■1* ' .^S** ■>' ,,.*:: ^mmmimM Cornell University Library QB 145.B29 Elements of natural philosophy 3 1924 004 964 981 Date ; Due - — — — V ' El\Sh ■ ■f-MT OTA^^f^C MENT blUiV »t\AI-» 'VS4' ■ 1 1 1 <^3S> 1 a W ^ Cornell University WM Library The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924004964981 BARTLETT'S SPHERICAL ASTRONOMY. ELEMENTS OS NATURAL PHILOSOPHY vt W. H. C. MBTLETT, LL.D., rBorsssoR of natukal asd gxpekihentai, ihilosopht ra ths bkited btats» MILITART AOADEUT AT WEST POINT, Avmos or " EtEMKKTO OF MECalAMIOS," "AiOOUSTICS," " OfTICS," AJTD "AJTALYTIOik]^ MBCHAmCS." IV.— SPHERICAL ASTRONOMY. THIRD EDITIONi B.KVI8ED AND CORRECTED. NEW YOEK: A. S. BARNES Dimensions and Distances of Pl.-vnets 90 vi CONTENTS. Inferior Planets — Superior Planets ^^ Synodic Revolutions — Geocentric Motions 91 Direct and Ketrograde Motions— Stations 92 Piloses of the Planets 93 Transits — Occnltations 9^ Masses and Densities of the Planets 9? Mercury 99 Venus : 100 Mars— Planetoids 102 Jupiter— Saturn , - 104 Urunus — Neptune 108 Secondary Bodies 109 The Moon— Lunar Orbit ■ HO Disturbing Forces US Librations , US Lunar Periods / 116 Lunar Phases 117 Eclipses of the Sun and Moon .' 118 Moon's Relative Geocentric Orbit 12S Ecliptic Limits 124 Number of Eclipses 125 TheSaros 126 Physical Constitution of the Moon 127 Satellites of Jupiter 130 Progressive Motion of Light 134 Satellites of Saturn 134 Satellites of Uranus 136 Satellites'of Neptune 137 Comets 137 Elements of the Orbits -of the Permanent Comets 140 S tars , m Elements of Stellar Orbits 156 Proper Motion of the Stars and of the Sun 158 NebuliE 159 Zodiacal Light ^ 162 AeroUtes — Meteors Ijg Ephemerides 155 Catalogue of Stars i j7y Applications ■ ; I74 Time of Conjunction and of Opposition I74 Angle of Position I75 Projection of a Solar Ecli!pse j75 Projection of a Lunar Eclipse 133 Time of Day , Ig^ .\ziiiiuths ,^ Ijjg Meridian Passages .^ ^ 191 lioduction to the Meridian I93 Turrustrial Latitude ig_;; Terrestrial Longitude 202 Calendar 229 CONTENTS vij rjLiim Appk.idix 1.— Elements of the Principal Planets 238 Apfkndix II. — Aistronomioal Instrunu .its 237 Clock and Chronometer 237 Vernier 243 Micrometer ' 245 Level 250 Beading Microscopes 252 Transit 255 CoUimating Telescope 266 Vertical Collimator 207 Collimating Eye-piece 268 Mural Circle 269 Altitude and Azimuth Instrument 276 Equatorial 282 Heliometer .' 293 Sextant 294 Artificial Horizon 298 Principle of Eepetition 300 Eeflecting Circle 301 Appendix III.— Atmospheric Refraction 305 Appendix IV. — Shape and Dimensions of the Earth 310 Appendix V.— The Earth's Orbit 813 Appendix VI. — Planets' Elements 318 Appendix VII. — Planets' Elements 817 Appendix VIII.— Planets' Elemenita 818 Appendix IX.— Planets' Elementis /. 319 Appendix X. — Geocentric Motion 331 Appendix XI. — Mr. Woolhouse on Eclipses, &c ; . . . 382 Appendix XII. — Equation of Equal Altitudes .' 424 Appendix XIII. — Correction for Ditference of Kcfraction 42S TABLES. Table I. — Mr. Ivory's Mean Refractions, with the Logarithms and their Differ- ences annexed 427 Table II. — Mr. Ivory's Refractions continued : showing the Logarithms of the corrections, on account of the state of the Thermometer and Baromietcr... 430 Taiile III. — Mr. Ivory's Ecfractious continued : showing the /urt/ier quantities by which the Refraction at low altitudes is to be corrected, on account of the state of the Thermometer and Barometer 431 Taisle IV.— For the Equation of Equal Altitudes of the Sun 4.32 Table V.— For the Reduction to the Meridian: showing the value of ^ = ^^1^ 440 sin 1" Table VI.— For the second part of the Reduction to the Meridian : showing the „ „ 2 sin* 4 }' „,, value of i?= — r-^rr *■"* sin 1" Tbioonobetbi"!al FonjiuLJi ^'^1 The Greek Alphabet is here inserted to aid-those who are not already familiar with it in reading the parts of the text in which its letters occur : L«t(«n. Nuaefc Lettere. Nimei. A a Alpha N V Nu B J8C Beta B g Xi r yf Gamma « OmicroD A s Delta n wr Pi £ { Epsilon r fS Rho z?? Zeta 2 e Phi Kx Kappa XX Chi A X Lambda r^ Psi M,* Mu n bi Omega ASTRONOMY. ASTRONOMY. § 1. The science which treats of the heavenly bodies is called Aslron- omy. It is divided into Physical and Spherical Astronomy. § 2. Physical Astronomy is a system of Mechanics, in which the forces .ire universal gravitation and inertia, and the objects the gigantic masses that move through indefinite space. It treats of the physical conditions of the heavenly bodies, their mutual actions on each other, and explains the causes of the celestial phenomena. § 3. Spherical Astronomy is mainly concerned with the appearances, magnitudes, motions, arrangements, and distances of the heavenly bodies ; and seeks to apply the deductions from these to the practical wants of society. It is a science of observation, and its principal means of investi- gation are Optical and Mathematical Insti-uments. This branch of As- tronomy will form the subject of the present volume. § 4. No subject calls more strongly upon the student to abandon firet impressions than Astronomy. All its conclusions are in striking contra- diction to those of superficial observation, and to what appears, at first view, the most positive evidence of the senses. § 5. Every student approaches it for the first time with a firm belief that he lives on something fixed, and, abating the inequalities of hill and valley, that this something is a flat surfa(^ of indefinite extent, composed of land and water ; and that the blue firmament which he sees around and above him in the distance is a stationary vault, upon the surface of which appear to be placed all objects out of contact with the ground. § 6. The Earth on which he stands is divested by Astronomy of its flattened shape and of its character of fixidity, and is shown to be a globular body turning swiftly about its centre, and moving onward through space with great rapidity. It teaches him that his vault has no existence 2 ASTBONOMT. in facf, and is but a mere illusion which comes fi'om looking through the indefinite space, extended without limit, in which he is moving. § 7. Were the Earth reduced to a mere point, and a spectator placed upon it, he would see around him at one view all the bodies which make up the visible universe ; and in the absence of any means of judging of their distances frona Uiin, would ref^ thejn in the direction in which they were seen from his station, to the concave surface of an imaginary sphere, having its centre at his eye and its surface at some vast and indefinite distance. , SOLAR SYSTEM. § 8. A little observation would lead him to conclude that by far the greater number of these bodies appear fixed while the ,rest seem ever on the move, continually shifting their positions witji respect to, thase which appear ^xed, and to each other. The former are called Fixed Stars : the latter compose what is called the Solab System, a group of bodies from whicli the fixed stars are so remote as to produce upon it no appre- ciable influence. , § 9. All bodies attract one another with intensities which are propor- tional to the (juantity of the attracting masses directly, and to the squares of the distances inversely, Analyt. Mech., §, 205. g 10. Bodies resist by their inertia all chaqge in their aotpal state of motion; this rissistance is exerted simnltaneonsly with tlfe change, andi» always equal in intensity, and contrary in direction, to the force which produces it. . § 11. The bodies of the solar system have motions that eariy them in directions oblique to the lines along which their, mutual attractions are exerted. The . attraotiye forces draw them aside from these directions ; inertia resists by an equal and, contrary reaction; and the bodies are forced into curvilinear paths, and made to revolve about the centre of inertia of the whole. § 12. Thus, the antagonistic fcrces of gravitation and of inertia are the simple but efficient causes which keep the bodies of the solar system to- gether as a single group, a,nd impress upon it a character of stability and perpetuity. But for the force of, grjivitatipn the bodies would separate more and more, and wander .through en41ess space ; and but for the force of inertia, that of gravitation w^uld pile thaqj together in one confused mass. § .13. Theforce,of,gravitation increases rapidly with a diminution, and .^decreases as.. rapidly with an augmentation, of distance. Those bodies *hich are nearest exert, therefore, the gi'eatest influence upon one another's SOLAR SYSTEM. 3 motions. Bodies composing an insulated group may perform their evolu- tions among each other undisturbed by the action of those -without, pro- vided the distances of the latter be very great in comparison to thoSe which separate the individuals of the group. § 14. This is a characteristic of the solar system. Its own dimensions, vast as they are when expressed in terms of any linear unit with which we are familiar, are utterly insignificant when compared with its distance from the fixed stars. Each of the latter, by virtue of this relatively great distance, acting upon all the bodies of the system equally and in parallel ' directions, the effect of the whole can only be to move the group collec- tively through space. § 15. The same thing takes place upon a smaller scale within the solar system itself. Some of its members are so close together, and at the sanie time so far removed from the others, as to be forced to revolve about one another, while the combined action of the rest cariies them as a sub-groap, so to speak, about the centre of inertia of the whole. § 16. The mass of the sun so far exceeds the sum of the masises of all the other bodies of the system, as to throw the centre of ineida of the whole group within the boundaiy of its own volume ; and although the centre of the sun actually revolves about this point, yet its motion te- comes so small, when viewed from the distance of the earth, that it is in- sensible except through the medium of the most refined instruments. ' All the other bodies are, therefore, said to revolve about the sun as a centre, and it is from this fact, and the controlling influence which this latter tody exerts over the motions of all the others, that the system takes its name. § l7. The same is true of the sub-groups ; the mass of one of the bodies in each being so much greater than the sum of the masses of the rest as to cause the latter to revolve approximately about its centre, while this centre revolves about the sun. § 18. The path a body describes about another as a principal source of attraction, is called an orbit. § 19. Those bodies which describe their orbits ahoiit the sin aie Called primary, and those which describe their orbits about the primaries' are Ciilled secondary lodies. These latter are also called Satellites. Of the primary bodies there are thiee distinct classes, differiiig' from each other mainly in the shape of their oriats, their densities, and gen- eral aspects. § 20. a! body subjected to the action of a central force, whose intensity varies as the square of the distance inversely, must describe one or other of 4 ASTRONOMY. the conic sections, defending upon the relation oetween its velocity and the intensity of the central force. The orbits that are known to belong to the solar system are ellipses. § 21. Those primaries which move in elliptical orbits of small eccentri- cities are called Planets. Those primaries having orbits of great eccentri- cities are called Comets. Comets are also distinguished from planets in having a degree of density so low as to give some the appearance more of a vapor than of a solid body. § 22. The solar system consists then of the Sun, Planets, Comets, and ^Satellites. Setting out from the sun, the known planets, with their names, occur in the following order, viz. : Mercury, Venus, the Earth, Mars, then a class called the Planetoids, of which seventy-one are known at the present time, Jupiter, Saturn, Uranus, and Neptune. See Plate I., Fig. 1. To these must be added a multitude of much smaller bodies of the nature of planetoids, whose existence is infen-ed from the fact that some of their number make their way now and then to the earth's surface under the name of meteors. § 23. It would be utterly impossible to give within the narrow limits of an octavo page a graphical representation of the relative dimensions of the solar system ; and to aid the conceptions of the student, Sir John Herschel has instituted the following illustration, viz. : On any well-levelled field place a globe two feet in diameter ; this will represent the sun ; Mercury will be represented by a grain of mustard-seed on the circumference of a circle 164 feet in diameter for its orbit ; Venus a pea on the circumference of a circle 284 feet in diameter; the Earth also a pea on the circumference of a circle 430 feet in diameter; Mars a rather large pin's head on the circumference of a circle of 654 feet diameter ; the Planetoids grains of sand on circular orbits varying from 1000 to 1200 feet in diameter; Jupiter a moderate sized orange on a circumference nearly half a mile in diameter ;" Saturn a small orange on the circumference of a circle four- fifths of a mile in diameter ; Uranus a full sized cherry on the circumfer- ence of a circle more than a mile and a half in diameter ; and Neptune a good sized plum on the circumference of a circle about two miles and a half in diameter. To illustrate the relative motions. Mercury must describe a portion of its orbit equal in length to its own diameter in 41 seconds ; Venus in 4 minutes and 14 seconds; the Earth in 7 minutes ; Mars in 4 minutes and 48 seconds ; Jupiter in 2 houre and 56 minutes ; Saturn in 3 hours and 13 minutes; Uranus in 2 hours and 16 minutes, and Neptune in 3 hours and 30 minutes. Now conceive the two feet globe to be in- creased till its diameter becomes 880,000 English miles, and suppose the SOLAR SYSTEM. 5 other bodies and their distances increased in the same proportion ; the re- sult will represent the dimensions of the solar system. It will give to the earth a diameter of nearly eight thousand miles, a distance from the sun equal to 95 millions of miles, and a velocity through space, around the sun, of 19 miles a second. , The orbits, although, referred to as circles, are in fact ellipses, but of ec- centricities so small as to justify the substitution for the mere purposes of the illustration. § 24. The fixed stars are self-luminous. The sun is regarded as one of this class of bodies, and by its greater proximity to the earth, becomes the principal source of heat and light to its inhabitants. § 25. The planets and satellites are opaque non-luminous bodies, and are visible only in consequence of light received from the sun and reflected to the earth. SPHEEICAL ASTEONOMY. MOTION, § 26. Motion signifies the condition of a body, in virtue of which it oc ciipies successively different places. But we can form po idea of place ex- cept by referring it to other places, and these again, to be known, must be referred to others, and so without limit ; so that place is, in its very nature, entirely relative. Motion is, from its definition, therefore, also relative. § 27. We judge of the rate of motion by the greater or less rapidity with which the object possessing it varies its distance from other objects assumed as origins. These origins may themselves be in motion, but if the circumstances of the spectator be such as to deceive him into the belief that they are at rest, he will attribute all change of distance to .1 motioa wholly in the object which he refers to them. And this is one of the most fruitful sources of the many erroneous notions with which students gener- erally commence the study of astronomy. § 28. If two objects be in motion, and they alone occupy the spectator's field of view, the effect to him will be the same if he suppose one fixed, and attribute the whole, of its motion to the other in a contrary direction ; for this will not alter the rate by which they approach to or recede from one another. PARALLACTIC MOTION AND PARALLAX. § 29. The real motion of a spectator gives rise to the appearance of motion among surrounding objects which are relatively at rest. Objects in front of him seem to separate from one another, those behind appear to approach one another, and those directly to the right and left; seem to move in a direction parallel to his own motion. A spectator, for example, travelling over a plain studded with trees or other objects will, on fixing his eyes upon a single object without with- drawing his attention from the general landscape, see or thmk he sees tho CELESTIAL SPHERE. Z' FiS- 1- latter in rotaiy motion about that object as a centre ; all objects betweeu it and himself apj)eariiig to inove backward, or contrary to his own niotiou, and all beyond it, forward or in the direction in which he moves. This apparent change in the relative places of objects, arising from a shifting of the point of view from which they are seen, is called pafaUacUc motion ; and tlie amount of angular change in the instance of any partic- ular object is called the parallax of that object. ' § 30; Let P be the place of an object, G and S the places from which it is Seen; and let its place be referred to some point Z', on the prolongation of the line GS, whicb joins the points of view. The angular change in the place of P as seen from C and /S will be Z' SP-Z' OP=SP (7=:the parallax of P. That is to say, the parallax of an object is the angle sub- tended at the object by the distance between the stations from which it is seen. Make CP=d; CSz=f,; the angle Z'SP^Z; the angle SPC^z Then from the triangle CSP,vie have sin z= ^. sm Z . (1) Whence the parallax increases with an increase of the spectator's change of place, with diminution of th6 object's distance, and also' with the approx- imation of ^ to 00°. § 31. All other things being equal, the parallax will be less as the ob- ject's distance is greater; and when the parallax is zero for any arbitrary value of Z, the factor ^ must be zero, and the change of the spectator's place must be utterly insignificant in comparison with the object's distance. CELESTIAL SPHERE. § 32. Now, when the heavens are examined it is foUnd that by far the greater number of the celestial bodies have no sensible parallax, while comparatively a few have. The first are the jfixed stars ; and they are so called from the fact that they always preserve the same angular distances from any assumed point and from each other, from whatever station on the earth they are viewed. ' The second are bodies of the solar system. § 33. The fixed stars are, therefore, beyond limits at which objects cease 8 SPHERICAL ASTRONOMT to be sensibly affected by parallax. The great concave of the heavens upon which the fixed stars appear to be situated, is called the celestial sphere. Not only, therefore, is the longest rectilineal dimension of the earth, hut also the distance between the points of its orbit about the sun most remote frorn each other— a distance, as we shall see in the sequel, equal to one hundred and ninety millions of miles — ^utterly insignificant when expressed in terms of the radius of the celestial sphere as unity. A sphere large enough to. contain the entire orbit of the earth is a mere point in comparison with the vast volume embraced by the celestial sphere. The centre of the earth may, therefore, always be regarded as the centre of tJie celestial sphere. SHAPE OF THE EARTH. § 34. The earth, being the station fi-om _ ^'S- ^ which all the other heavenly bodies are viewed, is the first to claim attention. It has been repeatedly circumnavigated in dif- erent directions, and the portions of its sur- face visible from elevated positions in the midst of extended planes or at sea, always appear as circles of which the spectator seems to occupy the centre. The apparent diameters of these' circles, measured by in- struments, are smaller in proportion as the points of view S are more elevated. The earth is, therefore, ylohular ; for to such figures alone belong the property of always presenting to the view a circular outline. § 35. By the figure of the earth is meant its general shape without regard to the irregularities of surface which form its hills and valleys. These are relatively insignificant and are disregarded in speating of the earth's form. They are less in proportion to the entire earth than tire protuberances and indentations on the surface of a smooth orange are to a large size specimen of that fruit. The earth is an oblate spheroid, and the 6perations and method of computations by which its precise magni tnde and< proportions are found, will be given presently. The shortest diameter of the earth is called its- axis. DIURNAL MOTION. DIURNAL MOTION. 36. The boundary of the visible portion of the earth's surface, sup perfectly smooth, is called the sensible horizon. The sensible horizon Fig. ^. is only seen at sea, or on extended plains. At most localities on land it is broken by hills, valleys, and other objects. § 37. The earth conceals from us that portion of space below our sen- sible horizon, while all above is exposed to view. It rotates upon its axis, and the period required to perform one entire revolution is called a day. § 38. Every spectator is carried about ^'S- *• the earth's axis in the circumference of a circle, and while the extent of the visible portion of space remains un- changed, different regions are continu- ally passing through the field of view. The horizon of a spectator will be ever depressing itself below those bodies which lie in the region of space towards which he is carried by the rotation, and elevating itself above those in the oppo- site quarter ; thus successively bringing into view the former and hiding the latter. § 39. The spectator being unconscious of his own motion, concludes, from first appearances, that his horizon is at rest, and attributes these changes to an actual motion in the objects themselves. Instead of his horizon approaching the bodies, he judges the bodies to approach his horizon ; and when it passes and hides them, he regards them as having sunk below it or set, while those it has just disclosed, and from which it is receding, he considers as having come up or risen. §1 40. One entire revolution about the axis being completed, the spec- tator returns to the place from which he commenced his observations, and he begins again to witness the same succession of phenomena and in the same order. All the heavenly bodies appear to occupy the same places in the concave sky which they did before. § 41. Thus the rotation of the earth about its axis produces the daily 10 SPHERICAL ASTRONOMY I rising and setting of the sun — the alternation of day and night ; also the rising and setting of the other heavenly bodies, their progress through the vault of the heavens, and their return to the same apparent places at short and definite intervals. § 42. The apparent motions with reference to the horizon by which these daily recurring phenomena are brought about, are called the diurnal motions of the heavenly bodies. The real motion is in the horizon, the origin of reference ; it is only apparent in the bodies themselves. DEFINITIONS. § 43. The axis of the celestial sphere is the axis of the earth produced. § 44. ThB poles of the earth arei the points in which its axis pierces its surface. The pole nearest to Greenland is called the north, \h% other the! south pole. § 45. The poles of the heavens are the points in which its axis pierces the celestial sphere. That above the north pole of the earth is called the north, the other the south pole. § 46. The earth's equator is the intersection of the earth's surface by a plane through its centre, and perpendicular to its axis. § 47. The equinoctial is the intersection of the surface of the celestial sphere by the same plane. § 48. A meridian line is the intersection of the earth's surface by a plane through its axis and the place of a spectator. ' ' ■ § 49. The celestial meridian is the intersection of the surface of the celestial sphere by the same plane. This is often called simply the me- ridian of the place. § 50. The poles of the celestial meridian are called the Hast and West points ; that towards which the spectator is mo\'ing by his diurnal motion being the East, that from which he is receding the West. § 51. The apparent zenith and apparent nadir are the points in which a plumb-line produced interaects the celestial sphere : that over head being the zenith. § 52. The rational horizon is the intersection of the celestial sphere by a plane through the earth's centre and perpendicular to the line of the zenith and nadir. The plumb-line being always noitnai to the earth's sur- face, the plane of the rational horizon is parallel to the plane tangent to the earth's surface at the spectator's place, and these planes intersect the celestial sphere sensifcly in the same great circle. § 53. The dip of the horizon is the angle which the elements of a DEFINITIOlfS. 11 visual cone, whose vertex is in the eye of. the spectator, and, whose surface is tangent to that of the earth along the sensible horizon, make with the tangent plane to the earth at the spectator's plsice. The dip is greater in proportion as the spectator's elevation above the earth is greater. When the eye is in the earth's surface, the dip is zero, and the visual cone be- comes the tangent plane. This coincidence will always be supposed to exist unless the contrary is specially noticed, § 54, The latitude of a place on the earth's surface is the arc of the celestial meridian from the equinoctial to the zenith of the place. It is always measured in degrees, minutes, seconds, and thil-da. Latitude is reckoned north or south ; that reckoned towards the north pole being called north latitude, that towards the south pole, south latitude. The greatest latitude a place can have is 90°, this being the latitude of the poles of the earth. „ § 55. Parallels of latitude are small circles on the earth's surface par- allel to the equator. All places on the same parallel have the same latitude. § 56. The longitvde of a place on the earth's surface is the arc of the equinoctial intercepted between the meridian of the place and that of some other place assumed as a first meridian. It is called East or West, according as it is reckoned in the direction fi-om the first meridian towards its east or west point. For the sake of uniformity, it will, in the text, al ways be reckoned in the latter direction, The English estimate longitude from the meridian of Greenwich, the French from that of Paris, and other nations from other meridians. In the United, States, for most geographical purposes, it is estimated from the meridian of Washington. § 67. A vertical circle is the intersection of the celestial sphere by a plane through the zenith and nadir. The prime vertical is the vertical circle whose plane is perpendicular to that of the meridian. § 58. The north and south pointi are the poles of the prime vertical ; that below the north pole being called the north point. § 59. The Azimuth of a body is the angle which a vertical circle through the body's centre makes with the meridian. It is measured on the horizon, and from the «outh towards the west, or from the north to- wards the west, according as the north or south pole is elevated above the horizon. It may vary from 0° to 360°. I 60. The zenith distance of aft object is the angular distance from the apparent zenith to the centre of the object, measured on a vertical circle. 12 SPHERICAL ASTRONOMY. § 61. The altitude of an object is the angular distance from the horizon to the object's centre, measured on a vertical circle. The azimuth and zenith distance are a species of polar co-ordinates for the designating an object's place in the heavens. By making the azimuth vary from zero to 360°, and the zenith distance from zero to 90°, every visible point of celestial space may be defined in position. § 62. A declination circle, or fiour circle, is the intersection of a plane through the axis of the heavens with the celestial sphere. § 63. The declination of an object is the angular distance of its centre fi'om the equinoctial, measured on a declination circle. The declination may be north or.south, and may vary from 0° to 90°. § 64. The polar distance of an object is the angular distance of its centre from the celestial pole, measured on a declination circle. § 65. The riffht ascension of an object is the angle which a declination circle through the object's centre makes with a declination circle through a certain point on the equinoctial, called the Vernal Equinox. This angle is measured upon the equinoctial, and eastwardly in direction. § 66. The polar distance and right ascension are also a kind of polai co-ordinates for defining the places of celestistl objects ; for this purpose it is only necessary to cause the right ascension to vary from 0° to 360°, and the polar distance to vary from 0° to 180°, to reach every point in the celestial sphere. § 67. The hx)ur angle of an object is the angle which its hour circlo makes with the meridian of the place. It is estimated from the meridian westwardly, and may vary from to 360°. The hour angle may be em- ployed, instead of the right ascension, with the polar distance to define an object's place. To illustrate, let the plane of the paper be that of the meridian ; the circle jETZ OiV its intersection with the celestial sphere; P P' the axis of the heavens ; P and P' the north and south poles respectively ; Z and iV the zenith and nadir respectively, and the earth a mere point at C; then will the circle QWQ'E, of which P and P' are the poles, be the equinoctial; HWOE, of which Z and N are the poles, the hori- zon ; E and W, the poles of the meridian, will be the east and west poin^ respectively; the arc ZQ will be the latitude, ZSA a vertical circle,- Fis. S, INSTRUMENTS. I3 Z S tlie zenith distance of the object S,ASHs altitude, and OWA its azimuth ; FS will be its polar distance, i) 5 its declination, ZPS, meas- ured by Q B, its hour angle, and if V be the vernal equinox, VD will be its right ascension. , INSTRUMENTS. § 68. Most of the data with which the practical astronomer labors, come from measurements made in the circles just referred to, by means of certain astronomical instruments. These instruments are described, and their theory, adjustments, and uses explained, in Appendix II. The student should study, in connection with short daily lessons of the text, from this point, the Clock, Chronometer, Transit, Mural Circle and Azimuth and Altitude Instrument. The others should he taken up where referred to, in the order of the text. PROPORTIONS OF LAND AND WATER.— THE ATMOSPHERE. § 69. To resume the consideration of the earth. About three-fourths of its surface are covered with water, and the greatest depth of the sea does not probably exceed the greatest elevation! of the continents. The earth is surrounded by a gaseous envelope, called the atmosphere, the actual thickness of which, were it reduced to a uniform density throughout, equal to that at the surface of the sea, would be about five miles. But owing to the law which regulates the pressure, density, and temperature of elastic bodies, it is much greater than this. The dif- ferent strata, being relieved from the weight of those below them, become more expanded in proportion as thfy are higher, and the place of the su- perior atmospheric limit must result from an equilibrium between the weight of the terminal stratum and the elastic force of that upon which it rests. The laws just referred to indicate that this limit cannot be much higher than 80 miles. § 70. The atmosphere is not perfectly transparent. The sun illumines its particles ; these scatter by reflection the light they receive, particularly the blue, in all directions, and produce that general illumination called daylight, and gives to the sky its bluish aspect. But for this diffusive power of the air, no object could be visible out of direct sunshine ; the shadow of every passing cloud would be pitchy darkness, the stars would bp visible all day, and every apartment into which the sun did not throw his direct rays would be involved in total obscurity. In ascending to 14 SPHERICAL ASTRONOMY. the summits of high mountains, the diffused light becomes less and less, the sky deepens in hue, and finally, at great altitudes, approaches to total blackness. § ll. The superior illumination of the athiosphere produced by the solar light obliterates, as it were by contrast, the light from almost all the other heavenly bodies, and few, if any, of the latter are seen when the sun is up. REFRACTION. § 72. Luminous waves which enter the atmosphere obliquely arfe, ac- cording to the liws of optics, deviated by the latter from their course, and made to exhibit the objects from which they proceed in positions different from those they actually occupy, and thus false impressions are produced in regard to true places of the heavenly bodies. Take, for example, a spectator fig. 86. - on the earth aiA; and ht LDL represent a section of the supe- rior limit of the atmosphere, and KA A' that of the earth's sur- face by a vertical plane. A star at S would, in the absence of the atmosphere, appear in tlie direction AS; but in reality, when the portion of the luminous wave moving on this line reaches the point 2>, it is turned down- ward, and made to come to the earth at some point A', pursuing a course such as to bring its suc- cessive positions normal to some curve, as J) A', whose curvature increases towards the earth's surface, in . consequence of the increasing density of the atmosphere in that direction. This part of the wave cannot therefore go to the spectator. Not so, how- ever, with a portion of the same general wave incident at some point as £>', nearer to the zenith ; this, after pursuing a path D'A similar to DA', will reach the spectator at A, and cause the body from which it originally proceeded to appear in the direction A S', tangent to the curve at the point A, the effect being the same as though the body had shifted its place towards the zenith by the angular distance S A S'. § V3. The air's refraction, therefore, diminishes apparently the zenith REFRACTION. 15 distances of all bodies, and increases their altitudes. Any body actually in the horizon will appear above it, and any body apparently in the horizon must be below it. § 74. It is also obvious that refraction can only take place in the ver- tical plane through the body, since this plane is always normal to the surfaces of the atmospheric strata, and divides them symmetrically. R«- fraction will not, therefore, in general, affect the azimuth of a body. § 75. This apparent angular displacement of a body from its true place, caused by the action of the atmosphere upon its luminous waves, is called refraction ; and various formulas have b^en constructed to compute its exact amount. One of the best of these is by Littrow, which has the merit of depending upon no special hypothesis in regard to the constitu- tion of the atmosphere, being constructed upon the most general princi pies, and from known and well-asoertained data, § 76. Make, Z = Z'A S' = observed zenith distance ; r ^ S A S' = corresponding refraction ; k = height of mercurial column, which the atmosphere supports ; t = temperature of the air and of the mercury ; a = coeflBcient of atmospheric expansion for each degree of Fahr.; jS = coeflBcient of expansion for mercury, same thermometiio scale. Then, Appendix No. III., r=57".82. — ■ ],,^^~''f .tanZ. A -0.0012517 seo'^+O.OOOOOlSgy^^^^ (2) 30 l+(<— 5U)a \ cos'Z / or, omitting the last term in the parenthesis as being insignificant for or- dinary zenitk distances, '■ = ''"■'' ■ 4 -ttI^- '^" ^ • (' - "•°°''''' ""''^^ ■ ■ ^') When h = 30, and t = 50, equation ( 3 ) becomes r„ = 57".82 tan Z (I — 0.0012517 seo" Z) = A . . (4) and the results given by this formula for diflferent values for Z are called mean refractions ; and for any other state of the thermometer and barometer, 1 and taking logarithms, _ _A^ 1 + (50 - t)l3 ^~ "ao'l-f (« — 50)a' logr = log^ + log- + log^-^-^-^. . . (5) 16 SPHERICAL ASTEjNOMY. Causing Z to vary from 0° to 90°, A from 28 to 31 inches, and t from 80° to 20°, the logarithms above may be computed and tabulated for future use, under the heads Z, t, and h. § 77. Causing Z to vary from 0° to 90°, in equation (4), we may construct Table I. ; causing t to vary from 80° to 20°, and h, to vary from 31 to 28, in the last two terms of equation (5), we may construct Table n. Returning to equation (2), resuming the quantity omitted to obtain equation (3), computing their values for zenith distances, varying from 75° to 90", on the supposition that /j=30 and <=50, an additional table may be computed to correct the refractions in low altitudes. Tables I., n., and in. are due to Mr. Ivory. § 78. For zenith distances exceeding 80°, refraction becomes very uncertain ; it then no longer depends solely upon the state of the atmo- sphere, which is indicated by the barometer and thermometer, being fie- quently found to vary at the same station some 3 to 4 minutes for the same indications of these instruments. Example. — The zenith distance of an object is observed to be i\° 26' 00", the barometer standing at 29.76 in., and the thermometer at 43° Fahr. : required the refraction. Table I. Mean refraction, log. 2.23609 Table 11. Barometer 29.76 " 9.99651 Table II. Thermometer 43° " 0.00668 Hefraction 2' 53".49 . . 2.23928 Observed zenith distance . . .71° 26' 00".00 Zenith dist. cleared from refraction 71° 28' 53".49 The refraction must always be added to the observed zenith distance, or subtracted from the observed altitude, to clear an observation from^ re- fraction. PARALLELISM OF THE EARTH'S AXIS, AND UNIFORMITY OF THE EARTH'S DIURNAL MOTION. § 79. Wherever upon the earth's surface the altitudes and instru- mental azimuths of a star are taken in the various points of its diurnal course, and the instrument is turned in azimuth, so as to read the half sum of two azimuths, corresponding to any two equal altitudes, the vertical plane through the line of coUimation is found to divide the path symmet- PARALLELISM OF THE EARTH'S AXIS. 17 rically ; and this plane of symmetry for any one star will, at the aame place of observation, also be a plane of symmetry for all the stars. In other woi'ds, the diurnal paths of the stars may be divided symmetrically by auy number of planes inclined to one another through the earth's centre — a condition which can only be fulfilled for paths upon the celes- tial sphere, when these paths are circles, of which the poles coincide, and the planes of symmetry pass through them. The diurnal motions of the stai-s are only apparent, and arise from an actual motion of the spectator about the earth's axis. This latter line preserves, therefore, its direction unchanged, and, in the motion of the earth around the sun, describes a cylindrical surface, of which the elements have their vanishing point in the poles of the celestial sphere. These poles are therefore the geometric poles of the diurnal paths of the stars, and the planes of symmetry are the meridian planes of the places of observation. § 80. Again, the interval of time during which a star is moving be tween any two given altitudes on one side of the plane of symmetry, is exactly equal to that during which it. is moving between the equal alti- tudes on the opposite side, which can only be true, for all positions of the observer, when the star's apparent, or tlie eartKs real motion about Hi axis, is uniform. § 81. The period of one revolution of the earth about its axis is called a day ; the day is divided into 24 equal parts called hcmrs ; the hours into 60 equal parts called minutes ; the minutes into 60 eqn^l piar^s. called seconds, and the seconds into 60 equal parts called thirds:^ § 82. The earth rotates therefore at the rate of 36,0^24=15° an hour ; 15' of space in 1 minute of time ; 15" of spaw i?, \ seooad of time, or 15'" of space in 1 third of time. § 83. Distances on the equinoctial may theifef /, and^<90°, cos P < 0, cos P > X- 1 ; P > 90°, P > 6" ; that is, all bodies between the elevated pole and the equinoctial, will b.e longer above than beloW the horizon. li p>l, and p > 90°, ^ cos P > 0, cos P < 1, P < 90°, P < 6" ; lliat is, if the body and the spectator be on opposite sides of the plane of the equinoctial, the semi-upper arc will be less than six hours, and the body will be a shorter time above than below the horizon. If 2? = 180° — I, then will tan ^ = — tan I, and cosP=l, P=0° = 0''; that is, when the body is at a distance from the depressed pole equal to the latitude of the place, the body will never rise above the horizon, but just graze it in the meridian. If ^ < 180° — I, then will tan ^ < — tan I, and cos P> 1, , „ , ^ ;, which is impossible. That is to say, if the body's distance from the de- pressed pole be less than the spectator's latitude, the body can never rise to the horizon, and must ever remain invisible. § 88. The act of a body's pas.sing the meridian, is called its culmina- tion. A body has its greatest or least altitude at the instant of its cul- mination. The altitude of a body when on the meridian is called its meridian altitude. 20 SPHEBICAL ASTRONOMY. TERRESTRIAL -LATITUDE AND LONGITUDE. § 89. Latitude.— "mim in Eq. ( 7 ) the angle Z P S" =P=18Q'' ; then will p = l; but in this case p is the polar distance of the point of the horizon of the same name as the elevated pole, and hence the latitude of the tpectator is always equal to the altitude of the elevated pole. § 90. This suggests an easy and accurate method of getting from obser- vation both the latitude of the specta- tor's place and the polar distance of a star. Let Z be the zenith, HJT the hoii- zon, Q Q' the equinoctial, P the eleva- ted and P' the depressed pole, and S' S, the diurnal path of a circumpolar star. Make I = HP = ZQ ,the latitude, p z= P S' = P S ,the polar distance of star, a' = IIS' , the greatest observed meridian altitude of star, a/ = HS , the least observed meridian altitude of star, ' r' and r^ , the refracti9n8 corresponding to the greatest and least meridian altitudes respectively. Then from the figure will _ a' — r' + a,— r, _ a' + a, — {r' -^ r,) Z = (8) a'— r'— (a^— r,) a'—a—{r'—r1) 2 ~ 2 (9) That is to say, the latitude of the observer's place is equal to the half sum of the greatest and least meridian altitudes of a circumpolar star ; and the polar distance of the star is equal to the half difference of its greatest and least meridian altitudes. Other methods for finding the latitude will be pven in another place. § 91. Longitude. — The uniform motion of the earth about its axis fur- nishes the means of finding the longitude of the spectator's place. Twenty-four perfect time-keepers, with dial-plates graduated to 24 hours, placed upon meridians 15° apart, and so regulated as to mark 24'" at the instant any one fixed star or other point of the heavens culminates, would, FIGURE OF THE EARTH. 