 7M1 of at the clock time 14* 13 m 10*.75. EXAMPLE 2. March 15, 1856, at the Cape of Good Hope, Latitude 33 56' S., Longitude 1* 13 m 56' E.; equal altitudes of Spica are observed with the sextant as below, the times being noted by a chronometer regulated to mean Greenwich time. The artificial horizon being employed, the altitudes recorded are double altitudes. East. 2 Alt. Spica. West. 1020- 0'.5 20 28. " 20 55. 104 0' 10 20 2 40" 38'. 40 10.5 " 39 42. Means 10 20 27 .83 2 40 10.17 10 20 27.83 Merid. Transit, by Chronom. = T = 12 30 19 .00 The chronometer being regulated to Greenwich time, we inust compute the Greenwich mean time of the star's transit at the Cape (Art. 52). We have * For the method of observing equal altitudes with the sextant, see Vol. II.. Sextant." 198 TIME. Local sidereal time of transit = a = 13* 17 m 37'.92 Longitude = 1 13 56. Greenwich sidereal time = 12 3 41 .92 March 15, sid. time of mean noon = 23 33 5 .37 Sid. interval from mean noon = 12 30 36.66 Reduction to mean time 2 2 .97 Mean Gr. time ef star's i local transit } = F = 12 28 33 ' 58 Chronometer time of do. = T = 12 30 19 .00 Chronometer correction = A T = 1 45 .42 140. (B). Equal altitudes of the sun before and after noon. If tlm declination of the sun were the same at both observations, the hour angles reckoned from the meridian east and west would be equal when the altitudes were equal, and the mean of the two clock times of observation would be the time by the clock at the instant of apparent noon, and we should find the clock cor- rection as in the case of a fixed star. To find the correction tor th change of declination, let

and 3 are north, give them the positive sign ; in the opposite cases they take the negative sign. The signs of A and B are given in the table ; A being negative only when t < 12 A and B positive when t < 6' 1 or > 18*. When we have applied A 7^ to the mean of the clock times (or the "middle time"), we have the time T = T 4- A T as shown by the clock at the instant of the sun's meridian transit. Then, computing the time 7 1 ', whether mean or sidereal, which the clock is required to show at that instant, we have the clock correction, as before, A T _ T' T EXAMPLE. March 5, 1856, at the U. S. Naval Academy, Lat. 38 59' K, Long. 5* 5 W 57*.5 W., the sun was observed at the same altitude, A.M. and P.M., by a chronometer regulated to mean Greenwich time ; the mean of the A.M. times was 1 A 8'" 26*.6, and of the P.M. times 8* 45"' 41".7 ; find the chronometer cor- rection at noon. We have first A.M. Chro. Time = 1* 8" 26'.6 P.M. " =8 45 41.7 time 2t =1 37 15.1 Middle time T = 4 57 4 .15 From the Ephemeris we find for the local apparent noon of March 5, 1856, d = 5 46' 22". 5 Equation of time = -f \\ m 35'. 11 A'($ = -f 58".10 For the utmost precision, we reduce A'# to the instant of local BY EQUAL ALTITUDES. 201 noon. With these quantities and

, that is, by changing the sigi\ of the first term ; but this noon at 202 TIME. the antipode is the same absolute instant as the midnight of the observer, and hence A T = A'3 . t tan y A'*, t tan 3 PEquation for! ._. 15 sin f 15 tan L midnight. J ^ ' and this is computed with the aid of the logarithms of A and B in Table IV. precisely as in (264), only changing the sign of A. The sign for this case is given in the table.* 142. To find the correction for small inequalities in the altitudes. If from a change in the condition of the atmosphere the re- fraction is different at the two observations, equal apparent alti- tudes will not give equal true altitudes. To find the change A/ in the hour angle / produced by a change A/I in the altitude h, we have only to differentiate the equation sin h = sin

, or, putting sin dy> for d COS 8 which gives t by a very simple logarithmic computation. From / we deduce, by Art. 55, the local time, which compared with the observed clock time gives the clock correction required. It is to be observed that the double sign belonging to the radical in (267) gives two values of sin | , the positive corre- sponding to a west and the negative to an east hour angle; since any given zenith distance may be observed on either side of the meridian. To distinguish the true solution, the observer must of course note on which side of the meridian he has observed. If the object observed is the sun, the moon, or a planet, its declination is to be taken from the Ephemeris, for the time of the observation (referred to the meridian of the Ephemeris); but, as this time is itself to be found from the observation, we must at first assume an approximate value of it, with which an approxi- mate declination is found. With this declination a first compu- 208 TIME. tation by the formula gives an approximate value of t, and hence a more accurate value of the time, and a new value of the decli- nation, with which a second computation by the formula gives a still more accurate value of t. Thus it appears that the solution of our problem is really indirect, and theoretically involves an infinite series of successive approximations; in practice, how- ever, the observer generally possesses a sufficiently precise value of his clock correction for the purpose of taking out the declina- tion of the sun or planets. The moon is never employed for determining the local time except at sea, and when no other object is available.* EXAMPLE. At the U. S. Naval Academy, in Latitude

, <5, and dt the corresponding error of t; A is the star's azimuth, q the parallactic angle, or angle at the star. If the zenith distance alone is erroneous, we have, by putting d

tan A by which we see that an error in the latitude also produces the least effect when the star is on the prime vertical, or when the observer is on the equator. Indeed, when the star is exactly in 212 TIME. the prime vertical, a small error in

sin A dt which gives, since A varies with /, but

sin A . A* -{ sin t 2 Since A and A< are here supposed to be expressed in parts of the radius, if we wish to express them in seconds of arc and of time respectively, we must substitute for them A sin V and 15 Ai sin 1", and the formula becomes cos

h + Pi the mean of which is

, - j>) (280) whence it appears that by this method the absolute values of p and ft are not required, but only their difference p l p. The change of a star's declination by precession and nutation is so small in 12* as usually to be neglected, but for extreme precision ought to be allowed for. This method, then, is free from any error in the declination of the star, and is, therefore, employed in all fixed observatories. EXAMPLE. With the meridian circle of the Naval Academy the upper and lower transits of Polaris were observed in 1853 Sept. 15 and 16, and the altitudes deduced were as below: Upper Transit. Lower Transit. Sept. 15, App. alt. 4028'25".42 Sept. 16, 87 81' 89". 76 Barom. 30.005 Att. Therm. 65.2 Ext. 63 .8 h h = 1 6 .34 Barom. Att. Therm Ext. 30.146 . 75 74 .6 I Ref. *,= Ji = 1 12 .45 40 1 27 19 28 26 .08 .04 37 1 30 28 27 .31 25 .86 38 58 53 C, 38 58 53 .17 53.04 Mean = 38 58 53 .11 In order to compare the results, each observation is carried out separately. By (280) we should have J (h -f A,) = 38 58' 53".20 Cfc-.P)=_ -09 f = 38 58 53 .11 This method is still subject to the whole error in the refraction, 228 LATITUDE. which, however, in the present state of the tables, will usually be very small. If the latitude is greater than 45, and the star's declination less than 45, the upper transit occurs on the opposite side of the zenith from the pole. In that case h must still represent the distance of the star from the point of the horizon below the pole, and will exceed 90. Thus, among the Greenwich observations we find 1837 June 14, Capclla ft, = 7 18' 7".94 h =95 39 7 .91 ) = sin A which determines D, and hence also ^>. For practical con- venience, however, put then, by eliminating d, the solution may be put under the follow- ing form : tan D = tan d sec / cos = sin A sin D cosec 3 (281) The first of these equations fully determines D, which will be taken numerically less than 90, positive or negative according to the sign of its tangent. As t should always be less than 90, or 6 /l , D will have the same sign as 3. The second equation is indeterminate as to the sign of 7-, since the cosine of + 7- and f are the same. Hence we obtain by the third equation two values of the latitude. Only one of these values, however, is admissible when the other is numerically greater than 90, which is the maximum limit of latitudes. When both values are within the limits + 90 and 90, the true solution is to be distinguished as that which agrees best with the approximate latitude, which is always suffi- ciently well known for this purpose, except in some peculiar cases at sea. EXAMPLE 1. 1856 March 27, in the assumed latitude 23 S. and longitude 43 14' W., the double altitude of the sun's lower ALTITUDE AT A GIVEN TIME. 231 limb observed with the sextant and artificial horizon was 114 40' 30" at 4* 21 m 15* by a Greenwich Chronometer, which wab fast 2 m 30'. Index Correction of Sextant = V 12", Barom. 29.72 inches, Att. Therm. 61 F., Ext. Therm. 61 F. Required the true latitude. Sextant reading = 114 40' 30" Index corr. App. alt. Q Semidiameter Ref. and par. = 1 12 114 39 18 = 57 19 39 = + 16 3 = 31 h= 57 35 11 5=4-2 51 30 log sec t 0.027360 log tan 3 8.698351 log tan D 8.725711 D= + 3 2' 38" Y = 25 58 49 D r = v = 22 56 11 Chronometer 4* 21 m 15' Correction 2 30 Gr. date, March 27, 4 18 45 Longitude = 2 52 56 Local mean t. = 1 25 49 Eq. of time = - 5 19 App. time, t = I 20 30 = 20 7' 30" log cosec S 1.302190 log sin D 8.725098 log sin h 9.926445 log cos Y 9.953733 EXAMPLE 2. 1856 Aug. 22; suppose the true altitude of Fomalhaut is found to be 29 10' 0" when the local sidereal time is 21* 49 m 44'; what is the latitude ? We have a = 22*49- 44% whence f= 1*0- <);* = 30 22'47".5; D= - 31 15' 13", Y = 60 0' 6", v = + 28 44' 53". The nega- tive value of Y here gives

BQCA = ^[sin (d + p) sin (d ?)] which shows that sec A diminishes as d increases. In order, therefore, to reduce the effect of an error in the declination REDUCTION TO THE MERIDIAN. 233 at the same time with that of errors of altitude and time, we should select a star as near the pole as possible, and observe it at or near its greatest elongation, on either side of the meridian. The proximity of the star to the pole enables us to facilitate the reduction of a series of observations, and we shall therefore treat specially of this case as our Fourth Method below. 167. When several altitudes not very far from the meridian are observed in succession, if we wish to use their mean as a single altitude, the correction for second differences (Art. 151) must be applied. It is, however, preferable to incur the labor of a sepa- rate reduction of each altitude, as we shall then be able to com- pare the several results, and to discuss the probable errors of the observations and of the final mean. When the observations are very near to the meridian, this separate reduction is readily effected, with but little additional labor, by the following method: THIRD METHOD. BY REDUCTION TO THE MERIDIAN WHEN THE TIME IS GIVEN. 168. To reduce an altitude, observed at a given time, to the meridian. This is done in various ways. (A.) If in the formula, employed in Art. 164, sin sin d -j- cos

v 234 LATITUDE. this equation, we then have the latitude as from a meridian observation by the formula 9 = 8 + fi> or 9 = ' 5 Ci according as the zenith is north or south of the star. When the star changes its declination, this method still holds if we take d for the time of observation, as is evident from our formulae, in which d is the declination at the instant when the true altitude is h. To compute the second member, a previous knowledge of the latitude is necessary. As the term cos

cos d cos

log cos rf log cosec Ci log A log log Am 9.890G 9.4193 Q.23'29 9.5428 1.7898 1.3326 REMARK 1. The reduction A t h increases as the -denominator of A decreases, that is, as the meridian zenith distance decreases. The preceding method, therefore, as it supposes the reduction to be small, should not be employed when the star passes very near the zenith, unless at the same time the observations are restricted to very small hour angles. It can be shown, however, from the more complete formulae to be given presently, that so long as the zenith distance is not less than 10, the reduction computed by this method may amount to 4' 30" without being in error more than V ; and this degree of accuracy suffices for even the best observations made with the sextant. 238 LATITUDE. REMARK 2. If in (284) we put sin \t = y sin 1". < (< being in seconds of time), we have = cos poos 3 ^25 gin r , <2 ^ cos Aj 2 in which a denotes the product of all the constant factors. It follows from this formula that near the meridian the altitude varies as the square of the hour angle, and not simply in proportion to the time. Hence it is that near the meridian we cannot reduce a number of altitudes by taking their mean to correspond to the mean of the times, as is done (in most cases without sensible error) when the observations are remote from the meridian. The method of reduction above exemplified amounts to sepa- rately reducing each altitude and then taking the mean of all the results. 171. (D.) Circummeridian altitudes more accurately reduced. The small correction which the preceding method requires will be obtained by developing into series the rigorous equation (282). This equation, when we put = 90 h = true zenith distance deduced from the observation, may be put under the form cos C = cos d 2 cos y cos S sin 2 J t which developed in series* gives, neglecting sixth and higher powers of sin $ t, * If we put y = 2 cos cos 6 sin 2 <, the equation to be developed is cos = cos j y (a) In which fj is constant and f may be regarded as a function of y ; so that by MAC- LAURIN'S Theorem in which (/), I J, &c. denote the values of fy and its differential coefficients when \ dy i y = 0. The equation (a) gives, by differentiation, rff < 1 sin C = 1 = - dy dy sin C cos C rf cot ~ sin 2 C dy ~ sin 2 f _ CIRCUMMEEIDIAN ALTITUDES. 239 cos y cos 3 2 sin* } t I cos y cos 3 v* 2 cot ; t sin* } < ^ Bind 8hTl~ \ sin C, / ' sin 1" By this formula, first given by DELAMBRE, the reduction to the meridian consists of two terms, the first of which is the same as that employed in the preceding method, and the second is the small correction which that method requires. These two terms will be designated as the " 1st Reduction" and " 2d Reduction." Putting 2 sin 1 \t 2 sin* \t m = - n = - sin 1" sin 1" sin d we have C, = C Am -f Bn (289) If a number of observations are taken, we have a number of equations of this form, the mean of which will be 'i = : - Am o + Bn in which is the arithmetical mean of the observed zenith dis- tances, m and n the arithmetical means of the values of m and n corresponding to the values of t. The values of n are also given in Table V. Having found p we have the latitude, as before, by the formula 9 = * + :, in which we must give t the negative sign when the zenith is south of the star, and it must be remembered that for the sun (or any object whose proper motion is sensible) 3 must be the mean of the declinations belonging to the several observations, But when y = we have, by (a), = $ v so that (6) becomes y y 2 cot L. V 5 C^Ci + - -- - - + i(l + Scoff,)-*- -Ac. (c) sin fj 2 sm j ^ sin 1 d Restoring the value of y, this gives the development used in the text, observing that as and t are supposed to be in seconds of arc, the terms of the series are divided by sin 1" to reduce them to the same unit. 240 LATITUDE. or, which is the same, the declination corresponding to the mean of the times of observation.* Finally, if the star is near the meridian below the pole, the hour angles should be reckoned from the instant of the lower transit. Recurring to the formula cos C = sin + 8) and cos t cos (

cos COS d Sun by mean time chron., A k' : Bin Bun by sidereal chron., A = k'i' Y [log? = 9.997625] for which log k will be taken from Table V. with the argument rate of the chronometer 3T; and log //from the same table CIRCUMMERIDIAN ALTITUDES. 243 with the argument dT dE ' = daily rate of the chronometer diminished by the daily increase of the equation of time. EXAMPLE. 1856 March 15, at a place assumed to be in lati- tude 37 49' K and longitude 122 24' W., suppose the fol- lowing zenith distances of the sun's lower limb to have been observed with an Ertel universal instrument,* Barom. 29.85 inches, Att. Therm. 65 F., Ext. Therm. 63 F. The chrono- meter, regulated to the local mean time, was, at noon, slow 11'" 20*.8, with a -daily losing rate of 6'.6. Obs'd zen. dist. Chronometer. t 771 n 40 8' 40".7 23* 37- 35'. 19 m 58.8 783".3 1' '.49 40 2 16 .5 42 3. 15 30 .8 472 .4 .54 39 57 28 .3 46 29.5 11 4 .3 240 .6 .14 39 54 17 .2 50 46.5 - 6 47 .3 90 .5 .02 39 52 33 . 55 16. 2 17 .8 10 .4 .00 39 52 34 .5 37.5 + 3 3 .7 18 .4 .00 39 54 28 .6 5 13. 7 39 2 115 .0 .03 39 58 9 .8 9 49.5 12 15 .7 295 .1 .21 40 3 .3 14 8. 16 34.2 538 .9 .70 40 9 36 . 18 31. 20 57 .2 861 .4 1 .80 iMeans 39 59 18 .5 t = +0 29.1 m = 342 .60 n = .49 The equation of time at the local noon being + 8 m 54*.6, we have Mean time of app. noon 0* 8 m 54'.6 Chronometer slow = 11 20 .8 Chr. time of app. noon = 23 57 33 .8 The difference between this and the observed chronometer times gives the hour angles t as above. The mean of the hour angles being + 29*.l, the declination is to be taken for the local apparent time O ft O m 29M, or for the Greenwich mean time March 15, S h IS" 1 59*.7; whence 3 = 1 48' 8".8 (Approximate)

* and, the chronometer rate being 3T= + 6'.6, we have d T dE -f- 24*. 0, with which as the argument "rate" in Table V. we find. log k' = 0.00024. The computation of the latitude is now carried out as follows: log cos f 9.89761 Mean observed zen. dist. = 39 59' 18".5 r p= + 41 .8 8= - 16 6 .5 Am = 7 4 .4 Bn -f .9 C,= 39 36 50 .3 3 = 1 48 8 .8 V= 37 48 41 .5 The assumed value of

sin (Jj sin 2 % t sin # The last term is extremely small, rarely affecting the value of by as much as 0".l; and since x is proportional to the hour angle, and therefore has opposite signs for observations on differ- ent sides of the meridian, the effect of this term will nearly or quite disappear from the mean of a series of observations pro- perly distributed before and after the meridian passage. Now, we have = 15 t SHI 1 3600 54000 Let 54000 cos y cos 3 t then, taking 15 t sin 1" = sin t + J sin s t we have sin* =(Bin t + i sin 3 *) sin * - C 8 * C 8 * sin (vdj and the formula for cos becomes, by omitting the last term, cos C = sin

cos d. 2 sin 2 i # 8 = 8 i+ rV : 7^ sm (

cos 8 l sin 1 i if This equation is of the same form as that from which (288) was obtained, and therefore when developed gives _ cos

cos ^ \* 2 cot C, sin* i if sin C t sin 1" \ sin C, / sin 1" in which ^ l = ( p d'. Putting then, as before, A = 8 ^ C 8 * l B = A* cot C, (292) sin C t and taking m and n from Table V., or their logarithms from Table VI., with the argument /', which is the hour angle reckoned CIRCUMMERIDIAX ALTITUDES. 247 from the instant the sun reaches its maximum altitude, we have Cj = C Am + En (293) Since t differs from the latitude by the constant quantity 3', its value found from each observation should be the same. Taking its mean value, we have r = c, -M' The angle #, being very small, may be found with the utmost precision by the formula =[9.40594] 810000 sin I" A A which gives & in seconds of the chronometer when A has been computed by the formula (292). The most simple method of finding the corrected hour angles t' will be to add # to the chronometer time of apparent noon, and then take the difference between this corrected time and each observed time. If we put d' = #j -f j/, we have (295? which requires only one new logarithm to be taken, namely, the value of log m from Table VI. with the argument $. We then have, finally, 9 = C, + \ + y EXAMPLE. The same as that of the preceding article. We have there employed the assumed latitude 37 49' ; but, since evec a rough computation of two or three observations will give a nearer value, let us suppose we have found as a first approxima- tion

, = sin h x cos h -\- p cos t cos h ^(x 2 2xpcost-{-p t ~)sin A-f& c - and from this we obtain the following general expression of the correction : * PI. Trig. (403) and (406). 254 LATITUDE. x = p cos t * (.r 4 2 xp cos t -f- /> 2 ; tan A -f (3* 3 .r 2 /> cos t -j- 3 jcp p z cos f ) + 24 O" 4 4 -^ P cos * + 6 &P* 4 - r / cos t + X) tan /i &c. (a) For a first approximation, we take x = p cos t (b) and, substituting this in the second term of (#), we find for a second approximation, neglecting the third powers of p and z, x = p cos t \ p 2 sin 2 tan A (c) Substituting this value in the second and third terms of (a), we find, as a third approximation, x = p cos t 1 2 p* sin 2 1 tan A -(- ? t p 3 cos < sin" < (W) This value, substituted in the second, third, and fourth terms of (a), gives, as a fourth approximation, x = p cos t $p* sin 2 1 tan h -f- i p 3 cos sin 2 f | /?* sin 4 1 tan 3 A + -^ P' ( 4 9 sin * sin ' 2 ' tan ^ (e) In these formulae, to obtain x in seconds when p is given in seconds, we must multiply the terms in p 2 , p 3 , and ^> 4 by sin 1", sin 2 1", sin 3 1", respectively. In order to determine the relative accuracy of these formulae, let us examine the several terms of the last, which embraces all the others. The value of t, which makes the last term of (e) a maximum, will be found by putting the differential coefficient of (4 9 sin 2 1) sin 2 1 equal to zero ; whence 4 sin t cos t (2 9 sin 2 1 ) = which is satisfied by t = 0, t = 90, or sin 2 1 = |, the last of which alone makes the second differential coefficient negative. The maximum value of the term is, then, ^ p* sin 3 1" tan A, which for p = 1 30' = 5400" is 0".0018 tan h. This can amount to 0".01 only when h is nearly 80, that is, when the latitude is nearly 80. This term, therefore, is wholly inappreciable in every practical case. BY THE POLE STAR. 255 The term \p* sin 3 V sin 4 1 tan 3 A is a maximum for sin t = 1, in which case, for p = 5400", it is 0".0121 tan 3 h. This amounts to 0".l when h = 64, and to 1". when h = 77. For the maximum of the term p 3 sin 2 V cos < sin 2 1 we have, by putting the differential coefficient of cos t sin 2 1 equal to zero, sin t (2 3 sin* t~) = which gives sin 2 t = f , and consequently cos t y'i I and hence the maximum value of this term is } p 3 am 2 V i/$ = 0".475. Since the maximum values of this and the following terms do not occur for the same value of t, their aggregate will evidently never amount to 1" in any practical case. Hence, to find the latitude by the pole star within 1", we have the formula 9 = h p cos f + $p 2 sin 1" sin 1 1 tan h (300) The hour angle t is to be deduced from the sidereal time of the observation and the star's right ascension a, by the formula t = To facilitate the computation of the formula (300), tables are given in every volume of the British Nautical Almanac and the Berlin Jahrbuch; but the formula is so simple that a direct computation consumes very little more time than the use of these tables, and it is certainly more accurate. EXAMPLE. (From the Nautical Almanac for 1861). On March 6, 1861, in Longitude 37 "W., at 7 A 43'" 35* mean time, suppose the altitude of Polaris, when corrected for the error of the in- strument, refraction, and dip of the horizon, to be 46 17' 28" : required the latitude. Mean time 7* 43- 35'. Sid. time mean noon, March 6, 22 56 47.9 Eeduction for 7* 43" 35' 1 16 .2 Reduction for Long. 2* 28" 24.3 Sidereal time 0= 6~42 3~A March 6, p = 1 25' 32".7 a = 1 7 39.0 t = 5 34 24.4 = 83 36' 6" 256 LATITUDE. ' Hence, by formula (300), log;? 3.71035 lo gj p 2 7.4207 log cos t 9.04704 log sin 2 1 9.9946 log 1st term 2.75739 log tan h 0.0196 log i sin 1" 4.3845 A = 46 17' 28" log 2d term 1.8194 1st term = 9 32 .0 2d = -f- 1 6 .0 y = 46 9 2 .0 By the Tables in the Almanac, y = 46 9' 1" 177. If we take all the terms of (e) except the last, which we have shown to be insignificant, we have the formula y = h p cos t -{- j;? 2 sin 1" sin 2 1 tan A I p s sin 2 1" cos t sin 2 1 -f p* sin 1" sin 4 1 tan* A (301) which is exact within 0".01 for all latitudes less than 75, and serves for the reduction of the most refined observations with first-class instruments. If we have taken a number of altitudes in succession, the separate reduction of each by this formula will be very laborious. To facilitate the operation, PETERSEN has computed very con- venient tables, which are given in SCHUMACHER'S Hiilfstafdn, edited by WARNSTORFF. These tables give the values of the following quantities : a =p cos t + I p * sin 2 !" cos t sin 2 1 ft = ^ p* sin 1" sin 2 1 X = I p (j9 2 j9 2 ) oin 1 1" cos t sin* t p. p sin 1 1" sin* t tan 3 A in which p = 1 30' = 5400". Then, putting P log A = logp 3.7323938 the formula (301) becomes 9 = h (A* + X) + A'P tan h + BY TWO ALTITUDES. 257 Putting then X = Aa -f /I y = A*0 tan h -f ^ we have or, when the zenith distance is observed, (302) # = 4a -f- /I 1 y = 40 cot C + / V (303) 90 ? = C + ;r y J The first table gives a with the argument t ; the second, /? with the argument <; the third, ^ with the arguments p and <; and the fourth, j* with the arguments y and p, ^ being used for h in so small a term. To reduce a series of altitudes or zenith distances by these tables, we take for h or the mean of the true altitudes or zenith distances ; for a and jj the means of the tabular numbers corresponding to the several hour angles, with which we find Aa and A 2 {3 cot . The mean values of the very small quanti- ties A and ft are sensibly the same as the values corresponding to the mean of the hour angles ; so that X is taken out but once for the arguments polar distance and mean hour angle, and p is taken with the arguments

l = f t = the difference of the hour angles,

cos f = cos 8' sin /i' sin 8' cos A' cos q cos ?> sin t' = cos A' sin q or, adapted for logarithmic computation, n sin j!V = sin A' n cos JV = cos A' cos ^ sin (f = n cos (JV- cos ^ cos f = n sin (JV cos y> sin H = cos A' sin q (307) 260 LATITUDE. The formulae (305) and (307) leave no doubt as to the quadrant in which the unknown quantities are to be taken. But we may take the radical in (306) with either the positive or the negative sign, and Q, therefore, either in the first or fourth quadrant. If we take J Q always in the first quadrant, the values of q will be q = P + Q and either value may be used in (307) ; whence two values of

sin C cos P = sin d sin a' . _. sin sin D = cos C (309) cos J (8 -f- d') sin J (<5 5') cos J> sin C which determine D and P after C has been found from (308). In precisely the same manner we derive from the triangles ZTS and ZTS' the equations sm H = 8n * (* + A') cos j ( - cos (7 cos = C 8 * (A + ^ 8i " * (A ~ * cos If sin C (310) Then in the triangle PTZ we have the angle PTZ, by the formula q = P-Q and hence the equations sin (f = sin D sin J3" -(- cos D cos S" cos g cos tf cos r = cos D sin H sin D cos H cos cos r cos /5 cos (D + 7-) I (312) cos sin r = sin /? j To find the hour angles t and f, let X = T - i (f + then, since >l = J (f /), we have * A -f # = r ^ the angle TPS, j A _ x = f T = the angle and from the triangles P !& and P7^' we have sin (U + .r) _ sin P sin (j a re) _ sin P sin C . cos 5 sin C cos 5' whence sin (j a -{- a?) sin (} a ap _ cos ^^ cos 8 sin (i A 4- a:) + sin (i x A 1 ) ~~ cos 3' + col* and, consequently, tan x = tan i ( by (323) (whkh in- volves very little additional labor, since the logarithms of sin ft and sin ^ have already occurred in the previous computation), and then we have the true latitude If we wish also to correct" the hour angle r found by this method, we have, from the second equation of (47) applied to LATITUDE. the triangle PTZ (taking 6 and c to denote the sides PT and Z T, which are here constant), cos H cos A AT = .AP COS (f in which A is the azimuth of the point T. But we have in the triangle PTZ cos H cos A = sin

cos T tan S \ sin | A \ cos J A / When this correction is added to r, we have the value that would be found by the rigorous formulae, and we have then to apply also the correction x according to (314). In the present case we have, by (313), x = A<5 tan 8 tan A and the complete formulae for the hour angles t and t' become t := T + AT x J A If = T + AT X + J >l Putting I/ = AT # we find, by substituting the above values of AT and x, /tan y cosr_JanM \ sin J >l tan * /I / and then we have ' (325) The corrections A^P and y are computed with sufficient accu- racy with four-place logarithms, and, therefore, add but little to the labor of the computation. Nevertheless, when both latitude and time are required with the greatest precision, it will be pre- ferable to compute by the rigorous formulae. BY TWO ALTITUDES. V&9 EXAMPLE. 1856 March 10, in Lat. 24 K, Long. 30 W., suppose we obtain two altitudes of the suri as follows, all correc- tions being applied : find the latitude. At app. time 30 h = 61 11' 38".3 (3 ) = 3 51' 52".8 4 30 h' = 18 46 35 .8 (/)= 3 47 57 .4 59 7 .1 3 = 3~49 ,55 .1 = -f 1' 57".7 = 2* 0- J ^ -f A') = 39 59 7 .1 = 30 0' i (A A') = 21 12 31 .3 The following is the form of computation by the formulae (316), (317), and (318), employed by BOAVDITCH in his Practical Navigator, the reciprocals of the equations (316) and of the second of (317) being used to avoid taking arithmetical comple- ments. This form is convenient when the tables give the secants and cosecants, as is usual in nautical works. cosec /I sec J 0.301030 0.000972 cosec nl. 175024 cosec C cos i (h 4 sin (A - 0.302002 cos 9.937854 cos - h') 9.884347 cosec 0.192065 D = 4 25' 21".3 cosec - h') 9.558428 sec 0.030459 9.937854 nl. 112878 sin/? 9.744777 cos 9.919829 cos sec 6T080207 y = 33 45 38 .0 9.919829 D + y = 29 20 16 .7 sin 9.690161 = 24 2' 23".2 sin 9.609990 If the approximate latitude had not been given, we might also have taken D f = 38 10' 59".3, and then we should have had cos ft 9.919829 sin (D Y) n9.791113

0.0394 *

sin A'dr in which A and A' denote the azimuths of the two stars, or of the same star at the two observations. 272 LATITUDE. Eliminating dr and dtp successively, we find da* sin A' dh \ 8in A dh' sin (A' A) C 8 A ' dh 1 sin (A' A) eoi -4 , sin (A' A) sin (A' A) (328) These equations show that, in order to reduce the effect of error? of altitude as much as possible, we must make sin (A' A) as great as possible, and hence A' A, the difference of the azi- muths, should be as nearly 90 as possible. If we suppose A' A = 90, we have d

will depend chiefly on the term sin A'dh. At the same time, cos A will be nearly unity and cos A' small, so that the error dr will depend chiefly on the term cosAdh'. Hence the accuracy of the resulting latitude will depend chiefly upon that of the altitude near the meridian ; and the accuracy of the time chiefly upon that of the altitude near the prime vertical. If the observations are taken upon the same star observed at equal distances from the meridian, we have A' = A, and the general expressions (328) become dh + dh' COS dr = 2 cos 4 dh dh' 2 sin A The most favorable condition for determining both latitude and time from equal altitudes is sin A = cos A, or A = 45. Errors in the observed clock times. An error in the observed time may be resolved into an error of altitude ; for if we say that dT is the error of T at the observation of the altitude h. it BY TWO ALTITUDES. 273 amounts to saying either that the time 7 1 dT corresponds to the altitude h, or that T corresponds to the altitude h -f- dh, dh being the increase of altitude in the interval dT. We may, therefore, consider the time T as correctly observed while h is in error by the quantity dh. Conversely, we may assume that the altitudes are correct while the times are erroneous. The discussion of the errors under the latter form, while it can lead to no new results, is, nevertheless, sufficiently interesting. We have, from the formula (304), dl = dT' dT The general equations (327), upon the supposition that h and h are correct, give = cos A d

and AT, we have sin A' cos q .^ , sin^coso' J!% . = --- *- dS -\- - *- dd' sin (A 1 A} sin (A' A) (332) cos A' cos q , cos A cos q' cos

This is but another expression of the correction (323). If, when the sun is observed, instead of employing the mean declination we employ the declination belonging to the .greater altitude, which we may suppose to be A, we shall have dd = 0, dd' = 2 A be numerically less than A^> ? First. If both observations* are on the same side of the meridian, the condition &'

gives 2 sin q cos q' < sin (q' -|- < A^, then, requires that 2 sin q cos q' < sin q' cos q cos q' sin q or tan q < ^ tan j' Therefore A ^>' will be greater than A^> OT^J/ when the observa- tions are on opposite sides of the meridian and tan q > tan q'. In cases where an approximate result suffices, as at sea, and the correction A^> is omitted to save computation, it will be expedient to employ the declination at the greater altitude, except in the single case just mentioned.* But to distinguish this case with accuracy we should have to compute the angles q and q' ; and therefore an approximate criterion must suffice. Since the parallactic angles increase with the hour angles, we may substi- tute for the condition tan q > % tan q' the more simple one t > $ I', which gives or (t and t' being only the numerical values of the hour angles) Hence we derive this very simple practical rule : employ the sun's declination at the greater altitude, except when the time of this altitude is farther from noon than the middle time, in which case employ the mean declination. The greatest error committed under this rule is (nearly) the value of A^> in (323), when T = t. But since in this case 3 = t', and i + t' = A, we have r = | ^, and therefore sin ft = cos

A sec $ L Since ^ will seldom exceed 6 A , A will not exceed 3', and the maximum error will not exceed 1'.6. In most cases the error will be under 1', a degree of approximation usually quite suffi- cient at sea. Nevertheless, the computation of the correction &y by our formula (323) is so simple that the careful navigator * BOWDITCH and navigators generally employ in all cases the mean declination ; but the above discussion proves that, if the cases are not to be distinguished, it will be better always to employ the declination at the greater altitude. BY TWO EQUAL ALTITUDES. 277 will prefer to employ the mean declination and to obtain the exact result by applying this correction in all cases. SIXTH METHOD. BY TWO ALTITUDES OF THE SAME OR DIFFERENT STARS, WITH THE DIFFERENCE OF THEIR AZIMUTHS. 184. Instead of noting the times corresponding to the observed altitudes, we may observe the azimuths, both altitude and azi- muth being obtained at once by the Altitude and Azimuth Instrument or the Universal Instrument. The instrument gives the difference of azimuths = L The computation of the latitude and the absolute azimuth A of one of the stars may then be performed by the formulae of the preceding method, only inter- changing altitudes and declinations and putting 180 A for t. When A has been found, we may also find t by the usual methods. SEVENTH METHOD. BY TWO DIFFERENT STARS OBSERVED AT THE SAME ALTITUDE WHEN THE TIME IS GIVEN. 185. By this method the latitude is found from the declinations of the two stars and their hour angles deduced from the times of observation, without employing the altitude itself, so that the result is free from constant errors (of graduation, &c.) of the instrument with which the altitude is observed. Let 0, 0' = the sidereal times of the observations, o. a' = the right ascensions of the stars, 3, 3' = the declinations " " t, t' = the hour angles " " h = the altitude of either star,

(sin 8' sin = wi cos [J (f -f- f) JW] (335) The equations (334) determine m and Jf, and then the latitude is found by (335) in a very simple manner. It is important to determine the conditions which must govern the selection of the stars and the time at which they are to be observed. For this purpose we differentiate the above expres- sions for sin h relatively to

26 ' the stars are observed. Let T, T', T" = the clock times of the observations, A T = the clock correction to sidereal time at the time T, 8T = the rate of the clock in a unit of clock .time, a, a', a" = the right ascensions of the stars, 8, 8', 8" = the declinations " " t, ?, t" = the hour angles " " h = the altitude,

l' = I 7 " J 7 -f 5 J 7 ( T" T] (a" a) The angles X and A' are thus found from the observed clock times, the clock rate, and the right ascensions of the stars. The hour angles of the stars being /, t + A, and t -f A', we have sin h = sin

cos 5' cos ( -f A) sin A sin

sin A' dT' cos sin A" dT" cos

. It is well to remark, also, for the purpose of verifica- tion, that the sum of the three coefficients in the formula for dtp must be = 0, and the sum of those in the formula for d T must be = - 1. The substitution of dX for dT' dT, and dX for dT" ' dT, will reduce the above expressions to a more simple form, which I leave to the reader. EXAMPLE. To illustrate the above method, GAUSS took the following observations, with a sextant and mercurial horizon, at Gottingen, August 27, 1808. The double altitude on the sextant was 105 18' 55". The time was noted by a sidereal clock whose rate was so small as not to require notice. BY THREE EQUAL ALTITUDES. 285 a Andromeda T = 21* 33 m 26* a Ursas Minoris T' = 21 47 30 a LyroR 1 " = 22 521 The apparent places of the stars were as follows : Andromeda: a = 23* 58 m 33'.33 Ursce Minoris a = 55 4 .70 Lyrce a" = 18 30 28 .96 3 = 28 2' 14" .8 8' = 88 17 5 .7 d" = 38-37 6 .6 Hence we find 1 A = 5 18' 25".28 i (d' 8)= 30 7 25 .45 !(' + 8} = 58 9 40 .25 log cot i OJ' 72 log tan (45 *) 9.8142617 i (2V' 2V) = 20 56 36 .24 log cot i (2V' N) nO.4171063 I-. } (JV' + L T ) = 59 35 14 .71 log tan [f -f- J (JV'+ JV)] nO.2313680 i = 20 34 23 .56 t = 39 51 .15 = 2* 36" 3*.41 o = 23 58 33 .33 t + a = = 21 22 29 .92 T= 21 33 26. Clock correction A T = 10 56 .08 Then, to find the latitude, we have 286 LATITUDE. t -f N= 38 38' 38".47 log cos (t -f JV) 9.8926738 log m 0.2072029 log tan

and h found by (348) are such that cos^> and sin h have opposite signs, we must substitute 180 + q for q and repeat the computation of these two equa- tions. In this repetition the same logarithms will occur as before, but differently placed. 2. If the values of

, /, and d. But the method is of no practical value, as the errors of observation have too much influence upon the result. NINTH METHOD. BY THE TRANSITS OF STARS OVER VERTICAL CIRCLES. 192. We may observe the time of transit of a star over any vertical circle with a transit instrument (or with an altitude and azimuth instrument, or common theodolite) ; for when the rota- tion axis is horizontal, the collimation axis will, as the instru- ment revolves, describe the plane of a vertical circle. For any want of horizontality of the rotation axis, or other defects of adjustment, corrections must be applied to the observed time of transit over the instrument to reduce it to the time of transit over the assumed vertical circle. These corrections will be treated of in their proper places in Vol. H. ; and I shall here assume that the observation has been corrected, and gives the clock time T of transit over some assumed vertical circle the azimuth of which is A. The clock correction A T being known, we have the star's hour angle by the formula t T -f A T G 2i*4 LATITUDE. and then, the declination of the star being given, we have the equation [from (14)] cos t sin if tan 3 cos

= dt dA -\ dd cos C sin A sin A cos C sin A from which it appears that sin A and cos must be as great as possible. The most favorable case is, therefore, that in which the assumed vertical circle is the prime vertical, and the star's declination differs but little from the latitude ; for we then have A = 90 and small. Indeed, these conditions not only increase the denominator of the coefficient of dt y but also diminish its numerator, since, by (10), we have cos q cos S = sin C sin

as the unknown quantities, we have, by eliminating them in succession, sin t sin d' cos 3 sin t f cos 8' sin 8 tan A sin > = tan A cos

, M= 90, which gives tan y = tana tan A = - OTt *<'+ '> (358) cos J (*' f) sin ? If the same star is observed twice on the prime vertical, we must have t' + t = 0, since tan A = oo ; and then, tan d tan d /nen ^ tan CP = - = - (359) cos } (f _ t) cos * which follows also from (354) when cot A = ; or, geometrically, from the right triangle formed by the zenith, the pole, and the star, as in Art. 19. If the latitude is given, we can find the time from the transits of two stars over any (undetermined) vertical circle by the second equation of (357), which gives sin [J (f + 9 - Jf] = i ^^ L sin [1 (t' - - Jf] tan d for the observation furnishes the elapsed time, and hence t' t; and this equation determines $(t' -f t), and hence both t and t'. If the latitude and time are given, we can find the declination of a star observed twice on the same vertical circle, by (358). When the observation is made in the prime vertical, this becomes one of the most perfect methods of determining declinations. See Vol. II., Transit Instrument in the Prime Vertical. 194. The following brief approximative methods of deter- mining the latitude may be found useful in certain cases. TENTH METHOD. BY ALTITUDES NEAR THE MERIDIAN WHEN THE TIME IS NOT KNOWN. 195. (A.) By two altitudes near the meridian and the chronometer J imes of the observations, when the rate of the chronometer is known, bat not its correction. Let h, h' = the true altitudes, T, T' = the chronometer times, T= J(T'- T) TWO ALTITUDES NEAR THE MERIDIAN. 297 then, I and t' being the (unknown) hour angles of the observations, we have, by (287), approximately, \ = h -f at* h 1 = h' + at* in which h v is the meridian altitude, and _ 225 sin V cos y cos 8 2 cos \ The mean of these equations is and their difference gives But we have r = j(T' T) == J(f f) in which we suppose the interval T 7 ' T 7 to be corrected for the rate of the chronometer. Hence which, substituted in the above expression for \, gives A, = \(h + A') + r + [*<*-*')]' (360) ar* According to this formula, the mean of the two altitudes is reduced to the meridian by adding two corrections: 1st, the quantity ar 2 , which is nothing more than the common "reduc- tion to the meridian" computed with the half elapsed time as the hour angle ; 2d, the square of one-fourth the difference of the altitudes divided by the first correction. If we employ the form (285) for the reduction, we have h t = i (h -f A') -f Am -f t^- -^- (361) JiW in which co8y>cos<5 _ 2 sin* i T cos A, sin 1" and m is taken from Table V. or log m from Table VI. 298 LATITUDE. EXAMPLE 1. From the observations in the example of Art. 171, I select the following, which are very near the meridian. Obsd. alts. Q 50 5' 42".8 h' 50 7 25 .5 h True alts. Q 50 21' 7".6 50 22 50 .4 25 .7 Chronometer. 23* 50 m 46'.5 _0 37.5 4 55.5 f* -f- /*') 50 21 59 .0 Am = -f 59 .0 log m 1.6778 2d corr. = -f 11 .2 log A 0.0930 A 4 = 50 23 9 .2 log J.m 1.7708 d = 39 36 50 .8 1og[KA-V) ]' 2.8198 <*i = 1 48 9 .2 log 2d corr. 1.0490 v = 37 48 41 .6 EXAMPLE 2. In the same example, the first and last observa- tions, which are quite remote from the meridian, are as follows : Obsd. alts; Q True alts. Q 49 51' 19^3 h = 50 6' 43".7 49 50 24 h' = 50 5 48 .4 J (h h') = 13 .8 Chronometer. 23* 37 m 35' 18 31 T= 20 28 which give Am = 16' 58", and the 2d corr. = 0".2, whence

cos C cos sin C cos A = sin (

8 = C t = the meridian zenith distance; and hence cos

cos S in which A J (/i A') T J = 24.2 * (/i + A') rt = 2".4 r 2 = 1st corr. [J (A A')] 2 = 625 ^ = 2d Merid. alt. Q Dip Semidiameter Refr. and par. = 1 40 25 = 50 10 50 58 = 50 = + 11 4 16 59 16 6 42 = 50 23 7 = 39 = 1 36 48 53 9 ~44 = 37 48 The accuracy of the result depends in a great degree upon the accuracy with which the difference of altitude is obtained. If in the above example this difference had been 2' 40", or V too great, we should have found \(h h') = 40", and the 2d correction = ^jp = 28" : consequently the resulting latitude would have been only 17" too small. Since the same causes of error, such as displacement of the sea horizon by extraordinary refraction, unknown instrumental errors, &c., affect both altitudes alike, the difference will usually be obtained, even at sea, within a quantity much less than 1'. The most favorable case is that THREE ALTITUDES NEAR THE MERIDIAN 309 in which the altitudes are equal and the 2d correction conse- quently, zero. It will be well, therefore, always to endeavor tc obtain altitudes on opposite sides of the meridian. We may also obtain the time, approximately, from the same observation ; for the mean of the hour angles is, Art. 195, which is the apparent time from noon at the middle instant between the observations, (in minutes, r being in minutes, h h' and a being in seconds) ; and this time will be before or after noon according as the second altitude is greater or less than the first. Thus, in our example, \ve have ** 25 ar 2.4 X 4.9 or the apparent time at the middle instant was 2"* 6* after noon. The first observation was, therefore, 2 49 s before noon, and the second 7"' 1* after noon. Fourth Method. By Three Altitudes near the Meridian when the Time is not known. 205. The method of Art. 196 does not require even the rate of the chronometer to be known ; but it is hardly simple enough for a common nautical method. But a very simple method will be obtained if we take three altitudes at equal intervals of time. Suppose the second altitude is observed at the (unknown) time jTfrom the meridian passage, the first at the time T x, the third at the time T+ x; then we have, by (363), h l= =h -f (T a;)' h=h' - a T 2 Subtracting the half sum of the first and third equations from the second, we deduce 310 LATITUDE AT SEA. The difference of the first and third gives which substituted in the second equation gives h r If then we put a for ax 2 , the computation is expressed by tne following simple formulae : a =h'- (377) EXAMPLE. The following three altitudes were observed at equal intervals of time near the meridian : h = 43 8' 20" A' = 43 15' 30" h" = 43 4' 0" J(A-f-A") =43 6 10 a = 9~20 = 560" J (h h") = 1 5 = 65 Hence the reduction of the middle altitude to the meridian is [K* - *")]' = 65 =yf a 560 which added to h' gives A, = 43 15' 38" Instead of equal intervals of time, we may employ equal inter- vals of azimuth (Art. 197), and still reduce the altitudes by (377); but this would be practicable only on land. Fifth Method. By a Single Altitude at a given Time. 206. This is the method of Art. 164, which, however, should be restricted, at sea, to altitudes taken not more than one hour from the meridian, as the time is always imperfectly known and ALTITUDE NEAR THE PRIME VERTICAL. 311 the error in the latitude produced by an error in the time increases very rapidly as the star leaves the meridian and ap- proaches the prime vertical (Art. 166), and the method fails altogether when the star is in the prime vertical. It may, how- ever, sometimes be very important to determine the latitude, at least approximately, when the sun is nearly east or west; and then the following method may be used. Sixth Method. By the change of Altitude near the Prime Vertical. 207. This is the method of Art. 199. In the morning, when the sun has arrived within 1 of the prime vertical as observed with the ship's compass, bring the image of the sun's upper limb, reflected by the sextant mirrors, into contact with the sea horizon, and note the time ; let the sextant reading remain un- changed, and note the time when the contact of the lower limb occurs. In the afternoon, begin with the lower limb. Then, taking the sun's semidiameter = S from the almanac, and put- ting the difference of the chronometer times = r, we have 00 C cos