21 § 82, when this regulating star or point comes to any one of these me- ridians, simultaneously mark the hours indicated by the natural numbers from one to twenty-four.inclusive; that 15° ^to the east of the regulating point marking 1'', that 30° to the east marking 2'', and so on to that 345° to the east, or 1 5° to the west, marking 23''. The timepieces to the east would be later and later, those to the west earlier and earlier. The times indicated on these several timepieces are called th& local times of their re- spective meridians. § 92. If now, without altering its hands or rate of motion, a traveller were to transport the time-keeper of any one of these meridians to that on any other, and note the difference of time indicated by the two, this differ- ence would be the difference of longitude of the two meridians, expressed in time ; and multiplied by 15 would give the same in degrees. § 93. If one of these meridians be the first meridian, this differen^o would be the longitude of the other. But if neither be the first meridian, this difference applied to the longitude of one, supposed known, would give the longitude of the other. § 94. The solution of the problem of longitude consists, tlierefore, in finding the difference of the local times which exist simultaneously on the first and required meridians. The various modes of doing this will be given in another place. FIGURE AND DIMENSIONS OF THE EARTH. § 95. A fluid mass .rotating about an axis, and of which "the particles attract one another with intensities varying inversely as the square of their distances apart, will assume the form of an oblate spheroid. Its axis of rotation will be both the shortest and a principal axis of figure. Where the angular velocity is such as to make the centrifugal force of the sur- iface elements small in comparison with their weight, due to the attraction of the whole mass, the figure of the meridian section will, ([§ 265, Analyt. Mechanics^ approach that of an ellipse of small eccentricity. § 96. The centrifugal force of a body at the equator of the earth, where it is greatest, is only about -j^-gth part of its weight. Observations upon the temperature of the strata composing the earth's crust, lead to the conclusion that at no great depth below its surface its materials are in a fluid state from excessive heat ; and the researches of geology make it more than probable that there was a time when the earth was without solid matter. Its present irregularities of surface, forming mountains, bills, valleys, the bed of the ocean, of seas, lakes and rivers, are due to 23 SPHERICAL ASTRONOMY. changes subsequent to the surface induration from cooling, and as the ver- tical dimensions of these are insignificant in comparison with the depth to the centre of the entire mass, it is concluded that the figure of the earth is one of fluid equilibrium due to its rotary motion. §■ 91. Assuming the meridian section of the earth to be an ellipse, its eccentricity and semi-axes are found, Appendix No. IV., from the relations c-e' «* — i «-_3 A = e sin' l^— c' sin' l'^ .,(l-e»8in'g^ (10) (11) (12) 1 - e»^ £ = A. -/l-e' in which e = the eccentricity of the meridian ; A = semi-transverse axis = equatorial radius of the earth ; B = semi-conjugate axis = polar radius of the earth ; c and c' = the linear dimensions of the arcs of the nferidian, whoso extremities differ in latitude by 1° ; l„ and l'„ = latitudes of the middle points of the arcs c and c' respec- tively. The quantities Z„, l'„, c, e' are ibund from observation and measure- ment. A method by which / and I' may be found is explained in § 90. § 98. To find c and c', a base lino AB is carefully measured on some extended plain, and a number of stations C, J), JE, F, H, &c., are se- lected in a northerly or southerly direction, and so that C may be seen from A and B, B from B and C, E from C and 2>, and so on to the end. The several stations being connected by right lines, a network of triangles is formed ; every angle in each triangle is carefully measured, and the instrumental azimuth of its vertex, and that of the meridian, as viewed'fi'om the other two, accurately noted, (§ 85). The angles being cleared from spherical excess, the sides of the triangles are then computed, beginning of course with the triangle of which the measured base is one of the sides. The difference between the instrumental azimuths of the several vertices and those of the meridian, gives the inclina- tion of the sides to the meridian line. The product of each side into the cosine of its inclination gives the projection of this side on the meridian,' and the sum of the projections of any one of the series ol ndes, as AB, BC, C D, D E, E H, and H F, connecting the most north- FIGURE OF THE EARTH 23 erly and southerly points, will give the linear meridional distance L L\ between the parallels of .atitude through the same points. Make a = the sura of these projections, expressed in miles ; l^ = the latitude of -4, supposed the most northerly ; I, = the latitude of F, supposed the most southerly ; then, % 1,-1. : r :: a : c, whenca ' a and The same operations being repeated in a diflferent locality considerably further north or south, the values of c' and l'„ are found, and hence from equations (10), (11), and (12), the dimensions of the earth. From the ares known as the Peruvian, Indian, French, English, Hano- verian, Danish, Prussian, Bussian, and Swedish, names derived from the countries in which the arcs were mostly measured, Bessel found, e' = 0.0068468, i A= 7925,604 miles, j 2 5 == '7899,114 miles, V (13) Polar compression = 26,490 miles. ) § 99. By theMlipticity of the earth is meant the difference between its equatorial and polar radii, expressed in terms of the equatorial radius as unity. Denoting the ellipticity by S, we have ^=-^=3^Tr. nearly (14) § 100. The length of a degree of latitude, denoted by ^, in any lati- tude I, is. Appendix No. IV, equation (?), given by ^^^ (1 - e' sin' l)^ The length of a degree, measured perpendicularly to the meridian, de- noted by j8i, is. Appendix No. IV., equation (w), given by o__2*_ . ./ 1 - e' sin' I , . '^'~360°^ ^^ l-«'(2-e')sinW ■ • ' ^ > 21 SPHERICAL ASTROiSOMT. and the length of a degree of longitude, denoted by a, measured ou a par- aller of latitude in the latitude I, is, App. No. IV., equation (o), given by __ 2* cos ? » ""360- Vl'-e'sin'/ ■ ■ ■ ■ • ^ ' § 101. The close agreement between the results of these formulas and those of actual measurement, at various and numerous places on the earth, justifies in the fullest manner the assumption in regard to its ellipsoidal figure. The equatorial oirouraferenoe of the eai'th is 24,899, say, for convenience of memory, 25,000 miles. The lengths of the degrees of latitude inerease ' from the equator to the poles. In the latitude of 50* the length is- abont 70 statute miles, and contains nearly as many thousand feet as the year contains days (365), and each second is equivalent to about 100 feet. GEOCENTRIC PARALLAX § 102, The bodies of the solai' system being comparatively near to the eaith, a change in a spectator's place on the earth's surface gives to them a sensible parallactic motion on the surface of the celestial sj^ere, and two observers at remote stations would not assign to these bodies the same places at the same time without first clearing their obseiTed co-ordinates of this source of discrepancy. The mode of correction is to refer all obser- vations to one common station, and this station is assumed, for conve- nience, to be at the centre of the earth, § 103. The place in which a body would appear, if viewed from the centre of the earth, is called its Geocentric Place. § 104. The apparent change of a body's place that would arise ft-om a change of the spectator's station from the surface to the centre of the earth is called Geocentric Parallax, . § 106, The transfer of station from the surface to the centre of the earth is sensibly in a vertical circle, and the geocentric parallax is there- fore in the same plane. § 106. The co-ordinates of i a body's place, as determined by observa- tion, coirected for geocentric parallax, are the geocentric co-ordinates ol the body. § 107. The point in which the radius of the earth produced through the spectator's place pierces the celestial sphere, is called the central zenith. The arc of the celestia^ meridian from the central zenith to the equinoc- GEOCENTRIC PARALLAX. 25 tial, is called the central latitude. The diflference between the latitude and the central latitude, is called the reduction of latitude. Thus BAB' A', being a meridian section of the terrestrial spheroid, and Z Q && arc of the celestial sphere in the same plane, M the spectator's place, Q the highest point of the equinoctial, MG the direction of the plumb-line, CM the radius of the earth ; then will Z' be the central zenith, Z' Q the cen- tral latitude, and ZMZ'=Z Q—Z'Q, the reduction of latitude. § 108. Denote in future the central latitude by I, the polar radius by y, and the latitude by /', then, Appen- dix 'So. IV., equation (j), ta.iil=f.ta,nl' (18) that is, the tangent of the central latitude is equal to the tangent of the latitude into the Square of the polar radius. Denote the radius of the earth drawn to the spectator's place by p, then. Appendix No. IV., equation (r), '=-r=zr= '*" Thus, the laiitude being found from observation (§ 90), the centrai latitude becomes known from equation (18), and hence the radius of the earth drawn to the spectator's place, equation (19). § 109. Let AB'A'B be a meridian Fig.4i. section of the earth's surface, A A' the equatorial diameter, M^ and Mi the places of two observers viewing the same body S. The observer at M^ would see the body projected upon the celestial sphere at S,, that at M^ would see it projected at S^, and to an observer at the centre it would appear at S3. The points Zi and Z^ are the central zeniths of the two ^observers ; Z, Si and Z2 Sj are the central zenith distances of S, as viewed from Mi and M^ respec- tively. The first diminished by *S^3 Si and the second by S3 S,, will ^ve 26 SPHERICAL ASTKONOMY. Zi S3 and Z, S, the central zenith dis- tances as they would appear from the centre. But S3 Si measures the angle SiS Si = MiS C, and 5, S3 the angle S3 S S3 = M'2 S C ; so that the angles M, S Cand M^SC are the corrections for parallax. § 110. Tlh^ parallax of a body is the angle at the body subtended by the earth's radius drawn to the spectator. When the body is above the horizon, or is in altitude, it is called the parallax in altitude. in the horizon, it is called the liorisontal parallax. Fig. 41 bla. When the body is Si.= Ml S C= parallax in altitude at J/l ; z,=:MiSC= paraUax in altitude at M3 ; Pi = horizontal parallax at Mi ; P, = horizontal parallax at M, ; r=: OS = distance of the body from the earth's centre ; Pi = Ml C = radius of the earth for Mi ; = radius of the earth for M3 ; = central zenith distance at M, ; = central zenith distance at M^ : l, = ZiCA= central latitude oi Mi ; Zj = Z3'0A= central latitude of M3. f3 = M,C Zi = Z, Si Z, = Zj S3 Then, in the triangle Jf, S O, whence sin Zi : sin Z, : : p, : r ; sin 2, = -^ . sin Z, ; r But because s, is always very small, we may write sin 8| = — ; u in which u is the number of second^ in radius, and g, is ■ame unit ; which substituted above gives expressed in the 2, = w . -!-l . sin Zi ; GEOCENTRIC PARALLAX. 27 when Zi becomes 90°, the body is in the horizon, and g, becomes P„ and we have -P' = "-7 (20) and this above gives «, = P, . sin Z, (21) Whence the parallax in altitude is equal to the horizontal parallax into the sine of the central zenith distance. ♦ § 111. If the observer be upon the" equator, then will p, become unity, Pi becomes the horizontal parallax on the equator, called the equatorial horizontal parallax ; designating this latter by P, we have, equation (20), and this in equation (20) gives i'. = i'.p. (22) that is- to say, the horizontal parallax of a body at any place, is equal to the product of the equatorial horizontal parallax of the body by the ra- dius of the earth at the place. The value of /"i in equation (21) gives g, = P p, . sin Z, (23) § 112. To find the equatorial horizontal parallax of any body, we have in the triangles Mi SO and M^SC Zi = P Pi sin Z, Z2= P f2 sin Z, adding but Si + 2i = -P (pi sin Z, + Pa sin Z,), 2, = Z,- ZtOS, Zg = ^2 — ^1 o 5 by addition g, +2, =Z, + Zj - (Z, OS + Z, CS)=z Zi+Z,- (Z, + 1,), which substituting above, and dividing by the coeflBcient of P, gives P _ ■^i + ■^i ~ (^1 + (24) pi sin Z| + p, sin Zj § 113. If the body be so remote that the difference between the radii of the earth, as viewed from it, be insignificant, which is the case with all 28 SPHERICAL ASTRONOMY. bodies except the moon, p, and p, may be regarded as equal to one an- other, and each equal to unity, and we shall have, equation (24), p__ Z,+Z3—{l, + 1,) -^g. sin Z| + sin Zj in which I, and Zj are the latitudes of the places M, and M, respectively. § 114. In all this the observers have been supposed to be on the same meridian ; but this is not necessary, nor would it, in general, be the case in practice. If on different meridians, make S — change of meridian zenith distance of the body in the interval between two consecutive culminations ; X = difference of longitude of the two observers, expressed in time ; J' = change in meridian zenith distance while passing from the first to the second meridian ; then 24^ : S : : \ : S' • whence X.S ^=^r^ (26) If the meridian zenith distance be increasing at the easterly station, S' is to be added to, if decreasing, subtracted from, the meridian zenith dis- tance at that station. This corrected meridian zenith distance will be that which the body would have to an obsei-ver on the meridian of the westerly station, and on the same parallel of latitude with the obseiTer on the easterly meridian, the reduction being in effect to bring the ob- servers to the same meridian. § 115. To recapitulate: the latitudes of two stations are first found from observation ; the central latitudes are found from equation {18); the radii of the earth at tlie two stations, from equation (19) ; the equatorial horizontal parallax, from equation (24) ; the horizontal parallax at any place, from equation (22) ; and the parallax in altitude, from equation (23). AUGMENTKD AND HORIZONTAL DIAMETERS. § 116. By the rotation of the earth upon its axis the spectator is con- tinually changing his distance from the heavenly bodies. A change of dis- tance gives rise to a change in the apparent dimensions of an object. A body seen in the horizon of a spectator would appear to him sensibly of the same size as if seen from the centre of the earth, the distances JI C and HM, for the nearest of the heavenly bodies, being sensibly the same. DIMENSIONS OF THE HEAVENLY BODIES. 29 The apparent semi-diameter of a body, is the angle at the observer subtended by the body's real semi-diameter, the latter being perpendicular to a visual ray drawn to one of its extemities. Make « = HMB = apparent semi-diameter of a body when in the horizon ; «' = LMB'z= apparent semi-diameter of the body when in altitude ; d = HB = i ^' = real semi-diameter of the body in linear units ; r = body's distance from the observer when in the horizon = distance from earth's centre ; r '= body's distance from the spectator when in altitude ; ^= Z' ML = the body's apparent zenith distance; z — ML 6'= the body's parallax in altitude. Then r . sin s = (f =: r' sin *' ; whence sin s' r sin ^ 1 sin s »•' C-^-^) COS 2 ' cos Z sin Z ■ sin 2 replacing sin z by its value — , and z by its value in Eq. (23), also writing - for sin s, and - for sin «', we find u w ^ 1 cos« — Pl-- ■cos Z (27) in which s' and s are expressed in seconds of arc. § 117. The apparent diameter 2 « of aTjody in the hoiizon, is called the horizontal diameter ; its apparent diameter 2 s' in altitude, is called the augmented diameter. DISTANCES AND DIMENSIONS OF THE HEAVENLY BODIES. § 1 18. Having found the horizontal parallax of a body, it is easy to find its distance from the earth's centre. From equation (20) we have P'-P, P-p (28) in which p and F are respectively the equatorial radius of the earth and 30 SPHERICAL ASTRONOMY. the equatorial horizontal parallax of the body ; and from which we con- clude that the distance of any body from the earth's centre, is equal to the equatorial radius of the earth repeated as many times as the number of seconds in the body's .equatorial horizontal parallax is contained in the number of seconds in radius. § 119. The horizontal parallax of a body is the apparent semi-diameter of the earth as seen from the body. The apparent semi-diameter of two Bbdies seen at the same distance are directly proportional to their real magnitudes. Make a = apparent semi-diameter of the body ; d = the real semi-diameter of the body in linear units, as miles t P=: the equatorial horizontal parallax of the body ; p = the equatorial radius of the earth ; then will F : s : : p : d; whence „, the greatest north and south declinations of sun respectively, and P^ I, and P^^^ I^^^ the greatest departure of the pole from the circle of illumination. Now I. i>, = p, e, = 90= = i,„ D,„ = p,„ e,,„ P.i>,=p.B, p,„D,„=p,„i)„r. and by subtraction that is, the radius of the zone of greatest polar illumination, or obscuration, is equal to the greatest declination of the sun. § 133. Two small circles parallel to the equinoctial, and at- a distance from the poles equal to the greatest declination of the sun, are called ^ofor circles ; that about the north pole is called the arctic, and that about the south the antarctic circle. The polar circles are the boundaries of the greatest zones of polar diurnal illumination and obscuration. § 134. When the intervals of time between three consecutive passages of a circumpolar star over the line of collimation of a transit or mural cir- cle are equal, these instruments are adjusted to the meridian. § 135. The diurnal motion brings the meridian of a place, in the couree of one revolution of the earth on its axis, into coincidence with the decli- nation circle of every body in the heavens. The difference of times between the meridian's passing the centres of any two bodies, is the difference of right ascension of these bodies. § 136. To find the time of the meridian's passing the centre of any body, find by the transit instrument and timepiece the time of, thi; merid- ian's passing the body's east and west limb, and take half the sum. u SPHERICAL ASTRONOMY. Fig. 44. § 137. To find the polar distance of a body's centre, take the reading of the mural circle when its line of collimation is upon the upper or lower limb; subtract from this the polar reading and correct the difference for refraction, parallax in altitude, and semi-diameter. The declination is ob- tained by subtracting the polar distance from 90*. § 138. The points in which the equinoctial intersects the ecliptic are called the equinoxes ; that by which the sun passes from the south to the north of the equinoctial is called the vernal equinox ; the other, or that by which the sun passes from the north to the south of the equinoctial, is called the autumnal equinox. § 139. The angle which the equinoctial makes-with the ■ecliptic is called the obliquity of the ecliptic. § 140. To find the place of the vernal equi- nox and the obliquity of the ecliptic, let VD^ be an arc of the equinoctial, VS^ of the eclip tic, V the vernal equinox, (S, and Si two places of the sun when on the meridian at different tiijaes, SxDj, S^D^ arcs of declina- tion circles ; and make ^1 = Di Si, the sun's declination at any meridian passage ; Si = Z>2 ^2, the same at some subsequent passage ; 2a = FZ>j — VD,, the corresponding difference of right ascension ; X = VDi, the right ascension of the sun at the time of fii-st meririan passage ; u = Sj VDi, the obliquity of the ecliptic. Then in the triangles 5, VDi and Si VD^, right-angled at i), and D,, sin X = tan 5, , cot u, sin {x -f- 2a) = tan 5, , cot u ; and by division sin {x -I- 2a) _ tan 5, sin x ~ tan S, ■STO. (30) adding unity and clearing the fraction, then subtracting unity and clear ing, and dividing one result by the other, we find sin (a; + 2a) + sin x tan S, + tan S, sin (x + 2a) — sin a; ~ tan ^'j — tan 5, ' tan(a!-t-a) sin (Jj -f- 5,) tan a sin (5, — 5,)" ECLIPTIC. 35 WLence . tan(:B4-a)='-!^^.tana . . . . (31) ^ ' sm ((5j — Si) ^ ' Also cot u = sin « . cot 5 (32) The value of the obliquity is thus found to be nearly 23° 27' 54", which is therefore the greatest north and south declination of the sun. The tropics are, therefore, 23° 21' 54" from the equinoctial, and the polar circles are at the same distance from the poles. § 141. The interval of time between the sun and a star crossing tlie meridian, applied to the right ascension of the sun, gives the right ascen- hion of the star. The declination of a star is found like that of the sun, except that there is no correction for parallax and semirdiameter, the only correction being for refraction. § 142. The right ascension and declination of one star being known, the differences of observed right ascensions and declinations, the latter being corrected for differences of refractions, give, when applied to the right as- cension and declination of the known star, the right ascension and decli- nation of other stars. Thus a list of the stars, together with their right ascensions and declinations, and arranged in the order of their right ascen- sions, furnishes the ground-work of what is called a catalogue of .stars, of which a fuller account will be given presently. § 143. A belt of the heavens extending on either side of the ecliptic, far enough fo embrace the paths of the planets, is called the zodiac. § 144. The ecliptic is divided into twelve equal parts, called signs. They commence at the vernal equinox, and are named in order, proceed- ing towards the east, Aries (f), Taurus (8), Gemini (n), Cancer (as), ZfCO (^), VirffO (^), Libra (===), Scorpio ('"l), Sagittarius (*), Capricor- nus (V?), Aquarius (~), and Pisces ()£). Motion in the order'of the signs is said to be direct ; the converse, retrograde. § 145. The points of the ecliptic in which the sun reaches his greatest north and south declination are called the solstitial points : that on thj| north is called the summer solstice, and that on the south the winter sol- stice. The sun when in these points appears to be stationary as regards tis apparent motion in declination. The solstitial colure is the declination circle through the solstitial points. The equinoctial colure is the declina- tion circle through the equinoctial points. The solstitial colure separates Gemini from Cancer, and Sagittarius from Capricornus; the equinoctial colure separates Aries from Pisces, and Virgo from Libra. § 146. A great circle of the celestial sphere passing through the poles of tbe p''''o*io is called a circle of latitude. 36 SPHEEICAL ASTEONOMT. § 147. The latitude of a body is the distance of the boay's centre from the ecliptic, measured on a circle of latitude. § 148. The longitude of a body is the distance from the vernal equinox to the circle of latitude through the body's centre, measured on the eclip- tic in the order of the signs. The longitude and latitude are co-ordinates that refer a body's place to the circle of latitude through the vernal equinox and to the ecliptic ; the longitude and ecliptic polar distance are polar co-ordinates that refer a body's place to the same circle of latitude and to the pole of the ecliptic- § 149. The longitude of the sun, as seen from the earth, is readilyob- tained from the obliquity of the ecliptic and either the right ascension or declination. For this purpose make u. = VSi, the longitude of the sun ; S — SiBi, his declination; a = VDi, his right ascension ; u = SiVDi, the obliquity of the ecliptic. g. 44 bis. Then will tan a = ■ sm « = tan a cos u sin 8 (33) (34) § 150. The place of the sun as seen from the earth, and that of the earth as seen from the sun, are at the opposite extremities of the same di- ameter of the ecliptic; and the longitude of the sun, increased by 180°, •will be the longitude of the earth as viewed from the sun, the centre of the earth's orbital motion. § 151. The sun appears in the vernal equinox on the 20th March, in the autumnal equinox on the 22d September, the summer solstice on the 21st June, and in the winter solstice on the 21st December. The poles of the ecliptic are at a distance from the nearest poles of the .equinoctial, equal to the obliquity of the ecliptic. § 152. The right ascension is obtained from observation by means of the clock and transit instrument, the declination by means of the mural drcle. From these and the obliquity of the ecliptic, the longitude and latitude are obtained from computation. Thus, let S be the body's place, V the vernal equinox, VB the body's right ascension, JDS its declina- PRECESSION AND NUTATION. 37 tion, VL its longitude, SL\\& latitude, and the *''B' <*■ angle L VD the obliquity of the ecliptic. A Make /V \ a = VB = right ascension ; "TzS^.^ \fi 6 = DS = declination ; / ^k \ l=VL= longitude ; ^^~~~""-^ a, whence .0 o , 1 — sin' a 1 — sin'' ip = tatf X . :-5 , ^ sin' a and solving with respect to sin a, tan X tan X &1U I* ^ — — — ■ ^ V 1 + tan' X — sin' ip V sec' X — sin' 9 and therefore, sin X sin a = — ; VI — cos' X. sin' 9 and this in equation (65) gives 20".246 sin X . - , (66) V 1 — cos' X . sin' 9 which is the polar equation of an ellipse, the pole being at the centre. 80 that, if the image of a fixed star were kept constantly on the cross wires of a telescope during one entire revolution of the earth in its orbit, the line of coUimation would trace upon the celestial sphere an ellipse of which the star would occupy the centre; the semi-transverse axis would l>e 20".246 and the eccentricity cos X. If the star were in the plane of the ecliptic, then would X = 0, cos X = 1, and the orbit would become a right line equal in length to 40".492. If the star were at either pole of the ecliptic, then would X = 90, cos X = 0, and the orbit would be a circle. Between these limits the eccentricity will vary from 1 to 0. The coeflScieM 20".246 is called the constant of aberration. § 219. Since the abeiTation is in the arc -4 5, its projection on AO will be the aberration in latitude. Denoting the latter by r', we hav€i HELIOCENTRIC PARALLAX. 53 20".246 . sin X . cos (p r' = —, ^ (67) V 1 - cos^ X . siu^ (p ^ -^ which is obviously the greatest when

, and if /, denote the temperature which would result at the unit's distance ' from the sun, and r the radius vector of the earth, we have frosd the law of difiiision, depending upon distance, whence /=J.cos0 (73) § 235. Kesuming Eq. ( 6 ), and making p = 90° — d, in which d de- notes the sun's declination, we have cos s = sin Z . sin d -f- cos I . cos d . cos P . . . ('74) which, in Eq. (73), gives / = -J . [sin I . sind + cos I . cos d . cos P] . . (75) This result is wholly inaependent of terrestrial longitude, and is only de- pendent on the latitude of the place, the sun's declina^on, and the place of the earth in its orbit. All places upon the same parallel are equally exposed, therefore, to the solar influence, and whatever difierences of mean temperature and of climate they may exhibit are due to local causes, such as the vicinity of mountains, extended plains, forests, deserts, or large bodies of water, upon all of which the sun is known to produce great va- riety of thermal effects. § 236. Making « = 90°, in Eq. (74), we have cos P = — tan f . tan rf (76) aid making P = 0, in Eq. (75), we have I=^cos.{l-d) (77) Eq. (76) gives the value of ihe semi-upper diurnal arc, or the time the sun is above the horizon, or the duration of calorific action ; and Eq. (77) the intensity of the solar influence when greatest. 58 SPHERICAL ASTEONOMY. § 237. In the course of the tropical year the declination varies nearly 47°, the sun being at one time about 23°.6 north, and at another about the same distance south of the equator. As long as the latitude and declination are of the same name, that is, _ both north or both south, the sun Tvill, Eq. (76), be longer than twelve hours above the horizon, and the place will receive more heat than it loses. And in proportion as the latitude and declination approach to equality, the intensity of the solar action will, Eq. (77), approach its maximum. This periodical variation in the daily average temperature of a place, caused by a change of the sun's declination, gives rise to the phenomena of the seasons. § 238. The interval of time during which the daily increment of tem- perature of a place is increasing is called its spring ; that during which this increment is decreasing is called its summer ; that during which the daily decrement is increasing is called its autumn or fall ; and that during which this decrement is decreasing is called its winter. § 239. Within the tropics CO' and J)D', and especially about the equator Q Q', the temperature is, Eqs. (76) and (77), nearly uniform, and always high, On this account the terrestrial belt bounded by the tropics is called the torrid zone. Between the tropica- and polar cir- cles A A' and JB3' the average daily temperature is much less uniform and always lower than in the torrid zone. The belts bounded by the tropics and polar circles are called temperate zones. Between the poles F and P' and polar circles, the variation of the av- erage daily temperature is the greatest possible and the temperature itself least. The portions t)f the earth's surface about the poles and bounded by the polar circles are called frigid zones. § 240. Places within the torrid zone may be said to have two of each of the seasons during a tropical year, and all places in the temperate and frigid zones but one. For all places in the north temperate and frigid zones, spring begins when the sun is on the equator and passing from south to north, or on the 20th March ; summer, when the sun reaches the tropic of Cancer, or on the 21st June ; autumn, when the sun returns to the equator in passing to TRADE WINDS. 59 the south, or 22d September; and winter, when the sun reaches the tropic of Capncoin, or 21st December. For all places in the south temperate and frigid zones the names of the seasons will be reversed — spring becomes autumn, and summer winter. § 241. The elliptic form of the earth's orbit causes the radius vector, and therefore, Eq. (77), the intensity of the solar heat, to vary. But the angular velocity of the earth about the sun also varies, and according to the same law, viz. : that of the inverse square of the earth's distance from the sun — Analytical Mechanics, Eq. (266). Equal amounts of heat will therefore be developed while the earth is describing equal arcs of longitude, and the supply will be the same during the description of any two seg- ments, equal or unequal, into which the entire orbit is divided by a line through the sun. The earth is nearer the sun while the latter is south of the equinoctial, or from the latter part of September to the latter part of March ; and it describes the corresponding part of its orbit in a time so much shortened as just to balance the increase of thermal intensity. But for this law of compensation, the effect would be to increase the difference of summer and winter temperature in the southern and to diminish it in the northern hemisphere. As it is, however, no such inequality is found to subsist, but an equal and impartial distribution of heat and light is ac- corded to both hemispheres. § 242. But it must not be inferred that the mean surface heat is con- stant throughout the year; for such is not the fact. 'By taking, at all sea- sons, the mean of the temperatures of places diametrically opposite to one another. Professor Dove finds the mean temperature of the whole earth's surface in June considerably greater than that in December.- This is due to the greater amount of land in that hemisphere which has its summer solstice' in June ; the thermal effect of the sun on land being greater than that on water. § 243. The variation of the radius vector amounts to about ^ of its mean value, and therefore the fluctuation of heat intensity to about -^j of its average measure — a circumstance which is manifested in a great excess of local heat in the interior of Australia during a southern, over that of the deserts of Africa during a northern summer. TKADE WINDS. § 244. A discussion of the trade winds, the eartKs magnetism, and the tides, belongs, in strictness, rather to terrestrial physics than to astronomy ; but the necessary connection of these phenomena with the earth's diurnal 60 SPHEEIOAL ASTRO'NOMT. rotation and the action of foreign bodies upon the earth, as well as their importance to navigation, make a sufiBcient apology for introducing them here. § 245. The surface of the torrid zone is most heated ; its excess of temperature is communicated to the superincumbent atmosphere; the latter is expanded, and becoming specifically lighter, is pressed upward by the colder portions on the north and south which move in and take its place. These, in their turn, are heated, expanded, and pressed upward, and a constantly ascending current is thus produced over an entire zone, of which the boundaries fluctuate with the varying declination of the sun and the proportion of land and water on the belt of the earth's crust lying immediately under the sun's diurnal path. The air thus accumu- lated at the summit of the ascending column, being unsupported on the north and south, flows oflf under the action of its own weight in either di- rection towards the poles, and, after cooling, descends again to the earth's surface in the higher latitudes of the temperate zones to supply the place and follow the course of that which has passed to the torrid zone. § 246. Two atmospheric rings, as it were, distinguished by peculiarities of internal circulation, are thus made to belt the earth on either side of the equator in directions paral- lel or nearly so to that great circle. On the lower side of these rings, in contact with the earth, the air moves towards the base of the ascending col- umn, and on the upper towards the poles. § 247. By the diurnal mo- tion of the earth, places on the equator have the greatest velo- city of rotation, and all other places less in the proportion of the radii of their respective parallels of latitude. The portions of the ascending column which flow towards the poles set out with the east- ward intertropical velocity, which they carry with them in part to the higher latitudes, where they descend to the earth's surface. To an ob- served situated in these latitudes, the air will have an apparent east- wardly motion, approaching to the excess of the intertropical velocity over that of the observer's parallel. Here westerly winds prevail. § 248. On parallels a few degrees lower, the tendency of the air is TRADE WINDS 61 towards the equator, and this combined with what remains of the apparent easterly component, just referred to, gives rise in the north- ern hemisphere to a northwesterly and in the southern to a southwesterly wind. § 249. In its onward course towards the equator, this same air crosses successively parallels of greater and greater velocity, and this, together with friction against the earth's surface, reduces the air's excess of easterly motion to zero, and here northerly winds prevail in the northern and southerly winds in the southern hemisphere. § 250. In latitu les still lower, the excess of rotation is in favor of the earth's sutface, and the air, unable to keep up, now lags behind, and ap- parently tends to the west ; and here, if the places be in the northern hemisphere, northeasterly, and if in the southern hemisphere southeasterly winds prevail. § 251. Nearer to the equator the radii of the parallels vary less rap- idljr, and the velocities of places on the same meridian are more nearly equal. In crossing these parallels the air in its onward course finds less variation in the velocity of the earth's surface, and friction, which now urges the air to the east, together with the easterly pressure below, arising from the westerly lagging in the summit of the ascending column, due to its decreasing angular motion as it recedes from the centre of rotation, soon brings the air and earth to relative rest. This occurs within the base of the ascending column where the currents of air, which are continually approaching each other from the directions of the poles, meet. This is, therefore, a region of calms. § 252. The aerial currents thus produced under the combined influence of solar heat and the diurnal motion of the earth, are called Trade winds ; and they are so called from the benefits they are continually conferring on trade dependent upon navigation. § 253. A voyage^ from the United States to northern Europe in a sailing vessel is on an average ten. days shorter than in the contrary direc- tion.' A sailing vessel on a passage from northern Europe to the southern coast of the United States would proceed to the Madeiras to take the east- erly trades, and returning would proceed to the Bermudas to catch west- erly trades. § 254. Within the region of calms the ascending column of air car- ries with it a large amount of aqueous vapor. In its ascent the air expands, its temperature is depressed, its aqueous vapor is first condensed into clouds, then into rain, and thus the region of calms is also a region of dense clouds and copious rains ; the former giving to the earth, as viewed from 62 SPHERICAL ASTRONOMY. a distance, the appearance of being girted by dark broken belts, arranged in zones parallel to the equator. § 255. The limits of the trades do not always occur in the same lati- tudes, but vary with the season. In December and January, when the sun is furthest south, the northern boundary of the northeast trades of the Atlantic is about 20° N"., whilst in the opposite season, from June to Sep- tember, it is 32° N. § 256. Owing to the great disparity in the effects of solar heat upon land and water, and to the influence of mountain ranges and valleys upon atmospheric currents, the regular trades only occur, as a general I'ule, at sea, though in some level countries, within or near the tropics, constant easterly winds prevail. This is remarkably the case over the vast plains drained by the Amazon and lower Orinoco. § 257. The trades of the ocean and of the land are separated by a belt, within which other and variable winds occur. This belt lies upoi\ the ocean, and extends along the coasts. When to the east of the trades, it is often a hundred miles wide, but when to the west its width is much smaller. The interruption of the trades, here referred to, is due to the difterence of temperature of the air on sea and land, which changes with the seasons. The air over the land in the higher latitudes is the warmer when the meridian zenith distance of the sun is least, and colder when greatest. During the first period the wind is from the sea to the Jand, and in the second from the land to the sea, thus giving rise to the period- ical winds called Monsoons, which occur even within the limits of the trades. A large island thus circumstanced is surrounded by a wind blow- ing from aH quarters at the same time. § 258. A similar difference of temperature, but which varies with the alternations of day and night, gives rise to what are called the sea and land breezes. TERRESTRIAL MAGNETISM. § 259. Another most important effect from the solar heat, combined with the diurnal motion of the earth, is the earth's viagnetism. § 260j a difference of temperature in different parts of any body form- ing a continuous circuit is ever accompanied by electrical waves, propa- gated from the hotter to the colder parts. If the circuit be composed of various materials, possessing different powers of conducting heat, this diffei-- ence may be maintained in greater degree and duration, and the effects of the electrical flow rendered more strikingly manifest. TERRESTRIAL MAGNETISM. 63 § 261. When the source of heat is moved gradually along the circuit, the electrical flow is in the direction of this motion, the colder portions always lying in advance and the wariner behind the moving source. § 262. A compass-needle, brought within the influence of such a cir- cuit, will arrange itself at right angles to the direction of the flow, and under the same circumstances the same end of the needle will always point in the same direction. All this is the result of observation and ex- periment. § 263. The earth's crust is one vast thermo-electrical circuit, and its source of heat is the sun. § 264. In the diurnal motion of the earth, the diflferent portions of its tropical regions are heated in succession by the sun during the day, and cooled by radiation during the succeeding night. The hotter portions will therefore lie to the east and the colder to the west of the sun's place. A perpetual flow of electricity is thus developed and maintained in and about the earth's crust from east to west, and gives rise to the earth's magnetic action. § 265. Were the materials of the earth all equally good electrical con- ductors, and the sun always in the equinoctial, the electrical flow would be parallel to that great circle, and the compass-needle would always point directly north and south. But neither of these conditions obtains. The materials vary greatly in conducting power, and the sun's declination is ever changing. § 266. The disparity of conducting power directs the electrical flow in paths of double curvature, of which the general direction is parallel to the equator, and the varying declinations of the sun are perpetually shift- ing their precise location and shape as well as changing the intensity of the flow. § 267. The position of stable equilibrium, assumed tiy a magnetic nee- dle reduced to its axis, freely suspended from its centre of gravity, and sub- jected alone to the directive action of the earth's magnetism, is called the magnetic position of the place. § 268. The intersection by a vertical plane through the magnetic posi- tion with the celestial sphere, is called the rnagnetic meridian. § 269. The angle made by the magnetic and the true meridian is called the magnetic declination, or simply declination. § 270. The inchnation of the magnetic position to the horizon is called the magnetic inclination or dip. § 271. The magnetic position at the same place is continually varying ■ It describes daily a conical surface, of which the place is the vertex, and 64 SPHERICAL ASTRONOMY. daily mean position the axis, while this axis itself describes a similar sur- face once a year about an annual mean position. § 272. The mean of all the declinations and of dips throughout any one day are the declination and dip for that day, and are called the diurnal declination and dip. The mean of all the diurnal declinations and dips for the different days throughout any given year, are the decli- nation and dip for that year, and are called the annual declination and dip. § 273. The daily and annual fluctuations here referred to are called periodic changes. The annual declination and dip also change, and these changes, which are found to take place in the same direction for a great many years, are called secular changes. § 274. The magnetic declination and dip vary, in general, with the locality. The line connecting those places where the declination is zero, is called the line of no declination ; and the line through the placeE where the dip is zero, is called the magnetic equator. .^M Fig. 5T. ^^^ m>- ^ m \ \\~ __j^kJ_^ ^X-^"^ ^^K^p \vVr 17 \K ~/^^y ^^ y \ -^y § 27.5. Accoiding to the Magnetic Atlas of Eansteen, constructed for 1787, the line of no declination is found on the parallel of 60° north, a little to the west of Hudson's Bay ; it proceeds in a southeasterly direc- tion, through British America, the northwestern lakes, the United States, and enters the Atlantic Ocean near Chesapeake Bay, passes near the An- tilles and Ca^e St. Koque, and continues on through the southern Atlantic till it cuts the meridian of Greenwich in south latitude 65°. It reappears in latitude 60° south, below New Holland, crosses that island through its centre, runs up through the Indian Archipelago with a double sinuosity, and crosses the equator three times — first to the east of Bonieo, then be- tween Sumatra and Borneo, and again south of Ceylon, from which it passes to the east through the Yellow Sea. It then stretches across the TifiEESTRIAL MAGNETISM. I 66 coast of China, making a semicircular sweep to the west till it reaches the parallel of 71° north, when it descends again to the south, and re- turns northward with a great semicircular bend, which terminates in the White Sea. On the magnetic chart this line is accompanied through all its windings by other lines upon which the declination is 5°, 10°, 15°, &c. ; the latter becoming more irregular as they recede from the line of no declination. The use of these lines is to point out to navigators sailing by compass, the boaiing of the true meridian from the magnetic. § 276. On the east of the American and west of the Asiatic branch of the line of no declination, the declination is west, while to the west of the American and east of the Asiatic branch the declinaticfti is east. § 277. The magnetic equator cuts the terrestrial equator, according to Hansteen, in four, and to Morlet in two points, called nodes, one of which is in the centre of Africa. § 278. Beginning at the African node the magnetic equator advances rapidly to the north, and quits Africa a little south of Cape Guardafui, and attains its gi'eatest north latitude, 12°, in 62° of east longitude from Green- wich. Between this meridian and 174° east, "the magnetic is constantly to the north of the terrestrial equator. It cuts the Indian peninsula a little to the north of Cape Comorin, traverses the Gulf of Bengal, making a slight advance to the terrestrial equator, from which it is only 8° distant at its entrance into the Gulf of Siaiji, It here turns again a little to the north, almost touches the north point of Borneo, traverses the straits be- tween the Philippines and the isle of Mindanao, and on the meridian of Naigion it again reaches the north latitude of 9°. From this point it traverses the archipelago of the Caroline Islands, and descends rapidly to the terrestrial equator, which it cuts, according to Morlet in 174°, and according to Hansteen in 187° east longitude. Its next point of contact with the equator is in west longitude 120°. Here, according to Morlet, it does not pass into the northern hemisphere, but bends again to the south, while Hansteen' makes it cross to the north, and continue there for a dis- tance of 1 5 °«of longitude, and then return southward and enter the south- ern hemisphere in longitude 108° west, or 23° from the west coast of America. Between this point and its intersection with the terrestrial equator in Africa, the magnetic equator lies wholly in the southern hemi- sphere, its greatest southern latitude being about 25°. § 279. The dip increases as the needle recedes on either side from the magnetic equator, the end of the needle which was uppermost in the northern being lowermost in the southern hemisphere. 66 SPHEEICAL ASTRONOMY. § 280. The points at which the magaetic needle is vertical are called the magnetic poles. Of these there are four, two in each hemisphere, their positions being indicated on the magnetic charts. § 281. On the magnetic charts, the magnetic equator is accom panied by curves of equal Jp as in the case of the lines of equal decli- nation. § 282. The line of no declination and the nodes of the magnetic equa- tor are found to have a slow westerly motion, thus causing the differ- ent lines of equal declination and dip to pass successively through the same place, and illustrating the utter worthlessness of all maps constructed from compass bearings unless the 'diurnal declinations of the needle are carefully ascertained and recorded thereon. § 283. The intensity of the earth's magnetic action increases with the proximity of the electrical paths to the needle and with the difference of temperature in their different parts ; and from changes in these, produced by the varying zenith distance of the sun during the day, and of his me- ridian zenith distance throughout the year, arise the daily and annual mutations of declination and dip ; while to changes of the earth's crust, produced by geological causes, and increased cultivation of the soil from the spread of civilization, are to be attributed the secular variations of the same elements. TEDES. § 284. Those periodical elevations and de- pressions of the ocean by which its waters are made to flow back and forth through, the estuaries that indent our coasts, are called Tides. § 285. Perpetual change in the weight of the waters of the ocean, due to the attraction of the sun and laoon upon the earth, and the diurnal rotation of the latter about its axis, cause and mairtvn the tides. § 286. Let ^C-B2> be a great circle of the earth, in a plane through the sun's centre at S. Draw SS through the earth's centre at E, and CD through the same point, and at right angles to S jE. Assume any unit of Tuass as that at & ; join G and S, and make TIDES. 67 d = S E = distance of sun from the e^rth ; f=:.EQ = radius of the earth ; z= SG = distance of G from sun ; (^ ■=■ AE G = angular distance of G from sun ; ^ = G SE= angle at sun subtended by radius p ; m = mass of sun ; k = the attraction of unit of mass at unit's distance. Then, since the attraction on unit of mass is proportional to the attract- mg mass directly, and the square of the distance inversely, the sun's action on G will be km or because ^ = d' + p' — 2d f COS (p, km d^ + p' — 2d p cos 9* But each unit of the earth's mass is acted upon by a centrifugal force equal and contrary to the centripetal force impressed upon the unit of mass to deflect it from its tangential into its orbital path. This latter is, by maldng p = 0, in the above km^ and applying this to ff in the direction Gil parallel to S E, we have all the action on G arising from the sun's attraction. Eesolving these forces into their components in the direction of the radius E G, and perpendicular thereto ; also making V = resultant of the components in direction of the radius, T= " " " " of tangent, and regarding the components which act towards the centre as positive and the contrary negative ; also the tangential components which act in the direction AG OB Aas positive and the contrary negative, we liave V = -jr- cos (p — ■ a -J . cos (

. . . . (80) Again, omitting i, in the last factor of equation CTQ), reducing to a common denominator and neglecting the terms of -which 4 is a factor, we have, after replacing cos (p . sin 9 by ^ sin 2 9, km p . , . '^= - -^ . sm 2 9 (81) § 287. Making 9 = 0° and 9 = 180° in equation (80), we have the effect on the waters at A and at £ ; and in both cases 2 km .p Again, making 9 = 90° and 9 = 270°, we have the effect on the wa- ters at Cand J) ; and in both cases km p v = — rr^. The values of v at .4 and B being negative and those at C and Z> posi- tive, and these being connected by a law of continuity, through equation (80), the effect of the sun's attraction is to increas6 the weight of the unit of mass, or, what is the same thing, the specific gravities of all bodies gradually, in both directioHs, from Atn G and JD, and to diminish them TIDES. 69 in like manner from C and D to B. Fis- 59. And this being true of all sections of ^a!'^^^^^^^T^^^^^<^ the earth through its centre and the y^ ^^v sun, the waters of the ocean on and /f \v , y/ \ near the circumference of a section /• \ 0/ A through ths earth's centre, and perpen- J/ \/ ;| dicular to these, will, by the principles U /\ %^ of hydraulics, press up those about A \ /oi \ jj and B till their increased height shall V / N^ j/ compensatef for their diminished speci- •^N^ 3*3' fie gravity, or till the weights of the ^*===*r=*'*^ , balancing columns become equal; so that the ocean surface will tend to assume, as its form of equilibrium, that of an oblongated ellipsoid, of which the longer axis is directed towards the sun. The difference of the longer and shorter semi-axes of this ellip- soid is about 23 inches. § 288. If the earth had no diurnal rotation about its axis, this ellipsoid of equilibrium would bd formed, and all would be permanent. But the earth's diurnal and orbital motion, together with the inertia of water, leave no sufficient time for this spheroid to be fully formed. Before the waters can take their level, these motions carry the line connecting the earth and sun westwardly, and the place of the vertex of the spheroid of equilibrium in the same direction, thus leaving that of the actual spheroid to the east of the sun, and forcing the ocean to be ever seeking a new bearing. The effect is to produce an immensely broad and excessively flat wave, which follows or endeavors to follow the apparent diurnal motion of the sun, and completes an entire circuit of the earth once in twenty-four solar hours, thus producing a rise and fall of the ocean level twice within this period on every meridian. § 289. The rising water is called the jlood, the felling the ehh tides, and the general swell of the ocean is called the primitive tide-wave, § 290. In the open ocean, where the water is deep, and therefore per- mits the free transmission of pressure from one remote point to another, the motion is one of oscillation in a vertical direction principally. But where the tide-wave approaches shoals, such as those along the coasts and the beds of estuaries, which intercept the free transmission of pressure, the water becomes piled up, as it were, on the side of the open ocean, without being able to press up any thing to its support on the land side. It there- fore flows inland, anj". produces what are called derivative flood tides. After the apex of the tide-wave has passed onward, and low-water sue- 70 SPHEEIOAL ASTRONOMY. ceeds, the ■want of support is transferred tc the side of the ocean, the water flows out to sea, and forms what are called derivative ebb tides. The lines on the earth's surface connecting thcee places at which high or low water, or any other corresponding phases of the tides, occur simulta- neously, are called cotidal lines. § 291. The earth and moon are so near to each other, and so remote from the sun, as to cause their mutual attractions greatly to predominate over the excess of the sun's attraction for one of them over his attraction for the other. They therefore revolve about their common centre of gravity, aind together move around the sun. The attraction of the moon for. the earth produces upon the ocean eflfects similar to those of the sun. § 292. The diminution of weight at A and B and increase at O and D vary directly as the attracting masses, and inversely as the cubes of their distance, equations (80) and (81), and the effects upon the tide- wave must be in the same proportion. The mass of the sun is 355000 x 88 that of the moon, and he is situated at 400 times the moon's distances Whence the effect of the moon at A and B being 2 A; p 7» ~~dr' that of the sun will be 2 ^p»t 3550 00 X 88 _ (ioofd^ ' and dividing the last by the first, we have 355000 X 88 (400)' = 0.488; 80 that the effect of the moon is more than double that of the sun. § 293. The lunar day exceeds the solar on an average about 60 min- utes ; the lunar tide must therefore move slower than the solar by about 12°.5 in 24 solar hours; and hence they must sometimes conspire and sometimes oppose one another. The former occurs when the angular dis- tance of the sun fi-om the moon, as seen from the earth, is 0° or 180°, and the latter when this distance is 90°. This alternate reinforcement and partial destruction of the lunar by the solar wave, produce what are called spring and neap tides ; the former being their sum, the latter their difference. § 294. The sun and moon, by virtue of the ellipticities of the terres- trial and lunar orbits, are alternately nearer to and further from the earth than their mean distances. TIDES. 71 If the mean distances of the sua and moon be substituted in Eq. (80), the corresponding ellipticities of the solar and lunar spheroids will be found to be 2 and 5 feet respectively ; so that the average spring tide will be to the average neap, as 6 + 2 to 5 — 2, or as 7 to 3. Substituting the greatest and least distance of the sun in the same equation, the resulting tides are called respectively apogean and perigean tides ; and representing the ellipticity of the solar spheroid at the mean distance by 20, the corresponding ellipticities become 19 and 21. In like manner the ellipticities of the lunar spheroid will be found to vary be- tween the limits 43 and 59. Hence, the highest spring tide will be to the lowest neap, as 59 + 21 is to 43 — 21, or as 10 to 2,8. § 295. The sun and moon act to form the apexes of their respective tide-waves at different places, depending upon their angular distances apart. This gives rise to a resultant wave, whose apex is at some inter- mediate place, and the actual tide day, or interval between the occurrences of two consecutive maxima of the resultant wave at the same place, will vary as the component waves approach to or recede from one another. This variation from uniformity in the length of the tide day is called the priming or lagging of the tides — the former indicating an acceleration and the latter a retardation of the recuiTence of high-water at the same place. The priming and lagging are particularly noticeable about the time the angular distance between the moon and sun is 0° or 180°, that is, as we shall presently see, about new or full moon. § 296. The effort of the attracting body being to form the nearest ver- tex of its aqueous spheroid immediately under it, the summit of the lunar and solar tide-waves follow the course of the moon and sun to the north and south of the equator, and this gives rise to a monthly and annual variation in the heights of the principal tides at a given place. § 297. But of all causes of difference in the heights of tides, local situation is the most influential. In some places, the tide-wave rushing up narrow channels becomes so compressed laterally as to be elevated to extra- ordinary heights. At Annapolis, in the Bay of Fundy, it is said to rise 120 feet. § 298. Were the waters of the ocean free from obstractions due to viscosity, friction, narrowness of channels leading to different ports, and the like, the time of high-water at a given place, would depend only upon the relative positions of the sun and moon, and their meridian passages. But all these causes tend to vary this time, and to postpone it unequally at different ports: This deviation of the time of actual from that of theoret- ical high-water at any place, is called the establishment of the port, and ia 72 SPHERICAL ASTRONOMY. an element of the highest maritime importance. When ascertained from obseiTation, it enables the mariner to know by simply noticing the places of the sun and moon with reference to the meridian, when ne may safely attempt the entrance of a port obstructed by shoals. § 299. In bays, rivers, and sounds, where tides arise from an actual flow of water, the time of " Slack water," or stagnation, must not be con- founded with that of high and low water. They may, indeed, coincide, but not of course. A river current, for instance, and another from the sea, uirfy neutrahze each other's flow, while both conspire to elevate the water suiface ; so, also, an ebbing current may continue its onward course after the more advanced part of a returning flood has put its surface on the rise by checking its velocity. The same of two currents meeting in a sound. § 300. Starting from ^ as an origin (Fig. 58), and proceeding in the direction oi A £ I) A, vie find the value of r, Eq. (81), negative in the 1st and 3d quadrants, and positive in the 2d and 4th ; so that the tangential components of the solar and lunar attractions conspire with the normal to increase the height of the great tide-waves by impressing upon the water a motion of translation towards their apexes. But before the inertia of the water will peiToit the latter to acquire much velocity, the rotary motion of the earth reverses the direction of the impelling forces, and the final efiect due to this cause is, in consequence, but small. TWILIGHT. / § 301. The curve along which a conical surface, tangent to the sun and earth, is in contact with the latter body, is called the circle of illuminalimi. It divides the dark from the enlightened portion of the earth's surface, and is ever shifting its place by the diurnal motion. § 302. The base of the earth's shadow, into which a spectator enters at sunset, and from which he emerges at sunrise, is inclosed by an atmospheric wall-like ring, illuminated by the direct light from the sun, immediately exterior to that which just grazes the earth's surface. The light is reflected from the particles of this ring into the shadow, and gives to the air about its boundary a secondary and partial illumination called Tvnlight. A co- nical surface through the summit of this ring, and tangent to the earth, determines, by its contact with the latter, a limit within which the twilight cannot sensibly enter, and twilight will only continue while the spectator is carried by the earth's diurnal motion across the zone of which this line is the inner, and the circle of illumination the exterior boundary. The TWILIGHT. 73 ^4wMiAiiii»** belt of the earth's suiface over which twilight is visible, is called the cre- puscular zone. Thus, let EOO'E' be a section ^ig-«»- of the earth's surface on the opposite side from the sun ; TAA' T' of the atmosphere by the same plane, the height of the air being exaggerated to avoid confusing the figure; and S A and S' A' two solar rays tan- gent to the earth's surface. The particles of air in EA T and E'A' T' will be illuminated, while those in the space EAA'E' will be in the shadow. The section will cut from the tangent cone the elements A V and A' V, which touch the earth at and 0', ref,pectively, and being revolved about the line connecting the centres of the earth and sun, the part EA T will generate the lumin- ous atmospheric inclosure and the points E and 0, the circle of illumina- tion and interior boundary of the crepuscular zone, respectively. § 303. To a spectator within the crepuscular zone a portion only of the illuminating ring will be visible, and will appear as a bright elliptical seg- ment, with its chord in the horizon, its vertex in the vertical circle through the sun, and its outline almost lost in the gradual decay of light produced by the diflFusive action of the air and the progressive thinning and conse- quent diminution in the number of reflecting particles towards the summit of the luminous ring. § 304. When the spectator is carried obliquely through the crepuscular, zone without crossing its smaller base, twilight will last all night. § 305. Resuming Eq. (74), that is cos 2 = sin Z sin d ■\- cos I cos d cos P ; substituting the latitude of the place for I, the declination of the sun for d, and the value of P, obtained by convertmg the observed time from noon to the end of twilight in the evening, or from the beginning of twilight in the morning till noon, into degrees, the average value of a number of de- terminations for a will be found to be about 108° ; so that at the end of evening or beginning of morning twilight the sun is 18° below the horizon. § 306. From the above equation we find 74 SPHERICAL ASTRONOMY, cos s — cos l.cosd . cos P sin I = sin d The angle P S Z, made by the hour circle P S and vertical circle Z S, is called the variation or the pai-allactic angle. Denote this by f, then from the triangle Z P S, will (1) . slnl=.sladeose + coidBlnis cob |. Equating the second members of this and the equation above, we have (2) . . . . cos I . cos P = cos g . cos d — sin z sin d . cos f ; and if the sun be in the horizon, then will e = 90°, P = P', and | = f , and (3) . . • . cos Z , cos P' = — sin cf . cos f. Also, from the same triangle, (4) . . . . cos L sin /* = sin g . sin I; and when the sun is in the horizon, (5) . . . . cos I . sin P' = sin |'. Multiply (2) by (3), also (4) by (5), and add the products, there will result, eoa" 1 , 009 (P— P') = — cosB cos d sin d cos f + sin z cos (f — f ) — cos' d sin « cos f cos f . From (1), we have cos 1 = sin I — sin (? , cos a cos c^ . sin s and for the sun in the horizon cos f = sin I cos d ' (82) (83) TWILIGHT. 75 ■wLicli substituted above, give cos' I . cos (P — P') = sin 2 . cos (f — |') — siu' I ; ■whence, because cos (P - i") = 1 - 2 sin' \{P - P'), we have • s 1 / n n/\ 1 — sin g . cos (f — |') ^ sm' i (P - P') = 2cos'^ 5* passing to the arc and making P - P' 15 we have 15 V 2 cos' / (84) which will give the time required for the sun, or other heavenly body, to pass from the horizon to a zenith distance z, or, conversely, from a zenith distance z to the horizon. Making g = 90° + 18° = 108°, Eq. (84) becomes = A...V^ cos 18° . cos {^ — g') 15 ■ — V 2 cos' I . • • (85) which will give the duration of twilight for any latitude and season of the year ; and for this purpose, the values of ^ and |' must be found from Eqs. (82) and (83), after making, in the former, z = 90° + 18°. The value of t, in Eq. (85), becomes a minimum when f = |', and for the duration of the shortest twilight, we have, after replacing 1 — cos 18° by its equal 2 sin' 9°, 2 < = — . sin-' (sin 9° . sec Z) . . . . . (86) Equating the second members of Eqs. (82) and (83) sin c^ = — tan 9° . sin Z (87) In a given latitude, Eq. (86) will make known the shortest twilight, and Eq. (8T) the season at which it will occur. * Ann Arbor Astronomical Notices, Nw 1. 76 SPHERICAL ASTEONOMT. § 308. The sign of the second member of Eq. {Si) shows that at the time of shortest twilight the spectator and the sun will be on opposite sides of the plane of the equinoctial. § 309. The depression of the lowest point Q' of the equinoctial below the ho- rizon HH', is 90° — i; and of the low- est point S of the sun's diurnal path, - when his declination is of the same name as the spectator's latitude, 90°— (l + d); and when 90° -{l + d) — 18°, the end of the evening will be the beginning of morning twilight, and the nocturnal path of the spectator will be tangent to the inner boundary of the crepuscular zone. THE SUN. § 310. The Sun, as before stated, is the central body of the solar sys- tem, and from this circumstance gives to the latter its name. It occupies one of the foci of all the elliptical orbits of the planets, and, of course, that of the earth. § 311. Distance and Dimensions of the Sun. — Its horizontal parallax denoted by F, and apparent semi-diameter denoted by «, vary inversely as the earth's radius vector. For the mean radius it is found, § 113-6, P = 8".6, and s = 16' 01".5 ; which in Eqs. (28) and (29) give 206264".8 r„=p ' F ■P 8".6 : 23984 . (88) , 16' 01".5 961".5 . « = P • — STT^— = P • ^77^ = 111-5 P 8".6 8".6 (89) From Eq. (88) it appears that the mean distance of the earth from the sun is 23984 times the earth's equatorial radius; and from Eq. (89) that the sun's diameter is 111.5 times that of the earth. The volumes of these bodies are as the cubes of their diameters, and hence the volume of the sun is 1384472 times that of the earth. THE SUN. 7Y § 312. If the equatorial radius p be replaced in Eqs. (88) and (89) by its value in miles, § 98, we find r„ = 95,043,800 miles, %d= 882,000 " ; that is to say, the mean distance of the earth from the sun is, in round numbers, about 95 millions of miles, and the diameter ol the sun is 882 thousand miles. The mean distance of the earth from the sun is assumed as the unit of linear dimensions in all celestial measurements. § 313. Mass of Sun. — In Analytical Mechanics, § 201, we find the equation , „ 2 * . a^ , , , ^=-7^' ^''^ in which T denotes the periodic time of a body revolving about a centre of attraction, a the mean distance of the body from the centre, it the ratio of the circumference to the diameter, and k the attraction on a unit of mass at the unit's distance. Let k become (jo in the case of the sun's action on the earth ; then will T become the sidereal year, and a the semi-transverse axis of the earth's orbit, and -(*=-yF- (90) and for the action of the earth upon the moon .■ = i4^' («) in which y.' denotes the attraction on the unit of mass at the unit's distance exerted by the earth. Now the attractions exerted by two bodies on the same mass at the same distance, are directly proportional to their masses respectively ; and denoting the mass of the sun by M, an^ that of the earth by iT, we have M'-'^'~T''-^' ^ ' But in Eqs. (62) and (88) T = 365''.25, and a = 23984 , p ; and we shall presently see that the moon revolves about the earth once in 2Y.5 days, at a mean distance , of 60 times the equatorial radius of the earth. Making, therefore, Y8 SPHEEICAL ASTROMOMT. T' = 27.5, and a' = 60 . p, and substituting above, we have M i- = 354936. M That is, the sun contains 354,936 times as much matter as the e^rth ; and as the common centre of inertia divides the line joining their respective centres of inertia into two parts, which are inversely proportional to their masses, the common centre of inertia of the sun and earth, about which both bodies would describe their respective orbits were they undisturbed by the other bodies of the system, is but 267 miles from the sun's centre, or about -jsVo*^ P^''' '^^ '*^ "^"^ dfameter. § 314. Denote by D the density of the sun, and by Fits volume; also by D' and F', respectively, the density and volume of the earth ; then. Analytical Mechanics, § 18, M' = D'r'\ and by division and substituting the ratio of the masses and of the volumes given above, Ve find Z> = 0.2543 .J)'; so that the sun is but a trifle more than one-quarter as dense as the earth. The latter is known, from the recent experiments of Mr. Francis Baily, to be 5.67, the density of water being unity; and this value substituted for D' above, makes the density of the sun not quite once and a half that of water. § 315. Surface Gravitation of the Sun. — By the laws of gravitation, the attraction of one body upon another varies as the quantity of matter in the attracting body directly, and the square of the distance through which the attraction is exerted, inversely. The distance is that between the cen- tres of gravity of the bodies. Denote by W and W the weights of the same body on the surfaces of the sun and earth, respectively ; then will W M „« whence ^=^^.1.; (93) THE SUN. 79 and substituting the values just found, W IP =28,5. That is, a body weigliing one pound at the equator of the earth -would weigh 28,5. pounds at that of the sun ; and acquire, therefore, during each second of its fall a velocity of 916,44 feet. § 316. Suri^s Rotation and Axis. — Through the telescope the sun's surface often exhibits dark spots which slowly change their places and figure. They cross the solar disk from east to west, and thus reveal' a rotary motion of the sun itself from west to east about an axis. • § SI'?. To find the time of rotation and the position of the axis, it will be necessary first -to find the heliocentric lon^tudes and latitudes of the same spot at different times. To do this, let S be the sun's centre, E that of the earth, P the spot, and N its projection upon the plane of the ecliptic. Make I = heliocentric longitude of the earth ; x=z " " " spot; y = P jSiV = heliocentric latitude of spot ; /3 = P HJf = geocentric latitude of spot ; e = SJEN" = difierence of geocentric longitude of the sun and the spot, ^ = sun's apparent semi-diameter. Then SP sin y = PJV= JSP sin ^ = SI! sin /3, because the difference between UP and SU'is insignificant in comparison with either ; whence ^ SU . _ sin|8 . ,, smy=^.sm/3=-j^ (94) SP . cos y : UP . cos j3 : : SIf : iV^, : : sin e : sin (l — x)\ Again whence ... . sin e . cos 13 UP '^ ' cos y SP sin e . cos /3 sin ^ . cos y ' and replacing cos y by its value, . ,, . sin e . cos B sm (< — a;) r- •v/sin' ^ — sin' ^ // 80 SPHEEICAL ASTEONOMT. or for logarithmic computation, sin(Z-.) = -^=a^^^tf===.. (95) ^ ' Vsin (^ + ;8) . sin (^ - /3) § 318. Position of the Sun's equator, and the time of the Sun's rota- tion. Let H be the pole of the ecliptic, P that of the sun's equator; A, A', and A" the heliocentric places of the same spot observed at three diflferent times ; and let HA, HA', EA", PA, PA', PA" be the arcs of great circles. The first three are known from Eq. (94), being the helio- centric colatitudes of the spot ; as also the angles AEA', AEA", and A' U A" from Eq. (95), being the differences of the he- liocentric longitudes — all deduced from ge- osurface ojservations of the spot's right ascension and declination, § 152 All the sides and angles of the triangles AEA', AEA", and 'A'EA may be found, two sides and the included angle in each being given; hence the sides A A', A' A", and A" A, and the angles A, A', and A", in the triangle A A' A", are known. Now P being the pole of the sun's equator, parallel to which the spot revolves, PA = PA'^ PA"; Make IS = A +A'+A" = 2PAR + 2PA'A + 2PA'A" = 2PAE + 2A': whence PAE = S-A', and PAR becomes known. If PP be perpendicular to A A", AR = \AA"; then in the right-angled triangle APR, the angle at A and the side AR being known, the side P^ is computed; and, •finally, in the triangle APE, the sides AP and AE, and the angle EAP = EAA" — PAA" . being known, P E is computed. § 319. The araEP is the heliocentric colatitude of the pole of the sun's equator, and the angle AE P, added to the heliocentric longitude of the spot at A, gives its heliocentric longitude. The position of the sun's equator becomes, therefore, known. The heliocentric latitude and longi- tude of its north pole at the beginning of the present century were, respec- tively, 82' 30' and 350° 21'. THE SUN. 81 Fig. 65. Fig. 66. From the triangle APR the angle APS becomes known, the double of which is A PA". Then, denoting by T the time of one rotation, and by t the interval between the observations on the spot at A and A", we have APA" : t :: 360° : T; whence T is known to be about 25.325 days, making the angular velocity of the sun around its axis about one twenty-fifth that of the earth. From this motion it is concluded that the sun is flattened at its poles. § 820. Physical constitution of Sun. — Ihe study of the solar spots has led to inter- esting conclusions in regard to the physical constitution of the sun itself. The spots are transient in character, variable in size, shape and number, and confined to two compara- tively narrow zones parallel to, and at no great distance from the sun's equator. They appear perfectly black, and suiTounded by a border less dark, called a penumbra. The black part and penumbra are distinctly de- fined in outline, and do not fade the one into the other. Sometimes this penumbra presents two or more shades, and in this case also there is no gradation, but well-marked out- line, indicating a total absence of blending. As the spots move towards the edge of the sun, the penumbra on the inner side gradually contracts, and with the black spot disappears before reaching the boundary of the disk ; the penum- bra on the outer side expands, and is the last visible remnant of the spot as it passes behind the sun. At its reappearance on the opposite edge of the sun, the spot exhibits .similar phenomena — the penumbra first appears, then the black portion on its in- ner side, the contraction of the penumbra in width, and its extension around the black till the latter is entirely surrounded. This is precisely the appearance that would be presented by a deep pit or excavation with a dark or non-luminous bottom. The rotation of the sun would bring the slanting surface leading from the inner edge of its mouth more and more in the direction of the spectator till it would be lo.st in the foreshortening, the inner edge would presently mask the bottom, and the surface of the opposite side would be turned so nearly perpendicu- 6 rig. 67. 82 SPHEEICAL ASTRONOMT. larly to the line of sight as to appear broadest just before passing behind, at disappearance, or at reappearance, to the front of the sun. § 321. The spots gradually- expand or contract, change their figure, vanish, and break out again at new places where none were before. When ■^ Fig. 68. ^^-"^ disappearing, the central black part contracts to a point and vanishes be- fore the penumbra ; and a single spot is sometimes seen to break up into two or more smaller ones. § 322. A circle of which the diameter is one second is the smallest vis- ible area. A single second at the earth is subtended at the sun by a dis tance of 461 miles, and the area of the least visible circle on the sun's surface is, therefore, 167,000 square miles. A spot whose diameter was 45,000 miles has been known to close up and disappear in course of siic weeks, thus causing the edges to approach one another at the rate of 1000 miles a day. Many spots distinctly visible have been observed to vanish in a few hours, indicating a degree of mobility inconsistent with the idea of solids and liquids. § 323. Light proceeding very obliquely from the surfaces of incandes- cent solids and liquids is always polarized, whereas that from gases under the same circumstances is not. The light from the edge of the solar disk THE SUN. 8S leaves the surface of the sun in a direction nearly coincident with the sur- face itself, and yet when examined by the usual tests exhibits no signs of polarization. § 324. From all of which it has been concluded that the main body of the sun is an opaque solid, that this solid is invested with at least two at- Fig. 69. mospheres, the one above the other, that the one next the sun is like that of the earth, non-luminous, and that the one above is self-luminous and the immediate source of light and heat. A temporary removal at any one place of both atmospheres, but more of the upper than . the lower, by upward currents arising from local causes, would produce all the phenomena of the solar spots. The projecting edges of the lower atmosphere illuminated from above would form the penumbra, while the uncovered portion of the opaque solid, deprived in great part of the light from above by the intercepting action of the interposed atmo- sphere, and reflecting proportionally less of that actually received, would form the black portion. It is highly probable that the black portions of the solar spots would themselves appear luminous could the light from the other parts of the sun be intercepted. They may be black only from con- trast. A piece of quicklime in a state of most active incandescence under the action of the compound blowpipe is, when projected upon the sun, as dark as the darkest part of the solar spots. § 325. But light and heat are the results of molecular agitation. What is the cause of that perpetual molecular vibration essential to the self-lumi- nosity of the upper solar envelope ? The solar system is believed to have resulted from the subsidence of a vast nebula ; the planets and satellites are detached fragments left behind in the progress of the general mass to- wards the centre ; the sun itself is the central accumulation. This nebula must have extended originally far beyond the orbit of Neptune, the exte- 84 SPHEEICAL ASTEONOMT. rior planet now known. The distance of this planet from the sun is more than thirty times that of the earth. The condensation has taken place under the action of weight impressed upon the elements by their reciprocal attractions for one another. The hving force with which so much matter would reach the terminus of a fall necessary to transfer it to its present abode could not fail to impress upon the condensed mass the most intense molecular agitation. This agitation, or molecular living force, can only be lost through the agency of the suiTounding medium which difiuses it through space, and the loss in a given time is determined by the density of the medium, being less as the density is less. The medium which per- vades the planetary space is so attenuated as to offer no siensible resistance to the denser bodies that move through it, nor could we be conscious of its existence at all but for the almost inconceivably small amount of hving force which it brings from the sun to impress upon us the sensations of light and heat. A process so slow would require countless ages to bring the solar molecules to rest, and convert the sun into a non-luminous mass. § 326. The luminous part of the sun is not uniformly bright, but pre- sents a mottled appearance, and immediately about the spots are often seen well-defined and branching streaks, called faeules, brighter than other parts of the surface ; among these, spots often make their appearance. They are best seen near the border of the disk. § 327. The brightness of the solar disk sensibly diminishes towards the borders ; and this fact has given rise to the supposition that the sun is surrounded by an atmosphere not perfectly transparent, and of great extent above the luminous envelope. The loss of light towards the borders would result from the greater absorption of the luminous waves in consequence of traversing a greater thickness of the atmosphere, in that direction PLANETS. § 328. Let us now resume the Planets. As before remarked, these bodies move in eUiptical curves, of which one of the foci of each is at the centre of the sun. A spectator on the earth views these bodies, therefore, from a station far removed from their centre of motion, and even from the planes of their orbits. Hence, their co-ordinates of place, measured by the aid of instruments, are affected with both geocentiic and hehocentric parallaxes. To eliminate these, and then from the resulting heliocentric co-ordinates to determine the elements of a conic section whose curve shall pass through the observed, places and have a focus at the sun's cen- tre, is the object of one of the mast important problems in Astronomy. PLANETS. 86 Three observed right ascensions and declinations, together with the inter- vals of time between the obseiTations, are suflBcient for its solution. § 329. The planes of the orbits passing through the sun, the orbits themselves will pierce the plane of the ecliptic in two points, called nodes. The node by which the body passes from the south to the north of thfe eclip- tic is called the ascending node ; the other is called the descending node. § 330. The angle which the plane of a body's orbit makes with that of the ecliptic or equinoctial, is called the inclination. § 331. The semi-transverse axis, called the mean distance, and eccen- tiicity, determine the size and shape of the conic section. § 332. The inclination, heliocentric longitude, or right ascension of the ascending node, and of the perihelion, fix the position of the orbit in space. § 333. The time of the body's being at perihelion, and its mean angu- lar velocity, called its mean motion, give the circumstances of the body's motion in the orbit. § 334. The orbit of a heavenly body is therefore completely deter- mined when the inclination, mean distance, eccentricity, longitude of the ascending node, longitude of the perihelion, epoch of the perihelion passage, and mean motion are tnown. These are calW the elements of an orbit. They are seven in number. § 335. Tofnd a planers elements. — The polar equation of the orbit is a (1 - e') r = -7-T (96) 1 -|- e cos V in which r is the radius vector of the planet, a the semi-transverse axis, called the planet's Tnean distance, e the eccentricity, and v the planet's an- gular distance, from perihelion, called the true anomaly ; the pole being at the sun. Making v = 90°, r becomes the semi-parameter, which denote by JJ, and we have, Eq. (96), and Analyt. Mechanics, § 200, L = a{l-^)=.^ (97) in which c denotes the area described by a radius vector in a unit of time ; and Eq. (96) may be written r = -r-T- (98) 1 -f- e cos V whence e cos v = 1 (99) r from which, denoting the planet's velocity in the direction of the radius rector by ¥„ we find, Appeniix VI., 3Q 81 HERICA], ASTRONOMY. L eski-rr — .F, (100) i C whicli divided bj Eq. (99) gives tsav = — .— — .F, . . . , . .(101) 2c Jj — f § 336. Denote by p, the perihelion distance, then, making v = 0,ia % (86), , , ^ T- p = a(l-e) (102) § 337. Denotuig by 2'the periodic time, we have, An. Mec. § 201, T=^^; (103) and denoting the mean motion by », ^ = -ji=-T vl04) a' § 338. Take an auxiliary angle, such that cos V = • (105) 1 — e cos M then, Appendix VII., »« = « — esinw (106) in which t denotes the time from perihelion, and n, as above, the mean motion. § 339. The product w / is the angular distance which the planet would be from perihelion had it moved from that point with its mean motion n, and is called the mean anomaly. § 340. The auxiliary angle u is called the eccentric anomaly, and dif- fers from n t only because of the eccentricity of the orbit ; for if the latter be zero, n t will equal u. § 341. From Eq. (105) we readily find tanl' = '/^tan^ (lOT) 2 ^ l+e 2 ^ ' § 342. Making in Eq. (103), * = (A, a = 1, and T= 366''.256, we find log ;* = 6.4711640 log -y/JA = 8.2355820 § 343. From the centre of the sun draw right lines respectively to the vernal equinox, intersection of the solstitial colure with the equinoctial, and north ceiestial pole, and take these as the axes x, y, and s. The planes of the equinoctial, of the equinoctial colure, and of the solstitial coluro, will be the co-ordinate planes xy,xz, and s y respectively. PLANETS. 87 Denote by V„ F„ and V, the component velocities of the planet in the direction of the axes, and by c', c", and c"' the projections of c on the co- ordinate planes xy,zy, and z x respectively ; then, Analytical Mechanics § 184, equations (249), will xV^-yV, = 2c', J yV,-zVy = 2c", V (108) zV.-xV, = 'i,c"',) and c» = c'" + c'" + e"" (109) § 344. Denote the inclination of the orbit to the plane of the equinoc- tial by t, then will cos «;= - (110) § 345. Also, and, Appendix VIII., ,^=ic»-f y« + s« r : + '-.v,+t. (Ill) (112) § 346. Let S be the sun, P the place of the planet, R that of the perihelion, B the vernal, equinox, E- the summer solstice, A the north celestial pole, BE the. ecliptic, B'P'N' the intersection of the plane of the planet's orbit with the ce- lestial sphere, N' the heliocen- tric place of the ascending node If on the equinoctial, A P'P'" and A B'B" quad- rants of great circles of the ce- lestial sphere. Make X = 5P"'=the planet's heliocentric right as- ^ cension. S z= A P' = the planet's heliocentric north polar distance. *j = N'P"'= distance in heliocentric right ascension from the node. s =z BJV' = heliocentric right ascension of ascending node. 9 == iV'/" = distance of the planet from the node. >/ =z N'B" = distance of perihelion in right ascension from- the asc. node. sar= B B" = heliocentric right ascension of the periheUon, 88 Then SPHERICAL ASTRONOMY. tan X z= (lis) tanP"S£ = tsmF'OP"' X and in the triangle AP'C, the side -4C being 90°, cot 5 = cos X. • - (1 14) Again, in the triangle P'P"'N', right-angled at P'", sin »i = cot 5 • cot i (115) e = X-») (116) tan(p = seci-tan(X— s) (117) In the triangle N'B'B", right- angled akR", tanX'=: cos i.tan {fi-\-v) (118) in which v is the true anomaly f'SR'; and hence ■a z= -K' +s (119) § 347. It thus appears that as soon as x, y, «, V„ V, and V, are found, all the elements become known ; and the preceding formulas, arranged in the order of sequence, will stand (1) (2) (3) (4) (5) (6) (7) r'= x' -\- if^ + z' ; '2c' =xV,-yV,; \2e" = yV.-zV,i .2c"'=zr.-xK; . c'=c'^+c"^+c"'\ r, = -.v.+ l.v,+-.v.. r r ^ r tan V = 2c L-r V;. = -^(--l); Eq.(99). cos V \r / ^ ^ ' PLANETS. (9) p = a(l-e). (10) « = ^. a' (11) cos« = ^^i?i±^;Eq.(105). l+ecosr ^ ^ ' /in\ « — esinw _ ,,„.^ (12) t = - ; Eq. (106). tit (13) . . . . i' cos t = -• c (14) tan X = ^. X (15) cot5 = cosX,-' X (16) simj = cot 5 . cot t. (17) ..... 6 = X - »,. (18) tan

S N the curtate distance of the planet. § 360. Draw (S F and E V io the vernal equinox ; they will be sensibly parallel. Also draw iViV^, and ^^, perpendicular to ^ F and S F, and make a = S N'= mean curtate distance ; p = EJV = earth's distance from the reduced place ; 1= VS]f= planet's heliocentric longitude ; n = ^ hourly change in the same ; Z = VS E = earth's heliocentric longitude ; X = VEN= planet's geocentric lon^tude; m = hourly change in the same. ./^f ..-^ ,.-T^ 92 SPHERICAL ASTRONOMY, Then, the mean distance of the earth from the sun being unity, will, Appendix X., m = F'[a!' + J — {a + a^).cos{L-l)].n. . .(124) in .which cos X a cos / — cos Ii and which will make known the rate and direction of the body's motion in geocentric longitude. § 361. Direct and Retrograde Motion; Stations. — When the planet is in apogean syzygy, then will 1/ — Z = 180°, cos (L — I) = — 1; and, Eq. (124), m = P\a.(a + l){l+a^).n (125) and m will always be positive ; that is, the geocentric motion of the planet will be direct. § 362. When the planet is in perigean syzygy, then will X — Z = ; cos (L—l) = l; and, Eq. (124), I m = F'.a{a-l){l—a^).n (126) and m will always be negative, whether a be greater or less than unity ; that is, the geocentric motion of the planet will be retrograde. § 363. In changing from direct to retrograde, and the converse, the body must appear stationary. This will make m = 0, and, Eq. (124), 1 IT n a-\- a"^ 1 cos (Z - ^ = = -^ J (127) 1 + a^ a= + o- 2 — 1 a quptity which is always less than unity, whether a be greater or less than unity; that is, all the planets must sometimes appear stationary. The condition expressed by Eq. (127), may always be satisfied for two val- ues oi L — I. The two places of a body, in which it appears stationary, are called stations, § 364. Let the value oi L — I for one of the stations be

the planet just touching the conical sur- i face, of which A£ and A' £' are sections, by a plane through the centres I of the three bodies. Make I SEA = S = sun's apparent semi-diameter; ' TEP = d = planet's apparent semi-diameter ; < EA B = * = sun's horizontal parallax ; ETB = it' z= planet's horizontal parallar; SEP = s = planet's elongation at the beginning of the transit; tnen will but whence s = 5 + d + AET, AET=ii' -k, s z=. h ■\- d ■\- 's' — * (132) ^ that is, when the elongation of an inferior planet is less than the sum of the apparent semi-diameter of the sun and planet, augmented by the differ- ence of their horizontal parallaxes, there will be a transit or an occultation of the planet, according as its horizontal parallax is greater or less than that of the sun. § 372. Let EE' be ^'s- "• an arc of the ecliptic, 0' an arc of the plan- et's orbit, and N the node. Parallel to 0', and at a distance from it equal to s, draw on either side a line cutting the ecliptic in S. Now, if at the time of inferior conjunction the difference between the geocentric longitudes of the sun and node be less than SN, there must be a transit ; if greater, there can be none. The distance SN is called a transit limit. To find its value, mate SNP = i — inclination of the planet's orbit ; IfS = Z = transit limit ; PLANETS. 97 "then, in the right-angled triangle SPN", ■ , sin s , ^ sin I = -; — r (133) sm » ^ ' The value of s is vsiriable, being a function of the radii vectors of the earth and planet at inferior conjunction. The inclination i is also slightly- variable. The greatest value of i and least value of s make I a minirnu7n limit ; the least value of i and greatest value of s make I a maximum limit. § 373. The earth returns sensibly to the same place of the heavens at intervals of a sidereal year. Any entire number ql sidereal years which will contain the synodic revolution of a planet a whole number of times, will bring the earth and planet to the same places they simultaneously oc- cupied before, and if a transit occur at one node, it will occur at the same node again at the expiration of this interval, provided the node be not carried by its proper motion beyond the transit limit. § 374. The bodies having returned to the places they previously occu- pied, will each have performed a whole number of entire revolutions, and making n = the number of the eaith's revolutions ; n'=z " « planet's " P = the length of the earth's sidereal year ; P'= » » planet's " " we shall have whence nP = n'P', n P' n'~ P (134) If P and P' be whole bumbers, and the second member be reduced to its simplest teims, the numerator will be the interval in sidereal years between the consecutive transits at the 'same node, and this interval will be constant. . But if P. and P' be not whole numbers, then will the numerators of the approximating fractions of the continued fiaction, which give 'the values of the second member within the transit limits, be the variable intervals, in sidereal years, between the transits at the same node. § "375. Masses and Densities of the Planets. — The masses of such of the planets as have satellites may easily be found by the process of § 313, as Boon as the periodic tin)e of the planet and that of its satellite are deter- mined by observation. But for such as have no satellites, recourse is had to a different process, which can be here indicated only in outline. A 7 98 SPHERIGAL ASTEOKOMT. planet uodisturbed by the action of the othera, would describe accurately its elliptical orbit about the common centre of inertia due to its own mass and that of Ae sun ; and from the elliptical elements already described, its futiu'e places are, as we shall see, predicted with the greatest precision. The difference between these places and those actually observed, give the effects of the disturbing action of the other planets. To compute thesti eflects, what are called perturhating functions are constructed upon the principles of mechanics. The masses of the perturhating or disturbing bodies enter these functions ; and from the observed amount of perturb- ations the value of the masses are computed. An. Mec, § 203, § 376. The masses and volumes being known, the densities result from the process of § 314. § 311. Botary motions. — All the planets whose surfaces exhibit through the telescope distinct marks, are found to have a rotary motion in the same direction as those of the sun and earth, viz., from west to east. § 378. Planetary Atmosphere. — The existence of an atmosphere about a planet is indicated by the apparent displacement it occasions in the geo- centric place of a star by refracting its light, when, by the motion of the earth and planet, the latter comes near the line of the star and observer. The atmosphere about a planet is in fact a vast spherical lens, of which the central part is deprived of its transparency by the opaque materials of the planet, but of which the outer portion is free from obstruction and acts upon the light which passes through it with an energy due to its refractive power and density. The height of the atmosphere is inferred from the greater or less angular distance between the star and planet when the displacement begins ; and the density, which must be regulated by the same laws that govern the equilibrium of heavy elastic fluids upon the earth, from the amount of dis- placement. § 379. In detailing the physical peculiarities of the planets, their mean distances and times of sidereal revolutions, Although contained in the sy- noptical table of elements, will be repeated ; and in all cases in which di- mensions or measures are given, they must be understood as expressed in the corresponding elements of the earth as unity. Thus, if it be the me.in distance, density, volume, solar heat and light, sidereal day, &c, those of the earth are the respective units. MERCURY. MERCURY. § 380. Proceeding ovtwards from the sun, Mercury is ihe first known planet. His mean distance is 0.3870985 ; sidereal year, 0.2408 ; true di- ameter, 0.398; volume, 0.003; mass, 0.175; density, 2.78,-greater than that of gold ; intensity of its attraction for a unit of mass on its surface, called surface gravitation, 1,15 ; solar heat and light, 6.68 ; time of rota- tion upon its axis, called, sidereal day, 1.20833. The eccentricity of his orbit being large, his greatest elongation varies from ,16° 12' to 28° 48'. The latter being his greatest apparent distance from the sun, he is generally lost to us in the light of that body, and it is difficult, therefore, to observe him. His arc of retrogradation varies from 9° 22' to 15° 44'. - § 381. When to the west of the sun he rises before, and when to the east he sets after that luminary. In the former position he is called a morning, and in the latter an evening star. § 382. The sun appears nearly seven times as large to the inhabitants of Mercury as to us ; and on the supposition that the intensity of solar light and heat varies inversely as the square of the distance, the solar il- lumination and temperature on Mercury would be 6.68 times that on the earth, as above. Heat and cold are, however, but relative terms, depending upon physical conditions as well as distance, and the Mercuriau surface may be as cold as the earth's ; the frosty summits of the Himalayas are nearer to the sun than the scorching plains of Hindostan. § 383. The changes of seasons on Mercury, depending, as they do, upon the inclination of his axis to that of his orbit, which has not been well determined, are not accurately known. If, as there are reasons to believe, this inclination have any considerable value, the mutations of Mer- cury's seasons must be very great ; his tropical year being only about one- fourth that of the earth, his seasons, if they follow the same proportion, can only be of some two or three weeks' duration. .