, which amounts to 3' when ^ = 68 30'. BY TWO ALTITUDES. 313 Chronometer 19* 12" 42' Correction Gr. M. T. Longitude Local M. T. 5 30 19 7 12 p = 1 27' 18" 10 = 87'.3 ~9 7 12 \ogp 1.9410 Sid. T. Gr. noon 23 13 23 log cos t n9.5234 Corr. for 19* 7 m h = 31 10'. 3 8 log p cos t nl.4645 pcoat = + 29.1 S= 8 23 43 a= 1 5 44 t= 7 17 59 = 109 29' 45" = BI 39.1 Eighth Method. By Two Altitudes with the elapsed Time between them. 209. This method may be successfully applied at sea, and is the most reliable of all methods, next to that of meridian or cir- cummeridian altitudes. The formulae fully discussed in Arts. 178 to 183 may be directly applied when the position of the ship has not changed between the observations. But, since there should be a considerable difference of azimuth between the observations, the change of the ship's position in the interval will generally be sufficiently great to require notice. All that is necessary is to apply a correction to the altitude ob- served at the first position of the ship, to reduce it to what it would have been if observed at the second position at the same instant. To obtain this correction, let Z', Fig. 28, be the zenith of the observer at the first observa- tion, S the star at that time ; Z his zenith at the second observation, and S' the star at that time. The first observation gives the zenith distance Z'S, the second the zenith distance ZS'. Joining the points S and S' with the pole P, it is evident that the hour angle SPS' is obtained from the observed difference of the times of observation precisely as if the observer had been at rest. We have, there- fore, only to find ZS in order to have all the data necessary for computing the latitude of Z\>y the general methods. The number of nautical miles run by the ship is the number of minutes in the arc ZZ'; and, since this will always be a suffi- Fig. 28. 314 LATITUDE AT SEA. ciently small number, if we draw ZA perpendicular to SZ', we may regard ZAZ' as a plane triangle, and take ZS = Z'S - AZ' or ZS = Z'S - ZZ' cos ZZ'S (380) The angle ZZ'S is the difference between the azimuth of the star at the first observation and the course of the ship; and this azimuth is obtained with sufficient accuracy by the compass.* Employing the zenith distance thus reduced and the other data as observed, the latitude computed by the general method will be that of the second place of observation. In the same manner we can reduce the second zenith distance to the place of the first, and then the latitude of the first place will be found. 210. The problem of finding the latitude from two altitudes is most frequently applied at sea in the case where the sun is the observed body, the observation of the meridian altitude having been lost. The computation is then best carried out by the formula (315), (316), (317), (318), employing for S the mean declination of the sun, i.e. the declination at the middle time between the two observations, and then applying to the result- ing latitude the correction A^> found by the formula (323). To save the navigator all consideration of the algebraic signs in computing this correction, it will be sufficient to observe the following rule : 1st. When the second altitude is the greater, apply this correction to the computed latitude as a northing when the sun is moving towards the north, and as a southing when the sun is moving towards the south ; 2d. When the first altitude is the greater, apply the correction as a southing when the sun is moving towards the north, and as a northing when the sun is moving towards the south. * If we wish a more rigorous process, we must consider the spherical triangle ZZ'S, in which we have the observed zenith distance Z'S = ( '), the required zenith distance ZS = C, the distance run by the ship Z'Z = d, the difference of the star's azimuth and the ship's course ZZ'S' a, and hence cos C = cos ' cos d -f- sin f ' sin d cos a which developed gives = f ' d cos a -f- J d 1 sin 1" cot ' sin" a the last term of which expresses the error of the formula given in the text. BY TWO ALTITUDES. 315 If the computer chooses to neglect this correction, he should employ the mean declination only when the middle time is nearer to noon than the time of the greater altitude. In all other cases he should employ the declination for the time of the greater altitude (Art. 183). 211. DOUWES'S method of "double altitudes."* This is a brief method of computing the latitude from two altitudes of the sun, which, though not always accurate, is yet sufficiently so when the interval between the observations is not more than 1 A , and one of them is less than 1 A from the meridian. Let h and h' be the true altitudes, S the declination at the middle time, 7" and T' the chronometer times of the observa- tions, t and t' the hour angles. The elapsed apparent time ^ is found from the times Tand T' by (322), but it is usually suffi- cient to take l=T'T. We then have t'=t-\-l\ and by the first of (14) we have sin h = sin

l If we put t = the middle time, or we deduce sin h sin h' 2 sm f = - (381) cos

>+ 2^+ r"+ r'r")- * and we find the erroneous longitude X' = a' (6) = J /?rY' Hence the error by simple interpolation, commencing with the station _B, is ctt." = PT'T" ; and the error in the mean of the two longitudes is an error which disappears altogether when the intervals r and r" are equal. Since the voyages are of very nearly equal duration, it follows that by computing the longitude, as proposed by STRUVE, commencing alternately at the two stations, the final result will be free from the effect of any regular acceleration or retardation of the chronometers. EXAMPLE. From the "Expedition Chronometrique" we take the following values for the chronometer "Hauth 31," being the combination next following after that given in the example of the preceding article, commencing now with the station J9, or Altona : At Altona (5), t = May 26, 10.72 b = 1 14 36'.77 " Pulkova 01), H tL " 31, .00 a = -f 7 9 .58 " Pulkova (A), f June 3, 5 .62 a' = -f 7 19 .36 "Altona (),<"'= " 7,20.52 b' = I 14 0.35 Here T = 4* 13.28 = 4".553 b' b = + 36'.42 T"=\ 14 .90 = 4.621 a ' a=+ 9.78 . 9Q4 4.553 + 4.621 9.174 14" 36'.77 13.22 (6) = 1 14 23 J>5 a = + Q 1 9.58 jl = a (6) = + 1 21 33.13 The mean of this result and that of Art. 217 is X = l h 21 m 33'.02. BY CHRONOMETERS. 329 219. Eelalive weight of the longitudes determined in different voyages by the same chronometer. From the above it appears that the problem of finding the longitude by chronometers is one of interpolation. If the irregularities of the chronometer are regarded as accidental, the mean error of an interpolated value of the correction ma y be expressed by the formula* where r and r' have the same signification as in the preceding article, and e is the mean (accidental) error in a unit of time. The weight of such an interpolated value of the correction, and, therefore, also the weight of a value of the longitude deduced from it, is inversely proportional to the square of this error, and may, therefore, be expressed under the form where k is a constant arbitrarily taken for the whole expedition, so as to give p convenient values, since it is only the relative weights of the different voyages which arc in question. But if the chronometer variations are no longer accidental, but follow some law though unknown, a special investigation may serve to give empirically a more suitable expression of the weight than the above. Thus, according to STRUVE'S investiga- tions in the case of certain clocks, the weight of an interpolated value of the correction for these clocks could be well expressed by the formulaf But even this expression he found could not be generally applied ; and he finally adopted the following form for the chronometric expedition : in which T is the duration of an entire voyage, including the * See Vol. II., "Chronometer." f Expedition CJtron., p. 102. 330 LONGITUDE. time of rest at one of the stations, r, r" are the travelling times of the voyage to and from a station, and K is an arbitrary constant. Although this is but an empirical formula, it represents well the several conditions of the problem. For,/rs*, the weight of a resulting longitude must decrease as the length of the voyage increases ; and, second, it must become greater as the difference between the two travelling times r, r" decreases, since (as is shown in Vol. II., " Chronometer") an interpolated value of a clock correction is probably most in error for the middle time between the two instants at which its corrections are given. 220. Combination of results obtained by the same chronometer, according to their weights. Let X', /", X"' be the several values of the longitude found by the same chronometer, according to the method of Arts. 217 and 218 ; and p', p", p f " their weights by formula (389) (or any other formula which may be found to represent the actual condition of the voyages) ; then, according to the method of least squares, the most probable value of the longitude by this chronometer is> L = * ^ * ^ r ~ -r (390) P' + /' + P'" + and if the difference between this value and each particular value be found, putting A' L = v', X'L = v", /T" L = v'", &c. n = the number of values of /I, c = the mean error of L, r = the probable error of L, then we shall have / r ruini (391) where [p~\ denotes the sum of p', p", &c., and [pvv] the sum of pW, p"v"v", &c. 221. Combination of the results obtained by different chronometers, according to their weights. The weights of the results by different BY CHRONOMETERS. 331 chronometers are inversely proportional to the squares of theii mean errors. The weight P of a longitude L will, therefore, be expressed generally by in which A' is arbitrary. For simplicity, we may assume k 1, and then by the above value of e we shall have p== If, then, Z/', L", L'" ..... are the values found by the several chronometers by (390), P', P", P'" ..... their weights by (392), the most probable final value of the longitude is P'L' + P"L"+P"'L'" + ..... L = P' + P" + P'" + ..... Then, putting L' L =V, L" L =V", L'" L Q =V" &c. N the number of values of L, E = the mean error of L , R = the probable error of L , we have 222. I propose to illustrate the preceding formulae by applying them to two chronometers of STRUVE'S expedition, namely, "Dent 1774" and "Hauth 31." In the following table the longitudes found by beginning at Pulkova are marked P, those found by beginning at Altona are marked A, and the numeral accent denotes the number of the voyage. The weights p in the second column are as given by STRUVE, who computed them by the formula (389), taking K= 34560 (the intervals T, r, r" being in hours), which is a convenient value, as it makes the weight of a voyage of nearly mean duration equal to unity ; namely, for T= 288*, T = T' = 120*. If we express T, r, r", in days, we' takt (24)' LONGITUDE. and we shall have STRUVE'S values of p by the formula P = (895) T\/TT" Thus, for the first voyage, we have, from the data in the example of Art. 217, T = t'" t = 11" 2 A .46 = 1K103 T =-. 5 d .047 T" = 4 d .553 whence, by (395), AA = 1.18 11.103 ^(5.047 X 4.553) The values of L' and L" are found by (390). In applying this formula, it is not necessary to multiply the entire longitudes by their weights, but only those figures which differ in the several values. Thus, by "Dent 1774" we have L > = - 30' 2-.51 X 1.10 + 2'.83 X 1-02 + 2-.09 X l.U + &c. 1.10 + 1.02 + 1.14 + &c. = 1* 21 m 30' + 2'.46 Weight. Longitudes by Longitudes by Chronometer V pvv Chronometer V pvv P Dent 1774. Hauth 31. P' 1.13 1* 21'" 32'.91 + 0-.30 0.102 A' 1.06 33.13 + 0.52 0.287 pii 1.10 1* 21"> 32'.51 + 0-.05 0.003 33.36 + 0.75 0.619 A" 1.02 32.83 + 0.37 0.140 33.12 + 0.51 0.265 piii 1.14 32.09 0.37 0.156 32.55 0.06 0.004 AM 1.05 32.25 0.21 0.046 31 .56 1 .05 1.158 piv 1.19 31.69 0.77 0.706 32.70 + 0.09 0.010 A iv 0.96 32.77 + 0.31 0.092 34.16 + 1.55 2.306 P" 1.09 32.79 + 0.33 0.119 32.23 0.38 0.157 A* 0.80 32.54 + 0.08 0.005 31.65 0.96 0.737 pvi 1.00 32.94 + 0.48 0.230 33.38 + 0.77 0.593 A vl 1.10 31 .93 0.53 0.309 31.97 0.64 0.451 pvll 1.20 32.34 0.12 0.017 33.16 + 0.55 0.363 Avll 1.09 32.95 + 0.49 0.262 31 .78 .83 0.751 plti 0.76 31.86 0.60 0.274 30.92 -1.69 2.171 M* 0.41 33.77 + 1.81 0.704 L' = 1* 21" 3.T.46 [pvv\ == 3.063 " = 1* 21" 32'. 61 [pvv^ 9.974 n = 14 [/>] = 13.91 n = 15 [p] = 15.69 P' 13 X 13-91 14 X 15.69 00 02 3.063 9.974 BY CHRONOMETERS. 333 Combining these two results, we have, by (393), L. = I" 21- 32- + +OMH X a = V 21 " S2 ' 501 with the probable error, by (394), R = 0'.067 This agrees very nearly with the final result from the sixty-eight chronometers. 223. In the preceding method, the sea rate is inferred from two comparisons of the chronometer made at the same place before and after the voyages to and from the second place ; and the correction of the chronometer on the time of the first place at the instant when it is compared with the time of the second place is interpolated upon the theory that the rate has changed uniformly. This theory is insufficient when the temperature to which the chronometer is exposed is not constant during the two voyages, or nearly so. I shall, therefore, add the method of introducing the correction for temperature in cases where circumstances may seem to require it. According to the experience of M. LIEUSSON, the rate m of a chronometer at a given temperature $ may be expressed by the formula (see Vol. II., "Chronometer") ro = m + k (0 # ) 2 k't (396) in which $ is the temperature for which the balance is compen- sated, m the rate determined at that temperature at the epoch i 0, t being the time from this epoch for which the rate m is required, k the constant coefficient of temperature, and k' that of acceleration of the chronometer resulting from thickening of the oil or other gradual changes which are supposed to be pro- portional to the time. It is evident that, since every change of temperature produces an increase of w, the term k($ $ a ) 2 will not disappear even when the mean value of & is the same as # . It is necessary, therefore, to determine the sum of the effects of all the changes. Let us, therefore, determine the accumulated rate for a given period of time r. Let m be the rate at the middle of this period, in which case we have in the formula t = 0. A strict theory requires that 334 LONGITUDE. we should know the temperature at every instant ; but, in default of this, let us assume that the period r is divided into sufficiently small intervals, and that the temperature is observed in each. Let us suppose n equal intervals whose sum is r, and denote the observed values of & by tf (1) , # (2) , # (3) ____ #<">. The rate in the 1st interval is [m + k (# (1) ) 2 ] X n 2d " [m + A-(#< 2 >-# ) 2 ] X ^ &c. &c. in the nth interval is \_m -f k (*"> *)] X and the accumulated rate in the time r is the sum of these quantities, where -T H ($ $ ) 2 denotes the sum of the n values of ($ $ ) 2 . To make this expression exact, we should have an infinite number of infinitesimal intervals, or we must put - = rfr, and substitute 71 the integral sign I for the summation symbol 2: thus, the exact expression for the whole rate in the time r is m T + kfJ(*-WdT (397) This integral cannot be found in general terms, since # cannot be expressed as a function of r ; but we can obtain an approxi- mate expression for it, as follows. Let # t be the mean of all the observed values of # ; then we have s n (* - *)= s n [(*, - * ) + (* - w in which ^ # is constant, and, therefore, for n values we have I n (#, 2 -f ^ (* - *:)' BY CHRONOMETERS. 335 Hence, also, or, for an infinite value of ft, /^ (* - *)' * - r (#, - *)* + / o T (* - Thus, the required integral depends upon the integral J ($ $i) 2 <&", which may be approximately found from the observed values of # by the theory of least squares. For, if we treat the values of $ $! as the errors of the observed values of $, and denote the mean error (according to the received acceptation of that term in the method of least squares) by e, we have ,=* in which n is the actual number of observed values of #. If we assume that a more extended series of values, or indeed an infi- nite series, would exhibit the same mean error (which will be the more nearly true the greater the number n), we assume the general relation *,(*-* I )=(J!r-i) in which N is any number. Hence, also, , l N and, making N infinite, / O r (*-*i)'* or [w c +k(9 l t? ) 2 + fo']r (400) from which it appears that m -\- k(& l $ ) 2 + ke* is the mean rate in a unit of time for the interval r, w? being the rate at the middle of the interval for a temperature $ = $ . For any subse- quent interval r', we must, according to (396), replace m by m k't, t being the interval from the middle of r to the middle of r'. 336 LONGITUDE. Now, let us suppose that the chronometer correction is obtained by astronomical observations at the station A, at the times 7^ and T v before starting upon the voyage, and again after reaching the station B, at the times T 3 and T 4 , these times being all reckoned at the same meridian. Let a v a 2 , a 3 , a v be the observed corrections, and put T 3 T t = T, T s T a = r', T 4 T 3 = T" so that r and T" are the shore intervals and r' the sea interval. Let the adopted epoch of the rate m be the middle of the sea interval r' ; then, by (400), with the correction k't, the accumu lated rates in the three intervals are a s - a, = [m + k (#/ - * )^ + he" ] r' \ (401) in which ^, ??/, ??/' are the mean temperatures in the intervals r, r', r", and e, e', e" are found by the formula (398). These three equations determine the three unknown quantities m m k', and L If we put o we have, from the first and third equations, which substituted in the second equation gives L If, however, we prefer to compute the approximate longitude without con- sidering the temperatures, and afterwards to correct for tempe- rature, we shall have TERRESTRIAL SIGNALS. S37 These formulae apply to a voyage in either direction ; but in the case of a voyage from west to east they give A with the negative sign. The term Jfc'(r" r) IT' in the first equation of (402) will not be rigorously obtained if the temperatures are neglected ; but it is usually an insensible term in practice, as r" and r are made as nearly equal as possible, and k' is always very small. In combining the results of different chronometers employed in the same voyage, the weight of each may be assigned accord ing to the regularity of the chronometer as determined from its observed rates from day to day.* SECOND METHOD. BY SIGNALS. 224. Terrestrial Signals. If the two stations are so near to each other that a signal made at either, or at an intermediate station, can be observed at both, the time may be noted simultaneously by the clocks of the two stations, and the difference of longitude at once inferred. The signals may be the sudden disappearance or reappearance of a fixed light, or flashes of gunpowder, &c. If the places are remote, they may be connected by interme- diate signals. For example : suppose four stations, A, B, C, JD, chosen from east to west, the first and last being the principal stations whose difference of longitude is required. At the in- termediate stations B, C let observers be stationed with good chronometers whose rates are known. Let signals be made at three points intermediate between A and B, B and C, C and _D, respectively. The signals must, by a preconcerted arrangement, be made successively, and so that the observers at the interme- diate stations may have their attention properly directed upon the appearance of the signal. If, then, at the first signal the observers at A and B have noted the times a and b; at the * Besides the papers already referred to, see the Report of the Superintendent of the U. S. Coast Survey for 1857, p. 314. VOL. I. 22 338 LONGITUDE. second signal the observers at B and C tlie times b' and c; at the third signal the observers at G^and D the times c' and d; it is evident that the time at A when the third signal is made is a + (6' b) + (c f c), at which instant the time at D is d: hence the difference of longitude of A and D is A = a -f (b' b~) -f- (c' c) d (403) and so on for any number of intermediate stations. It is re- quired of the intermediate chronometers only that they should give correctly the differences b' 6, c' c, for which purpose only their rates must be accurately known. The daily rates are obtained by a comparison of the instants of the signals on suc- cessive days. Small errors in the rates will be eliminated by making the signals both from west to east and from east to west, and taking the mean of the results. The intervals given by the intermediate chronometers should, of course, be reduced to sidereal intervals, if the clocks at the extreme stations are regulated to sidereal time. EXAMPLE. From the Description Geometrique de la France (PUISSANT). On the 25th of August, 1824, signals were observed between Paris and Strasburg as follows : Paris. Intermediate Stations. Strasburg. A ^ *^^___-'^ B ** "^ ^^> C D > 6" 20'.3 8* 49" 48'.2 8 54 10.8 9* 16" 0-.2 9 30 37.8 19 46" 51'.4 The correction of the Paris clock on Paris sidereal time was 36'.2 ; that of the Strasburg clock on Strasburg sidereal time was 27*.7. The chronometers at B and Cwere regulated to mean time, and their daily rates were so small as not to be sensible in the short intervals which occurred. We have V b= 4-22'.6 c' c = 14 37.6 Mean interval =19 .2 Eed. to sid. int. = -j- 3 .1 Sid. interval = 19 L3 CELESTIAL SIGNALS. 339 Paris clock 19* G m 2O.8 Strasburg clock 19 46" 51'. 4 Correction 3G .2 Correction 27 .7 Paris sid. time 19 5 44 .1 Strasburg sid. time 19 46 23 .7 Sid. interval +19 3 .3 Paris sid. time of the ) lt signal [192447.4 Strasbunr do. 19 46 23 .7 A = 0* 21 36'.3 In the survey of the boundary between the United States and Mexico, Major "W. H. EMORY, in 1852, employed flashes of gun- powder as signals in determining the diff. of long, of Frontera and San Elciario.* The signals may be given by the heliotrope of GAUSS, by which an image of the sun is reflected constantly in a given direction towards the distant observer. Either the sudden eclipse of the light, or its reappearance, may be taken as the signal ; the eclipse is usually preferred. Among the methods by terrestrial signals may be included that in which the signal is given by means of an electro-tele- graphic wire connecting the two stations; but this important and exceedingly accurate method will be separately considered below. 225. Celestial Signals. Certain celestial phenomena which are visible at the same absolute instant by observers in various parts of the globe, may be used instead of the terrestrial signals of the preceding article : among these we may note a. The bursting of a meteor, and the appearance or disappear- ance of a shooting star. The difficulty of identifying these objects at remote stations prevents the extended use of this method. b. The instant of beginning or ending of an eclipse of the moon. This instant, however, cannot be accurately observed, on account of the imperfect definition of the earth's shadow. A rude approximation to the difference of longitude is all that can be expected by this method. c. The eclipses of Jupiter's satellites by the shadow of that planet. The Greenwich times of the disappearance of each * Proceedings of 8th Meeting of Am. Association, p. 64. 340 LONGITUDE. satellite, and of its reappearance, are accurately given in the Ephemeris : so that an observer who has noted one of these phenomena has only to take the difference between this observed local time of its occurrence and the Greenwich time given in the Ephemeris, to have his absolute longitude. With telescopes of different powers, however, the instant of a satellite's disappear- ance must evidently vary, since the eclipse of the satellite takes place gradually, and the more powerful the telescope the longer will it continue to show the satellite. If the disappearance and reappearance are bath observed w r ith the same telescope, the mean of the results obtained will be nearly free from this error. The first satellite is to be preferred, as its eclipses occur more frequently and also more suddenly. Observers who wish to deduce their difference of longitude .by these eclipses should use telescopes of the same power, and observe under the same atmospheric conditions, as nearly as possible. But in no case can extreme precision be attained by this method. d. The occultations of Jupiter's satellites by the body of the planet. The approximate Greenwich times of the disappearance behind the disc, and the reappearance of each satellite, are given in the Ephemeris. These predicted times serve only to enable the observers to direct their attention to the phenomenon at the proper moment. e. The transits of the satellites over Jupiter's disc. The ap- proximate Greenwich times of "ingress" and "egress," or the first and last instants when the satellite appears projected on the planet's disc, are given in the Ephemeris. /. The transits of the shadows of the satellites over Jupiter's disc. The Greenwich times of "ingress" and "egress" of the shadow are also approximately given in the Ephemeris. Among the celestial signals we may include also eclipses of the sun, or occultations of stars and planets by the moon, or, in general, the arrival of the moon at any given position in the heavens; but, in consequence of the moon's parallax, these eclipses and occultations do not occur at the same absolute in- stant for all observers, and, in general, the moon's apparent position in the heavens is affected by both parallax and refrac- tion. The methods of employing these phenomena as signals, therefore, involve special computations, and will be hereafter treated of. See the general theory of eclipses, and the method of lunar distances BY THE ELECTRIC TELEGRAPH. 341 THIRD METHOD. BY THE ELECTRIC TELEGRAPH. 226. It is evident that the clocks at two stations, A and B, may be compared by means of signals communicated through an electro-telegraphic wire which connects the stations. Sup- pose at a time T by the clock at A, a signal is made which is perceived at B at the time T' by the clock at that station. Let A T and A T' be the clock corrections on the times at these sta- tions respectively (both being solar or both sidereal). Let x be the time required by the electric current to pass over the wire; then, A being the more easterly station, we have the difference of longitude A by the formula A!T) (T'+ATO+tf^ + a: Since x is unknown, we must endeavor to eliminate it. For this purpose, let a signal be made at B at the clock time T", which is perceived at A at the clock time T'" ; then we have /I = ( T'" + A T'"} ( T"+ A T") x = l. 2 x In these formulae ^ and A 2 denote the approximate values of the difference of longitude, found by signals east-west and west-east respectively, when the transmission time x is disregarded; and the true value is Such is the simple and obvious application of the telegraph to the determination of longitudes; but the degree of accuracy of the result depends greatly more than at first appears upon the manner in which the signals are communicated and received. Suppose the observer at A taps upon a signal key* at an exact second by his clock, thereby producing an audible click of the armature of the electro-magnet at B. The observer at B may not only determine the nearest second by his clock when he hears this click, but may also estimate the fraction o a second; and it would seem that we ought in this way to be able to deter- mine a longitude within one-tenth of a second. But, before even this degree of accuracy can be secured, we have yet to eliminate, or reduce to a minimum, the following sources of error: * See Vol. II., " Chronograph," for the details of the apparatus here alluded ta. 342 LONGITUDE. 1st. The personal error of the observer who gives the signal; 2d. The personal error of the observer who receives the signal and estimates the fraction of a second by the ear; 3d. The small fraction of time required to complete the galvanic circuit after the finger touches the signal key; 4th. The armature time, or the time required by the armature at the station where the signal is received, to move through the space in which it plays, and to give the audible click; 5th. The errors of the supposed clock corrections, which involve errors of observation, and errors in the right ascensions of the stars employed. For the means of contending successfully with these sources of error we are indebted to our Coast Survey, which, under the superintendence of Prof. Bache, not only called into existence the chronographic instruments, but has given us the most effi- cient method of using them. The "method of star signals," as it is called, was originally suggested by the distinguished astro- nomer Mr. S. C. Walker, but its full development in the form now employed in the Coast Survey is due to Dr. B. A. Gould. 227. Method of Star Signals. The difference of longitude be- tween the two stations is merely the time required by a star to pass from one meridian to the other, and this interval may be measured by means of a single clock placed at either station,* but in the main galvanic circuit extending from one station to the other. Two chronographs, one at each station, are also in the circuit, and, when the wires are suitably connected, the clock seconds are recorded upon both. A good transit instrument is carefully mounted at each station. When the star enters the field of the transit instrument at A (the eastern station), the observer, by a preconcerted signal with his signal key, gives notice to the assistants at both A and , who at once set the chronographs in motion, and the clock then records its seconds upon both. The instants of the star's tran- sits over the several threads of the reticule are also recorded upon both chronographs by the taps of the observer upon his signal key. When the star has passed all the threads, the ob- * The clock may, indeed, be at any place which is in telegraphic connection with the two stations whose difference oflongitude is to be found. BY THE ELECTRIC TELEGRAPH. 343 server indicates it by another preconcerted signal, the chrono- graphs are stopped, and the record is suitably marked with date, name of the star, and place of observation, to be subsequently identified and read off accurately by a scale. When the star arrives at the meridian of -6, the transit is recorded in the same manner upon both chronographs. Suitable observations having been made by each observer to determine the errors of his transit instrument and the rate of the clock, let us put T l the mean of the clock times of the eastern transit of the star overall the threads, as read from the chrono- graph at A, T a = the same, as read from the chronograph at B, 7y = the mean of the clock times of the western transit of the star over all the threads, as read from the chrono- graph at A, T a ' = the same as read from the chronograph at 5, e, e' = the personal equations of the observers at A and 2> respectively, r,r'= the corrections of T, and T,' (or of T,and T 9 ') foi the state of the transit instruments at A and S, or the respective "reductions to the meridian" (Vol. II., Transit Inst.), 9T = the correction for clock rate in the interval T 7 ,' T lt x = the transmission time of the electric current between A and B, X. = the difference of longitude ; then it is easily seen that we have, from the chronographic records at A, and from the chronographic records at B, * = I 7 ,' + 5T+ r' + 4 + x (T a + T + e) and the mean of these values is ' = H ( *V + 2V) + O - ( T, + r a ) + r] + 8 T + e> - e (404) which we may briefly express thus : /I = ^ -f- e f e 344 LONGITUDE. in which Aj the approximate difference of longitude found by the exchange of star signals, when the personal equations of the observers are neglected. This equation would be final if e' e, or the relative personal equation of the observers, were known : however, if the observers now exchange stations and repeat the above process, we shall have, provided the relative personal equation is constant, /I = ; 3 -f e e' In which X 2 is the approximate difference of longitude found as before ; and hence the final value is I have not here introduced any consideration of the armature time, because it affects clock signals and star signals in the same manner; and therefore the time read from the chronographic fillet or sheet is the same as if the armature acted instanta- neously.* It is necessary, however, that this time should be constant from the first observation at the first station to the last observation at the second, and therefore it is important that no changes should be made in the adjustments of the apparatus during the interval. As the observer has only to tap the transits of the star over the threads, the latter may be placed very close together. The reticules prepared by Mr. W. WURDEMANN for the Coast Survey have generally contained twenty-five threads, in groups or "tal- lies" of five, the equatorial intervals between the threads, of a group being 2*. 5, and those between the groups 5* ; with an ad- ditional thread on each side at the distance of 10* for use in ob- servations by "eye and ear." Except when clouds intervene and render it necessary to take whatever threads may be avail- able, only the three middle tallies, or fifteen threads, are used. The use of more has been found to add less to the accuracy of a * Dr. B. A. GOULD thinks that the armature time varies with the strength of the battery and the distance (and consequent weakness) of the signal ; being thus liable to be confounded with the transmission time. The effect upon the difference of longitude will be inappreciable if the batteries are maintained at nearly the same strength BY THE ELECTRIC TELEGRAPH. 345 determination than is lost in consequence of the greater fatigue from concentrating the attention for nearly twice as long. A large number of stars may thus be observed on the same night ; and it will be well to record half of them by the clock at one station, and the other half by the clock at the other station, upon the general principle of varying the circumstances under which several determinations are made, whenever practi- cable, without a sacrifice of the integrity of the method. For this reason, also, the transit instruments should be reversed during a night's work at least once, an equal number of stars being observed in each position, whereby the results will be freed from any undetermipjd errors of collimation and inequality of pivots. Before and after the exchange of the star signals, each observer should take at least two circumpolar stars to determine the instrumental constants upon which r and -' depend. This part of the work must be carried out with the greatest precision, employing only standard stars, as the errors of r and r' come directly into the difference of longitude. The right ascensions of the "signal stars" do not enter into the computation, and the result is, therefore, wholly free from any error in their tabular places : hence any of the stars of the larger catalogues may be used as signal stars, and it will always be possible to select a sufficient number which culminate at moderate zenith distances at both stations, (unless the difference of latitude is unusually great), so that instrumental errors will have the minimum effect. A single night's work, however, is not to be regarded as con- clusive, although a large number of stars may have been ob- served and the results appear very accordant ; for experience shows that there are always errors which are constant, or nearly so, for the same night, and which do not appear to be represented in the corrections computed and applied. Their existence is proved when the mean results of different nights are compared. Moreover, it is necessary to interchange the observers in Border to eliminate their personal equations. The rule of the Coast Survey has been that when fifty stars have been exchanged on not less than three nights, the observers exchange stations, and fifty stars are again exchanged on not less than three nights. The observers should also meet and determine their relative personal equation, if possible, before and after each series, as it may prove that this equation is not absolutely constant. 846 LONGITUDE. Before entering upon a series of star signals, each observer will be provided with a list of the stars to be employed. The preparation of this list requires a knowledge of the approximate difference of longitude in order that the stars may be so selected that transits at the two stations may not occur simultaneously. EXAMPLE. For the purpose of finding the difference of longi- tude between the Seaton Station of the U. S. Coast Survey and Raleigh, a list of stars was prepared, from which I extract the following for illustration. The latitudes are Seaton Station (Washington)

1 and L 2 are A l + e and A 2 + e, the error of the Ephemeris being supposed to be sensibly constant for a few hours ; but their difference is (A 9 + e)-(A l + e) = A.-A l so that the computed difference of right ascension is the same as if the Ephemeris were correct. If now the observed differ- ence a 2 a t is the same as this computed difference, the as- sumed difference of longitude, or L 2 L v is correct;* but, if this is not the case, put r ^(a a - 0l )-(^ a -A) (^08) and kL = the correction of the uncertain longitude, which we will suppose to be _Z/ a then f is the change of the right ascension while the moon is describing the small arc of longitude AZ/ ; and for this small difference we may apply the solution of the preceding article, so that we have at once &L = (in hours) (409) H or QAAA A = r X - (in seconds) (409*) H * It should be observed, however, that one of the assumed longitudes must be nearly correct, for it is evident that the sanie difference of right ascension will not exactly correspond to the same difference of longitude if we increase or decrease both longitudes by the same quantity. BY MOON CULMINATIONS. 355 In which the value of H must be that which belongs to the uncertain meridian Z/ 2 , or, more strictly, H must be taken for the mean longitude between Z/ 2 and L 2 + &L; but, as &L is generally very small, great precision in H is here superfluous. However, if in any case &L is large, we can first find H for the meridian Z/ 2 , and with this value an approximate value of AZ/; then, interpolating H for the meridian Z/ 2 + AZ/, a more correct value of A will be found.* EXAMPLE. The following observations were made May 15, 1851, at Santiago and Greenwich : Object. Santiago. Greenwich. # Librae 15*46" 3'.37 15* 45" 22'.37 Moon II Limb 16 21 36 .84 16 9 39 .41 B.A. C. 5579 16 33 40.12 16 32 59.17 We assume here, as in the preceding example, for Santiago Z/ 2 = 4* 42'" 19*, and for Greenwich we have L { = 0. The places of the stars being as in the preceding article, we find for Greenwich, o, == 16* 9" 39'.54 Santiago, <* a = 16 20 55 .99 a a a, = 11 16 .45 The computed right ascension for Greenwich is in this case simply that given in the Ephemeris for May 15 ; the increase to the meridian 4 A 42" 1 19*.0 has been found in our example of in- terpolation, Art. 71, to be and hence r = + 0-.61 We find, moreover, for the longitude 4* 42" 1 19*, H = 143'.77 whence = + 15--28 By these observations we have, therefore, Longitude of Santiago = 4* 42" 34'.28 *This method of reducing moon culminations was developed by WALKER, Tram- actions of the American Philosophical Society, new series, Vol. V. 356 LONGITUDE. 234. Reduction of moon culminations by the hourly Ephemeris. The method of reduction given in the preceding article is per- fectly exact; but the interpolation of the moon's place to fourth differences is laborious. The hourly Ephemeris, however, requires the use of second differences only. The sidereal time of the transit of the moon's centre at the meridian L^ is = the observed right ascension of the centre = a r If then we put Tj = the mean Greenwich time corresponding to o, as found by the hourly Ephemeris, 0j = the Greenwich sidereal time corresponding to T v we have at once, if the Ephemeris is correct, L 1 = Q i o, (410) This, indeed, was one of the earliest methods proposed, but was abandoned on account of the imperfection of the Ephemeris. The substitution of corresponding observations, however, does not require a departure from this simple process; for we shall have in the same manner, from the observations made at another meridian (which may be the meridian of the Ephemeris), A = 9 -a 3 and hence A = L t - L t = (0, - e a ) - (a, - ,) (411) and it is evident that the difference (0 t ) of the Greenwich times will be correct, although the absolute right ascension of the Ephemeris is in error, provided the hourly motion is correct. The correctness of the hourly motion must be assumed in all methods of reducing moon culminations ; and in the present state of the lunar theory there can be no error in it which can be sensible in the time required by the moon to pass from one meridian to another. In this method a is the right ascension of the moon's centre at the instant of the transit of the centre ; which may be de- duced from the time of transit of the limb by adding or sub- tractingthe " sidereal time of semidiameter passing the meridian," given in the table of moon culminations in the Ephemeris.* To find T^ corresponding to a t , we may proceed as in Art. 64, * If we wish to be altogether independent of the moon-culminating table, we can compute the sidereal time of semidiameter passing the meridian by the formula (see Vol. II;, Transit Instrument), BY MOON CULMINATIONS. 35t or as follows: Let T and T -\- 1* be the two Greenwich hours between which a t falls, and put Aa = the increase of right ascension in l m of mean time at the time r o , da, = the increase of Aa in 1*, a = the right ascension of the Ephemeris at the hour T OJ then, by the method of interpolation by second differences, we have T: ; ](^) iii which the interval 7\ T is supposed to be expressed in seconds. This gives 60 (,-.) 1 ~ , * T,-T Ao -\ r 2 3600 and in the second member an approximate value of T x may be used, deduced from the local time of the observation and an approximate longitude. A still more convenient form, which dispenses with finding an approximate value of T v is obtained as follows : Put T^T 9 +x then we have 8 15(1 _ A) cos 6 in which S = the moon's semidiameter, A = the increase of the moon's right ascen- sion in one sidereal second, and 6 = the moon's declination, which are to be taken for the Greenwich time of the observation, approximately known from the local time and the approximate longitude. Or we may apply to the sidereal time (= # x ) of the transit of the limb the quantity S 15 cos 6 and the resulting a t $, =t T J 5 S sec r We then find the Greenwich time 9j corre- sponding to Uj as in the text, and we have 858 LONGITUDE. 60 (a, a ) 60 fo-.,)/ a? a.v-1 Aa \ ^ 7200 Aa / or, with sufficient accuracy, 7200 * Putting then _ . (412) 7200 AO we have, very nearly, x = x' + x" (413) As a practical rule for the computer, we may observe that x" will be a positive quantity when AOC is decreasing, and negative when Aa is increasing. The method of this article will be found particularly conve- nient when the observation is compared directly with the Ephemeris, the latter being corrected by the following process. See page 362. 235. Peirce's method of correcting the Ephemeris.* The accuracy of the longitude found by a moon culmination depends upon that of the observed difference of right ascension. When this difference is obtained from two corresponding observations, if the probable errors of the observed right ascensions at the two meridians are e l and e 2 , the probable error of the difference will be = i/( i 2 H~ 2 2 )- [Appendix]. But if instead of an actual ob- servation at Z/ 2 we had a perfect Ephemeris, or e 2 = 0, the probable error of the observed difference would be reduced to L ; and if we have an Ephemeris the probable error of which is less than that of an observation, the error of the observed difference is reduced. At the same time, we shall gain the additional advantage that every observation taken at the meridian whose longitude is required will become available, even when no corre- sponding observation has been taken on the same day ; and * Report of the Superintendent of the U. S Coast Survey for 1854, Appendix, p. 115*. BY MOON CULMINATIONS. 369 experience has shown that, when we depend on corresponding observations alone, about one-third of the observations are lost. The defects of the lunar theory, according to PEIRCE, are involved in several terms which for each lunation may be principally combined into tw T o, of which one is constant and the other has a period of about half a lunation, and he finds that for all practical purposes we may put the correction of the Ephemeris for each semi-lunation under the form X=A + Bt+CP (414) in which J., B, and C are constants to be determined from the observations made at the principal observatories during the semi-lunation, and t denotes the time reckoned from any assumed epoch, which it will be convenient to take near the mean of the observations. The value of t is expressed in days ; and small fractions of a day may be neglected. Let Oj, a a , a 3 , &c. =the right ascension observed at any observa- tory at the dates t v t a , t a , &c., from the assumed epoch, o/ja/jttg'j&c. = the right ascension at the same instant found from the Ephemeris, and put ' n a = a * ~ a '> w a = a s tt s'> &c - then Wp n v n y &c. are the corrections which (according to the observations) the Ephemeris requires on the given dates, and hence we have the equations of condition A + Bt^ + Of* n, = A + Bt, + CtJ n a =0 A -f Bt a -f Ct^ n 3 = &c. In order to eliminate constant errors peculiar to any observa- tory, when the observation is not made at Greenwich, the ob- served right ascension is to be increased by the average excess for the year (determined by simultaneous observations) of the right ascensions of the moon's limb made at Greenwich above those made at the actual place of observation. 360 LONGITUDE. If now we put m = the number of observations the number of equations of condition, T the algebraic sum of the values of t, T a == the sum of the squares of t, T a the algebraic sum of the third powers of t, T 4 = the sum of the fourth powers of t, N = the algebraic sum of the values of n, N l = the algebraic sum of the products of n multiplied by t, N a = the algebraic sum of the products of n multiplied by *, the normal equations, according to the method of least squares, will be m A + TB + r a <7 N = >v TA + T,B + T 3 CN,= I (415) T a A+ T 3 B -f T t C N a = ) The solution of these equations by the method of successive substitution, according to the forms given in the Appendix, may be expressed as follows : /T7J rpl m * J. j = J. 2 m f a 4 ?>l T 4 " ,S = ft' N;-T>C TN _ N T a C TB m (416) Then, to find the mean error of the corrected Ephemeris, we observe that this error is simply that of the function X, which is to be found by the method of the Appendix, according to which we first find the coefficients k w k v k 2 by the following formulae: mk = 1 mk + Tl k t = t and then, putting BY MOON CULMINATIONS. 361 we have (417) in which denotes the mean error of a single observation and (e X} the mean error of the corrected Ephemeris ; or, if e denotes the prohable error of an observation, (eJf) denotes the probable error of the corrected Ephemeris. (Appendix.) If the values of , k v and k z are substituted in JHf, we shall have It will generally happen, where a sufficient number of observa- T' tions are combined, that -^ is a small fraction which may be *$ neglected without sensibly affecting the estimation of a probable error, and we may then take (-!) , (*'- lyi 4 " J (418*) According to PEIRCE, the probable error of a standard observa- tion of the moon's transit is OM04 (found from the discussion of a large number of Greenwich, Cambridge, Edinburgh, and Wash- ington observations) ; so that the probable error of the corrected Ephemeris will be equal to M. (0*.104). EXAMPLE. At the Washington Observatory, the following right ascensions of the moon were obtained from the transits over twenty-five threads, observed with the electro-chronograph : Approx. Green. Mean Time. R. A. of 3 II Limb. Sid. time semid. passing merid. R. A. of 5 centre = i- 1859, Aug. 16, 19 " 17, 20 " 18, 21 0* 8"53'.40 54 33.57 1 42 48.53 62-.06 63.54 65.77 0* 7-5K34 53 30.03 1 41 42.76 The sidereal time of the semidiameter passing the meridian is here taken from the British Almanac, as we propose to reduce the observations by means of the Greenwich observations which are reduced by this almanac. We thus avoid any error in the semi- diameter. During the semi-lunation from Aug. 13 to Aug. 27, the Greenwich observations, also made with the electro-chronograph, 362 LONGITUDE. gave the following corrections ( n) of the Nautical Almanac right ascensions of the moon : Approx. Greenwich Mean Time. n t 1859. Aug. 14, 13* - 0'.39 -3. " 15, 14 -0.26 -1.9 16, 14 -0.49 -0.9 18, 16 0.63 + 1.2 19, 17 -1.04 + 2.2 20, 17 1.08 + 3.2 Let us employ these observations to determine by Peirce's method the most probable correction of the Ephemeris on the dates of the Washington observations. Adopting as the epoch Aug. 17th 12* or 17 d .5, the values of t are approximately as above given. The correction of the Ephemeris being sensibly constant for at least one hour, these values are sufficiently exact. We find then T k = 29.94 T 3 = 10.556 ST* = 225.045 T; = 29.83 T 3 ' = 6.564 r; = 75.644 T 4 " = 74.200 N = 3'.89 Nl , = 4.41 N* = 21-.85 1 = 3.89 N 2 '= - 2.44 N," = 1-.58 and hence, by (416), C= 0-.02135 B = 0'.1257 A = 0-.525 The correction of the Ephemeris for any given date /, reckoning from Aug. 17.5, is, therefore, = 0'.525 O'.02135 2 Consequently, for the dates of the Washington observations. the correction and the probable error (Ms) of the correction, found by (418) or (418*), are as follows: Aug. 16, 19* t = 0.7 17, 20 t = + 0.3 18, 21 t = -j- 1.4 = 0.56 Me = 0'.05 Ms = .04 Me = .04 The longitude of the Washington Observatory may now be found by the hourly Ephemeris (after applying these correc- tions), by the method of Art. 234. Taking the observation of Aug. 16, we have 13Y MOON CULMINATIONS. Aug. 16, T = 19*, R. A. of Ephemeris = o* 4,'.56 X=_ -0.45 Aa = 1.8122 da = + 0.0023 o = 6 47 .11 a t =:0 7 51.34 4.23 log (a, - ) 1.80774 ar. co. log Aa 9.74179 log 60 1.77815 log x f 3.32768 x> = 35 m 26'.57 x" = - .80 x =35 25.77 log a-' 2 log da ar. co. log Aa log r&v 6.6554 7.3617 9.7418 6 - 142 ? log x" n9.9016 Hence, Greenwich mean time = T -\- x = 19* 35" 25'.77 Sidereal time mean noon Correction for 19* 35- 25'.77 Greenwich sidereal time Local sidereal time = a. = 9 24.18 13 .09 3.04 51.34 Longitude 5 8 11 .70 The observations of the 17th and 18th being reduced in tb same manner, the three results are Probable error.* Weight. Aug. 16, 5* 8 m 11-.70 3'.5 1. "~ 17, 12 .50 3.1 1.3 " 18, 11.10 2.9 1.5 Mean by weights = 5 8 11.74 1.8 236. Combination of moon culminations by 'weights. When some of the transits either of the moon or of the comparison stars are incomplete, one or more of the threads being lost, such observa tions should evidently have less weight than complete ones, if we wish to combine them strictly according to the theory of probabilities. Besides, other things being equal, a determina- tion of the longitude will have more or less weight according to the greater or less rapidity of the moon's motion in right ascen- sion. For the computation of the probable error and weight, see the following article. 864 LONGITUDE. If the weight of a transit either of the moon or a star were simply proportional to the number of observed threads, as lias been assumed by those who have heretofore treated of this sub- ject,* the methods which they have given, and which are obvious applications of the method of least squares, would be quite suffi- cient. But the subject, strictly considered, is by uo means so simple. Let us first consider the formula a, = a! -f t *' or, rather in which $ t and &' are the observed sidereal times of the transit of the moon and star, respectively; a' is the tabular right ascen- sion of the star, and a t is the deduced right ascension of the moon. The probable error of a t is composed of the probable errors of # t and of a' #', which belong respectively to the moon and the star. We may here disregard the clock errors, as well as the unknown instrumental errors, since they affect $, and #' in the same manner, very nearly, and are sensibly elimi- nated in the difference #1 #'. The probable error of the quantity a' &' is composed of the errors of a/ and &'. The probable error of the tabular right ascension of the moon-culmi- nating stars is not only very small, but in the case of correspond- ing observations is wholly eliminated ; and even when we use a corrected Ephemeris it will have but little effect, since the ob- served right ascension of the moon at the principal observatories always depends (or at least should depend) chiefly upon these stars. We may, therefore, consider the error of a' #' as sim- ply the error of &'. We have here to deal with those errors only which do not necessarily affect #' and #, in the same manner, and of these the chief and only ones that need be considered here are 1st, the culmination error produced by the peculiar con- ditions of the atmosphere at the time of the star's transit, which are constant, or nearly so, during the transit, but are different for different stars and on different days; and, 2d, the accidental error of observation. It is only the latter which can be diminished * NICOLAI, in the Aslronomische Nachrichten, No. 26; and S. C. WALKER, Transac tions of the American Philosophical Society, Vol. VI. p. 253. BY MOON CULMINATIONS. 365 by increasing the number of threads. In Vol. II. (Transit In- strument) I shall show that the probable error of a single deter- mination of the right ascension of an equatorial star (and this may embrace the moon ^nfminating stars) at the Greenwich Observatory is 0".06, whereas, if the culmination error did not exist it would be only 0*.03, the probable error of a single thread being = s . 08, and the number of threads 7. Hence, putting c = the probable culmination error for a star, we deduce* c = V/(0.06) 2 (0.03) 2 = 0.052 If, then, we put e = the probable accidental error of the transit of a star over a single thread, n = the number of threads on which the star is observed, the probable error of #', and, consequently, also of a'-- #', is and the weight of a' &' for each star may be found by tho formula in which E is the probable error of an observation of the weight unity, which is, of course, arbitrary. If we make p = 1 when n = 7, we have E O s .06. Substituting this value, and also c = s . 052, e = s . 08, the formula may be reduced to the fol- lowing : 100 + The value of a, is to be deduced by adding to #, the mean * The value of c thus found involves other errors besides the culmination error proper, such as unknown irregularities of the clock and transit instrument, &c These cannot readily be separated from c, nor is it necessary for our present purpose. 366 LONGITUDE. according to weights of all the values of a x # x given by the several stars, or ^ f tefefcj*21 (420) where the rectangular brackets are employed to express the sum of all the quantities of the same form. The probable error of the last term will be E 0'.06 If now we put Cl = the probable error of a l5 c t = the culmination error for the moon, kc = the probable accidental error of the transit of the moon's limb over a single thread, n l = the number of threads on which the moon is observed, the probable error of ^ will be = A /c, 2 -f ^ ^, and hence \ n t To determine c t I shall employ the values of the other quantities in this equation which have been found from the Greenwich observations. Professor PEIRCE gives e,= OM04, and in the cases which I examined I found the mean value k = 1.3. As- suming [p] = 4 as the average number of stars upon which a t depends in the Greenwich series, we have whence c t = O'.OQl and the formula for the probable error of a v observed at the meridian L is ' (422) H l IP] In the case of corresponding observations at a second meridian _L 2 , the probable error 2 is also to be found by this formula, and then the probable error of the deduced difference of right ascen- sion will be BY MOON CULMINATIONS. 367 and the probable error of the deduced longitude will be = h |A?Ttf (423) where, H being the increase of the moon's right ascension in 1* of longitude, we have But if the observation at the meridian L { is compared with a corrected Ephemeris (Art. 235) the probable error of which is Jlf (0*.104), the probable error of the deduced longitude will be = h !/,' -j- M* (0.104) 8 (425) Finally, all the different values of the longitude will be com- bined by giving them weights reciprocally proportional to the squares of their probable errors. The preponderating influence of the constant error represented by the first term of (422) is such that a very precise evaluation of the other terms is quite unimportant. It is also evident that we shall add very little to the accuracy of an observation by increasing the number of threads of the reticule beyond five or seven. For example, suppose, as in the Washington observations used in Art. 235, that twenty-five threads are taken, and that four stars are compared with the moon ; we have for each star, by (419), p= 184 238 =1.22 and hence whereas for seven threads we have e, = OM04, and therefore the increase of the number of threads has not diminished the probable error by so much as O'.Ol. For the observations of 1859 August 16, 17, 18, Art. 235, the values of h are respectively 32.1 30.8 and 28.8 and, taking Me M (O.104) as given in that article, namely, 0'.05 0'.04 and 0-.04 368 LONGITUDE. with the value of e, = 0*.097 above found, we deduce the proba- ble errors of the three values of the longitude, by (425), 3'.5 3'.1 and 2'.9 The reciprocals of the squares of these errors are very nearly in the proportion of the numbers 1, 1.3, 1.5, which were used as the weights in combining the three values. 237. The advantage of employing a corrected Ephemeris instead of corresponding observations can now be determined by the above equations. If the observations are all standard observations (represented by ^ = 1 and \_p] = 4), we shall have e,= e a OM04, and the probable error of the longitude will be by corresponding observations = he v j/2 by the corrected Ephemeris = Ae t j/1 -f- M * The latter will, therefore, be preferable when M < 1, which will always be the case except when very few observations have been taken at the principal observatories. But experience has shown that when we depend wholly on corresponding observations we lose about one-third of the observations, and, consequently, the probable error of the final longitude from a series of observations is greater than it would be were all available in the ratio of i/3 : |/2. Hence the proba- ble errors of the final results obtained by corresponding observa- tions exclusively, and by employing the corrected Ephemeris by which all the observations are rendered available, are in the ratio ]/3 : j/1 -f J/ 2 , and, the average value of M being about 0.6, this is as 1 : 0.67. If, however, on the date of any given observation at the meri- dian to be determined, we can find corresponding observations at two principal observatories, the probable error of the longitude found by comparing their mean with the given observation will be only ta,.j/1.5, which is so little greater than the average error in the use of the corrected Ephemeris, that it will hardly be worth while to incur the labor attending the latter. If there should be three corresponding observations, the error will be reduced to hs 1 ;/1.33, and, therefore, less than the average error of the corrected Ephemeris. BY MOON CULMINATIONS. 369 The advantage of the new method will, therefore, be felt chiefly in cases where either no corresponding observation, or but one, has been taken at any of the principal observatories. 238. The mean value of h is about = 27, and therefore a probable error of OM in the observed right ascension, supposing the Ephemeris perfect, will produce a mean probable error of 2*.7 in the longitude. If the probable error diminished without limit in proportion to the square root of the number of observa- tions, as is assumed in the theory of least squares, we should only have to accumulate observations to obtain a result of any given degree of accuracy. But all experience proves the fallacy of this law when it is extended to minute errors which must wholly escape the most delicate observation. The remarks of Professor PEIRCE on this point, in the report above cited, are of the highest importance. He says : " If the law of error embodied in the method of least squares were the sole law to which human error is subject, it would happen that by a sufficient accumulation of observations any imagined degree of accuracy would be attainable in the determination of a constant; and the evanescent influence of minute increments of error would have the effect of exalting man's power of exact observation to an unlimited extent. I believe that the careful examination of observations reveals another law of error, which is involved in the popular statement that ' man cannot measure what he cannot see.' The small errors which are beyond the limits of human perception are not distributed according to the mode recognized by the method of least squares, but either with the uniformity which is the ordinary characteristic of matters of chance, or more frequently in some arbitrary form dependent upon individual peculiarities, such, for instance, as an habitual inclination to the use of certain numbers. On this account, it is in vain to attempt the comparison of the distribution of errors with the law of least squares to too great a degree of minuteness ; and on this account, there is in every species of observation an ultimate limit of accuracy beyond which no mass of accumulated observations can ever penetrate. A wise observer, when he perceives that he is approaching this limit, will apply his powers to improving the methods, rather than to increasing the number of observations. This principle will thus serve to stimulate, and not to paralyze, effort ; and ita VOL. L 24 <370 LONGITUDE. vivifying influence will prevent science from stagnating into mere mechanical drudgery. " In approaching the ultimate limit of accuracy, the probable error ceases to diminish proportion ably to the increase of the number of observations, so that the accuracy of the mean of several determinations does not surpass that of the single deter- minations as much as it should do in conformity with the law of least squares : thus it appears that the probable error of the mean of the determinations of the longitude of the Harvard Observatory, deduced from the moon-culminating observations of 1845, 1846, and 1847, is 1'.28 instead of being I'.OO, to which it should have been reduced conformably to the accuracy of the separate determinations of those years. " One of the fundamental principles of the doctrine of proba- bilities is, that the probability of an hypothesis is proportionate to its agreement with observation. But any supposed computed lunar epoch may be changed by several hundredths of a second without perceptibly ati'ecting the comparison with observation, provided the comparison is restricted within its legitimate limits of tenths of a second. Observation, therefore, gives no informa- tion which is opposed to such a change." The ultimate limit of accuracy in the determination of a longitude by moon culminations, according to the same distin- guished authority, is not less than one second of time. This limit can probably be reached by the observations of two or three years, if all the possible ones are taken ; and a longer continuance of them would be a waste of time and labor. From these considerations it follows that the method ot moon culminations, when the transits of the limb are employed, cannot come into competition with the methods by chronometers and occultations where the latter are practicable.* * In consequence of the uncertainty attending the observation of the transit of the moon's limb, it has been proposed by MAEDLER (Astron. Nach. No. 337) to sub- stitute the transit of a well-defined lunar spot. The only attempt to carry out this suggestion, I think, is that of the U. S. Coast Survey, a report upon which by Mr. PETERS will be found in the Report of the Superintendent for 1856, p. 198. The varying character of a spot as seen in telescopes of different powers presents, it geeins to me, a very formidable obstacle to the successful application of this method. BY AZIMUTHS OF THE MOON. 371 FIFTH METHOD. BY AZIMUTHS OF THE MOON, OR TRANSITS OF THE MOON AND A STAR OVER THE SAME VERTICAL CIRCLE. 239. The travelling observer, pressed for time, will not unfre- quently tind it expedient to mount his transit instrument in the vertical circle of a circumpolar star, without waiting for the meri- dian passage of such a star. The methods of determining the local time and the instrumental constants in this case are given in Vol. II. He may then also observe the transit of the moon and a neighboring star, and hence deduce the right ascension of the moon, which may be used for determining his longitude precisely as the culminations are used in Art. 234. 240. But if the local time is previously determined, we may dispense with all observations except those of the moon and the neighboring star, and then we can repeat the observation several times on the same night by setting the instrument successively in ditto rent azimuths on each side of the meridian. It will not be advisable to extend the observations to azimuths of more than 15 on either side. The altitude and azimuth instrument is peculiarly adapted for such observations, as its horizontal circle enables us to set it at any assumed azimuth when the direction of the meridian is approximately known. The zenith telescope will also answer the same purpose. But as the horizontal circle reading is not required further than for setting the instrument, it is not indis- pensable, and therefore the ordinary portable transit instrument may be employed, though it will not be so easy to identify the comparison star. The comparison star should be one of the well-determined moon-culminating stars, as nearly as possible in the same parallel with the moon, and not far distant in right ascension, either preceding or following. The chronometer correction and rate must be determined, with all possible precision, by observations either before or after the moon observations, or both. An approximate value of the cor- rection should be known before commencing the 'observations, as it will be expedient to compute the hour angles and zenith distances of the two objects for the several azimuths at which it is proposed to observe, in order to point tho instrument properly and thus avoid observing the wrong star. 372 LONGITUDE. To secure the greatest degree of accuracy, the observations should be conducted substantially as follows : 1st. The instrument being supposed to have a horizontal circle, let the telescope be directed to some terrestrial object, the azimuth of which is known (or to a circurnpolar star in the meri- dian), and read the circle. The reading for an object in the meridian will then be known ; denote it by a. 2d. The first assumed azimuth at which the transits are to be observed being A, set the horizontal circle to the reading A-\- a, and the vertical circle to the computed zenith distance of the moon or the star (whichever precedes). This must be done a few minutes before the computed time of the first transit 3d. Observe the inclination of the horizontal axis with the spirit level. 4th. Observe the transit of the first object over the several threads. 5th. If there is time, observe the inclination of the horizontal axis. 6th. Set the vertical circle for the zenith distance of the second object, and observe its transit. 7th. Observe the inclination of the horizontal axis with the spirit level. The instrument must not be disturbed in azimuth during these operations, which constitute one complete observation. Now set upon a new azimuth, sufficiently greater to bring the instrument in advance of the preceding object, and repeat the observation. It will often be possible to obtain in this way four or six observations, two or three on each side of the meridian, but the value of the result will not be much increased by taking more than one observation on each side of the meridian. The collimation constant is supposed to be known; but, in order to eliminate any error in it, as well as inequality of pivots, one-half the observations should be taken in each position of the rotation axis. The azimuth of the instrument at each observation is only known from the local time, and hence the following indirect method of computation will be found more convenient than the usual method of reducing extra-meridian transits; but the reader will find it easy to adapt the methods given in Vol. II. for such purpose to the present case. BY AZIMUTHS OF THE MOON. 373 We shall make use of the following notation : T, T' = the mean of the chronometer times of transit of the moon's limb and the star, respectively, over the several threads,* AT, AT' = the corresponding chronometer corrections, b, b' = the inclinations of the horizontal axis at the times T and T', c = the collimation constant for the mean of the threads. , a' = the moon's and the star's right ascensions, " " " declinations, " " " hour angles, f, ' = " " u true zenith distances, q, q' = " " " parallactic angles, A,A' = " " " azimuths, Aa = the increase of the moon's right ascension in ono minute of mean time, A f ur decimal places will suffice. The following formulae for this purpose result from a combination of (16) and (20) : For the moon. For the star. *=T-fAT a f=T'+AT' a' tan M = tan d sec t } f tan M ' = tan 8' sec f tan t cos M I th 8 f J tan t' cos M' tan A = C decimals; | tan A' = . sm (?> jlf ) J sin (p JIT) (426) tan N = cot ^ cos t tan sin N with four decimals ; tan q' tan C' = cot