§ 384. Mercuiy's nodes are, and will for ages continue, in that part o^f the ecliptic which the earth passes in May and November, and his transits os-er the sun must occur in those months. His periodic time = 8Y''.9'7, and that of the earth = 366''.256, in Eq. (134), give the approximating fractions, 7 13 33 29' 54' 137' So that the intervals between the transits which may be expected at the same node are seven, thirteen, &c., years. The great inclination of Mercu- ^00 SPHERICAL ASTRONOMY. ry's orbil makes his transit limits, Eq. (133), small, and the ahove interva* will not therefore always be those which separate the actual recurrence » the transits. The last transit occurred at the ascending node in 1848,. the next will occur in 1861. VENUS. § 385. Venus follows Mercury in the order from the centre. Her mean distance is 0.T233317; sidereal year, 0.6152; true diameter, 0.975; vol-^ iime, 0.927 ; mass, 0.885 ; density, 0.95 ; smface gravitation, 0.93 ; solar heat and light, 1.91 ; sidereal day, 0.97315. ' § 386. Venus is the brightest of the planets, her light being of a bril- liant white, and at times so intense as to cause a shadow. The elotigations of her stations vary but little from 29°. Her phases are finely exhibited through the telescope. The southern horn of her crescent varies its shape, being alternately sharp and blunt, and the changes are attributed to the periodical interposition of high mountains by an axial rotation of Venus so as to intercept the solar light she at other times reflects to the earth from her southern surface. From these changes her sidereal day has been determined. § 387. Her axis is inclined to that of her ecliptic under an angle of 75°, thus placing her tropics at the distance of 15° from her poles, and her polar circles at the same distance from her equator. Her seasons suc- ceed each other, therefore, very rapidly, there being two summers and two winters in each of her annual revolutions. Her atmosphere resembles in extent and density that of the earth; § 388. Her synodical revolution is 583.92 days. Venus is, therefore, about 292 days continuously to the east, and as long to the west of the sun. In the former position she sets after the sun, and is called an evening star ; in the latter, she rises before the sun, and is called a morning star. Her greatest elongation is about 45°, and she is brightest when on her way from the east to the west of the sun, and at an elongation on either side, of 'about 40°. § 389. The line of Venus's nodes lies in that part of the ecliptic through which the earth passes in June and December, and her transits occur in 'those months. The periodic time of Venus = 224''.7, and that of tjie earth = 365''.256, which, in Eq. (l34), give the approximating fractioss, _8_ 235 713 13' 382' 1159' ■ VENUS. 101 ^fc=^ and the transits at the same node may be expected at inteivals of eight, two hundred and thirty-five, &c., years. Two transits, separated by an interval of eight years, will occur at one node, and then at the opposite node after an interval of one hundred and five, or one himdred and twenty- , two years, between the last of the first pair and first of the second pair. As astronomical phenomena the transits of Venus are of the highest ira portance. They afibrd the best means of ascertaining the sun's horizontal parallax, and therefore the earth's distance from the sun, and the dimen- sions of the solar systern, expressed in terms of some known terrestrial measure. § 390. The principle on which the sun's horizontal parallax is found from a transit of Venus may be thus illustrated. Conceive two observers situ- ated at the opposite extremities A and B of the earth's diameter, which is perpendicular to the plane of the planet's orbit. To the observer A, the planet would "" ■ appear to transit the sun's disk along the chord m n, and to the observer £, along the chord p q, being the intersections of the solar disk by two planes through the portion D of Venus's orbit, described during the transit, and each observer. A third plane through the observers and Ve- nus's centre would cut from the other two the lines A a and B b, and froni the sun's disk the perpendicular distance a b between the chords. Now, because the angle A V B is equal to the angle a Vb, A B will be to a 6 as Venus's distance from the earth is to her distance from the sun ; that is § 385, as 1—0.723 : 0.723, or as 1 to 2.61 nearly ; and the radius of the earth, half of u4 J? is to a 6, as 1 to 5.22 nearly. The apparent mag nitudes of two objects, viewed at the same distance, being directly pro- portional to the true magnitudes, the ra- dius of the earth viewed at the distance of the sun, in other words, the sun's hori- zontal parallax, is equal to the angular distance between the chords divided by 6.22. § 391. The relative geocentric motion of the sun and planet into the observed durations of the transit at the two stations will give the chords m n and p g. The chords being known, as also the apparent Fig. T9. 102 SPHERICAL ASTKONOMY. semi-diametei-s Sq and 5 m, the distances Sa2aad.Sb become known, and therefore their difference a b. § 392. The general result of all the observations made on the transit of 17C9 gives 8".5776 for the sun's horizontal parallax. The next two transits of Venus will occur on Dec, 8th, 1874, and Deo. 6th, 1882. MASS. § 393. Mars is the fii'st of the superior planets. His mean distance is 15237; sidereal year, 1.8807; true diameter, 0.517; volume 0.138G ; density, 0.95 ; equatorial gravitation, 0.493 ; solar heat and light, 0.43 ; sidereal day, 1.02694 ; oblateness, about -^g- ; and the inclination of his axis to that of his ecliptic 30° 18' 10".8. § 394. He has a dense atmosphere of moderate height. His surface (Plate I., 'Fig. 2) exhibits through the telescope outlines of what are deemed to be continents and seas, the former being distinguished by a ruddy color, which is characteristic of this planet, and indicates an oohry tinge in the soil, contrasted with which the seas appear of a greenish hue. These markings are not always equally distinct ; and the variation is attributed to the formation of clouds and mists in the planet's atmosphere. Brilliant white spots sometimes appear at that pole which is just emerging from the long night of its polar winter, and are attributed to extensive snow-fields that push their borders to an average distance of some six de- grees from either pole. PLANETOIDS. § 395. Next to Mars come the class of small planets, which, on account of their comparatively diminutive size, are called planetoids. Little is known of them beyond their orbit elements, but they are interesting on account of their history and the speculations connected with their discov- ery, which began with the present century. § 396. If the mean distance of Mercury be, taken from the mean dis- tances of the other planets, the remainders will form a series of numbers doubling upon each other in proceeding outward from the sun. To this law there was a remarkable exception in the distance between the orbits of Mercury and Jupiter as compared with that between Mercury and Mars, the former being so large as to require the interpolation of another body between Mars and Jupiter. § 397. Although th» law is strictly empirical and wholly inexplicable PLANETOIDS. 103 a priori upon any known physical hypothesis, yet the coincidence was so remarkable as to induce the prediction that by proper search a planet would be found in the interpolated place. § 398. This body was only to be recognized by its proper motion. Tt detect this, an examination of the telescopic stars of the Zodiac was com- menced, their places were carefully mapped, and on the first day of the present century, the prediction was verified by the addition of Geres to the system. Her mean distance is 2.76692, and the hiatus was filled. § 399. But the discovery of Ceres was soon followed by that of Pallas, at the mean distance of 2.7728 — nearly the same as that of Ceres — and the law was again broken. § 400. The points in which the paths of the new planets are intersected, ya. either side of the sun, by the line common to the planes of both orbits, are not very far apart, and it was suggested that Ceres and Pallas were but fragments of a larger planet that once revolved at an average distance, and which had been broken to pieces by some disraptive force. But where were the other fragments ? § 401. A number of bodies projected in difierent directions from a com- mon point, would each describe about the sun an hyperbola, a parabola, or an ellipse, depending upon the relations between the velocity of projection and the intensity of the sun's attraction upon the unit of mass, and in the case of elliptical orbits, the bodies would, abating the effects of the pertur- bating action of the other planets, return at fixed intervals to the place of departure. § 402. The opposite points of the heavens, in which the orbits of Ceres and Pallas approached most nearly each other, were therefore regarded as the common haunts of the suspected fragments, and the places especially to be watched, to detect their existence. A constant scrutiny of these points, and diligent revision of the maps of the zodiac, have resulted in the discovery, to the present time, of 71 of these little bodies. § 403. The mean distances of the planetoids vary about from 2.2 to 3.6, and periodic times about from 3.3 to 6.9. Their small size makes it diffi- cult to determine their true dimensions, the diametoT of the same individ- ual, as given by the best authorities, varying from 0.02 to 0.20. They exhibit considerable variety of color ; some have shown signs of possessing atmospheres, and those who regard them as debris of a single body, find evidence of an angular or fragmental figure in sudden changes of illumina- tion, which have been ooser>Jed, and which are attributed to the shifting of their bounding planes by a diurnal or axial rotation. 104 SPHERICAL ASTRONOMY. JUPITER. § 404. Jupiter is the largest, and except Venus, which he sometimes surpasses in this respect, the brightest of the planets. His mean distance is 5.202 ; sidereal year, 11.86; diameter, 11,2 ; volume, 1280.9 ; mass, 331.57 ; density, 0.24 — but little greater than that of water ; equatorial gravitation, 2.716; solar heat and light, 0.037; sidereal day, 0.41376; oblateness, -^-g ; inclination of axis to that of his ecliptic, 3° 5' 30". § 405. The disk of Jupiter is always crossed, in a direction parallel to his equator, by dart bands or belts, presenting the appearance indicated in Plate I., fig. 3, which was taken by Sir John Herschel. These belts are not always the same, but vary in breadth and situation, though never in direction. They have sometimes been seen broken up and distributed over the whole face of the planet. From their parallelism to Jupiter's equator, their occasional variation and the appearance of spots upon them, it is in- ferred that they exist in the planet's atmosphere, and are composed of extensive tracts of clouds, formed by his trade-winds, which, from the great size of Jupiter, and the rapidity of his axial rotation, are much more de- cided and regular than those of the earth. § 406. The great oblateness of this planet is due to the shortness ^ of his, sidereal day, and its amount agrees with that assigned by theory to give him a figure of fluid equilibrium. § 407. From the small inclination of his axis to that of his ecliptic, there can be but httle variation in the length of his days and nights, each of. which is less than five of our hours; and changes of seasons must be. almost, if not quite unknown to his inhabitants. § 408. Jupiter is attended in his circuit about the sun hy four satellites or moons, which revolve about him from west to east, and present a min- iature system analogous to. that of which Jupiter himself is but a single in- dividual, thus afibrding a most striking illustration of the efiects of gravi- tation and of distance in grouping, as well as shaping the courses of the. heavenly bodies. These satellites will be noticed under the head of Sec- ondary Planeti. SATURN. § 409. Saturn is the next in order of size as he is of distance to Ju- piter. His mean distance is 9.538850'; sidereal year, 29.46 ; true diam- eter, 9.982 ; volume, 995.00; mass, 101.068 ; density, 0.102 — little more than half that of water; equatorial gravitation, 1.014; solar heat and SATURN. 105 light, 0.011 ; sidereal day, 0.43701 ; oblateness, -fV 5 inclination of axis to that of orbit, 26° 49', and to that of our ecliptic, 28° 11'. § 410. Saturn is the most curious and interesting body of the system, being attended by eight satellites or moons, and surrounded (Plate II.,Fig. 4), according to some authorities by two, and others by four, broad flat and extremely thin rings, concentric with each other and with the planet. § 411. ,The dimensions of the rings and planet, and the intervals as given by the advocates of but two rings, are, ,1 miles. Exterior diameter of exterior ring .... 40.095 = 176,418 Interior " " " . . . . 35.289 = 155,2'72 Exterior diameter of interior ring .... 34.475 = 151,690 Interior " " « . . . . 26.668 = 117,339 Equatorial diameter of planet 17.991= 79,160 Interval between the planet and interior ring 4.339 = 19,090 Interval between the rings 0.408 = 1,791 Thickness of ring not exceeding 230 § 412. The evidence of recent observations with very powerful instru ments seems, however, in favor of a division of the outer ring, as just given, at a distance less than half its width from the exterior edge, and of the existence of a dusky ring still nearer the body of the planet, and composed of materials partially transparent, and possessing but feeble powers of re- flection, resembling in these particulars a sheet of water. And there seem good reasons for believing that the rings are not precisely in the same plane. The disk of the planet is crossed by parallel belts, similar to those of Jupiter ; these are supposed to be due to Saturn's trade-winds. From the parallelism of .the belts to the plane of the rings, it is inferred that the planet's axis of rotation is perpendicular to that plane, and this is con- firmed by the' occasional appearance of extensive dusky spots on his sur- face, which, when carefully watched, give the time of his rotation about an axis having that direction. § 413. By watching the ditferent shades of illumination on different portions of the rings, the latter are found to complete a revolution in their own plane once in 10'' 32°' 15", thus making their sidereal day 0.43906, which exceeds that of the planet itself by 0.00205. § 414. That the rings are opaque and non-luminous is shown by their throwing a shadow on the body of the planet on the side nearest th^ sun, and bv.the other side receiving that of the planet as shown in the figure. 106 SPHERICAL ASTRONOMY Fig. 80. § 415. The axes of the planet and rings preserve their directions un- changed during their orbital motion. The plane of the rings, which is inclined to that of the ecliptic under an angle of 31° 19', intersects the latter plane in a line which makes with the line of the equinoxes an angle equal to 167° 31', so that the nodes of the ring lie in longitudes 167° 31' and 347° 31'. § 416. The orbital motion of the planet causes this intersection to oscil- late, as it were, parallel to itself, in the plane of the ecliptic, through a distance on either side of the sun equal to the radius vector of Saturn's orbit ; and the period of a semi-oscillation is one-half of the planet's pe- riod, or about 15 years. Within this period the plane of the ring must pass once through the sun, and from once to thrice through the earth, depend- ing upon the initial position or place of the latter when the trace of the plane on the ecliptic touches the earth's orbit at the time o^ nearing the sun. § 417. Thus, let S be the sun, jEE'E"E"> the earth's orbit^ P P' an arc of Saturn's orbit projected upon the plane of the ecliptic, P E and P' E" the traces of the "plane of the rings on the same, and tangent to the earth's orbit, and suppose the motion of the earth and of Saturn to take place in the direction indicated by the arrow-heads. Draw SB parallel ia P E and P' E", and make r = S P = the mean distance of Saturn ; ^r'= SE = " " " of earth; az= P S P' = the angle at the sun subtended by PP' : then, since the angle P S B = S P E, we have r' 1 sin -1 a = — = — :- = 0.1082, * r 9.54 ' whence a = 12° 2', SATURN. 107 which dividjd by 2' 0".6, the mean motion of Saturn, gives 359.46 days, wanting only 5.8 days of a complete year ; that is to say, the earth de- scribes nearly one entire revolution in the time during which the earth's orbit is traversed by the plane of the ring. § 418. The rings are invisible when their plane passes between the sun and earth, their enlightened fece being then turned from the latter body ; and the interval of non-appearance will be that between any two epochs at which the plane passes the sun and earth, and of which the effect of one is to throw these bodies on opposite and the other to restore them to the same side of this plane. § 410. If the initial place of the earth be at E", nearly three days in advaiice of B", then will the plane itself pass the sun and earth at the same time, the earth being at B', and these bodies could not be on oppo- site sides of the plane of the' rings during its present visit to the earth's orbit. If the initial position^f the earth be at -E'', nearly three days in advance of Ej it will be at E" when the plane passes the sun ; the rings will then disappear, and continue invisible till the earth meets and passes their advancing plane, which it will do somewhere in the quadrant E"B' ; they will then reappear, and continue visible for the next fifteen years. If the earth's initial place be at E'", some days in advance of B\ it will meet and pass the plane in the same quadrant, the rings will disappear and continue invisible till their plane is overtaken and passed again by the earth somewhere in the quadrant ^^"; when the plane passes the sun the earth will be in the quadrant B"E", and the rings will again disap- pear, and again become visible only when their plane is reorossed by the- earth in the quadrant E"B'. Thus, with this initial place, the earth will cross the plane of the rings three times in one year, and there will be two disappearances. § 420. When the plane of the ring passes through the sun, the edge of the ling alone is enlightened, and can only appear as a straight line of light projecting from opposite sides of the planet in the plane of his equa- tor, and parallel to his belts. This phase of the ring has been seen, but it requires the most powerful telescopes ; and from the fact of its non-ap- pearance in a telescope which would measure a line of light one-twentieth of a second in breadth, of which the subtense at Saturn's distance is 230 miles, it is inferred that the thickness of the ring cannot exceed this latter dimension. § 421. When the dark side of the ring is turned to the earth, the planet appears as a bright round disk with its belts, and crossed equato- rially by a narrow and perfectly black line. This can only happen when IQ^ SPHERICAL ASTRONOMY. the planet is less than 6° 1' from the node of his rings. Generally the northern side is enlightened when the heliocentric longitude of Saturn is between 172° 32' and 341° 30', and the southern when between 353° 32' and 161° 30'. The greatest opening occurs when the heliocentric longi- tude of the planet is 11° 81' or 257° 31'. t URANUS. § 422. Uranus is one of the more recently discovered planets, being only recognized as a planet for the first time in 1781, though it had often been seen before and mistaken for a fixed star. Of this planet nothing can be seen but a small round uniformly illumi- nated disk without rings, belts, or discernible spots. His mean distance is 19.18239 ; sidereal year, 84.01 ; true diameter, 4.36 ; volume, 82.91 ; mass, 14.25 ; density, 0.17 ; equatorial gravitation, 0.75 ; solar heat and light, 0.003. He is attended by six satellites, which will be noticed presently. NEPTUNE. § 423. Neptune is the last known planet in the order of distance, and third in size. Its discovery dates only from 1846, though its existence had been suspected from certain irregularities in the motion of Uranus, which could only be attributed to the disturbing action of some body exterior to itself. The departures, of Uranus from places assigned by the combined action of the known bodies of the system, and certain assumed conditions in re- gard to position and shape of orbit, direction of motion, and mean distance, rendered highly probable by analogy, were the data from which, by the methods of physical astronomy, was wrought out in the closet in Paris, the place of a new planet whose disturbing action would account for the unexplained waywardness of Uranus. The result was sent to an observer in Berlin, and in the evening of the very day of its receipt in the latter city, Neptune was added to the known system by actual observation. It was found within 52' of the place assigned, and its discovery, in all its circumstances, must ever be regarded as one of the greatest triumphs of modern scibnce. § 424. Neptune's mean distance is 30.0367 ; petiodic time, 164.6181 ; real diameter, 4.5 ; volume, 91.125 ; mass, 18.219 ; density, 0.208 ; equa- torial gravitation, 0.9035 ; solar heat and light, 0.0011. The apparent size of the sun as seen from the earth, bears to that as seen SEOONDAKY BODIES. 10& from Nepl.une, about the relation of an ordinary orange to a commou duck- shot. § 425. Neptune has at least one satellite, and certain appearances have indicated a second, and also a ring, but of these there are yet doubts. General RemarJe. § 426. In the foregoing enumeration of the physical peculiarities of the planets, one is impressed by the great differences in their respective sup- plies of heat and light from the sun ; in the relations 'which the inertia of matter bears to its weight at their surfaces ; and in the nature of the ma- terials of which they are composed, as inferred from variety of mean density. The intensity of solar radiation is nearly seven times greater on Mercury than on the earth, and on Neptune 900 times less, giving a range of which the extremes have the ratio of 6300 to 1. The eflBcacy of weight in counteracting muscular effort and repressing animal activity on the earth, is less than half that on Jupiter, more than twice that on Mars, and probably more than twenty times that on the planetoids, making a range of which the limits are as 40 to 1. Lastly, the density of Saturn does not exceed that of common cork. Now, under the various combinations of elements so important as these, what an immense diversity must exist in the conditions of animal life, if the planets, like our earth, which teems with living beings in every corner, be inhabited ! A globe whose surface is seven times hotter than ours or 900 times colder, on which a man might by a single muscular effort spring fifty feet high, or with diflBculty lift his foot from the ground ; where his veins would burst from deficiency or col- lapse trom excess of atmospheric pressure,, affords to our ideas an inhospi- table abode for animated beings. But we should remember that heat and cold, light and darkness, strength and weakness, weight and levity, are but relative terms ; and to the very conditions which convey to our minds only images of gloom and horror, may be adjusted an animal and intellectual existence which make them the most perfect displays of wisdom and be- neficence. SECONDARY BODIES. § 427. The secondary bodies are those which revolve about the planets, and accompany them around the sun. Of these, twenty are known at the present time. One belongs to the earth, /owr to Jupiter, eight to Saturn, dx to Uranus, and one to Neptune. ° They are commonly called satellites, and sometimes moms, but this latter appellation is more particularly ap- plied to the earth's secondary. 110 SPHERICAL ASTKONOMT THE MOON. § 428. The moon revolves in an elliptical orbit, of which -one of tha toci is at the earth's centre. Its motion is -from west to east, and its an- gular velocity about the earth is much greater than that of the earth around the sun. The moon appears, therefore, to move among the fixed stars in the same direction as the sun, but more rapidly ; and from the axial motion of the earth she has, like other heavenly bodies, an apparent diurnal motion, by which she rises in the east, passes the meridian, and sets in the west. § 429. The oblateness of the earth would be quite appreciable to an ob server at the distance of the moon. Her equatorial horizontal parallax is therefore found from Eq. (24) ; her distance from Eq. (28) ; her true diam- eter from Eq. (29) ; and her mass from her effects in producing precession and nutation. Lunar Orbit. § 430. The elements of the moon's orbit may be found from four ob- served right ascensions and declinations, coriected for refraction, parallax, and semi-diameter. Let i> C be an arc in which Fig. 82. the plane of the oi'bit cuts the celestial sphere ; V£ an arc of the ecliptic, and VA of the equinoctial ; V the vernal equi- nox, iV the ascending node, P the perigee, and M„ M,, M,, Mf the geocentric places of the moon. First convert the geocentric right ascensions and declina- tions into geocentric longitudes ;tnd latitudes, and make V = VJV = longitude of node ; i = ClfJB = inclination of orbit ; lt= VOt = longitude of Jf, ; l,= VO, = « M,; X, = J/, 0, = latitude of JT, ; \ = M,Oi = « M,; a-X. THE MOON. Ill then in the right-angled triangles Mi iV 0, and M^ JV 0,, we have sin (li — v) = cot i . tan X, 1 sin (Ji — v) = cot i . tan Xj ) ^ ' and by division sin (^1 — v) tan X, sin (Zj — v) tan Xj' Adding unity to both members, reducing to common denominator, then subtracting each member from unity, reducing as before, and finally divi- ding one result by the other, we find sin (?j — v) + sin (Z, — v) tan Xg -f tan X, _ sin {^2 — v) — sin (i, — v) ~ tan X, — tan X, ' replacing the members by their equals, we have Also, from first of Eqs. (135), we have coti = !^^ (m) tan X, ^ ' whencev and i are known. The longitude of the ascending node, increased by the angular dis- tance of a body from the same node, is called the Orbit Longitude. Make V, = VEN + NEMi, = orbit longitude of Jf, ; p=VEN+NEP = " " perigee; 9 = PEMi = Vx—p = true anomaly of Jlf, ; e = eccentricity of orbit ; m = mean motion of moon in orbit ; t, = time since epoch for J/", ; L = mean orbit longitude. Then resuming Eq. (48), we have L + mti = Vi — 2e sin (vj — p) . . . . (138) in which ,. = v-htan-^^B_(^^ .... (139) Four values for the geicentric longitudes denoted by /„ /j, l„ It, in Eq. (139), give four values for v, viz., v„ Vj, v,, and Vf ; and these, and the timei 112 SPHERICAL ASTRONOMY. of observation ti, t„ tj, and t,, in Eq. (138), give four equations involving the four unknown quantities L, m, e, and p ; whence these become known precisely as in § 197, employing for the purpose Eqs. (50), (51), (53), and (54). § 431. Denoting the ecliptic longitude FO of the perigee by^„we have, in the triangle NF 0, right-angled at 0, tan i\rO = tan (^ — v) . cos i, and ^, = V + tan""' [tan (^ — v) . cos J] . . .. (140) § 432. In the same way, denoting the mean ecliptic longitude of the moon at the iepoch by X,, £, = V + tan"' [tan (Z — v) . cos J] .... (141) § 433. The passage of the moon through one entire circuit of 360° around the earth, is called a sidereal revolution. The interval of time re- quired to perform a sidereal revolution is called a sidereal period. Denote the sidereal period by s, then will 360° , , « = (142) The equation of the orbit, the centre of the earth being the pole, is ' a(l-e") r = * 1 4- e cos {v — p) ' fln(l~ the value of r being found by means of Eq. (28), that of the mean distance a will result, and every thing in regard to the moon's path be- comes known. § 434. At the epoch January 1st, 1801, the elements of the lunar orbit were Mean a = 59.96435000 of the earth's equatorial radius; " s = 27.321661418 mean solar days; « e= 0.054844200; " v= 13°S3'iV".7; " pi = 266° 10' 07".5 ; « i= 5° 08' 47".9 ; " Z, = 118° 17' 08".3. § 435. The moon's true diameter, Eq. (29), is 0.27280, or about 2153 miles; volume, 0.0204; mass, 0.011309; density, 0.5657; and surface gravitation, 0.1666. § 436. Comparing the lunar elements which depend upon the orbit as THE MOON 113 determined at different times, they are all found to vaiy. The nodes have a retrograde and the perigee a cSrect motion, the former performing a com- plete revolution in 18.6, and the latter in 8.854 years. The inclination fluctuates between 4° 57' 22" and 5° 20' 06" ; the mean distance has a secular variation, and it is at the present time diminishing ; the same is true of the sidereal revolution, and the mean motion of the moon is increas- ing. All these changes are due to the disturbing action of the other bod- ies of the system, but principally of the sun. The action of the protuberant ring of matter about the equator of the earth also has its effect. Disturbing Forces. § 437. To illustrate the way in which ^'ft- M. these changes of the lunar orbit are brought about, let H be the earth, S -the sun, M the moon, moving in her orbit in the directionif-DiV^'iV; iV and JV^' be- ing the nodes, and H V the direction of the vernal equinox. Then, resuming equations (80) and (81), making p =- HM, the radius vector of the moon, and employing in all other respects the nota- tion of § 286, V becomes the change which the sun's attraction causes in the weight of a unit of the moon's mass due to the earth's attraction, and t the change which the sun's attraction causes in the force normal to the radius vector and in the plane passing through the sun, earth, and moon. This latter force being in general oblique to the plane of the lu- nar orbit, urges the moon out of that plane, and causes her to describe a curve of double curvature, while the former has no such action. Resolve t into two components, one perpendicular to the radius vector and in the plane of the orbit, the other normal to this latter plane. For this purpose conceive a sphere of which the centre is at that of the earth, and radius, the radius vector p = MM, of the moon. Its surface will be cut by the plane of the ecliptic in AN^BN^^ by that of the lunar orbit 8 114 SPHEEICAL ASTEONOMY in JV, MN,,, arid by ihat of the sun, earth, and moon 'va.AM,B. Make O z= VU S = sun's longitude ; Si = VEJV,=z long, of moon's node; i = MN,A— inclination of lunar or- bit; X = N,MA = inclination of orbit to plane of sun, earth, and moon ; cr = r . cos X = component of r in plane of lunar orbit ; = r . sin X = component normal to ' plane of orbit. In the triangle N, MA, the arc If^ A = (O — ft) and AM= (p, and we have Fie.8SbigL sin X = sin j. sin (O — SJ)_ cos' X = 1 — sm

' and bringing forward Eq. (80), and replacing t by its value in Eq. (81),. there will result V = -T^ . (2 cos' (p — sm' whence :.NSz=mt- -nt = {m -n)t; (p is tan 9 = then known. 9 m — n • The angle §475. Denoting S Mi by J, we SJ^= .^ _ sm ip have Also ^V(m — n 9 f + g' (156) (157) NO SN~m-n' whence, substituting the value of S N, m ■)/\nt — nf + g^ i—n' y N0 = ^. (158) ECLIPSES. 125 Making ^ equal to that given in Eq. (149), the moon will just touch the earth's shadow, and N will become what is called the ecliptic limit ; that is, the least difference of longitude thai can exist letween the moon, and her nearest node at full, to avoid an eclipse of the moon. Taking the greatest value for ji, W 0\& found to be 12° 24', and least value it is found to be 9°. The first is called the greatest and the secoud the least lunar ecliptic limit. If therefore at the time of full moon the difference between the longitude of the moon and her nearest node exceed 12° 24', there cannot be an eclipse; if less than 9°, there must be one ; if less than 12° 24' and greater than 9°, there may or may not, depend- ing upon the inclination of the relative orbit and actual value of ^. To solve the doubt, we have the given difference of longitude between 12° 24' and 9°, and the inclination (p, to find S M-^. If this latter be greater than ^, there can be no eclipse ; if less, there must be one. § 476. Again, making ^ equal to that given in Eq. (155), and pro- ceeding exactly as above, we find the greater and lesser solar ecliptic limr its. The first is 18° 36' and the latter 15° 25'. Number of Eclipses. § 477. 'UtNHN'H' be the ecliptic, N and N' the moon's nodes. Take NLi, NL„ N'Li, and N'Li, each equal to 18°.6, the greatest ecliptic limit. Then will Ly Li and L, L^ be each equal to 37°.2, and the number of new moons that can happen while the sun is appar rently describing these arcs, will deter- mine the number of solar eclipses that can occur in a single year. The mean daily motion of the nioon's node is — 0°.055 ; the mean apparent daily motion of the sun is 0°.985, and hence the apparent relative motion of the sun and node is 0°.985— (— 0°.055) = 1°.04, say cme degree. Any arc, therefore, estimated from the node and expressed in degrees, ipay be taken to express also the num-. ber of days during which the sun is in this arc. The sun will be 37.2 days in Z, L^ and as long in L3 Z4. A lunation is 29.53 days, and 360° -7- 29.53 gives 12 lunations and 5.64 days over. Take Z, Jf, = 7° and My M^ = 6°.64 ; then, while the sun is apparently describing Jf, HH'M^ there will be time enough for 12 lunations exactly, and if the sun begin to describe this arc with the 126 SPHERICAL, ASTRO JS'OMr. moon in conjunction at il/"i, it will end Fig. gows. it with the moon in conjunction at Jfj. '' — Moreover, L^ Xj — Z, if, = 3Y°.2 - 7° = 30°.2 ; to describe which the sun would require more tipie than the moon would to make a lunation, so that there must be another new moon between N and ij, making three within the arc Z, A- Besides, Z, A —L^M^= 180° — 7° = 173 ; and 173 H- 29.53 gives 5 lunations and 25.35 days over, and a new moon will occur at Jfj, at the distance of 29.53 — 25.35 = 4°.l8 from A. T;he value of M^ L^ is equal to 37°.20 — 4°.18 = 33°.02, and the time requisite to describe this being greater than a lunatioii, there must be another new moon between N' and Lf, making five in the course of the year, and as many solar eclipses within the same period. ^Again, M^N = 18°.6 — 7° = 11°.6 ; and in a semi-lunation or 14.76 days the sun will have passed the node N only by the distfince 14.76 — 11.6 = 3°.16; so that at opposition or full moon, the moon will be within the lunar ecliptic hmit, and there will be an eclipse , of the moon. When conjunction took place at Jlfajthe sun was 18°.60 — 4°. 18 = 14°.42 from the node N' ; at the expiration of 14.76 days, or half a lunation af- terwards, the moon will be full, the sun will be within 0°.34 of the node N\ and there will be another eclipse of the moon, making two in one year. It appears, then, that there may be seven eclipses in one year, and when this is the case five will be of the sun and two of the moon. 478. The least solar ecliptic limits being 15° 25', the arc Zr, L^ must be at least 30° 50' ; and as it requires longer than a lunation for the sun to pass over this arc, there must be at least one eclipse of the sun both in Lx Li and L, L^, so that there must always be at least two eclipses of the sun each year. , The sun is less than a lunation in passing through the lunar ecliptic limits ; therefore there can be no more than one eclipse at each node, and there may be none. In other words, there ?»ay be two lunar eclipses in a year, and there may be none. The Saros. § 479. The synodic period of the moon is 29.53058, and that of the moon's node 346.6196 days. These numbers are to one another as 19 to 223 nearly. If, therefore, the moon and her nodes be in syzygy at the CONSTITUTION OF THE MOON. 127 same time, they will be so again after 19 revolutions of the node, or 223 lunations ; so that the eclipses will recur again very nearly in the same order within the same period, which is about 18.027 years. This period is known as the Chaldean Saws. There are generally IQ echpses in the saros, of which 29 are lunar and 41 solar. PHYSICAL CONSTITUTION OF THE MOON. § 480. Telescopes disclose certain varieties of illumination oi the moon's surface, which can only arise from mountains and valleys. The shadows cast by the former lie in directions and are of lengtha re- quired by the inclination of the solar rays to that portion of the moon's surface on which the mountains stand. The convex outhne of the moon turned towards the sun is always circular and nearly smooth ; but the op- posite or elliptical border of the illuminated part is extremely ragged, and indented with deep recesses and prominent points. To places along this line the sun is just rising, and the neighboring mountains cast long black shadows on the plains below. As the sun rises these shadows shorten ; and at full moon, when the solar light penetrates the mountain valleys and shines on every point of the field of view, no shadows are seen. § 481. The summits of the lunar mountains often appear as small bright points, or islands of light, beyond the edge of the illuminated part, as they catch the sunbeams before the intervening plains. As the sun ad- vances in altitude, these luminous patches expand, and finally unite with the general illumination, and the- mountains appear as projections from its elliptical border. . '^ § 482. To compute the ^'g-«i- height of a lunar mountain, let U, M, and S be the cen- tres of the earth, moon, and sun respectively; ACBD and DOC sections (3f the general surface ,of the moon by planes respectively perpen- dicular to JEM and MS;. then will the visible illumi- nated part of the disk be contained between CBJ) and the projection oi D on the section A CBD. Also let m be the top of a mountain just catching the solar rays that graze th/3 general surface of the moon at ; B OFih.Q arc of a 128 SPHERICAL ASTRONOMY. great circle of this surface, and of which the plane passes through the top of the moun- tain and centres of the sun and moon ; n the point in which this arc is cut by the line Mm drawn from the top of the mountain to the moon's centre. Make r = Mn = raflius of the moon ; s = FCO=:.EMS = MEA'= exterior angle of elongation ; y = Om = distance of m from ; x = m,n = height of mountain ; a = the projection of y on the plane AC BB. Then, since the ray /S' m is perpendicular to the section D C,\i is m- clined to the section AC BD, under an angle equal to the complement oiFCO=Q0° — s; and we have o = y . cos (90° — s) = y . sin e\ whence sm £ Also and, therefore, y = Va(2 »• + «); Vx(2r + x) = -^^\ neglecting x in comparison with 2 r, we have 1 a' - = — . cosec' « (1^9) S ^ T a being the observed distance of the bright spot from the boundary of the illumination, may be measured by means of the micrometer. § 483. The heights of many of the lunar mountains have been thus computed, and they range through all elevations up to 23,000 English feet. § 484. The lunar mountains are strikingly uniform in aspect. They are very numerous, especially towards the southern border, occupjring by far the larger portion of the surface. They present almost universally a circular or cup-shaped form in ground plan, which becomes foreshortened into an ellipse towards the limb. The larger of these cups have for the most CONSTITUTION OF THE MOON. 12U ^art flat bottoms, from each of which rises centrally a small, steep, conical hill, presenting in all respects the tnie volcanic character as exhibited by Fig. 92. like districts on the earth, but 'with this peculiarity, viz. : that the bottoms are so deep as to lie below the general surface of the moon, the internal depth being often twice or thrice the external height. § 485. The heights of mountains in the immediate vicinity of each other being proportional to the length of their respective shadows, the depths of the pits or craters are easily computed from the heights of the edges above the general level, and the lengths of the shadows they cast internally and externally. § 486. Through the Rosse telescope, the flat bottom of the crater called Albategnius, is seen to be strewed with blocks not visible through inferior instruments ; and the exterior of another, called Aristillus, is hatched over with deep gullies, radiating from a centre. § 487. There are also extensive tracts of the lunar surface which are perfectly level, and present decided indications of an alluvial character, and yet there is a total absence of all appearances of deep water. § 488. There are no clouds, or other indications of an atmosphere. A lunar atmosphere of a mean density equal to 1980th that of the earth, would give a horizontal refraction of 1", and cause the diameter of the moon, measured with a micrometer and estimated by the interval of a star's disappearance in an occi^ltation, to diflfer ; would cause the limb of the moon, during a solar eclipse, to appear beyond the cusps externally to the sun's disk as a narrow line of light, extending for some distance along the edge ; and would extinguish very faint stars before oecultations. But none of these phenomena are seen. During the continuance of a total lunar eclipse, when the light of the moon is so deadened as not to obliter- ate by contrast the feeble light of the smaller stars, the latter are seen to come up to the moon's limb and undergo sudden extinction, without any apparent displacement. § 489. The light from -the moon developes but feeble heat, for even 9 130 SPHERICAL ASTRONOMY. when collected into the foci of large reflectors, it affects but little the thermometer ; and there are no appearances indicating the slightest change of surface, such as -would result from the periodical growth and decay of vegetation which accompany a change of seasons. § 490. To an inhabitant- of the moon, if there be such a thing, the earth must present the appearance of a moon 2° in diameter, exhibiting phases complementary to those the moon presents to us, but fixed in the sky, while the stars seem to pass slowly beside and behind it. It must appear clouded with variable spots, and belted with zones corresponding to our trade-winds. During a solar eclipse our atmosphere will appear as a narrow, bright ring, of a ruddy color where it rests on the earth, gradually passing into faint blue, encircling the whole or part of the earth's disk. SATELLITES OF JUPITEE. § 491. The satellites of Jupiter, four in number, revolve about their pri- mary from west to east in planes nearly coincident with that of the planet's equator, and but slightly inclined to the ecliptic. § 492. Their orbits appear, therefore, projected very nearly into straight lines, in which they oscillate to and fro, sometimes passing between the sun and Jupiter, causing an eclipse of the sun to the latter, sometimes en- tering the planet's shadow and being themselves eclipsed, and sometimes disappearing either behind the body of Jupiter or in transiting his disk. § 493. Thus, let S be the sun ; ^, the earth, of which the orbit is EFGH\ J, Jupiter ; and e/a 6, the orbit of a satellite. The cone of Ju- piter's shadow will have its vertex at X, far beyond the orbit of the satel- rig. 9S. SATELLITES OF JUPITER. 131 lite, and the penumbra, owing to the great distance of the sun and conse- quent smallness of the angle at Jupiter subtended by his disk, will extend but little beyond the shadow within the limits of the satellite's orbit. The satellite revolving from west to east, will cast a shadow upon Jupiter while passing from m to n, will transit his disk from e to /, enter his shadow at a, emerge from it at b, and disappear behind the body of the planet while passing from c to d. § 494. The shadows of the satellites are frequently seen crossing the disk of Jupiter. While in the act of transiting, the satellite generally dis- appeai's, its light being confounded with that of the planet, unless it hap- pens to be projected upon a dark belt, in which case it is visible. Under these circumstances it occasionally appears as a dark spot smaller than its shadow, which has led to the conclusion that certain of the satellites have now and then on their own bodies, or within their atmospheres, obscure ' spots of great extent. § 495. From the eclipses of, the satellites are obtained all the data for the determination of the laws of their motions. These eclipses are in gen- eral analogous to those of the moon, but in their details they differ con- siderably. The great distance of Jupiter from the sun and his great size, make his shadow much larger and longer than that of the earth. The sat- eUites are much smaller in proportion to their primary, and their orbits less inclined to his ecliptic, than in the case of the moon. From these causes the three interior satellites enter the shadow at every revolution, and are totally eclipsed ; and although the fourth, from the greater inclination and distance of its orbit, sometimes escapes eclipse, yet it does so seldom. § 496. Besides, these eclipses are not seen by us from the centre of mo- tion, as are those of the moon, but from some remote station, of which the place with respect to the 'shadow is ever changing. And while this cir- cumstance makes no difference in the time of the eclipses, it yet affects materially the visibility and the apparent relative situations of the planet and satellites at the instant of the latter's entering and quitting the shadow § 497. A satellite never enters the shadow suddenly because of its sen sible diameter, and the time from the first perceptible loss of hght to its total extinction will be that required by the satellite to describe about Ju- piter an angle equal to its apparent diameter as seen from the planet's cen- tre. The same is tme of the emergence. Owing to the difference in tel- escopes and eyes, this becomes a source of discrepancy in the times assigned by different observers for the beginning and ending of an eclipse. But if both the immersion and emersion be observed by the same person and with the same telescope, the. half sum of the two times, as given by a properly 132 SPHERICAL ASTEOliTOMY. regulated time-keeper, will be that of apparent opposititm measurably free from error. § 498. The intervals between the oppositions give the synodic period, which, in Eq. (146), will give the mean motion, knowing that of Jupiter, and hence the sidereal period. Eq. (142). The satellites are named first, second, third, and fourth, according to their order of distance from Jupiter. The elements of the satellites' orbits will be found in the following Table. Sat. Sidereal period. Inclination of Mean 1 orbit to a distance, j flxed plane Ead.ofJ=l.I"-''I»"»«»'"' Inclination of the fixed plane to Jupiter^s equator. Eetrograde revolution of nodes on fixed plane. Mass: tbat of Jupiter 1,000,000,000. Ist., 2d. 3d. 4th. d. h. m. 8. 1 18 27 33.506 3 13 14 36.393 7 03 42 33.362 16 16 31 49.702 6.04853 9.62347 15.3*5024 26.99835 ' " 27 50 ■ 12 20 14 58 o '. " 6 1 5 5 2 24 4 Tears. 29.9142 141.7390 531.0000 17328 23235 88497 42659 It will assist in forming some idea of the relative dimensions of Jupitei and his satellites to examine the following Table. Mean apparent diameter as seen from earth. Mean apparent diameter as seen from Jupiter. Diameter in miles. Maes. Jupiter. 38.327 ' " 87000 1.0000000 Ist sat; 1.017 33 11 2508 0.0000173 2d " 0.911 17 35 2068 0.0000232 3d " 1.488 18 00 3377 0.0000885 4th '• 1.273 8 46 2890 0.0000427 From which it follows that the first satellite appears to a spectator on Jupiter as large as our moon to us ; the second and third nearly equal to each other, and somewhat more than half the size of the first ; and the fourth about a quarter of that size. They frequently eclipse each other. The apparent diameters of the planet as seen from the satellites are 19° 49'; 12° 29'; '7°4'7'; 4° 25'. § 499. Figure 93 shows that the eclipses take place to the west of Ju- piter, while the latter is moving fi'om conjunction to opposition, and to the.. SATELLITES OP JUPITEE. 133 east from opposition to conjunction. As Jupiter approaches to opposition, the line of sight from the earth becomes more nearly coincident with the direction of the shadow, and the place of the eclipse will be nearer and nearer to the body of the planet. When the earth comes to F, fi'om which a hne drawn tangent to the body of the planet will pass through h, the emersion will cease to be visible, and will, up to the time of opposition, take plaee behind the planet. Similarly, from opposition up to the time when the 6arth arrives at K, the immersion will be concealed from view. These remarks apply particularly to the third and fourth satellites, the proximity of the others to the planet being so great as to make it impossi- , ble ever to see the immersion and emersion both at the same ecitpse. § 500. The mean motions of the satellites are connected oy this re- markable law, viz. : If the mean angular velocity of the first ^tellite be added to twice that of the third, the sum will equal three times ;Jiat of the second. If, therefore, from the mean longitude of the first sriellite, in- creased by twice that of the third, three times the mean longitu-te of the second be subtracted, the remainder will be a constant quantity, and this constant is found to be equal to 180°. This Laplace has showi. to be a consequence of the mutual attractions of the satellites for on( another. The first three satellites cannot, therefore, be eclipsed at the same vime. § 501. While, however, the satellites cannot all be eclipsed at once, they may be, and, indeed, occasionally are, all invisible by the simultaneous eclipse of some, occultations of others, and transits of the rest. § 502. The orbits of the satellites are but slightly eccentric, tht iwo in- ferior- ones not at all so, so far as observation is capable of revealing? eccen- tricity. Their mutual attractions produce in them perturbations analogous to those of the planets about the sun. These are investigated in physical astronomy. § 503. By careful observations the satellites are found to exhibit iu.arked fluctuations in respect to brightness. These fluctuations happen periodi- cally, and appear connected with the position of the satelUtes with r««pect to the sun ; from which it is inferred that they revolve upon their axe? "^ke our moon, each once in its sidereal period. § 504. At one time the eclipses of Jupiter's satellites were much L-ed in the determination of terrestrial longitude, but more modern methots, free from the objections referred to in § 497, have in a measure supplas- "^ th»'" 134 SPHERICAL ASTRONOMjr. Progressive Motion of Light, § ^05. 1'j these eclipses science is indebted for the discovery of the suc- cessive propagation and velocity of light. The earth's orbit being concentric with that of Jupiter and interior to it, the distance of these bodies is contin^ually varying, the variation extending from the sum to the diflFerence of the radii of the two orbits, making .the excess of the greatest over the least distance equal to the diameter of the earth's orbit. Now, it was observed by Roenier, a Danish astronomer, on comparing together the eclipses during many successive years, that those which took place about opposition were observed earlier, and those about conjunction later than an average or mean tiine of occurrence. And con- necting the observed acceleration in the one case and retardation in the other with the variation of Jupiter's distance below and above its average value, he found the difference fully and accurately accounted for by allow- ing le™ 26'.6 for light to traverse the diameter of the earth's orbit. In other words, using the figure of a cord moving in the direction oT its length from the satellite to the earth to illustrate the flow of luminous waves in the same direction, if the cord were severed at the edge of Jupiter's shadow, the severed end would be IS"" 26".6 longer in reaching the earth when the planet is in conjunction than in opposition, having a greater distance to travel in the first case by the diameter of the earth's orbit = 190,000,000, miles, than in the second. The satellite is seen long after it has entered the shadow, and is invisible long after it has emerged from it. Dividing the diameter of the earth's orbit by 16™ 26".6 reduced to seconds, the ve- locity of light is found to be 192,000 miles a second. SATELLITES OF SATUEW. ■ § 506. Eight satellites are known to accompany Saturn. They revolve about him from west to east, and in planes nearly coincident with that of the planet's ring, except the eighth, whose orbit is inclined to this latter plane under an angle of about 12° 14'. This satellite is also distinguished from the others by its remoteness from the planet, its distance being 2.3 times that of the most distant of the others, and equal to 64 times the equatorial radius of Saturn, resembhug in this respect our own moon. It is also remarkable for the exhibition of greater variety of illumination in different parts of its orbit than any other known secondary. Indeed, so feeble is the light which it reflects to the earth when to the east of Saturn tliat it becomes invisible through ordinary telescopes ; and from this defi- SATELLITES OP SATURN. 135 ciency of light occurring constantly on the same side of Saturn, as -seen from the earth, it is inferred that this satellite revolves on its axis once during its sidereal period. § 607. The next in order, |)roceeding inwardly, is so obscure as to have eluded the observations of astronomers until very recently. It was dis- covered simultaneously by Mr. Bond, of Cambridge, U. S., and Mr. Lassell, of Liverpool, England, in 1848. § 508. The next in order, proceeding in the same direction, is by far the largest and most conspicuous of all, and probably not inferior to Mais § 509. The next three in order are very small, and require pretty pow- erful telescopes to see them, while the two interior, which just skir4; the edge of the ring, can only be seen with telescopes of extraordinary power and perfection, and under the most favorable atmospheric circumstances. When first discovered, they appeared to thread the excessively thin film of light reflected from the edge of the ring then turned towards the earth, and for a short time to advance off at either end, speedily to return again. § 510. Owing to the obliquity of their orbits to the plane of Saturn's ecliptic, there are no eclipses, occultations, or transits of the satellites, or shadows on the disk of the primary, except at uie time when the ring is seen edgewise, and their observation is attended with too much difficulty to be of any practical use, like the corresponding phenomena of Jupiter's satellites, for the determination of terrestrial longitude. § 511. The names and elements of Saturn's satellites are given in the following Table. Named and Order of Satellites. Sidereal Period. Mean Distance. Epoch of Ele- ment. Mean Longi- tude at the Epoch. Eocentri- city. Perisatar- num. 1. Mimas . . . d. h. m. 8. 22 37 22.9 3.3607 1790.0 ' " 256 58 48 2. Enceladus 1 08 53 06.7 4.3125 1836.0 67 41 36 3. Tethys... 1 21 18 25.7 5.3396 it 313 43 48 0.04? 54°! 4. Dione. . . . 2 17 41 08.9 6,8398 l( 327 40 48 0.02? 42 ? 5. Rhea 4 12 25 10.8 9.5528 (t 353 44 00 0.02? 95 ? 6. Titan .... 15 22 41 25.2 22.1450 1830.0 137 21 24 0.029314 256° 38'.11 1. Hyperion. 22 12 ? ? 28. ± 8. lapetus . . 79 07 53 40.4 64.3590 1790.0 1 269 37 48 The longitudes are reckoiied in the plane of the ring from its descend- ing node on the ecliptic. The apsides of Titan have a direct motion of 30' 23" per annum in longitude on the ecliptic. 136 SPHERICAL ASTRONOMY. § 512. The periodic times of tlie first four satellites in order of distance from Saturn are connected by this law, viz. : The period of the third is double that of the first, and the period of the fourth is double that of the second; the coincidence being exact to within -jJo part of the larger period. SATELLITES OP URANUS. § 513. Uranus is believed to have six satellites, which levolve about-the primary from east to west, in orbits nearly, if not quite, circular, and which make with the ecliptic an angle of 78° 58'. They thus difier from all the other known bodies of the solar system both in the direction of their mo- tion and inchnation of their orbits, which latter, as well as the places of the nodes, have undergone no sensible change, during at least one-half of the planet's period around the sun. The elements of these satellites, as far as known, are given in this Table. Sat Sidereal Eevolution. Mean Distance. Epoch of passing Ascend- ing Node. Nodes and Inclination. 1 2 3 4 .5 6 4djh 8 Iff' se^SKs 10 23? 13 11 07 12.6 38 2 f 107 12 f 17 19 8? 22 8 45 5? 91 0? Gr. T. 1787. Feb. 16, Oi" 10™ 1787* Jan. 1, 0" aS™ Inclination of orbits to tbe ecliptic, 78° 58' ; ascending node in longitude, 165° 30'. (Equinox of 1798.) Motion retrograde, and orbits nearly circular. § 514. The satellites of Uranus require very powerful and perfect tele- scopes for their observation. The second and fourth are far the most con- spicuous, and their periods and distance have been ascertained with toler- able certainty. The first and third have also been observed since their original announcement, but of the existence of the fifth and sixth we have not the same evidence. Sir John Herschel is of opinion that if future observations should assign them places, they would be exterior to that of the fourth. 515. When the earth is in the plane of the orbits or nearly so, the ap- paient paths of the satellites are straight lines or very elongated ellipses, in which case these secondaries become invisible long before they come up to the disk of the planet, in consequence of the superior light of the latter, so that it is not possible to observe their occultations, eclipses, anc' transits. COMETS. 137 SATELLITES OF NEPTUNE. § 516. If the observation of the satellites of Uranus be difficult, those of Neptune, owing to the great distance of this plauet, ' must offer still greater difficulties. Of the existence of one satellite there remains no doubt. Its sidereal period about the planet is nearly 5.9 days ; its mean distance is fourteen times Neptune's semi-diameter ; and its orbit is in- clined to the plane of the ecliptic under an angle of about 35°. COMETS. § 517. Comets differ from all the primary bodies -with which we have thus far been concerned, in their appearance, the shape and inclination of their orbits, and in following no rule, as a class, with regard to the direc- tion of their motions. They are of various sizes, some being visible to the naked eye even in daytime, while others require the aid of telescopes even at night to see them. § 518. The larger consist for the most part of an ill-defined mass, called the head, from which, in a direction opposite the sun, proceeds a train, of gi'eater or less extent, called the tail. Fig. 94. § 619. The head is much brighter towards its centre. Sometimes this increase of illumination terminates in a bright spot, called a nucleus, the surrounding haze which makes up the rest of the head being called the § 520. The tail appears to consist of two streams of luminous matter which, starting from a point near the head, and on the side towards the sun, pa-ss suddenly to the opposite side, and grow broader and more dif- fused as they increase in length; they commonly unite at a little distancei from the head, but sometimes continue distinct for the greater part of their course. This appendage has been known to attain the enormous length of forty-one millions of miles, and to stretch over 104 degrees of the celes- tial sphere. § 621. The tail is not, however, an invariable appendage of comets, 138 SPHERICAL ASTRONOMY. many of .the brightest having been seen with little or none, and others as round and well-defined as Jupiter. § 522. On the other hand, there are instances of comets with many tails or streamers, spreading out like an immense fan, and extending to the distance of some 30 degrees of the celestial vault. One is recorded as having two tails, mating with each other an angle of 160°, the fainter being turned towards, the other from the sun. The tails are often curved, bending, in general, towards that part of space which the comet has left, as if retarded by the opposition of some renting medium. § 523. The smaller comets, such as are only visible through telescopes, and which are by far the most numerous, present no appearance of a tail, and seem as round or oval vaporous masses, more luminous towards the centre, where, in some instances, a small stellar point has been seen, but without any distinct nucleus or other signs of a solid body. Stars of the smallest magnitude, such as would be obliterated by a moderate fog, are seen through their brightest part. § 524. A comet never exhibits the least signs of phases ; but, on the contrary, appears as a mass of thin vapor, either self-luminous, or easily penetrated by the luminous waves from the sun, which are reflected from its interior parts as from its exterior surface. § 525. The tail, where it comes up and surrounds the head, is yet sep- arate from the latter by an interval less luminous, as if sustained and kept from contact by a transparent stratum of atmosphere ; and seems to be a kind of hollow envelope of a parabolic form, inclosing the head near its vertex. § 626. The number of recorded comets is very great, amounting to sev- eral hundred ; and when it is considered that in the earlier stages of as- tronomy, before the invention of the telescope, only large and conspicuous ones could be noticed, and that, since due attention has been paid to the subject, scarcely a year passes without the observation of one gr two of these bodies, and sometimes two or three have appeared at once, it may very reasonably be supposed that many thousands exist Multitudes must escape observation by reason of their paths traversing only that part of the heavens which is above the horizon in daytime. Comets so circumstanced can only become visible during, a total eclipse of the sun — a coincidence which is related to have, taken place sixty years before Christ, when a a large comet was observed near the sun. § 527. The motion of comets is characterized by the greatest irregular- ity. Sometimes they appear in sight for a few days only, at others for COMETS. 139 many months. Some move very slowly, others with vast velocity ; and not unfrequently the two extremes of speed are exhibited by the same in- dividual in different parts, of its path. Some pursue a direct, others a ret- rograde, and othere a tortuous and very in-egular course ; nor are they confined, like the planets, to any particular region of the heavens, but traverse indifferently every part alike. § 528. Their variations in apparent size, while visible, are equally re- markable ; sometimes they make their appearance as faint, slow-moving objects, with little or no tail ; by degrees they accelerate their speed, en- large and extend their tail, which increases in length and brightness till they approach the sun near enough to be lost in his light. After a time they again emerge on the opposite side, receding from the sun. It is now for the most part they shine forth in all their splendor, and display their tails in greatest length and development. As they continue to recede from the sun their motion diminishes, their tails subside about the head, which grows continually feebler till lost in the distance, from which by far the greater number have never returned ; thus indicating their paths to be along the parabola or hyperbola. § 529. These seemingly irregular and capncious movements are ftillj' explained by the doctrine of universal gravitation, and are no other than consequeuces of the laws of elliptic, parabolic, or hyperbolic motions. But the physical changes of the head, the process by which it builds up the enormous tail, takes it down again, and wraps it as a mantle about itself; the position of the tail as regards the direction of the sun, the multijjlicity of tails, and other physical phenomena to be noticed presently, remain without satisfactory solution. . § 530. The elements of a comet's orbit are readily computed from three observed places, exactly as in the case of a planet;' and the comet usnally takes the name of the computer who thus first defines its track through the heavens. The elements of a few now reckoned among the permanent members of the solar system, will be found in the following table : 140 SPHERICAL ASTRONOMY. O D H a ?3 O n H o S aj J? -o o oa bP -^ d o o d i •13 TS -o n;:! 1 p e^ , ^«3 o CO CO 5 in OS in in t^ i-H CD r-l CT CO Oi Oi CO r^ CD in CM CO ■^ ■^ 05 CT CO l-~ Ir^ lo »n i> CO <* CD ^ lO iO I— 1 m C5 00 r-. m CO t^ O o d d d d ID o CT Ol CD •-H as ■^ oc !>• -"^ M IS- u: 1-1 Oi o 00 c ■-* -^ CT -* in " •* o CO CM m in °S CO C( ,_i CT o I-H 1— CO , Oi CO 00 m f-1 CO - in CO in r-l CO CO a: 05 CO Ci CTl c:5 c§ - o iC « TP CO .^ i^ ■^ in o: SO tM O lO CO -fl o CO O CO ©^ Ol V (N rH t~- CTi in •* - m o rf CO -^ in rr< _H 00 I! " cr -^ o CO CO Ql •rt 1-" o- (T- Ol CO O O u; c ^ ^ p: CO '"' ^S (= CO M in r— 1 ir in CO 95 CT CT CO CO C Tf CO rH JS ir o Cf _< OS ■—I J"" O" •— 1 t> CT in K "Oi-i ■^ CT ^ -^ £ 4- c C Sept. Feb. ir ;£ cc cr -•t CD cr :3^ Tf ■-+ Tf CX OO 00 00 QC p-H t— 1— I-H 1 c W > 1 a , t C 4, ^ « > j2 .5 c .a g ■* ^ o tc 1 ^ PC Pt P f= .1 •§= 03 .r. R ^ ca hn a C S!i n o n & cS O T-l T) o !3 -{ bn fin H c •n o CS J3 COMETS. 141 § 531. By far the most interesting of these comets is that of Ilalley. Its last return took place according to prediction in 1835. While yet remote from the sun in its approach to that luminary, its appearance was that of an oval nebula without tail, and having a minute point of concen- trated hght eccentrically situated within. Soon its tail began to be devel- oped, and increased rapidly till it reached its greatest length, about 20 degrees, when it decreased with such haste as to disappear entirely before perihelion passage. "When the tail first began to form, the nucleus be- came much brighter, and threw out a jet or stream of light towards the sun. This ejection continued, with occasional intermission, as long as the tail continued visible. . Both the form and direction of this luminous stream underwent singular and capricious . alterations, the difierent phases succeeding one another with such rapidity that no two successive nights presented the same appearance. At one time the jet was single, at others fan-shaped, while at others two, three, or more jets were darted forth in different directions, the principal one oscillating to and fi-o on either side of the line drawn to the sun. These jets, though very bright at their point of emanation from the nucleus, faded away, and became diffused as they expanded into the coma, at the same time curving backward as if thrown against a resisting medium. After its perihelion passage, the comet was not seen for two months, and at its reappearance presented itself under a new aspect. There was no longer a vestige of tail ; it seemed to the naked eye a hazy star of the fourth magnitude, and through a powerful telescope a small round well-defined disk, rather more than 2' in diameter, surrounded by a nebulous coma of much greater extent. Within the disk, and somewhat removed from its centre, appeared a mi- nute but bright nucleus, from which extended, in a direction opposite the sun, a short vivid luminous ray. As the comet receded from the sun, the coma* disappeared, as if absorbed into the disk, which increased so rap- idly as in one week to augment its volume in the ratio of 40 to 1. And so it continued to swell out, with undiminished rate, until from this cause alone it ceased to be visible, the illumination becoming fainter as the magnitude increased. While this increase of dimensions proceeded, the fonn of the disk passed, by gradual and successive additions to its length in the direction oi)posite to the sun, to that of a paraboloid, the side towards the sun preserving its planetary sharpness, but the base being so faint and ill-defined, as to indicate that if the process had been continued with suffi- cient light to render it visible, a tail would ultimately have been observed. The parabolic envelope finally disappeared, and the comet took its leave as it came — a small round nebula, with a bright point in or near the 142 SPHERICAL ASTRONOMY. centre. Figures 5 to 10 inclusive, of plates, taken in order, show some of the successive aspects of this comet at its last appearance. , § 532. Many other great comets are recorded, all aflFording peculiarities more or less interesting. § 533. On comparing the intervals between the successive returns of Encke's comet, its periods arei found to be continually shortening ; that is, its mean distance from the sun, or semi-major axis of its orbit, diminishes by slow and regular degrees, and at the rate of about O'^.H during each revolution. This is attributed to the resistance of the ethereal medium which fills the planetary space, and serves as the medium for the transmis- sion of light. This resistance checks the velocity, diminishes the centrifu- gal force, and gives to the sun more effect in drawing the comet towards itself. It will probably ultimately fall into that body. Like the comet of Halle}', its apparent diameter is found to diminish as it approaches to, and to increase as it recedes from the sun. It has no tail, and presents to tlie view only a small ill-defined nucleus, eccentrically situated within a more or less elongated oval mass of vapors, being nearest to that vertex which is towards the sun. § 634. Biela's comet is scarcely visible to the naked eye ; its orbit nearly intersects that of the earth, and had the latter, at the time of its passage in 1832, been a month in advance of its actual place, it would have passed through the comet. At its last appearance it separated itself into two parts, which contin- ued to journey along together, side by side, through an arc of 70 degrees of their orbit, keeping all the while within the same field of view of a tel- escope directed towards them. Both had nuclei, both had short tails par- allel to one another, and perpendicular to their line of junction. At first the new comet was extremely small and faint in comparison with the old : the difference both in light and size diminished till they became equal ; after which the new comet gained the superiority of light, presenting, ac- cording to Lieut. Maury, the appearance of a diamond spark. The old comet soon, however, recovered its superiority, and the new one began to fade, till finally the comet was seen single before it disappeared. While this interchange of light was going on, the new comet threw out a faint bridge-like arch of light, which extended from one to the other. "When the original comet recovered its superior brightness, it in its turn threw forth additional rays, so as to present the appearance of a comet with three tails, forming with one another angles of about 1 20°. The distance be- tween the comets at one time was about 39 times the equatorial radius of the earth, or less than two-thirds the distance of the moon from the earth. COMETS. 143 § 535. The orbits of comets being very eccentric, and inclined under all sorts of angles to tlie ecliptic, these bodies must pass near to the planets, and be more or less affected by their .disturbing action. One passed Jupiter at the distance of j'j- of the radius of that planet's orbit, and the earth, three years afterwards, at seven times the moon's dis- tance. This comet was found by Lexell to have passed its perihelion in an elliptical orbit, of which the eccentricity was O.YSSS, and with a pe- riodic time of about five and a half years, having, in all probability, been drawn into this path by the perturbating action of Jupiter and the earth at its previous visits. Its next return could not be observed by reason of the relative places of its perihelion and of the earth, and before another revo- lution could be accomplished, it passed within the orbit of Jupiter's fourth satellite, and has never been seen since. The action of Jupiter doubtless changed its orbit into an extremely elongated ellipse, or perchance into a parabola or hyperbola ; and what is most remarkable, none of Jupiter'? satellites suffered any perceptible derangement — a sufficient proof of tho smallness of the comet's mass. § 536. The gi'eat number of comets which appear to move in para- bolic orbits, or elliptical orbits so elongated as not to be distinguished from them, has given rise to an impression that these bodies are extraneous to our system, and that our elliptic comets owe their permanent denizenship within the sphere of the sun's dominant attraction to the retarding action of one or other of the planets near which they may have passed, and by which their velocity was reduced to compatibility with elliptic motion. A similar disturbing cause, acting to increase the velocity, would give rise to a parabolic or hyperbolic orbit, so that it is not impossible for a comet to be drawn into our system, retained during many revolutions about the sun, and finally expelled from it, never more to return, as was probably the case with that of Lexell. § 537. The fact that all the planets and nearly all the satellites move in one direction about the sun, while retrograde comets are very common, would go far to assign them an extraneous origin. From a consideration of all the cometary orbits known in the early part of the present century, Laplace found that the average situation of their planes was so nearly per- pendicular to the ecliptic as to afibrd no presumption of any cause biasing their inclinations. And yet as the planes of the eUiptical orbits approach that of the ecliptic, the number of direct comets increases ; and a plane of motion coincident with that of the earth, and periodicity of return, are d& cidedly favorable to, direct motion. 144 SPHERICAL ASTROlfOMT. STABS. § 638. Besides the bodies composing the solar sj'stem, there are a couniless multitude of others which, because they retain their relative places sensibly unchanged are called, though improperly, fixed stars. Like our sun they are poised in space, are self-luminous, and in all probability are centres of planetary systems. § 639. Among these stars, which at first view seem scattered over the celestial vault at random, appears, every evening, a bright band, called the milky way, that stretches from horizon to horizon and forms a zone com- pletely encircling the heavens. It divides in one part of its course into two branches, which unite again after remaining separate for 1-50° of their course. § 540. The most refined observations have been able to assign to none of the stars a sensible geocentric, and to but veiy few only an exceeding small and uncertain annual parallax ; while the most powerful magnifieiB have thus far failed to reveal an appreciable disk. But little can, therefore, be known of their distances, nothing .at all of their real dimensions, and the only means by which one may be distin- guished from another are in the character and intensity of their illumina- tion. § 641. It is usual to arrange the stars into classes called magnitudes, and this without reference to their location in the heavens. The brightest are said to be of the first magnitude, those which fall so far short of the first degree of brightness as to make a strongly marked distinction, are classed in the second, and so on down to the sixth or seventh, which com- prise the smallest stars visible to the naked eye in the clearest and darkest night. § 542. Beyond this, however, telescopes continue the range of visibility down to the 16th ; nor does there seem any reason to assign a limit to the progression, for every increase in the dimensions and pcwver of telescopes has brought into view multitudes innumerable of objects invisible before ; and, for any thing experience has taught us, the number of stars may, to our powers of enumeration, be regarded as absolutely without limit. § 643. The mode of classification into orders is entirely arbitrary. Of a multitude of bright objects, differing in all probability intrinsically both in size and splendor and arranged at unequal distances, one must appear the brightest, another next below it, and so on. An order of succession must exist, and when it is gradual in degree and indefinite in extent, to draw a line of demarkation is matter of pure convention. STARS. 145 § 544. Sir John Herschel proposes to make the scale of decreasing brightness of the stars which head the several orders of magnitudes, to 1 X 1 I 1 vary inversely as the squares of the natural numbers, or as 1, j, ^, y^, &c. ; that is, the brightest star of the first magnitude shall be four times that of the brightest of the second, nine times that of the brightest of the third, and so on: stars of intermediate brightness to be expressed deci- mally. Thus a star half way in brightness between the brightest of the third and of the fourth magnitudes would be expressed by , , On the hypothesis that all the stars possess the same intrinsic brightness, coupled with the fact that the distance of the same luminous object varies inversely as the square root of its apparent brightness, the mere mention of the magnitudes of the stars would suggest, according to this classification, their relative distribution through space. § 545. To accomplish this rig. 95. photometrical classification, he proposes to receive the light from the planet Jupiter, at A, on the first face of a triangular prism, so as to fall on the second face at under an angle of total reflection; this light, on its emergence from the third face, being received upon a convex lens D, would form an image of Jupiter's disk at F. An eye placed at E, within the field of the diverging waves, would re- ceive the light from this image and that from a star proceeding along the line BH. The ap- parent brightness of Jupiter's image would vary inversely as the square of FF, because this planet has no sensible phases, and under the same atmospheric circum- stances is of a constant brightness, while that of the star would be constant for all positions of the eye, and by altering the place of the latter the stai- and the image may be made to appear equally bright. The value of UF being ascertained for different stars, their relative brightness becomes known. ' 10 146 SPHERICAL ASTRONOMY. § 546. Astronomers have generally agreed to restrict the first magni- tude to about 23 or 24 stars, the second to 60 or 60, the third to about 200, and so on, their numbers increasing rapidly as we proceed in the order of decreasing brightness, the number of stars registered to include the sev- enth magnitude being from 12 to 15 thousand. § 547. Stars of the first three or four magnitudes are distributed pretty uniformly over the celestial sphere, the number being somewhat greater, however, especially in the southern hemisphere, along a zone following the course of a great circle through the stars called s Ononis and a Crusis. But when the whole number visible to the naked eye are considered, they increase greatly towards the borders of the milky way. And if the tele- scopic stars be included, they will be found crowded beyond imaginatiou along the entire extent of that remarkable belt and its branches. Indeed, its whole light is composed of stars of every magnitude from such as are visible to the naked eye to the smallest point perceptible through the be«t telescopes. § 548. The general course of the milky way, neglecting occasional de- viations and following the greatest brightness, is that of a great circle in- clined to the equinoctial under an angle of 63°, and cutting that circle in right ascension O"" 47" and 12'' 47™, so that its northern and southern poles are respectively in right ascension 18' 47" and 6'' 47"- § 549. This great circle of the celestial sphere with which the general course of the milky way most nearly coincides, is called the gallactic circle. To count the number of stars of all magnitudes visible in a single field of a telescope, and to alter the field so as to take in successively the entire celestial sphere, is to gauge the lieavens. § 550. A comparison of many different gauges has given the average number of stars in a single field of 1 5' diameter, within zones encircling the poles of the gallactic circle, found in the following TabU. s of North Gallactic Polar distance. Average Number of Starf In field of 15'. 0° to 15° . 4.32 15 to 30 5.42 30 to 45 8.21 45 to 60 13.61 60 to 76 2f09 16 to 90 63.43 STARS. 147 s of South Gallactio Polar distance. Average Nnmber of Stan in field of 15'. 0° to 15° . 6.05 15 to 30 6.62 30 to 45 9.08 45 to 60 13.49 60 to 15 26.29 ' 75 to 90 59.06 Fig. 96. § 661. This shows that the stars of our firmament, instead of being scattered in all dii'ections indilFerently through space, form a stratum of which the thickness is small in comparison with its length and breadth, and that our sun occupies a place somewhere about, the middle of the thickness, and near the point where it subdivides into two prin- cipal laminae, inclined under a small angle to one another. For to an eye so situated, the appai'ent density of stars, supposing them pretty equally scattered through the space they occupy, would be least in the direction A S, perpendicular to the laminae, and greatest in that of its breadth SB, SO, or SD ; in- creasing rapidly in passing from one direction to the other. § 552. For convenience of reference and of mapping, the stars are sep- arated into groups by conceiving inclosing lines drawn upon the celestial sphere after the manner of geographical boundaries on the earth. The groups of stars wjthin such boundaries are called constellations. The brightest star in each constellation is designated by the first letter of the Greek alphabet, the next brightest by the second, and so on till this alpha- bet is exhausted, when recourse is had to the Roman alphabet, and then to numerals. A star will be known from the name of the constellation and the letter or numeral : thus, a Centauri, 61 Cygni. Many of the bright- est stars have also proper names, as Sirius, Arcturus, Polaris, &c. § 553. If, in Eq. (28), p denote the radius of the earth's orbit, * becomes the annual parallax, d the star's distance, and u as before the number of seconds in radius unity. That equation gives ^- = '± (160) / p * § 554. A line connecting the earth and a star would in the course of a year describe the entire surface of a cone of which the vertex would be the 148 SPHERICAL ASTRONOMY. star, and the base the orbit of the earth. The intersection of the nappe of this cone beyond the star with the celestial sphere would be an ellipse, and the apparent orbit of the star, arising from heliocentric parallax. The greater axis of this ellipse would be double the annual parallax. § 555. The stars floating, as it were, in space, and beings subjected to the laws of universal gravitation, must each have a proper motion. In con- sequence of their vast distances from one another this motion may be com- paratively slow, and their excessive distance from us almost conceals it, re- quiring years to describe spaces sufficiently great to subtend sensible angles at the earth. By comparing the relative places of stars at remote periods this proper motion has been detected and measured in a great many in- stances. § 556. Stars having the greatest proper motion are inferred to be near- ■est to us, and this has determined the selection of certain stars in preference to others in the efforts which have been made to ascertain their paral- kxes. § 557. Two methods have been pursued. First, to find by careful me- ridional observations of right ascensions and declinations, cleared from re- fraction, nutation, aberration, and proper motion, the places of the star throughout the year, and thence the distance between those places most remote from one another. This is double the annual parallax. Second, after selecting two stars very near to one another, and of which one has an obvious proper motion and the other not, to measure with the heliometer or micrometer their apparent distances apart, and to note the corresponding positions of the line joining them throughout the year ; then to construct therefrom, after coiTecting for proper motion, the annual path of the moving star. Its longer axis will be double the annual parallax. This second is greatly the preferable method. The stars being separated by a few seconds only, they will be equally affected by refraction, nutation, and aberration, none of these depending upon actual distance. The method supposes the apparently immovable star to be immensely distant beyond the movable one. By the first method Professor Henderson found the parallax of a Cen- tauri to be 0".9ia ; and by the second M. Bessel that of 61 Cyghi to be 0".348. § 558. Assuming the parallax of a Centauri = 1", to avoid multiplicity of figures, substituting it for * in Eq. (160), and writing the numerical value of w, we have d u , , - = - r= 208265 (161) STARS. • 149 and in this proportion at least must the distance of the fixed stars exceed the distance of the sun from the earth. Substituting for p its value, say in round numbers 95,000,000 of miles, and we have d = 206265 X 95000000 — 19595175000000™, or about twenty billions of miles. § 559. Denoting the velocity of light by v, the time required for it to traverse the distance which separates the star from the earth by t, we have first, § 505, V = 192000'", and d t = -= 3''.23 ; V that is to say, it would require light three years and a quarter to come from the nearest fixed star to the earth. And as this is the inferior limit which it is already ascertained that even the brightest and therefore, in the absence of all other indications, the nearest stars exceed, what is to be al- lowed for the distances of those innumerable stara of the smaller magni- tudes which the nftst powerful telescopes disclose in the remote regions of the milky way ? § 560. The space penetrating power of a telescope, or the comparative distance to which a star would require to be removed in order that it may appear of the same brightness through the telescope as it did before to the naked eye, may be calculated from the aperture of the telescope as com- pared with that of the pupil of the eye, and from its power of reflecting or , of transmitting incident light The space penetrating power of the tele- scope employed on the gauge stars referred to in § 560 was 75. A star of the 6th magnitude removed to 75 times its distance would therefore still be visible, as a star, through that instrument, and admitting such a star to have 100th part the light of a standard star of the 1st magnitude, it will follow, from the law of illumination and distance, that such standard star if removed 75 X 10 = 750 times its distance would excite in the eye, when viewed through the telescope, the same impression as a star of the 6th magnitude does in the naked eye. Among the infinite number of stars in the remoter regions of the -milky way it is but reasonable to con- clude that there are many individuals intrinsically as bright as those which immediately surround us. The light of such stars must, therefore, have occupied 750 X 3.25 = 2437.5 years in travelling over the distance which separates them from our own system. And it follows that when we observe the places and note the appearances of such stars, we are only 150 SPHERICAL ASTRONOMY. 0.913 Henderson 0.348 Bessel. 0.261 Sti'uve. 0.280 Henderson 0.226 Peters. 0.133 (1 0.127 u 0.067 " 0.046 s reading their history more than two thousand years before. Nor is thia conclusion, startling as it may appear, to be avoided without attributing an inferiority of intrinsic illumination to all the stars of the milky way — an alternative much less in harmony, as we shall see presently, with astro nomical facts (ionnected with other sidereal systems, revealed by the tele- scope, than are the views just taken. § 561. Of some of the stare whose parallaxes have been determined, the values of the parallaxes, and the names of the discoverers, are given in this Table. a, Centauri 61 Cygni a Lyra Sirius 1831 Groombridge I UrsiE Majoris Arcturus Polaris Capella § 562. As remarked in the beginning of this chapter, the very best telescopes aflford only negative information respecting the apparent diam- eteiB of the stars. The round and well-defined planetaiy disks which good telescopes exhibit are mere optical illusions, the-se disks diminishing more and more in proportion as the aperture and power of the instrument are increased. And the strongest evidence of a total absence of perceptible dimensions is the fact, that in occultations of the stars by the moon, the extinctions are absolutely instantaneous. If our sun were removed to the distance of a Centauri, its apparent di- ameter of 32' 3" waild be reduced to only 0".0093, a quantity which no improvement of our present instruments can ever show with an apprecia- ble disk. § 563. The star a Centauri has been directly compared with the moon by the method of § 545. By eleven such comparisons, after making due allowances for known sources of error, it was found that the light of the full moon exceeded that pf the star in the proportion of 27408 to 1. WoUaston found the proportion of the sun's light to that of the moon to be as 801072 to 1. Combining these results, the light we receive from the sun is to that from a Centauri as 21,955,000,000, or about twenty- two thousand millions to one. Hence, the illumination being inversely as STARS. 151 the square of the distance, the intrinsic splendor of this star is to that of the sun as 2.324'7 to 1. The light of Sinus is four times that of a, Cen- lauri, and its parallax only 0".230, which give to Sirius a splendor equal to 63.02 times that of the sun. § 564. Periodical Stars. — Many of the stars, which in other respects are no way distinguished from the rest, undergo periodical increase and diminution of brightness, involving in one or two instances complete ex- tinction and renovation. These are called periodical stars. § 565. The most remarkable star in this respect is o Ceti, sometimes called Mira. It appears at variable intervals, of which the mean is 331'' 15'' I"- It retains its greatest brightness for a fortnight, being on some occasions equal to a large star of the second magnitude ; decreases for about three months, becoming completely invisible to the naked eye for about five months, and inqreases for the remainder of the period. Such is the general course of its phases. It does not always return to' the same degree of brightness, nor increase nor decrease by the same gradations, neither are the succeissive intervals of maxima equal. The mean interval is subject to a cyclical fluctuation embracing eighty-eight such intervals, and having the effect to shorten and lengthen the same about 25 days one way and the. other. § 566. Another very remarkable periodical star is that called j8 Persei, and also frequently called jilgol. It is usually visible as a star of the second magnitude, and as such continues for 2'' 13''.5, when it suddenly begins to diminish in splendor, and in about 3''.5 is reduced to the fourth magnitude, at which it continues for about IS"". It then begins to in- crease, and in 3''.5 is restored to its usual brightness, going through all its changes in 2'' 20'" 48'° 58".5. Eecent observations indicate that this period is on the decrease, and not uniformly, but with an accelerated rapidity, indicating that it too has its cyclical period, and that instead of continuing to decrease, it will after a while be found to increase. § 567. The star S Cepheus is also a periodical star. Its period from minimum to minimum is 5'' 8'' 47"° 39'.5. The extent of its variations is from the fifth to between the third and fourth magnitudes. Its increase is more rapid than its diminution — the former occupying 1'' 14'', and the latter 3" 19^ § 568. The periodical star /3 Lyra has a period of 12'' 21'' 53'° 10', within which a double maxima and minima take place, the maxima being ' about equal, but the minima not. The maxima are about 3.4, and the minima 4.3 and 4.5. Here agSiin the period is subject to change, which ;b itself peiiodical. 152 SPHERICAL ASTRONOMY. § 669. Numerous other periodical stare are recorded. These remark- able variations of brightness, and the laws of their periodicity, have sug- gested the revolution of some opaque body or bodies around the stars thus distinguished, which, becoming interposed at inferior conjunction, would intercept a greater or less portion of the light on its way to the earth. Or the stars may possess very different degrees of intrinsic illumination on different portions of their surfaces, which, being subject to periodical changes and presented to the earth by an axial rotation of the stars, would produce the phenomena in question. § 570. Temporary Stars. — The irregularities above referred to may afford an explanation of other stellar phenomena, which have hitherto been regarded as altogether casual. Stars have appeared from time to time in different parts of the heavens blazing forth with extraordinary splendor, and after remaining a while, apparently immovable, have faded away and disappeared. These are called temporary stars. One of these stars is said to have appeared about the year 125 b. c, and with such brightness as to be visible in the daytime. Another appeared in a. d. 389, near a Aquila, remaining for three weeks as bright as Venus,, and disap- pearing entirely. Also in 945, 1264, and 1572, brilliant stars appeared between Cepheus and Cassiopeia, which are supposed t'o be one and the same periodical star, with a period of 312, or perhaps 156 years. The appearance in 1572 was very sudden. The star was then as bright as Sirius ; it continued to increase till it suipassed Jupiter, and was visible at mid-day. It began to diminish in December of the same year, and in March, 1574, it had entirely disappeared. So, also, on the 10th of Octo- ber, 1604, a star not less. brilliant burst forth in the constellation Serpen tarius, which continued visible till October, 1605. § 571. Similar phenomena, though of less splendor, have taken place more recently. A star of the fifth magnitude, or 5.4, very conspicuous to the naked eye, suddenly appeared in the constellation Ophiuchus. From ihe time it was first seen it continued to diminish, without alteration of place, and before the advance of the season put an end to the observations upon it, had become almost extinct. Its color was ruddy, which was thought to have undergone many remarkable changes. § 572. The alternations of brightness of ■>) Argus are very remarkable. In 1677 it appeared as a star of the fourth, in 1751 of the second, in 1811 and 1815 of the fourth, in 1822 and 1826 of the second, in 1827 of the first, and in 1837 of the second magnitude. AH at once, in 1838, it sud- denly increased in lustre so as to surpass all the stars of the first magnitude except Sirius, Canopus, and a. Centauri. Then it again diminished, but not STARS. 153 below the first magnitude, till Apiil, 1843, when it had" increased so as to surpass Garutpas, and nearly equal Sirius. § 573. On careful re-examination of the heavens, and comparison of catalogues, many stars are missing. § 574. Double (Stars.