' is the reduction of the latitude, and p is the terrestrial radius for the latitude and then the reduction of the true azimuth to the instrumental azimuth (see Vol. II., Altitude and Azimuth Instrument) is c b for the moon, + for the star, q= Ci tan c b' tan the upper or lower sign being used according as the vertical circle is on the left or the right of the observer. The computed instrumental azimuths are, therefore, S , p-(')sinl"sinvl' c =A -' "sin sin C sin d tan (star) A l ' = A'+ 7 = F-- sin C/ tan (427) If now the longitude and other elements of the computation are correct, we shall find A l and AJ to be equal : otherwise, put x = A l ~A l l (428) then we are to find how the required correction &L depends on x, supposing here that all the elements which do not involve the longitude are correct. Now, we have taken a and d from the Ephemeris for the Greenwich sidereal time T + A T + L', when they should be taken for the time T + A T + L' + *L. Hence, if \ and /3 denote the increments of the moon's right ascension and declination in one sidereal second, both expressed in seconds of arc, /I = [9.39675] Aa we find that a requires the correction ^ . &L 9 " " ft.*L t " A . A L and these corrections must produce the correction x in the moon 's azimuth. The relations between the corrections of the azimuth, the hour angle, and the declination, where these are so small as to be treated as differentials, is, by (51), 376 LONGITUDE. dA= sin C sin C that is, _ x = _C08*^, l Bin sin C sin C Hence, if we put a= ,. _,. sin C sin C we have Ai = (431) and hence, finally, the true longitude Z/-f A-- 241. In order to determine the relative advantages of this method and that of meridian transits, let us investigate a formula which shall exhibit the eifect of every source of error. Let da, 3d, 3n, dS = the corrections of the elements taken from the Ephemeris of the moon, da, dd' = the corrections of the star's place, dT, dT' = the corrections for error in the obs'd time, 3&T= the correction of A T, dy> = the correction of y. If, when the corrected values of all the elements that of the longitude included are substituted in the above computation, A l and Af become A l + dA l and AJ + dAJ, we ought to find, rigorously, A l + dA l = A,' + dAi' which compared with (428) gives x = dA l -f dAi' (432) We have, therefore, to find expressions for dA l and dA^ in terms of the above corrections and of A.L. We have, first, by differentiating (427), dA l =dA^L + ^-^* [n We neglect errors in c and b which are practically eliminated by comparing the moon with a star of nearly the same declina- tion, and combining observations in the reverse positions of the BY AZIMUTHS OF THE MOON. 877 The total differential of A is, by (51), after reducing dl to arc, dA = sinC sinC consequently, also, cos<5'cos<7' , sin 7' ., ' - d<5' cot C sin A'd? ' Since * = T + A r a, we have where dT and d&Tmay he at once exchanged for dT&nd 8&T; hut da is composed of two parts : 1st, the correction of the Ephemeris, and 2d, l(*L + dT + d*T), which results from our having taken a for the unconnected time. Hence we have, in arc, 15 dt = 15 we have put A = A'. By the aid of this equation we can now trace the effect of each source of error. 1st. The coefficients of 3d, II Limb. T = 16* 11" 30M7 b = + 2".2 c = 0.0 j Vertical circle ff Scorpii T'=IQ 27 49.83 6'=-f2.2 I left. These times are the means of three threads. The chronometer correction, found by transits of stars in the meridian, was 55 m 9M6 at 13* sidereal time, and its hourly rate 0*.32. The assumed latitude and longitude were

COB d t after which the moon's right ascension is found by the formula a = t (437) and hence tne Greenwich time and the longitude as above stated. But since we have taken d for an approximate Greenwich time depending on the assumed longitude, the first computation of t will not be quite correct ; a second one with a corrected value of 8 will give a nearer approximation ; and thus by successive approximations the true value of t and of the longitude will at last be found. But instead of these successive approximations we may obtain at once the correction of the assumed longitude, as follows. We have taken d for the Greenwich time -f- _L', when we should have taken it for the time -f L' + A.L. Hence, putting j3 = the increase of S in a unit of time, it follows that d requires the correction fi&L; and therefore, by (51), the correction of the computed hour angle will be cos 8 tan q in which q is the parallactic angle. Since a = t, the com- puted right ascension requires the correction (in seconds of time) 15 cos d tan q Therefore, if we put ^ = the increase of a in a unit of time, the computed Greenwich time and, consequently, also the longi- tude derived from it requires the correction 15 /* cos d tang BY ALTITUDES OF TIIK MOON. 385 Hence, denoting the longitude computed from the right ascen* sion a = t by L" ', we have True longitude = L' L" ^ 15 A cos d tan q whence L"L' 1 -f-Jl-8ec TT = 56' 3".l j ATT (Tab. XHI.) = + 4 .4 W, sin ' ir, = 56 7 A) = 64 40 7 0" = _ u 57 5 33 21.6 56 0. 2)64 25 3 =. 32 12 31 ~Z Approx. Or. time = 10 39 21 .6 (For wliich time we take ir, S, and from the Nautical Almanac ) = + 3 47' 47".6 z jr,sin

- 39" 48'.7, Increase of o in l ra = /I = -f- 2'.014 " 3 in 1 = 13 = -f 10".01 and hence, by (438) and (439), a = 0.3317 *L = -f 40*.6 L = L'+ *L = 5* 6 TO 40'.6 244. The result thus obtained involves the errors of the tabular right ascension and declination and the instrumental error. The tabular errors are removed by means of observations of the same data made at some of the principal observatories, as in the case of moon culminations. The instrumental error will be nearly eliminated by determining the local time from a star at the same altitude and as nearly as possible the same declina- tion ; for the instrumental error will then produce the same error in both and t, and, therefore, will be eliminated from their difference I = a. The error in the longitude will then be no greater than the error in 0. But to give complete effect to this mode of eliminating the error, an instrument, such as the zenith telescope, should be employed, which is capable of indicating the same altitude with great certainty and does not involve the errors of graduation of divided circles. A very different method of observation and computation must then be resorted to, which I proceed to consider. 245. (B.) By equal altitudes of the moon and a star, observed with the zenith telescope. The reticule of this instrument should for these observations be provided with a system of fixed horizontal threads : nevertheless, we may dispense with them, and employ only the single movable micrometer thread, by setting it suc- cessively at convenient intervals. BY ALTITUDES OF THE MOON. 387 Having selected a well determined star as nearly as possible in the moon's path and differing but little in right ascension, a preliminary computation of the approximate time when each body will arrive at some assumed altitude (not less than 10) must be made, as well as of their approximate azimuths, in order to point the instrument properly. The instrument being pointed for the first object, the level is clamped so that the bubble plays near the middle of the tube, and is then not to be moved between the observation of the moon and the star. After the object enters the field, and before it reaches the first thread, it may be necessary to move the instrument in azimuth in order that the transits over the horizontal threads may all be observed without moving the instrument daring these transits. The times by chronometer of the several transits are then noted, and the level is read off. The instrument is then set upon the azimuth of the second object, the observation of which is made in th same manner, and then the level is again read off. This com- pletes one observation. The instrument may then be set for another assumed altitude, and a second observation may be taken in the same manner.* Each observation is then to be separately reduced as follows : Let i, i', i", &c. = the distances in arc of the several threads from their mean, m, m' = the mean of the values of i for the observed threads, in the case of the moon and star respectively, /, I' = the level readings, in arc, for the moon and star, 6, 0'= the mean of the sidereal times of the observed transits of the moon and star; then the excess of the observed zenith distance of the moon's limb at the time above that of the star at the time 0' isf m m' -f- I I' the quantities m and I being supposed to increase with increasing zenith distance. * The same method of observation may be followed with the ordinary universal instrument, but, as the level is generally much smaller than that of the zenith tele- scope, the same degree of accuracy will not be possible. f When the micrometer is set successively upon assumed readings, m and m' will be the means of these readings, converted into arc, with the known value of the screw. 388 LONGITUDE. Also, let a, S, t, C, A, q = the R. A., decl., hour angle, geocentric zenith distance, azimuth, and parallactic angle of the moon's centre at the time 0; of, 8', if, C', A', q' = the same for the star at the time 0'; TT, /S> = the moon's equatorial hor. parallax and semidiameter; A = the increase of a in 1* of sid. time ; cos t tan t sin N COS (d + N) with four decimals; cos A' = tan (? M ') cot t: ' tan N' = cot y> cos f tan o' = tan f sin JV 7 cos((5' The zenith distance thus computed will not strictly correspond to the time unless the assumed longitude is correct. Let its true value be + d. Also put C, = the observed zenith distance of the moon's limb, C,' = the observed zenith distance of the star, r, r' = the refraction for C, and C', BY ALTITUDES OF THE MOON. 389 then Putting then and, by Art. (136) f = {(f (p'*) cos A sin^=:/>sin7rsin(C" k = p q= S q= %(p T $) sin p sin $ (441) the { J^P 61 " 1 sign being used for the moon's / l^P 61 " } limb, we I lower j ( lower J have This equation determines d. We have, therefore, only to determine the relation between d and A.L. Now, we have taken a and 3 for the Greenwich sidereal time -f L ', when we should have taken them for the time -f- L ' -\- &L ' : hence a requires the correction d " " t " " and then, by (51), d* = cos q . fib.L sin q cos 8 . 15 X &L Hence, putting x = d, or x = C - C" + k and a = 15 ^ sin q cos 5 -f p cos = _Z> = L' -f- ^-^ The solution of the problem, upon the supposition that all the data are correct, is completely expressed by the equations (440), (441), and (442). 246. The quantity x is in fact produced not only by the error in the assumed longitude, but also by the errors of observation and of the Ephemeris. In order to obtain a general expression 390 LONGITUDE. in which the effect of every source of error may be represented, let T, T'= the chronometer times of observation of the moon and star, AT = the assumed chronometer correction, 8T, ST'= the corrections of T and T' for errors of observation, fa T = the correction of A T, rfo, 88, 8n, dS = the corrections of the elements taken from the Ephemeris, 8 d* 1 '= 15 sin q' cos 8' df cos q' d8' -f- cos -4'rf^p Since < = T+ AT 1 a, we have where dT and ^A7"may be exchanged for <57"and 3&T, but da is composed of two parts : 1st, of the actual correction of the Ephemeris; and 2d, of l(L + 7"+ d^T) resulting from our having taken a for the uncorrected time : hence we have dt = 9T + tiTfa A(A + 8T + The correction dd is also composed of two parts, so that d8 = 88 + /9(Ai + 3T + 8&.T) Further, we have simply dS' = dd', and df= ST' + JAT ia.' in which for at the time T' is assumed to be the same as at the BY ALTITUDES OF THE MOON. 391 time Tj an error in the rate of chronometer being insensible in the brief interval between the observations of the moon and the star. Again, we have, from (441), cos p dp = p cos n sin (C" ?} dn -\- p sin it cos (C" f) d'" dk = dp + dS or, with sufficient accuracy, dk sin C' fa =f 88 + sin TT cos C' dl' Now, substituting in d and d' the values of dt, dd, &c., and then substituting the values of d and d' thus found, in (443), together with the value of dk, we obtain the final equation desired, which may be written as follows:* (444) x = a*L+f.8a + cosq.33 (f a) dT mf .da' m cos q' 88' -f mf . 8T' SS sin C ' 3* (f mf' a (cos A m cos A'} 8

= 9.999983 Since the same fixed threads were used for both moon and star, we have m = m', and hence also sensibly r r' ; therefore^ by (441), we find C"= 53 13' 59".30 Y = 54".5 p = 46' 21".25 : C"= 62' 9".86 k = + 62' 9".17 Hence, by (442), x = 0".69 a = + 0.5575 *L = 1'.24 The longitude by this observation, if the Ephemeris is correct, is therefore L = L' + A/V = - 7" 7" 38'.24 BY LUNAR DISTANCES. 393 If we compute all the terms of (444), we shall find A = P.24 24.84 da 0.27 dd -f 23.84 d T 24.24 8 T' 0.44 S* T + 24.28 &'+ 0.29 88'+ 1.79 dS + 1.44 fa 0.04 %> This shows clearly the effect of each source of error; but in prac tice it will usually be sufficient to compute only the coefficients of da, and dd. In the present example, therefore, we should take A = 1'.24 24.84 cos m si" (. ni A') smMtf = -- ^-rr - T7 - sinlQ= - ^-77^ -- =r^ sin cr cos A sin cr cos xf and then the apparent distance by the formula d'=d".(sf AS cos 2 + ^ C 8 * (h ' + H '~ d "> cos h' cos H ' and, then, in the triangle MZS we shall have given the angle Z with the sides 90 A, and 90 JET,, whence the side MS = ^ will be found by the formula [Sph. Trig. (17)], sin 1 J 1 J rr^ COS ^1 COS H, sin 2 i d, cos 2 i (A, -f- jff,) cos m cos (m d ) COH /i' cos .77' Let the auxiliary angle M be determined by the equation sin* M = ' . - cos A' cos J7 ' cos 2 J (A, + If,) then we have* sin i d, = cos i (A, + #,) cos 3/ (449) Finally, to reduce the distance from the point to the centre Fig. so. of the earth, let P (Fig. 30) be the north pole of the heavens, M l the moon's place as seen from the point 0, M the moon's geocentric place, S the sun's place (which is sensibly the same for either point). The point being in the axis of the celestial sphere, the points M l and M evidently lie in the same declination circle PM^M. Hence, putting d = the geocentric distance of the moon and sun = SM, d, = SM lt S = the moon's geocentric declination = 90 PM, d l = the declination reduced to the point = 90 PM^ J == the sun's declination = 90 PS, we have, in the triangles PMS and M^MS, _ cos d, cos ( cos A cos Ji The quantities r' and R' computed from the mean values of the 1 refraction are given in Table XIV. under the name " Mean Reduced Refraction for Lunars." The numbers of the table are corrected for the height of the barometer and thermometer by means of Table XIV. A and B. These tables are computed from BESSEL'S refraction table, assuming the attached ther- mometer of the barometer, and the external thermometer, to indicate the same temperature, which is allowable in our present problem.* By the introduction of r' and R', we obtain and the coefficients of formula (g) become * If it is desired to compute r' and R' with the utmost rigor, it can be done bj Table II., by taking (Art. 107) sin A' sin H' The tables XIV. and XIV. A and B give the correct values to the nearest second in all practical cases. BY LUNAR DISTANCES. 407 A, = (*! - r') (1 -1- A) sin (h f + J A/I) (7, = (E' P) sin (H' 2 cos A' The term A l C l sin V cosd' is very small, its maximum being only 1". It is easy to obtain an approximate expression for it and to combine it with the term A l cosd f . In so small a term we may take <7, sin V'= R' sin 1" sin H' = sin R tan H' = k and hence A l AtCi sin 1" = A l (1 -f- k~) = (! r') (1 -f A) 1 sin (A' -)- i AA) If now we put A=(l+*)'-^^ (455) _ sin (2 h' H- AA) sin 2 A' and .4' = (rfj r') A sin A' cot d' B'= (jr t r') 5 sin #' coscc d' C'= (R' P) C sin JT cot d' D'= (R'P)Dsinh r cosecd' the formula (#) becomes, when divided by sin rf', sin a the first member of which may be put under the form / 2 sin J Atf cos (d' -|- J Arf) \ Aal 1 -\ : I ^ \ sin d' / 408 LONGITUDE. so that if we put AJ ,. . 77 cos h sin d [Jut, by (456), sin H' B' sin h' cos d' A' cos A' sin d' B(~i r') cos h' cos h' sin d' A (~ t r') cos h' so that r, r') cos h' If we put A = 1 and B = 1, which are approximate values, we shall have A'+ B' ~ (461) ilO LONGITUDE. In order to ascertain the degree of accuracy of this formula, we observe that the errors in cos q produced by the assumption A = 1, B = 1, are e = (A - 1) ^^ e f = (l - B) si " H ' tan d cos h sin d the errors in cos 2 q are 2ecos q 2e'cos q and the errors in A 5 are, therefore, _ 2AS, (A 1) tan h' cos q , 2AS, (1 B) sin H' cos q tan d' cos A' sin d' In order to represent extreme cases, let us suppose q = and H f = 90, which will give e v and ej their greatest values; then we shall find for the different values of h' the following errors : h' e l tan d' e,' sin rf' 5 0".45 0".02 10 .16 .00 15 .08 .00 80 .02 .00 50 .00 .00 It can only be for very small values of d' that the error t can be important, even for h' = 5; and, as these small values of the distance are always avoided in practice, our formula (461) may be considered quite perfect. In the same manner, we shall find which is even more accurate than (461). These formulae are put into tables as follows. For the moon, Table XVII. A, with the arguments h' and ^ r f , gives the value of _ _ AS, ^^ (*,-/)' cos 2 A' X/ where / is an arbitrary factor ( 18000000) employed to give g convenient integral values. Then Table XVII.B, with the argu- ments g and A' -\- B', gives BY LUNAR DISTANCES. 411 For the sun, Table XVIILA, with the arguments H 1 and R' P, gives the value of in which F= -; and Table XVIII.B gives In these tables A' '-f- -B' is called the "whole correction of the moon," and C' + D' the "whole correction of the sun." As these quantities are furnished by the previous computation of the true distance, the required corrections are taken from the tables without any additional computation. The values of AS and A$ are applied to the distance as follows : when the limb of the moon nearest to the star or planet is observed, AS is to be subtracted, and when the farthest limb is observed, AS is to be added ; when the sun is observed, both AS and A are to be subtracted from d. In strictness, these corrections should be applied to the dis tance d', and the distance thus corrected should be employed in computing the values of A', B', C", and D'. This would require a repetition of the computation after AS and A had been found by a first computation ; but this repetition will rarely change the result by 0".5. In the extreme and improbable case when the distance is only 20 and one body is at the altitude 5 and the other directly above it in the same vertical circle (so that the entire contraction of the vertical semidiameter comes into account), such a repetition would change the result only 1".8 ; and even this error is much less than the probable error of sextant observations at this small altitude, where the sun and moon already cease to present perfectly defined discs. 250. I shall now recapitulate the steps of this method. 1st. The local mean time of the observation being T 7 , and the assumed longitude L, take from the Ephemeris, for the npproxi- 412 LONGITUDE. mate Greenwich time T -f _L, the quantities s, S, x, P, d, and J. (For the sun we may always take P = 8". 5 ; for a star, S = 0, P=0.) 2d. If A", #", d" denote the observed altitudes and distance of the limbs, find s' = s -f correction of Table XII., ?r l = -f correction of Table XIII., and the apparent altitudes and distance of the centres, h'=h"=fs', H'=H"+S, d'=d"s'S upper signs for upper and nearest limbs, lower signs for lower and farthest limbs. For the altitudes h' and H', take the "reduced refractions" r' and R f from Table XIV., correcting them by Table XIV.A and B for the barometer and thermometer. Then compute the quantities A' = (^r'^Asinh'cotd' C' = (R r P) CsinH'cotd' B' = (TT, r 1 ) sin H' cosec d' J>' = (R'P)Dsmh'cosecd' for which the logarithms of A, -B, (7, and D are taken from Table XV. In this table the argument TT I r' is called the " reduced parallax and refraction of the moon," and R' P the " reduced refraction and parallax of the sun (or planet) or star." For a star this argument is simply R'. When rf'> 90, the signs of A' and C" will be reversed. It may be convenient for the computer to determine the signs by referring to the following table : -4 S' C" D' 90 -I- + 3d. The terms A' and B', which depend upon the moon's parallax and refraction, may be called the first and second parts of the moon's correction, and the sum A' + B' the " whole cor- rection of the moon." In like manner, C' and .D'may be called the first and second parts of the sun's, planet's, or star's correc- BY LUNAR DISTANCES. 413 tion, and the sum C" -f- D 1 the " whole correction of the sun, planet, or stir." The sum of these corrections A' -f B' + C' + D' may he called the "first correction of the distance." Taking it as' the upper argument in Tahle XVI., find the second correction x, the sign of which is indicated in the tahle. 4th. Take from Table XVII.A and B the contraction of its inclined semidiameter = AS. If the sun is the other body, take also the contraction from Tahle XVIII. A and B, = &S. The sign of either of these corrections will be positive when the farthest limb is observed, and negative when the nearest limb is observed. 5th. The correction for the compression of the earth is = N sin is negative. All the corrections being applied to rf', we have the geocen- tric distance d; and hence the corresponding Greenwich time and the longitude. EXAMPLE. Let us take the example of the preceding article (p. 399), in which the observation gives 1856, March 9th, 4, = 35. T = 5* 14 ra 6' 3) h" = 52 34' 0" Barom. 29.5 in. Assumed L = 10 00 Qff"= 8 5623 Therm. 58 F. Approx. Gr. T. = 15 14 6 ]* d" =44 3658.6 By the Ephemeris, we have s = 16' 23".l * = 60' 1".9 S = 16' 8".0 P= 8".6 Table XII. + 1* -0 Tab. XIII. + 3 .9 8 = + U A= 4 The computation may be arranged as follows: 414 LONGITUDE. A"=r 5234'.0 '= + 16.6 A'= 52 50.6 Table XIV. A. 1 . B. - 1 . r'= 1 11 .1 ir.= 60 5 .8 *-, r'= 5854 .7 (Table XV.) log A 0.0019 log (TT, r') 3.5484 log sin A' 9.9015 log cot d' _9._9975 log A' 3.4493 ^' = _j_ 46'53".9 7/"r= 856'.4 5 = 16.1 77 = 12.6 5' 49". 6 6 . - 6 . R'= 5 37 .6 P= 8 .6 JfP= 529 .0 (Table XV.) log C 9.9978 log (K - P) 2.5172 log sin //' 9.2042 log cot d' 9.9975 log C' nl.7167 C' = 52". 1 (Table XV.) log B 9.9981 log (rr, - r') 3.5484 log sin 77' 9.2042 log cosec d' 0.1493 (Table XV.) log D 9.9987 log (R' P) 2.5172 log sin h' 9.9015 log cosec d' 0.1493 log B' n2.9000 ' = 13'14".3 A' + B'= +3339 .6 log D' 2.5667 D'= +6' 8". 7 C" + D'= +516 .6 1st (Table XVI.) 2d ' = 16 37 .1 S 16 8 .0 d' = 45 9 43 .7 Table XIX. 1st Part of N= - 6 2d = _8. 0=35. 1st corr. = -f 38' 56 ".2 2d corr. = 18 .5 (Table XVII.) A* = . (Table XVIII.) &S= - 9 . If sin = 46 d = 45 48 12 This result agrees with that found by the rigorous method on p. 401, within 1". To find the longitude, we now have, by the American Ephe- meris for March 9, =15* 0" 0' (d) = 45 40' 54" $=0.2510 J= t = Table XX. T = 15~ /T R 13 3 1 L = 9 58 56 d = 45 48 13 7 19 log = 2.6425 log t =2.8935 BY LUNAR DISTANCES. 41f> 251. Ill consequence of the neglect of the fractions of a second in several parts of the above method, it is possible that Jie computed distance may he in error several seconds, hut it is easily seen that the error from this cause will he most sensible in cases where the distance is small ; and, since the lunar distances are given in the Ephemeris for a number of objects, the observer can rarely be obliged to employ a small distance. If he confines himself to distances greater than 45 (as he may readily do), the method will rarely be in error so much as 2", especially if he also avoids altitudes less than 10. When we remember that the least count of the sextant reading is 10", and that to the probable error of observation we must add the errors of gradua- tion, of eccentricity, and of the index correction, it must be con- ceded that we cannot hope to reduce the probable error of an observed distance below 5", if indeed we can reduce it below 10". Our approximate method is, therefore, for all practical purposes, a perfect method, in relation to our present means of observation. 252. If the altitudes have not been observed, they may be computed from the hour angles and declinations of the bodies, the hour angles being found from the local time and the right ascensions. But the declination and right ascension of the moon will be taken from the Ephemeris for the approximate Green- wich time found with the assumed longitude. If, then, the assumed longitude is greatly in error, a repetition of the computation may be necessary, starting from the Greenwich time furnished by the first. As a practical rule, we may be satisfied with the first computation when the error in the assumed longitude is not more than 30*. In the determination of the longitude of a fixed point on land, it will be advisable to omit the observation of the altitudes, as thereby the observer gains time to multiply the observations of the distance. But at sea, w r here an immediate result is required with the least expenditure of figures, the alti- tudes should be observed. 253. At sea, the observation is noted by a chronometer regu- lated to Greenwich time, and the most direct employment of the resulting Greenwich time will then be to determine the true correction of the chronometer. This proceeding has the advan 416 LONGITUDE. tage of not requiring an exact determination of the local time at the instant of the observation. For example, suppose the observation in the example above computed had been noted by a Greenwich mean time chrono- meter which gave 15 7 ' 10"* 0*, and was supposed to be slow 4 m 6". The true Greenwich time according to the lunar observation was 15 A 13'" s , and hence the true correction was -|- 3 m 0*. "With this correction we may at any convenient time afterwards deter- mine the longitude by the chronometer (Art. 214). In this way the navigator may from time to time during a voyage determine the correction of the chronometer, and, by taking the mean of all his results, obtain a very reliable correc- tion to be used when approaching the land. He may even determine the rate of the chronometer with considerable accu- racy by comparing the mean of a number of observations in the first part of the voyage with a similar mean in the latter part of it. 254. To correct the longitude found by a lunar distance for errors of the Ephemeris. In relation to the degree of accuracy of the observation, we may in the present state of the Ephemeris regard all its errors as insensible except those which aftect the moon's place. If, therefore, the longitude of a fixed point has been found by a lunar distance on a certain date, the corrections of the moon's right ascension and declination are first to be found for that date from the observations at one or more of the prin- cipal observatories, and then the correction of the longitude will be found as follows. Let o, S = the right ascension and declination of the moon given in the Ephemeris for the date of the observation, A, A = those of the sun, planet, or star, S a ,8d= the corrections of the moon's right ascension and declination, Sd = the corresponding correction of the lunar distance, 8L = the corresponding correction of the computed longi- tude; In Fig. 30, M and S being the geocentric places of the two bodies, as given in the Ephemeris, and d denoting the distance MS, we have cos d = sin d sin J -|- cos S cos J cos (a A~) (463) BY LUNAR DISTANCES. 417 by differentiating which we find cos 3 cos J sin (a A} sin d cos d sin A sin d cos J cos (a A") . : * . 33 (464) sin cZ If then v = the change of distance in 3*, we shall have 3L = 3d x - (465) in computing which we employ the proportional logarithm of the 3* Ephemeris, Q = log , reduced to the time of the observation. EXAMPLE. At the time of the observation computed in Art. 250, we have Moon, a = 2* 11- 14' * = + 14 18'.4 Sun, A = 23 22 25 A = - - 4 3 .1 a A= 2 49 19 d= 45 48.2 = 42 19'.8 with which we find, by (464), 3d = 0.908 8a + 0.350 S3- and hence, by (465), with log Q = 0.2511, 3L = 1.62 Sa 0.62 33 Suppose then we find from the Greenwich observations da 0'.38 = - 5".7 and dd = - 4".0, the correction of the longi- tude above found will be 3L = + ll'J 255. To find the, longitude by a lunar distance not given in the Ephemeris. The regular lunar-distance stars mentioned in Art. 247 are selected nearly in the moon's path, and are therefore in general most favorable for the accurate determination of the Greenwich time. Nevertheless, it may occasionally be found expedient to employ other stars, not too far from the ecliptic. Sometimes, too, a different star may have been observed by mistake, and it may be important to make use of the observation, VOL. I. 27 418 LONGITUDE. The true distance d is to be found from the observed distance by the preceding methods, as in any other case. Let the local time of the observation be T 7 , and the assumed longitude L. Take from the Ephemeris the moon's right ascension a and de- clination d for the Greenwich time T -\- _L, and also the star's right ascension A and declination J ; with which the correspond- ing true distance d ti is found by the formula cos d = sin d sin J -f- cos d cos J cos (a. A) Then, if d = d w the assumed longitude is correct ; if otherwise, put />. = the increase of o in one minute of mean time, /9 = the increase of 3 " " " " f = the increase of d " " " " then we have, by (464), cos d cos J sin (o A) cos 3 sin J sin 8 cos A cos (a A) sin d sin d and hence the correction of the assumed longitude in seconds of time, For computation Ly logarithms, these formulae may be ar- ranged as follows : tan J tanJlf = cos 8 cos J sin (a r = * : COS (a A) sin A cos (8 M ) (466) EXAMPLE. Suppose an observer has measured the distance of the moon from Arcturus, at the local mean time 1856 March 16, T 10* 30"' s , in the assumed longitude L = 6 A O m O 8 , and, reducing his observation, finds the true distance d = 73 55' 10" what is the true longitude ? BY LUNAR DISTANCES. 419 For the Greenwich time T + L = 16* 30" 1 we find a = 8* 47" 6v54 8 = + 23 12' 7".l * = + 31".40 A = 14 9 7 .04 A = + 19 55 44 .8 0= 8 .62 a _4 = _ 5* 22 0-.50 - - 80 30' 7".5 with which we find by (466), d = 73 55' 35". r = 25".59 d d = 25" d = + 58.6 and therefore the longitude is 6* O" 1 58'.6. 256. In order to eliminate as far as possible any constant errors of the instrument used in measuring the distance, we should observe distances from stars both east and west of the moon. If the index correction of the sextant is in error, the errors produced in the computed Greenwich time, and conse- quently in the longitude, will have different signs for the two observations, and will be very nearly equal numerically : they will therefore be nearly eliminated in the mean. If, moreover, the distances are nearly equal, the eccentricity of the sextant will have nearly the same effect upon each distance, and will there- fore be eliminated at the same time with the index error. Since even the best sextants are liable to an error of eccentricity of as much as 20", according to the confession of the most skilful makers, and this error is not readily determined, it is important to eliminate it in this manner whenever practicable. If a circle of reflexion is employed which is read oft' by two opposite verniers, the eccentricity is eliminated from each observation ; but even with such an instrument the same method of observa- tion should be followed, in order to eliminate other constant errors. It has been stated by some writers that by observing distances of stars on opposite sides of the moon we also eliminate a con- stant error of observation, such, for example, as arises from a faulty habit of the observer in making the contact of the moon's limb with the star. This, however, is a mistake; for if the habit of the observer is to make the contact too close, that is, to bring the reflected image of the moon's limb somewhat over the star, the effect will be to increase a distance on one side of the moon while it diminishes that on the opposite side, and the effect upon the deduced Greenwich time will be the same in 420 LONGITUDE. both cases. This will be evident from the following diagram, (Fig. 31). Suppose a and b Fig ', 31 ' are the two stars, M the moon's limb. If the observer - a judges a contact to exist when the star appears within the moon's disc as at c, the distance ac is too small and the distance be too great. But, supposing the moon to be moving in the direc- tion from a to 6, each distance will give too early a Greenwich time, for each will give the time when the moon's limb was actually at c. If, however, we observe the sun in both positions, this kind of error, if really constant, will be eliminated ; for, the moon's bright limb being always turned towards the sun, the error will increase both distances, and will produce errors of opposite sign in the Greenwich time. Hence, if a series of lunar distances from the sun has been observed, it will be advisable to form two distinct means, one, of all the results obtained from increasing distances, the other, of all those obtained from decreasing dis- tances : the mean of these means will be nearly or quite free from a constant error of observation, and also from constant in- strumental errors. FINDING THE LONGITUDE AT SEA. 257. By chronometers. This method is now in almost universal use. The form under which it is applied at sea differs very slightly from that given in Art. 214. The correction of the chronometer on the time of the first meridian (that of Green- wich among American and English navigators) is found at any place whose longitude is known, and at the same time also its daily rate is to be established with all possible care. The rate being duly allowed for from day to day during the voyage, the Greenwich time is constantly known, and therefore at any instant when the local time is obtained by observation, the lon- gitude of the ship is determined. The local time on shipboard is always found from an altitude of some celestial object, observed with the sextant from the sea horizon. (Art. 156.) The computation of the hour angle is then made by (268), and the resulting local time is compared directly with the Greenwich time given by the chronometer at A'l 1 SEA. 421 the instant of the observation. The data from the Ephemeris required in computing the local time are taken for the Greenwich time given by the chronometer. EXAMPLE. A ship being about to sail from New York, the master determined the correction on Greenwich time and the rate of his chronometer by observations on two dates, as follows: 1860 April 22, at Greenwich noon, chron. correction = -{- 3 m lO'.O 30, " " " " == H- 3 43 - 6 Eate in 8 days = -f 33 .6 Daily rate = -f 4 .2 On May 18 following, about 7* 30 m A.M., the ship being in lati- tude 41 33' N., three altitudes of the sun's lower limb were observed from the sea horizon as below. The correction of the chronometer on that day is found from the correction on April 30 by adding the rate for 18 days. (It will not usually be worth while to regard the fraction of a day in computing the total rate at sea.) The record of the observation and the whole computa- tion may be arranged as follows : I860 May 18. $ = 41 33' Chronometer 9* 37 m 21' " 37 53. " 38 20, Q 29 40 7 10" ~ " 46 " 50 50 Barom. 30.32"'. Therm. 59 F. Mean = 9 37 51 , Correction = -f 4 59 3 Mean = 29 45 40 ,2 Index corr. = 1 10 Gr. date = May 17, 21 42 50 .5 Dip =42 for which time we take from the 29 40 28 Ephemeris the quantities Semid. = -f- 15 50 Q's 6 = 19 38' 39" Refraction = 1 42 Semidiameter = 15' 50" Parallax = -f 8 Equation of time 3" 49 .8 h = 29 54 44 = 41 33 P = 70 21 21 sec 0.12588 cosec 0.02604 cos 9.51464 sin 9.81692 a = 70 54 33 a h = 40 59 49 Apparent time = 7*32 6.3 9.48348 srn 9.74174 Local mean time = 19 28 Gr. " =21 42 Longitude 16.5 50.5 = 2 14 34 = 33 38'.5 W. In this observation, the sun was near the prime vertical, a posi- tion most favorable to accuracy (Art. 149). 422 LONGITUDE. The method by equal altitudes may also be used for finding the time at sea in low latitudes, as in Arts. 158, 159. 258. In order that the longitude thus found shall be worthy of confidence, the greatest care must be bestowed upon the determination of the rate. As a single chronometer might deviate very greatly without being distrusted by the navigator, it is well to have at least three chronometers, and to take the mean of the longitudes which they severally give in every case. But, whatever care may have been taken in determining the rate on shore, the sea rate will generally be found to differ from it more or less, as the instrument is affected by the motion of the ship ; and, since a cause which accelerates or retards one chro- nometer may produce the same effect upon the others, the agree, ment of even three chronometers is not an absolutely certain proof of their correctness. The sea rate may be found by determining the chronometer correction at two ports whose difference of longitude is well known, although the absolute longitudes of both ports may be somewhat uncertain. For this purpose, a "Table of Chronometric Differences of Longitude" is given in RAPER'S Practice of 'Navigation, the use of which is illustrated in the following example. EXAMPLE. At St. Helena, May 2, the correction of a chro- nometer on the local time was A 23"* 10*. 3. At the Cape of Good Hope, May 17, the correction on the local time was -f l h 14 m 28'. 6 ; what was the sea rate ? We have Corr. at St. Helena, May 2d = 0* 23" 10'.3 Chron. diff. of long, from Eaper = -f 1 36 45 . Corr. for Cape of G. H., May 2d = -f 1 13 34 .7 " 17th = -J- 1 14 28.6 Eate in 15 days = -f- 53 .9 Daily sea rate = -f- 3 .59 259. By lunar distances. Chronometers, however perfectly made, are liable to derangement, and cannot be implicitly relied upon in a long voyage. The method of lunar distances (Arts. 247-256) is, therefore, employed as an occasional check upon the chronometers even where the latter are used for finding the longitude from day to day. When there is no chronometer on AT SEA. 423 board, the method of lunar distances is the only regularly avail- able method for finding the longitude at sea, at once sufficient!} accurate and sufficiently simple. As a check upon the chronometer, the result of a lunar distance is used as in Art. 253. In long voyages an assiduous observer may determine the sea rates of his chronometers with considerable precision. For this purpose, it is expedient to combine observations taken at various times during a lunation in such a manner as to eliminate as far as possible constant errors of the sextant and of the observer (Art. 256). Suppose distances of the sun are employed exclusively. Let two chronometer corrections be found from two nearly equal distances measured on opposite sides of the sun on two different dates, in the first and second half of the lunation respectively. The mean of these corrections will be the correction for the mean date, very nearly free from constant instrumental and personal errors. In like manner, any number of pairs of equal, or nearly equal, distances may be combined, and a mean chro- nometer correction determined for a mean date from all the observations of the lunation. The sea rate will be found by comparing two corrections thus determined in two different lunations. This method has been successfully applied in voyages between England and India. 260. By the eclipses of Jupiter's satellites. An observed eclipse of one of Jupiter's satellites furnishes immediately the Green- wich time without any computation (Art. 225.) But the eclipse is not sufficiently instantaneous to give great accuracy ; for, with the ordinary spy-glass with which the eclipse may be observed on board ship, the time of the disappearance of the satellite may precede the true time of total eclipse by even a whole minute. The time of disappearance will also vary with the clearness of the atmosphere. Since, however, the same causes which accele- rate the disappearance will retard the reappearance, if both phenomena are observed on the same evening under nearly the same atmospheric conditions, the mean of the two resulting longitudes will be nearly correct. Still, the method has not the advantage possessed by lunar distances of being almost always available at times suited to the convenience of the navigator. 261. By the moon's altitude. This method, as given in Art. 243, 424 CIKCLES OF POSITION. may be used at sea in low latitudes; but, on account of the unavoidable inaccuracy of an altitude observed from the sea horizon, it is even less accurate than the method of the preceding article, and always far inferior to the method of lunar distances, although on shore it is one which admits of a high degree of precision when carried out as in Art. 245. 262. By occultations of stars by the moon. This method, which will be treated of in the chapter on eclipses, may be successfully used at sea, as the disappearance of a star behind the moon's limb may be observed with a common spy-glass at sea with nearly as great a degree of precision as on shore ; but, on account of the length of the preliminary computations as well as of the subsequent reduction of the observation, it is seldom that a navigator would think of resorting to it as a substitute for the convenient method of lunar distances. CHAPTER VIII. FINDING A SHIPS PLACE AT SEA BY CIRCLES OF POSITION. 263. IN the preceding two chapters we have treated of methods of finding the position of a point on the earth's surface by the. two co-ordinates latitude and longitude; and therefore in all these methods the required position is determined by the inter- section of two circles, one a parallel of latitude and the other a meridian. In the following method it is determined by circles oblique to the parallels of latitude and the meridians. The prin- ciple which underlies the method has often been applied ; but its value as a practical nautical method was first clearly shown by Capt. THOMAS II. SUMNER.* Let an altitude of the sun (or any other object) be observed at any time, the time being noted by a chronometer regulated to Greenwich time. Suppose that at this Greenwich time the sun * A new and accurate method of finding a ship's position at sea by projection on Merca- tor's chart: by Capt. THOMAS H. SUMNER. Boston, 1843. SUMNER'S METHOD. 425 is vertical to an observer at the point M of the globe (Fig 32). Let a small circle AA'A" be described on the globe from M as a pole, with a polar dis- Fig.jj2. tance MA equal to the zenith distance, or complement of the observed altitude, of the sun. It is evident that at all places within this circle an observer would at the given time observe a smaller zenith distance, and at all places without this circle a greater zenith distance ; and therefore the observa- tion fully determines the observer to be on the circumference of the small circle AA'A". If, then, the navigator can project this small circle upon an artificial globe or a chart, the knowledge that he is upon this circle will be just as valuable to him in enabling him to avoid dangers as the knowledge of either his latitude alone or his longitude alone; since one of the latter elements only determines a point to be in a certain circle, without fixing upon any particular point of that circle. The small circle of the globe described from the projection 01 the celestial object as a pole we shall call a circle of position. 264. To find the place on the globe at which the sun is vertical (or the sun's projection on the globe) at a given Greenwich time. The sun's hour angle from the Greenwich meridian is the Greenwich apparent time. The diurnal motion of the earth brings the sun into the zenith of all the places whose latitude is just equal to the sun's declination. Hence the required projection of the sun is a place whose longitude (reckoned westward from Green- wich from 0* to 24*) is equal to the Greenwich apparent time, and whose latitude is equal to the sun's declination at that time. 265. From an altitude of the sun taken at a given Greenwich time, to find the circle of position of the observer, by projection on an artificial globe. Find the Greenwich apparent time and the sun's declina- tion, and put down on the globe the sun's projection by the preceding article. From this point as a pole, describe a small circle with a circular radius equal to the true zenith distance deduced from the observation. This will be the required circle of position. 266. The preceding problem may be extended to any celestial 426 CIRCLES OF POSITION. object. The pole of the circle of position will always be the place whose west longitude is the Greenwich hour angle of the object (reckoned from 0* to 24 A ) and whose latitude is the decli- nation of the object. The hour angle is found by Art. 54. 267. To find both the latitude and the longitude of a ship by circles of position projected on an artificial globe. First. Take the altitudes of two different objects at the same time by the Greenwich chronometer. Put down on the globe, by the preceding problem, their two circles of position. The observer, being in the circum- ference of each of these circles, must be at one of their two points of intersection ; which of the two, he can generally determine from an approximate knowledge of his position. . Second. Let the same object be observed at two different times, and project a circle of position for each. Their intersection gives the position of the ship as before. If between the observa- tions the ship has moved, the first altitude must be reduced to the second place of observation by applying the correction of Art. 209, formula (380). The projection then gives the ship's position at the second observation. 268. From an altitude of a celestial body taken at a given Greenwich time, to find the circle of position of the observer, by projection on a Mercator chart. The scale upon which the largest artificial globes are constructed is much smaller than that of the working charts used by navigators. But on the Mercator chart a circle of position will be distorted, and can only Fig. 33. be laid down by points. Let L, L', L" / (Fig. 33) be any parallels of latitude L> , crossed by the required circle. For each of these latitudes, with the true altitude L found from the observation and the polar distance of the celestial body taken for the Greenwich time, compute the local time, and hence the longitude, " by chro-, nometer" (Art. 257). Let I, I', I" be the longitudes thus found. Let A, A', A" be the points whose latitudes and longitudes are, respectively, L, 1; L', I' ; L", I" ; these are evidently points of the required circle. The ship is consequently in the curve AA'A", traced through these points. SUMNER'S METHOD. 427 In practice it is generally sufficient to lay down only two points ; for, the approximate position of the ship being known, if L and L' are two latitudes between which the ship may be assumed to be, her position is known to be on the curve AA' somewhere between A and A'. When the difference between L and L' is small, the arc AA' will appear on the chart as x. straight line. 269. To find the latitude and longitude of a ship by circles of position projected on a Mercator chart. First. Let the altitudes of two objects be taken at the same time. Assume two latitudes em- bracing between them the ship's probable position, and find two points of each of their two circles of position by the preceding problem, and project these points on the chart. Each pair of points being joined by a straight line, the intersection of the two lines is B> ' A> very nearly the ship's position. Thus, if one object gives the points A, A' (Fig. 34) corresponding to the lati- tudes Z/, Z/, and the other object the points B, B' corresponding to the same latitudes, the ship's position is the point C, the intersection of AA' and BB'. It is, of course, not essential that the same latitudes should be used in computing the points of the two circles ; but it is more convenient, and saves some logarithms. If greater accuracy is desired, the circles may be more "fully laid down by three or more points of each. Second. The altitude of the same object may be taken at two different times, and the circles laid down as before ; the usual reduction of the first altitude being applied when the ship changes her position between the observations. It is evident from the nature of the above projection that the most favorable case for the accurate determination of the inter- section C is that in which the circles of position intersect at right angles. Hence the two objects observed, or the two positions of the same object, should, if possible, differ about 90 in azimuth. This agrees with the results of the analytical discussion of the method of finding the latitude by two altitudes, Art. 183. If the chronometer does not give the true Greenwich time, the only effect of the error will be to shift the point C towards the east or the west, without changing its latitude, unless the error is 428 CIRCLES OF POSITION. so groat as to affect sensibly the declination which is taken from the Ephemeris for tiie time given by the chronometer. This method is, therefore, a convenient substitute for the usual method of find- ing the latitude at sea by two altitudes, a projection on the sailing chart being always sufficient for the purposes of the navigator. Instead of reducing the first altitude for the change of the ship's position between the observations, we may put down the circle of position for each observation and afterwards shift one of them by a quantity due to the ship's run. ff _ ' "A- 'a' _ Thus, let the first observation give the position line AA' (Fig. 35), and let Aa represent, in direction and length, the ship's course and distance sailed be- a tween the observations. Draw aa' parallel to AA' . Then, BB' being the position line by the second observation, its intersection C with aa' is the required position of the ship at the second observation. 270. If the latitude is desired by computation, independently of the projection, it is readily found as follows. Let l v I t = the longitudes (of A and /?) found from the first and second altitudes respectively with the latitude L, IJ, l{ = the longitudes (of A' and B'~) found from the same altitudes with the latitude L', L = the latitude of C. From Fig. 34 we have, by the similarity of the triangles ABC euidA'B'C, Z/_ // : 1^-1,= B'C : BC whence (*;-,') + (*,-*.) : 1,-l^BB': BC = L' - L : L u - L --VL (467) This formula reduces SUMNER'S method of " double altitudes" to that given long ago by LALANDE (Astronomic, Art. 39P2, and Abreye de Navigation, p. 68). The distinctive feature of SUMNER'S process, however, is that a single altitude taken at any timo is made available for determining a line of the globe on ivhich the ship is situated. MERIDIAN LINE. 429 271. To find the azimuth of the sun by a position line projected on the chart. Let AA' (Fig. 36) be a position line on the chart, derived from an observed altitude by * Art. 268. At any point C of this line draw CM perpendicular to AA', and let NC8 be the meri- dian passing through (7; then SCM is evidently the sun's azimuth. The line CM is, of course, drawn on that side of the meridian NS upon / which the sun was known to be at the time of A the observation. The solution is but approximate, since AA' should be a curve' line, and the azimuth of the normal CM would be different for different points of AA'. It is, however, quite accurate enough for the purpose of determining the variation of the compass al sea, which is the only practical application of this problem. CHAPTER IX. THE MERIDIAN LINE AND VARIATION OF THE COMPASS. 272. THE meridian line is the intersection of the plane of the meridian with the plane of the horizon. Some of the most use- ful methods of finding the direction of this line will here be briefly treated of; but the full discussion of the subject belongs to geodesy. 273. By the meridian passage of a star. If the precise instant when a star arrives at its greatest altitude could be accurately distinguished, the direction of the star at that instant, referred to the horizon, would give the direction of the meridian line ; but the altitude varies so slowly near the meridian that this method only serves to give a first approximation. 274. By shadows. A good approximation may be made as follows. Plant a stake upon a level piece of ground, and give it a vertical position by means of a plumb line. Describe one or 430 MERIDIAN LINE. more concentric circles on the ground from the foot of the stake as a centre. At the two instants before and after noon when the shadow of the stake extends to the same circle, the azimuths of the shadow east and west are equal. The points of the circle at which the shadow terminates at these instants being marked, let the included arc be bisected ; the point of bisection and the centre of the stake then determine the meridian line. Theoretically, a small correction should be made for the sun's change of declina- tion, but it would be quite superfluous in this method. 275. By single, altitudes. TVith an altitude and azimuth instru- ment, observe the altitude of a star at the instant of its passage over the middle vertical thread (at any time), arid read the horizontal circle. Correct the observed altitude for refraction. Then, if h = the true altitude,