^rMany of the stars when examined through the telescope appear double, that is, to consist of two individuals close to- gether. They are divided into classes according to the proximity of their component individuals. The first class comprises those only of which the distance does not exceed 1" ; the second those in which.it exceeds 1", but falls short of 2" ; the third those in which it ranges from 2" to 4" ; the fourth from 4" to 8" ; the fifth from 8" to 12" ; the sixth from 12" to 16" ; the seventh from 16" to 24" ; and the eighth from 24" to 32". Each of these classes is subdivided into two others, called respectively Mnspicuous and residuary double stars. The first comprehends those in which both individuals exceed the 8.25 magnitude, and are therefore sep- arately bright enough to be seen with telescopes of veiy moderate capa- city ; the second embraces those which are below this limit of visibility, . Specimens of each class will be found in the following Table. y Coronse Bor. y Centauri. y Lupi. E Arietis. 5 Heroulis. y Ciroini. i Cygni. t ChaniEeleontis Class 1 — 0" to 1" V Coronse. ti Hercalis. X Cassiopeia;. X Ophiuchi. ■a Lnpi. 7 Ophiuehi. ^ Draconis. TJrsiE Majoris. ;^ Aquilse. Ill Leonis Class II.— 1" to 2". ? Bootis. I Ursa Majoris. I Cassiopeise. i3 Cancri. TT Aquilse. 0- CoronaB Bor. Atlas Pleiadum- 4 Aquarii. 42 Comse. 52 Arietis. 66 Pisoiam. 2 Camelopardi. S2 Orionis. 52 Orionis. a Pisclum. Hydrse. y Ceti. y Leonis. y Coronffi Aus. a Crusis. a Heroulis. a Geminomm. i Geminoram. 5 Coronse Bor. Class III.— 2" to 4". V Virginis. 5 Aqnarii. i Serpentia. $ Orionis. e Bootis. I Leonis. E Draeonis. ' Triangnli. e Hydras. a Leporis. Cla^ IV, e Phosniois. «c Cephei. X Orionis. /I Cygiii. f Bootis. .—4" TO 8". f Cephei. » Bootis. p Caprioorni. c Argus. H Aurigse, fi Draeonis. /I Canis. p Heroulis. a Cassiopeije. 44 Bootis. fi Eridani. 70 Ophiuehi. 12 Eridani. 32 Eridani. 95 Heroulis. 154 SPHERICAL ASTEONOMY. fi OrioniB. Y Arietia. y Delphini. s Centauri. fi Cephei. $ Soorpii. a Canum Yen. ( Kormse. { Fjscium. S Heroulia. I, Lyra. I Cancri. Class V.— 8" to 12"- p Antilse. B Eridani. Class VL— 12" to 16'' V Volantis. 17 Lupi. f TJrssB Majoris. Class VII— 16" to 24". B Serpentis. K OoTonee Aus. X Tauri. Class VIII.— 24" to 32" K Herculis. K Cephei. i/i Dracouis. i Ononis. / Eridani. 2 Canum Veo. It Bootis. 8 Monocerotis. 61 Cygni. 24 Comse. 41 Draconia. 61 Ophiuohi. X Cancri. 23 Orionis. § 575. Triple, Quadruple, and Multiple Stars. — Stars which answer to these designations also occur, and of them the most remarkable are, 1. AndromedsB. B Orionis. ( Soorpii. i Lyra. /I Lapi. 11 Monocerotis, 5 Cancri. It Bootis. 12 Lyncis. Of these, a, Andromedce, (* Bootis, and (* I/iipi, appear through telescopes of considerable optical power only as ordinary double stars ; and it is only when excellent instruments are used that their companions are subdivided and found to be extremely close double stars, e Li/ra offers the remarka- ble example of a double-double star. In telescopes of low power it ap- pears as a coarse double star, but on increasing the power, each individual is perceived to be double, the one pair being about 2".5, the other about 3" apart. Each of the stars ^ Cancri, | Scorpii, 11 Monocerotis, and 12 Lyncis, consists of a principal star closSly double and a smaller and more distant attendant; while ^ Orionis, (Fig. 11, Plate III,) presents four briUiant principal stare of the 4th, 6th, 7th, and 8th magnitudes, forming a trapezium, of which the longest diameter is 24". 4, and accompanied by two excessively minute and very close companions, to perceive both of which is one of the severest tests that can be applied to a telescope. § 576. Of the delicate subclass of double stars, or those consisting of very large and conspicuous double stars, accompanied by very minute companions, the following are specimens, viz. : STARS. 155 ai Cancri. a Polaris. /t Circini. Virginia. ai Capricorni. p Aquarii. k Geminorum. ' x Efidani. a Indi. Y HydrsB. /i Persei. 16 Auriga. a Lyra. i Ursa Major. 7 Bootis. 91 Ceti. § 577. Binary Stars. — ^Many of the double stars are physically con- lected in such proximity to one another as to revolve about their cornmon jentre of gravity in regular orbits. These are called binary stars. They liffer from what are called ordinarily " double stars" in being so near to one another as to be kept asunder only by a rotary motion about a com- mon centre ; whereas the individuals of a double star are separated by a vast distance, and appear double only in consequence of one being almost directly behind the other as seen from the earth. § 578. The position micrometer gives from time to time the apparent distance between the places into -which the stars of a binary system are projected upon the celestial sphere, and also the angle which the arc of a great circle, drawn from one to the other, mates with the meridian passing through either, assumed as the central body ; from these polar co-ordinates, the apparent orbit, as pro- jected upon the celestial sphere, is easily traced. § 579. The relation which is found to connect the distances with the angular velocities shows the stars to be under the control of a central force, and the elliptical form of the orbit, with the eccentric position of the central star, is proof that this force can be no other than that of grav- ■taticto. § 580. Thus, the same principle which, under the influence of distance, directs the satellites about their primaries, and the primaries about our sun, also wheels distant suns around suns, each, perhaps, carrying with it its system of planets, and each planet a group of satellites. § 581. From the micrometrical measurements above referred to, and the intervals of time between them, the elements of the actual stellar orbits are easily computed.* A number of sets are given in the following table : * See Memoirs of Royal ABtronomical Scciety, voL v., p. 171. 156 SPHERICAL ASTRONOMY. i A 4 Irs » s T "w '3" -rr Si 3 ^ ig -§ O l-l pC 1^ R .2 1-i •S o ts <^ w a w a w a w ;^ ffl s w l« o m r* m ro ^ s nn CO CO •-< 00 t^ CO CO o CO o 00 o o ^O CT « CT i? 00 o !>: -* cc 00 » < CO r- o? t- OS (M CO Ift CO CTl CO CO « pH n CM tN u; a rr nr f7) ex CD a a 00 oc 22 H h f-H 1— t 1-1 2 Tl* CO o» o O l« o o o o •* o o o o o CD CO cr; CO Tt -.n c c« cc O) W5 tN o || cS oa Cvt •<*. lO 00 CO OD t- iH l> ^ CD I-) (M ■«*' en m nn cr> <-! CN CO o CT Tft t^ 00 OT (M Ol CM 00 o m c- c: lo u l^ nr' O) a 00 ^ v. er iH rH iH Ot IN i eo on r^ o CO CO cc Ui 'm •H rH »jO CO CO eo nn Tf cs Oi j> CD S.fc '*o rH ■* "^ CO • U) ui lO S -* Tf ■^ CO CO 00 T* CT t- t^ Irt lO Ot t^ O CO eo t^ t- CD Tt* Ci CO T-4 U ca CO CO 00 Tf CO o ■rr nri iT. cc 05 (?« Ol Cl l-l r-1 r-l i-l r-l I-l r^ -H (M CO ■ CO T-t (M **-o nn (D rrt 1^ CT ^ (N c» ifi CO ,0» -ef CO CO in ■^ t- CO rH t^ Qi r-< (?t CM Tj< liO lO CO CM (M Oi -rfl 1-1 lO r^ OD m h- r^ ^ in m -^ in 00 CO ,_( ic rH r^ CD oa m O) a CI w ■ai CC CT r- lO Cv Ifl (M I- (M c^ I— CO iH f-* p-l pH 00 rH , •^ o CO o o O GO r^ o O 00 Tf t^ (M O lO (-1 no <-) r*. u: C£ or Tf W3 cr o 00 ir. wo (? o 1 CO cr o co cc ■g g. g s? 3 p: ^ S g ■^ ? ^ s s GO I^ s CM CO CM s no « o o o o o o o o o o o o o O O o O o o o o ^^ O! m CT 1^ nn t- J- nn w « IT CO r- o cc nr {- nn ■^ IT 'H M ' o r- ph ir or a- Ol IT CD .- QO GC f— f-H rr 1 ■—J CO cr r-. OI 1/5 00 UO C o Ci- Oi rH CM m :% iH i-H n cc |i "« TjJ IT CO- CO CO 1> .00 a o - " ^ r-l c p: STARS. 157 § 582. If the annual parallax of the system, the apparent semi-axis of the stellar orbits, and the earth's radius vector, be substituted respec- tively for P, s, and p, in Eq. (29), d will become the number of linear units in the mean distance between the stars. Assuming the data of the table, selecting a Centauri, and making « = 15".5 and P = 0.913, we have ^= ^-P^ 16.977. p (162) whence the stellar orbits of u. Centauri are (§ 422) about nine-tenths that of Uranus. § 583. Denoting by T the periodic time of a body about its centre, we have, Andlyt. Mechanics, § 201, ys __ k in which a is the mean distance, * the ratio of the circumference to diam- eijer, and k the intensity of the central attraction at the unit's distance. For a second body , 4^«a" whence y/s . but from the laws of gravitation h and k' are directly proportional to the attracting masses, and we have a' a" M : M' :: — : ■—. Making a = p, a' = «? = 16.977 p ; T — 1 year, and T'— 17 years ; then will M denote the mass of the sun and M' that of the central star of a Centauri, and we have from the above proportion that is, the mass of the central star is a little over eight-tenths that of our sun. § 584. Color of Double Stars. — Many of the double stars present the curious phenomena of complementary colors. In such instances the larger star is usually of a ruddy or orange hue, while the smaller one appears blue or green. The double star i Cancri presents the beautiful contrast of 158 SPHERICAL ASTRONOMY. yellow and 6Zm« ; y AnSiromedm, crimson and green. Where there is great difference in the magnitudes of the individuals, the krger is usually white, while the smaller may be colored ; thus, i] Cassiopeice exhibits the beautiful combination of a large white star and a small one of a rich ruddy purple. If this be not the mere optical effect of contrast of brightdess, what variety of illumination two suns — a red and a blue one, a crimson and green one — must afford to the inhabitant" of planets that circulate around them, having sometimes both suns above their horizon at once and at others each in succession, thus producing an alternation of red and blue, crimson and green days ! Insulated stars of a rec! color, almost as deep as blood, occur in many parts of the heavens. § 585. Proper Motions of the Stars. — As might be expected from theii- mutual attractions, however enfeebled by distance and opposing attractions from- opposite quarters, the stars are found to have a proper motion, which in thejapse of time has produced a sensible change of internal arrange- ment. Thus, from the time of Hipparchus,' 130 years b. c, to a. d. 1717, eighteen- hundred and forty-seven years, the conspicuous stars Sinus, Arc- turus, and Aldebaran, are found to have changed their latitudes respect- ively 37', 42', and 33', in a southerly direction. Besides, the observations of modern astronomy prove that such motions do really exist. The two stars 61 Cygni are found to have retained sensibly unchanged their dis- tance apart for the last fifty years, while they have shifted their places ir. the heavens in the same interval no less than 4' 23", giving an annual proper motion to each of 5".3. Of the stars not double, and no way dif- feiing from the rest in any other sensible particular, s Indi and (i, Cassio- peice have the greatest proper motions, amounting annually to 7".74 and 3".74 respectively. § 586. Proper' Motion of the Sun. — The inevitable consequence of a pioper motion in our sun, if not equally participated in by the rest, must be a slow average apparent tendency of all the Stars to the point of the celestial sphere from which the sun is moving, and a corresponding retro- cession from the opposite point — and this, however greatly individual star." may differ from such average by reason of their own peculiar proper mo- tion. This is the necessary effect of parallax, and has ^een detected bj observation. By properly treating the observations on the stars of the northern hemi- sphere, the solar apex, as it is called, or the point towards which the sun was moving at the epoch of 1790, was in right ascension 250° 09', and north polar distance 55° 23'. The sout.hern stars gave, by a similar mode of treatment, right ascension 260° 01', and n^rth polar distance 55° 37': NEBULA. 159 results so nearly identical as to remove all doubt of the sun's proper motion. § 587. All analogy would lead to the conclusion that the sun is de- scribing an orbit of vast extent about the centre of gravity of the group of stars of which it forms a single member, and of which the milky way is to us but the distant trace, while this group may itself be moving as a single system around some other and vastly distant centre. A line drawn tangent to the solar orbit in 1Y90 pierced the celestial sphere near the stars ir Herculis and a Columba, the sun being then moving towards the former and from the latter. And the result of calculations thus far gives to the sun a velocity of 422,000 miles a day, or little more than one-fourth the earth's rate of annual motion in its orbit. NEBULA. § 588. Besides the stars which appear as shining points, there are cloud-like patches of light to be seen scattered here and there over the celestial vault. These are called nebulce. They present themselves under great variety of shapes and sizes, as exemplified in Figs. 12, 13, 14, Plates in. andlV., and exhibit in the telescope different characters of internal structure with every increase of optical power. They are very unequally distributed over the heavens. In the northern hemisphere, the hours 3, 4, 5, 16, 17, and 18 of right ascension are singularly poor, while the horn's 10, 11, and 1 2, especially the latter, are exceedingly rich in these objects. In the southern hemisphere a much greater uniformity prevails, with two remark- able exceptions, to be noticed presently. They have no decided tendency to any particular region. § 589. When viewed through the telescope, many nebulae are resolved into stars, and the number that thus yield their cloud-like aspect increase.s with every augmentation of instrumental power. Nebulae are therefore classified, with reference to their appearance through the telescope, into resolvable, irresolvable, planetary, and stellar nehulce, and nebular stars. § 590. Resolvable Nebuloe. — These are usually called clusters of stars. Some are very broken in outline, while others are so regular as to suggest the prevalence of, some internal action productive of symmetrical arrange- ment among their internal parts, § 591. Irregular clusters are much less rich in stars, and much less condensed towards the centre. In some the stars are nearly of the same size, in others very different. The group called the Pleiades, in which six 160 SPHERICAL ASTEOIfOMY. or seven stars may be counted with the naked eye, and fifty or sixty with the telescope, is cue of the most obvious examples of this class. Coma Berenices, represented in Fig. 15, Plate IV, is another such gi5)up. § 592. Globular Clusters.— '-These take their name from their round appearance. They are much more difficult of resolution, and some have frequently been mistaken for comets without tails. When viewed through the telescope, they are found to be composed of stars so crowded together as to occupy an almost definite outline, and to run up to a blaze of light towards the centre, where their condensation is greatest. It would be vain . to attempt to count the stars in these clusters ; some have been estimated to contain five thousand, within an area not greater than the tenth part of the lunar disk. § 593. Elliptic NehulcB. — The figure ''here again suggests the name. They are of all degrees of eccentricity, from moderately oval to elongations so great as to be almost linear. In all, the density increases towards the centre, and generally their internal strata approach more nearly the spheri- cal form than their external. Their resolvability is greater in the central parts ; in some the condensation is slight and gradual, in others great and sudden. The largest and finest specimen of elliptic nebulse is in the Girdle of Andromeda, given m Fig. 12, Plate III. § 594. Annular nebulce also exist, but are veiy rare. The most con- spicuous of this class is found between j8 and y Lyra;, and may be seen through a telescope of moderate power. The central vacuity. Fig. 16, Plate IV, is not quite dark, but appears as a light-colored gauze stretched over a hoop. The powerful telescope of Lord Rosse resolves this nebula into excessively minute stars, and shows filaments of stars hanging to its edge. § 595. Spiral Nehulm. — These are most curious objects. Their dis- covery is but very recent, and is due to the powerful instrument of Lord Eosse. As their name indicates, they appear to consist of a spiral or vor- ticose arrangement of stars diverging from a centre, and suggest the idea of a vast self-luminous mass of matter, travelling to a common destination along separate curvilinear paths. Their form and general appearance are represented in Figs. \1 and 18, Plates IV aii^ V. § 596. Planetary Nehuloe. — These take their name from the planet- like disk which they present. In some instances they bear a perfect re- semblance to a planet in this respect, being round or slightly oval, and quite sharply terminated. In some the illumination is perfectly equable ; in others mottled, and of a peculiar texture, as if curdled. They are com- NEBULA 161 parativelv rare, not above four or five and twenty having been observed, and of these nearly three-fourths are in the southoj'n hemisphere. They are sometimes colored; one, whose right ascension in 1830 was 10'' 16" 36', and north polar distance 107° 47', is of a sky-blue; it is slightly el- liptical, and has an apparent diameter of 30". One of the largest of these objects is near /3 Ursa Majoris; its apparent diameter is 2' 40", which, supposing its distance not greater than 61 Cygni, would imply a linear one seven times greater than Neptune's orbit. The light of this stupen- dous globe is perfectly equable, except at the edge, where it is slightly softened. The Rosse telescope is rapidly transfening the planetary to the annular and spiral nebula. § 597. Double Nehulce. — These occasionally occur, and the constituents are most commonly spherical, and belong, most probably, to the globular clusters. Figs. 19 and 20, Plate V. § 598. Nebulous Stars. — These consist of stars surrounded concentiic- ally by a perfectly circular disk or atmosphere of faint light, Figs. 21 and , 22, Plate V ; in some cases diminished gradually on all sides, and in others suddenly terminated. Ifi right ascensions 7'' 19"" 8', 3'' 58"" 36", and north polar distances 68° 45', 59° 40', are stars of the eighth magnitude sui- rounded by photospheres of the kind just described, respectively 12" and 25" in diameter. § 599. Nebulous Double Stars. — In Fig. 23, Plate VI, is represented an elliptical nebula having its longer axis about 50", in which, symmetrically placed and rather nearer the vertices than the foci, are the equal individuals of a double star, each of the 10th magnitude. In a similar connection, Fig. 24, Plate VI, represents two unequal stars situated at the extremities of the major axis. In light ascension 13'' 47"", north polar distance 129° 9', is an oval nebula of 2' diameter, having near its centre a close double star of the 9.10 magnitude, not more than 2" asunder. Other instances might be adduced of nebulas uniting great peculiarities of shape with regularity of outline. § 600. Irregular Nebulce. — Besides the nebulae just described, there is a class totally different in character, being of great extent, utterly devoid of all symmetry of form, and most remarkable for the extent of their con- volutions and the distribution of their light. No two of them can be said to present any similarity of aspect, and the only thing in common is their locality, which is in or near the borders of the milky way, the most remote being that in the sword handle of Oiion, about 20° from the gal- lactic circle, and represented in Fig. 13, Plate III. But even this i» m the prolongation of a faint offset of the milky way. 11 162 SPHEEICAL ASTRONOMY. These nebulae may be grouped into four great masses, whict occupy the regions of Orion, of A^o, of Sagittarius, and of Cygnus. § 601. The Magellanic Clouds, or the Nuhemlce (Major and' Minor), as they are called in celestial maps and charts, are two nebulous or cloudy masses of light conspicuously visible to the naked eye in the southern hemisphere, and in appearance and brightness resemble portions of the milky way of the same size. They are in shape somewhat oval, the larger deviating most from the circular form. The larger is situated between the hour circles 4'" 40"" and 6'" 40", the parallels 156° and 162° north polar distance, and occupies an area of about 42 square degrees. The lesser, which is between the hour circles O"" 28'" and 1'" 15™, and the parallels of 162° and 165° north polar distance, covers about ten square degrees. The general ground of both consists of large tracts of nebulosity in every stage of resolubility, from light irresolvable up to perfectly separated stars like the milky way, including groups sufficiently insulated and condensed to come under the designation of irregular and globular clusters, the latter being in every stage of condensation. In addition they contain nebular objects quite peculiar, and which have no analogy in any other part of the heavens. Globular clusters, except in one region of small extent, and neb- ula; of regular elliptic forms are comparatively rare in the milky way, but .ire congregated in greatest abundance in parts of the heavens the most re- mote possible from the gallactic circle ; whereas in the Magellanic Clouds they are indiscriminately mixed with the general starry ground. § 602. Regarding the nubeculce as spherical in form, and not as vastly long vistas foreshortened by having their ends turned towards the earth — which would be improbable seeing there are two of them close together — the brightness of objects in their nearer portions cannot be much exagger- ated, nor those in its remoter much enfeebled by diflference of distance. It must, therefore, be an admitted fact that stars of the 1th. and 8th mag- nitudes and irresolvable nebulse may coexist within limits of distance com- paratively small, and that all inferences in regard to relative distance drawn from relative magnitudes must be received with caution. § 603. Our Sun a Nebulous Star. — ^Various phenomena indicate that our sun is itself a nebulous star. The chief is that called the zodiacal light, which may be seen on any clear evening soon after sunset about the months of March, April, and May, and at the opposite seasons of the year just before sunrise, as a OQuically-shaped light, extending from the hori- zon upwards in the directioii of the sun's equator. The apparent angular distance of its vertex Y from the sun S varies from 40° to 90°, and its breadth at its base, perpepdicjilarly to its length, frcm 8° to 30°. Every NEBULA. jg3 drcumstance connected with it indicates it to be ^'s- »«• a lenticularly-formed envelope surrounding the ^ sun, and extending beyond the orbits of Mercury JRcrizon • \ and Venus and even to the Earth, its vertex having been seen 90° from the sun in a great "V ^\ circle. Different parts of the heavens furnish '_, examples of similar forms. Figs. 25, 26, 27, '•; Plate VI. § 604. Aerolites. — Nothing prevents that the particles of this vast ma- terial envelope may have tangible size and be at great distances apart, and yet compared with the planets, so called, be but as dust floating in the sunbeam. It is an established fact that masses of stone and lumps oi iron, called Aerolites, do occasionally fall upon the earth from the upper regions of the atmosphere, and that they have done so since the earliest records. On the 26th April, 1803, one of these bodies fell in the imme- diate vicinity of the town of L'Aigle, in Normandy, and by its explosion into fragments, scattered thousands of stones over an area of' thirty square miles. Four instances are recorded of persons having been killed by the descent of such bodies, and after every vain attempt to account for them as coming originally from the earth, and even from the moon, by volcanic projections, their planetary nature is now generally admitted. Their heat when fallen, the igneous .phenomena which accompany them, their explo- sion on reaching the denser regions of our atmosphere, are accounted for by the condensation in front of them created by their enormous velocity, and by the relations of air, in a highly attenuated state, to heat. § 605. Meteors. — ^Besides these more solid bodies, others of much less density appear also to be circulating around the sun at the distance of the earth from that luminary. These on coming within the atmosphere ap- pear as shooting siars, followed by trains of light, and are called Meteors. They appear now and then as great fiery balls, traversing the upper re- gions of the atmosphere, sometimes leaving long luminous trains behind them, sometimes bursting with a loud explosion, and sometimes becomitjg quietly extinct Among these latter may be mentioned the remarkable meteor of August 18th, 1783, which traversed the whole of Europe, from Shetland to Rome, with a velocity of 30 miles a second, at a height of 50 miles above the earth, with a light greatly surpassing that of a full moon, and diameter quite half a mile. It changed its form visibly and quietly, separated into several distinct parts, which proceeded in parallel direc- tions, each followed by a train. § 606. On several occasions meteors have appeared in astonishing 164 SPHERICAL ASTRONOMY. I numbers, falling like a shower of rockets or' flakes of sn>.w, illuminating at once whole continents and oceans, even in both hemispheres. And it is significant that these displays have occurred between the 12th and 14th Novemberand 9th and 11th A'ngust. In November they are much more brilliant, but their returns less certain than in August, when numerous large and brilliant shooting-stars with trains are almost sure to be seen. § 607. Annual periodicity, iiTespective of geographical location, points at once to the place of the earth in its orbit as a necessary concomitant, and leads to the conclusion that at that place the earth enters a stratum, or annular stream of meteoric planets, in their progress of circulation around the sun. The earth plunging in its annual course into a ring of these bodies, and of such thickness as to be traversed in a day or two, their motions, referred to the earth as at rest, would be sensibly uniform, recti- linear, and parallel. Viewed from the centre of the earth, or irom any point on its surface, neglecting the diurnal as being insignificant in com- parison with the annual motion, their paths wctfild appear to diverge from a common point on the celestial sphere. Now this is precisely what hap- pens. The vast majority of the November meteors appear to describe arcs 6f great circles passing through y Leonis, and those of August appear to move along paths having a common point in (3 Camelopardi. § 608. As the ring may have any position and be of an elliptical fig- ure having any reasonable eccentricity, both the velocity and direction of each meteor may differ to any extent from those of the earth, so there is nothing in the great difference of latitude of these meteoric apices at all opposed to the foregoing conclusion. § 609. If the meteoric planets were uniformly distributed in the sup- posed ring, the earth's annual encounter with them would be certain if it occurred once; but if such ring be broken, and the bodies revolve in groups, mth periods differing from that of the earth, years may pass with- out rencontre, and when such happen, they may differ to any extent in intensity of character, according as the groups encountered are richer oi poorer in the number of their elements. § 610. From careful observations, made at the extremities of a base ■ 50,000 feet long, it has been inferred that the heights of meteors at the instant of first appearance and disappearance, vary from 16 to 140 miles, and their relative velocities from 18 to 36 miles a second. "Altitudes and velocities so great as these clearly indicate an independent planetai-y circulation round the sun. § 611. It is not impossible that some of these bodies may have been converted by the superior attraction of the earth, arising from greater prox- EPHEMERIDES. 165 imity, into permanent satellites ; and there aio those who believe in the existence of at least one of these bodies, which completes its circuit about the earth in about S"" 20"", and therefore at a mean distance of about '5000 miles. EPHEMERIDES. § 612. The facts and principles now explained enable us to predict the aspect of the heavens, or positions of the heavenly bodies, for all future time. This prediction is usually drawn up in the condensed form of tables, which are called ephemerides. The table relating to any one body is called the epheneris of that body, as the ephemeris of the sun, of the moon, &a. § 613. Ephemerides are prepared in advance to subserve the wants and Jiromote the interests of navigation, geography, and chronology, as well as of future astronomical discovery and research. § 614. To facilitate the computation of the ephemeris of a body, it is usual first to construct what are called its tables ; and the manner of doing this may best be explained by taking a particular example, say that of the sun, or rather the earth, since this is the moving body ; but as the place of the sun, as seen from the earth, differs from that of the earth as seen from the sun by the constant 180°", we shall speak of the sun. § 615. We have seen, § 197, how the mean longitude of the sun, his mean motion, longitude of the perigee, and eccentricity, may be found from observation and computation. These elements being found at epochs widely separated from one another, the changes which take place in the last three, and the rate of motion of the perigee, are ascertained. § 616. Having fixed upon any epoch, say mean noon or midnight, 1st January, 1800, any interval of time, either after or before the epoch, mul- tiplied by the mean motion of the sun in longitude, will give the increase of mean longitude during that interval, and being added to the mean lon- gitude at the epoch and the suni divided by 360°, the remainder will give the mean longitude at the beginning of the interval, if it be before, or end, if it be after the epoch. These longitudes, with the corresponding dates, being tabulated, give what is called a table ofepoclis, which tells by simple inspection the mean longitude on any given day, hour, minute, and second. § 617. The same process being performed with reference to the longi- tude of the perigee and its rate of change, gives a corresponding table in jvhich the longitude of the perigee is found. § 618. Resuming Eq. (o), Appendix No. V., and causing m t', which is the mean anomaly, to vary from 0° to 360°, corresponding equations of 166 SPjiEKI0 4.L ASTltONOMY. the centre will result, and these properly arranged form a table of equations of the centre, of which the arguments, as they are called, "are the mean anomalies. Then causing the eccentricity to vary according to ascertained rates, the same equation gives the elements of an additional table by which the equations of the centre may be corrected from time to time. § 619. Nutation causes the true equinox to oscillate about a mean place, its distance therefrom being equal to the algebraic sum of two func- tions, of which one depends upon the longitude of the moon's node, the other upon the longitude of the sun, and both upon the obliquity of the ecliptic. Tables containing the values of these functions for assumed places of the moon's node and of the sub, give the numbers whose sum is equal to the equation of the equinoxes in Imc/itude. § 620. In addition, the larger of the planets, especially Venus and Ju- piter, disturb the earth's orbit. These perturbations are computed by pro- cesses in physical astronomy, and their values arranged under heads that, give the angular distances of the disturbing planets from the earth as seen from the sun, and, together with the place of the moon's node, furnish the arguments with which other tables are entered that give the corresponding eiFects upon the sun's longitude. § 621. Lastly, as the purpose is to find the place where the sun's centre is to be seen, provision is made for the effect of aberration. This in the case of the sun is nearly constant, and equal to — 20".25, because of the small eccentricity of the earth's orbit, the greatest variati6n there- from being less then 0".35. This constant is included in the epoch tables. § 622. Ephemerii of the Sun. — We are now prepared to find where the sun has been and where he will be on the celestial sphere throughout time. For this purpose, enter the table of epochs with the date, take out his mean longitude and the longitude of the perigee ; the difference will be the mean anomaly, with which enter the table of the equations of the centre and take out the coiresponding equation ; add this to or subtract it from the mean longitude according to its sign, and the result will be the true lon- gitude of the sun as affected by nutation and perturbations. Take these latter from the appropriate tables, and we have True longitude of sun = mean longitude -\- equation of the centre -j- nuiation or equation of equinoxes in longitude + perturbations. § 623. With the true longitude and obliquity of the ecliptic, we pass, by spherical trigonometry, § 149, to right ascension and declination. § 624. The mean anomaly in Eq. (n), Appendix V., gives the corres- ponding true anomaly ; and the latter in Eq. (c), same Appendix, gives the EPHEMEEIDBS. 167 radius vector r, which in equations (28) and (29) give the correspond- ing horizontal parallax and apparent diameter. § 625. The mean longitude corrected for the equation of the equi- noxes in right ascension, and diminished by the right ascension, gives the equation of time. § 626. These and other elements being determined &,t different epochs, say for every noon on some nxed meridian, their consecutive differences, di^'ided by the number of hours between the epochs, give the homly changes, and therefore the means of finding the value of the elements themselves for any other meridian. The elements with their hourly changes make up, when properly tabu- lated, an ephemeris of the sun. § 627. Ephemeris of the Moon. — The motion of the moon is altogether more irregular and complicated than the apparent motion of the sun, owing mainly to the disturbing action of this latter body. But these and other perturbations have been computed and tabulated, and from these tables, including those of the node and inclination, the places of the moon in her orbit are found in much the same way as those of the sun in the ecliptic. The mean orhit longitude of the moon and of her perigee are first found and corrected ; theirdifference gives her mean anomaly, opposite to which in the appropriate table is found the equation of the centre, and this being applied with its proper sign to the me&n oi;bit longitude gives the true orbit longitude. § 628. Let E be the earth, if' the moon, Fig- M- V the vernal equinox, VM' an arc of the j^r^ 1^' ecliptic, VQ of the equinoctial, and MM'oi a .--■'' \ p-. ^ circle of latitude; then will MM' be the ,,--' \ / yQ latitude and Fil/"' the longitude of the moon, S;'- \/ / VN the longitude of the node and VEN \ "yi / + NEM the orbit longitude of the moon. "--, /^J[ Subtracting from the orbit longitude of '■y^ / the moon the longitude of the node, the re- mainder NM will be the moon's angular distance from her node. This and the inclination MNM' will give, in the right-angled triangle MNM', the latitude MM' and the side NM', which latter added to the longitude of the node N V gives the longitude VM'. The latitude and longitude, together with the obliquity of the ecliptic, give, § 153, the right ascension and declination. The radius vector, equatorial horizontal parallax, apparent diameter, &c., are computed as in the case of the sun. And thus an ephemeris of the moon is constructed. 168 SrHERICAL ASTKONOMY. Fig. too. § 629. Ep%emeris of a Planet. — From tables of a planet its true orbit longitude as seenfrom the sun is found, as in the case of the moon as seen from the earth. From the heliocentric orbit longitude, heliocentric longi- tude of the node, and inclination, the heliocentric longitude and latitude, together with the radius vector, are found ; just as the corresponding geo- centric elements of the moon are found from similar data relating to the lunar orbit ; and from the heliocen- tric longitude, latitude, and radius vector, we pass to the geocentric, thus: § 630. Let P be the planet, E the earth, S the sun, and the projection of the planet upon the plane of the ecliptic. Draw from S and E the parallels S V and EV to the vernal equinox, and r = ES = radius vector of earth ; r' = SP = radius vector of planet; X = FS = heliocentric longitude of planet ; X' = V EO = geocentric longitude of planet; 6 = P S = heliocentric latitude of planet ; &' =z P E = geocentric latitude of planet; S = S E = commutation : = 80E = heliocentric parallax ; E— SEO = elongation ; = V'ES = longitude of sun. Then and because we have SO = r' cos 6; rST= V'ES ^SGO°- O, S= 180°— (360°— ©) — X= — 180° — X; whence the commutation is known. Then in the plane triangle OES, r' cosd + r : r' cosd — r : : tan i {^-1- 0) : tan i {E— 0); S+ 0-f -£'=180°, but whence ^(^+O) = 90°-| . (163) EPHEM-ERIDES. Substituting this above, we have and making tan| {E -0) = tany: : cot ^ ;S , ^ r' cos 6 — > cos cos + r' 169 tan \{E-0) = coi\S. ^^"^ \ = cot-^ 5tan (^ - 45°) . . (164) Knowing from Eq. (163) the half sum of ^and 0, and from Eq. (164) their half difierenoe, E and become known. And we have X' = ^-(360°-,©) = ^+ O - 360° . . . (165) § 631. Again; FO:=EO.U.n6' = SO.tan6; whence tan^' _ S" _ sin ^_ tana "~ Ed ~ sinS' and ^ ,, ^ , sin ^ tan a' = tan a . -^ — ^ (166) sm (5 ^ ' From equations (165) and (166) the geocentric longitude and latitude be- come known. § 632. Denote by r" the distance EF of the planet from the earth; then will ^0 = r" cos a' and >S = r' cos «; and in the triangle ES r" cos a' : r'cos 4 : : sin (S : sin E\ whence , cos a sin S cos a' ' sin E (167) The right ascension, declination, horizontal parallax, and apparent diam- eter, are found as in the case of the sun and moon. § 633. The ephemerides most commonly used in this country are those computed for the meridian of Greenwich, England, and published several years in advance under the title, " Nauticai Almanac and Astkonom- iCAL Ephemeris." 170 SPHERICAL ASTRONOMY. CATALOGUE OF STAES. § 634. Another important, indeed indispensable auxiliary to practica. astronomy, is a catalogue of stars. This consists of a list of certain stars arranged in the order of their right ascensions, with the means of obtaining the right ascensions and declinations of the places in which they appear at any given epoch. § 035. By precession, § 157, nutation, § 156, and aberration, § 215, the right ascension and declination of a star are ever varying. The place of a star referred to the mean equinoctial and mean equinox is called its mean place; that referred to the true equinoctial and true equinox, its true place ; and that in which it is seen referred to the true ^ equinoctial and true equinox, its apparent place. The true place is equal to the mean, corrected for nutation ; and the apparent place is equal to the true, corrected for aberration. The true and mean places are found from the apparent, by applying the same correc- tions, with their signs changed. § 636. The apparent places of the stars are used as points of reference on the celestial sphere ; and knowing the right ascensions and declinations of these places, those of the apparent place of any other object become known also when the distance of the latter in right ascension and declina- tion from one or more stars is found by instrumental measurement. § 637. Annual Precession,. — The annual precession for any year is, Luni-solar = 50".37572 —yx 0".0002435.890, General = 50".21129 + y X 0".0002442966 ; in which y denotes the number of the year from 1750, minus when lefore that epoch. § 638. The epoch of the catalogue which will be refeired to hereafter, that of the British Association, is January 1st, 1850. Making y = 100, and denoting the nutation of obliquity by ^ u, we have A u = 9".2600 COS Q, - 0'.0903 cos 2 Q -1- 0".0900 cos 2 J 40".5447 cos 2 ; in which SI denotes the mean longitude of the moon's node, D the true longitude of the moon, and O the longitude of the sun. § 639. And assuming the mean obliquity of the ecliptic for 1850 equal to ij = 23° 27' 31", we have then for the nutation in longitude, denoted by^X, A £= - 17".3017 sin Q, -\- 0".2081 sin 2 SJ - 0",2074 sin 2 ]) - 1".2552 sin 2 3. CATALOGUE OP STABS. 171 § 640. Denoting the equation of the equinoxes in right ascension hy ^ A, we have aA = — 16".872 Bin JJ + 0".192 sin 2 S3 — 0".190 sin 2 D - 1".B00 sin 2 0. § 641. Denoting the right ascension and declination of any hody by a and 5 respectively, and by p and p', its change in the same due to an- nual precession, then will p = 46".05910 + 20".05472 sin a . tan 5 . . . (168) p' = 20".054'72 cos a (169) § 642. The change in right ascension and declination for any fractional portion of the year will be found by multiplying the above by t — = 0.00273785 xd .... (170) 365.25 ^ '' In which d denotes the number of days from the beginning of the year to the end of the fraction. § 643. Denoting by ^a,, and ^S, the change in right ascension and dec- lination arising from nutation, then, omitting terms involving sin 2D, will La, = — (15".872 -|- 6".888 sin a . tan S) . sin Q — 9".250 cos o . tan 5 . cos fl ) -f (o".191-+0".083 sin a . tan i) . sin 2a-f 0".090 cos a . tan i . cos iQ, V (171) — {l".151-|-0".50O . sin a . tan i) . sin 20— 0".545 cos a.ta.nS. cos 20 ) A 5, = 9".250 . sin a . cos Q, — 6".888 cos a . sin ft ) — 0".090 sin a cos 2 a + 0".083 cos a . sin 2 SJ > . (172) -f- 0".645 sin a . cos 2 O — 0".500 cos a . sin 2 ) § 644. Aberration. — Denoting by ^ a.^ and J S^ the change in right ascension and declination arising from aberration, disregarding the eccen- tricity of the earth's orbit, A as =^ — (20''.4200 sin © . sin a -f- 18".7322 cos cos aj.secS . . (173) A 52 = — (20''.4200 sin © . cos a — 18".7322 cos © sin a) sin I ) .^ , — 8".1289 cos© cosi ) § 645. Multiplying Eq. (171) by Eq. (170), adding together the prod- uct and equations (171) and (173), and denoting the apparent right as- cension by a and the mean by a', there will result, after suitable reduction, ■ _a = Aa = (<— 0.348 8ln J3-l-0.004 6ln2JJ — O.C25 5in2©)x(46".059-l-20".056Bliiatim5) — (9".250co3 ^— O''.090cos 2S3-]-0".545 cos 2 ©). cos a . tan i — 20".420 sin © . sin a . sec S — 18".732 cos . cos a . sec i — 0".0530 sin Q, + 0".000 sin 2 ft — 0".0039 sin 2 ©. 172- SPHERICAL ASTRONOMY. Muitiplying Eq. (1'72) by Eq. (1'70), adding together tlie product and equations (171) and (174), and denoting the apparent declination by S and tlie mean by S', we also have, after reduction, I' -S=Ai = (t- 0.343 Bin ft + 0.004 sin 2 ft - 0.025 sin 2 ) X 20''.055 cos a + (9".250 cos Si — 0".090 cos 2 ft + 0".545 cos 2 0) sin a — 20".420 sin O .• cos o . tan i5 — 18".'732 cos (tan u . cos S — sin a . sin i). Neglecting the three last terms in the value for A a as insignificant, and making A= — 18".732 cos ©, B— — 20".420 sin ©, C=t — 0.025 sin 2 O — 0.343 sin Si 4- 0.004 sin 2 Si, D= — 0".545 cos 2 — 9".250 cos Si .+ 0".090 cos 2 Si, a = cos a, . sec S, h = sin a . sec S, c = 46".059 + 20".055 sin a . tan 5, d = cos a tan S, a' =: tan to . cos S — sin a . sin S, V = cos a . tan 5, c/ = 20".055 cos a, d' = — sin a ; the above become Aa. = a.A + b.B + c.C+d.I) .... (175) A5 = a'.A + b'.B + c'.C+d'.3 .... (176) § 646. Proper Motion. — To the foregoing must be added the proper motion of the star when it is known with sufficient accuracy, and is of sufficient magnitude to be taken into the account. Equations (173) and (174) enable us to pass from the apparent to the true, or from the true to the apparent right ascension and declination of a star. § 647. Since the motion of the equinoxes is very slow, the values of the functions a, b, c, d, a', b', c', and d^ will be sensibly constant for a number of years, particularly when the stars are not very near the poles, while those of the functions A, B, C, and D vary sensibly from day to day. These latter are, therefore, computed for every day in the year, and their logarithms recorded in the astronomical epheraeris ; the others are. com- puted for the epoch of the catalogue, and their logarithms recorded oppo- site each star in the catalogue. CATALOGUE Oi" STARS. 173 § 648. Construction of tlie Catalogue. — ^The elements relating to each star occupy a portion of the two pages exposed to view on opening the catalogue. On the left-hand page will be found every thing relating to right ascension, and on the right, to declination. The l^ft-hand page con- sists of eleven vertical columns : in the first is placed the number of the star, in the order of its right ascension ; in the second, the name of the con- stellation in which it is situated, with its letter or number; in the third, its magnitude; in the fourth, its mean right ascension, January 1st, 1850, in time ; in the fifth, its mean annual precession in right ascension, Eq. (168), reduced to time ; in the sixth, its secular variation, reduced to time ; in the seventh, its proper motion in right ascension, reduced to time ; and in the eighth, ninth, tenth, and eleventh, the logarithms of the functions a, 6, f, and rf, reduced to time, respectively, each preceded by the sign of the fufiction to which it belongs. The right-hand page consists of fifteen vertical columns, in the first of which the number of the star is repeated ; the second contains the mean north polar distance, January 1st, 1850 ; the third, fourth, and fifth, the annual precession, secular variation, and proper motion in north polar distance, respectively ; the sixth, seventh, eighth, and ninth, the logarithms of the functions a', 6', c', and d! re- spectively, each preceded by the sign of the function to which it be- longs ; the remaining columns contain the numbers by which the star is recognized in the catalogTies of the several authors, whose names are at the top. Example. — Required the apparent right ascension and dechnation ol y Orionis, February 5th, 1854. Mean a January 1st, 1850 . 4 years' prec. and pr. motion Mean a January lat, 1854 . h. m. s. o ' " 5 17 05.33 Mean N. P. D. . . . 83 47 26.7 -f- 1 2.88 4 y'rs' prec. and pr. motion — 14.9 5 17 18.21 Mean N. P. D. . . . . 83 47 10.8 a A , and jP* ^ perpendicular to US, and we have AC = p. sin (l — d); B I) = p .sin {I + d) ; £J = p. cos [l — d); JEI> = p.cosll + d). Also, Eq. (28), BS=p.l; EM^p.^; whence ^^='(M)='- P* From the figure, SC=ES-EG=p.-- - p . COS (/ — d), Then in the triangles ^ C^ and SMA', SO : SM :: AC : A'M, and by substitution A'M-p. -^(^-'^). 1 . COS (« - P-* • p 1 -d) also, SD = I!S + ED = p.- + p.cos{l + d); and in the same way as above, from the triangles S B B and S M B\ B'M=p.-^l±^—.^r l + -.cos(Z + rf) But * can never exceed 9", and to is equal to 206264".8, so that the terms into which * -=- u enters as a factor may be neglected, and we have MA' = p.&m{l-d).—~ (181) MB' = p.sm.{l + d).