M) Now, let 0, Fig. 37, be the apparent position of the terrestrial point, projected upon the celestial sphere; $ the apparent place of the sun, Z the zenith, P the pole ; and put MERIDIAN LINE. 433 D the apparent angular distance of the Fig. 37. sun's centre from the terrestrial point = the observed distance increased by the sun's semidiameter, H= the apparent altitude of the point, .V = the sun's apparent altitude, a = the difference of the azimuth of the sun and the point, A' = the azimuth of the point. The apparent altitude h' will be deduced from the true altitude by adding the refraction and subtracting the parallax. Then in the triangle SZO we have given the three sides ZS = 90 A', ZO = 90 If, SO = D, and hence the angle SZO = a can be found by the formula sin (s jff) sin (s A') tan 2 a = cos s cos (s D) in which (472) Then we have A' = A a (473) and the proper sign of a to be used in this equation must be determined by the position of the sun with respect to the object at the time of the observation. If the altitude of the sun is observed, we can dispense with the computation of (471), and compute A by the formula (468). The chronometer will not then be required, but an approximate knowledge of the local .time and the longitude is necessary in order to find d from the Ephemeris. If the terrestrial object is very remote, it will often suffice to regard its altitude as zero, and then we shall find that (472) reduces to tan \ a = ^[tan } (D + A') tan 1 (D A')] (474) This method is frequently used in hydrographic surveying to determine the meridian line of the chart. EXAMPLE. From a certain point B in a survey the azimuth of a point is required from the following observation : Chronometer time = 4* 12 m 12' Chronom. correction 2 Local mean time = 4 10 12 Equation of time = 4 10.9 Local app. time, t = 4 6 1 .1 VOL. I. 28 Altitude of C=H= 30' 20" Distance of the nearest limb of the sun from the point C == 48 17' 10" Semidiameter = 10 1 D = 48 33 11 434 MERIDIAN LINE. The sun's declination was S = -f 4 16' 55", the latitude was p ~ + 38 58' 50" ; and hence, by (4T1), we find A = 74 36' 36" h = 24 37' 58" Refraction and parallax = 1 54 h' = 24 39 52 and, by (472), a = 43 35' 6" N^ow, the sun was on the right of the object, and hence A'= A a = 31 1'30" Therefore, a line drawn on the chart from B on the left of the line BC, making with it the angle 31 1' 30", will represent the meridian. 280. By two measures o/ the distance of the sun from a terrestrial object. In the practice of the preceding method with the sextant, it is not always practicable to measure the apparent altitude of the terrestrial object. We may then measure the distance of the sun from the object at two different times, and, first com- puting the altitude and azimuth of the sun at each observation, we may from these data compute the altitude of the object and the difference between its azimuth and that of the sun at either observation, by formulae entirely analogous to those employed in computing the latitude and time from two altitudes, Art. 178, (304), (305), (306), and (307). 281. By the azimuth of a star at a given time. When the time is known, the azimuth of the star is found by (471) : hence we have only to direct the telescope of an altitude and azimuth instrument to the star at any time, and then compare the read- ing of its horizontal circle with the computed azimuth. This method will be very accurate if a star near the pole is employed, since in that case an error in the time will produce a comparatively small error in the azimuth. It will be most accu- rate if the star is observed at its greatest elongation, as in the following article. 282. By the greatest elongation of a circumpolar star. At the instant of the greatest elongation we have, by Art. 1 8, cos 8 sin A = MERIDIAN LINE. 435 in which A is the azimuth reckoned from the elevated pole. At this instant the star's azimuth reaches its maximum, and for a certain small interval of time appears to be stationary, so that the observer has time to set his instrument accurately upon th star. In order to be prepared for the observation, the time of the elongation must be (at least approximately) known. The hour angle of the star is found by the formula tan

vil principles ; and the investigation of solar eclipses, with which v 1 34' 53", and doubtful between these limits. For the doubtful cases we must apply (478), or for greater precision (477), using the actual values of ;:, ', 5, s', A, and / for the date. EXAMPLE. On July 18, 1860, the conjunction of the moon and sun in longitude occurs at 2* 19'".2 Greenwich mean time: will an eclipse occur ? We find at this time, from the Ephemeris, = 33' 18".6 which, being within the limit 1 23' 15", renders an eclipse cer- tain at this time. Having thus found that an eclipse will be visible in some part of the earth, we can proceed to the exact computation of the phenomenon. The method here adopted is a modified form of BESSEL'S,* which is at once rigorous in theory and simple in practice. For the sake of clearness, I shall develop it in a series of problems. Fundamental Equations of the Theory of Eclipses. 288. To investigate the condition of the beginning or ending of a solar eclipse at a given place on the earth's surface. The observer sees the limbs of the sun and moon in apparent contact when he is situated in the surface of a cone which envelops and is in contact with the two bodies. We may have two such cones : * See Astronomische Nachrichten, Nos. 151, 152, and, for the full development of the method with the utmost rigor, BESSEI/S Axtronomische Untersuchungen, Vol. II. HANSEN'S development, based upon the same fundamental equations, but theoreti- cally less accurate, may also be consulted with advantage: it is given in Astronom. Nach., Nos. 339-342. 440 SOLAR ECLIPSES. First. The cone whose vertex falls between the sun and the moon, as at V, Fig. 39, and which is called the penumbral cone. An observer at C\ in one of the elements CB V of the cone, sees the points A and B of the limbs of the sun and moon in apparent exterior contact, which is either the first or the last contact ; that is, either the beginning or the ending of the whole eclipse. Fig. 39. Fig. 40. 7 Second. The cone whose vertex is beyond the moon (in the direction of the earth), as at F, Fig. 40, and which is called the umbral cone, or cone of total shadow. An observer at (7, in the element CVBA, sees the points A and _B of the limbs of the sun and moon in apparent interior contact, which is the beginning or the ending of annular eclipse in case the observer is farther from the moon than the vertex of the cone (as in the figure), and which is either the beginning or the ending of total eclipse in case the observer is between the vertex of the cone and the moon. If now a plane is passed through the point C\ at right angles to the axis SVD of the cone, its intersection with the cone will FUNDAMENTAL EQUATIONS. 441 be a circle (the sun and moon being regarded as spherical) whose radius, CD, we shall call the radius of the shadow (penumbral or umbral) for that point. The condition of the occurrence of one of the above phases to an observer is, then, that the distance of the point of observation from the axis of the shadow is equal to the radius of the shadow for that point. The problems which follov* will enable us to translate this condition into analytical language 289. To find for any given time the position of the axis of the shadow. The axis of the cone of shadow produced to the celes- tial sphere meets it in that point in which the sun would he projected upon the sphere by an observer at the centre of the moon. Let 0, Fig. 41, be the centre of the earth ; S, that of the sun ; Jf, that of the moon. The line MS produced to the infinite celestial sphere meets it in the common vanishing point of all lines parallel to MS; that is, in the point Z, in which the line OZ, drawn through the centre of the earth parallel to MS, meets the sphere. The position of the axis of the cone will be determined by the right ascension and declination of the point Z. In order to determine the point Z, let the positions of the sun and moon be expressed by rectangular co-ordinates (Art. 32), of w l iich the axis of x is the straight line drawn through the centre ot the earth and the equinoctial points, the axis of y the inter- action of the planes of the equator and solstitial colure, and the axis of z the axis of the equator. Let x be taken as positive towards the vernal equinox ; y as positive towards the point of the equator whose right ascension is 90 ; z as positive towards the north. Let a, d, r = the right ascension, declination, and distance from the centre of the earth, respectively, of the moon's centre, a', (479) g sin d = sin 8' b sin 8 j FUNDAMENTAL EQUATIONS. 443 where the second members, besides the right ascensions and declinations, involve only the quantity 6, which may be expressed in terms of the parallaxes as follows : Let TT = the moon's equatorial horizontal parallax, ' = the sun's " " " then we have (Art. 89) r sin TT' If, further, x = the sun's mean horizontal parallax, and r' is expressed in terms of the sun's mean distance from the earth, we have, as in (146), and hence , sin T sin d = b = m*- (480) r' sin it which is the most convenient form for computing 6, because r' and n are given in the Ephemeris, and TT O is a constant. 290. The equations (479) are rigorously exact, but as 6 is onlj about -fa, and a a' at the time of an eclipse cannot exceed 1 43', a a' is a small arc never exceeding 17", which may be found by a brief approximative process with great precision The quotient of the first equation divided by the second gives b cos <5 sec 3' sin (a a') tan (a a') 1 b cos <5 sec d' cos (a a') where the denominator differs from unity by the small quantity b cos d sec d' cos (a a') ; and, since d and d' are nearly equal, this small difference may be put equal to b, and we may then write the formula thus :* i COS d SCO , fi = the given sidereal time; then, if in Fig. 41 we had taken M for the place of observation, M' would have been the geocentric zenith with the right ascen- sion p and declination y>', and, the distance of the place from the origin being p, we should have found = p cos ' cos d cos 9?' sin d cos (^ a)] > (483) = p [sin ' sin (fi a) ~) d) V(483*) C = A cos OB d) J The equations (482) might be similarly treated; but the most accurate form for tljeir computation is (482*). The quantity // a is the hour angle of the point Z for the meridian of the given place. To facilitate its computation, it is convenient to find first its value for the Greenwich meridian- Thus, if we put for any given Greenwich mean time T jtij = the hour angle of the point Zat the Greenwich meridian, w = the longitude of the given place, FUNDAMENTAL EQUATIONS. 447 we have // a = fJ^ ta To find fo we have only to convert the Greenwich mean time T into sidereal time and to subtract . By means of the formulae (482) and (483) the co-ordinates of the moon and of the place of observation can be accurately com- puted for any given time. Now, the co-ordinates x and y of the moon are also those of every point of the axis of the shadow : so that if we put A = the distance of the place of observation from the axis of the shadow, we have, evidently, J=(*-*)+(y-i0 (484) [The co-ordinates z and have also been found, as they will be required hereafter.] 292. The distance J may be determined under another form, which we shall hereafter tind useful. Let M', Fig 42 Fig. 42, be the apparent position of the moon's centre in the celestial sphere as seen from the place of observation ; P the north pole ; Z the point where the axis of the cone of shadow meets the sphere, as in Fig. 41 ; M Y , C v the projections of the moon's centre and of the place of observation on the principal plane. The distance C l M l is equal to J, and is the projection of the line joining the place of ' <*" observation and the moon's centre. The plane by which this line is projected contains the axis of the cone of shadow, and its intersection with the celestial sphere is, therefore, a great circle which passes through Z, and of which ZM' is a portion. Hence it follows that C l M l makes the same angle with the axis of y that M'Z makes with PZ: so that if we draw (\N and parallel to the axes of y and x respectively, and put Q = PZM' = we have, from the right triangle A sin Q = x c J G o S Q = y-v the sum of the squares of which gives again the formula (484). 448 SOLAR ECLIPSES. 293. To find the radius of the shadow on the principal plane, or on any given plane parallel to the principal plane. This radius is evi- dently equal to the distance of the vertex of the cone of shadow from the given plane, multiplied by the tangent of the angle of the cone. In Figs. 39 and 40, p. 440, let EF be the radius of the shadow on the principal plane, CD the radius on a parallel plane drawn through C. Let H = the apparent semidiameter of the sun at its mean dis- tance, k = the ratio of the moon's radius to the earth's equatorial radius, / = the angle of the cone = EVF, c = the distance of the vertex of the cone above the princi- pal plane = VF, C = the distance of the given parallel plane above the prin- cipal plane = DF, I = the radius of the shadow on the principal plane = EF, L = the radius of the shadow on the parallel plane = CD. If the mean distance of the sun from the earth is taken as unity, we have the earth's radius = sin TT O , the moon's radius = k sin - = MB, the sun's radius = sin H = SA, and, remembering that G = r'g found by (481) is the distance MS, we easily deduce from the figures r'g (4861 in which the upper sign corresponds to the penumbral and the lower to the umbral cone. The numerator of this expression involves only constant quan- tities. According to BESSEL, H= 959". 788 ; ENCKE found r = 8".57116; and the value of A", found by BURCKHARDT from eclipses and occultations, is k = 0.27227 ;* whence we have log [sin H + k sin TT O ] 7.6688033 for exterior contacts, log [sin If k sin TT O ] = 7.6666913 for interior contacts. * The value of k here adopted is precisely that which the more recent investiga- tion of OODEMANS (Astron. Nach., Vol. LI. p. 30) gives for eclipses of the sun. For occultations, a slightly increased value seems to be required. FUNDAMENTAL EQUATIONS. 449 Now, taking the earth's equatorial radius as unity, we have VM=JL. sin/ MF = z (Art. 291) and hence c = 2 (487) sin/ the upper sign being used for the penumbra and the lower for the umbra. We have, then, I = c tan/ = z tan / k sec/ =Z Ctan/ For the penumbral cone, c is always positive, and there- fore L is positive also. For the umbral cone, c is negative when the vertex of the cone falls below the plane of the observer, and in this case we have total eclipse : therefore for the case of total eclipse we shall have L = (c ) tan /a negative quantity. It is usual to regard the radius of the shadow as a positive quantity, and therefore to change its sign for this case ; but the analytical dis- cussion of our equations will be more general if we preserve the negative sign of L as the characteristic of total eclipse. When the vertex of the umbral cone falls above the plane of the observer, L is positive, and we have the case of annular eclipse. For brevity we shall put f = tan/ 1 l = ic V (489) L=l X ) 294. The analytical expression of the condition of beginning or ending of eclipse is A=L or, by (484) and (489), (x - *) + (y - ,) = (i- i:y (490) It is convenient, however, to substitute the two equations (485) for this single one, after putting L for J, so that VOL. I. 2 450 SOLAR ECLIPSES. may be taken as the conditions which determine the beginning or ending of an eclipse at a given place. The equation (490), which is only expressed in a different form by (491), is to be regarded as the fundamental equation of the theory of eclipses. 295. By Art. 292, so long as A is regarded as a positive quan- tity, Q is the position angle of the moon's centre at the point Z; and since the arc joining the point Z and the centre of the moon also passes through the centre of the sun, Q is the common position angle of both bodies. Again, since in the case of a contact of the limbs the arc joining the centres passes through the point of contact, Q will also be the position angle of this point when all three points sun's centre, moon's centre, and point of contact lie on the same side of Z. In the case of total eclipse, however, the point of contact and the moon's centre evidently lie on opposite sides of the point Z; and if I z> in (490) were a positive quantity, the angle Q which would satisfy these equa- tions would still be the position angle of the moon's centre, but would differ 180 from the position angle of the point of con- tact. But, since we shall preserve the negative sign of I i for total eclipse (Art. 293), (and thereby give Q values which differ 180 from those which follow from a positive value), the angle Q will in all cases be the position angle of the point of contact. 296. The quantities , d, x, y, /, and i may be computed by the formulae (480), (481), (482), (486), (487), (488), for any given time at the first meridian, since they are all independent of the place of observation. In order to facilitate the application of the equations (490) and (491), it is therefore expedient to com- pute these general quantities for several equidistant instants preceding and following the time of conjunction of the sun and moon, and to arrange them in tables from which their values for any time may be readily found by interpolation. The quantities x and y do not vary uniformly ; and in order to obtain their values with accuracy from the tables for any time, we should employ the second and even the third differences in the interpolation. This is effected in the most simple manner by the following process. Let the times for which x and y have been computed be denoted by T Q 2*, T 9 1*, T , T + 1*, FUNDAMENTAL EQUATIONS. 451 T + 2*, the interval being one hour of mean time ; and let the values of x and y for these times be denoted by x_ 2 , x_i, &c., y_2, 3/-i, &c. Let the mean hourly changes of x and y from the epoch T to any time T= T + r be denoted by x' and y' '. Then the values of x' and / for the instants T 2*, T 1 A , &c. will be formed as in the following scheme, where c denotes the third difference of the values of x as found from the series x_ 2 , x_ 1? &c. according to the form in Art. 69, and the difference for the instant T is found by the first formula of (77). The form for computing y' in the same. Time. X x' f 2* TO 1* X~\ -K- 1 o *'-) T .r ](^j .r.O |c T o+ 1* x l a:, # 7;+ 2* X, A (:r, j; ) If then we require x and y for a time T 7^ + r, we take r' and j/' from the table for this time, and we have x = x + x'r y = y, + y' r 297. EXAMPLE. Compute the elements of the solar eclipse of July 18, 1860 The mean Greenwich time of conjunction of the sun and n toon in right ascension is July 18, 2* 8 m 56'. The computation of the elements will therefore be made for the Greenwich hours v, 1, 2, 3, 4, and 5. For these hours we take the following quantities from the American Ephemeris : For the Moon. Greenwich mean time. a 6 7T July 18, 0* 116 44' 24".30 21 52' 20".3 59' 45".80 1 117 21 59 .10 42 32 .8 47 .13 2 3 117 59 30 .45 118 36 58 .35 32 36 .4 22 31 .2 48 .44 49 .72 4 119 14 22 .65 12 17 .2 50 .98 5 119 51 43 .35 1 54 .6 52 .22 452 SOLAR ECLIPSES. For the Sun. Greenwich mean time. a' 6' log r' July 18, 0* 117 59' 41".85 20 57' 56".20 0.0069675 1 118 2 12 .50 57 29 .42 61 2 118 4 43 .14 57 2 .60 47 3 118 7 13 .77 56 35 .75 33 4 118 9 44 .39 56 8 .86 19 5 118 12 15 .00 55 41 .94 05 The formulae to be employed will be here recapitulated, for convenient reference. I. For the elements of the point Z: b = sin TT O log sin r = 5.61894 a = a cos 8 sec 8' (a a') or, nearly, a = a! b (a c-'j lb b l b II. The moon's co-ordinates : x = r cos 8 sin (a a) y = r sin (8 d) cos 2 J (a a) -\- r sin (5 -\- d) sin 2 (a a ) z = r cos (d d) cos 2 i (a a} r cos (8 -f- d) sin 2 (a a) m. The angle of the cone of shadow and the radius of the shadow : For penumbra : or exterior contacts. [7.668803] sin / = )r umbra: or interior contacts B . n/= [7.666691] fg , log k = 9.435000, c = z sin/ sin/ i = tan / l=ic i = tan f lisfe FUNDAMENTAL EQUATIONS. 453 IV. The values of a, d, x, y, log z, and I, will then he tabulated and the differences x' and y' formed according to Art. 296. I give the computation for the three hours 1*, 2 A , and 3 A , in extenso. I. Elements of the point Z. 2* 3* a a' 40' 13". 40 5'12".69 +0 29' 44".58 66' + 45 3 .38 + 35 33 .80 + 25 55 .45 log cosec TT = log r 1.7596999 1.7595414 1.7593865 ar. co. log r' 9.9930339 9.9930353 9.9930367 (!) Constant log sin TT O log b 5.61894 7.37167 7.37152 7.37136 (2) ar. co. log (1 b) 0.001023 0.001023 0.001022 log cos 6 9.96805 9.96855 9.96905 log sec 6' 0.02973 0.02970 0.02968 log (a a' 3. 38263 2.49511 3.25154 log (a a') 0.75310 9.86590 nO. 62265 a a' + 5". 66 + 0".73 4". 19 (1) -} - (2) 7.37269 7.37254 7.37238 log(iJ 5') 3.43191 3.32915 3.19185 log (d - 6') 0.804GO nO.70169 nO.60423 d-6' 6". 38 5".03 3".67 a 118 2' 18".16 118 4'43".87 118 7' 9".58 d 20 57 23 .04 20 56 57 .57 20 56 32 .08 log (1-6)= log g 9.998977 9.998977 9.998978 II. Co-ordinates x, y y and z. a a 40' 19". 06 5'13".42+029'48".77 6 d + 45 9 .76 + 35 38 .83 + 25 59 .12 <* + d 42 39 55 .84 42 29 33 .97 42 19 3 .28 log sin (a a) n8.0692116 nl. 181 701 4 7.9381239 log cos 6 9.9680502 9.9685481 9.9690490 log r cos 'cos d p cos ' cos d cos # J The five equations in (493) and (494) involve the six variables , ^, , , which being dependent upon 1, and after an approximate value of sin

= -- r- p sin

cos y. = - - - j/(l ee sin 2 j) we shall have sin

,) = - T_L- - '- T/(l ee sin 2 ? ) or COS l = p COS tf>' j/(l ee} sin ' tan Hence the equations (494) become = cos >j sin 9 ij sin ^ cos rf |/(1 ee) cos fj sin ^ cos 9 C = sin ^ sin rf |/(1 ee) -f cos ^i cos ^ cos A Put p t sin rfj sin d p a sin d 2 = sin <^ |/(1 ee} Pj cos d l = cos d j/(l ee) p 2 cos d 2 = cos d The quantities p v d v p 2 , d v may be computed for the same times as the other quantities in the tables of the eclipse, and hence obtained by interpolation for the given time. The factors (\ and /> 2 will be sensibly constant for the whole eclipse. Wo now have = cos , sin # \ 7j l = sin l sin rf, cos # > (497) C, sin ^ sin d l -f- cos y>, cos d l cos # J The quantity t diifers so little from that we may in practice substitute one for the other in the small term i ; but if theo- retical accuracy is desired we can readily find when i* known ; for the second and third of (497) give cos

) sin = x - I ) (i-f: i )co 8 e = y -/>,,, V (499) e 2 + v + :, 2 = i J which for each assumed value of Q determine , ^, and r Then we have co? ^ sin 9 = f cos 0>j cos # = ij, sin d l -f- C, cos d t V (500) j= )jj cos rf t -j- C t sin d, 1 V J which determine ^ t and #. Then the latitude and longitude of a point of the required outline are found by the equations (501) To solve (499), let and f be found by the equations sin /9 sin f = x I sin Q = a V I cos Q sm cos f = i- - - = b then we have = sin /9 sin y -f- *, 8 i n 4^ ijj = sin ,9 cos y -f- z'Cj cos ^ OUTLINE OF THE SHADOW. 459 where we have omitted />, as a divisor of the small term /, cos Q, since we have very nearly ^Oj 1. Substituting these values in the last equation of (499), we find :, == cos z /3 2iC, sin cos ( 7-) (i^ Neglecting the terms involving i 2 as practically insensible, this gives C, [cos ,9 i sin /? cos (Q p)] In order to remove the ambiguity of the double sign, let us put Z = the zenith distance of the point Z(Art. 289) ; then, since & = // a is the hour angle of this point, we have cos Z = sin if sin d -f cos n the earth's surface, we must, in general, have Z less than 90 ; that is, cos Z must be positive, and therefore , must be taken only with the positive sign. The negative sign would give a second point on the surface of the earth from which, if the earth were not opaque, the same phase of the eclipse would also be observed at the given time. In fact, every element of the cone of shadow which intersects the earth's surface at all, intersects it in two points, and our solution gives both points. If we put t = ,-C08(g- r ) sin 1" we have C, = cos /? sin /? sin e or, with sufficient accuracy, * C, = cos (0 + e) (505) Thus, /? and ? being determined by (502), , is determined by (504) and (505) : hence also $ and 7, by the equations *=*? + #, si Q ,,= 6 + 1^008$ 460 SOLAR ECLIPSES. The problem is, therefore, fully resolved ; but, for the conve- nience of logarithmic computation, let c and C be determined by the equations ^ 1(507) ^ then the equations (500) become cos

l = c sin (C -\- dj j The curve thus determined will be the intersection of the penumbral cone, or that of the umbral cone, with the earth's surface, according as we employ the value of I for the one or the other. 299. The above solution is direct, though theoretically but approximate, since we have neglected terms of the order of i 2 . It can, however, readily be made quite exact as follows. We have, by substituting the values of ; and 3^ in (498), and neg- lecting the term involving the product i sin (d^ cQ, which is of the same order as i 2 , C = /o 2 cos (/? -f- e) p. 2 sin ,3 cos y sin (d^ d t ) and, putting e' = (d l d 2 ~) cos Y we have, within terms of the order i 2 , C = ftC08(/? + e + e') (509) The substitution of this value of in the term i involves only an error of the order 3 , which is altogether insensible. The exact solution of the problem is, therefore, as follows. Find /3 and 7- for each assumed value of Q, by the equations sin sin f = x I sin Q = a y I cosO sin /9 cos f = -- -= b P* Pi then e and e' by the equations i cos ( Q y) e - k^ - if. sin 1" OUTLINE OF THE SHADOW. 461 Find ft' and f' by the equations sin /3' sin / = a -f 1> 2 cos (/? -f e -f- e') sin $ = $ , ?>, COS (/? + e 4- e') COS Q sin /?' cos /= b + - -S- s= ^ then we have, rigorously, C 1= = cos/9' and these values of , ft, and t may then be substituted in (500) } which can be adapted for logarithmic computation as before.* 300. It remains to be determined whether the eclipse is begin- ning or ending at the places thus found. A point on the earth's surface which at a given time T is upon the surface of the cone of shadow will at the next consecutive instant T + d T be within or without the cone according as the eclipse is beginning or ending at the time T; the former or the latter, according as the distance J = \/\_( x ~) 2 "^~ (# ^Y] becomes at the time T -\- o 1 less or greater than the radius of the shadow I ?'. In the case of total eclipse I i is a negative "quantity, but by compari. g J 2 with (I if we shall obtain the required criterion for ul cases ; and, therefore, the criterion of beginning or ending, either of partial or of total eclipse, will be the negative or positive value of the differential coefficient, relatively to the time, of the quantity (a-O'+Cy-'rt'-Cf-tt) 1 or the negative or positive value of the quantity A-iUfr-WA^AV-oA^r** dT ifl \dT dT! \dT dT * In this problem, as well as in most of the subsequent ones, I have not followed BESSEL'S methods of solution, which, being mathematically rigorous, though as simple as such methods can possibly be, are too laborious for the practical purposes of mere prediction. As a refined and exhaustive disquisition upon the whole theory, BESSEL'S Analyse der Finsternisse, in his Astronomische Untersuchungen, stands alone. On the other hand, the approximate solutions heretofore in common use are mostly quite imperfect; the compression of the earth, as well as the augmentation of the moon's semidiameter, being neglected, or only taken into account by repeating the whole computation, which renders them as laborious as a rigorous and direct method. I have endeavored to remedy this, by so arranging the successive approximations, when these are necessary, that only a small part of the whole computation is to be repeated, and by taking the compression of the earth into account, in all cases, from the commencement of the computation. In this manner, even the first approxima- tions by my method are rendered more accurate than the common methods. 462 SOLAR ECLIPSES. where we emjt the insensible variation of i. For brevity, let us write x', y', &c. for- , - ^-, &e. and denote the above quantity by P; ihen, after substituting the values of x = (I i) sin Q, y fj = (I i) cos , we have P=L [(x' - r) sin Q + (y'- r/) cos - (V :')] in which _L i. If we put P' = (>' - ') sin + (/_,') cos q-(V- i:') (510) we shall have P=LP' The quantity P will be positive or negative according as L and P' have like signs or different signs. For exterior contacts, and for interior contacts in annular eclipse, L is positive (Art. 293), and hence for these cases the eclipse is beginning or ending according as P' is negative or positive ; but for total eclipse, L being negative, we have beginning or ending according as P' is positive or negative. We must now develop the quantity P'. Taking one hour as the unit of time, x', ?/', ', 6', ^', ', will denote the hourly changes of the several quantities. The first three of these may be derived from the general tables of the eclipse for the given time; but ', r/, ' are obtained by differentiating the equations (494), in which the latitude and longitude of the point on the earth's surface are to be taken as constant. Since & = ^ w, we shall ^ have j-= = r=f ; and hence, putting i dft. . dd . .... ft' ==. * sm 1" d' = sin 1" cos ^'cos # =r p.' ( ij sin d -f- C cos d) = // [ y sin d -(- C cos d -{- (I i'C) sin d cos Q] I/ = IJL ? sin d d'Z = I*' [x sin d (I i'C) sin d sin Q] d'C C' = // f cos d -\- d'j] = (j! lx cos d + (f tf) cos d sin Q] _|_ (5H) d x' -\- fi' y sin d -f- // z7 cos d J The values of these quantities may be computed for the same times as the other quantities in the eclipse tables, and their values for any given time will then be readily found by interpo- lation. For any assumed value of Q, therefore, and with the value of found by (509), the value of P' may be computed, and its sign will determine whether the eclipse is beginning or ending. In most cases, a mere inspection of the tabulated values of a', 6', and e', combined with a consideration of the value of Q, will suffice to determine the sign of P' ; but when the place is tiear the northern or southern limits of the shadow, an accu- raxe computation of P' will be necessary; and, since other appli- cations of this quantity will be made hereafter, it will be propcT to give it a more convenient form for logarithmic computation. Put e*inE = V f*mF=d' = = f ^ then we have P'=a'+ esin (Q E) :/sin (Q F*) (513) Since a' and .Pare both very small quantities, and a very precise computation of P' will seldom be necessary when its algebraic sign is alone required, it will be sufficient in most cases to neglect these quantities, and also to put t for , and then we shall have the following simple criterion for the case of partial or annular eclipse : If e sin (Q E) < C t / sin Q, the eclipse is beginning. If e sin (Q E) > C,/ sin Q, the eclipse is ending. For total eclipse, reverse these conditions. 301. In order to facilitate the application of the preceding as well as the subsequent problems, it is expedient to prepare the values of d v log ? d v log /> 2 , a', 6', e', e, E, /, F, and to arrange them in tables. 464 SOLAR ECLIPSES. For our example of the eclipse of July 18, 1860, with the 1 values of d given on p. 454, we form the following table by the equations (495) : d i lo g Pi ^ !og ft 0* 21 T 39".5 9.9987324 20 53' 58".0 9.9998143 1 1 14 .0 23 53 32 .6 45 2 48 .5 22 53 7 .3 46 3 22 .9 21 52 41 .8 47 4 20 59 57 .4 20 52 16 .4 48 5 59 31 .8 19 51 50 .9 50 The values of #', y', and V, required in (511), derived also from the eclipse tables on p. 454, by the method of Art. 75, are as follows : x' ?/' f 0* -f 0.545277 - 0.160108 - 0.000038 1 5312 0486 061 2 5310 0846 084 3 5256 1188 107 4 5134 1512 130 5 4928 1818 154 Hence, by (511) we find the values of ', b', c' to be as follows. The values for interior contacts are seldom required. For exterior contacts. For interior contacts. a' V c* a' V c' 0* 1 2 3 4 5 + 0.001356 + 0.000766 -f 0.000175 0.000415 0.001005 0.001595 + 0.050342 + 0.101816 + 0.153241 + 0.204612 + 0.255925 + 0.307171 + 0.631779 + 0.616776 + 0.601711 + 0.586571 -f 0.571342 + 0.556010 -f 0.001350 + 0.000762 + 0.000175 0.000413 0.001000 0.001586 +' 0.050342 + 0.101816 + 0.153241 + 0.204612 + 0.255925 + 0.307171 + 0.631165 + 0.616162 + 0.601097 + 0.585957 -f 0.570728 + 0.555395 The values of e, E, f, F, for exterior contacts, deduced from these values of b' and c', and from d' = - 25".5 sin 1", by (512), are as follows: OUTLINE OF THE SHADOW. E log e /' log/ i 0" 4 33' 21" 9.801939 1' 44" 9.388244 1 9 22 25 .795965 tt 264 2 14 17 17 .793034 it 285 3 19 13 48 .793255 305 4 24 7 46 .796604 u 326 5 28 55 7 .802923 347 302. To illustrate the preceding formulae, let us find some points of the outline of the penumbra on the earth's surface at the time T= 2 A 8'" 12 s . For this time, we have x = _ 0.00672 log Pl = 9.99873 log / = 7.66287 y = 4- 0.57409 j sin 9 n9.61782 9.65779 log c cos (9 9.86803 9.82560 log tan * n9.74979 9.832-19 log cos

, / sin Q, and the eclipse is ending. At the second point, we have e aiu(Q E} < Ci/ 8m Q, an d the eclipse is beginning. Rising and Setting Limits. 303. To find the, rising and setting limits of the eclipse. By these limits we mean the curves upon which are situated all those points of the earth's surface where the eclipse begins or ends with the sun in the horizon. It will be quite sufficient for all practical purposes to determine these limits by the condition that the point Z is in the horizon. This gives in (503) cosZ= 0, or d 0, and, consequently, by (496), we have *+V= 1 ( 514 ) as the condition which the co-ordinates of the required points must satisfy. Now, let it be required to find the place where this equation is satisfied at a given time T. Let x and y be taken for this time, then we have, by putting , = in (499), I sin Q = x I cos Q = y T) Let p cos y = m cos M = y then, from the equations I sin Q = m sin M p sin Y I cos Q m cos M p cos y we deduce, by adding their squares, /" = m 1 2mp cos ( M y) + p* } (515) - r} = l - cos (M- r ) = RISING AND SETTING LIMITS. 467 If then we put /I = M Y, we have sin A = . /I ^ i *^J -^^ I \L 4mp J (ol7) in which yl may always be taken less than 90, but the double sign must be used to obtain the two points on the surface of the earth which satisfy the conditions at the given time. In this formula, m, M, and I are accurately known for the given time, but p is unknown. It is evident, however, from (514) and (515), that we have nearly p = 1, and this value may be used in (517) for a first approximation. To obtain a more correct value of 7-, let us put = sin f'\ then, by (514), we have y 1= = cos Y', and, consequently, since rj= p^ v p sin Y = sin / p cos Y = p l cos Y' Hence we have tan Y' = f>i tan = sin/ __ f , cos r ' sin y cos Y (518) and with this value of p the second computation of (517) will give a very exact value of f. With this second value of 7- a still more correct value of p could be found ; but the second approxi- mation is always sufficient. With the second value of 7-, therefore, we find the final value of f' by the formula tan Y = />, tan Y and then, substituting the values = sin 7-', ^ cos p', , = 0, in (500), we have, for finding the latitude and longitude of the required points, the form ul re cos . = cos / cos d, = fi. > tan a> = (519) tan In the second approximation, we must compute >l and Y by (517) separately for each place. 468 SOLAR ECLIPSES. 304. The sun is rising or setting at the given time at the places thus determined, according as & (which is the hour angle of the point Z] is between 180 and 360 or between and 180. To determine whether the eclipse is beginning or ending, we may have recourse to the sign of P' (513); and it will usually be sufficient for the present problem to put both a' and = in that expression, and then the eclipse is beginning or ending according as sin (Q E] is negative or positive. Now, by (516), we find I sin (Q E) = m sin (M E~) p sin (y E) Hence, for points in the rising or setting limits, , If m sin (M E} <^p sin (/ E*), the eclipse is beginning, If m sin (M E) > p sin (j E), the eclipse is ending. 305. In order to apply the preceding method of determining the rising and setting limits, it is necessary first to find the extreme times between which the time T is to be assumed, or those limits of T between which the solution is possible. The two solutions given by (517) must reduce to a single one when the surface of the cone of shadow has but a single point in common with the earth's surface, i.e. in the case of tangency of the cone and the terrestrial spheroid. Now, the two solutions reduce to one only when X = 0, and both values of f become = M ; but if X = 0, the numerator of the value of sin ^l must also be zero; and hence the points of contact are determined by the conditions I -\- m p = and I m -{- P = or by the conditions m = p -f- I and m = p I- There may be four cases of contact, two of exterior and two of interior contact. The two exterior contacts are the first and last, pr the beginning and the end of the eclipse generally; the axis of the shadow is then without the earth, and therefore we must have for these cases m = \/x* -f y 2 = p + I. The first interior contact corresponds to the last point on the earth's surface where the eclipse ends at sunrise ; the second, to the first point where it begins at sunset. But these interior RISING AND SETTING LIMITS. 469 contacts can occur only when the whole of the shadow on the principal plane falls within the earth, and for these cases, there- fore, we must have m = p I. For the beginning and end generally we have, therefore, by (515), (P + 8 i n M = x Let The the time when these conditions are satisfied, and put T= T + r in which !F is the epoch of the eclipse tables, for which the values of x and y are x and y . Then, x' and y' being the mean hourly changes of x and y for the time T, we have x = x + r.r' y = y -I- f]f Putting m sin M = x n sin N = .x' \ m cos M y n cos N = y' ) the above conditions become T (P + cos m >n o cos -^o ~r~ T n CO8 -^ Vvhence (p -f. r> sin (JW JV) = W sin ( Jf JV") (p -f cos (Jf aV) = w cos (M N) 4- nr so that, if we put M N = i^, we have = T = T + r in whicn cos ^ may be taken with either the negative or the positive sign ; and it is evident that the first will give the beginning and the second the end of the eclipse generally. For the two interior contacts we have m n sin (M n sin 4, = -J p I ' ?> T = cos 4, - cos n n .(522J 470 SOLAR ECLIPSES. These interior contacts cannot occur when p I is less than m sin (M N), which would give impossible values of sin ^. , In these formulae we at first assume p = 1, and, after finding an approximate value of ^ we have, by (517), in which /I 0, f = M, and in the present problem M N -f- ^: therefore r = N+t (523) with which p is found by (518), and the second computation of (521) or (522) will then give the required times. We must employ in (523) the two values of ^ found by taking cos ^ with the positive and the negative sign ; and therefore different values ofp will be found for beginning and ending, so that in the second approximation separate computations will be necessary for the two cases. In the first approximation the mean values of x', ?/, and I may be used, or those for the middle of the eclipse. With the approximate values of r thus found, the true values of x', y', and I for the time T= T + r may be taken for the second approximation. After finding the corrected value of $, we then have also the true value of f = N + ^ f r eacn point? and hence also the true value of f' by (518), with which the latitude and longitude of the points will be computed by (519). For the local apparent time of the phenomenon at each place we may take the value of # in time, which is very nearly the sun's hour angle. 306. When the interior contacts exist, the rising and setting limits form two distinct enclosed curves on the earth's surface. If we denote the times of beginning and ending generally, de- termined by (521), by T r and T v and the times of interior con- tact, determined by (522), by TJ and T 2 ', a series of points on the rising limit will be found by Art, 303, for a series of times assumed between T Y and T 7 /, and points of the setting limit for times assumed between T 2 f and T r When the interior contacts do not exist, the rising and setting limits meet and form a single curve extending through the whole eclipse. The form of this curve may be compared to that of the figure 8 much distorted. A series of points upon it will be found by assuming times between T v and T r 307. EXAMPLE. Let us find the rising and setting limits of the eclipse of July 18, 1860. RISING AND SETTING LIMITS. 471 First. To find the beginning and ending on the earth gene rally, we have for the assumed epoch T = 2*, page 455, m sin M = x = 0.081244 m cos M = y = -f 0.596075 which give log ro = 9.77930 M = 352 14' 19" log m a sin (M JV) == n9. 73938 =x'= + 0.5453 ='= 0.1608 log n = 9.75474 N= 106 25'.8 ^ cos ( M N) = 0.433b V W / For a first approximation, taking p = 1, we find, by (521), p + I = 1.5367 log sin 4 = COB -1 ~ n9 .5528 F 2.525 f 2.434 71 Approx. beginning T l = 11 end T, = 23.909 (July 17) 4 .959 (July 18) Taking cos -^ negative for beginning and positive for ending, we have then, by (518) and (523), Beginning. End. 4 200 55'.4 339 4'.6 + + = r 307 21.2 85 30.4 log tan p nO.11732 1.10466 lo g Pi 9.99873 9.99873 log tan / nO. 11605 1.10339 log sin / 9.89985 9.99865,5 log sin Y 9.90032 9.99866,3 log;> 9.99953 9.99999 P 0.99892 0.99998 I 0.53687 0.53640 P + I 1.53579 1.53638 For the above computed times we further find log x' = log n sin N log y' = log n cos N loff n JT 9.73664 n9.20538 9.75467 106 23' 50" 9>.73654 n9.20774 9.75477 106 29' 8" 472 SOLAR ECLIPSES. For a second approximation, therefore, recomputing (521), we now find log sin 4, log cos 4 T log tan / n9.55316 W9.97032 23*.9098 200 56' 27" 307 20 17 nO.11629 n9.55269 9.97039 4V9587 339 4' 58" 85 34 6 1.10942 and by (518) : Then, for the latitude and longitude of the points, we have, by (519), 21 T42" 357 9 57 254 38 57 102 31 34 38 34 20 59' 33' 72 54 8 91 35 43 341 18 25 4 9 46 Therefore the eclipse begins on the earth generally on July 17, 23 A 54 m .5 Greenwich mean time, in west longitude 102 31' 0" and latitude 34 38' 34", and ends July 18, 4'' 57'".5 in longitude 341 18' 25" and latitude 4 9' 46". It is evident that for practical purposes the first approximation, which gives the times within a few seconds, is quite sufficient, especially since the effect of refraction has not yet been taken into account. (See Art. 327.) Secondly. "We now pass to the computation of the curve which contains all the points where the eclipse begins or ends at sun- rise or sunset. In the present example, this curve extends through the whole eclipse, since we have m sin (M N) > 1 I: hence the required points will be found for Greenwich times assumed between July 17, 23\91 and July 18, 4*.96. Let us take the series T, 0, 0\2, 0*.4, 0.6, 0\8 4\6, 4.8 The computation being carried on for all the points at once, the regular progression of the corresponding numbers for the suc- cessive times furnishes at each step a verification of its correct- ness. To illustrate the use of the formulae, I give the computa- tion for T= 2 A .0 nearly in full. For this time, we find, from p. 454 and p. 464, x = m sin M = 0.08124 y = m cos M = 4- 0.59608 1 = 0.53675 ^=210'49" log f ) 1 = 9 99873 RISING AND SETTING LIMITS. 473 and hence M = 352 14' 21" log m = 9.77931 Then, by (517), taking p = 1, we have m = 0.60160 ar. co. log 4m/) 9.61863 I + m p = 0.13835 .......... log 9.14098 I _ m + j> = 0.93515 .......... log 9.97088 /I = 26 49' log sin 2 J A 8.73049 With this first approximate value of A we find the value of p for each of the two points, by (518), as follows : M /I = Y 19 3' 325 25 log tan Y 9.53820 9.83849 log f>i tan Y = log tan / 9.53693 9.83722 log f) l COS Y' 9.99887 9.99914 P 0.99740 0.99802 Repeating (517) with these values of p : ar. co. log 4 mp 9.61976 9.61949 log (1 -\- m p) 9.14907 9.14715 log (1 m + j) 9.96967 9.96996 log sin 2 A 8.73850 8.73660 -f- /I _j_27 4' 4" 27 0' 26" Jf Jl = 7- 19 18 25 325 13 55 log tan Y 9.54448 n9.84148 log tan ^' 9.54321 n9.84021 Hence, by (519), .9 135 45' 4" 242 36' 45" For r= 2*. (p. 455), ^ 28 31 12 28 31 12 /Jlj ? = U) 252 46 8 145 54 27