^^- (182) From which we see that the length of the projection of any dimension at the earth, and parallel to the circle of projection, is found by multiply- ing this dimension by (P — *) -;- P. § 660. Denoting the conjugate axis A'B' by 2 6, we have 2 6 = MB' - MA' ; 12 178 SPHERICAL ASTEONOMY. and by substitution, Also, i = p . cos I .sin d . FA: P-* (183) p . cos ? ; and because that diameter of the parallel of latitude, which is perpendic- ular to A B, is parallel to the plane of projection, we have, denoting the semi-transverse axis of the ellipse by a, , P — It , V a = p . cos i . — = — ...... (184) And denoting the distance M F' from the centre of the circle of projec- tion to that of the ellipse by Y, we have, taking half sum of equations (ISl) and (182) P — It 7"= p , sin Z . cos c? . — ^ — (185) § 661. Revolve the parallel of latitude about AB till it coincides with the meridian section. When the observer is at A, it is to him apparent noon ; when at B, apparent midnight; when at 0, the angle OF A is the apparent hour angle of the sun, and therefore local apparent time. , Fig. 102 bis. Draw T perpendicular to A B, and S L through the point T. The projeotion oi F L will give the distance of the sun from the transverse, and that of This distance from the conjugate axis of his elliptical path. Denote the first by y, the second by ar, and the hour angle F A by A. Then FO = FA=f.(i(^l; T =^ f . cos 2 . sin A ; F T = p , cos / . cos A ; PROJECTIOlir OF A SOLAR ECLIPSE. 179 and since FL T is sensibly a right angle, the value cX. E STJ, which is much greater than E ST^nnMw exceeding Q" ; and because FTL = UE P = d,yie hsive , FL = ^ 7'. sin rf = p . cos Z . cos A . sin rf ; and projecting FZ and T on the circb of projection, there will result = p . cos I .siad . cos , a; = p . cos I .sin h . P-* (186) (187) But p -^ P is the linear subtense of the unit of arc in which P is ex- pressed — say one minute. Calling this distance unity, equations (183), (184), (185), (186), and (187) may be written b = cosl.sind{P'—«') (188) a = cosi.(P'— *') (189) F=sinZ.cosc?.(P'-*') (190) y — cos I . sin d . cos h . {P' — If') . . . (191) a; = cos Z . sin A . (P' — *') (192) § 662. Let Cbe the centre of the circle of projection, CiVthe trace of a circle. of declination through the sun's centre on the plane of projec- tion. Take the distance Ci?' equal to I^, Eq. (190); through F draw A A' perpendicular to CJf, and make FA = FA' = a, Eq. (189) ; take FB — FB' = b, Eq. (188) ; and making, successively, h equal to 15°, 30°, 45°, &e., in Eqs. (191) and (192), construct the corresponding hour 180 SPHERICAL ASTRONOMY. Fig. 103 bis. points of the parallactic path of the sun's centre. The geometrical con- struction of this path is indicated in the figure. § 663. Moov!s geocentric relative orhit. — Substitute in Eq. (180) the values oizs and S, and make the angle NC L equal to the resulting value of S ; the line CL will he the trace of a circle ,of latitude on the circle of projection. Make D equal to the moon's latitude at conjunction, and draw D E perpendicular to L and equal to the excess of the moon's hourly motion in longitude over that of the sun ; draw ^JT perpendicular to H D and make it equal to the moon's hourly motion in latitude ; through H and D draw an indefinite straight line ; this line will represent the moon's geocentric relative orbit on the plane of the circle of projection. § 664. Scale of time on the Moon's geocentric relative orbit. — Make m = moon's hourly motion in longitude ; n— "■ " " latitude; s = sun's hourly motion in longitude ; i = Jj J) IIz= inclination of the geocentric relative orbit to circle of latitude through the moon at conjunction. m' = moon's hourly motion on relative orbit : then coti= (193) m— s ^ ' m' = (w — s) . cosec I (194'^ From the ephemeris time of conjunction take the lon^tude of the place in time, the remainder will be the local time at which the moon's centra PROJECTION OF A SOLAR ECLIPSE. 181 is at D. Let e denote its excess above the next preceding whole hour, an t = ' (213) 1 + s ^ ' 695. If a planet with retrograde motion, m would change its sign, and '=^4;^^ M § 696. If the sidereal time were asked for, then would -4,= 0, » = 0, and t = -T^ (215) 1 — m ^ ' and if the body be a star, then m = 0, and t = A. REDUCTION TO THE MERIDIAN. § 697. Some of the most important astronomical determinations de- pend upon the measured zenith distances or altitudes of a body when on the meridian ; but these measurements it is not always convenient nor possible to make, and besides it is desirable to multiply measurements as much as possible to secure the advantages of a general average in elim- inating errors of observations. The purpose of the next proposition is, therefore, to pass from a measured zenith distance or altitude taken when the body is off the meridian to what it would have been had the body been on that circle. The difference between any two zenith distances, applied with the proper sign to either, will give the other ; and when one is the meridian zenith distance, this difference is called the reduction to the meridian. § 698. Reduction to the Meridian. — To find the reduction to the me ridian. REDUCTION TO THE MEEIDIAIf. 193 Let P be the pole, Z the zenith, S a star, SM an arc of the star's diurnal circle cutting the meridian in. M,S the arc of a horizontal circle through the star, and cutting the meiidian in 0. Make !c=:ZS-ZM=Z 0- ZM=re- duction to meridian ; I = latitude of place ; d = declination, of star ; F = hour angle Z P S ; s = zenith distance Z S. Then because P Z = 90° - I; P S = 90" - d; we have in the triangle P Z S cos s = sin Z . sin rf + cos I .cosd .co& P; but cosP= ] -2.sin'iP; and substituting this we get cos 2 = sin Z . sin «Z + cos Z . cos (^ — 2 cos Z . cos d sin' J P, = cos (Z — rf) — 2 cos Z . cos d , sin' \ P- But Z M=l — d; and ^:=x + l — d; and therefore, cos 2 = cos X , cos (i — d) — sin x sin (l — d) ; also, cos a; == 1 — ^- a:' + , &c. ; and if the observations be made near the meridian, x will be very small, and its powers higher than the second may be neglected. Making this supposition, writing the arc for its sine, and substituting the value of cos X above, we have cos 2 = (1 — ^ x^) . cos {I — d) — X . sin (I — d). Equating these values of cos e, there will result lx'.coa(l — d)+xsia{l — d)=z2cosl. cos cJ. sin' ^P . . (216) In consequence of the small value of x, it will be sufiBcient for all prac- tical pui-poses to make an approximate solution of this equation ; for this purpose write it 2 cos I . cos d x = sin (I — d) . sin* i P - cotan (Z - cZ> . i ar" (217) 13 194 SPHEEICAL ASTRONOMY, neglecting the term involving the second power of x, 2cos / . coad . , , „ /„,„» x = —^j ^r-.sin'^P (218) sm (I — d) ' ^ ' and this in Eq. (217) gives cos I . cos d ■"■■i^-^'c-o-iSp^y-^-'iP, ~ sin (Z — S) " * ^ ' ' \sin (I — d) ^ and mating, in order to find x in seconds of arc, 2 sin* 4 P k 2 sin' i P sin 1" COS I. cosd sin (Z — d) m = ' "'■" ',' .... (219) sin 1" ^ ' ,. /cos Z . COS dV -m.cotil-d).(.^-^-^j .(220) sin 1" a; = & § 699. Now P is to be found from the time when the stai is on the meridian and that of observation, being equal to the difference of the two converted into arc. These times are to be taken from a time-piece, and this never runs accurately to sidereal or mean solar time. If the, time- keeper run too slow, the difference of its indications would be less than the corresponding difference of true hour angles — if too fast, the contrary ; and P, in the formula, must be corrected. Let the time-piece lose r seconds a day ; then while the true day will be equal to 88400', the clock or watch day will be 86400* — r, and any two coiTesponding hour angles, one being the true and the other that indicated by the time-keeper, denoted respectively by P and P', will bear the re- lation P : P' :: 86400 : 86400 — r ; tvhence 86400 ^ ■ 86400 — r making 86400 / = — ^ — = 0.000011. r (221) 86400 ^ ' developing the fraction, and neglecting the higher powers of r', P = P'{l + r>) = P' + P'r', and sin ^P = sin J P' cos | r'i" + cos |P' sin ^r'P' ; making cos \r' P' = 1, squaring and rejecting the term containing the second power of sin \ r' P'y we find TERRESTRIAL LATITUDE. I95 Bin" ^P = sin'i ^ P' + 2 sin 1 P' cos i P' . sin Ir' P' ; but 2 sin ^ P' t cos I P' = sin i", and since P' and r' are both small, sin P' — 1 sin | P', s^n^r' P' —1-' sm.\P'; which substituted above give sin" A P = sin= A P' + 2 ?•' sin^ A P' = (1 + 2 r') sin= | P' ; and finally making t 1= I + 2 »•' = 1 + 0.000022 r . . . . (222) and substituting in Eq. (220) we have . , cos Z. cose? fooil.cosdy , x = i.k.~^—-- — — %\m.(ioi{l—d).[^—- — ) (223) sm(/— rf) '' '' \s\n[l — d)) ^ ' in which it will be recollected that r, in the value of i, is the rate of the time-keeper, minus when the latter gains and plus when it loses on side- real time. § 700. The first term in the second member of Eq. (223) will always be suflBcient when the observations are made within five or ten minutes of the meridian. And it is important to remark, in view of the use presently to be made of the value of a;, that the latter will not be sensibly affected by a small error in the value of Z, and that an approximate latitude may therefore be substituted therefor. The values of h and w are computed for- all values of P'from to 35", and inserted in Tables V. and VI. TERRESTRLAL LATITUDE AND LONGITUDE. § 701. The determinations of terrestrial latitude and longitude by means of astronomical observations and ephemerides, are among the most important of the objects of practical astronomy. All appreciate the value of these determinations in navigation and geography, and we now proceed to consider them in the order named. Terrestrial Latitude. § 702. The zenith distance of the pole is always the complement of the latitude of the place, and when known the latitude is known from the relation ?v :rr 90° - I, 196 SPHERICAL ASTRONOMY. in which \ denotes the zenith distance _ z of the pole, and I the latitude of the ^ — °\^^^ 1 -9^ \ \ place. / -^^.^.^ \ N. § 703. The zenith distance of the pole / ^\ \ V forms one side Z P of a spherical triangle, / ^ \ of which the two other sides, Z S and [-^-_____ L— -^^ P S, form, respectively, the zenith and \ / polar distances of some heavenly body, \ , / of which the angle at the pole is the \^ */ hour angle, or distance of the body fi'om ^ --^'^ the meridian. And the determination of latitude consists in the solu- tion of this triangle, the data for this purpose being the true zenith distance Z S determined from observation, the polar distance P'S found from the ephemeris, and the hour angle Z P S, which is always equal to the sidereal time of observation, diminished by the body's right ascension at the same instant. Having, then, found the true zenith distance by cor- recting the observed for refraction, parallax, and semi-diameter when ne- cessary, and the body's true hour angle and polar- distance from the time of observation, the ordinary formulas for the solution of spherical triangles will do the rest. § 704. Latitude hy Meridian Zenith Distance of a Body. — But it is desirable, in practice, to select those moments for observations which will give most accurate results, and these are when the hour angle is 0° or 180° ; in other words, when the body is on or near the meridian, for then it has the least change in zenith distance for a given interval of time. Make z =. Z S = true zenith distance of body ; d — 90° — P S = the body's declination ; P = Z P S = hour angle of the body ; A = PZ S= 180°— the body's azimuthal angle. Then in the triangle ZPS, cos s = sin Z . sin c? -J- cos ^ . cos rf . cos P . . . (224) sin d = sin Z . cos z -f cos / . sin z . cos ^ . . . (225) § 705. Making P — 0°, the body will be on the meridian some- where between the poles on the side of the zenith, and A will be 0° or 180°. In the first case, the body will be between the zenith and elevated pole cos ^ ■= 1, and Eq. (225) will become .iM wnence and TERRESTRIAL LATITUDE. I97 sin rf = sin Z . cos g + cos Z . sin 2 = sin (I + s), d = l + z, l = d — z (226) Fig. 114 In the second case, the body will be on the opposite side of the zenith from the elevated J)ole, cos^ = — 1; and if the latitude and declination bo of the same name, sin d and sin I will hare the same sign, and Eq. (225) ^ves sin rf = sin Z . cos g — cos Z . sin s = sin (l — s) ; whence d = 1 — z, and l=d+z ....... (227) Fig. 115. I^ in the second case, the declination and latitude be not of same name, the body will be beloV the equinoctial ; sin d and sin I will have contrary signs, and Eq. (225) gives whence and sin (— (^ = sin Z . cos z — cos i . sin g — sin {I — s) ; — d-=l — z, lz=z — d 198 SPHEKICAL ASTRONOMY. If P = 180°, the body will be on the meridian below the elevated pole, and A= 0° ; cos P = — 1, and, Eq. (224), cos z = sin / , Bin (2 — cos / . cos d= — cos {l-{-d); whence z= 180° - {I + cl), and Fig. 116. l=180° — s + d (229) § 706. Latitude by Circum-meridian Altitudes. — Thus it is easy to find the latitude when the meridian zenith distance and declination of a heav enly body are known. The declination is found from the ephemeris, if the body belong to the solar system, or from the catalogue, if it be a star. The meridian zenith distance is best determined by the method of circum-meridian altitudes, which consists in measuring with an instrument a number of altitudes of the body just before and after its meridian pas- sage, noting the corresponding times ; reducing to the meridian, taking an average valOe of the results, and subtracting this from 90°. § 707. Denote by Ai, As, 'A3, &c., the measured altitudes; »•„ r^, r^, &o., the corresponding refractions ; pi, p^, p^, &c., the parallaxes ; A the ap- parent semi-diameter ; «„ x^, x,, &a., the reductions to the meridian ; n the number of observations ; and IT the average meridian altitude ; then will ff = hi — rt+pt-]-Xi-\-hs — rs-[-ps+Xi-\- &c. ±A (230) the upper sign corresponding to the lower limb, and vice versa. Denote by P„ Pj, P3, &c., the watch hour angler of the body ; that is, the differ- ence between the watch time of meridian passage and those of obse^va- tions. These, with tables, give ki, k^, k,, &c., m,, Wj, rrii, (fee, Eq. (223); and making 2 A = A, -f A;, -t- *3 4-, &c. ; 2 »i = mi -)- Wj -)- wis -f , ]/ — i . sin 4' (235)' Now, A is a small angle, not more than 1° 40'; and replacing cos A and TEERESTEIAL LATITUDE. 20L tan A by their values in terms of A, equations (234) become, omitting the powers of A above the third, a = 1 — i A= + (A — ^A') . cot A . cos P, 6 = (1 - ^ A=) . cot A — (A — -l-A') cos P. Let ■^ = Aa + £a'+Ca^+,&o. .... (236) be the development of 4- according to the ascending powers of A, in which there can be no independent term ; since, when A = 0, then will 4' = 0- Whence cos 4/ = 1 — ^A'A' — ABa^, sin ■i = AA + BA'' + {0-lA')A\ Substituting the values of a, b, cos 4', and sin 4', in Eq. (235), we have the identical equations, cot A . cos P — -4 . cot A = 0, - ^ (1 + ^=) + ^ cos P - 5 cot A = 0, A^-|(l+3^»)cosP-(C-^^') = 0, Whence -4 = cos P ; ^ = — ^ sin ''P . tan A ; (7= JcosP.sin'P; which in Eq. (236)' give 4. = A . cos P — ^ sin "ip . tan A . A= + ^ cos P , sin "P , A». To express 4^ and •^ in seconds, write 4^ sin l" for 4' and A sin 1" for A, and make m = ^ sin 1", m = 5- sin' 1", then will 4. = A cosP — m (A . sinP)'. tan A + » . (A . cos P) . (A . sinP)» (237) , This value applied with its proper sign, to the observed altitude, cor- rected for refraction, will give the latitude. It is best to take some half dozen altitudes, and to note the corresponding times in pretty rapid suc- cession-; a mean of the altitudes corrected for refraction will give A, and a mean of the sidereal times diminished by the right ascension of the star, and the remainder multiplied by 15, will give P. § 711. This method is of such practical utility as to have caused the insertion into the English Astronomical Ephemeris and Nautical Almanac of three tables, of which the first contains the value of A cos P for every 10 minutes, sidereal time, for a mean and constant value of A; the second contains the values of — w . (A . sin. F)' . tan A ; and the third contains 202 SPHEKICAL ASTRONOMY. * correotions to be applied to the values ia the second tible. The secona and third tables are arranged in the form of double entry, the arguments for the former being the sidereal time and altitude, and in the latter side- real time and date. The third term of Eq. (237) is neglected as being insignificant. lionffitude. § 712. The longitude of a place is the angle made by its meridian with some assumed meridian taken as an origin of reference. The problem of longitude is much more complex than that of latitude, and its solution consists, as we have seen, § 94, in finding the difference of local times that exist simultaneously on the required and first meridian. § 713. Longitude hy Chronometers. — Could the motion of a time-piece be made perfectly uniform, and the angular velocity of its hour-hand equal to that of the earth's axial rotation, without the risk of variation, the de- termination of longitude would be a simple matter. It would then only be necessary to put the time-keeper in motion ; on a given meridian ascer- tain, by the methods explained, its error on the local time of this meridian ; transport it to the unknown meridian, determine its error on local time there, and take the difference of these errors ; this difference would be the difference of longitude of the meridians in time. But such time-pieces cannot be made. The results to which they would lead may, however, be approached within limits all-sufficient for practical purposes. It is only necessary that the time-keeper shall run uniformly, a condition which chronometers have been made so nearly to attain as tQ vary their rate but half a second in 31536000 seconds. § 714. By daily observations find the error of a chronometer ; fi-om the variation of the error during the intervals between the observations, find that for 24 chronometer hours. This will be the rate. Make e = error on local time at given meridian ; plus when the chronome- ter is too slow, minus when too fast ; e, = error on local time at required meridian ; e^i = error on local time of given meridian at same instant ; r = rate ; -f- when gaining, — when losing ; i = interval of chronometer time between the observations, which gave the errore e and e^ • I = difference of longitude. Then I = e,— e„ ; «// = « + »•»•; TERRESTRIAL LONGITUDE. 203 whence I =: 6/ — e + i.r (238) § 716. Longitude hy Lunar Distances. — The moon has a rapid motion m longitude. Her geocentric angular distances from the sun, planets, and fixed stars that lie in and about her path through the heavens, are com- puted in advance and inserted into the Nautical Almanac. From these hours and distances is readily found, by interpolation, the Greenwich time corresponding to any given distance not in the Almanac, and the diflerenoe between this interpolated time and the local time on any other meridian at which the moon is found from observation to have this given distance, is the longitude of the meridian on which the observation is made. § 716. Measure the altitude of the star, and that of the upper or lower bright limb of the moon ; also measure the angular distance from the star to the bright limb of the moon, and note the local time of this measure- ment ; coiTect the altitude of the limb and measured distance for semi- diameter ; then correct the altitude of the star for refraction, and that of the moon for refraction and parallax. Let Z be the zenith, Z S smA ZM the arcs ^* "8. of vertical circles, the first passing through the star >S' and the second through the moon's cen- tre M. The efiect of refraction being to ele- vate and that of parallax to depress, and the parallax of the moon being always greater than her refraction, the star will appear at S' above its true place, and the moon at M' below her true place. Make itr"^ S h = 90° — Z M' = observed altitude of moon's limb corrected for semi-diameter ; h' = 90° — Z S' = observed altitude of star ; ^' = M'S' = observed distance corrected for semi-diameter of the moon ; II ==. 90° — Z M = true altitude of moon's centre ; W = 90° — Z S = true altitude of star ; ^ = M S = true or geocentric distance between the moon'a centre and the star ; 2 = MZ S' = angle at Z. Then in the triangle M'Z S', cos A' — sin A . sin h' cos 2 = cos h . cos h' ' 204 SPHERICAL ASTRONOMY. and in triangle M Z S, cos A — sin H . sin H' _ cos Z ^ ^ ^^j } equating these values of cos «, cos A' — sin A . sin h' _ cos A — sin H.sm.H' cos A. cos A' ~ cosiT.cosJT' ' adding unity to both memhers and reducing, cos A' + cos (A + h') _ cos A + cos (E ■>(■ H') cos h . cos h' ~ cos ff. cos ff h + h' + A' = 2m (239) ■whence cos (h + h') = cos (2 TO — A') ; substituting this above and reducing, we find , R+H' . ,A , . ,. cos'' . — sin"' -— cos m . cos (m — A ) 2 2 cos h . cos h' cos IT . cos Jf' ' whence . , i / , , ,rr . ms COS IT. COS IT' , ,^ ' sin ^ A = y cos" \ {H+ H!) - ,_-^_^. cos m . cos (m - A'), and making, to adapt the foregoing to logarithmic computation, COS if' 7" . cos wi . cos (m — A ) COS h . cos h' , . "■^^^ cos I [11 +JI') • • (240) then will result sin i A = cos 1 (^ + jy') . cos

= i>' + i,.A,+^yif^[A, + &c; supposing the second difierences constant, which we may do without sen- sible error, and solving with respect to first power of t, t = ^^ ^3" (242) Neglecting the second difference, we have JD — D' t=- =-3^ (243) A, which in the denominator of the preceding equation gives D — B' t= =- 3h. . . . (244) A.-lA, + i(i)-i)')-^ and replacing t by its value T — T', we have finally B — B' T=T' -^ =- Z^ . . . (245) A-iA,+ i(i>-i)')-|; § 720. A single observer begins by taking with his sextant an altitude of the star, then an altitude of the moon's bright limb, then the distance be- tween the star and moon's limb, then the altitude of the moon's bright limb, th'en the altitude of star, carefully noting the time of taking the distance. A mean of the altitudes of the moon and star will give the approximate altitudes which the moon and star had when the distance was measured. 206 SPHERICAL ASTRONOMY. § 721. It is scarcely necessary to^dd, that if the sun or a planet be taken instead of a star, corrections for semi-diameter and parallax must be added to that of refraction. § 722. Longitude hy Lunar Culminations. — If the change in right- ascension of a point of the moon, in its passage from one meridian to another, be known, the distance between the meridians becomes known from the point's rate of motion in right ascension. Make c, =: the point's right ascensipn w-hen on any upper first meridian ; c, = its right ascension when on an upper known meridian to the west. Zr=: longitude of this known meridian, west. I = approximate longitude of any unknown meridian between these. L = true longitude of the same. e = L-l. a = right ascension of the point when on the upper meridian, of which the longitude is L. § 723. — 1st Approximation. Then, were point's motion in right ascension uniform, Cj — C] : II : : a. — c, : I or H I =z --— (a - c) § 724. — id Approximation. But the moon's motion in right ascen- sion is not uniform, and the above will in general be erroneous, and by the quantity e, which is a small arc of longitude ; and we have L — l + e The arc e being small, the moon's , motion in right ascension will be sensibly uniform while between the meridians, through its extremities. Make «! = the lunar point's right ascension when on the meridian, of which the longitude is /, V — the point's rate of motion in right ascension ; and let this be measured by the distance in right ascension over which the point would move, with this rate constant, while between the meridians of which the distance apart is H. Then by the principle above, writing e for I, v for Cj — Cj, and aj for c„ we have rr e = -.(a_«j); and this in the abov i gives L = l + ^.(^-a,) TERRESTRIAL LONGITUDE. 207 § 725. Now, these equations will be equally true from whatever point of the equinoctial, taken as an origin, the right ascension be estimated. For convenience, take the origin at the declination circle through the lunar point at its last passage over the first, or meridian of the Ephemeris. Then will c, = 0, Cj = change of right ascension between the known > meridians^ a = increase of right ascension from the first to •the intermediate or required meridian, aj = increase of right ascension from the first to the approximate meridian I. vWith this new notation the above equations become ^ = ?-« (246) L ~ I -\ • (a — a, ) V (247) § 726. In the Nautical Almanac and Astronomical Ephemeris are given the right ascension of the point of the bright limb at which a declination circle is tangent to the lunar disc, and also the right ascen- sions of one or more stars, at the instant of passing the upper and lower meridian of Greenwich" for every day in the year. The stars are so situated as to lie about the moon's parallel of declination, and not far from her in right ascension. § 727. — 1. Interpolation. Take the following scheme : I F ^, ^2 ■^3 ^4 ^5 t'" a"' h" ■ t" a" V c" d' t< a' b . c' d e' / t, «/ h ", d. «/ t., a„ K "n ';.; «/// in which the column I contains the independent variable, or argument, as time, terrestrial longitude, degrees, and the like ; F the value of a 208 SPHEEICAL ASTEONOMT. function of this variable, as found in any set of tables ; Aj, A^, Ag, etc., the first, second, third, etc., orders of differences of these functions. Make s = the interpolated valu'e of the function correspond- ing to any given value t, of the argument be- tween t' and tj ; t = t.-t' t. —V A, = 6, c' + c, A3 = 4 e' + e. (248) Then, limiting the operation to the fourth order of differences, will s = a' ^- At-\- Bt^-V Cfi + DC + - a' = At + Bf^ + Of i-Bt* (249) in which (250) ^ = Aj - 1 Aj + T-V A3 4- tV A4. 1 ^ = ^A2-iA3-^A„ I Also taking first differential coefficient of the function (249) ' V = A + 2Bt+ SO' + iBfi .... (251) which would be the increment of the function for an increment of t equal to unity,' were the function to increase uniformly and at the rate ^ it had for any arbitrary value for t,. § 728. — 2. Observation. Make S' = true sidereal time of 5 's bright limb passing the meridian, II' = time indicated by time-piece of the same passing middle wire of transit. e' = error of time-piece on sidereal time, t' =: error of transit, in time, for altitude equal that of moon j . TEKKESTKIAL LONGITUDE. 209 jhen will S' = H' + e' + i\ Also, make s' = true sideral time of star's passing the meridian at the alti tude of 5 , h' = time indicated by the same time-piece as before, e" = error of time-piece at same instant. Then, the declinations, and therefore the altitudes, of the moon and star being about equal, will s' = h' + e" + t' ; and subtracting this from the preceding equation, S' -s' = H' -h' + (e' - e") ^nd making e" = e' T 54" 57", 41 7 10 54,36 8 o3 31 , 44 3)23 09 i3 , 21 7 23 04 , 40 Then, Eq. (246), ff = 12'' op°>00', a = 00 10 34 , 53 Nautical Almanac ■;, = 00 25 41 , 18 I = 4 56 38, Log a = A = . . . . 4,6354837 . . 2 , 8034530 a.c, 6 , 8121918 . . 4,250127s Next, interpolate change of right ascension for I; 4I1 56'>' 28", Log ( = t,-f _ 4'' 56'° 38' t,-f~ 12 0* l5<>>i7>,83 -o,o3 04 43,36 o* io<»34', 53 . 4 , 3501375 " a. e. 5 , 3645i63 " . . 9,6146436 Nautical Almanac. Feb. 17, L. C. 7'' oi" 55', 37 " 18, TJ.C. 7 27 47.66 «' " L.C. 7 53 38, « 19, TJ.C. 8 18 59 , 56 23" 01", og I ,66 ( — 10", 31 fl,= 35 41,18 A, = iS.{ A,= -o',J» ,84 I — 10 , 46 A = a5"4i', 18 + o5', 17 — o>, 02 = 25"» 46', 33 £=—05,17+00,06 =— o5,n C= — 00,04 ^ TERBESTEIAL LONGITUDE. 211 Then, Eq. (249), A . . Log . . 3 , 1893022 t . . " . . 9, 6146433 2 , 8039460 Nos . . i36.,7» S . . Log • • 0,7079957 t> . . « . . 9 , 2292B76 Nos . * 9,9372833 . - 0,86 C . . Log . . 8 , 6190933 « / «» . . " . . 8 , 8439314 Nos . ■7,4630247 — 0,, oo3 635 , 86 io'°34', 63 = a — Oi = ... 634,53 . - I', 33 Again, Eq. (251), A . . . . Nog . . 25- 46', 33 ^ . . Log • • 0,7079957 t . . " . . 9,6146438 a . . " . .0 , 3oio3oo Nos . , 6236695 . -4, so ff . . Log . . 8,6190933 t* . . " . . 9,2292876 3 . . " . . , 477'2l3 Nos . 8 , 3255022 V = — 0,0a . s5">4a%ll TheB,aast term of Eq. (247), ^ . . Log . . 4,6354837 a-a, . . " , . , 1205739 V . . "a ;. c. 6,8119128 = — 36',g8 . . " . . I ', 5679704 Nos . . 00 36,98 Eq. (247), Z = 4" Se™ 28" - 36", 98 = 4'' 55" 5i>, 02 212 SPHERICAL ASTRONOMY. ^ § 729. It frequently happens that the moon cannot be observed on the middle wire, in which case she is far enough from the meridian to have a sensible parallax in right ascension ; and as it may be very desirable not to lose the observation, this parallax must be computed and applied to the apparent hour angle from the middle wire, which is siippoaed to be nearly coincident with the meridian. Denoting the hour angle by h, the parallax in hour angle by A h, the geocentric latitude by I, the moon's declination by D, and her horizontal paralLix by P, then. Appendix XI., p. 379, Ah =^ p .cos I . sin P . sin h. sec D ; and to make this applicable to the case before us, h will denote the equa- torial interval, in sidereal time, from the lateral to the central wire. This angle being small, its arc, expressed in seconds of time, may be taken for its sine, in which case, A h will be in time-seconds, and tlie true distance of the moon's limb from the central wire, denoted by A,, will be A, =: A . (1 — pi. cos Z . sin P . sec D) ; and the reduction to the meridian, denoted by r, in time-seconds, h 1 — p . cos I .sin P . sec D *" ~ cosi> ■ 1 — 0,00277 ."to ' in which m is the moon's daily motion in right ascension. The upper sign, when the observation is before the middle wire. The quantities p and I are found from tables on pp. 336, 337. § 730. It also often happens that two observers do not use the same luimber of wires, or if they do, that the same stars are not observed at the same number. Such observations are not of equal weight. To find the relative value with which such observations should enter into the final determination. Professor Gauss has given the following formula, dedtlced from the principle of least squares. Let the number of wires on which the moon is observed at one place be denoted by n, and" at the other by n' ; and let the number of wires at which the stars are observed at the first place be a, b, c, &o., and at the other be a', 6', c', Ac. Make 71 + m' (233) . = «., **' /3. r^ = r.&c- • • (254) a + a'~ ' 6 + 6' ' ' c + c' tf = a + /3 + y + i + 4') ~3^ ~ " A* «)s(Z)) ' TERRESTRIAL LONGITUDE. 217 which, subtracted from to, will give the ephemeris time of observation. Denote this time by t, and we have / = (0 + («o) — {A) A sin (i) +4^) A (-0) (259) The longitude, from the meridian of the ephemeris, is found by the difference between this time and that of observation, previously making both apparent, or both mean time, by applying the equation of time ; and it will be west or east, according as the ephemeris time is greater or less than that of observation. To find Aa, and A^, take Eqs. (2), Appendix XI., p. 379, and write therein Aa for Ah, P for sin P, AS for AJD, d for D and i>', unity for cos ^Ah, and substitute for h its value h'—Ah; we find „ cos / . . A« = p • i* f • sm n, , ^ msS ' |. (260) a5 = p • P • (sin I ■ cos S — cos / • sin f cos (A — ^^Aa) ; . in which I denotes the central latitude ; and, employing the method of solution in Appendix XI, page 381, we have „ cos Z . , Aa = p • P • = • sm ft, ^ cos (k) = h — ^Aa, tan ^ = cos (A) • cot I, tan M = tan g = tan (4 + 5) • cos M, aS = f • P • cos M- cos s. sin 6 cos {6 + S) tan h, (261) To find Atf, resume Eq. (27), substituting therein rf for s, i' for »', cos (90° — e) for cos Z, unity for cos z ; and we have «" — .. .1 — 1^. ■ ■ u — p • P ■ sin fi 218 SPHERICAI. ASTRONOMY, f t subtracting unity from both members, clearing the fraction, writing P — * for P, and then P' for p (P — *), we have tf . i^ ■ sin e i = hourly var. {D) — hourly var. {S) in arc ; ('»o)= («) + Aa; (5,)= (5) + A 5. ^- ^^ K)-(^) . «o = (<) + »! [3.56630]; Ai J)„ = {D)+m.J),; k = D^-{S^); n = [1.11 609] At cos {B); Di , Jc cos »i tan i\=z ; cos U^ = — - — . n 'A A Corresponding Greenwich mean time ^ 1^ + [3.55630] - sin (rj ^ 4'); 1} to have a different sign from Z>j : upper under ) . , ( immersion ) . , , )■ sign when an ■( . }■ is observed. ) ( emersion ) II. — Occultation of a Star by the Moon. 6. With the estimated longitude find the corresponding Greenwich time, and thence take out the moon's horizontal parallax P, and her declination D, roughly to the minute ; also,' sid. time = apparent time + 0's right ascension ; or, sid. time =; mean time + sid. time mean noon, from p. III. of ephemeris ; + accel. on Greenwich mean time ; h = sid. time — a, in arc ; P' = pi'; a being the star's right ascension. 1 V p = P' coslamh; aA in min. = [7.92082] —^; (A) = A — AA> x = P' sin looaS] x' = P' cos I sin 5 cos (A) ; S^ = S + x — x' \ A a in time = [8.823911 ^. ; a. = a + A a. ■- ■' cos Og 8. In the hourly ephemeris of the moon fix on a convenient time (t) at which the moon's right ascension is near to a.g, and for this tipe take out TEREESTKIAL LONGITUDE. 221 the right ascension (A), the declination {D), and their houi'ly variations Ai, Bi. Then, m = "°~^ ^ ; |c. Eeduce the latitude of the place by subtracting the coiTection found in the table in Appendix XL, p. 336, for which the nearest correction found in the table will be suflBcient. To the proportional logarithm of the moon's horizontal ftarallax, add the correction answering to the latitude in the following series : ooo aoooo ooooo^oo o. lat. . 11 19 24 29 34 38 42 46 60 54 69 64 69 11 90 Corr. . 1 2 3 4 5 6 T 8 9 10 11 12 13 14 To the proportional logarithm of the horizontal parallax, so corrected, add the log. secant of the reduced latitude and the log. cosecant of the hour angle. To the sum (S,) add the log. cosine of the moon's declination and the constant log. 0.3010. The result will be the prop. log. of an arc, which, subtracted fi'om the hour angle, will give the hour angle corrected. To the corrected prop. log. of tihe horizontal parallax, add the log. secant 222 SPHERICAL ASTRONOMY. of the *'s declination, and the log. cosecant of the reduced latitude. To the same log. add the log. cosecant of the *'s declination, the log. secant of the reduced latitude, and the log. secant of the hour angle corrected. These sums will be the prop. logs, of two arcs. ^ The former arc to have the same name as the latitude. The latter to have ) ( less ) > the dec. when the h. angle is -j [■ than 90°. a different name from ) ,, , , ^, i i • ( less ) the same name as The sum of these two arcs, having regard to their names, will give the correction to be applied to the *'s declination to get the declination corrected. To the sum (/S,) add the constant log. 1.1761, and the log. cosine of the *'s declination corrected; the sum will be the prop. log. of an arc in time, to be added to subtracted from J- the *'s R. A., when it is •! ,. fof the meridian. to get the *'s right ascension corrected. In the hourly ephemeris of the moon, fix on a convenient time at which her right ascension is near to that of the star corrected ; and, for this time, take out the right ascension, the declination, and their hourly- variations. Subtract the common log. of the difference between the corrected right ascension of the stai; and the right ascension of the moon, from the com- mon log. of the hourly motion in right ascension ; to the remainder add the constant log. 0.4771 ; to the same remainder add the prop. log. of the hourly motion in declination. The former sum will be the prop. log. of a time to be added to subtracted from vihe assumed time when ^'sR-A-is-]^ t than D's R. A. to get the tipie corrected. The latter will be the prop. log. of a correction of the 5 's declinatiot, to be applied with I hourly var. when *'8 R. A. is | ^^* *'l than > 'a R. A. the same name as a different name from To the common log. of the hourly motion in right ascension, add tiie log. cosine of the P's corrected declination; and to the sum (/Sj) add the. prop. log. of the hourly motion in declination and the constant log. 7.1427. TEE.EESTEIAL LONGITUDE. 223 The result will be the log. cotangent of the first orbital inclination,* and must take the same name as a different name from > hourly motion in dec. -when * i% ■< (■ of J) . To the prop. log. of the difference between the star's declination cor- rected and the moon's declination corrected, add the constant log. 9.4354, and the log. secant of the preceding orbital inclination ; and from the sum' deduct the prop. log. of the horizontal parallax. The remainder will be the log. secant of the second orbital inclination,f which must have the name S. 1 , ^, , ,. . ( immersion ,._ > when the observation is an ■( JM. ) ( emersion. Add together the two orbital inclinations, having proper regard to their names ; and to the log. cosecant of this sum add the preceding sum (Si), the prop. log. of the horizontal parallax, and the constant log. 8.1844. The sum will be the prop. log. of a correction to be applied to the time corrected to get the mean time at Greenwich : it must be . ^ ^ , > when the sum of the orbital inclinations is ■< „ ' subtracted ) ( S. By applying the equation of time from p. II. of the ephemeris, there will result the Greenwich apparent time, and the difference between it and the apparent time of observation will show the longitude of the place from Greenwich ; it will be ' > when the Greenwich time is [• ? [■ than the observed. Examples. I. SOLAS ECLIPSE. For a solar eclipee, take the example directly calculated in Appendix XL, page 412: Suppose the beginning of the solar eclipse On May 16, 1836, to be observed to take place at i' 36"" 35'-6 p. h., apparent time, in latitude 55° 5?' ao" N., and longitude about 12™ W. * With the parallel of declination. f With the moon's \aah. 224 SPHERICAL ASTRONOMY Here we have Observed apparent time Longitude .... Greenwich apparent time Equation of time (xreenwich mean time . Il m. I 36.6 I2.0 I 48-6 3-9 I 44-7 * = +!" je" 35».6 =+ s4° 8'.9 We hence take from the ephemeris, a = 3'' 29" 19', i = +i8° 67' .6, , = i5' 49"-9, i) = +i9° 19', P = 54' 24"-4, t = 8".5, P — »=:54' iS"-^. Latitude + 55° 57' 20" Reduction 10 28 p = 9.99902 I . +55 46 52 P— IT 3.51267 e ■ 9-999°^ P' . 3.51169 e+3i 50-7 tan 9 +9.79316 G l+iS 57-6 3.51169 cos I 9.75001 em fl +9.72231 sinA+9'6ii83 9+i+5o 48-3 cos +9.80069 JS cos (A) +9.96060 . +9.96060 cot I +9-83256 cos I +9-75001 . • +9-71061 +9-78899 p +2.87353 (I) cos Z> 9-97484 ^ +2-89869 A +24 8*9 '=°°'*- 7-9'°8^ aA+ 6-6 +0-81951 +9-92162 check +9-92162 tan (A)+9-64936 tan Jf +9 -57098 cos jif +9-97180 . +9-97180 tan (9+J)+o. 08861 cose +9.81719 (A) + 24 2-3 i+i8 57-6 + 33-3 «+48° 58'-3 taut +o.o6o4i £ P' A« + 33' 18" -4 +9.78899 +3.51169 +3-3oo68 » . i5' 49" -9 Ao II -6 P . 3-5i38o ~Z T^ Z const. 9.43537 ITg . 13 JO -d f 1_ s . i4 49 -6 , 2-94917 A . 3o 27 '9 i„+i9 3o-9 cos 9-97430 (2) +2.89923 (i) — (2) const. 8.82391 (log. . +i-723i4 I Aa + oi> o™ 52»-86 a 3 29 19 o^ 3 3o 12 By inspecting the hourly ephemeria of the moon's right ascension on May ISth with o„ = 3'' 30"° 1 2', the most eligible time to assume is evidently (<) = 3'' o" o* ; at this time we have {^) =3'' 3o™ 42'-84, {AO =2" o'-68, {D) = + 19° 3i' 34"-o, (A) = +9' 55". 2, (a)=3i' 29"' 3i"-57, (a,) = + 9"-89, (i) = + i8° 58' 2i".4, (il) = + 34" -8 : with these we proceed as follows : . 2 0-68 9-89 (A) . (^0 . . +955.2 . + 34-8 A^ . . I 5o-79 Dr . . + 9 20-4 TERRESTRIAL LONGITUDE, 225 K)- ( log- • h. m. 8. . 3 29 3i .57 + 52.86 + + ' " 18 58 21-4 33 i8-4 . 3 3o 24-43 . 3 3o 42-84 (-So) + 19 3i 39-8 (-4) . — i8-4i m . const log. . — i-265o5 2-o445o . — 9-22o55 . 3-55630 . — 2.77685 A. (log. ^ . + 2-74850 (I) — 9-22055 — 1-96905 0° t' 33"- 1 (0 — oi" 9" 58'-2 3 + 2 5o 1-8 (J>) + 19 3i 34 -0 19 3o -9 [9 3i 39 -8 k COS {JD) A. . const. . - I 38 -9 9.97428 2 - o445o I • I 7609 n . 3.19487 (2) « . . — 19 4i-2 tan q . cos 1 . k . A . — 9^55363 (i)- + 9-97384 — I -99520 — 1-96904 3-26196 -w ^ . . + 92 55-2 cos t// . — 8-70708 ■^ • . —112 36-4 . . sin A . const. . — 9.96528 3.26196 3-55630 — 6.78354 (3) corr. — I" 4"" 38»-5 . . . — 3.58867 (3)- -w 1<'3 dec. . . . o-oi5o . cosec 0-5876 cosec. red. lat. . . . 0-1037 • sec. . . . . , o-2io3 >T ° /' T)'^ T, T -■ ^, sec. corr^.hour angle o-4i84 N. o 42 33-0 P. L. 0-6264 S. o 3 23-9 P- 1' 1-7240 corr". . . W. o 39 9-1 sum (Si) .... o-75i3 sic's dec. . K. i4 58 38.8 const, log. . . . 1-1761 ^'a dec. corr''. N. i5 87 47'9 cos 9-9836 corr". ... o'' 2m 12'.56 P. L. corr". . . . 1-9110 ;|c's R. A. . . 10 23 26 -39 >1<'8 R. A. corr''. 10 21 i3 -83 On referring with the if/t'a corrected R. A. to the hourly ephemeris of the moon, it will evidently be most convenient to take out the data at 11''; for this time we liave B 's R. A. = 10'' 20"" 58"-47, hourly motion > 'b R. A. =: a™ 2'-g, J 's dec. = N. i5° 47' ii"'0, hourly motion J's dec. =S. 11' 4i"'5. TERRESTRIAL LONGITUDE 227 :J('a corrJ. E. A. K'sR. A. . . ( diffi . . . li. m. B. lo 21 i3-83 lo 20 58-47 o i5-36 ( common log, . . i . 1 864 com. log. h. m. D 's R. A. 2 • 0896 Re const, log. mi under , 0, .9082 ■ 4771 • P. L. h. m, . 5 's dee. • • ■ I ■ 9o3i •1874 corr". . . Time assumed 1 1 7 29.9 P.L 1 .38o3 5 corr". 's dec. • S. K". , I i5 47 27 II' •7 ■0 P.L. 2 ■ 0906 Time corr''. . 11 7 29-9 D'a dec. corr''. N. i5 45 43-3 com. log. h. m. > 's R. A. . 2-0896 ^'s corr''. dee. . cos. 5 's corr''. dee. ... 9 • 9834 5 's " " . . Bum{S,) 2.0730 fdiff. (jtcS. of J) P. L. h. m. > 's dec. . . . 1-1874 (P.L. . . . const, log. 7-1427 const, log. IfitOrb. incl. N. 21° 34' cot. o-4o3i . sec ... . P. L. ]) 's hor. par. 2nd Orb. incl. S. 61 9 ...'.. sec ... . emn ... S. 89 35 cosec. sum (Si) P. L. D 's hor. par. const, log. h. m. B. corr". o 19 44-5 Time corr''. 11 7 29-9 Greenwich mean time 10 47 45-4 Equation of time . . 6 3i-o P.L. Greenwich app. time Observed " " Longitude 10 4i i4-4 10 39 22-4 I 52-0 W S. i5 37 47'9 N. i5 45 43-3 7 55-4 . TT3563 . 9-4354 . o - o3 1 5 0-8232 0-5668 o-3i64 0-1957 2 - 0730 o-5o68. 8.1844 0:9699 P. S. — The principle of reversing the effect of the relative horizontal parallax on the position of the sun, instead of using the actual effect on the position of the moon, may be advantageously employed in the direct calculation of an eclipse for a particular place. It will only be necessary to use the parallaxes for the sun viewed as an apparent position, and to diminish the semi-diameter by the amount derived from the table on page 360. Thus, it appears, at the beginning of the eclipse, for instance, that the contact may be mathematically t6sted in two ways. First, we may apply the actual effects of the 'parallax to the true position of the moon, then augment her semi-diameter, and thus establish a contact of the limbs. But, if we reverse the operation, and consider the sun to be an apparent body under the influence of the relative parallax, then clearing it from this supposed 228. SPHERICAL ASTRONOMY. influence by reversing the parallax, and diminishing the semi-diameter, a contact will similarlj be established with the true limb of the moon ; and this principle, in its application to solar eclipses, possesses an advantage similar to that dsriived in the case of an occultation, by considering the star as an apparent place. (See Appendix XI, page 399 )* The formulse, Nos. 2, 3, 4, and 5, pp. 406, 407, may, according to this tnethod, be supplied by the following : 2. P' = p(P — If); m = P'cosZ; Q, = [9.4180] ; Qi = [9.4180] msinS; s = [9.43537] F. 3. , m cos D ' A A in minutes = [7.92082] A sin A ; {h)=h — Ah; tan 6 = cos (A) cot Z ; G = cos (A) cos I ; tan M== 7-. Tv tan (h) ■ tan s = tan (^ + 8) cos M: cos{s + 6) ^ ' ■ ^ ' check B = cos Jf cos s ; sin d O cos(a + d) B' Ad = B .F'; (/(J = (f — diminution for s ; (partial 1 „t,_ a' _ i * + ''o ( total or annular f ( s /^ tf 4. Kg = J- ; A a = *3 sm A ; cos 5 ' A ai = Qi k^ cos A ; A 5, = ^j sin (k). S;--i 5q = 5 + a5; a' = « — Aa; y = (a -^ A a) cos 2> ; y, = (a, — A a,) cos D ; x= (I) + a' corr.) — 5^', Xi = Di — A 5,. * This was inadvertently ascribed to Carlini. Professor Henderson, by whom a paper has appeared upon this very point in the Quarterly Journal for 1828, page 411, informs me that the method has been long in practice, and that it wsa employed at an early period by Br. Maskelyne, CALENDAR. 229 § 734. LfOngitude hy Eclipses of Jupiter's! Satellites. — ^The eclipses of Jupiter's satellites are computed" ia advance, and the times of occurrence inserted in the Nautical Almanac, to facilitate the determination of terres- trial longitude. After ascertaining, by inspection, about the time an eclipse begins and ends, the satellites are watched with a good telescope, and the precise local time of entrance into and departure from the shadow noted as nearly as possible. The time given in the Almanac, diminished by this observed local time, is the longitude ; west, when the difference is positive, east when negative. This method for finding longitude is defec- tive, for reasons stated in § 497. CALENDAR. § 735. To divide and measure time and to note the occurrence of events in a way to give a distinct idea of their order of succession and the intervals bf time between them, is the purpose of Chronology. § 736. All measurements require standard units. These units are, for the most part, purely arbitrary, and are equally convenient in practice. But such is not the case in chronology. Time is divided and marked. by- phenomena which are beyond our control, and which indeed regulate our wants and occupations. The alternation of day and night forces upon us the soiar day as a natural unit of time. § 737. To avoid the use of numerous figures in the expression of great magnitudes, all measurements must have their scales of large and' small units, and usually the selection of the larger is as arbitrary as the smaller ; but here the phenomena of nature again interpose, and the periodical return of the seasons, upon which all the more important arrangements a.nd business transactions of life depend, prescribes the tropical year as an- other and higher order of unit in chronology. § 738. But the solar day and tropical year are both variable, and are therefore wanting in all the essential qualities of standards. Neither are they commensurable the one with the other ; they are on this account unfit units for the same scale. In the measurement of space, for instance, each unit is constant, and one is an aliquot part of another — a yard is equivalent to three feet, a foot to twelve inches, &c. But a year is no exact number of days, nor an integer number and any exact fraction, as a third or a fourth, even ; but the surplus is an incommensurable fraction which possesses the same kind of inconvenience in the reckoning of time that would arise in that of money with gold coins of 101 dimes and odd cents, and a fraction over. For this there would be no remedy but to 230 SPHERICAL ASTB.ONOMY. keep an accurate register of the surplus fractions, and when they amount to a whole unit, to cast them over to the integer account. To do this in the simplest and most convenient manner in the reckoning df time, is the object of the calendar. § 739. A calendar is, therefore, a classification of the natural and other divisions of time, with such rules for their application to chronology as shall take into account every portion of duration without recording any one portion twice. These divisions are years, months, weeks, days, and certain periods, to be noticed presently, and which are chiefly important in the use made of them in fixing upon a common epoch or origin of reference. § 740. Julian Calendar. — The years are denominated as pears current, not as years past, from the midnight between the 31st of December and 1st of January, immediately subsequent to the birth of Christ, according to the chronological determination of that event, and this origin is desig- nated by the letters A. D. or B. C, according as the year is subsequent or previous. Every year whose number is not divisible by four without a re- mainder, consists of 365 days, and every year which is so divisible of 366. The additional day in every fourth year is called the Intercalary day. The years which consist of 365 days are called Common ypars ; those which consist of 366 days are called Bissextile, years, and frequently Leap years. The mean length of the year by this rule is obviously 865J days, and the mode of reckoning time by this unit in the way just described is called the Julian Calendar. § 741. The year is divided into 12 months of unequal length. They are named, ia order of succession, January, February, March, April, May, June, July, August, September, October, November, and December. January, March, May, July, Aug-ust, October, and December, have each 31 days ; each of the others except February has 30, and February has in a common year 28 and in a bissextile year 29 ; so that the intercalary day is added to February. The weeks consist of seven days, named in order, Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, and Saturday. § 742. Gregorian Calendar. — The Julian year consists of 365.25 days; the tropical year of 365.24224, making the Julian longer than the tropical by 0.00776 of a day, and causing the seasons to begin earlier and earlier every year as designated by the Julian dates. In process of time the seasons would therefore correspond to opposite dates of the year, and as this was likely to interfere with the times of holding certain church fes- tivals. Pope Gregory XIII. determined upon a reformation of the Julian calendar. CALENDAR. 