/- *') sin Q + Q/'- ,') cos Q The equation L= I i gives *?- = - dT and, therefore, V i :' - (*' *') sin Q (y' ,') cos Q = (525) or, by (510), P'=0 (526) This is, therefore, the general condition which characterizes the maximum of the eclipse at a given time. In the present problem we have also the condition that the sun is in the horizon, for which we may, as in Art. 303, substitute the condition t = 0., Since, however, the instant of greatest obscuration is not subject to any nice observation, a very precise solution of the problem is quite unimportant, and we may be satisfied with the approxi- mate solution obtained by supposing = 0, and at the same time neglecting the small quantity a' in P'. The condition (526) will then be satisfied when in (513) we have sin (Q E} = that is, when Q = E or Q = 180 + E CURVE OF MAXIMUM IN THE HORIZON. 477 Hence, for any given time, the conditions (524) become A sin E = x ' J cos E = y i) which with the condition ? + i* = I must determine the required points of our curve. The angle E is here known for the given time, being directly obtained from its tabulated values, but J is unknown. Putting, as in the preceding problem, m sin M = x p sin f = m cos M = y p cos y = y we have A sin E = m sin M p sin f A cos E = m cos M p cos f whence == m sin (M E) p sin ( r E) A = m cos (M E) p cos (/ E] Therefore, putting ^ = T ~ Q we have m sin (Jf E} am 4- = (527) J = m cos (M E} p cos 4. The first of these equations will give two values of ^, since we may take cos ^ with the positive or the negative sign ; but, as only those places satisfy the problem which are actually within the shadow, we must have J < , or, at least, J not greater than I That value of ^ which would give J> I must, therefore, be excluded : so that in general we shall have at a given time but one solution. It will be quite accurate enough, considering the degree of precision above assigned, to employ in (527) a mean value of p, or, since p falls between />, and unity, to take log p = Jlog p r But, if we wish a more correct value, we have only to take Y = 4 + E (528) and then find p as in (518) ; after which (527) must be recom- puted. 478 SOLAR ECLIPSES. Having found the true value of ^ by (527), and of 7- by (528), we then have f' by the equation and the latitude and longitude of each point of the curve by (519). The limiting times between which the solution is possible will be known from the computation of the rising and setting limits, in Avhich we have already employed the quantity m sin (M J); and the present curve will be computed only for those times for which m sin (M E] < I. These limiting times are also the same as those for the northern and southern limiting curves, which will be determined in Art. 313. 310. The degree of obscuration is usually expressed by the fraction of the sun's apparent diameter which is covered by the moon's disc. When the place is so far immersed in the penumbra as to be on the edge of the total shadow, the obscuration is total ; in this case the distance of the place from the edge of the penumbra is equal to the absolute difference of the radii of the penumbra and the umbra, that is, to the algebraic sum L + Z/,, , denoting the radius of the umbra (which is, by Art. 293, negative); but in any other case the distance of the place within the penumbra is L J: hence, if D denotes th: degree of obscuration expressed as a fraction of the sun'^ apparent diameter, we shall have, very nearly, (529) This formula may also be used when the eclipse is annular, in which case L v is essentially positive ; and even when J is zero, and the eclipse consequently central, the value of D given by the formula will be less than unity, as it should be, since in that case there is no total obscuration. In the present problem we have in which I and ^ are the radii of the penumbra and umbra on the principal plane, as found by (488). EXAMPLE. In the eclipse of July 18, 1860, compute the curve on which the maximum of the eclipse is seen in the horizon. CURVE OF MAXIMUM IN THE HORIZON. 479 In the computation of the rising and setting limits, the quantity m sin (M E~) was less than unity only from T= 0*.6 to T= 4 A .2: so that the present curve may be computed for the series of times A .6, 0*.8 4*.0, 4 A .2. For an approximate computation we may take log p = log p^= 9.9994, and employ only four decimal places in the logarithms throughout. The computation for T = 2 A is as follows. For this time we have already found (p. 473) log m M E Hence, by (527), M E log m sin ( M E) \ogp log sin 4 log cos 4 log p cos 4 log m cos (M E} mcos(Jf E) 9.7793 352 14'.4 14 17.3 337 57.1 n9.3538 9.9994 n9.3544 9.9886 9.9880 9.7463 + 0.5575 + 0.9727 0.4152 Here, if cos ^ were taken with the negative sign we should find J= 1.5302, which is greater than L Taking it, therefore, with the positive sign only, we have log Pl = 9.9987 with which we find, by (519), log tan log tan App. time = # in time 13 4'.3 -f 1 13. 8.3271 8.3258 176 37'.2 28 31.2 211 54 69 1 IP 46-5 Sunset. To express the degree of obscuration according to (529*) we have, taking the mean, values of I and ^ (p. 454), I = 0.5366 J, = 0.0092 I + l t = 0.5274 I A = 0.1214 0.1214 D = 0.5274 = 0.23 In the same manner all the following results are obtained : 480 SOLAR ECLIPSES. SOLAR ECLIPSE, July 18, I860. CURVE OF MAXIMUM OF THE ECLIPSE IN THE HORIZON. Greenwich Mean T. Latitude. * Long. W. from Greenwich. ti App. Local Time. tf Degree of Obscuration. D 0.6 .8 .0 .2 .4 _j_ 24 44' 37 47 47 3 54 31 60 38 107 41' 117 47 127 49 139 1 152 24 17* 19-.3 16 50 .9 16 22 .8 15 50 .0 15 8 .5 0.30 .76 .97 .74 .56 .6 .8 2 .0 2 .2 2 .4 65 20 68 16 69 1 67 34 64 20 169 189 16 211 54 233 32 251 42 14 14 .1 13 5 .0 11 46 .5 10 31 .9 9 31 .3 .41 .31 .23 .18 .17 2 .6 2 .8 3.0 3 .2 3.4 59 55 54 41 48 52 42 35 35 49 266 11 277 50 287 31 295 56 303 30 8 45 .3 8 10 .8 7 44 .0 7 22 .4 7 4 .1 .17 .21 .28 .37 .50 3.6 3 .8 4.0 4.2 28 28 20 21 + 11 2 45 310 33 317 22 324 15 331 14 6 47 .9 6 32 .6 6 17 .2 6 1 .1 .67 .89 .87 .48 Northern and Southern Limiting Curves. 311. To find the northern and southern limits of the eclipse on the earth's surface. These limits are the curves in which are situated all the points of the surface of the earth from which only a single contact of the discs of the sun and moon can be observed, the moon appearing to pass either wholly south or wholly north of the sun. They may also be defined as curves to which the out- line of the shadow is at all times in contact during its progress across the earth. The solution of this problem is derived from the consideration that the simple contact is here the maximum of the eclipse, so that we must have, as in (526), and consequently, by (513), a' + e sin (Q - E~) = Zf sin (Q - F) (530) NORTHERN AND SOUTHERN LIMITS. 481 For any given time T t therefore, we are to find that point of the outline of the shadow on the surface of the earth for which the value of Q and its corresponding satisfy this equation. This can be effected only indirectly, or by successive approxima- tions. For this purpose, we must know at the outset an approxi- mate value of Q; and therefore, before proceeding any further, we must show how such an approximate value may be found. We can readily determine sufficiently narrow limits between which Q may be assumed. For this purpose, neglecting a' in (530), as well as F, which are always very small, we have, approximately, The extreme values of are = and 1. The first gives sin (Q E} = 0, and therefore for a first limit we have Q = E or Q = 180 4- E The second gives eam(Q E)=fsm Q whence tan (Q \ E} = e -*- Put then the equation tan (Q %E] = tan ^ gives for our second limits Q = \E+* or C = 180 To compute ^ readily, put tan v then tan 4 = tan (45 4- v~) tan J E and Q is to be assumed between E and E 4- 4- or between 180 + E and 180 4- i E -f VOL. I. 31 (531) 482 SOLAR ECLIPSES. These limits may be computed in advance for the principal hours of the eclipse from the previously tabulated values of jE 1 , e, and /, and an approximate value of Q may then be easily inferred for a given time with sufficient precision for a first approximation. When the shadow passes wholly within the earth, there are two limiting curves, northern and southern. For one of these Q is to be taken between E and \ E + ^ ; for the other, between 180 + E and 180 + $E-\- $ Since E is always an acute angle, positive or negative, it fellows that when Q is taken between E and $ E + ij/, its cosine is in general positive, while it is nega- tive in the other case. The equation y = y (I i) cos Q shows that y will be less in the first case and greater in the second, and hence the values of Q between E and % E + ^ belong to the southern limit, and the values of Q between 180 -f- E and 180 -f E + 4- belong to the northern limit. There is only one limit, northern or southern, when one of the series of values of Q would give impossible values of in the computation of the outline of the shadow by Art. 298. But when the rising and setting limits have been determined, the question of the existence of one or both of the northern and southern limits is already settled ; for if the rising and setting limits extend through the whole eclipse in north latitude, only the southern limiting curve of our present problem exists, and vice versa; while if the rising and setting limits form two distinct curves, we have both a northern and southern limiting curve ; and the latter must evidently connect the extreme northern and southern points respectively of the two enclosed rising and setting curves. In our example of the eclipse of July 18, 1860, there exists only the southern limiting curve of the present problem, the penum bral shadow passing over and beyond the north pole of the earth. Having assumed a value of , we find \ by the equations (502), (504) and (505), and then by (509). This computed value of and the assumed value of Q being substituted in (530), this equa- tion will be satisfied only when the true value of Q has been assumed. To find the correction of Q, let us suppose that when the equation has been computed logarithmically we find log C/sin (Q F) log [a' + e sin (Q ^)] = x If then dQ and d? are the corrections which Q and require in NORTHERN AND SOUTHERN LIMITS. 485 order to reduce x to zero, we have, by differentiating this equation. cot -* a' -H esln (Q J) A A: in which A is the reciprocal of the modulus of common logarithms. In this differential equation we may neglect a' without sensibly affecting the rate of approximation. If then we put 9 = we shall have dQ = cot (Q - E) - cot (Q-F) + g This value of dQ is yet to be reduced to seconds by multiplying it by cosec V or 206265". To find g, we may take, as a sufficiently exact expression for computing dQ, and by differentiating (502) (omitting the factor p v which will not sensibly affect g), cos ft sin Y d{3 -f- sin ft cos y df = I cos Q dQ cos ft cos f d,3 sin ft sin Y dy = I sin Q dQ whence, by eliminating rfy, dft ^sin(Q- r ) dQ cos ft By (505) a sufficiently exact value of , for our present pur- pose is C, = cos ft \vhence d' ^dft * = sin ft dQ dQ g = I Bin ft see 2 ft Bm(Q y} (532) Putting, finally, g 484 SOLAR ECLIPSES. we have de = [6fwg. & + 9 in which 5.67664 is the logarithm of A X 206265". When the true value of Q has thus been found, the corre- sponding latitude and longitude on the earth's surface are found as in Art. 298. 312. The preceding solution of this problem (which is com- monly regarded as one of the most intricate problems in the theory of eclipses) is very precise, and the successive approxi- mations converge rapidly to the final result. For practical pur- poses, however, an extremely precise determination of the limit- ing curves of the penumbra is of little importance, since no valuable observations are made near these limits. I shall, there- fore, now show how the process may be abridged without making F . 43 any important sacrifice of accuracy. In the first place, it is to be observed that great precision in the angle Q is unnecessary. If LM, Fig. 43, is the limiting curve which is tangent at A to the shadow whose axis is at M C, and if Q is in error by the quan- tity ACA'j the point determined will be (nearly) A' instead of A. Now, although A' may be at some distance from A, it is evident that it will still be at a proportionally small distance from the limiting curve. In fact, we may admit an error of several minutes in the value of Q without sensibly removing the computed point from the curve. The equation (530), which determines Q, may, therefore, without practical error be written under the approximate form and in this we may employ for t the value C, = cos y5 Hence, having found /9 from (502) by employing the first assumed value of Q, we then have 8 in (Q E~) sin NORTHERN AND SOUTHERN LIMITS. 485 whence tan( IE) = ^ J -'-tan IE e f cos ,9 by which a second and more correct value of Q can be found. This equation will be readily computed under the following form : tan v' = cos 8 1 V (535) tan (Q J E) = tan (45 + /) tan \ E ) The value of Q thus determined may be regarded as final, and we may then proceed to compute the latitude and longitude by the equations (502) to (508). In this approximate method, loga- rithms of four decimal places will be found quite sufficient. 313. For the computation of a series of points by the preceding method, it is necessary first to determine the extreme times between which the solution is possible. It is evident that the first and last points of the curve are those for which L = 0, and, consequently, Q = E, or Q = 180 + E. It is easily seen that these points are also the first and last points of the curve of maximum in the horizon (Art. 309), and, therefore, the limiting times are here the same as for that curve. If, however, we wish to determine these limiting times independently (that is, when the rising and setting limits have not been previously computed), the following approximative process will give them with all the precision necessary. Since Q = E, or = 180 -j- E, we have, at the required time, = y q= I cos E j together with the condition (514), for which we may here employ ? + ^ = 1 If we put = sin 7% this condition gives 37 = cos 7-. We have, by (512), sin E = cos E e e and we may here regard e as constant. Let the required time be denoted by T= T + r, T being an assumed time near the middle of the eclipse. Let 6 ', e/, be the values of b' ad c f for 486 SOLAR ECLIPSES. the time T w and denote their hourly changes by b" and c" ; then we have, for the time T, and hence, E being the tabulated value of E for the time T w b" c" sin E = sin E -\ T cos E = cos E Q -\ T If, also, z , y , are the values of x and y for the time T , x' and y 1 their hourly changes, we have X = X -f X 1 r y^y^y'r and the equations (536) become sin r = x T I sin E n + / x' + L b" \ r cos Y = y =p I cos 1? 4- 1 y' q= c" I r \ e / Let w?, J[f, 7i, ^\T, be determined by the equations m sin M = x =p I sin E m cos M = y ^ I cos J / sin JV ^ q= - b" n cos JV = v' T " e (537) in which the upper sign is to be used for the southern and the lower sign for the northern limit ; then, from the equations sin Y = ni sin M -|- n sin N. T cos y = m cos M -f- n cos N . T we derive sin (Y JV) = m sin (Jlf AT) cos O' JV) = m cos (Jf JV) -f- nr Hence, putting f N=^ sin 4 = m sin (M JV^ cos ^ m cos (Jf-.., (53g) It is evident that cos ^ is to be taken with the negative sign for the first point and with the positive sign for the last point of the curve. NORTHERN AND SOUTHERN LIMITS. 487 To find the latitude and longitude of the extreme points, we take f = JV-f 1^, tan f' = p l tan f, and proceed by (519). EXAMPLE. To find the southern limit of the eclipse of Jul} 18, 1860. First. To find the extreme times. Taking T 9 2*, we have. from our tables, pp. 454, 455, and pp. 464, 465, x = 0.0812 tf= + 0.5452 y = -f 0.5961 y' 0.1610 I = 0.5367 E = 14 17' b" = -f 0.0514 log e = 9.7977 c" = 0.0151 where we take mean values of x', y f , &c. From these we find by (537), taking the upper signs in the formulae, log m = 9.3555 log n = 9.7182 Hence, by (538), log sin ( Jf JV) = n8.7354 log sin 4 = n8.0909 log cos 4 = 0.0000 Jlf=28935' N= 106 28 M ^V=183 7 log cos (M JV) = n9.9994 *~ 2=*i.n n T = 1 .480 or T = + 2 .346 Therefore, for the first and last points of the curve we have, respectively, the times T t = 2* 1*.480 = 0.520 T a = 2 +2 .346 = 4 .346 To find the latitude and longitude of the extreme points corre- spending to these times, we have log A = 9.9987 First Point. Last Point. 4 180 42' 042' = N 4- 4 287 10 105 46 log tan f nO.5102 nO.5492 log tan / nO.5089 nO.5479 rf i 21 1'.4 20 59'.8 ^ 6 19.2 63 42.7 488 Heuce, by (519), SOLAR ECLIPSES. 102 40' 16 5 339 30' 14 47 Second. To find a series of points on the curve. We begin by computing the limits of Q for the hours 0*, 1*, 2 A , 3*, 4*, 5*. Thus, for 0* we have, from the table p. 465, and by (531), T o* . log/ 9.3882 loge 9.8019 log tan v 9.5863 V 21 5'.6 %E 2 16.7 log tan (45 + v) 0.3533 log tan I E 8.5997 log tan ^ 8.9530 4 5 7'.7 E-\- 4 7 24.4 For the southern limiting curve, Q falls between Jand i.e., for O ft , between 4 33' and 7 24'. In the same manner we form the other numbers of the following table : T Lower limit of Q. Upper limit of Q. 0* 4 33' 7 24' 1 9 22 15 18 2 14 17 23 13 3 19 14 30 53 4 24 8 38 4 5 28 55 44 36 The points of the curve are to be computed for times between 0\520 and 4\346, and we shall, therefore, assume for T the series 0\6, 0' 1 .8, l ft .O 4*.0, 4*.2, which, with the extreme points above computed, will embrace the whole curve. Instead of determining Q for each of these times by the method of Art. 312, it will be sufficient to determine it for the hours l ft , 2 A , 3 A , 4 A , and, hence, to infer its values for the inter- vening times. Thus, for T= 1 A , assuming Q = 12, which is a NORTHERN AND SOUTHERN LIMITS. 489 mean between its two limiting values, we proceed by the equa- tions (502), for which we can here use as follows : For T=l sin /? sin y = x I sin Q sin ft cos f = y I cos Q X I Assume Q a = x I sin Q b = y I cos Q log a = log sin /? sin f log b = log sin IS cos y log sin /3 We thus find, for T= 1* Q = 11 55', 0.6266 log cos /S 9.7396 + 0.9170 lo * 9.5923 0.5368 12 log tan v' 9.3319 0.7382 v' 12 7'.1 + 0.3920 \E 4 41.2 n9.8682 logtan(45+v/) 0.1894 9.5933 tan \E 8.9137 9.9221 tan( \E) 9.1031 Q IE 7 13'.5 Q 11 54.7 2* 3* 4* 22 20', 30 16', 32 17'. From these numbers we obtain by simple interpolation suffi- ciently exact values of Q for our whole series of points. And since it is plain from Art. 312, that even an error of half a degree in Q will not remove the computed point from the true curve by any important amount, we may be content to employ the following series of values as final : T Q T Q T Q T Q 0\6 8 1*.6 18 2.6 28 3.6 31 .8 10 1 .8 20 2.8 29 3 .8 32 1 .0 12 2 .0 22 3 .0 30 4.0 32 1 .2 14 2.2 24 3.2 30 4.2 32 .5 1 .4 16 2 .4 26 3.4 31 For each time !Twe now take ar, y, and , from the tables of the eclipse, and, with the value of Q for the same time, deter- mine the required po'.nt on the outline of the shadow by the 490 SOLAR ECLIPSES. complete equations (502) to (508) inclusive, the use of which has already been exemplified in Art, 302. Employing only four decimal places in the logarithms, we shall find that the curve may be traced through the points given in the following table : SOLAR ECLIPSE, July 18, I860. SOUTHERN LIMIT. Greenwich Mean Time. Latitude. Long. W. from Greenwich. u 0\520 + 16 5' 102 40' .6 21 32 88 31 0.8 25 6 76 37 .0 26 36 69 2 .2 27 17 63 9 .4 27 27 58 14 .6 27 15 53 57 .8 26 47 50 9 2 .0 26 4 46 43 2 .2 25 9 43 33 2 .4 24 3 40 34 2 .6 22 48 37 45 2 .8 21 5 34 33 3 .0 19 9 31 25 3 .2 16 41 27 50 3 .4 14 14 24 39 3.6 11 9 20 44 3 .8 8 5 16 55 4 .0 + 4 3 11 46 4 .2 - 39 5 17 4 .346 -14 47 339 30 314. "We have applied the preceding method only to the deter- mination of the extreme limits of the penumbra, which may be designated as the extreme limits of partial eclipse. The same method will determine the northern and southern limits of total or annular eclipse, by employing the value of I for the total shadow that is, for interior contacts. The latter are, indeed, more important, practically, than the former, and therefore in CURVE OP CENTRAL ECLIPSE. 491 special cases somewhat greater precision might be desired than has been observed in the preceding example. In any such case, recourse may be had to the rigorous method of Art. 311 Since the limits of total or annular eclipse often include but a very narrow belt of the earth's surface, extending nearly equal distances north and south of the curve of central eclipse, they may be derived, with sufficient accuracy for most purposes,. from this curve, by a method which will be given in Art. 320. The curve upon which any given degree of obscuration can be observed may also be computed by the preceding method. It is only necessary to substitute J for I, and to give J a value cor- responding to D according to the equation (529). All the curves thus found begin and end upon the curve of maximum in- the horizon. Curve of Central Eclipse. 315. To find the curve of central eclipse upon the surface of th( earth. This curve contains all those points of the surface of the earth through which the axis of the cone of shadow passes. The problem becomes the same as that of Art. 298 upon the suppo- sition that the shadow is reduced to a point that is, when I i = 0, and, consequently, by (493), 5 = x rj=y Hence, putting the equations (502) to (508) are reduced to fhe following ex- tremely simple ones, which are rigorously exact: sin /? sin f = x sin ft cos f = y, c sin C = y l c cos C = cos {3 cos ?! sin = x cos i cos & = c cos (C -f dj sin fpj = c sin ((7 -f- dj tan . tan

tan

,= 9.99873 we form, from the values of y given Vn *h and with the values of j^f , = rf^ cos d l -f rfCj sin d l whence, by eliminating d v and substituting ^ for its value given by the third equation of (497), we find C t cos , cos i? sin t sin # ^ l d<(> l = rfx sin # sin rf t -f- dy l cos * Hence, substituting cos ft for cos/3 (cos 9 sin sin rf, -f sin & cos 0) tan I i cos ft . ~ - sin Q cos d. cos /? - (sin # sin Q sin d, cos # cos cos /? 500 SOLAR ECLIPSES. These values are yet to be divided by sin 1' to reduce them to minutes of arc. It will be convenient to put sin 1' sin 1' _ ., cos /9 sin 1' cos /9 (548) In which /', z y , and X will be expressed in minutes. We may in practice substitute -dtp for ,, within the limits of accuracy we have adopted ; for we find, from the equations o?i p. 457, d

, but with opposite signs ; and therefore we may com- pute the equations (549) with only the acute value of Q, and then the longitude and latitude of a point on one of the limits are ta -f- dot,

(550 j C=Acos( d) ) Let ', y' denote the hourly increments of and jy ; then, assuming that these increments also are uniform, the values of the co-ordi- nates at the time T are + 'r and ^ + -/r. The values of ' and rf are found by the formulae (p. 462) f = p! p cos cos = 4 44 30 Local time of max. obscur. = t 2u 23 54 For the amount of greatest obscuration we have, also, from the first approximation, by (557) and (558), L = 0.5340 log L = 9.7275 k = 0.2723 log sin 4, = n9.6955 L k= 0.2617 log J = 9.4230 2(i A-) = 0.5234 A = 0.2649 7. --- 1 = 0.2691 = 0.514 0.5234 Or, by (559), taking as constant the value of e found by emplo3- ing the mean value I = 0.5367, i.e. e = 1.015 we have e sin 4, = 0.503 D = 0.512 which is quite accurate enough. 325. Prediction for a given place by the method of the American Ephemeris. This method is based upon a transformation of BESSEL'S formula suggested by T. HENRY SAFFORD, Jr., and, with the aid of the extended tables in the Ephemeris, is somewhat more convenient than the preceding. The fundamental equa- tion (490) gives, by transposition, (x - )*=(* - C tan/)' - (y - ,) the second member of which may be resolved into the factors or, by (494), b = I -J- y p sin ' (sin d cos d tan /) cos # c = I y -\- p sin tp' (cos d sin d tan /) p cos ; and the value of /*, is also given in the Ephemeris for the Washington meridian. If now for any assumed time T we take from the Ephemeris the values of these auxiliaries, and, after computing , 6, and c by (560), find that a differs from \/bc, the assumed time requires to be corrected; and the correction is found by the following process. Put m = i/6c, a', b', m' = the changes of a, 6, w, in one second, T = the required correction of the assumed time; then at the time of beginning or ending of the eclipse we must have a -f- a'r = m -f- rn'r whence m a To find a' we have, by differentiating the value of a and de- noting the derivatives by accents, a' = A' (JL'P cos ?' COB # (56 1) VOL. I. 33 514 SOLAR ECLIPSES. in which //' denotes the change of ^ in one second, and is the same as the p.' of our former method divided by 3600. To find m' we have, following the same notation, and neglect- ing the small changes of E, F, G, H, I, and /, B'= y' = C" b' = B' pi G p cos b C V Examples of the application of this method are given in every volume of the American Ephemeris. CORRECTION FOR REFRACTION. 515 326. The preceding articles embrace all that is important in relation to the prediction of solar eclipses. Since absolute rigor is not required in mere predictions, I have thus far said nothing of the effect of refraction, which, though extremely small, must be treated of before we proceed to the application of observed eclipses, where the greatest possible degree of precision is to be sought. CORRECTION FOR ATMOSPHERIC REFRACTION IN ECLIPSES. 327. That the refraction varies for bodies at different distances from the earth has already been noticed in Art. 106 ; but the difference is so small that it is disregarded in all problems in which the absolute position of a single body is considered. Here, however, where two points at very different distances from the earth are observed in apparent contact, it is worth while to inquire how far the difference in question may affect our results. Let SMDA, Fig. 44, be the path of the ray of light from the sun's limb to the observer at A, which touches the moon's limb at M ; 8MB the straight line which coincides with this path between Sand M, but when produced intersects the vertical line of the observer in B. It is evident that the observer at A sees an ap- parent contact of the limbs at the instant when an observer at B would see a true contact if there were no refraction. Hence, if we substitute the point B for the point A in the formula of the eclipse, we shall fully take into account the effect of refraction. For the purpose of determining the position of the point -B, whose distance from A is very small, it will suffice to regard the earth as a sphere with the radius ft = CA. It is one of the pro- perties of the path of a ray of light in the atmosphere that the product q/y. sin i is constant (Art. 108), q denoting the normal to any infinitesimal stratum of the atmosphere at the point in which the ray intersects the stratum, // the index of refraction of that stratum, and i the angle which the ray makes with the normal. 516 SOLAR ECLIPSES. If, then, />, [JL W Z' denote the values of 9.53, and the tabular correction less than .000001. From the zenith distance 70 to 90 the correction increases rapidly, and should not be neglected. CORRECTION FOR THE HEIGHT OF THE OBSERVER ABOVE THE LEVEL OF THE SEA. 328. If s' is the height of the observer above the level of the sea, it is only necessary to put p + s' for p in the general formulae of the eclipse ; and this will be accomplished by adding to log , log 27, and log the value of log 1 1 + I which is (M being the modulus of common logarithms) But s' is always so small in comparison with p that we may 518 SOLAR ECLIPSES. neglect all but the first term of this formula ; and hence, by taking a mean value of p (for latitude 45) and supposing s' to be expressed in English feet, we find Correction of log |, log ij, log C = 0.00000002079 s' (565) For example, if the point of observation is 1000 feet above the level of the sea, we must increase the logarithms of c, 37, and C by 0.0000208. If s'is expressed in metres, the correction becomes 0.000000064 s'. APPLICATION OF OBSERVED ECLIPSES TO THE DETERMINATION OF TER- RESTRIAL LONGITUDES AND THE CORRECTION OF THE ELEMENTS OF THE COMPUTATION. 329. To find the longitude of a place from the observation of an eclipse of the sun. The observation gives simply the local times of the contacts of the discs of the sun and moon : in the case of partial eclipse, two exterior contacts only ; in the case of total or annular eclipse, also two interior contacts. Let - T (566) the values of x and y at the time t -\-M (which is the time at the first meridian when the contact was observed) are The values of x' and y' to be employed in these expressions may be taken for the time t + to obtained by employing the LONGITUDE. 519 approximate value of w, and will be sufficiently precise unless the longitude is very greatly in error. The quantities I and t change so slowly that their values taken for the approximate time t + to will not differ sensibly from the true ones. For the same reason, the quantities a and d taken for this time will be sufficiently precise : so that, the latitude being given, the co-ordinates , 37, C of the place of observation may be correctly found by the formulae (483). Since, then, at the instant of contact the equation (490) or (491) must be exactly satisfied, we have, putting L= I i, (567) in which r is the only unknown quantity. Let the auxiliaries m, M, n, jVbe determined by the equations m sin M = x n sin N = x' m cos M = y -q n cos N = y' then, from the equations L sin Q = m sin M -f- n sin N . r L cos Q = m cos M -}- n cos N . T by putting ^ = Q N, we obtain m sin (M N) sin 4, = - L cos 4 m cos ( M N~) n n __ m sin (M N 4,) w sin 4 where the second form for T will be the more convenient except when sin ^ is very small. As in the similar formulae (553), the angle ^ must be so taken that L cos ^ shall be negative for first contacts and positive for last contacts, remembering that in the case of total eclipse L is a negative quantity. Having found r, the longitude becomes known by (566), which gives a, = T - t + T (570) 520 SOLAR ECLIPSES. If the observed local time is sidereal, let /* be the sidereal time at the first meridian, corresponding to !T ; then, r being reduced to sidereal seconds, we shall have <" = j" f- -f T and this process will be free from the theoretical inaccuracy arising from employing an approximate longitude in converting (JL into t. The unit of T in (569) is one mean hour ; but, if we write _ h L cos 4 hm cos ( M W) n n _ m 8\n(M N 4) n sin 4 we shall find r in mean or sidereal seconds, according as we take h = 3600, or A = 3609.856. 330. The rule given in the preceding article for determining the sign of cos ^ (which is that usually given by writers on this subject) is not without exception in theory, although in practice it will be applicable in all cases where the observations are suitable for finding the longitude with precision ; and, were an exceptional case to occur in practice, a knowledge of the approxi- mate longitude would remove all doubt as to the sign of the term co . But it is is easy to deduce the mathematical condition n for this case. At the instant of contact, the quantity (:r e -* + a'T)+(y i -*+y'T) is equal to L 2 . At the next following instant, when r becomes r + dr, it is less or greater than L 2 according as the eclipse is beginning or ending. If then we regard L 2 as sensibly constant, the differential coefficient of this quantity relatively to the time must be negative for first and positive for last contacts. The half of this coefficient is (x - + afr) (of - *') + (y. - -n + y' r ) (y'~ V) (where the derivatives of and y are denoted by c' and 37'), or, by (567), putting N+ ^ for Q, I, [sin (N + 4) ' cos (/z a) r/ = // sin d and putting n'sin N'=sf ' n' COB N' = y> y' the above expression becomes Hence, when Z/ is positive, that is, for exterior contacts and interior contacts in annular eclipse, oj/ must be so taken that cos(N N f + 4) shall be negative for first and positive for last contact. That is, for first contact ^ must be taken between N' N+ 90 and N' N -f 270 ; and for last contact between N' N + 90 and N' N 90. For total eclipse, invert these conditions. In Art. 322, we have N = N', and hence the rule given for the case there considered is always correct. 331. To investigate the correction of the longitude found from an observed solar eclipse, for errors in the elements of the computation. Let AX, Ay, &L = the corrections of x, y, and L, respectively, for errors of the Ephemeris, A, AIJ = the corrections of and ^ for errors in p and J) cos $ + n cos (Q N} . Ar and substituting for Q its value JV+ <4/, ^ sin (JV+ 4) ,cosJV4 Ai AT -= (AX A?) - - (Ay AT?) n cos 4 n cos 4. n cos 4, 522 SOLAR ECLIPSES. or AT =. (A.T sin N-\- Ay cos TV) -j ( A.r cos .A r 4- Ay sin JV) tan 4 /I 71 -f- - (A sin N+ by cos JV) ( A? cos iV 4- AJJ sin TV) tan 4 + ^=-^- (571) which is at once the correction of r and of the longitude, since we have, by (570), AW = Ar. 332. In this expression for Ar, the corrections AX, Ay, &c. have particular values belonging to the given instant of observation or to the given place. In order to render it available for deter- mining the corrections of the original elements of computation, we must endeavor to reduce it to a function of quantities which are constant during the whole eclipse and independent of the place of observation. For this purpose, let us first consider those parts of Ar which involve AX and Ay. For any time T v at the first meridian, we have x = x + n sin N ( T, T ~) whence x sin N -f y cos TV x sin TV + y cos TV + n ( ^ T ) x cos TV -f- y sin TV = x cos TV -f y sin ^V The last of these expressions, being independent of the time, is constant. If we denote it by x ; that is, put x = x cos TV -f- y sin TV = x cos TV -}- y sin TV (572) we shall find from the two expressions -f yy = xx -f [x sin TV -f- y cos TV -f- n ( T 2T )]* (573) xx This equation shows that the quantity \/xx -f yy, which is the distance of the axis of the shadow from the centre of the earth, can never be less than the constant x, and it attains this minimum value when the second term vanishes, that is, when x sin TV + y a cos TV + n ( T v T ~) = and hence when T, = T ^ (x sin TV -f y cos TV) (574) LONGITUDE. 523 which formula, tharefore, gives the time T v of nearest approach of the axis of the shadow to the centre of the earth, while (572) gives the value of the distance of the axis from the centre of the earth at this time. By the introduction of the auxiliary quanti- ties 7\ and x, we can express the corrections involving AX and AJ/ in their simplest form ; for we have now, for the time of obser- vation t -f- o, x sin N -\- y cos N = x n sin N -(- y Q cos N -\- n (t -f- <" ^>) and if AW, AT 1 , and AX are the corrections of w, T^ and x on account of errors in the elements, we have AJC cos N -f- AZ/ sin .ZV = AX / These expressions reduce those parts of Ar which involve AX ana AJ/ to functions of A^, A, and AX, which may be regarded as constant quantities for the same eclipse. We proceed to consider those parts of Ar which involve A and A^. These corrections we shall regard as depending only upon the correction of the eccentricity of the terrestrial meridian ; for the latitude itself may always be supposed to be correct, since it is easily obtained with all the precision required for the calculation of an eclipse ; the values of a and d depend chiefly on the sun's place, which we assume to be correctly given in the Epherneris ; and // is derived directly from observation. Now, we have (Art. 82), e being the eccentricity of the meridian, cos (p , (1 ee~) sin

' sin d cos (ju a) we deduce and hence Af sin N -f A? cos N = J/3/3 ( ? sin JV + TI cos JV) A? 3 coa d cos ^V Aee Af cos 2V + A? sin N = $38 ( f cos N -{- TI sin N) Aee 3 coa d sin JV Ae<> The values of $ and 37 may be put under the forms = x (X Q ) = x wi sin Jf 7 = y G/o 7) y m cos -M" oy which the second members of the preceding expressions are changed respectively into J 33 [ z sin JV+ y cosJV m cos (M N)~\ bee 3 coa d coa N A and J j8/J [ x cos JV -f y sin N + TO sin (M JV)] Ae 8 coa d sin N bee or, by (574) and (572), into \QQ[n(T TJ m cos (If JV)] Aee /3 cos d cos JV A< and J /?/? [ x + m sin (If JV)] A<- p coa d sin JV Af# or, again, by (569) and (570), into i 88 [n (< + w T 7 ,) Z cos 4,] Aee 3 cos d cos JV Ae and J /3/3 [ * + Z sin 4] Aee 3 coa d sin ^V Aee Hence, that part of Ar which depends upon Aee is equal to W [n ( + - TJ - * tan * - i sec ^] A* - ** C 8 * C 8 ( ^ + 4) 2n n cos 4- When these substitutions are made in (571), we have Ar = Au == AAT 7 , + h tan 4 . h (t + u 7\) + A sec 4 . f i/3/3 [n (i + - T l} - * tan 4 - I sec 4 ] - Aw (577) LONGITUDE. 525 where we have multiplied by h to reduce to seconds. The unit is either one second of mean or one second of sidereal time, according as r is in mean or sidereal time. If the former, we take h = 3600; if the latter, h = 3610. 333. The transformations of the preceding article have led us to an expression hi which the corrections &T V AX, An, and &ee are all constants for the earth generally, and which, therefore, have the same values in all the equations of condition formed from the observations in various places. But a still further transform- ation is necessary if we wish the equation to express the rela- tion between the longitude and the corrections of the Ephemeris. so that we may finally be enabled not only to correct the longi tudes, but also the Ephemeris. Since A T v AX, A?* are constant for the whole eclipse, we can determine them for any assumed time, as the time T r itself. For this time we have x sin N -f y cos N = x cos N -\- y sin N = x A.r sin N -(- AZ/ cos N = n A T l A.r cos N -\- A# sin N = AX The general values of x and y (482) may be thus expressed: X Y .7 = - - V = - - sm TT sin TT where X = cos 3 ain (a ) Y sin 8 cos d cos 8 sin d cos (a a] From these we deduce A- A Y A?r .E = . -- X sin?: whence (578) A.E = . -- X AM = - -- V sin?: tan n sin* ^ tan TT A* sin N + Ay cos N = AJsin 3T+ AFcoa A" _ ^ y ^ _^_ sin TT tan n and for the time 1\ these become, according to (578), ATT AX = - r -- . -- X - sm TT tan ?r 526 SOLAR ECLIPSES. Again, by differentiating the values of X and F, we have A X =. cos 3 cos (a a) A (a a) sin ^ sin (a a) A d A Y =. [cos 8 cos d -J- sin 3 sin d cos (a a)] Ad [sin d sin d -f cos d cos d cos (a a)] Ad -|- cos 5 sin d sin (a a) A (a ) But for the time of nearest approach we may take a = a and put cos ($ c?) = 1, whence A ^ = COS f is a constant for all the places of observation, and combines with o>, so that it cannot be determined unless the longitude of at least one of the places is known. If then we put Q = ta' -\- vf a = v tan % the equations of condition will assume the form Q _ a & ta = Suppose, for the sake of completeness, that the four contacts of a total or annular eclipse have been observed at any one place, and that the values of the longitude found from the several con- tacts by Art. 329 are o v o> 2 , o> 3 , a) 4 . We then have the four equa- tions [1] Q a t # w, = [2] Q a a 9 a> 9 = [3] fl.-a 8 *-a, 8 = [4] Q a 4 * a> 4 = LONGITUDE. 520 where the numerals may be assumed to express the order in which the contacts are observed ; [1] and [4] being exterior, and [2] and [3] interior. In a partial eclipse we should have but the 1st and 4th of these equations. Since exterior contacts cannot (in most cases) be observed with as much precision as interior ones, let us assign different weightf to the observations, and denote them by p v p z , p z , p v respectively. Combining the four equations according to the method of least squares, \ve form the two normal OQUCtfacfU [p ] Q [pa ] [p 5 , o> 6 , w r and that, having put Q'= o>" + 1/7-, we have formed the three equations [5] Q' a 5 * w & = with the weight p & [6] Q' 6 # w 8 = " " p [7] ff-a,*-, = p, The subtraction of each of the first two from those which follow gives the three equations (a, e ) * + w 5 w 6 = (a s - a 7 ) * + <<>* - <*>, = (a, a 7 ) * + w 6 a, 7 = of which the weights will be respectively, according to the above forms, PsP* P;Pi PePi Ps+Pt + P, Ps+Pt+Pi Ps+Pt + Pi and the combination of these three equations, according to weights, will give a normal equation of the form which gives a value of & with the weight P'. Now, suppose that this method applied to all the observations at all the places has given us the series of equations in #, P + Q = P'* + Q' = P"* + #"=0, &c.j then, since P, P', P", &c. are the weights of these several deter- minations, the final normal equation for determining #, derived from all the observations, is P * = LONGITUDE 531 that is, it is simply the sum of all the individual equations in $ formed for the places severally. The same reasoning is applicable to any of the terms which follow the term in & in (584) ; so that if we suppose all the terms to be retained, this process gives an equation in $ for each place, in which besides the term P& there will be terms in A/.:, A//, c., and from all the equations, by addition, a final normal equation (still called the equation in $) as before. In the same manner, final normal equations in A&, A^/, &c. will be formed. Thus we shall obtain five normal equations involving the five unknown quantities #, A&, A//, A/T, ACC, which are then determined by solving the equations in the usual manner. But, unless the eclipse has been observed at places widely distant in longitude, it will not be possible to determine satisfactorily the value of ATT, much less that of Aee. It will be advisable to retain these terms in our equations, however, in order to show what effect an error in TT or ee may produce upon the resulting longitudes. When #, &c. have been found, we find J2, Q', &c. from the equations [1], [2] [5], [6] The final value of Q will be the mean of its values [1 4] taken with regard to the weights ; and so of ', &c. Hence we shall know the several differences of longitude to' w" Q' : ID' to"' Q Q", &c. If one of the longitudes, as for instance cos

take , rf, I, and log i from the eclipse tables, and compute the co-ordinates of the place and the radius of the shadow by the formula? A sin B = p sin ip' % = P cos ' s ^ n (/* a ) A cos B = p cos = T t + r If the local sidereal time p. was observed, take h = 3609.856, in the preceding formulae ; then, /^ being the sidereal time at the first meridian corresponding to T , we have The longitudes thus found will be the true ones only when all the elements of the computation are correct. IV. To form the equations of condition for the correction of these longitudes, when the eclipse has been observed at a suffi- cient number of places, compute the time T^ of nearest approach, and the minimum distance x, by the formulae T i= T o (#o sin N + y cos JV) x = x cos N + y sin N * The values of JV and log n being nearly constant, it will be expedient, where many observations are to be reduced, to compute them for the several integral hours at the first meridian, and to deduce their values for any given time by simple interpolation. 534 SOLAR ECLIPSES. Take n for the time 7 1 ,, and compute the logarithm of h ' nn the same value of h being used here as before. For each observation at each place compute the coefficients v tan ^, v sec ^ and TT E = vn (t -f- 7^ is one mean hour, J J# [v( + - !) - . tan 4 - L sec 4] - in which Jf = 959".788 log If = 2.98218 p * ln *' log (I -)= 9.99709 Then, "+v^, at" de- noting the true longitude of Washington : 2.681 ATT + 0.722 7rAe + 0.509 1.323 2.3927TA* 2.392 '^_6 7 21.9 2.406 +2.959 +2.959 More observations would be necessary in order to determine all the corrections ; but I shall retain all the terms in order to illustrate the general method. Subtracting each of the Konigs- berg equations from each of those which follow it, we obtain the six equations, 540 SOLAR ECLIPSES -f 1'.2 0.689 & 0.071 TTA& + 3.531 -f O.C11 ATT 0.315 TA = -f .7 + 0.781 + 3.694 0.234 + 0.517 -f 0.433 = + 6 .4 + 0.022 + 3.463 -f- 3.463 + 1.071 -f 0.060 = .5 + 1.470 + 3.765 3.765 0.094 + 0.748 = + 6 .2 + 0.711 + 3.534 0.068 + 0.460 + 0.375 = + 5 .7 0.769 0.231 + 3.697 + 0.554 0.373 where the weight in each case is the quotient obtained by dividing the product of the two weights of the equations whose difference is taken, by the sum of the weights of the four original equations (Art. 334). The same method, applied in the case of the two Washington equations, gives the single equation I- (B') i = 8'.0 4.066 & + 5.351 TA& + 5.351 ^ + 3.190 ATT 2.055 n&ee From the equations (A 7 ) and (B ') are formed the following final equations, having regard to their weights, in the usual manner : = + 15.495 + 10.426tf - 5.300 TA& - 16.377 ^- 6.609 AT + 5.281 ^ee = 12.445 5.300 + 34.506 -f 6.135 + 10.040 2.575 =-- - 8.19116.377 + 6.135 +34.505 +10.740 8.214 = 9.371 6.609 + 10.040 + 10.740 + 5.672 3.316 + 7.951 + 6.281 - 2.575 - 8.214 - 3.316 + 2.675 As we cannot expect a satisfactory determination of A;T and ~&ce from these observations, we disregard the last two equations, and then, solving the first three, we obtain $, TTA&, and ^in terms of ATT and TrAee, as follows : # 4".36 -f- 0.375 ATT 0.525 nbee **k = + .02 0.216 ATT 0.004 i^ee ^JL = \ .83 0.095 ATT 0.010 TTiee These values substituted in the equations (A) give Q _ p 22 44.38 + 0.651 A* -f 0.432 Q= l 22 46 .64 -f 0.684 + 0.443 fl = 1 22 46 .58 + 0.685 -f 0.442 fl = 1 22 44 .34 + 0.653 -f 0.432 LONGITUDE. 541 the mean of which, giving the second and third double weight, is (A") Q = 1 22" 45'.86 + 0.674 A* + 0.439 mee The equations (B) become Q'= 5* 7" 26'.99 1.314 A* 0.116 xtee Q'= 5 7 27.03 1.314 -0.101 the mean of which is (B") Q'= 5 7- 27'.01 1.314 A* 0.109 x*ee Now, if we assume the longitude of Konigsberg to be well determined, we have Q = of -f vr = 1 22* Ov4 -j- v r which, with the equation (A"), gives vy= 45*.46 -f 0.674 AJT -f 0.439 x&ee Hence, by (B "), the true longitude of Washington is ta"= Q'v Y = 5* '8- 12'.47 1.988 AJT 0.548 If the longitude of Washington were also previously well estab. lished, this last equation would give us a condition for deter- mining the correction of the moon's parallax. Thus, if we adopt o/'=5* 8 m 12'.34, which results from the U.S. Coast Survey Chronometric Expeditions of 1849, '50, '51, and '55, this equation gives = + 0.13 1.988 ATT 0.548 xtee whence ATT = -f- 0".07 0.276 rAee The probable value of Aee, according to BESSEL, is within 0.0001, so that the last term cannot here exceed 0".10. If, therefore, the above observations are reliable and the supposed longitudes exact, the probable correction of the parallax indi- cated scarcely exceeds O'M, a quantity too small to be established by so small a number of observations. Nevertheless, the example proves both that the adopted parallax is very nearly perfect, and that a large number of observations at various well determined places in the two hemispheres may furnish a good determination of the correction which it yet requires. 642 LUNAR ECLIPSES. Finally, the corrections of the Ephemeris in right ascension and declination, according to the above determination of f and #, are found by (586) to be (putting a' for a and d' for cf) 28".42 + 0.469 A* + 0.187 rAee _ o .48 -f 0.314 A* 0.556 mee This large correction in right ascension agrees with the results of the best meridian observations on and near the date of this eclipse. Since that time the Ephemerides have been greatly Improved. LUNAR ECLIPSES. 338. To find whether near a given opposition of the moon and sun Fig 45 a l umr eclipse will occur. The solution of this prob- lem is similar to that of Art. 287, except that for the sun's semidiameter there must be substituted the apparent semidiameter of the earth's shadow at the distance of the moon ; and also that the apparent distance of the centres of the moon and the shadow will not be affected by parallax, since when the earth's shadow falls upon the moon, an eclipse occurs for all observers who have the moon above their horizon. If $, Fig. 45, is the sun's centre, E that of the earth, LM the semidiameter of the earth's shadow at the moon, we have Apparent scmidiamctcr of Hie total shadow = LEM = BLEEVL = BLE (AES- EA V} TT s' -f r' 63' 53", and doubtful between these limits. The doubtful cases can be examined by (589), or still more exactly by (588), employing the actual values of TT, /r', 5, s', at the time, and computing /' by (475). These limits are for the total shadow. For the penumbra we have ADP. semid. of penumbra = (ir-{-s'+ *') (590) DU 544 LUNAR ECLIPSES. so that the condition (588) may be employed to determine whether any portion of the penumbra will pass over the moon, by substituting -f s r for s f . It will be worth while to make this examination only when it has been found that the total shadow does not fall upon the moon. 339. To fold the tim.e when a given phase of a lunar eclipse will occur. The solution of this problem may be derived from the general formulae given for solar eclipses, by interchanging the moon and earth and regarding the lunar eclipse as an eclipse of the sun seen from the moon ; but the following direct investigation is even more simple. Let S, Fig. 46, be the point of the celestial sphere which is opposite the sun, or the appar- ent geocentric position of the centre of the earth's shadow; M, the geocentric place of the centre of the moon ; P, the north pole. If we put o = the right ascension of the moon, o' = the right ascension of the point S, = B. A. of the sun + 180, 8 = the declination of the moon, d' = the declination of the sun, Q = the angle PSM, L = SM, we have d'= the declination of S, and the triangle PSM gives sin L sin Q = cos 3 sin (a a') ") ,-<,, sin L cos Q = cos 3' sin 8 -j- sin 8' cos 8 cos (a a') ) The eclipse begins or ends when the arc SM is exactly equal to the sum of the apparent semidiameters of the moon and the shadow. The figure of the shadow will diifer a little from a circle, as the earth is a spheroid ; but it will be sufficiently accu- rate to regard the earth as a sphere with a mean radius, or that for the latitude 45. This is equivalent to substituting for jr in (587) and (590) the parallax reduced to the latitude 45, which may be found by the formula LUNAR ECLIPSES. 545 7r t = [9.99929] * (592) where the factor in brackets is given by its logarithm. Hence the first and last contacts of the moon with the pe- numbra occur when we have i^^fX + s'+iO-fs (593) ou For the first and last contacts with the total shadow, L = ll^-s'+O + s (594) For the first and second internal contacts with the penumbra, For the first and second internal contacts with the total shadow, or the beginning and end of total eclipse, & = !(,-'+*')- (596) The solution of our problem consists in finding the time at which the equations (591) are satisfied when the proper value of L is substituted in them. A very precise computation would. however, be superfluous, as the contacts cannot be observed with accuracy, on account of the indefinite character of the outline both of the penumbra and of the total shadow. It will be suffi- cient to write for (591) the following approximate formulae, easily deduced from them : L sin Q = (a a') cos 8 \ sin 1" Let us put e sin 2J* sin* j (a ') sin 1" X = (a a') COS d (598) x f , y'= the hourly increase of x and y , I / then, if the values of x and y are computed for several successive VOL. I 35 546 LUNAR ECLIPSES, hours near the time of full moon, we shall also have x' and y' from their differences ; and if x and y denote the values of x and y for an assumed epoch T^ near the time of opposition, we shall have for the required time of contact T= T -\- T the equations L sin Q = x -f- x'r = y -\-y'r from which r is obtained by the process already frequently employed in the preceding problems. Thus, computing the auxiliaries m, M, ?*, N t by the equations m sin M=x t n sin N = x' j (599) m cos M = y n cos N y' ) we shall have m sin (M N) sin 4 = _ L cos 4. wi cos (M T= (600) in which we take cos ^ with the negative sign for the first contact and with the positive sign for the last contact. The angle Q = N-\- $ is very nearly the supplement of the angle PMS, Fig. 46 ; from which we infer that the angle, of posi- tion of the point of contact reckoned on the moon's limb from the north point of the limb towards the east = 180 -f N+ ^. The time of greatest obscuration is found, as in Art. 324, to be T = T - mc ( M - N ) ( 6oi) n which is also the middle of the eclipse. The least distance of the centres of the shadow and of the moon being denoted by J, we have, as in Art. 324, J = m sin (M N) (602) the sign being taken so that J shall be positive. If then we put D = the magnitude of the eclipse, the moon's diameter being unity, LUNAR ECLIPSES. 547 we evidently have D = L- A 2s (603) in which the value of L for total shadow from (594) is to be employed. The small correction e in (598) may usually be omitted, but its value may be taken at once from the following table : Value of e. y o a' 0" 1000" 2000" 3000" 4000" 5000" 6000" 0" 0" 0" 0" 0" 0" 0" 5 1 2 3 5 8 10 2 4 7 10 15 15 1 2 6 10 15 22 20 1 3 7 13 19 28 25 1 4 8 15 23 33 30 1 4 9 17 26 38 The quantit} 7 e has the same sign as *', and is to be subtracted algebraically from S + 3'. EXAMPLE. Compute the lunar eclipse of April 19, 1856. The Greenwich mean time of full moon is April 19, 21 /l 5 ro .5. We therefore compute the co-ordinates x and y for the Greenwich times April 19, 18 A , 21*, 24''. D E. A. =o 18* 21* 24* 13* 46- 36'.62 13*52- 9'.81 13* 57" 45'. 12 H.A. + 180 = a a o' a a' (in arc) 13 52 52.98 6 16.36 5645" 13 53 20.93 1 11.12 1067" 13 53 48.88 + 3 56.24 + 3544" 3) Dccl. = S 11 27' 0".2 12 6'43".7 -12 46' 5".5 =8' e +11 35 49 .4 15 . +11 38' 22 .8 0. +11 40 56 .6 6. y + 514" 1701" 3915" log (a a') log cos d log x ??3.75166 9.99127 n3.74293 *3.02816 9.99022 7*3.01838 3.54949 9.98913 3.53862 548 LUNAR ECLIPSES. Hence we have the following table : Diff. Diff. = 3x' y = 3,' 18* 21 ; 5533" -1043 + 4490 + 4499 x' = + 1498 + 514" 1701 -2215 2214 i/'= - 738 24 + 3456 3915 To find L, we have, by (593) and (594), * = 54' 32" JT, = 3267' s' = 957 *>= 9 s' + TT' = 2319 46 = 891 IT -J_ c' _L ir' J.9Q3" **l I * I * =rr 4"<5" 1 ^_ [ ' I '\ ttfv 5^_891_ Z for shadow = 3256 i for penumbra = 5209 Assuming the time T Q = 21 ft , we proceed by (599) and (600) : x -=- m sin M y = m cos M M log m 1043 1701 210 ai'.o 3.3000 x'= n sin JV j + 1498 y' n cos N \ - 738 116 13'.7 log n 3.2227 - cos (M N} = + 0\089 n T n = 21 j = Time of middle of eclipse = il .089 Shadow. log sin .L cos 9'' 9.7861 T l ft .543 21 .089 9.5820 =F 2\883 21 .089 n T t Beginning End 19 .546 22 .633 18 .207 23 .972 For the magnitude of the eclipse, we have, by (602) and (603) OCCULTATIONS OF FIXED STARS. 549 m sin (M N) = J = 1990" = 3256 1266 2s -1782 For the position of the points of contact with the shadow, we have, from the above value of log sin

a') A cos B = p cos 71 is one mean hour, and also in which ft ^/ log (1 _ e e) = 9.99709 A "~ 66 Then, a*' denoting the true longitude, a>'= u) vf -p v tan 4 . # -j- v sec 4- . rrAA' -j- E . Arr -j- F . in which f and $ have the signification Y = sin .ZV cos 8 A( a a') -f- cos W A ('* O # = cos N cos d A(O a') -f sin N A(<* I*) The discussion of the equations of condition thus formed may then be carried out precisely as in Art. 334, taking 7-, $, TTA^, ATT, and 7TAC6 as the unknown quantities. EXAMPLE. The occultation of Aldebarasi, April 15, 1850, was observed at Cambridge, Mass., and Konigsberg, as follows:* At Cambridge,

= 4* 44" 30'. Immersion, 2* l m 52'.45 Mean time. Emersion, 3 1 38.35 " 4^ Konigsberg j y> = 54 42' 50".4, , = _! 22" 0*.4 Immersion, 10* 57* 43'.66 Sidereal time. Emersion, 11 47 47 .60 " " I. The Greenwich mean time of conjunction of the moon and star was about l h 30 m , and hence we take our data from the Nautical Almanac as follows : * Astronomical Journal, Vol. I., pp. 139 and 175. LONGITUDE. 553 I860 April 15. a 4 7T 6* 65 56' 21".16 + 16 40' 0".05 58' 55".22 7 66 32 32 .06 16 46 30 .53 58 55 .87 8 67 8 46 .02 16 52 54 .77 58 56 .49 9 67 45 3 .02 16 59 12 .76 58 57 .10 The position of Aldebaran for the same date was a' 66 49' 33".9 3'= -J- 16 12' 1".7 Hence, by I. of the preceding article, we form the following table : Gr.T. X z' y y' 6 0.86519 4- 0.58849 4- 0.47664 4- 0.10871 7 0.27671 47 .58531 63 8 + 0.31176 42 .69390 56 9 -f- 0.90014 32 .80243 48 II. The sidereal time of Greenwich Mean Noon, April 15, 1860, was 1* 33 W 8'.96. With this number, converting the Konigsberg times into mean times for the sake of uniformity, we find Cambridge. Konigsberg. Immersion. Emersion. Immersion. Emersion. 2* 152.45 3* 1" 38*. 35 9 23 15-.64 10* 13 m 1P.38 6 46 22.45 7 46 8.35 8 1 15.24 8 51 10.98 54 2' 2". 55 69 0' 58". 35 164 25' 54". 90 176 56' 64".00 347 12 28 .65 2 11 24.45 97 36 21 .00 110 7 20 .10 9.826441 9.909898 9.869121 9.762639 n9.214324 8.451362 9.758801 9.735287 9.646065 9.641159 9.904038 9.922175 9.944427 9.952794 9.185091 8.540726 ft a.' log p sin 0' log p cos ' bfl log TI logC The value of log has been found in order to find the correc- tion for refraction. This correction is here quite sensible in the case of the Konigsberg observations which were made at a great 554 OCCULTATIONS OF FIXED STARS. zenith distance. By the table on p. 517, we find that the loga rithms of and y must be increased by 0.000006 for immersion, and by 0.000041 for emersion. Applying these corrections, the values of the co-ordinates are as follows : f 1 0.16380 + 0.02827 II + 0.57386 + 0.54366 T, \ + 0.44266 + 0.43768 II +0.80175 + 0.83602 III. Assuming convenient times not far from t + o>, v/c ave Assumed T 6*. 8 7*. 8 8*.0 8*. 85 *o 0.39440 + 0.19406 + 0.31176 -;- 0.81188 y -f 0.56358 + 0.67218 + 0.69390 + 0.78615 z c f = m sin M 0.23460 + 0.16579 0.26210 + 0.26822 y, *i = mcosM + 0.12092 + 0.23450 - 0.10785 0.04987 M 297 40' 16".5 35 15' 36". 1 247 38' 1".0 100 31' 57".7 log m 9.415608 9.458164 9.452433 9.435871 x' = n sin N + 0.58847 4- 0.58843 + 0.58842 + 0.58836 y' =n cosJV + 0.10865 + 0.10857 + 0.10856 + 0.10849 N 79 32' 21".l 79 32' 45". 8 79 32' 48". 5 79 33' 8". 5 4 216 11 35 .9 312 33 59 .0 167 35 28 .5 21 1 28 .1 (h = 3600) T 89-. 74 -128*.82 68'. 63 - 3'. 52 u 4* 44" 37'.81 4* 44 12.83 l*22 m 7'. 01 _1*22- 4'.90 IV. For the equations of condition, taking T = 7\8, T t = 7.2772 n = 3536" x = + .6258 log v = 0.2308 and putting l = the true longitude of Cambridge, / " " Konigsberg, we find, neglecting terms in Aee, o, = 4* 44 37'.81 v r + 1.245 A 2.108 mk 1.293 A* *, = 4 44 12 .83 vf 1.852 A + 2.515 n*k + 1.660 A* !/== 1 22 7 .01 vr 0.374 1.742 TTA* + 0.991 ATT w t '= I 22 14 .90 v r +- 0.654 * + 1.822 TTA* + 1.195 ATT whence the two equations = + 24'.98 -+ 3.097 * 4.623 r^k 2.953 AT = + 7 .89 1.028 # 3.564 TTA* 0.204 Ar If we determine $ and ir&k in terms of ATT, these equations give A = 3".33 + 0.607 ATT TTAA- = + 3 .17 0.232 ATT LONGITUDE. 555 and then we find w l = 4 44 26'.98 vr 0.048 A* <= 1 22 11.29 v r -f- 1.169 ATT Assuming >/= l ft 22 m 0'.4 as well determined, the last equa tion gives vr = - 10.89 + 1.169 ATT which substituted in the value of w l gives ,, = 4* 44 37'.87 1.217 A* Finally, adopting the correction of the parallax for this date as given in Mr. ADAMS'S table (Appendix to the Nautical Almanac for 1856), ATI = + 5".l, this last value becomes w l = 4 44 31.66 which agrees almost perfectly with the longitude of Cambridge found by the chronometric expeditions, which is 4 A 44 m 31 S .95. With the same value of ATT we find r = 2".90 = 0".23 7TAA: = -(- V.99 and hence, by (586), the corrections of the Ephemeris on this date, according to these observations, are A(O a') = 2".93 A(<5 3') = 0".77 The value x*k = + 1".99 gives AA; = 0.00056, and hence the corrected value k = 0.27227 + 0.00056 = 0.27283, which is not very different from OUDEMANS'S value. (See p. 551). 342. When a number of occultations have been observed at a place for the determination of its longitude, it will usually be found that but few of the same occultations have been observed at other places. If, then, we were to depend altogether upon corresponding observations at other places, we should lose the greater part of our own. In order to employ all our data, we may in such case find for each date the corrections of the moon's place from meridian observations (see Art. 235), and, employing the corrected right ascension and declination in the computation of x and y, our equations of condition will involve only terms in 7TAA; and ATT. The value of AT: will, however, be different on 556 OCCULTATIONS OF FIXED STARS. different dates, and, therefore, if we wish to retain this term, we must introduce in its stead the correction of the mean parallax which is the constant of parallax in the lunar tables. If this constant is denoted by TT O , we shall have, very nearly, 7T ATT = where TT is the parallax for the given date. The equations of condition will then be of the form ut l = ta -f a . 7TAA -j- b . ATT O where a = vsec4 b = E In PEIRCE'S Lunar Tables, now employed in the construction of our Ephemeris, TT O = 3422".06. 343. The passage of the moon through a well determined group of stars, such as the Pleiades, affords a peculiarly favorable opportunity for determining the correction of the moon's semi- diameter as well as of the moon's relative place, of the relative positions of the stars themselves, and also (if observations are made at distant but well determined places) of the parallax. Prof. PEIRCE has arranged the formulae of computation, with a view to this special application, for the use of the U. S. Coast Survey. See Proceedings of the American Association for the Adv. of Science, 9th meeting, p. 97. 344. When an isolated observation of either an immersion or an emersion is to be computed, with no corresponding observa- tions at other places, it will not be necessary to compute the values of x and y for a number of hours. It will be sufficient to compute them for the time t + to (t being the observed local mean time, and 10 the assumed longitude) ; and, as the correction of this time will always be small, the hourly changes may be found with sufficient precision by the approximate formulae, easily deduced from (482), da. dd X? COS 3 y'= PREDICTION FOR A GIVEN PLACE. 557 where da and dd denote the hourly increa8e of a and 3 respect- ively. 345. To predict an occupation of a given star by (he moon for a f/iren place on the earth. We here suppose that it is already known that the star is to be occulted at the given place on a certain date, and that we wish to determine approximately the time of immersion and emersion in order to be prepared to observe it. The limiting parallels of latitude between which the occultation can be observed will be determined in the next article. For a precise computation we proceed by Art. 322, making the modifications indicated in Art. 340. But, for a sufficient approximation in preparing for the obser- vation, the process may be abridged by assuming that the moon's right ascension and declination vary uniformly during the time of occultation, and neglecting the small variation of the parallax. It is then no longer necessary to compute the co-ordinates x and y directly for several different times at the first meridian, but only for any one assumed time, and then to deduce their values for any other time by means of their uniform changes. It will be most simple to find them for the time of true conjunction of the moon and star in right ascension, which is readily obtained by the aid of the hourly Ephemeris of the moon. Let this time be denoted by T . We have at this time x = 0, and the value of y will be found with sufficient accuracy by the formula in which d, ~, are the moon's declination and horizontal parallax at the time T , and d' is the star's declination. Let AOI (in seconds of arc) and A here denote the hourly changes of the moon's right ascension and declination for the time T . Then we have, nearly, Aa Ad X? = COS (J M = TT rr Let 7\ be any assumed time (which, in a first approximation, may be the time T itself). Then the values of the co-ordinates at this time are 558 OCCULTATIONS OF FIXED STARS. and to find the time (T) of contact of the star and the moon's limb, we shall, according to Art. 322, have the following formulae : in which fa is the sidereal time at the first meridian corresponding to T v a f is the star's right ascension, and /;, we liave the impossible value sin ^ > 1, which shows that the star is not occulted at the given place. If we wish to know how far the star is from the moon's limb at the time of nearest approach, we have (Art. 324) A = m sin (M N~) the sign being taken so that J shall be positive. This is the linear distance of the moon's centre from the line drawn from the place of observation to the star, and therefore the angular distance as seen from the earth is TT J. The apparent semidiameter of the moon is irk, and hence the apparent distance of the star from the moon's limb is ^(J />;).* EXAMPLE. Find the times of immersion and emersion in the occupation of Aldebaran, April 15, 1850, at Cambridge, Mass. The elements of this occultation have been found on p. 553, with which an accurate computation may be made by the method of Art. 322 ; but, according to the preceding approximate method, we proceed as follows. The Greenwich time when the moon's right ascension was equal to that of the star is found, from the values of a on p. 553, to be T = 7*.47 = 7* 28 12'. For this time we have A O = + 2174" d = + 16 49' 31".l AJ = + 384 d'= 16 12 1 .7 TT= 3536 d S'=+ 2249" whence, by the above formulae, y o = -[- 0.6360 x'= + 0.5886 y' = + 0.1086 Then the computation for Cambridge, tp = 42 22' 49", to = 4* 44 OT 30", will be as follows. For the first approximation, we assume T v = 7J, and hence we have * More exactly, allowing for '.he augmentation of the moon's semidiameter, it it f (J *) (1 + C sm 7r )> where we have f = A cos (B (5'). 560 OCCULTATIONS OF FIXED STABS. T t = 7* 28 12V Sid. time Gr. noon = 1 33 90 Reduction for T t = 1 13 .6 At, =nr~2 34.6 a.'= 4 27 18.3 w = 4 44 30 At, a' at = .V = 23 50 46 .3 = 357 41'.6 with which we find the following results : x = 0. y = -f 0.6360 f = _ 0.0298 , = + 0.4377 , w sin M = + 0.0298 m cos Jf = -f- 0.1983 M = 8 32'.4 log w = 9.3021 0^== + 0.5886 /= + 0.1086 I' = -f 0.1940 V= 0.0022 w sin .flT = -f- 0.3946 w COB N = -f 0.1108 j\r= 74 19M logn= 9.6127 log sin 4 = w9.8395 log cos 4 = 9.8590 n = _ o^ ^ For immersion. For emersion. r = _ 0\6491 r = + 0.3111 T, = 7 .4700 r, = 7 .4700 T= 6.8209 T= 7.7811 T = 6* 49" 15' T = 7* 46" 52- ^ = 4 44 30 w = 4 44 30 Local time = 2 4 45 Local time = 3 2 22 These times are nearly correct enough ; but, for a more accurate time of emersion, we now assume 7^= 7*.7811, with which we find x = x'(T 1 T ) = + 0.1831 \f ( T; r o ) = 4. 0.0338 y = -f 0.6360 y _|_ 0.6698 and to find the new value of & we have r = -\- O ft .3111 = 18* 40*, the sidereal equivalent of which is 18 nl 43M, or in arc 4 40'.8. This, added to the above value of $, gives the corrected value t? = 2 22'.4. Repeating the computation with these new values of x, y, and #, we find LIMITING PARALLELS. 561 mco 8 (^-.V)^_ . 5()82 l = 'll% n -tV = I -* oo . 4- .4901 ___ 212 17 = -0.0181 r= 3 33 7 .7811 208 34 = 7* 45"- 47' Local time = 3 1 17 The star reappears at 212 17' from the north point, or 208 34' | from the vertex, of the moon's | limb. This time agrees within 21* with the actually observed time of emersion (given on p. 552). The principal part of the difference is due to the error of the Ephemeris on this date. 346. To find the limiting parallels of latitude on the earth for a given occultation. The limiting curves within which the occulta- tion of a given star is visible may be found by the general method given for solar eclipses, Art. 311, which, of course, may be much abridged in such an application. But, on account of the great number of stars which may be occulted, it is not pos- sible to make even this abridged computation for all of them. The extreme parallels of latitude are, however, found by very simple formulae, and may be used for each star. For a point on the limiting curve, the least value of J in Art. 324 is in a solar eclipse L, but in an occultation it is = k. Hence we have, by (557), the condition m sin (M N) = k or, restoring the values of m sin M= x , m cos M = y 37, (x ) cos N -- (y iy) sin N k The angle Nie here determined by the equations (552); but, for an approximate determination of the limits quite sufficient for our present purpose, we may neglect the changes of and 37, and take n sin N = x' n cos N = y' Let x and y be the values of x and y for the assumed epoch T ; then for any time T= T^ + r we have x = x 9 -|- n sin N . r y = y -j- n cos N . r VOL. L 3 562 OCCULT ATIONS OP FIXED STARS. which reduce the above condition to (x ) cos N (y if) sin N = k By the last equation of (500), we have, by neglecting the com- pression of the earth, sin

, which fulfil these conditions. Let us put a = cos N -f- ^ sin jV b = sin JV r -f- T) cos J\T from which follow = a cos jV" -j- 6 sin N T) = a sin _ZV -f- 6 cos JV c = Then we also have, by our first condition, a = x cos N -f- y sin N k which is a constant quantity, since we may here assume x' and y f to be constant. Since we have 2 + 6 2 + 2 1, we can assume 7- and e so as to satisfy the equations cos Y = a sin f cos e = b sin r sin e = C in which sin f may be restricted to positive values. The formula for

f One of the points thus determined may, however, be upon that side of the earth which is farthest from the moon, since we have not restricted the sign of , and our general equations express the condition that the point of observation lies in a line drawn from the star tangent to the moon's limb, which line intersects the surface of the earth in two points, for one of which is' positive and for the other negative. But, taking only with the positive sign, we must also have sin e positive. For the northern limit, therefore, when A e, sin ^ must be positive, which, according to the equation cos /9 sin A = sin =/3 f gives the southern limit only when the star is in south declina- tion. The second limit of visibility in each case must evidently be one of the points in which the general northern or southern limiting curve meets the rising and setting limits, namely, the points where = 0, and consequently, also, sin e = 0, cos e = 1, which conditions reduce the general formula for sin

l will be the northern limit and y> 2 the southern ; and the reverse when the declination is south. The angle A T is here supposed to be less than 90, and is found by the formula always considering y' as well as x' to be positive. When the cylindrical shadow extends beyond the earth, north or south, we shall obtain imaginary values for ft or Tz- The following obvious precepts must then be observed : 1st. When cos ft is imaginary, the occultation is visible beyond the pole which is elevated above the principal plane of reference, and, therefore, we must put for the extreme limit

2 = 3' 90, or ^ 2 = 1 = /? ft may exceed 90, in which case the true value is either

, p' = the distances of the centre of the planet and the point on its surface from the observer, we shall have (Arts. 32 and 33)f p cos y? cos ^ = sin yl = TJ \ (605) p' COS y9' COS A' = X p' cos/5' sin X y y J> (606) p' sin [f = z C * The method of investigation here adopted, so far as relates to the apparent form of the disc, is chiefly derived from BESSEL, Astronomische Untersuchungen, Vol. I. Art. VI. f The group (606) may be deduced by supposing for a moment that the position of the observer is referred to a system of planes parallel to the first, but having its origin at the point on the surface of the planet. The co-ordinates in this system are equal to those in the first increased respectively by x, y, and z. The negative sign in the second members of both groups results from the consideration that the longi- tude of the observer as seen from the planet is 180+ A, or 180 -f- A'; and hi* latitude, /?, or /?'. Compare Art. 98. FORM OF A PLANET S DISC. 567 Now, let and C, Fig. 47, be the apparent Fi & places of the planet's centre and the point on its surface, projected upon the celestial sphere; Q the pole of the planet's equator; P the pole of the earth's equator ; and let 5- = the apparent distance of C from O = the arc OC, p'= the position angle of C reckoned at 0, from the declination circle OP towards the east, = POC, p= the position angle of the pole of the planet = POQ; then, in the triangle QOC, we have sin s' sin (p 1 p) = cos /?' sin (A' A) sin s' cos (p' />) = cos /3 sin ft' sin ft cos ft' cos (A' A) Multiplying these by p', and substituting the expressions (605) and (606), we obtain p' sin s' sin (p 1 p) = x sin A -f- y cos A // sin s' cos (p' p~) = x sin ft cos A y sin /? sin /I -f- 2 cos ft or, since s' is very small and p f sin 5' or p's' differs insensibly from p sin s' or /?s', / p) = a; sin A -\- y cos A sn ps' cos (p' p) = x sin /9 cos X y sin /? sin ( 307 > These equations apply to any point on the surface of the planet. If we apply them to those points in which the visual line of the observer is tangent to that surface, they will determine the curve which forms the apparent disc. The equation of an ellipsoid of revolution whose axes are a and 6, of which b is the axis of revolution, is T-Y titi ?y (608) = xx j/y zz aa aa bb and the equation of a tangent line passing through the point whose co-ordinates are , 77, and is = _i_ _i_ aa aa bb (609) The distances , r t , and are very great in comparison with x, 568 OCCULTATIONS OF PLANETS. K ~ r y, and z. If we divide (609) by />, the quotients -, -, - will be X V Z of the same order as -, -, p but the quotient will be inappre- ciable in relation to the quotients , , Performing this aa aa bb division, therefore, and substituting the values of c, ' p) v = s' cos ( y p) the equations (607) and (610) will enable us to determine z, y, and z in terms of u and v. Putting bb = lee aa the three equations become pu = x sin A -f y cos >l pv = (x cos A -j- y sin /t) sin /9 -\- z cos ,3 = (x cos X -\- y sin A) (1 e) cos /5 -f- 2 sin from which we derive x sin A -f y cos A = /m x cos A y sin A = pv 1 ee cos 2 /? (1 ee) cos ,3 z = pv *- - - - - 1 eeco&fi Substituting these values in (608) and putting s = - = the greatest apparent semidiaraeter of the planet, c = i/ (I ee cos 2 >5) FORM OF A PLANET'S DISC. 569 we find ss = uu + - (611) which is the equation of the outline of the planet as projected upon the celestial sphere, or upon a plane passed through the centre of the planet at right angles to the line of vision. It represents an ellipse whose axes are 2s and 2s ]/(! ee cos 2 /9), e being the eccentricity of the planet's meridians. The minor axis (OB, Fig. 47) lies in the direction of the great circle drawn to the pole of the planet's equator. We next proceed to determine what portion of this ellipse is illuminated and visible from the earth. 349. To find the apparent curve of illumination of a planers surface. If the sun be regarded as a point (which will produce no sensible error in this problem), the curve of illumination of the planet, as seen, from the sun, can be determined by conditions quite similar to those employed in the preceding problem ; for we have only to substitute the co-ordinates expressing the sun's position with reference to the planet, instead of those of the observer. If, therefore, we put A, B = the heliocentric longitude and latitude of the centre of the planet referred to the great circle of the planet's equator, the equation of the tangent line from the sun to the planet, being of the same form as (610), will be x COB B cos, A ycos-Bsinvl zs'mS If each point which satisfies this condition be projected upon the celestial sphere by a line from the observer on the earth, and u and v again denote the co-ordinates of the projected curve, we have here, also, to satisfy the equations /m^-tfsinA+ycosA i pv = (x cos /I + y sin /I) sin /? + z cos /3 / in which A and ft have the same signification as in the preceding article. The values of x, ?/, and z, determined by the three equations (612), (613), being substituted in the equation of the ellipsoid, we obtain the relation between u and v, or the equation 570 OCCULTATIONS OF PLANETS. of the required curve of illumination as seen from the earth. In order to facilitate the substitution, let us put x l = x sin A -j- y cos X y l = x cos A -j- y sin X from which follow x= #1 sin >l -f- y, cos A y Xi cos ^ -f- #1 sin A At the same time, let us introduce the auxiliaries & and B^ dependent upon ft and B by the assumed relations - cos ft cos /9 ^ cos B t = cos B 1 fl . 1 . -f. Q, . _ - sin ft = - sin /9 sin B^= r sin 5 a b (TV (614) 9 Then the three equations become = x l cos .Bj sin (A ,1) -f y x cos B t cos (J X) -\- -z sin 5, jDtt = #, - gr/>y = y l sin ft -j- - z cos ft from which we derive Ny l = pu coa ft cos 5, sin (A /I) - gpv sin B, jVr 2 = /ou sin ft cos .fijsin (A A) -\- rgpv cos 5 1 cos(yl ,*) where, for brevity, A r isput for sin /9j sin B v + cos^ cos .B, cos (yi /). Before substituting these expressions in the equation of the ellipsoid, it will be well to consider the geometrical signification of the quantities /?, and B r If we draw straight lines from the centre of the planet to the earth and to the sun, the latitudes of the points in which these lines intersect the surface of the planet will be ft and B. If these points be projected upon the surface of a sphere circumscribed about the ellipsoid, by perpendiculars to its equator, the latitudes of the projected points will be ft l and B t ; and g and (rwill be the corresponding radii of the ellipsoid. If now these projected points are referred to the celestial sphere, by lines from the planet's centre, they will form with the pole Q of the planet's equator a spherical triangle QOS, in which the FORM OF A PLANET'S DISC. 571 angle Q will be A A ; and the sides including this angle will he 90 ft = QO, 90 B l = QS. Denoting the angle at by w, and the side OS by V, we shall have cos V = sin ft sin B l -f cos /? t cos B v cos (/I J) ^ sin F cos w = cos ft sin S l sin ^ cos B cos (yl ^) V (615) sin Fsin w = cosj5 t sin (A A) J in which V is very nearly the angular distance between the sun and the earth as seen from the planet. This triangle also gives sin 5j cos Fsin /? t -f- sin Fcos /3j cos w cos 5j cos (A A) cos F cos ^ sin F sin /^ cos M; cos 5, sin (yl X) ~ sin F sin w By these equations the above expressions for x v y v and z are reduced to cos F. x t = pu cos F cos F. j/j = jM sin F sin M? cos /9, - gpv (cos F sin /?, -(- sin F cos /3 t cos u>) cos F- - z = pu sin F sin w sin /3 Z -\-j-gpv (cos F cos ^ sin F sin /? t cos w?) Substituting these in (608), observing that xx + yy = x^ + y^ v we have PP -\-\(u &m w -\- -gv cos i/?) sin F cos /Jj -|- - gv cos F sin /3, 1 -f- J (M sin w -j- - ^y cos to) sin F sin /9j - gv cos F cos /3j I Developing the squares in the second member, and putting s for -, and also c = ;/(l ee cos 2 /3) = - we shall find / sin w \* / cos w \* ss = I M cos w; u- -I -{- (u sin w -\- v J sec 2 F (616) 572 OCCULTATIONS OF PLANETS. which is the required equation of the curve of illumination, as seen from the earth, projected upon the celestial sphere. It represents an ellipse whose centre is at the origin but whose axes are, in general, inclined to the axes of co-ordinates, and, consequently, to the axes of the ellipse of equation (611). The equation (611) is only the particular case of (616) which corre- sponds to V 0, or the case of full illumination. Fig. 48. 350. We have yet to determine what portions of the apparent disc are bounded by the two curves respectively. If ABA'B', Fig. 48, is the ellipse of (611), which I shall call theirs* ellipse, and CDC'D' that of (616), which I shall distinguish as the second ellipse, the visible outline of the planet is composed of one- half the first and one-half the second curve, and these halves either begin or end at the points C'aml C', which are the common points of tangency of the two curves. These points satisfy both equations; and, therefore, putting w, and v i for the co-ordinates of either point, and subtracting (611) from (616), we find which is satisfied, in general, by taking cos w Denoting the position angle corresponding to u^ v 1? by /> we have Wi = Sj sin (/>, ;>), v l = s l coa(pi-p). Substituting these values, and also putting c, cos w, = (617) c, sin w 1 = sin w the preceding condition becomes c l s l cos (p l p u?,) = whence Pl = p + u>, q= 90 (618) which expresses the position angles of both C and C". If we draw the arc ODO', Fig. 48, making the angle BOO' = w, and FORM OF A PLANET'S DISC. 573 take 00' = V, the point 0' will be nearly the position of the planet as seen from the sun, and the arc Vwill be the measure of the angular distance between the sun and the earth as viewed from the planet. If we assume sin w to be positive in equations (615), as we are at liberty to do, the arc Fwill be reckoned from the planet eastward from to 360. Now, so long as Vis less than 180, the west limb will evidently be the full limb, and when Vis greater than 180, the east limb will be the full limb. Hence we infer that a point whose given position angle is p f is on the east limb when p' > p J|_ Wl 90 and < p + w^ -f- 90 but on the west limb when P'

P + MI + 90 When V > 90 and < 270, the planet is crescent ; but when V> 270 arid < 90, it is gibbous. In the case of a crescent planet there are two points, one on the full and the other on the crescent limb, corresponding to the same position angle : hence in observations of a crescent planet the point of observation on the limb will not be sufficiently determined by the position angle alone ; it will be necessary for the observer to distinguish the crescent from the full limb in his record. 351. In order to apply the preceding theory, it is necessary to find the quantities p, /9, ^, B, A. The direction of the axis of x in Art. 348 was left indeterminate, and may be assumed at pleasure, but it is most convenient to let it pass through the ascending node of the planet's equator on the equinoctial, so that X and A will be reckoned from this node. The position of the node must, therefore, be known, and this we derive from the researches of physical astronomers. If we put n =. the longitude of the ascending node of the planet's equator on the equinoctial, i =. the inclination of the planet's equator to the equi- noctial, we have at any given time t, for the planets Jupiter and Saturn, the only ones whose figures are sensibly spheroidal, 574 OCCULTATIONS OF PLANETS. _ T ( n = 357 56' 25" + 3".59 (t 1850) For Jup.ter. | . = ^ ^ ^ J ^ ForSati n f n = 12513'54" + 128".76(<- 1850) +0".0605(i-1850)' " 15".08( 1850) + 0".0035(Z 1850)* in which is expressed in years.* The values for Saturn apply either to its equator or the rings, which are sensibly in the same plane. If now we put a', 8' = the right ascension and declination of the planet, we can convert a' and o f into / and /? by Art. 23 ; we shall merely have to substitute in (29) or (31) a' n fora, d' for <5, and i for e. The angle p is here the position angle of the pole of the planet reckoned from the declination circle of the planet towards the east; but in Art. 25 the angle y is the position angle reckoned towards the west, and, therefore, we shall have to put y = 360 p in (33). Hence we obtain the following formulae for , ^, and p: f\nF=tsii\8' f sin A cos (F i) / cosF = sin (a' n) /' cos A = cos F cot (a' n) tan /? = sin A tan (F t) sin F' cot (a' n) tan F = tan i sin (V n) tan = cos OF' d') (619) To find A and 5, we avail ourselves of the heliocentric longi tude and latitude of the planets given in the British Almanac, and as these quantities are referred to the ecliptic, while A and B are referred to the planet's equator, we must know the rela- tive position of these circles. Putting JV'= the longitude of the node of the planet's equator on the ecliptic, I' = the inclination of the planet's equator to the ecliptic t N = the arc of the planet's equator between the equi- noctial and the ecliptic, * These values I have deduced from the data given in DAMOISEAU'S Tables tiqnes des Satellites de Jupiter, Paris, 1836; and BESSEL'S Bestimmung der Lage vn, 0", Fig. 50, be the three places above referred to, and P the pole of the equinoctial. Draw O'Q perpen- dicular to the great circle S'0'0". This perpendicular passes through the adopted pole of the planet, and we have PO'Q=p, or PO'S'= 90 /?, and S'0'= ?. Hence, denoting by d' and D the declination of the planet and the sun, and by a' and A their right ascensions respectively, the spherical triangle PS'O 1 gives cos Y = sin 8' sin D -f cos 8' cos D cos (a' A) *\ sin Y sin p cos 5' sin D sin 5' cos D cos (a' 4) i (622) sin Y cos p = cos Z> sin (a' A*) ) Hence, introducing an auxiliary to facilitate the computation, both p and F will be found by the following formulae : tan F = tan D sec (a' A) tan p = cot (a' A) sin (F 8') sec F (623) sn = R' ' cosp In this method of finding F we do not determine whether it is VOL. I.. -37 578 OCCULTATIONS OF PLANETS. greater or less than 90. This is of no importance in computing an actual observation, but only in predicting the phase of the planet, whether crescent or gibbous. For the latter purpose we must have recourse to the triangle 8EO of Fig. 49, the three sides of which are given in the Ephemeris. The value of V being found, the equation (616) will be used to determine the apparent outline after substituting c = 1 and w 90, whereby it becomes s z == v 1 -f u 2 sec 2 V The value of s in our equations is supposed to be given. It will be most convenient to deduce it from the apparent semi- diameter of the planet when at a distance from the earth equal to the earth's mean distance from the sun, which is the unit employed in expressing their geocentric distances in the Ephe- meris. Thus, denoting the mean semidiameter by s , and the geocentric distance by r', we have (Art. 128) (624) and s may be taken from the following table : o Authority. MERCURY 3". 34 LE VERRIER, Theory of Mercury. VENUS 8 55 MAES 5 05 99 70 STRUVE Astr Nach No 139 SATURN 81 36 BESSEL, Astr Nach., No. 275. SATURN'S RINGS Outer semi-major axis of outer ring Inner " " " " Outer " " inner " Inner " " ' " 187 .56 165 .07 161 .27 124 .75 ( STRUVE, Astr. Nach.,No. 139, reduced to agree with BES- SEL'S measures of the outer >- diameter of the outerrin$ 353. To find the longitude of a place from the observed contact of the moon's limb with the limb of a planet. In the following investi- gation, it is assumed that the quantities p, w, V, c, are known for the time of the occultation. They may be computed by the above methods for the time of conjunction of the moon and planet, and regarded as constant for the same occultation over the earth in general. LONGITUbE. 579 Let 0, Fig. 51, be the apparent centre of the planet, and C the point of contact of its limb with *hat of the moon. Let OM be drawn from towards the moon's centre, in- tersecting the moon's limb in D. Since the apparent semidiameter of any of the planets is never greater than 31", it is evident that no appreciable error can result from our assuming that the small portion CD of the moon's limb coincides sensibly with the common tangent to the two bodies drawn at C. If, then, the planet were a spherical body with the radius 0Z>, the observed time of contact would not be changed. We may, therefore, reduce the occupation of a planet to the general case of eclipse of one spherical body by another, by substituting the perpen- dicular OD for the radius of the disc of the eclipsed body. Let s" denote this perpendicular; let OA and OQ be the axes of u and v respectively, to which the curve of illumination is referred by the equation (616) ; and let $ be the angle QOD which the perpendicular s" makes with the axis of i\ The equation of the tangent line CD referred to these axes is u sin # -f- v c H # = s" (625) We have also in the curve dv 5- = tan # du Differentiating the equation (616), therefore, we have (v sin w\l , tan # sin w' u cos w 1 1 cos w -f- / v cos w \ I . tan 9 cos w \ -f- I u sin w -\ 1 1 sin w I sec 2 V = By means of this equation, together with (616) and (625), we can eliminate u and y, and thus obtain the relation between 5 and s". To abbreviate, put v sin w X = U COS W y = u sin w -j- c v cos to 530 OCCULTATIONS OF PLANETS. and also then the three equations become x cos (#' 10) y sin (#' w) sec 2 V = x 3 + z/ 2 sec 2 V = s 2 a: sin (#' 10) -|- y cos (*' w) = -, CC From the first and second of these we find s sin (*' ti?) yll cos 2 (*' 10) sin 2 F] 5 COS (#' 10) COS 2 F ^ = !/[! co8(#' ic) sin 2 F] which substituted in the third give s" =scc'i/[l cos 2 ('9' ic) sin 2 F] Hence, if we put sin % = cos (?' 10) sin F ~) we have [ (627) s" = s .cc' cos % ) We have seen (Art. 352) that in all practical cases we may take w = 90, and, therefore, instead of (626) and (627) we may employ the following : tan tan #' = c sin / = sin #' sin F s sin 9 cos x (628) If the occultation of a cusp of Venus or Mercury is observed, we have at once s" = s cos & (for the axis of v coincides with the line joining the cusps), and we do not require V. The value of s" is to be substituted in (486) for the apparent semidiameter of the eclipsed body. In that formula, H denotes the apparent semidiameter at the distance unity : therefore, we must now substitute the value sin H ' = r' sin s" LONGITUDE. 581 or, by (624) and (628), sin H = 8ing o sin * c08 ;r (629) sin *' Since / is here very small, we may put tan / = sin /, and the formula for L (488) becomes L = (z C) sin / k r'g r'g Hence, putting * = * + -0^p (630) we have L = (z-^^-fK ( 631 > WTien the angle & is known, therefore, the preceding formulae will determine L, with which the computation will be carried out in precisely the same form as in the case of a solar eclipse, Art. 329. To find #, let OP, Fig. 51, be drawn in the direction of the pole of the equinoctial ; then we have POQ = p, and, denoting POM by Q, #=Q-P and Q has here the same signification as in the general equations (567), as shown in Art. 295 : so that when N and ^ have been found by (568) and (569), we have Q = N+ ^, or $=N+*p (632) But to compute ^ by (569) we must know L, and this involves H, which depends upon ft. The problem can, therefore, be solved only by successive approximations; but this is a very slight objection in the present case, since the only formulae to be repeated are those for L and ^ and the second approximation will mostly be final. It can only be in a case such as the occul- tation of Saturn's ring, where the outline of the eclipsed body is very elliptical, and especially when the contact occurs near the northern or southern limb of the moon, that it maybe necessary (for extreme accuracy) to compute H a second time and, conse- quently, -\J, a third time. The formula (629) is adapted to the general case of an ellip- 582 OCCULTATIONS OF PLANETS. soidal body partially illuminated, the point of contact being on the defective limb. When the point of contact is on the full limb, we have only to put T^= 0, and the formula becomes Bing= 8in oa " J ^ (633) tun* and for the full limb of a spherical planet (Venus, Mercury, and Mars) we have H = s . In the first approximation we may take L = k. 354. Sometimes it may not be known from the record of the observation whether the point of contact is on the full or the defective limb of the planet. This might be determined by the method of Art. 350 ; but, since that method supposes the position angle p f to be given, which we do not here employ, the following more direct and simple process may be used. In that article the common point of tangency of the two curves of the full and defective limbs was determined by the condition cos w M. sin w -\- v. = c in which w t and v l denotes the co-ordinates of the point of tan- gency. In the notation of Art. 353 this is simply y l = ; and since we have s cos (#, w) cos 2 V 1 |/[1 co8 2 (*j to) sin'F] it follows that we must have cos (#, w) = or #j = w =f 90 Hence, when, as in our present application, we take w = 90, we have ^ = or ^ = 180 Hence a point is to be regarded as on the east limb for values of & between and 180, and on the west limb for values of & between 180 and 360 ; and (Art. 350) the east or the west limb is defective accord- ing as Vis between and 180 or between 180 and 360. But, since sin #' and sin # have the same sign, we deduce from this a still more simple rule ; for we have sin % = sin &' sin V, whence it follows that the observed point is on the defective limb when sin y is positive, and on the full limb when sin % is negative. LONGITUDE. 583 355. In the cases of the planets Neptune, Uranus, and the asteroids, the occupation of their centres will be observed, and it will be most convenient to compute by the method for a fixed star, only substituting for x the difference of the moon's and planet's horizontal parallaxes that is, the relative parallax in the formulae for x and y, Art. 341. This artifice of using the relative parallax may also be used with advantage for Jupiter and Saturn. Having thus found x and y as for a fixed star, we shall have, in the preceding method, (634) the other formulae remaining unchanged. EXAMPLE 1. Several occultations of Saturn's Ring were ob- served by Dr. KANE at Van Rensselaer Harbor on the northwest coast of Greenland during the second Grinnell Expedition in search of Sir JOHN FRANKLIN.* The first of these was as follows : 1853 December 12th, Van Rensselaer Mean Time Immersion, contact of last point of ring, . . . 14* 20 48*.8 Emersion, " " "... 14 54 18.3 The assumed longitude of the place of observation was * = qi k) ar. co. log i nO.56441 0.56441 log sin 4. 9.92708 n9.91980 *4 57 43'.2 303 45'.5 A" p 74 14.5 74 16.6 & + 4- p = * 131 57.7 18 0.1 log tan # nO.0462 9.5118 logc 9.6094 9.6094 log tan >' nO.4368 9.9024 log sin ft n9.8713 9.4900 ar. co. log sin ft' nO.0273 0.2047 log 1 - j sin s 7.8465 7.8467 flog a 7.7451 7.5414 a 0.00556 0.00348 rp A 0.27264 + 0.27264 a+ k = L 0.26708 + 0.27612 log I, n9.42664 9.44110 Applying the difference between log L and log k to log sin 4/, we find, for our second approximation, Corrected log sin 4. 9.93603 9.91429 " 4 59 39'.6 304 49'.5 ii ? Q 133 54.1 19 4.1 log tan ft wO.0167 9.5387 log tan >' nO.4073 9.9293 log sin ft n9.8577 9.5141 ar. co. log sin ft' nO.0310 0.1887 7.8465 7.8467 Corrected log a 7.7352 7.5495 " a 0.00543 0.00354 " L 0.26721 -f 0.27618 " log L n9.42685 9.44119 Final value of log sin 4, 9.93582 n9.91420 log cos 4- 9.70403 9.75688 * The angle 4. is to be taken so that L cos 4, shall be negative for immersion and positive for emersion, Art. 329. f Putting a = (z f ) ^ = ^5 . * ~ ^ . *, ^ *' -j :_ .a' _' " LONGITUDE. 587 Immersion. Emersion. 3600 n2.94455 3.01171 log c km cos (M N) n2.95956 3.00741 10g n b - 880M + 1027'.3 c - 911.1 + 1017 .2 bC = T + 31.0 + 10.1 Gr. Time of obs. = T + T = T 19* 4-43M) 19* 37 M 58.l Tt = w 4 43 54.2 4 43 39.8 If now we wish to form the equations of condition for deter- mining the effect of errors in the data, we proceed precisely as in the case of a solar eclipse, page 533, and find log v tan 4 log v sec 4 Immersion. 0.5341 0.5983 Emersion. nO.4596 0.5454 3600 where log v = log = 0.3023. Hence, neglecting the terms depending on the correction of the parallax and of the eccen- tricity of the meridian, the equations of condition are (Im.) ^ = 4* 43 m 54'.2 2.001 7- -f 3.421 * 3.965 - k (Em.) o>, = 4 43 39 .8 2.001 ? 2.881 * + 3.511 JTA* Eliminating # from these equations, we have w l = 4* 43" 46'.4 2.001 r + 0.092 -A* An error of 1" in the moon's semidiameter (represented by -t^k, would, therefore, have no sensible effect upon this combined result; and since 7- must also be very small, as we have corrected the places of the moon and planet by the Greenwich observations, we can adopt, as the definite result from this observation, It will be observed that in this example OUDEMANS'S value, f< = 0.27264, has been employed ; but our final equation shows that the result would have been sensibly the same if we had taken the usual value 0.27227 ; for the reduction of the result to that which the latter value of k would have given is only 0.092 X 3247 X (- 0.00037) = - OM1. 588 OCCULTATIONS OF PLANETS. EXAMPLE 2. The occultation of Venus, April 24, 1860, was observed at the U. S. Military Academy, West Point (ft>=4* 55 m 51",

= 42 39' 49". 5), as follows : Immersion. First contact, planet's full limb Disappearance of cusp The observations were made with the large refractors of the West Point and Dudley observatories. I. To find p for the cusp observations, we have for the Green- wich time 13\478, which is the mean of the times of the obser- *>' log p cos 0' f 9.818064 9.875814 + 0.745828 + 0.551616 + 0.37 59.82 + 0.746178 + 0.552909 + 0.37 59.82 9.828792 9.867157 + 0.730013 + 0.563428 + 0.38 59.81 + 0.730378 + 0.564641 + 0.38 59.81 IV. Assuming T = 13*.45, we find, for this time, x o + 0.558390 y + 0.773921 *- = m sin M 0.187438 0.187788 0.171623 ^0.171988 y n m cos M + 0.222305 + 0.221012 + 0.210493 + 0.209280 M 319 51' 50" 319 38' 47" 320 48' 30" 320 35' 11" log m 9.463563 9.462425 9.433915 9.432783 N 89 29' 6" log n 9.727480 Then, for the observations of the full limb, we have for both places, by (631), putting H = s , log (z - 1.7768 0820 1.7768 00820 k = 0.27264 constant 5.0542 log sin * 5.6176 0.00082 . log 6.9130 K =0.27346 00299 loe (11 7 4763 L= 0.27645 590 OCCULTATIOXS OF PLANETS. West Point. Albany. MN 230 22' 44" 231 19' 24" i 234 6 57 230 4 55 T + 2- 37'.7 52'.7 T 13*27 0'. 13* 27" 0-. T 13 29 37.7 13 26 7.3 T-t = a> 4 55 54.0 4 55 5.4 For the observations of the cusps we can employ the preceding values of ^ as a first approximation ; and hence we proceed as follows : West Point. Albany. 2V-f-4-j=# log cos ft log(l) 331 3'.4 9.9421 7.4763 327 1'.3 9.9237 7.4763 k' 7.4184 0.00262 0.27346 7.4000 0.00251 0.27346 L MN log sin (MN) logm ar. co. log L 0.27084 230 9' 41" n9.885278 9.462425 0.567287 0.27095 231 6' 5" n9.891124 9.432783 0.567111 log sin ^ 4 Corrected * log cos # log (1) n9.914990 235 18'.5 332 14.9 9.9469 7.4763 w9.891018 231 5'.0 328 1.4 9.9285 7.4763 Corrected L ar. co. log L Corrected log sin 4- T T + r = T Tt = a 7.4232 0.00265 0.27081 0.567335 n9.915038 + 3 33'.7 13* 30" 33.7 4 55 55.7 7.4048 0.00254 0.27092 0.567159 9.891066 0'.4 13* 26 m 59'.6 4 55 5.4 Finally, if we wish to form the equations of condition for correcting these results for errors in the data, including an error in the planet's semidiametcr, we proceed as for an eclipse of the TRANSITS OF VENUS AND MERCURY. 591 sun, p. 533. For the full limb we have only to substitute A.9 for &H; but for the cusp we must evidently substitute A.? O cos # for H. It will be more accurate to restore r'g in the place of r', since g here differs sensibly from unity. We shall thus find / = 4*55" 54'.0 1.967 ? + 2.720 * 3.358 * A* 4.061 AS O w' = 4 55 55 .7 1.967 r + 2.844 3.459 * AA + 8.697 AS O <,."= 4 55 5 .4 1.967 r + 2.352 3.067 r A* 3.704 AS O ."= 4 55 5 .4 1.967 r + 2.438 * 3.134 r A* + 3.349 AS O where to' and o>" denote the true longitudes. Hence, also, .' i" = -f 48'.6 -f- 0.368 # 0.291 AA- 0.357 AS O ' - /' = + 50 .3 -f- 0.406 * 0.325 - AA -f- 0.348 AS O and the mean is w' a>" = -j- 49.5 -f 0.387 * 0.308 r AA- 0.005 AS O The effect of an error in s upon the difference of longitude of the two places is, therefore, insensible ; but, to eliminate & and 7TAA", observations of the emersion should also be used. The effect of f and $ upon co' and to" can only be eliminated by means of observations of the moon's place at a standard observa- tory on the day of the observation, as we have already shown in other examples. TRANSITS OF VENUS AND MERCURY. 356. The transits of Venus and Mercury may be computed by the method for solar eclipses, substituting the planet for the moon. In the formulae (486), (487), &c., we must employ for Venus, k = 0.9975 for Mercury, A- = 0.3897 which are the values which result from the apparent semi- diameters of these planets adopted on p. 578. Since 6 is no longer a small quantity, it will be necessary to employ the exact formulae (479) instead of (481). The longitude of a place at which the transit is observed may be computed from each of the four contacts of the limb of the sun and planet, by the formulae ot Art. 329. These observations, however, are of little use in determining an unknown longitude, on account of the great effect of small errors in the assumed 592 TRANSITS OF VENUS AND MERCURY. parallax upon the computed time ; but, on the other hand, when the longitude is previously known, each observation furnishes an equation of condition of the form (584) for determining the correction of the parallax. In developing this equation, however, we supposed g = 1, in the formula (486), and we must, therefore, here restore the true value. "We may take in which TT and it' are the assumed horizontal parallaxes of the planet and sun respectively at the time of the observation. Instead of the form for I employed on p. 449, we shall now take the more correct form l= * * r'g* g If we denote the sun's semidiameter at the time of the obser- JT vation by s', that of the planet by s, we have s' = , s xk, and hence _ s ' s 9* and instead of (581) we shall have Omitting the term depending upon &ee, which can never be appreciable in the transits of the planets, the equation (582) will now become w' T~) x tan 4- s s 8ec 4,") ATT (635) 9 K J where f and t? have the signification (583); to' is the true longi- tude, and CD that which is computed from the observation. Since, by KEPLER'S laws, the ratio of the mean distances of any two planets is accurately known from their periods, the ratio is also known, and will not be changed by substituting the cor- rected values TT -f- ATT and 7r -(- ATT O : in other words we shall have A?r TT r or AJT O I=: "ATT ATT O TT TT TRANSITS OF VENUS AND MERCURY. 593 The discussion of all the equations of condition of the form (635) will, therefore, give not only the correction A~ of the planet's parallax, but also, by the last-mentioned relation, that of the solar parallax.* The transits of Venus will afford a far more accurate deter- mination of this parallax than those of Mercury ; for, on account of its greater proximity to the earth, the difference in the dura- tion of the transit at different places will be much greater, and the coefficient of AT: in the final equations proportionally great. Although the general method for eclipses may also be ex- tended to the prediction of the transits of the planets (by Art. 322), yet it is more convenient in practice to follow a special method in which advantage is taken of the circumstance that the parallaxes of both bodies are so small that their squares and higher powers may be neglected. LAGRANGE'S method for this purpose is the most simple, and, in the improved form which I shall give to it in the following article, most accurate. 357. To predict the times of ingress and egress for a given place. ~ We first find the times of ingress and egress for the centre of the earth, from which the times for any place on the surface are readily deduced. Let a, 3, a/, o' be the right ascensions and declinations of the planet and the sun for an assumed time T w at the first meridian, near the time of con- junction. Let m denote the apparent dis- tance of the centres at this time. Let S' and S, Fig. 52, be the geocentric places of the centres of the sun and planet, P the pole ; then, denoting the angles PS'S and PSS' by P' and 180 - P, the triangle PSS 1 gives sn m sn - = sn sin j m cos (P + P') = cos } (a a') sin i ( we mav regard the following equations as practically exact: rosin2lf=:(a-a')coBa o \ mcosM=: 5-3' f in which d (8 + d'). Now, let the required time of contact be T 1\ + r, and put a = the relative hourly motion of the two bodies in right ascension, = the planet's hourly motion the sun's, d = the relative hourly motion in declination, then at the time T the differences of right ascension and de- clination are a a' -f r and d d' -j- dr. If further we put s, s' = the apparent semidiameters of the planet and sun, respectively, the apparent distance of the centres at the time T is s' s, the lower sign being employed for inner contacts ; and if the value of M at this time is Q, we have (' ?) sin Q = (o o') cos <5 -f- a cos 3 9 . ? (s f s) cos Q = 3 d' -f dr Putting, therefore, n sin 2V = a cos . 1 (637) n cos 2V = d f we have (s' 5) sin Q = m sin M -\- n sin 2V. T (s' db s) cos Q = wi cos M -j- n cos 2V. r which, solved in the usual manner, give sin 4 S' 1C S s' s m (638) T = cos 4 cos (M 2V) v where cos ^ is to be taken with the negative sign for ingress and with the positive sign for egress. The angle Q is (as in t'clipses) the position angle of the point of contact. TRANSITS OF VENUS AND MERCURY. 595 The formula (636), (637), and (638) serve for the complete prediction for the centre of the earth. To find the time of a contact for any point of the surface of the earth, let m be the geocentric apparent distance of the centres of the two bodies at any given time ; m' the apparent distance, at the same time, as seen from a point on the earth's surface in latitude

(642) sin D = sin <5 sin Y -f- cos ^ cos y cos M ) or, adapted for logarithms, fs\nF=s\T\y cos D Bin (A a ) = cospsinJlf ^ /cos F= cosy cos M coBDcos(Aa ) = fsin(d o -}-F) > (642*) For any given time T 7 , therefore, we can find m and M by (636), then r and g by (641), and hence the values of A and D by (642). Now, let IJL be the sidereal time (at the first meridian) corresponding to T 7 , and put = ,t _ A then, in the triangle PGZ, we have the angle GPZ M, and hence, and, therefore, - = n cos * dt dm which gives in which the values of n and ^ found in the computation for the centre of the earth are to be employed. The value of A to be employed must be that which results from the preceding formulte at the time T. Now, at this time the value of the angle 598 TRANSITS OF VENUS AND MERCURY. M is Q, which is found by (638), and this value is to be employed in (642), while in (641) we take m = s' s. The formula for T' will be T'=T+ *~ r \j> sin f' sin D -f- p cos cos ' cos (101 1'.6 o) Egress, T' = 16 12 23 .8 + 48 .10 p sin <$ + 26.23 p cos 0' cos (303 20.5 w) or, in a more convenient form, giving the logarithms of the constant factors, Ingress, T' = 12* 8 m 47'.8 [1.2217] p sin 0' + [1.7176] p cos ' cos (u + 78 58'.4) Egress, T' = 16 12 28 .8 + [1.6821] p sin 0' + [1.4187] p cos ' cos (u + 56 39.5) To determine whether the phenomenon is visible at the given place, we have only to determine whether the sun is above the horizon at the computed time. All the places at which it will be visible will be readily found by the aid of an artificial terres- trial globe, by taking that point where the sun is in the zenith at the time 7 1 , and describing a great circle from this point as a pole. All places within the hemisphere containing this pole evidently have the sun above the horizon. In the present example this point at ingress is in latitude 17 43' and longi- tude 186 2' west from Washington ; and at egress it is in lati- tude 17 46' and longitude 247 4'. The whole transit is invisible in the United States, and in Europe only the egress is visible. For the egress at Altona,

from which we can find i^, II, and ^, as follows. In the triangle JV^^ we have = V t NV a # = F, F 2 180 = NV,V l n + 4=^F 1 i =NV^V. t n + 4 t = A T F, and hence, by the GAUSSIAN equations [Sph. Trig. (44)] * DB. C. A. f. PETERS, Numerus Consians Nutationis, pp. 66 et 71. The observa- tions at Dorpat give 0".4645 for the annual diminution of the obliquity, and this is adopted in the American Epherneris instead of 0".4738, which results from theory and is subject to an error in the estimated mass of Venus. The difference, however, is so small that either number will serve to represent the actually observed obliquity for half a century within 0".5. I have here adopted the precession constant (50".3798) given by PETERS, rather for the convenience of the reader (this being employed in the English and American Almanacs) than on account of its superior accuracy. Recent researches rather confirm BESSEL'S constant (50".36354). See MADLEK'S Die Eigenbeweyungen der Fizsterne, Dorpat, 1856, p. 11. PRECESSION. 607 cos $ TT sin i (4 4i) = sin * * CO8 K e + ; ) cos i JT cos i (4- 4,) cos J # cos i (e e^ sin JTT sin (n + *4 + i 4^ = sin H sin * (e + e t ) sin *rr COS (n -f 4 + Hj) = COSi#sin J (e e,) (647) The angles J $ and (e e t ) are so small that their cosines may always be put equal to unity, and, consequently, also those of ;r and J (^ a//,) ; while for their sines we may substitute the arcs. We thus obtain at once, from the first two equations, 4 ^=-#008*0 + e,) where we can take, with sufficient accuracy, cos J (e + e,) =- cos (e c 0".2369 <) cos e + 0".2369 sin 1" sin e and hence, by substituting the values of & and e from (646), 4 _ +i = 0".1387* 0".0002218< J 4, = 50".241U -j- 0". 0001134 " (648) The sum of the squares of the last two equations of (647) given * = # sin' J ( + e,) + (e e^ 1 m which we may take sin J (e + e,) = sin 2 e 0".2369< sin 1" sin 2e and then, substituting the values of #, e , and e e v we obtain r f = 0".228111< 2 0".0000033234< and, by extracting the root, * = 0".4776< 0".0000035f (649) The quotient of the third equation of (647) divided by the fourth gives tan (n + *4 + H t ) e -^7 Bin J( + O in which we have T? _ 0.15119< 0.00024186^ e e, ~ -0.4738^ 0.00000875^ = 0.3191 + 0.00051636^ 0.00020595 < cos- n. = sin 1" 608 PKECESSION. and sin * (s -f e t ) = sin e 0".2369f sin 1" cos e whence tan (n + H + Hj) = 0.127062 + 0.00020595 f If, then, we put tan n = 0.127062 or n =r 172 45' 31" and also n 1 =n-f-i4-|-i4'i we have tan n, tan n = (n t n ) sin 1" sec 2 n = 0.00020595 t whence n 1 = n4-J+4-Hi= 172 45' 31" -f 41".805# and, subtracting from this the quantity J* 4- H! = 50".310 we have, finally, n = 172 45' 31" 8".505# The equation (648) determines the general precession, and (649) and (650) the position of the mean ecliptic. 369. To find the precession in longitude and latitude of a given star, from the epoch 1800. Let LNB (Fig. 54) be the fixed ecliptic of 1800; L l NB l the mean ecliptic at the given time 1800 + *; P and P! the poles of these circles respectively. The node N is the pole of the great circle PPj-Lj joining P and Pj. Let S be the star, and put L = the star's given mean longi- tude for 1800, reckoned from the mean equinox of that year, = the star's given mean lati- tude for 1800, ^, /9 = the mean longitude and lati- tude for 1800 + 1. We have in the figure (as in Fig. 53) PRECESSION. 609 and in the triangle P&F\ we have PS = 90 B SPP, = BL = 90 -f L n SP l P = 180 L& = 180 (90 -f A n ^) = 90 - (A - n - +l ) so that, by the fundamental equations of Sph. Trig., cos cos (A - n + f ) = cos cos (I/ n) ^ cos p sin (A n 4,,) = cos B sin (L n) cos TT -f- sin 5 sin TT > (651) sin /3 = cos 5 sin (_L n) sin n -{- sin B cos TT j Instead of these rigorous formulae, we may deduce approximate ones, which will be sufficient in all practical cases, as follows. Neglecting the square of it (that is, putting cos n = 1), let the first equation be multiplied by sin (L II), the second by cos (L II) ; the difference of the products is cos /5 sin (A L 4 t ) = sin n sin B cos (L n) The sum of the products obtained by multiplying the same equations by cos (L II) and sin (L II), respectively, is cos p cos (A L 4,) = cos B -f- sin TT sin B sin (L n) and the quotient of these last equations is tan 1 -f sin TT tan.fi sin (L n) which developed in series (PL Trig., Art. 257) gives /I L ^ I =.-K tan B cos(L n) J it* tan 2 J? sin 2(L n) &c. where, however, since we here neglect the square of x, the first term of the series suffices : so that we have A L = 4 t -f TT tan B cos (L n) (652) Here ^ appears as the precession in longitude common to all the stars, and the term TT tan 1? cos (L II) as that which varies with the star. The last equation of (651) gives sin $ sin B = sin n cos B sin (L n) VOL. I. S 610 PRECESSION. whence, neglecting ,T 2 as before, B = it sin (L n) (653) The values of 4^ TT, and II being found for the time 1800 -f- t, by means of (648), (649), and (650), the formulae (652) and (653) determine the required precession in the longitude arid latitude, and, consequently, also the mean place of the star for the given tlate. 370. To find the precession in longitude and latitude between any two given dates. Suppose A and ft are given for 1800 + *, and A' and ft' are required for 1800 + t f . Denoting by L and J5 the longitude and latitude for 1800, we shall have, by (652), /I - L 4 t -f- it tan B cos (L n ) X L 4/ -f- Tt f tan B cos (L n') where ^/, n', IT are the quantities given by (648), (649), and (650) when f is substituted for L If we subtract the first of these equations from the second, and at the same time introduce the auxiliaries a and A, determined by the conditions a sin A = (it 1 -f- ?r) sin (n' n) a cos A = (n' TT) cos J (n' n) we find jrlaB^'-.-vf Q and in the same manner, from (653), For the values of A and a we have tan A * ' * tan i (n' n) = ., . tan J (n' n) It 7T t I or, by (650), '-m^- * m so that cos ^1 may be put equal to unity, and therefore we have a = n' it PRECESSION. 611 We may also put tan $ instead of tan B in the above formulae, since the error in X' ^ thus produced will be only a term in /r 2 ; and for L we may take ^ ^ : so that if we put and then substitute the numerical values of our constants, we shall have the following formulae for computing the precession from 1800 + t to 1800 -f t' : M = 172 45' 31" 4- t . 50".241 (/ -f f) 8".505 l'~l= (T f)[50".241 1 + (T+f) 0".0001134] (654) 4. (f _f) [0".4776 (t+ f) 0".0000035] cos (A M )tan p'p = (t' t) [0".4776 - (f 4 f) 0".0000035] sin (A M ) These are the same as BESSEL'S formulae in the Tabulae Eegiomon- tance, except that we have here employed the constants given by PETERS, and the epoch to which t and t' are referred is 1800. To find the annual precession in longitude for a given date. If we divide the equations (654) by t 1 t, the quotients if -t will express the mean annual precession between the two dates ; and if we then suppose t' and t to differ by an infinitesimal .quantity, or put t f = t, these quotients will become the differen- tial coefficients which express the annual precession for the in- stant 1800 + t ; namely, ^ = 50".2411 -f 0".0002268 4- [0".4776 0.0000070 1] cos (-1 Jf ) tan ^ = _ |0".4776 - 0.0000070*] sin (A - Jf) at in which M = 172 45' 31" -f 33".23 * (655) EXAMPLE. For the star Spica, we have, for the beginning of the year 1800, the mean longitude, L 201 3' 5" .97 the mean latitude, B = 2 2' 22".64 612 PRECESSION. Find its mean longitude and latitude for the beginning of the year 1860. First. By the direct formula; (652) and (653). We find, by (648), (649), and (650), for i = 60, 4, = 50' 14".874 T. = 28".6434 n = 172 37' 1" whence L n = 28 26' 5" TT tan B cos (L n) = 0".897 7i sin (L n) = -f 13".639 and hence, by (652) and (653), the precession is J L = 50' 14".874 0".897 = 50' 13".977 ft B = - 13".639 and the mean longitude and latitude for 1860.0 are >l = 201 53' 20".95 13 = 2 2'36".28 Second. By the use of the annual precession. The mean annual precession for the sixty years from 1800 to 1860 is the annual precession for 1830. Hence, by taking t = 30 in (655), and denoting by ^ and /9 the longitude and latitude for 1830, ^ == 50".2479 -j- 0".4774 cos (A M) tan ,? ^ = _0".4774sin(/l - M) M= 173 2' 8". To compute these, we can employ approximate values of >1 and /9 , found by adding the general precession for thirty years to -L, and neglecting the terms in TC ; namely, ; =201 28'.2 /?= -2 2'.6 and hence >? - M = 28 26'.1, ^ = 50".2329 ? = 0".2274 dt at These multiplied by 60 give the whole precession from 1800 to 1860, /I L = 50' 13".97 /? B = 13".64 agreeing with the values found above. PRECESSION. 613 371. Given the mean right ascension and declination of a star for any date 1800 -j- , to find the mean rigid ascension and declination for any other date 1800 +V. Let V l F/ (Fig. 55) be the fixed ecliptic of 1800, V^Q the mean equator of 1800 -f t, Vi'Q the mean equator of 1800 4- <', Q the intersection of these circles (or the ascending node of the second upon the first). The position of the point Q is found as follows. The arc V, F/ is the luni-solar precession for the in- terval t' t: so that, distinguishing by accents the quantities obtained by (646) when t' is put for t, we have, in the triangle QVM* V, V\' = 4' - 4, Q V, F/ = 180 - e 1} Q F/ F, = <, and putting (656; we find, by GAUSS'S equations of Sph. Trig., cos J sin J (2* + 2) = sin J (4' 4) cos \ (e/ + e,) cos J cos J (^ + r) cos \ (4' 4) cos \ (e/ e,) sin \ sin \ (z 1 2) = cos J (4' 4) sin J (E/ e,) sin $0 cos | (z 1 2) = sin J (4' 4) sin J (e/ -f- ej which determine 0, 2, and 2 r in a rigorous manner. But, since (e,' e,) is exceedingly small, we can always put unity for its cosine, and the arc for the sine, and, consequently, the same may be done in the case of the arc $(z f z); we thus obtain the following simple but accurate formulae : tan 1 (^ + 2) = tan J (4' 4) cos J (e/ + sin J0 r= sin J (4' 4) sin J (e/ + (657) If Fj and T^' are the positions of the mean equinox in 1800 + f and 1800 -f- t', FI V 2 is the planetary precession for the first and Vi F 2 ' that for the second of these times, which being denoted by & and #' we have F a Q = 90 2 '9 614 PRECESSION. If then we put o,, 8 = the given mean right ascension and declination of a star S, for 1800 + t, 0,',$'= those required for 1800 -j- t', \ve have a = V 2 D, and a' = V a 'J>, wi-i, consequently, QD = V a DV a Q = *+z+ 90, Qiy = V t 'jy V,'Q = a' z' -(- *' 90, Now, let P and P' (Fig. 56) be the poles of the equator at the times ID 1800 + *, 1800 + V, AQD, A'QD', the two positions of the equator at these times, as in Fig. 55 ; S the star. Q is the pole of the great circle PP'A' joining the poles P and P', and, therefore, PP' = A A' = AQA' 0, and in the triangle PP'S we have PS 90 d, P'S = 90 8', PP' = SPP'= AD= 90 -f QD = a + z + * SP'P= 180 A'jy= 90 Qjy = 180 (a' z' -f *') Hence, by the fundamental equations of Spherical Trigonometry, cos^'sin (a' z'-|-#') = cos5sin ( a sin^sin V (658) sin <5'=coscJcos(a-f ^+9)sin0 + sin5cos@ J We have thus a rigorous and direct solution of our problem by finding, first, 0, z, and z' from (656), and hence a' and d' by (658), employing the values of e, ij/, # for the time 1800 + <, and of e', 4/, #' for the time 1800 + *', as given by (646) for the two dates. 372. The formulfe (658) may be adapted for logarithmic com- putation by the introduction of an auxiliary angle in the usual manner ; or we may employ the GAUSSIAN equations, which, if we denote the angle at the star by C7, and for the sake of brevity put A = + z + # A' =*' *'+#' (659) PRECESSION. 615 give cos i (90 + is necessarily the same for both epochs. It is evident, moreover, that we have y' = % + 7-, and hence, if p and ^ have been found for one epoch, it is only necessary to compute f to obtain the reduction to another epoci; v ior we then have, by (665), cos d' da,' = p sin (^ -|~ r) P 8 ^ n x' d8' = p cos (x -f- Y~) = p cos/ 624 NUTATION. NUTATION. 381. By the luni-solar precession, combined with the diminu- tion of the obliquity of the ecliptic, the mean pole of the equator is carried around the pole of the ecliptic, but gradually approach- ing it. But the true pole of the equator has at the same time a small subordinate motion around the mean pole, which is called nutation. This motion, if it existed alone, would be nearly in an ellipse whose major axis would be 18". 5 and minor axis 13".7, the major axis being directed towards the pole of the ecliptic ; and a revolution of the true around the mean pole would be completed in a period of about nineteen years. This period is the time of a complete revolution of the moon's ascending node on the ecliptic,, upon the position of which the principal terms of the nutation depend. This periodic nutation of the pole involves a corresponding nutation of the obliquity of the ecliptic AS, and a nutation of the equinox in longitude, or, briefly, a nutation in longitude = A^, which are expressed by the following formulne* for the year 1800: Ae = 9".2231 COB & 0".0897 cos 2 & + 0".088G cos 2< + 0". 551 Ocos2O + 0". 0093 cos (Q -f r) (666) AA = 17".2405 sin &-f- 0".2073sin2 0".2041sin 2([ -f 0".OG77 sin [ r ) l".2694sin2Q+0".1279sin(O r) 0".0213sin(O+/') in which & = the longitude of the ascending node of the moon's orbit, referred to the mean equinox, C = the moon's true longitude, Q = the sun's true longitude, F = the true longitude of the sun's perigee, F' the true longitude of the moon's perigee. The quantity A^ is also called the equation of the equinoxes. The coefficient of cos in the formulae for AS is called the constant of nutation. The coefficient of sin & in the formula for A^ is equal to this constant multiplied by 2 cot 2e , in which * PETERS, Numerus Constant Nutationis, p. 46. Some exceedingly small terms, and others of short period, are here omitted, as, even if they are not altogether insensible in a single observation, their effect disappears in the mean of a number of observa- tions. NUTATION. W25 e = 23 27' 54".2. These coefficients, however, are not absolutely constant : so that, according to PETERS, the formulae for 1900 will b Ae = 9". 2240 cos & 0". 0896 cos 2^ -fO".0885cos2<[; -fO".5507cos2Q-|-0".0092cos(O + .r) (667) AA = 17".2577 sin +0". 2073 sin 2 0". 2041 sin2<[ + 0".0677 sin (C /") l".2695sin2O + 0".1275sin(Q r) 0".0213sin(0+r) Since the attractions of the sun and moon upon the earth do not disturb the position of the ecliptic, but only that of the equator and its intersection with the ecliptic, the nutation does not affect the latitudes of stars, and its effect upon their longitudes is simply to increase them all by the same quantity &.L 382. To find the nutation in right ascension and declination for a given star at a given time. Let a and d denote the mean right ascension and declination of the star at the given time ; a' and 8' the true right ascension and declination at this time, or the mean place corrected for the nutation. Let the coefficients of the formulae for AS and &X be found for the given year by interpola- tion between the values for 1800 and 1900, and then, taking &> ^ be the pogition of the object \ and eye end of the telescope at the instant t; \ . A', -B', their positions at the instant t' ; BB', the V \ motion of the earth in the interval t' t, in \\o \ which the ray SAB' from the star is describing \ VA the line AB'. Then it is evident that, while B'A \ \\\ is the true direction of the star, B'A f is the ap- \ \ \ parent -direction as given by the telescope.* B, B b B' Moreover, supposing the motion of the earth for * GAUSS : Theoria Motus Corporum Coelestium, p. 6& ABERRATION. 629 8O small an interval to be rectilinear and uniform, and the motion of light to be uniform, the lines BA and E'A' are parallel, and the ray of light during its progress from A to J3', is constantly in the axis of the telescope ; for instance, when the telescope is in the position 6a, the ray will have reached the point a, and we have Aai Bb = AB': SB' The difference of apparent direction thus caused by the motion of the earth combined with that of light is called the aberration of the fixed stars. When we also take into account the motion of the luminous body, as in the case of the planets, another species of aberration occurs, which will be considered hereafter, under the name of the planetary aberration. The whole displacement of the star produced by aberration is in the plane passed through the star and the line of the observer's motion, and the star appears to be thrown forward in this plane in the direction of that motion. Thus, in the figure the whole aberration is the angle SB' A' ; and, if we conceive the plane of the lines SB' and BB' to be produced to the celestial sphere, this plane will be that of a great circle drawn through the place of the star and the points of the sphere in which the line BB' meets it. The displacement of the star will be the arc of this circle subtending the angle SB' A' and measured from the star towards that point of the sphere towards which the observer is moving. 385. To find the aberration of a star in the direction of the observer's motion. Let # = AB'B l = the true direction of the star referred to the line B'B V = the arc of a great circle of the sphere joining the star's true place and the point from which the observer is moving, #' = the apparent direction of the star referred to the same line, = ABB V F the velocity of light, v = the velocity of the observer; then the aberration in the plane of motion is the angle A'B'A = B'AB &' #, and the triangle ABB' gives 8 Jn ' BB' v 630 ABERRATION. As &' & is very small, we may put the arc for the sine : aud if we then also put * = TSP (669 > we shall have & # = A sin #' (670) where the constant k may be regarded as known from the velo- cities of light and of the observer. 386. The motion of the observer on the surface of the earth is the resultant of the motion of the earth in its orbit and its rota- tion on its axis ; that is, of its annual and diurnal motions. These may be separately considered. The annual aberration is that part of the total aberration which results from the earth's annual motion. It may be called the aberration for the earth's centre. The diurnal aberration is that part of the total aberration which results from the earth's diurnal motion. It obviously varies with the position of the observer 011 the earth's surface, and vanishes for an observer at the poles. 387. To find the annual aberration of a star in longitude and lati- tude. Let yl, {3 = the true longitude and latitude of the star, A', /?' = the apparent longitude and latitude (affected by aberration), O = the true longitude of the sun. The point of the sphere from which the earth appears to be moving is a point in the ecliptic whose longitude is 90 -f O (the eccentricity of the earth's orbit being here neglected), and the mean velocity of the earth in its orbit may be supposed to be substituted in (669) : so that k is known. If, then, BE (Fig. 58) is an arc of the ecliptic, JS'the point from which the earth is moving, 8 the true place of the star, and if SB is drawn per- pendicular to BE, we have, in the right triangle SBJE, SB = 0, BE= 90 + O A, SE = *, ABERRATION. 631 and hence, if we denote the angle E by f, we have sin # sin y = sin ft ^ sin # cos Y = cos ft cos (O *) > (671) cos * = cos ft sin (O X) ) The apparent place of the star is on the great circle ES at the distance &' from S: so that, if we now suppose S to be the apparent place, the angle f is not changed, and we have sin #' sin ? = sin ft' ~\ sin #' cos Y= cos ft' cos (0 /) V (672) cos *' = cos ft' sin (O *') J If, then, the true place of the star is given, the equations (671) may be used to determine ? and & ; then #' will be found from (670), and, finally, X' and p' will be found from (672). This is the direct and rigorous solution of the problem ; but a more convenient solution is obtained by eliminating & and 7- as follows. We find, from the equations (671) and (672), sin #' cos # cos f = cos ft cos ft' sin (O X) cos (O X) sin # cos tf'cos f = cos ft cos ft' cos (O *) sin (O A') the difference of which is sin ( #' *) cos Y = cos ft cos ft' sin (A' A) whence , _ __ (#' #) cos r _ A; sin ' cos ^ cos ft cos /S' cos /? cos ft' (673) COS ft Again, we find, from our equations, cot Y = cot ft' cos (O A') = cot ft cos (O A) by which ft' can be found from ft after X' has been found by (673), or we may find the difference between 0' and ft thus : - CO s (Q A') cos (Q tan ft' tan ft = tan ft'\ cos (O A') of __ 9^ 2 8in * (*' A ) 8in CO * (^ + ^)] 8in ^' COS ^ cos (O - *') whence, taking 2 sin J(A' A) sin (^ r X), we obtain, by means of (673), ft' ft = k sin [O - JO' + *)] sin 0' (674) 682 ABERRATION. The equations (673) and (674) are almost rigorously exact ; but, since the value of k is only about 20", a sufficient degree of accuracy will be obtained if in the second members we put A and ft for X' and ft'. The formulae for the annual aberration in longitude and latitude thus become A'_A = -Aco8(0->l)sec/? , ft' p= A sin (O A) sin/? / in which the value of the constant, according to STRUVE,* is k = 20".4451 These last formulae may be directly deduced by differentiating the equations (671). If we retain terms of the second order in developing (673) and (674), we shall find that the following quantities will be added to the second members of (675) : - \ rt'sin I" sin 2 (O A) sec 2 /? and | A' 1 sin 1" tan /9 | A 2 sin 1" cos 2 (O A) tan ft But the term J^sin l"tan/3 being constant may be omitted, since it will be included in the expression of the star's mean place, which (Art. 361) involves the non-periodic elements of the star's position. Retaining, therefore, only the periodic terms namely, those involving O the more complete formulae will be V A = 20". 4451 cos (Q *) sec ft 0".0010133 sin 2 (Q 3L) see' ft 1 , -,, p' ft = 20".4451 sin (Q A) sin ft 0".0005067 cos 2 (Q A) tan ft / l The last terms will be sensible only for stars very near the pole. Terms of the second order not multiplied by tan ft or sec ft are wholly insensible, and have been disregarded in the deduction of the above formulae. 388. It is easy to prove, from the equations (675), that the effect of the aberration is the same as if the star actually moved in a circle parallel to the plane of the ecliptic ; the diameter of the circle being equal to the distance of the star multiplied by sin k. This circle will be seen projected upon a plane tangent to the sphere at the mean place of the star, as an ellipse whose major axis is sin A; and minor axis sin k sin ft, the radius of the * Attron. Nach., No. 484. ABERRATION. 683 sphere being unity. The period in which a star appears to describe this ellipse is a sidereal year. 389. To find the annual aberration in righi ascension and declina- tion. Let A,D = the right ascension and declination of the point E (from which the earth is moving) ; then, in the triangle formed by the point E, the star, and the pole of the equator, the sides are 90 D, 90 d, and $ ; and the angle opposite to & is A a. If then we suppose the side & to vary, the corresponding variations of the angle A a and the side 90 d may be directly deduced by the differential formulae of Art. 34. The angle at E and the side 90 D being constant, we find cos 8 . da. = d$ sin C d8 = d& cos C where C denotes the angle at the star. For determining (7, our triangle gives sin # sin C = cosD sin (J. a) sin # cos C = cos 8 sinD sin 8 cosD cos (A a) In (670) we may employ sin & for sin #' : so that, putting a' at, and $' $ for da and d&, we find a' a = k sec 8 cos D sin (A a) 1 8' 8 = k [cos 8 sinZ> sin 8 cosD cos (A a)] / The quantities A and D are found from the right triangle formed by the equator, the ecliptic, and the decimation circle drawn through E, by the formulae, cosD cos A = sin O cosD sin ^4. = cos Q cos ) (677) = cos O sin e If we substitute these values in the formulae for a' a and 3, after developing sin (A a) and cos (A a), we obtain a' a = k sec 8 (cos O cos e cos a -f- sin O sin a) V 8' 8 = k cos O (sin e cos 8 cos e sin 8 sin a) > (678) k sin O sin 8 cos a J 634 ABERRATION. If we retain the terms of the second order, (omitting, however, those which do not involve 0, or the non-periodic terms), we find that the aberration in right ascension obtains the additional terms | If sin 1"(1 4- cos 8 e) cos 2 sin 2 a sec' d -f A 2 sin 1" cos e sin 2 cos 2 a sec 2 d and the aberration in declination the terms -I- | A 2 sin I"[sm 2 e (1 + cos 2 e) cos 20 cos 2 a] tan 8 \k* sin 1" cos e sin 2 sin 2 a tan S Substituting the value of k in these terms, together with = 23 27' 30" (for 1850), and omitting insensible quantities, the corrections of the formulae (678) will be . in (<*' a), 0".000931 sin 2 ( a) sec 2 d in (s' _ a), _ 0".000466 cos 2 (0 a) tan 8 EXAMPLE 1. The mean longitude and latitude of Spica for January 10, 1860, are >l = 201 53' 22".33 ft = 2 2' 36".29 and the sun's longitude is = 289 30' Hence, we find, by (675), the aberration in longitude and latitude, ;'; = 0".85 ft' /3 = -f 0".73 The corresponding mean right ascension and declination are a = 13* 17"* 49.62 9 = 10 25' 44".9 whence, by (678), taking e = 23 27'.4, we find the aberration in right ascension and declination, ' a = 0".53 = 0*.035 3' 8 = + 0".99 EXAMPLE 2. The mean place of Polaris for 1820.0 was o = 57" 1'.505 == 14 15' 22".57 d = 88 20' 54".27 and for this date, Q=280(/ e = 2327'.8 ABERRATION. 635 with which the aberration in right ascension and decimation is found, by (678), to be a' - a = + 62".51 = + 4M67 *' 8 = + 20".27 The additional terms of (678*) are in this case 0".158 = O'.Oll and + 0".016, and the more correct values are, there- fore, a' a = -f 4M56 8' 8 = -f- 20".29 390. Gauss's Tables for computing the aberration in right ascension and declination. If we determine a and A by the conditions a sin (O + A) = k sin a cos (Q + A) = k cos O cos e the formulae (678) may be expressed as follows : o' a = a sec 3 cos(Q -f A a) d' d = a sin S sin (Q + A a) k cos Q cos d sin e = a sin d sin (Q -{-A a) J k sin e cos (O + <*) # sin e cos (O fl) The first of the tables proposed by GAUSS* gives A and log a with the argument sun's longitude, and with these quantities we readily compute the aberration in right ascension and the first part of the aberration in declination. The second and third parts of the aberration in declination are taken directly from the second table with the arguments O + 9 and O d. The tables have been recomputed by NICOLAI with the constant k = 20".4451, and are given in WARNSTORFF'S edition of SCHU- MACHER'S Hulfstafeln. The value of e for 1850 is employed in computing these tables. The rate of change of e is so slow that the tables will answer for the whole of the present century, unless more than usual precision is desired. 391. In the preceding investigation of the aberration formulae we have, for greater simplicity, assumed the earth's orbit to be a circle and its motion in the orbit uniform. Let us now inquire what correction these formulae will require when the true ellip- tical motion is employed. * Monatliche Correspondent, XVII. p. 312. 636 ABERRATION. If u is the true anomaly of the earth in the orbit, reckoned from the perihelion, at the time t from the perihelion passage, r the radius vector, a the mean distance of the earth from the sun, or the semi-major axis of the ellipse, we have r = <*(!-**) 1 -f- e cos M The true direction of the earth's motion at any time is not, as in the circular orbit, at right angles to the direction of the sun, but in that of the tangent to the curve. If we denote the angle which the tangent makes with the radius vector by 90 i, we have, by the theory of curves, cot (90 i)=!.- r du whence, by the above equation of the ellipse, e sin u tan i = - - 1 -f- e cos u and the true direction of the earth's motion will be taken into account in our formulae (675), if for O we substitute Q i. If v l denotes the true velocity of the earth in its orbit at the time <, we have .du and if / is the area described by the radius vector in the time t, F the whole area of the ellipse described in the period T, we have, by KEPLER'S first law, f F t T or df = F dt T We also have, by the theory of the ellipse, Jt~ ~2"dt and hence du __ 2 ita* |/(1 e*) ~dt ~ Tr* ABERRATION. 637 which, substituted in the above value of v v together with the value of r, gives Vl= i/(T^)T' (1 + e c 8 M)sect The mean value of this velocity is that value which it would have if the small periodic terms depending on u and i were omitted (Art. 361) ; thus, denoting the mean velocity by i>, we have a 2 v = --- , (679) v l =v(l + e cos M) sec i (680) If. then, V is the velocity of light, and we put we can at once adapt our equations (675) to the case of the elliptical orbit, by introducing k v for k and O i for O, so that we have X ). = k (1 -f- e cos M) cos (0 A i) sec t sec ft ft' /3 = k (1 -j- e cos t/) sin (Q * sec sin ft Developing the sine and cosine of (O ^) i, we have cos (O A i) sec i = cos (O X) + sin (O *) tan t' sin (Q /I ') sec i = sin (Q A) cos (O <0 tan i and substituting the value of tan z, we find A' A = k cos (O >0 sec ^ Ae cos (O w A) sec ; /? = k sin (O X) sin y9 ke sin (Q M ^) sin ^ The longitude of the sun's perigee is by the introduction of which we have, finally, A' A = k cos (Q A) sec ft ke cos (F A) sec ft ) 0' ft = & sin (O >0 sin ft ke sin (T A) sin / (681) These formulae differ from (675) only by the second terms, which therefore are the corrections for the eccentricity of the 638 ABERRATION. orbit. But we observe that these terms involve only quantities which for a fixed star are very nearly constant, so that for the same star they will have, sensibly, the same values for very long periods : the corrections themselves being exceedingly small, since e = 0.01677, and hence ke = 0".3429. They may, there- fore, be regarded either as constant corrections, or as corrections having only a slow secular change ; and in either case they will be combined with the mean place of the star, and may be altogether disregarded in the correction for the annual aberra- tion.* The formulae (675), derived from the circular orbit, will therefore be considered as complete (for the fixed stars), and, consequently, also (678), which are derived from the same hypo- thesis. 392. The sun's aberration. Since /? is less than 1", there is no sensible aberration in latitude. The aberration in longitude must be found by the complete formula (681), for in the case of the sun A is variable. Hence, writing O for ^, the aberration of the sun is found by the formula O' O = 20".4451 0".3429 cos (7 1 0) (682) in which for this century we may employ F = 280 without an error of 0".01. We could derive, from this, formulae for the sun's aberration in right ascension and declination ; but the practical method is to treat the sun as a planet, and to employ the planetary aberration which is given in a subsequent article. 393. To find the diurnal aberration in right ascension and declina- tion. Let v' = the velocity of a point of the terrestrial equator, arising from the rotation of the earth, g=_ f - = *- Fsin 1" v The diurnal aberration in the places of stars, as observed from a point on the equator, may be investigated in the same manner as the annual aberration, by substituting the equator for the ecliptic, and, consequently, right ascensions and declinations for * BESSEL, Tabulse Regiomontanx, XIX. ABERRATION. 6&* I ititudes and longitudes. The nadir of the point of observation i then to be substituted in the place of the sun :* so that if we put - the right ascension of the zenith, or the sidereal time, the formulae (675) are rendered immediately applicable to the present case by putting 180 -f 0, a, d, and k r for Q> *> & and k; whence we have, for a point on the terrestrial equator, of a = k COS (0 a) S6C 8 8' d = k sin (0 a) sin 8 Since every point on the surface of the earth moves in a plane parallel to the equator, and this plane is to be regarded as coin- cident with the plane of the celestial equator, the same formulae are applicable to every point, provided we introduce into the ex- pression of k' the actual velocity of the point. This velocity varies directly with the circumference of the parallel of latitude, or with its radius ; and this radius for the latitude y is p cos ^>',

f without sensible error; and hence the diurnal aberration may be found by the formulae a' a = 0".311 COS

Fig. 60. 399. To find the heliocentric parallax of a star in longitude and latitude at a given time, the annual parallax being given. Let T (Fig. 60) be the place of the earth in its orbit, H that of the sun. Conceive a plane to be passed through the line HT and a star S; the intersection of this plane with the plane of the ecliptic is the line HT, which, produced to the celestial sphere, meets it in a point E of the ecliptic whose longitude is the earth's heliocentric longitude, or 180 + O (putting O for the geocentric longitude of the sun at the given time). If then we also put r = the distance of the earth from the sun at the given time, # = the angle SHE, tf= " 8TE, the triangle SET gives or (#' 0) = sin *' # =pr sin (689) This formula corresponds to the formula (670) for the aberra- tion reckoned in a direction from a point (E) of the ecliptic, only in the present case this point is in longitude 180 -f O, while in the case of the aberration it was in longitude 90 + Q . The formulae for the aberration may therefore be immediately applied to the parallax if we put pr for k, and 180 + O for 90 + O, or 90 -f Q for O. We thus find, by (675), Jf /I = pr sin (A O ) sec P' P = pr cos (A O) sin (690) REDUCTION OF STARS' PLACES. 645 400. To find the heliocentric parallax of a star in right ascension and declination, the annual parallax being given. By (678), putting pr for k, and 90 + O for Q , we have, at once, a' a = pr sec d (cos Q sin a - sin Q cos e cos a) ^ . MEAN AND APPARENT PLACES OF STARS. 402. The formulae above given enable us to derive the appar- ent from the mean place, or the mean from the apparent place ; 346 REDUCTION OF STARS* PLACES. but in their present form their computation is exceedingly trouble- some. We owe to BESSEL a very simple arrangement by which their application is facilitated. In all catalogues of stars the mean places only can be given, and these only for a certain epoch. For each star there is given also the annual precession in right ascension and declination : so that the mean place for any time after or before the epoch of the catalogue is readily obtained, as in the example of Art, 374. But, since the annual precession is variable, there is generally added its secular variation, which is the variation of the precession in one hundred years. Finally, there is given the star's proper motion. If the epoch of the catalogue is t , and the mean place is re- quired for the time t. and we denote by p. the precession for the epoch f , AJ?, its secular variation, H, the proper motion, then, since in computing the whole precession for the interval t t Q we must employ the annual precession for the middle of the interval, the reduction of the mean place to the time / will be This form applies both to the right ascension and the declination.* In this way the mean place is brought up to the beginning of any given year. If then we wish the apparent place for a time r from the beginning of the year, r being expressed in fractional parts of the year, we have first to obtain the mean place for the given date by adding the precession and proper motion for the interval r, and then the apparent place, by further adding the nutation and aberration. Hence, denoting the mean right ascen- sion and declination at the beginning of the year by a and <5, the apparent right ascension and declination for the given time T by * The annual proper motions being also variable (Art. 379), it would seem that their values given for the epoch of the catalogue could not be carried forward to another time without correction. But, to avoid the necessity for this separate correction, it may be included in the secular variation of the precession, as is done by ARGELAN- DKE in his catalogue, "DLX Slellantm Fixarum Positions Medite, ineunte anno 1830." REDUCTION OF STARS PLACES. 647 a' and ', tne annual proper motions in right ascension and de- clination by fi and //, we have, by (663), (668), and (678), a,' -. a 4- T (m 4- n sin a tan 6) 4- r/u. (Precession and proper motion.) (15".8148 4- 6".865U sin a tan 6) sin ft for 18 15 .8321 6 .8682 19 + ( .1902 + .0825 sin a tan 6) sin 2ft ( .1872 4- .0813 sin a tan 6) sin 2 + ( .0621 + .0270 sin a tan 6) sin (( F') ( 1 .1644 4- .5055 sin a tan 6) sin 2Q -f ( .1173 4- .0509 sin a tan 6) sin (0 /*) ( .0195 4- .0085 sin a tan *) sin (Q + f) (Nutation.) 9".2231 cos a tan 6 cos ft 9 .2240 4- .0897 cos a tan 6 cos 2 ft .0886 cos a tan 6 cos 2( .5510 cos o tan 6 cos 20 - .0093 cos a tan 6 cos (Q -f -T) 20". 4451 cos e cos Q cos a sec 6 20 .4451 sin sin a sec 6 1800 1900 (Aberration, j 6' = 6 -\- T . n cos a -(- r/z' (Precession and proper motion.) 6".8650 cos a sin ft -f 9".2231 sin a cos ft for 1800 6 .8682 9 .2240 1 4- .0825 cos a sin 2 ft .0897 sin a cos 2 ft .0813 cos a sin 2([ 4-0 .0886 sin a cos 2([ -f .0270 cos o sin (C 7") .5055 cos o sin 2 4- .5510 sin a cos 2 4- .0509 cos a sin (0 r) .0085 cos a sin (Q 4- r) 4- .0093 sin a cos (0 + - 20". 4451 cos e cos (tan e cos 6 sin a sin A) 20 .4451 sin cos a sin 6 (Nutation.) } (Aberration.) it is to be remarked that the two numerical coefficients of sin ft, sin 2ft, sin 23), &c. in the formula for a' are in each case very nearly in the ratio of m to n;* and hence, if, according to the method of BESSEL, we put 6".8650 = ni 15".8148 = mi -f h S .8682 15 .8321 .0825 = ni' .1902 = mi' + K .0813 = ni" .1872 = mi" + h" .0270 = ni'" .0621 = mi'" 4- A'", .5055 == ni 1 * 1 .1644 = mi 1 ' 4- A |T .0509 = ni" .1173 == mi* 4- A T .0085 = ni" .0195 = mi" + h" 1 * This relation is not accidental, but results from the general theory of nutation, which, the student will remember, is only the periodical part of the precession. 648 REDUCTION OF STARS' PLACES. we shall have *' = a -f [r sin & + f sin 2 fain 2<[ + f'sin ( r') i lv sin 2Q -f i T sin(O r) *sin (Q +/^)] |> + n sin a tan d] [9".2231 cos & 0".0897 cos 2 -f- 0".0886 cos 2 9 .2240 + 0".5510 cos 2 + 0".0093 cos (Q + T 7 )] cog o t*n 6 20". 4451 cos e cos Q cos o sec d 20 .4451 sin Q sin o sec 6 + Tp. h sin -f A' sin 2& A" sin 2+ A'" sin /") and sin (([ r') > sin 2 Q + > sin (Q F) t>i sin (Q + r )] n cos a + [9".2231 cos ^ 0".0897 cos 2 ^ + 0".0886 cos 2(C 9 .2240 + 0".5510 cos 2Q + 0".0093 cos (Q -f r)] sin a 20".4451 cos e cos Q (ta n f cos d sin a sin 6) 20 .4451 sin Q cos a sin d Putting then, in accordance with BESSEL'S original notation, as employed in the American Ephemeris for 1865 and subse- quent years, A r i sin & + t'sin 2& ."sin 2([ + ""sin C /*) -t^sin2O + v sin(Q r) V sin ( Q + JT) 5 = 9". 2231 cos ^ + 0".0897 cos 2& 0".0886 cos 2 9 .2240 0". 5510 cos 2Q 0".0093 cos (Q -f r) C 20". 4451 cos e cos Q D = 20 .4451 sin Q E= h sin Q + A' sin 2& A" sin 2< -f h'" sin (([ /*) A iT sin 2 + * T sin (Q r) h* sin (Q + r) which quantities are dependent on the time, and are wholly inde- pendent of the star's place ; and also a = m -\- n sin a tan d a' = n cos o b = cos a tan d b' = sin a c = cos a sec 8 d = tan e cos d sin o sin d d = sin a sec d d' = cos a sin d REDUCTION OF STARS PLACES. 649 which depend on the star's place, we have a' = a -f- Aa -f Bb + Cc 4- Dd -f E Cc' + Dd' -f V } (692) The logarithms of A, B, (7, J9 are given in the Ephemeris for every day of the year. The residual E never exceeds 0".0o, and may usually be omitted. The logarithms of a, 6, c, d, a', b f , c', d' are usually given in the catalogues, but where not given are readily computed by the above formulae. When the right ascen- sion is expressed in time, the values of a, 6, c, d, above given, are to be divided by 15. 403. If we substitute the values of m and n, namely, for 1800, m = 46".0623 1900, m == 46 .0908 we find the following values of z, i v , &c. : n = 20".0607 n = 20 .0521 t f t" t'" jlT t> t>i 1800 1900 0.34221 0.34252 0.00411 0.00405 0.00135 0.02520 0.02521 0.00254 0.00042 h A and h', h", h'", h v , h vl inappreciable. 1800 1900 + 0".052 + .045 + 0".004 + 0".003 The terms in i v and i v[ in the expression of A may be combined in a single term ; for, putting have jcosJ= (i v .; sin J = (i v + cos sin I T sin (O n i vi sin (O = j sin (0 + J) and taking for 1800, F= 279 30' 8"; and for 1900, P= 281 12' 42", we find 1800 1900 4- 0.00294 4- 0.00293 83 10' 81 55 650 REDUCTION OF STARS' PLACES. Hence the values of A, B, and E may be expressed as follows : A T 0.34221 sin 0.02520 sin 2Q + 0.00294 sin (Q + 83 10') for 1800 0.34252 0.00293 81 55 "1900 + 0.00411 sin 2^ 0.00405 sin 2 + 0.00135 sin ( /") B - 9".2231 cos 0".5510 cos 2Q 0".0093 cos (Q + 279 30'.)" 1800 9 .2240 281 13 " 1900 + .0897 cos 2 Q .0886 cos 2< tf 0".052 sin 0".004 sin2Q " 1800 .045 .003 " 1900 These values agree (within quantities practically inappreciable) with those given by Dr. PETERS (Numerus Constans Nutationis, pp. 75, 76). It is necessary to remark that in the British Associa- tion Catalogue and the British Nautical Almanac, the preceding values of C and D are denoted by A and B, and vice versa.* See p. 94. 404. When the catalogue does not give the logs of a, b, c, &c., another form of reduction, also proposed by BESSEL, may some- times be preferred. By putting / = mA + E i = C tan c g cos G = n A h cos If = D g sin G = B h sin H = C we find ) tan 5 -f f r=5-f icosd-f T/i'-f- cos(-f a) +/t cos (H -fa) sin 3 ) ( ^ The values of /, log g, G, log A, 77", log i, and log r are given in the Ephemeris for every day of the ye^r. 405. A star's apparent place may be reduced to its mean place and referred to the mean equinox of any given date by reversing the signs of the reductions as above determined. By the apparent place of a star we here mean the apparent geocentric place. The observed place (that seen from the surface of the earth) differs *Thia interchange of letter?, most unnecessarily introduced by BAILY in the British Association Catalogue, produces considerable inconvenience, as in most of the Knn>- pean catalogues of stars BESSEL'S notation is preserved, while in the English Almanac BAJLV'S notation is followed. In the American Ephemeris for 1865 mid subsequent years the notation of BESSEL has been restored: an example which will doubtless N ollowed by the British Almanac at an early day. FICTITIOUS YEAR. 651 from this by the diurnal aberration and the refraction ; but the first of these corrections depends on the latitude of the observer and the star's hour angle, and the second upon the star's zenith distance : so that neither of them can be brought into the com- putation of a star's position until the place of observation and the'local time are given. 406. The fictitious year. In the preceding investigations, we have used the expression " beginning of the year," without giving it a definite signification. For the purpose of introducing uniformity and accuracy in the reduction of stars' places, BESSEL proposed a fictitious year, to begin at the instant when the sun's mean longitude is 280. This instant does not correspond to the beginning of the tropical year on the meridian of Greenwich ; that is, the (mean) sun is not at this instant on the meridian of Greenwich, but on a meridian whose distance from that of Greenwich can always be determined by allowing for the sun's mean motion. This meridian at which the fictitious year begins will vary in different years; but, since the sun's mean right ascension is equal to his mean longitude (Art. 41), the sidereal time at this meridian when the fictitious year begins is always 18 A 40 m (= 280). By the employment of this epoch, therefore, the reckoning of sidereal time from the beginning of the year is simplified, and, accordingly, it is now generally adopted as the epoch of the catalogues of stars. In the value of log A, which involves the fraction of a year (r), the same origin of time must be used ; and this is attended to in the computation of the Ephe- merides, which now give not only the logarithms of A, B, C, and .D, but also the value of r (or its logarithm) reckoned from the beginning of the fictitious year and reduced to decimal parts of the mean tropical year. For all the purposes of reduction of modern observations, the computer need not enter further into this subject, and may depend upon the Ephemerides.* But, as the subject is inti- * The reduction of observations made between 1750 and 1850 will be most con- veniently performed by the aid of the Tabula Regiomontanx of BESSEI,. The con- stants used by BESSEL differ materially from those now adopted in the American and British Almanacs. Professor HUBBARD has given a very simple table by which the values of log A, log B, log C, and log D as given in the Tab. Reg. may be reduced to those which follow from the use of PETEKS'S constants, in the Astronomical Journal, Vol. IV. p. 142. Ttie special and general tables for the reduction of stars' places. 652 LENGTH OF THE YEAR. mately connected with that of time in general, I shall prosecute it a little further. 407. The sun's mean motion, and the length of the year. Accord- ing to BESSEL,* the sun's mean longitude at mean noon at Paris in 1800, January 0, is 279 54' 1".36 and the sun's sidereal motion in 365.25 mean days is 360 22".617656 (By January is denoted the noon of December 31 in the com- mon mode of expressing the date ; and, consequently, Jan. 1, 2, &c. denote 1 day, 2 days, &c. from the epoch, while in the com- mon mode they mean the beginning of the 1st day, 2d day, &c.) The sidereal motion is referred to a fixed point of the ecliptic; but the mean longitude is referred to the moving vernal equinox. Hence the change of the mean longitude in any time is the sidereal motion in that time plus the general precession ; and therefore, adopting here BESSEL'S precession constant,! in order to follow his computations, Sid. motion in 365".25 = 360 22".617656 General precession -f 50 .22350 -f- 0".OQ0244361 1 Mean motion in 365 d .25 = 360 -f 27 .605844 -j- .00024436H and, dividing by 365.25, Mean daily motion == 59' 8".3302-f 0".0000006902 vvhere t is the number of years after 1800. The secular change of the mean motion, expressed by the second term, brings with it a secular change of the length of the tropical year. This year given in the Washington Astronomical Observations, Vol. III., Appendix C, are also adapted to the new constants. For the reduction of observations from 1850 to 1880, the Tab. Reg. have been continued by WOLFERS and ZECH (Tabulse Reductionum Observationum Astronomic a rum Annis 1860 usque ad 1880 respondents, auctore J. PH. WOLFERS : Additse sunt, Tabula Regiomontanse annis 1850 usque ad 1860 respondenles ab ILL. ZECH continuatse. Berlin, 1850). In the continuation by ZECH, which extends from 1850 to 1860, all the constants are the same as those used by BESSEL ; in the continuation by WOLFERS, from 1860 to 1880, BESSEL'S precession constant is retained, but PETERS'S nutation constant is adopted. *Astnn. Nach., No. 133. t Ibid - LENGTH OF THE YEAR. 653 is the time in which the sun changes his mean longitude exactly 360, and is, therefore, found by dividing 360 by the mean daily motion : thus, if we put Y = the length of the tropical year in mean solar days, we find Y= 365*242220027 (K00000006886* where the value of the second term fort = 100 is 0'.595, which is the diminution of the length of the tropical year in a century. The length of the sidereal year is invariable, and is readily found by adding to 365.25 the time required by the sun to move through 22". 617656 at the rate of his sidereal motion; or, putting Y' = the length of the sidereal year, by the proportion 360 22".617656 : 360 = 365<*.25 : Y' which gives Y' = 365.256374416 mean solar days, = 366.256374416 sidereal days. 408. The epoch of the sun's mean longitude. This term denotes the instant at which the common year begins. The value of the longitude itself at this instant is frequently called "the epoch," and is denoted by E. Its value for January of any year, 1800 + I, is found by adding the motion in 365 days for each year not a leap year, and the motion in 366 days for each leap year. The motion in 365 days is found from the above value for 365.25 days by deducting one-fourth the mean daily motion, or 14' 47". 083: so that, if / denotes the remainder after the division of t by 4, we have, for the epoch of 1800 + t, Jan. 0, at Paris, E= 279 54' 1".36 + 27".605844 + 0".0001221805< - (14' 47".083)/ (693) To extend this formula to years preceding 1800, we must put / 4 in the place of/: so that the multiplier of (~ 14' 47".083) will be, for example, 1, 2, 3, 4, 1, &c. for the years 1799, '98, '97, '96, '95, &c. But these rules for / will give the 654 THE YEAR. mean longitude at the beginning of the leap years too great by the motion in one day (since the additional day is not added until the end of February) ; and therefore the epoch for these years is January 1 instead ot January 0. A general table containing the mean longitude at mean noon for every day of the year, computed from the mean longitude for Jan. by the formula, will be applicable to leap years if in the months of January and February we increase the argument of the table by one day, as in Table VI. of the Tab. Meg. 409. To find the beginning of the fictitious year. Denoting the mean time from the beginning of the fictitious year to Jan. of any year by k, we have k _ E 280 mean daily motion whence, taking the daily motion = 59' 8". 3302, we find, in deci- mals parts of a mean day, k= 0.10107289 -f- 0.0077799535 1 -\f + 0.000000034433 1> The quantity k is evidently equal to the east longitude from Paris of that terrestrial meridian on which the fictitious year begins (Art. 406). 410. In the Tabula Regiomontance the argument is the reduced date as it would be reckoned at the meridian in the east longitude A, the beginning of the fictitious year being always denoted by January 0. If then d is the west longitude from Paris of any place, the instant of mean noon at this place corresponds to the instant k -f d of the fictitious meridian, and therefore k + d ia the reduction to apply to the mean time at the place to obtain the argument with which to enter those tables. But, if the sidereal time at the place d is given, it is most ex- pedient to reckon the time at once in sidereal days from the beginning of the fictitious year. Accordingly, in the tables con- taining the values of log A, log S, &c. for the reduction of stars, the argument is the sidereal date at the fictitious meridian. To obtain this date, it is to be observed, first, that the tables are im- mediately available on the fictitious meridian for the sidereal time 18 A 40 n , without any reduction of the date. For any other SIDEREAL TIME. 655 meridian, at the sidereal time 18 A 40 m the argument of the table will be the reduced date ; but at any other sidereal time g the argument must be this reduced date increased by a 18*40 24* which must be always taken < 1 and positive ; or by the quantity ,^g 24* omitting one whole day if g + 5 A 20 m > 24\ Now, in order that the local date may correspond with that supposed in the tables, the day must be supposed to begin at the instant when that point is on the meridian whose right ascension is 18 ; ' 40. Therefore, whenever the right ascension of the sun is as great as 18* 40 m , so that the point in question culminates before the sun, one day must be added to the common reckoning. Hence the formula for preparing the argument of the tables will be Argument = Eeduced date -f- g' -j- i; in which we must take i = from the beginning of the year to the time when the sun's E. A. = g, and i = -f- 1 after this time. The values of g' are given on p. 16 of the Tab. Reg. for given values of g. The values of k are given in Table I. The values of log A, log B, log (7, log D are also given in the Berlin Jahrbuch for the fictitious date ; and the constants of pre- cession, nutation, and aberration are the same as those employed by BESSEL in the Tab. Eeg. 411. Conversion of mean into sidereal time, and vice versa. In the explanation of this subject in Chapter II. we said nothing of the effect of nutation, which we will now consider. Let us go back to the definitions and state them more precisely. 1st. The first mean sun, which may be denoted by O u moves uniformly in the ecliptic, returning to the perigee with the true Bun. The longitude of this fictitious sun referred to the mean equinox is called the sun's mean longitude. 2d. The second mean sun, which may be denoted by O 2 , moves 656 SIDEREAL TIME. uniformly in the equator, returning to the mean equinox with the first mean sun. 3d. The sidereal day begins with the transit of the true equi- nox ; and the sidereal time is the hour angle of the true equinox. Hence it follows that the mean E. A. of O a the mean l n g- f Oi = the sun's mean longitude; and since when O 2 is on the meridian, its E. A. reckoned from the true equinox is also the hour angle of the true equinox, it also follows that F = the sidereal time at mean noon. = true K.A. of O a = mean K. A. of Q a + nutation of the equinox in E. A. = sun's mean longitude -}- nutation of the equinox in E. A. The nutation of the equinox in E, A. is found from the first equation on p. 626 by putting a = 0, d = 0, whence nutation of equinox in E. A. = A/ cos e which is the quantity given in the Nautical Almanac as the "equation of the equinoxes in right ascension." Since the nutation is contained in the value of V given in the Almanac for each mean noon, no further attention to it is needed when that work is consulted ; and the rules given in Chapter II. are therefore practically complete. For the conversion of time between 1750 and 1850, the Tab., Reg. furnish the following facilities : Table VI. gives the right ascension of the second mean sun, corrected for the solar nuta- tion of the equinox, for every mean noon at the fictitious meri- dian A:. Since the fictitious year always begins with the same mean longitude of the sun (or right ascension of Q 2 ), the num- bers of this table are general, and may be used for every year. The number taken from this table for any given date (which must be the reduced date above explained) are then corrected for the lunar nutation of the equinox in right ascension, which is given in Table IV. for all dates between 1750 and 1850. We thus obtain the sidereal time at mean noon ( T^) at the fictitious meridian on the given day. Any given mean time at another meridian is then converted into the corresponding sidereal time. REDUCTION OF A PLANET'S PLACE. t>57 or vice verm, according to the rules in Chapter II., employing the V Q for the fictitious meridian precisely as it was there employed for the meridian of Greenwich. The longitude of the place to be used here is k + d, d being the west longitude of the place from Paris, and k the east longitude of the fictitious meridian from Paris given in Table I. REDUCTION OF THE APPARENT PLACE OF A PLANET OR COMET. 412. The observed place of a planet (or comet) being freed from the effect of refraction, diurnal aberration, and geocentric parallax, we have the apparent geocentric place, referred to the true equator and equinox of the time of observation, and aifected by the planetary aberration. For the calculation of a planet's orbit from three or more observations at different times, it is necessary to refer its places at these times to the same common fixed planes, which is most readily effected by reducing all the places to the equinox of the beginning of the year in which the observations are made, or, when the observations extend beyond one year, to the beginning of any assumed year. To effect this, we must apply to each apparent geocentric place 1st. The aber- ration (687), with its sign reversed, in computing which the posi- tion of the observer on the surface of the earth may be con- sidered by taking r' equal to the actual distance of the planet from the observer at the time of observation. This distance is found from the geocentric distance at the same time with the parallax, by the equation (137). 2d. The nutation for the date of the observation, with its sign reversed. 3d. The precession from the date of the observation to the assumed epoch, which will be subtracted or added according as the epoch precedes or follows the date. But the nutation and precession are most conveniently com- puted together by the aid of the constants A and B used for the fixed stars. These constants being taken for the date, a, 6, a', and V are to be computed as in Art. 402, with the right ascen- sion and declination of the planet; and then to the a and , we then have jy log tan D' log cosec A log tan e June 16 23 21' 65".84 9.6355081 0.0018927 9.6374008 23 27' 23".61 " 19 26 28 .38 .6370823 03278 4101 25 .22 " 20 27 7 .11 .6373056 00930 3986 23 .23 " 23 26 40 .56 .6371524 02478 4002 23 .61 " 27 20 18 ,56 .6349452 24592 4044 24 .25 Apparent obliquity = 23 27 23 .96 664 OBLIQUITY OF THE ECLIPTIC. For the sake of comparison, I add the results of the computa- tion by the series (697), which, however, will be far less con- venient than the above direct computation, unless a table of the reduction is used. M D Red. to solstice. Red. for lat. June 16 + 27 m 22'.87 23 21' 66". 02 + 5' 27".77 0".18 23 27' 23".61 '< 19 + 8 54.23 26 28 .19 56 .84 + 0.19 25 .22 " 20 + 4 44.56 27 6 .79 16 .13 + .32 23 .24 " 23 7 44.44 26 39 .92 42 .96 + .63 23 .51 " 27 -24 22.00 20 17 .84 7 5 .69 + 0.72 24 .25 Apparent obliquity = 23 27 23 .96 Nutation* = + 8 !24 Reduction to Jan. 0. 1846 = 0".4645 X 0-469 = + .22 Mean obliquity 1846.0 = 23 27 32 .42 In the same manner, for the southern solstice we have : D Red. to solstice. Red. for Olat. | Dec. 14 + 33" 7'.27 23 14' 17".26 13' 6".48 0".35 23 27' 24".09 " 15 + 28 41.57 17 33 .82 9 50 .24 0. 46 24 .62 " 16 + 24 15.62 20 22 .94 7 1 .98 -0.57 25 .49 " 18 + 15 23.09 24 32 .69 2 49 .70 0. 72 23 .11 21 + 2 3.31 27 20 .43 3 .03 .70 24.16 " 22 2 23.39 27 19 .64 4 .09 -0 .64 24 .37 " 23 - 6 50.09 26 49 .82 33 .49 0.50 23 .81 | " 29 33 28.11 Reduct 14 1 .20 13 23 .05 + .19 24 .06 ] Apparent obliquity = 23 27 24 .20 Nutation = + 8 .98 ion to Jan. 0. 1846 = 0".4645 X 0.971 = + .45 Mean obliquity 1846.0 = 23 27 33 .63 The results from the two solstices being combined in order to * The nutation for 1846 is found by the formula (Art. 381) As = 9". 2235 cos & 0".0897 cos 2& + 0".0886 cos 2([ + .5509 cos 2Q + .0093 cos (Q + r) For the northern solstice June 21, 9*, I have taken = 214 27', <[ = 69, Q = 90, F= 280; for the southern solstice, Dec. 21. 16*, Q = 204 45', ([ = 319, Q = 270, r= 280. To proceed with theoretical rigor, the nutation should be found for the time of each observation. EQUINOCTIAL POINTS. 665 eliminate the error of the assumed latitude of Washington,* we have, finally, Mean obliquity for 1846. from observation = 23 27' 33".03 The same by PETERS'S formula (646) with ") the annual decrease 0".4645 j = 420. The secular variation of the obliquity is found by com- paring its values at very distant epochs. The observations of BRADLEY from 1753 to 1760 gave for 1757.295 the mean obliquity 23 28' 14".055. The observations at the Dorpat Observatory gave for 1825.0 the mean obliquity 23 27' 42".607. Heiice Annual var. = 31 "' 448 = 0".4645 67.705 BESSEL found 0".457 by comparing BRADLEY'S observations with his own. The secular variation is also found in Physical Astronomy, theoretically. The value thus obtained by PETERS in his Nurne- rus Constans Nutationis is 0".4738, as given in the formulae (646). 421. Determination of the equinoctial points, and the absolute right ascension and declination of the fixed stars. The declinations of the fixed stars are either directly measured by the fixed instruments of the observatory, or deduced immediately from their observed meridian zenith distances (corrected for refraction) by the formula 8 = y . The practical details, which depend on the instru- ment employed, will be given in Vol. II. Here we have only to observe that the immediate result of such a measurement is the apparent declination at the time of observation, which must then be reduced to the mean declination for some assumed epoch by the formulae of the preceding chapter. The position of the equinoctial points is determined as soon as we have found the right ascension of one fixed star ; and this is done by deducing from observation the difference between the * The latitude employed in deducing the declinations was 38 53' 39". 25. The latitude given by the culminations of Polaris is 38 53' 39". 52 ( Washington Aitr. Obs., Vol. I., App. p. 113). If we adopt the latter value, the obliquity derived from the northern solstice will be increased by 0".27, and that derived from the southern solstice will be diminished by the same quantity ; and the difference then remaining between the two results will be only 0".67. 6()6 EQUINOCTIAL POINTS. sun's right ascension and that of the star at the time the sun \s at the equinoctial points. For this purpose a bright star is selected, which can be observed in the daytime and at either equinox, and which is not far from the equator. On a day near the equinox the times of transit of the sun and the star are noted by the sidereal clock ; and at the time of the sun's transit his declination is also measured. Let T = the clock time of the sun's transit, t = " " star's " A, J), ft the sun's apparent right ascension, declination, and latitude at the time T, a := the star's apparent right ascension at the time t, e = the apparent obliquity of the ecliptic at the time T; /hen, correcting the sun's declination by the formula (695), or, D'= D ,9 sec cos D we have, by (694), sin A = tan D' cot e (690) Thus A becomes known, and hence, also, a by the formula a = A + (t T) (700) in which t T is the true sidereal interval between the observa- tions corrected for the clock rate. The observation is to be repeated on a number of days pre- ceding and following each equinox. The star's apparent right ascension is in each case to be freed from the. effects of aberra- tion, nutation, and precession (also proper motion and annual parallax, if known). Each observation thus furnishes a value of the star's mean right ascension at tue epoch to which the re duction is made. In order to learn what combination of these values will best eliminate constant errors in the elements upon which A depends, let us examine the effects of these errors. We speak only of constant errors ; the accidental errors of obser- vation being reduced to their minimum effect by taking the mean of a large number of observations. The correction which the assumed value of the obliquity requires being denoted by ds, the corresponding correction of A is found, by differentiating (699), to be EQUINOCTIAL POINTS. 667 2 tan A dA = sin 2e The correction of the declination D' is composed of the cor- rections in the latitude ^>, and the zenith distance ; since, by the formula D =

, rfoc , and d/9 . THE CONSTANT OF SOLAR PARALLAX. 425. The constant of solar parallax is the sun's mean equatorial horizontal parallax, or its horizontal parallax when its distance from the earth is equal to the semi-major axis of the earth's orbit. The constant of parallax of any planet is also its parallax when its distance from the earth is equal to the semi-major axis of the earth's orbit: so that the constant of solar parallax belongs to the whole solar system. The relative dimensions of the orbits of the planets are known from the periodic times of their revolutions about the sun, since, by KEPLER'S third law, the squares of their periodic times are proportional to the cubes of their mean distances from the sun, VOL. I. 43 074 CONSTANT OF SOLAR PARALLAX. that is, to the cubes of the semi-major axes of their orbits. The ratios of these distances are therefore known. Again, the form and position of each orbit are known from Physical Astronomy;* and therefore the ratio of the planet's distance from the earth at any given time to the earth's mean distance is also known. According to these principles, if the distance of any planet from the earth can be found at any time, the dimensions of all the orbits are also found : in other words, when we have found the parallax of one planet we have also found that of all the planets, as well as that of the sun. 426. To find a planet's, or the sun's, parallax by meridian observa- tions. Let the meridian zenith distance of the planet's centre be observed on the same day at two places nearly on the same meridian, but in very different latitudes. After correcting the observed quantities for refraction, let C', C/ = the apparent zenith distances at the north and south places of observation, respectively, C, Cj = the true (geocentric) zenith distances, p f p l the parallax for the zenith distances C and C t , TT, K I = the equatorial horizontal parallax at the respective times of observation, J, Jj = the geocentric distances of the planet at these times. d, , f) 1 = the radii of the terrestrial spheroid for these latitudes. We have ERR sin TT = sin n. = sm TT O = 4 \ 4, * They are found from three complete observations of the right ascension and declination of each planet at three different times (GAUSS, Theoria Motus Corporum Ceelestium), and therefore from the observed directions of the planet, the absolute distance being unknown. CONSTANT OF SOLAR PARALLAX. 675 and therefore sin TT = - sin it. sin >:. = sin JT. J 4 The quantities J and J t are to be found from the planetary tables, or directly from the Nautical Almanac, where they are expressed in terms of J as the unit : so that their values there j J given are the values of the ratios and '. Hence we shall put J = 1 in the preceding formulae, and also put the arcs for their sines (since the greatest planetary parallax is only 35") : so that we have Then, by (114), p = p x sin [' (^> y>')] i - sin [C' (y Pi == P\^\ ^i^ Ei' (^i ?*/)] :== ^~^~ ^ C/ (^i i But we also have C = / (D -f- A^)] (706) and then, as before, A great number of such corresponding observations will be necessary in order to determine TT O with accuracy ; and all the equations of the form just given are to be combined by the method of least squares. Thus, from the equations a'7T = n', a"- = n", &c. we obtain the final equation [aa] TT O = [an] or ^=11 in which [a] = aa + a'a' -\- a"a" -\- &c., and [an] = an -f a'n' [- a"n" + &c. 427. To find the solar parallax by extra-meridian observations of a planet. The preceding process will require but a slight modifi- cation. The difference of apparent declination of the planet and a neighboring star is measured at both stations with a micrometer attached to an equatorial telescope, and is to be corrected for refraction. The quantity n will then be found by (705). -The coefficient a will now be the difference of the coefficients of parallax in declination, computed by the formulae (143), accord- ing to which, if we put tan w' tan tp' tan f = tan ?. = - COS (8 a) COS (0 X BI ) 678 CONSTANT OF SOLAR PARALLAX. we shall have a _ p sin j = the geocentric declinations of the moon's centre at the respective times of observation, <5) \jp sin (C y} qp K] sin P where k is the constant ratio of the radii of the moon and the earth, for which the value 0.272956 may be assumed ; and the upper or lower sign of k is to be used according as the upper or lower limb is observed. At the southern station we have C/ = d i ~ Vi Ci = A fi and hence, taking the reduction f l as a positive quantity, sin (D, fl,) = [>j sin (C/ y,) A] sin P, where the sign of k is reversed, since the same limb will be an upper limb at one station and a lower limb at the other. For brevity, put m = p sin (C' f ) + k m 1 = / o 1 sinCC, ft) * then, from the equations sin (Z> /J) = m sin P sin (D t 5,) = m, sin P, r-82 CONSTANT OF LUNAR PARALLAX. we derive,* neglecting powers of sin P above the third, _ msinP , , wi 3 sin 3 P JJ o = 7) _ g _ -i i \ r ' '~ sin 1" ~~ e ' sin 1 If now the times of the two observations reckoned at the same first meridian are T and T v and for the middle time t = $(T+ TJ we deduce from the lunar tables the hourly in- crease of the moon's declination, or , we shall have, with at regard to second differences, Again, if we denote the moon's horizontal parallax at the time t by p, and compute from the tables its hourly increase fur thia time, or -j-> we shall have sin P = sin p + cos p sin 1" ( T t) ~ sin Pj = sin p -f cos p sin 1" (T l t}-^- Taking the difference of the above values of D d and D l d lf we obtain, therefore, = [( T, - T) d / - (*, - 5)] sin 1" + (m + m/) ^ + cos p sin 1" I? [m (T - f) + m, (T, - 0] + (m + m,) sin ^ (708) The parallax is sufficiently well known for the accurate compu- tation of the terms in sin 3 p and 2E : so that the only unknown quantity in this equation is the last term. In this term we have m + m t = p sin (:' r } + Pl sin (:/ n ) (709) * By the formula, [PI. Trig. (413)], x = sin x -f- ^ ^in s x -(- &c. Where the second member is to be reduced to seconds by dividing it by sin 1". CONSTANT OF LUNAR PARALLAX. 683 which is independent of A', and thus free from any error in that quantity. Small errors in k will not appreciably affect the other terms of the equation. Thus every pair of corresponding observations gives an equa- tion of the form = n -f a sin p (710) from which the parallax p at the mean time of each pair of observations could be derived. But, in order to combine all these equations, we must introduce in the place of the variable p the constant mean parallax, which is effected as follows. Let TT = the horizontal parallax taken from the lunar tables for the time t, TT O = the constant mean parallax of the tables, ^> the true value of this constant. The form of the moon's orbit is well known : so that for any given time the ratio of the radius vector to the semi-major axis, as employed in the tables, is to be regarded as correct ; that is, the ratio " = ^ cm, derived from the tables, is to be regarded as the ratio between the true parallax at the given time and the trite constant: so that we have also sin p or sin p = u. sin p. and the equation (710) becomes - = (712) The quantities , which are then to be solved by the method of least squares. 433. The quantities p and 7-, which enter into the coefficient m, will be computed for an assumed value of the compression of the earth. But, in order to see the effect of the compression, we 681 CONSTANT OF LUNAR PARALLAX. may isolate the terms which involve it, as follows. Neglecting tR fourth powers of the eccentricity e, we have, hy (84) and (83), P = 1 i e 2 sin 2

u sin ^ [sin 2 ^ sin f '-}- sin 2 sin C' + sin 2 $P cos C' -f sin 7

It is here to be observed that we have taken f\ as a positive quantity even for the southern station : so that sin 2 ^> l must be taken positively in computing b. Let us now suppose we have obtained from a large number of such corresponding observations the equations = n + x (a cb} = n' + x (a' cb') = w"-f #(a" c6") &c. Multiplying these respectively by a, a', a", &c., and then forming their sum, we have = [an] -f [aa] x [ab] ex where [aw] = an -f a'w' -f &c., [aa] = aa + o!a' + &c., &c. The last term is very small : so that an approximate value of x may be found by neglecting it, whence [an] 00=- [aa] which value may then be employed with sufficient accuracy in the term [6] ex ; we thus find the complete value [on] [on] [06] [aa] T [aa] [aa] 686 CONSTANT OF LUNAR PARALLAX. This is essentially the method by which OLUFSEN* has dis- cussed the observations made by LACAILLE in the years 1751, 1752, and 1753, at the Cape of Good Hope, and the correspond- ing observations made at Paris, Bologna, Berlin, and Greenwich. He found from all the observations the final equation x = 0.01651233 -f 0.02449201 c Consequently, if we take the most probable value of c = , 299.1528 there results x = si np = 0.01659420 The parallax given by the lunar tables of BURCKHARDT and DAMOISEAU is properly the sine of the parallax reduced to seconds. In order to compare this determination with the constants of these tables, we therefore take The constant of BURCKHARDT'S tables is 3420". 5; that of DAMOISEAU'S, 3420". 9 ; that of HANSEN'S new tables, 3422".06. This last value, which is derived from theory, agrees remarkably with that which is derived from direct observation ; for the determination by HENDERSON from corresponding observations at Greenwich and the Cape of Good Hope isf 3421".8, and the mean between this and OLUFSEN'S value is 3422". 3. 434. The correction of the moon's parallax may also be found from the observations of a solar eclipse at two places whose dif- ference of longitude is great, as is shown in the chapter on eclipses, p. 541. It is also possible to determine the moon's parallax by com- paring the different zenith distances of the moon observed at one and the same place between her rising and setting, since the eflect of so great a parallax is easily traced from its maximum when the moon is in the horizon to its minimum when at the least zenith distance. But this very obvious method, by which, in fact, HIPPARCHUS discovered the moon's parallax, depends too much upon the measurement of the absolute zenith distances to admit of any great degree of accuracy. * Astronomische Nachrichten, No. 326. f Ibid, No. 338. PLANETS MEAN SEMIDIAMETERS. t)87 THE MEAN SEMIDIAMETERS OF THE PLANETS. 435. The apparent equatorial semidiameter of a planet when its distance from the earth is equal to the earth's mean distance from the sun is the constant from which its apparent semidiameter at any other distance can be found by the formula = J (715) in which s is the mean semidiameter and J the actual distance of the planet from the earth, the semi-major axis of the earth's orbit being unity. To find the value of s from the values of s observed at different times, we have then only to take the mean of all its values found by the formula s = sA (716) taking J from the tables of the planet for each observation. But here it is to be remarked that, in micrometric measures of the apparent diameter of a planet, different values will be obtained by different observers or with different instruments. The spurious enlargement of the apparent disc arising from imperfect definition of the limb, or from the irradiation resulting from the vivid impression of light upon the eye, will vary with the telescope, and may also vary for the same telescope when eye pieces of different powers are employed. The irradiation may be assumed to consist of two parts, one of which is constant and the other proportional to the semidiameter. Those errors of the observer which are not accidental may also be supposed to consist of two parts, one constant and the other proportional to the semidiameter ; the first arising from a faulty judgment of i contact of a micrometer thread with the limb of the planet, the second, from the variations in this judgment depending on the magnitude of the disc observed, and possibly also upon any peculiarity of his eye by which the irradiation is for him not the same quantity as for other observers. With the errors proportional to the semidiameter will be combined also any error in the sup- posed value of a revolution of the micrometer. The errors of the two kinds will, however, be all represented in the formula s = (s -j- x -f sy) J (717) where x is the sum of all the constant corrections which the 688 CONSTANT OF ABERRATION. observed value 5 requires, and sy is the sum of all those which are proportional to s. Now, let s^ = an assumed value of s , ds = the unknown correction of this value then the above equation may be written sA s l -f- xA -f syA rfs t But syJ will be sensibly the same as s$. It will, therefore, be constant, and will combine with ds r We shall, therefore, put z for syJ dsv and then, putting n = sJ -Sj our equations of condition will be of the form xA -f 2 + n = (718) from all of which x and 2 may be found by the method of least squares. But it will be impossible to separate the quantity ds l from z ; we can only put Oo) = s , ^ whereas we have, for the true value, s a Sj -j- ds l = s l z -f- s y or and then, if any independent means of finding y are discovered, the true value of s can be computed. THE ABERRATION CONSTANT AND THE ANNUAL PARALLAX OF FIXED STARS. 436. The constant of aberration is found by (669) when we know the velocity of light and the mean velocity of the earth in its orbit. The progressive motion of light was discovered by ROEMER, in the year 1675, from the discrepancies between the predicted and observed times of the eclipses of Jupiter's satellites. He found that when the planet was nearest to the earth the eclipses occurred about 8 OT earlier than the predicted times, and when farthest from the earth about 8'" later than the predicted CONSTANT OF ABERRATION. 689 times. The planet was nearer the earth in the first position than in the second by the diameter of the earth's orbit; and hence ROEMER was led to the true explanation of the discrepancy, namely, that light was progressive and traversed a distance equal to the diameter of the earth's orbit in about 16 m . More recently, DELAMBRE, from a discussion of several thousand of the observed eclipses, found S m 13'. 2 for the time in which light describes the mean distance of the earth from the sun. From this quantity, which is denoted by > Art. 395, we obtain the aberration ' constant by the formula * = ^ (720) V nTsinl'Vl c f Hence, with the values -^ = 493'.2, T= 366.256, n = 86164, e = 0.01677, we find k = 20".260. DELAMBRE gives 20". 255, which would result from the above formula if we omitted the factor i/l e 2 , as was done by DELAMBRE. On account of the uncertainty of the observations of these eclipses (resulting from the gradual instead of the instantaneous extinction of the light reflected by the satellite), more confidence is placed in the value derived from direct observation of the apparent places of the fixed stars. 437. To find the aberration constant by observations of fixed stars. Observations of the right ascension of a star near the pole are especially suitable for this purpose, because the effect of the aberration upon the right ascension is rendered the more evident by the large factor seed with which in (678) the constant is multiplied. The apparent right ascension should be directly observed at different times during at least one year, in which time the aberration obtains all its values, from its greatest positive to its greatest negative value. If we suppose but two observa- tions made at the two instants when the aberration reaches its maximum and its minimum, the earth at these times being in opposite points of its orbit, and if a' and a" are the apparent right ascensions at these times (freed from the effects of the nutation and the precession in the interval between the observa- tions), we shall have k = \ (a! a") cos d 690 CONSTANT OF ABERRATION. But, not to limit the observations to these two instants, let us take, for any time, a = the assumed mean right ascension of the star -}- the nutation -f- proper motion, a' the observed right ascension, and, further, let Aa = the correction of the assumed mean right ascension, A k = the correction of the assumed aberration constant, then, by (678), we have a' = a -f Ac* (k -f- A*) (COS Q COS e COS a -f- Sin sin a) 860 i or, putting m sin M sin a m cos M = cos a cos s a' = a -f Aa (k -f A/0 m COS (0 M} S6C d (721) Hence, collecting the known quantities, and putting a = m cos (O -M") sec d n = a -)- ak a' we have the equation of condition and by the formulas P sin y A'O cos d 1 p COS = A'<5 J Let r be the time of any observation reckoned from the assumed epoch and expressed in fractional parts of a year. In the above diagram, if AA' now represents the proper motion on a great circle in the time r, then AA f = rp] and, if the effect of the proper motion upon the distance is denoted by A'S, we have also A'B = s + A'S, A'AB = P , and the triangle AA'B gives cos (s -f- A'S) = cos (r/>) cos s -f sin (r/>) sin s cos (P /) Developing this equation, and retaining only second powers of r/>, we find fT> \ , OY>) J sin J (P %) A'S =-= - r/> cos (P-/) + ^ - - 696 PARALLAX OF A FIXED STAR. in whic 1 : r is the only variable. Taking then for the constants (728) the computation of the correction for each observation is readily made by the formula The assumed proper motion may, however, be in error ; and there may also be errors in the observed distances which are proportional to the time (such as any progressive change in the value of the micrometer screw, &c.). The correction for all such errors may be expressed by a single unknown correction y or the coefficient/, so that we shall take *'=(/ 4- *)*+/' (729) The corrections of micrometric measures for the effects of aberration and refraction* are treated of in Vol. II. Chapter X. We shall, therefore, suppose these corrections to have been applied, and shall take s' = the observed distance at the time r, corrected for differ- ential aberration and refraction, and then we shall have s' s -{- AS -f A'S (730) This equation involves three unknown quantities, namely, the distance s, the parallax involved in AS, and the correction y in- volved in A'S. Let s be an assumed value of s nearly equal to the mean of the values of .s', and put S = S -\- X The substitution of this in our equations of condition will in- troduce the small unknown quantity x in the place of the larger * These effects are only differential, and so small that the errors in the total refrac- tion and aberration may safely be assumed to have no sensible influence. It is also an advantage of this method of finding the parallax of a star, that it is free from the errors of the nutation and precession, which, being only changes in the position of the circles of reference, have no effect whatever upon the apparent distance of iwo stars. PARALLAX OF A FIXED STAR. 697 one 5, and will thus facilitate the computations. When all the substitutions are made in the expression of s', we obtain the following equation : = s s' + fr -f- f're -4- x -f ry -f- prm cos (O M) To put this in the usual form, let us take n = s -s'-\-fr+frr c rm cos (Q M ) then each observation gives the equation x + ?y-\-cp-\-n = Q (731) and from all these equations we find, by the method of least squares, the most probable values of x, y, and p. In the determination of so small a quantity as p, it is neces- sary to give to the micrometric measures the greatest possible precision. It is particularly important to find the effects oi tem- perature upon the micrometer screw; for these effects, depending on the season, have a period of one year, like the parallax itself, and may in some cases so combine with it as completely to defeat the object of the observations. At the time BESSEL pub- lished his discussion of his observations on 61 Cygni, he had not completed his investigations of the effect of temperature upon the screw, and therefore introduced an indeterminate quantity k into his equations of condition, by which the effect upon the parallax might be subsequently taken into account when the correction for temperature was definitively ascertained. This was done as follows. He had assumed the correction of a measured distance for the temperature of the micrometer screw to be A"S = 0".0003912 s (t 49.2) in which t is the temperature by Fahrenheit's scale, and s is ex- pressed in revolutions of the screw. If the coefficient 0". 0003912 should be changed by subsequent investigations to 0". 0003912 X (1 -f- A-), each observed distance would receive the correction &"s.&, the quantity n in the equations of condition would become n A"s.A;, and the equations would take the form = 4- ry -f- cp A"S . k + n = (732} 698 THE NUTATION CONSTANT. The quantity k being left indeterminate, x, y, and p were found as functions of it. The value of p was thus found to be = 0".3483 0".0533 k, with the mean error 0".0141 The final result of his investigation of the micrometer gives* k = 0.4893 with the mean error 0.0903 and hence the corrected value of the parallax = 0".3744 with the mean error 0".0149 If this result had been deduced by comparison with but one star, it could only be received as the relative parallax. BESSEL, however, employed two stars whose directions from \Cygni were nearly at right angles to each other, and found nearly the same parallax from both; whence it follows either that both these stars have the same sensible parallax, or, which is more probable, that both are so distant as to exhibit no sensible parallax. This conclusion would be confirmed if a comparison with other surrounding stars gave the same parallax, especially if these were of different magnitudes ; for it would be in the highest degree improbable that all these stars were at the same distance from our solar system. THE NUTATION CONSTANT. 442. To find the constant of nutation from the observed right ascen- sions or declinations of a fixed star. In Art. 437 it was assumed that the observations by which the aberration constant was de- termined extended over only a year or two : so that the nutation affected all the observations by quantities which differed so little that any error in the total nutation would not sensibly affect the determination. When the observations are extended over a longer period, we may introduce into the equations of condition an additional term for the correction of the nutation. As before, let the mean right ascensions and declinations be reduced to their apparent values at the time of each observation by means * According to PETERS in the Astron. Nach. Ery'dnzungs-heft, p. 55 ; derived from BESSBL'S Attronomische Untersuchungen, Vol. I. p. 125. THE NUTATION CONSTANT. 699 of an assumed aberration and nutation, and denote these apparent values by a and <5, and put A = the correction of the nutation constant, a', d' = the observed right ascension and declination; then a = a -f- Aa -f- a A # + bp -f- C Av d' = 3 -j- A in which a and d are taken for the mean epoch (^ -f- <,). And from the m and n thus found we have, by (661), -37- sin e. = n dt (737) in which is the annual luni-solar precession (or the precession constant), and the annual planetary precession. But is very accurately obtained theoretically by substituting the known masses' of the planets in the general formula deduced from the theory of gravitation : so that a value of the precession may be derived both from m and from n. In these formulae, the value of j is to be employed as given by (646) for the epoch Having thus obtained a preliminary value of the precession, the quantities m -f- n sin a tan d and n cos a , computed from it for each star, can be compared with the a and b found by (735), and the differences which exceed the probable errors of observa- tion may be regarded as resulting from the proper motion of the star. Those stars which are found to have a very large proper motion are then to be excluded from the investigation ; and from the remaining ones a more accurate value of the precession will be obtained. In this way, BESSEL, from 2300 stars whose places were deter- mined by BRADLEY for 1755 and by PIAZZI for 1800, found the precession constant for the year 1750 to be 50". 37572, and for 1800, 50". 36354.* In this investigation those stars were ex- cluded which in the preliminary computation exhibited annual proper motions exceeding 0".3. See also Article 445. * Fundamenta Astronomic, p. 297, where the value 50".340499 is found ; and Astron. Nach., No. 92, where the value is increased to 50". 37572. MOTION OF THE SUN IN SPACE. 703 THE MOTION OF THE SUN IN SPACE. 444. With a knowledge of the precession we are enabled to distinguish proper motions in a large number of stars. Upon comparing these proper motions, Sir "W. HEESCHEL was the first to observe that they were not without law, that they did not occur indiscriminately in all directions, but that, in general, the stars were apparently moving towards the same point of the sphere, or from the diametrically opposite point. The latter point he located near the star ^ Herculis. This common apparent motion he ascribed to a real motion of our solar system, a con- clusion which has since been fully confirmed. Nevertheless, there are many stars whose proper motions are exceptions to this law : these must be regarded as motions com- pounded of the real motions of the stars themselves and that of our sun. These real motions must, doubtless, also be connected by some law which the future progress of astronomy may develop ;* but thus far they present themselves in so many direc- tions that (like the whole proper motion in relation to the precession) they may be provisionally treated as accidental in relation to the common motion. Hence, for the purpose of determining the common point from which the stars appear to be moving, and towards which our sun is really moving, we may employ all the observed proper motions, upon the presumption that the real motions of the stars, having the characteristics of accidental errors of observation and combining with them, will be eliminated in the combination. Nevertheless, in order that the errors of observation may not have too great an influence, it will be advisable to employ only those proper motions which are large in comparison with their probable errors. The direction in which a star appears to move in consequence of the sun's motion lies in the great circle drawn through the star and the point towards which the sun is moving. -Let this point be here designated as the point 0. If the great circle in which each star is observed to move were drawn upon an artificial globe, * The law which we naturally expect to find is that of a revolution of all the stars of our system around their common centre of gravity. MADLER, conceiving that our knowledge of the proper motions is already sufficient for the purpose, has attempted to assign the position of this centre. He has fixed upon Alcyone, the principal star of the Pleiades, as the central sun. Astron. Nach., No. 566. Die Eigmbewegungen der Fixsternein ihrer Beziehung zum Gesammt 'system, von J. H. MADLER, Dorpat, 1866. 704 MOTION OF THE SUN IN SPACE. all these circles would intersect in the same point 0, if the obser- vations were perfect and the stars had no real motion of their own. But, the latter conditions failing, the intersections which would actually occur would form a group of points whose mathe- matical centre of gravity would, according to the theory of proba- bilities, be the point from which, or towards which, the common motions were directed. Thus, an approximate first solution might be obtained by a purely graphic process. Let us then assume that an approximate solution has been found, and put A, D = the assumed approximate right ascension and decli- nation of the point 0. It is then required to find a more exact solution by determining the corrections &A and AJD which A and D require. Let P (Fig. 62) be the pole of the equator, and S a star whose apparent motion resulting from the sun's motion is in the great circle OSS'. The angle PSS' = %, w r hich this great circle makes with the declina- tion circle (reckoned in the usual manner from the north towards the east), is the supplement of the angle PSO. Hence, if a and d are the right ascension and declination of the star, and ^ the arc SO joining the star and the point 0, we have, in the triangle POS, sin ^ sin / = sin (a A) cos D ) sin ^ cos x = cos (a A) cos D sin 8 sin D cos 3 j by which / and / are found for each star. The angle thus computed will be equal to the observed angle which the path of the star makes with the declination circle only when A and D are correctly assumed. Let #' be the observed angle, or that which results from the equations > sin = Aa cos Mtdix meunte anno 1830. J Fundaments Astrnnomix. VOL. I. 45 706 MOTION OF THE SUN IN SPACE. three classes according to their proper motions, and found, for the epoch 1792.5, From Whose annual proper motion was A = D = 23 stars 50 " 319 greater than 1".0 between 0".5 and 1 .0 .2 " .5 256 25M 255 9 .7 261 10.7 + 38 37'.2 + 38 34 .3 + 30 58 .1 and, combining these results with regard to their respective weights, A = 259 51'.8 D = + 32 29M As supplementary to this computation, LUNDAHL compared 147 of BRADLEY'S stars not contained in ARGELANDER'S catalogue with POND'S catalogue of 1112 stars for 1830, and found* A = 252 24'.4 D = + 14 26M which ARQELANDER combined with his former results and found, for 1800, A = 257 54' D = -f 28 49' OTTO STRUVE, employing 400 stars, mostly identical, however, with ARGELANDER'S and LUNDAHL'S stars, and determining their proper motions from the Dorpat observations compared with BRADLEY'S, found, for 1790, A = 261 21'.8 D = 37 36'.0 GALLOWAY, from the southern stars observed by JOHNSON at St. Helena and HENDERSON at the Cape of Good Hope (for 1830), and by LACAILLE at the Cape of Good Hope (for 1750), found A = 260 1' D = -\- 34 23' Finally, MADLER, recomputing the proper motions of a large number of stars, with the aid of the best modern observations, has found, for 1800, f From Whose proper motion is A = D = 227 stars 663 1273 greater than 0".25 between 0".l and .25 .04 " .01 262 38'.8 261 14.4 261 32.2 + 39 25'.2 + 37 53 .6 + 42 21 .9 Astron. Nach., No. 398. f- Die Eigenbewegungen der Fixtterne, p. 227. MOTION OF THE SUN IN SPACE. 707 and by combination, having regard to the number of stars in each class, A = 261 38' 8 D = + 39 53'.9 445. It would at first sight seem that the existence of any law in the proper motions of the stars would vitiate the value of the precession constant found by BESSEL according to the method of Art. 443. Accordingly, OTTO STRUVE has attempted to determine both the precession constant and the motion of the solar system from equations of condition involving both. In order to accomplish this it was necessary to introduce into the equations the magnitude as well as the direction of the proper motions. But since the apparent angular motion of a star, so far as it depends upon the motion of our sun, is a function of the star's distance from us, it became necessary also to make an hypothesis as to the relative distances of the stars of different orders of magnitude. Thus, the new value of the precession constant given by him, and which we have (provisionally) adopted on page 606, is also exposed to the objection that it rests upon an hypothesis. Astronomers have, therefore, been led to re-examine the grounds upon which BESSEL' s determination rests. It is to be observed that the method which he employed would give a re- sult entirely free from the effects of the sun's motion, if the stars employed were uniformly distributed over the sphere, and if the average distance of these stars in all directions from the sun were the same. MADLER, in the work above quoted, has shown that for 2139 stars distributed with tolerable uniformity, BESSEL'S constant gives proper motions in ngnt ascension the mean of which is only 0".0003. If now this quantity were applied to BESSEL'S value of m and the proper motions again computed, their mean would come out exactly zero. Hence he concludes that these stars fully confirm BESSEL'S constant, since the correc- tion 0".0003 is insignificant. It appears, however, that, in drawing this inference without reservation, he has left out of view the second conclusion above stated, that the average dis- tance of the stars on all sides of us should be the same. For, if the sun's motion produces greater apparent motions in stars near to us than in those more remote, a want of uniformity in the dis- tances, notwithstanding the equal distribution of the stars, would produce a greater amount of proper motion in one hemisphere 708 MOTION OF THE SUN IN SPACE. than in the other; and the aggregate of all the proper motions, having regard to their signs, would not be zero. Since it is probable that the average distance of stars of the same magnitude is the same on all sides of us (although there are not a few individual exceptions of small stars with large proper motions and large stars with small ones), a more satisfactory determination of the precession constant may result from future investigations in which not only all the stars employed shall be uniformly distributed, but those of each order of apparent magni- tude shall be so distributed. It will be impossible to secure this condition if the larger stars are retained ; for their distribution is too unequal. By confining the investigation to the small stars, there will also be obtained the additional advantage that the amount of the proper motions themselves will probably be very small, and thus have very little influence upon the precession constant, even if they are not wholly eliminated. The formation of accurate catalogues of the small stars is therefore essential to the future progress of astronomy in this direction. END OF VOL. 1. ENGINEERING AND MATHEMATICAL SCIENCES LIBRARY (213) 825-4951 University of California, Los Angeles Please return to the above library NOT LATER THAN DUE DATE stamped below. PSD 2340 9/77 A 000427715 8 SOUTHERN UNIVERSITY OF CALIFORNIA, LIBRARY, iLOS ANGELES, CALIF,