231 § 'J43^In A. D. 825, the seasons, festivals, and Julian dates corres- ponded with one another, according to church rule. The reformation was effected in 1582. Now (1582 — 325) X 0\00116 = 9''.6243. Again, , 1— 1 ' :? p M o ^ 1 ^ w ^ W EH Ph (In 6l O tin rn Plh < g < s i^^ 13 ^ ■^ :^ ^ ^ ^5 1 2 T— 1 •o» '«•» ■w» •e» i lO 05 o CD OT CO 00 C5S ^ ..■ (N UO CO CJ in s g *■ o CO o .—1 00 OS CO CO 00 O ■^ * (N -* CO t?4 CO "^ CO Oi CT ,—1 OS t- l>- m Oi CO pH 00 ■co ■^ rH CO ^H b* CO M 00 CT 00 ITS in 4 B CD CT CO fM o OS O 5 o CD in tf to 00 CJ i-H CO l^ H lO to CD CO 00 CO to 00 ^ O O <-H Oi Tt* IT. Tf< o H (N O o o o o o o O o o o d d d d M irt t- sj CO i-H rH t^ o CD to OS CO in o CO CD t- oo ■<:** < S5 t^ o CT 00 CM U S 00 o o CO 00 ■-3 a DQ c I 3 » Of_ ® *o ^ *o» s fej APPENDIX IL • 237 APPENDIX II. ASTKONOMICAL INSTRUMENTS. Astronomical Clock and Chronometer. 1 — The order and succession of celestial phenomena make time a most important element in astronomy, and accordingly the utmost scientific and mechanical skill has been devoted to the perfection of instruments to indicate and measure its lapse ; § 37. The best time-keepers now in use are the Clock and Chronometer. Both consist essentially of a motor, a combination of ■wheel-work to transmit and qualify the motion it impresses, and a check, alternately to arrest and liberate the movement, and thus to mark an interval designed to be some aliquot part of a day, the natural unit of duration. 2. — The Clock.— In the clock, the motor is a weight A suspended from a cord wound about the drum ^ of a wheel C, and the check is the anchor escapement N, controlled by the vibrations of a pendulum P, whose rod is geared to an arm projecting from the axis 0, with which the anchor is firmly connected. The weight A turns the drum B and its wheel C;_the wheel C turns the pinion D and its wheel E\ the latter turns the pinion F and its wheel Q, and so on to the pinion L and its wheel M, called the scape-wheel, of which the teeth are considerably under- cut, so as to turn their points in the direction of the motion. The flukes of the anchor are turned inward, forming two projections called pallets. The distance between the ends of the pallets is less than that between the points of two teeth that lie nearest the line drawn from one pallet to the other ; and no two teeth can, therefore, pass the sarne pallet without the wheel being arrested by the contact of a tooth on the opposite side with the other. With the swing of the pendulum the anchor oscillates, and one pallet is thus made to approach while the other recedes from the wheel. As soon as the receding pallet disengages itself from- a tooth, the wheel is turned 238 SPHERICAL ASTRONOMY Fig; 6. by the motor and intermediate machinery till arrested by the approaching pallet, now interposed between its teeth on the opposite side. The re- turning swing of the pendulum reverses the pallet motion, liberates the wheel long enough for another tooth to pass, and again arrests it, and so on. Thus, by regulating the length of the pendulum and number of teeth on the scape-wheel, an index or hand connected with the arbor of the latter may be made to travel by successive leaps, as it were, around the circumference of a circle on the dial-plate in any given time. 3. — If the anchor be connected with the seconds pendulum, and there be sixty teeth on the wheel, each leap will mark a second. The APPENDIX II. -239 motions of the minute and hour hands are regulated by suitably propor- tioning the relative dimensions of the intermediate wheels with whose arbors these hands are connected. 4. — The scape-wheel being in a state of constant tension by the incessant action of the motor, its teeth must act upon the pallets first by a blow and then by a pressure during the time of contact. Now, the bearing surfaces of the pallets are so cut that a normal to them at the point of action passes clear of the axis of the anchor, and on that side which will cause the blow and subsequent pressure to act for a part of the time in favor of the pendulum's swing, and thus to restore whatever ot its arc of vibration may have been lost by friction and atmospheric re- sistance. 5. — The pendulum bob possesses the principle of compensation. It consist? of a cylindrical glass vessel resting upon a plate at the end of the pendulum rod. This vessel is filled with mercury to a depth so adjusted to the length of the rod as to elevate by its expansion or depress by its contraction the centre of oscillation just as much as this centre is depressed by the expansion or elevated by the contraction of the rod during a change of temperature. The distance between the axes of suspension and of oscillation being thus made invariable, the time of vibration will con- tinue constant, and the check be interposed at equal intervals. 6. — Chronometer. — The chronometer is an accurately constructed balance watch, uniting great portability with extreme accuracy. It is of various sizes, the larger having dial-plates from three to four inches in diameter, and running from two to eight days between the windings. The larger kind are suspended upon gimbals to secure uniformity of position, are mounted in boxes, and are called hox chronometers. The smaller kind resemble in shape and size a common watch, are worn in the pocket, and are called pocket chronometers. 1. — The motor is an elastic spiral spring inclosed in a short cylin- drical box A, called the barrel, one end being permanently fastened to a stationary axis H, about which the barrel freely turns, and the other to the inner surface of the barrel. The barrel being turned in the direction of the coils of the spring, the elastic force of the latter is brought more and more into play, and its variable action thus produced is communicated by means of a chain £ to a variable lever C, called a. fusee, whose office is to modify and transmit it uniformly to the works of the instrument. The fiisee is a conical solid having its surface broken into a spiral shoulder, running from one end to the other, the curve being so regulated 24(J SPHERICAL ASTRONOMY that the distance of any one of its points from the axis of the fusee's mo- tion multiplied into the force of the spring, acting through the intermedium of the chain, shall be a constant quantity ; and as the main wheel 2), which Fig. 7. f ■ ^t#^ gives motion to the rest, is firmly secured to the fusee, the motion is made to act uniformly upon the instrument. 8. — The swings of the pendulum by which the check wag alternately interposed between and withdrawn from the teeth of the scape-wheel in the clock, are, in the chronometer, replaced by the vibrations of what is called the balance. This consists of a wheel B, freely movable about an axis G, and a thin spiral spring S, one end of which is securely fastened to the hub of the wheel, and the other to a fixed support A. If when the spting is free fi-om tension, the wheel Fij. 8. be brought to rest it will remain so, ' ^ just as a pendulum bob brought to rest at its lowest point will remain im- movable. If from this position of the wheel it be turned in either direction about its axis, the spring will wind or unwind, the elastic force of the spring will be called into play, and will, when the wheel is unobstructed, carry it back to its position of equilibrium. But having reached this position, its living force car>ies it beyond ; the action of the spring is reversed, and, after destroying the living force, will reverse the motion ; the wheel will return to its posititin of equilibrium, which it will reach with a living force equal to that it had before at the same place, but in a contrary direction. The wheel will pass on, the action of the spring be reversed, the wheel will return as before, and thus the vibrations be continued forever, as in the case of the pendulum, but for the waste of living force from friction, atmospheric resistance, and absence of perfect elasticity in the spring. 9. — The angular acceleration impressed upon the balance by the spring APPENLXX II. 3^1 is measured by the moment of its elastic force divided by the moment of inertia of the entire balance. When the temperature is increased, the spring is lengthened and the elastic force it exerts lessened ; the wheel is expanded, its matter thrown further from the axis of motion, and the mo- ment of inertia consequently increased. On both accounts the angular acceleration is diminished, and the balance will vibrato slower, and the intervals between the checks be increased. The effect is just reversed when the temperature is diminished. This is the source of greatest difiBculty with all portable time-keepers, and renders the common watch worthless for any thing beyond an approximate indicator of the time. 10. — To remedy this defect, the common wheel is replaced by what is called an expansion balance, which is re- presented in the figure. ^ ^ is a bar which receives the end of the arbor into an aperture at its middle point. To the ends of the bar are securely attached two compound metallic curves C C, composed of two concentric strips, one of steel and the other of brass, the latter being on the convex side ; these are soldered or burned together throughout their entire length. Each of these curved pieces carries a heavy mass D D, movable from one end to the other, but capable of being secured in any one place by means of a small clamp-screw shown in the drawing. ' No\«:when the temperature increases, the exterior brass expanding more than the interior steel, the ends C C are thrown inward towards the arboi-, while the ends of the bar are thrown outward, but through a much leas distance ; and thus by properly adjusting the places of the masses D D, the moment of inertia of the balance may be made to vary dii'ectly as the raoment of the elastic force^of the spring ; in which case the angular ac- celeration becomes constant, and the intervals between the interposition of t!ie checks equal. ' 11. — I'o regulate the rate, two large-headed screws B B, called menn- time screws, are inserted, one into each end of the bar. If the chronometer run too slow, the moment of inertia is too great for that of the spring, and these screws must be screwed up, which has the efiect to lessen the dis- tance of their heads from the axis of motion, and thus to lessen the mo- ment of inertia, and increase the angular acceleration. If the chronometer run too fast, the screws must be unscrewed, the efl'ect of which must be obvious. 16 242 SPHERICAL ASTRONOMY. 12. — The escapement is of the kind usually called the detached, from the fact that except at certain instants of time, the whole appendage of the balance-spring is relieved from the action of the scape-whee!. Fig. 10. The scape-wheel is represented at M; it is urged by the motor, acting through the wheel-work, to move in the direction of the arrow-head. A steel roller C, called the main pallet, is firmly fixed to the arbor of the bal- ance. In the pallet is a notch i, having one of its faces considerably un- dercut, and covered with an agate or riiby plate to receive the action of the teeth of the scape-wheel. Securely fixed to one of the frame-plates of the chronometer is a stud B, and to this is attached a spring A, called the detent ; this spring is extremely thin and weak at the stud B. Attached to the detent is a stud D. A ruby pin projects from the detent at c, which receives a tooth of the scape-wheel when one escapes from the pallet bear- ing i. From the stud D proceeds a very delicate spring E, called the lifting spring, which rests upon and extends beyond a projection F from the end of the detent ; this projection being so made that the lifting' spring cannot move in the direction /row the scape-wheel without taking the detent with it, and thus lifting, as it were, the pin c from the tooth with which it is in contact;, while it leaves the lifting spring free to move towards the scape- wheel without disturbing the detent. Concentric with the main pallet, at- tached to and just above jt, is a small projecting stud a, called the lifting pallet, which is flattened on the fac6 turned from the scape-wheel and rounded on the other. The flattened is called the lifting face. 13. — Mode of Action. — In the position of the figure, the main pallet, under the action of the balance-spring, is moving in the direction of the arrow-head/, and the lifting pallet is coming with its lifting face in contact with the lifting spring E, which it lifts with the detent so as to raise the pin c clear of the tooth of the ^cape-wheel with which it is in contact. * APPENDIX II. 243 By the time the wheel is free from the pin c, the main pallet has advanced far enough to receive an impulse from the tooth t upon its jewelled surface t, and before this tooth escapes, the lifting pallet a parts vfith the lifting spring E, and the detent returns to its place of rest and interposes the pin c to receive the tooth t^ as soon as the tooth t has been liberated by the onward movement of the main pallet from its face i. The balance having performed a vibration ly the impulse given to the main pallet, returns by the action of the balance-spring, and with it the lifting pal)et tr, whose rounded fi^ce, pressing against the lifting spring E, raises it and passes, first the detent without disturbing the latter, then the lifting spring, and moves on till the balance has completed the vibration, when it returns to the po- sition indicated in the figure, and the same evolution is performed again ; the balaiice thus making two vibrations for every impulse. The Vernier. 1 . — This is a device by which the value of any portion of the linear distance between two divisions of a graduated scale of equal parts may be found in t^rms of the space itself. It consists of a scale whose length is equal to any assumed number of parts of that to be subdivided, and is divided into equal parts of which the number is one greater or one less than the number of the primary scale taken for the length of the vernier. • ^- » I — i — T — I — I I I I ~l I -^ C y be any scale of equal parts, and denote by s the length of n— 1 of these parts ; then will s be the value of the unit of the scale. Take a vernier BE oi equal length s, and suppose it divided into n equal .parts, then will be the length of one of its parts, and the difference of length between m, paits of the scale and an equal number of parts of the vernier, will be ms ms m.s 244 SPHERICAL ASTKONOMT. But is the value of the unit of the scale, and n the whole number of w— 1 divisions of the vernier ; denoting the firet by V, this difference may bo ^vl•itten - V. n Nov,', the length of a part on the scale is greater than that on the vernier, and the number of parts on the vernier is greater by one than the number in an equal length on the scale; hence, if the wi"" intermediate division of the vernier coincide with anyone division on the scale, the zero of the ver- nier will fall between two divisions of the scale, and be in advance of that bearing the smaller figure by the distance expressed above ; so that, taking the zero of the vernier as the index or pointer, its distance from the zero of the scale will be the number of units denoted by the figure on the division next preceding, plus the — th part of the unit of the scale. Thus, in the figure, A being the zero of the scale, B that of the vernier and therefore the pointer, the distance 6f the latter from the former will be Aa+aB; and because ti—lO, and the division frof the scale coincides witli the ,4th 4 • of the vernier, »i=4, and the distance A B^Aa + — .fe. 2.-^ The least value that may be read with certainty is obtained by making to=1, which will give, V n' Whence we have this rule for finding the lowest reading by means of the vernier, viz. : Divide the lowest count, or unit of the scale, by the number of divisions on the vernier. If the scale be tenths of inches, and we make w=10, then will n~ IQ ' ~ 100 ' in which case the subdivisions will be carried to hundredths of inches, 3. — The vernier is equally applicable to all kinds of scales, to circular as well as rectilinear : the only condition being that the different parts shall be equal. Suppose each degi'ee on the circumference of a circle is divided into 6 ■ equal parts, and that tlie number of parts on the vernier is 60, then will APPENDIX II. 245 F=10'=z600" and V 600" — = =rlO". n 60 So that the read! ng of angles with an instrument having such a circle may be canied to ten seconds. Micrometer. 1. — The Micrometer is an instrument employed to make minute measurements, and is applicable alike to time and linear distance. It has s-arious forms. 1*. — The Reticle. — He who views a distant object through a telescope, does not look at the object but at its image within the tube of the instru- ment. The image of a point is always in a plane through the focus of the leas conjugate to the point itself, and perpendicular to the tube of the tel- escope. The visible portion of this plane is called the field of view. Some point in the field of view is arbitrarily assumed as an origin of reference, and marked by the intersection of a pair of cross wires. The line thjough this point and the optical*centre of the field lens, is called- the line of collimation. 2. — If the telescope be at rest and an object in motion, the image of any one of its points will when visible pass across the field of view ; and one of the opaque wires being made to coincide with its path, the image will move directly towards the line of collimation, and the exact instant of its reaching it may be noted. But every such observation is liable to error. To increase the chances of avoiding this error, the wires marking the line of collimation are made perpendicular to one another, and an equal number of equidistant and parallel wires added on either side of that which is per- pendicular to the path of the image. When the motion of the image is uniform, an average of the times of passing the parallel wires will, accord- ing to the doctrine of chances, give a time of passing tiie line of collima- tion more free from error than the single observation. 3. — This simple form of the micrometer is call- ed a reticle. The wires or spider lines are stretched across a circular metallic diaphragm pierced by a large concentnc opening. On the edge of the diaphragm, and in the prolongation of the single wire, two studs project at right angles to its plane; and these, with two antagonistic screws ^^, hold the reticle in po- Fig. 11. 2m SPHERICAL ASTRONOMY. si don ; the screws, for this purpose, passing tlirough the tube of the teles cope and leaving the heads exposed for purposes of adjustment. 4. — Position Filar Micrometer. — The purpose of this instrument is to measure the angles at the observer, subtended by the distances between objects that appear very close together, and to determine the positions of the planes of these angles. It consists of two parts, viz. : One to measure the angle between the objects ; and the other, the inclination of the plane of the objects and observer to some co-ordinate plane. 5. — The first is represented in the figure, a and c are two fine par- allel wires, which are made to move at right angles to their lengths by means of screws firmly connected with th^ forks A and C, to whose prongs they ai'c attached. The screws have fifty threads to the inch, and are Fig. 12. moved by nuts so mounted as to admit of a motion of rotation without translation, so that by turning the nuts a motion of translation is commu- nicated to the wires in either direction, depending upon the direction of the rotation. The outer surfaces of the nuts are cylindrical, and enter fric- tion tight the central perforations of two circular wheels whose planes are perpendicular to the lengths of the screws, and which are large enough to admit of their circumferences being divided into 100 equal parts, which parts are marked and numbered. Each wheel is provided with a station- ary pointer or index. A third and stationary wire, perpendicular to the first two, is supported by a diaphragm-disconnected from the forks. Upon one of the' interior edges of this diaphragm, and parallel to its wire, is a graduated scale in the shape of a comb, having 50 teeth to the inch, so that one revolution of a nut will carry its movable wire from the centre of one valley between the teeth to that of the next. Near the central valley of the scale is a small hole to mark the zero of the comb-scale, from which the scale is estimated in eitlier direction. It is easily seen that a turn of the nut-head through one of its divisions will move its wire through a linear distance equf.l to j-Jj of i'S °'' 4^0 7 ^^ -^^ '^^^^ ! ^°<^ having ascertained by the measurement of some small distance on the circumference of a great circle of the celestial sphere, or by the process in Appendix No. I., its equivalent in arc, this, the M '.a APPENDIX II. 347 micrometer part of the arrangement, is readily applied to the dele.'mination of small angles. 6. — The second and position part consists of a circular plate A A, called the position circle, some three or four inches in diameter, having its circumference divided into 360°, which are again subdivided to any convenient extent. The central part is cut away, and the micrometer aiTangement so attached, with its wires parallel to the position circle, as to admit of a free motion Fig. is. of rotation about an axis /iCtti^ through its centre, and per- pendicular to the plane of the wires. To the revolving plate of the micrometer part are attached two verniers V V, and motion is com- municated to the latter by a ratchet an'd pinion, of which latter the head is seen at 0. The microscope by which the wires and comb-scale are magnified, and which serves also for the eye-glass of the telescope, is represented at £!. By means of a screw cut upon a projecting ring ;"vround the large and central aperture of the position circle, the instrument, as represented in the figure, is attached to the tail end of the telescope. 7. — To measure the angular distance between two objects in the field of view, turn the head till the fixed wire passes through their images, then bisect the images by the movable wires ; note the reading on the comb-scale and upon the h^ads; take their surn or difference according as the wires are on opposite sides, or same side of the zero of the eomb-scale. This reduced to arc will be the measure sought. Note also the reading of the position circle ; this will give the inclination of the plane of the angle to the plane through the zero of the position circle. A second angle being measured in the same waj', the difference between the second and first reading of the position circle will give the inclination of the planes of the two angles. Micrometer Revolution. The micrometer being supposed in place, and the eye-piece pressed for- ward far enough to obtain a distinct view of the wires, the. telescope is directed to some distant object, and adjusted to distinct vision. An image of the object will be formed on the plane of the wires, and any one of ius 248 SPHERICAL ASTRONOMY. linear dimensions may be measured by turning the position circle till the stationary wire coincides with, and the movable wires pass through the extremities of its image. The number of entire comb-teeth between the movable wires, multiplied by 100, and this product increased by the sum of the readings of the screw-heads, will give the linear dimensions of the image expressed in units of the screw-head. The value of the latter is, in the case we have taken, g-^gg of an inch. To find the angle subtended by the object, we must know the angular value of the unit on the screw- head. It is demonstrated ( Optics, § 60) that the optical image of any jjoint of an object, is on a right line drawn through the point and the optical ceu- tie of the lens by which the image is formed. The angles, at the optical centre, subtended by an object and its image, are therefore equal, and if the images of objects which subtend equal angles were at the same dis- tance from the optical centre, they would be of the same size. The lineal dimensions of the images at the same distance from the optical centre, would therefore be proportional to the angles subtended by their respective objects, and to find the angular value in question, it would be sufficient io^ cause the image of some well-defined object, whose distance and dimen- sions are known, to he embraced by the 'wires, and to divide the angle which ilie object subtends, expressed in seconds, as determined trigcmometriealhj, by the number of units of the screw-heads, which indicate their separation. But the distances and therefore the dimensions of images, whose objecis subtend the same angle, are variable, being dependent on the distance of the objects, and from the value found by the. above process must be. de- duced that which would have resulted had the image been formed at some constant distance, which is that of the principal focus. Let / and /" denote the distances respectively of the object and its image from the optical centre, and F^^ the principal focal distance of the object- glass, supposed convex. Then, Optics, § 44, Eq. (40), f"~ f ' r.iid denoting by n and N, the number of units of the screw-heads when the imige is embraced at the distances /" and F,, respectively, we shall. Uhvc, Optics, § 64, Eq. (58), /" :F„::n:N- whenc ) F f—F TV— ZIL n — — n ■' '—H APPENDIX II. 249 and calling «, the number of seconds in the angle subtended by the object^ we have, by the rule just given, > a _ a.f . . N--n.{f-F^) W Example. — ^The length of the object measured in a direction perpen- dicular to the line of sight was 3 feet ; the distance from the object-glass, 261.9 yards ; the principal focal length, 4o.'75 inches ; and the sum of the divisions on the screw-heads indicating the separation of the wires, 1819. Then /= 261.9'""- ; F^^ = 45.75'"' = 1.2'708''"- ; n = 1819. /— F^^ = 260.6292>"'\ H . O.S""- ^ tan ^- a ■. whence *""- 261.9""'"'""'^ jlL tllC Avg. lO t.^UKJKJUtJ a^ 13' 0l".5l = = 181".51. Log. a . . , 2.8962892 " / • • 2.4181355 •' n a comp. . -4.7401673 . —3.5839923 " 4-=0"-4351 iv . -1.6385843 Now, to measure the angle subtended by the distance between any tw9 points, direct the telescope so as to get the images of the jjoints in the field, and turn the micrometer till the stationary wire apparently passes through them, and by a motion of the scrow-heads bring the movable wires to the images — the, number of units of 'Jie screw-head, which indi- cate the separation of the wires, multiplied by tho decimal 0".4351, will give the number of seconds in the angle. The value of -^, being a function of F^^, Eq. (a), will of course vary with the object-glass, but is perfectly independent of the eye-glass. If the distance/ be so great that F,j may be neglected in comparison, than will Eq. (a) give which will be the case when the ang'ilar value is determined from astro- nomical objects. 250 SPHERICAL APTEONOMY. Spirit-Zevel. 1. — This is an instrument used to adjust a line to a given position in reference to the horizon. It consists of a cylindrical glass tube A A, whose axis is the arc of a circle. This tube is filled nearly full with some one of the more perfect fluids, such as alcohol or naphthalic ether, leaving a small portion of air, seen at B, called the air-bubble, and hermetically sealed at both ends. It is then usually set in a metallic tube C, very much cut away on one side from the middle towards the ends, so as to exhibit the bubble and fluid when in a horizontal position. This metallic tube is connected with a plate of metal F Khy a hinge £ and screw D, the axis of the hinge being perpendicular, and that of the screw parallel to the plane of the circular axis of the level. 2. — A scale of equal parts is cut either upon the upper surface of the glass tube dr upon a slip of ivory and metal lying in the plane of the tube's curve, as represented at G O. The divisions of the scale being numbered, the value of the spaces in arc is readily ascertained by attaching the level to the face of a vertical graduated circle, and turning the latter sufiiciently to cause the air-bubble to pass from one end of the scale to the other. The angular space passed over by the circle reduced to seconds, divided by the number of units on the scale traversed by the bubble, will give the value of the unit in some njultiple of the second. '3. — Use. — The suiface of the fluid being always horizontal, the line connecting the ends of the bubble will be a level chord of the level's are, and the radius passing through the point of the scale midway between the ends of the bubble will be vertical. Now, suppose any line of an instrument with which, the level is used to be made parallel either to the radius passing through the zero of the scale, or to the chord whose ends are marked by fbe same numbers ; then, to make this line vertical in the first case, or horizontal in the second, move the instrument, the level being securely attached, till the ends of the bubble are equally distant from the zero. If the ends of the bubble be not at the same distance from the zero, the inclinal'on x of the fine in question to the vertical or horizontal APPENDIX II. 251 direction is thus found : Let a denote the semi-.ength of the buhble, m and n the numbers of the scale at its extremities, then will a-\-x=m, a—x-=n'; whence rh — n , , . ^ = -y- = ^ (2) This value of x being independent of the length of the bubble, which is indeed a variable quantity, even in the same level, because of its varying temperature, gives the inclination of the line under consideration to its proper position, when the level is adjusted to the instrument. If the lower surface of the plate jF'JF' be parallel to the chords of equal numbers, the inclination of any given line or plane may be ascertained by laying this plate upon it and applying the above rule. But if the lower surface of the plate be not parallel to the chords of equal numbers, its inclination to them and that of the' plane or line in question to the horizontal or vertical direction may nevertheless be found thus : Denoting the first by y, and the latter, as before, by x, and usmg the notation of equation (2), we have for one position of the level, x=l-y, and for the reversed position of the plate with its level, x=l'+y, whence l+V m — n-\-m' — n' X = — — = , I — V m — n — m' — n' y = -J" ~ 4 ^ ■ K the given surface or line be provided with adjusting screws, as la the case in all astronomical instruments, the ends of the bubble may be brought to the same reading in the first position of the level, in which case, we have »i=», and m'—n' ,. x = —^-=-y (3) The angle y is called the error of the level, and the angle x the error in level of the instrument, and the above equation gives this rule for finding and correcting these errore, viz. : 252 SPHEPICAL ASTRONOMY. The level heinp placed over (he given line, bring, by means of the adjusting screws of the instrument, the bubble to read thi same at both ends ; then reverse the level, or turn it end for end, and take one fourth of the difference of the new readings ; add this to the lesser of the read- ings, and turn the screw D till the end of the bubble nearest the zero reach the numbi r answering to this sum, to which add again the same quantity, and bring the end of the bubble to this new reading by the adjusting screws of the instrument. The ends ^of the bubble will stand at the same numbers, and both errors will be destroyed. Heading Microscope. 1. — This instrument, like the vernier, has for its object to read and subdivide the space between two consecutive divisions of any scale of equal parts, and is the most perfect yet devised for this purpose. It is a compound microscope, whose object-glass forms an enlarged image of the space to be divided. This image is thrown upon the plane of two spider-lines or wires, arranged ia the form of a St. Andrew's cross, and so placed that a line bisecting its smaller angles is parallel to the cuts or division marks of the scale. The cross is attached to a diaphragm, which is moved by a micrometer screw in the direction of its plane, per- pendicular to the . axis of the microscope. The head of the screw is divided into any number of equal parts, depending upon the nature of the scale and the extent to which the subdivisions are to be carried-. The numbers on the head are so placed that when the screw is turned in the direction to bring them in the order of their increase to a fixed pointer, the cross shall move along the image-scale in the direction in which its numbers decrease. Within the barrel of the microscope is a stationary comb-scale, like that in the position micrometer. Its plane is "i^Urallel to that of the cross, and the distance between the centres of two valleys, separated by a single tooth, is equal to the space over which the cross is moved by a single revolution of the screw. Every fifth valley is cut deeper than the others to facilitate the reading ; and near the bottom of the central valley of the comb is a small circular aperture, to mark the zero position of the pointer or index, which is a small wire attached to the movable diaphragm, and so placed that its prolongation shall bisect the smaller angles of the cross. In (1), A A is the main tube of the microscope, passing through a collar or support B, where it is firmly held by two milled nuts g g, which act upon a screw cut upon the outer surface of the tube. These nuts also serve to change the distance of the whole microscope from the scale to APPENDIX II. 253 be read ; h is tlie object-glass placed in a smaller tube, upon whoso outer surface is also a screw, by which this glass may be mo/ed independently Fig. 15. Fig. 16. 2 of the main tube ; the diaphragm of the cross is in a working box,^ whose edge is seen at a ; c is the graduated head, firmly attached by a fnction clamp to the nut 6 of the micrometer screw ; / is a pointer a1;tached to the working box ; d is the eye-glass, which moves freely in the direction of the axis of the microscope by a sliding tube ; at c' is represented the head of a small screw, which supports and gives motion to the comb-scale within the working box, and S S represents the edge of the scale to be subdivided. In (2) is represented the field of view, as seen when the eye is apphed at d, in which m m' is the image of the scale, with one of its outs bisecting the smaller angles of the cross, and e the wire index at its zero position, as indicated by its being seen through the centre of the circular aperture of the comb. In this position of the pointer, the zero of the graduated head e is brought to the index /, by holding the nut h firmly in the hand, and turning the head, which is only held in its place, as before stated, by the action of the friction nut. 2. — The quotient arising from dividing the length of the image space by that over which the wires move in one revolution of the screw-head, as given by the comb-scale and head, is called the run of the micrometer. For convenience, the run should be an entire number. 3. — The image-scale must be accurately in the plane of the wireb, otherwise there would be a parallactic motion, which would shift the position of the wires on the image-scale at every change in the position of the eye, and thus vitiate the measurement. This parallactic motion is 'Easily detected by slightly shifting the position of the eye when looking through the eye-glas.";. There are, then, two adjustments for the reading microscope, viz., that for the run and that for parallax. 254: SPHERICAL ASTEONOMY. 4. — The size of the image of an object, and its distance from the lens by which it is formed, are dependent upon the distance of the object from the lens, being greater in proportion as this distance is less, and less as it is greater. If the distance of the object-glass of the microscope from the scale be changed by means of the screw on the tube at h, the size of the image space will be altered, and may, therefore, be made of such dimensions that the cross will move from one division to the next in ordei', by a given entire number of revolutions; and if by this operation, the image be thrown off the plane of the wires, as it in general will, it 'is restored by changing the distance of the whole body of the • microscope from the scale by means of the milled nuts gg. By two or three efforts cautiously conducted, the adjustments may be made without difBculty. To illustrate, let the scale be that of the sexagesimal division of the circle, and suppose each degree divided into twelve equal parts, e;ich space will, be equal to five minutes; if we make the run five, each toot.li on the comb will be equal to one minute, and if the screw-head be divided into sixty equal parts, each of its spaces will be equal to one second; so that the circle may be read to seconds. Now suppose on examining the run, which is done by turning fl-.e screw-head till the cross moves from one division to the next in order, it be found 5' 10" ; it is too great. Move the object-glass h from the piano of the circle by screwing in its tube, the image will decrease, and, if it were before on the plane of the wires, it will now pass to some position between that plane and the object-glass h. Move the whole body of the microscope by means of the milled nuts gg towards the circle ; the image will be restored to its proper position, with less dimensions tEan it had before. By one or two repetitions of this process the adjustments are made. 5. — The wire pointer at its zero position on the comb-scale is the index of the circle or instrument scale. When the pointer, in this po- sition, is immediately opposite a division mark of the circle scale, say the third after that marked 2'7°, which is indicated by the angles of the cross being bisected by the image of that division mark,, the reading ia 27° 15' 00" ; but if the intersection of the cross wires falls between the third and fourth divisions after that marked 27°, then will the reading be greater than that above by the value of the distance from the cross wires to the division mark to which the cioss will move by turning the screw- head in the order of its increasing numbers. To find this value, turn the Bcrew-head in the direction just indicated till the angles of the cross are APPENDIX II. 255 bisected by the division mark in question, and count the entire number of comb teeth between the aperture and pointer, then note the reading on the screw-head ; suppose the former to be 3 and the latter 41, the true reading will be27°18'41". T/ie Transit. , 1. — The transit is an instrument which is used in connection with a time-piece to ascertain the precise instant of a body's passing the me- 256 SPHEEICAL ASTRONOMY. ridian of a place. It consists of a telescope T T, usually of considerable power, permanently fixed to a substantial axis A A, at light angles to its length. The axis terminates at each end in a steel pivot, accurately turned with a diamond point, to a cylindrical shape. The pivots are of equal diameters, received into notches cut in two blocis of metal, called Ys, which rest in metallic boxes, the latter being imbedded in metallic or stone piers, accoi-ding as the instrument is intended to be portable or fixed. 2. — Perrrianently attached to the tail or eye end of -the telescope, on opposite sides, are two small gi-aduated circles, called finders. The planes of these circles are perpendicular to the axis of the transit, and each circle has an index-arm, which carries a small spirit-level and two verniers, one at each end. The index-arms are movable about the centres of their respective circles^ and are, as well as the axis of the transit, provided with a clamping and tangent screw arrangement, thus afibrdlng, with the aid of the level and vernier^, the means of giving the telescope any de- sired inclination to the horizon. 3. — At the solar focus of the object-glass of the telescope is a reticle, Fig. 11, iii which the single is replaced by a double wire, with small interval, and so placed as to be parallel to the axis of the transit. These are called axis wires. Those wires of the reticle which are at right angles to these are called the normal ■wires. To the fixed wires of the reticle a movable one is added ; it is always parallel to the normal wires, indeed, is itself a normal wire, and is put in motion in the direction of the axis wires by means of a micrometer screw, with graduated head, shown at m. 4.' — The small tube -containing the eye-piece of the telescope is attached to a sliding-frame, connected with a screw e, by which the eye- piece is carried fiom one side of the field of view to the otlier, in the direction of the axial wires. S. — The axis is hollow throughout, and the pivots are perforated at the ends to admit the light from a lamp L, supported upon one of the piers. This light js received by a reflector within the tube of the tel- escope, atd inclined to its axis undef an angle of 45°, and is reflected to -the eye-glass, thus illuminatifig the field of view, and exhibiting the wires of the reticle. The reflector is perforated by an elliptical opening in its centre, to permit the direct light from any external object to pass freely to the eye end of the telescope. When the illumination is through the other end of the axis, the jeflector is revolved through an angle of 90°, by means of a milled-headed wire, with which it is permanently con- nected. The head is shown at r. APPENDIX II. 257 Pig. 18. Fig. 19. 6. ■=— The boxes -which support the Ys are large enough to permit a slight play in the latter; one in a horizontal, Fig. 18, and the other in a vertical direction, Fig. 19, the motions being effected by antagonistic screws. By the first of these motions, the line of collimation is brought to the meridian, after the rougher ap- proximations to that plane are made by other means, and by the second the axis is made horizontal by the aid of a large and delicate spirit- level. Fig. 20, mounted upon in- verted Ys, far enough apart to rest upon the pivots. Adjustments. 7. — The transit is adjusted within itself when its hne of collimation is perpendicular to its axis ; and it is in position, when its axis is perpen- dicular to the meridian. Its finders are adjusted, if the air-bubbles at their levels indicate the same reading at both ends, when the verniers indicate the true inclination of tbe line of collimation to the vertical or horizon. 8. — It is by no means niscessary, or even desirable, to aim at' perfect adjustment. It will, in general, be much safer to reduce tbe en-ors of adjustment to narrow limits, then to determine their amount, and eliminate their effect from observation, in the manner to be described presently. 9. — Line of Collimation. — Direct the telescope to some . small, distant, and well-defined terrestrial object. Bring it apparently between the horizontal wires, and measure its distance from the central vertical wire by means of the micrometer and movable wire j denote' this dis- It 258 ' SPHERICAL ASTRONOMY. tance by c'. Lift the transit from its Ys, turn the axis end for end, and meiisure, as before, the apparent distance of the same object from the middle wire, and denote this distance by c". Place the movable wire at the distance, of c'+c" on the side of the object from the middle wire, and move the whole reticle by the antagonistic adjusting screws, which he in the direction of the axial wires, till the object appears' on the movable wire ; the line of collimation will be adjusted. 10. — Errofr of this adjustment. — If n denote the value in arc of the micrometer's unit, then will the angle which the line of collimation makes with its proper position, before moving the diaphragm, be (4) and the line of collimation will describe, when the telescope is moved, a conical surface, whose intersection with the celestial sphere will be a small circle. Example. — When the telescope is pointing to the south, let the middle wire appear to be 326.3 revolutions to the right hand of the object ; when the axis is reversed, let it appear 318.7 tp the right, then will 326.3 — 318.7 n . =c=3.8?2, Ji and if one revolution of the micrometer correspond to the space an equa- torial star would pass over in three seconds of time, then will 3s.xl5 „,, ^^ 100 ' and c=3.8xO".45=l".7l. 11. — The axis. — This must first be levelled, then moved in azimuth till it is perpendicular to the meridian. Mount the level with its inverted Ys upon the pivots, bring the bubble to the same reading at each end by the adjusting screw of the level ; reverse the level, and bring the bubble again to the same readings— half by the iscrew of the level and half by the vertical antagonist screws of the Y, which admits of vertical motion. Eepeat the operation once or twice, and the thing is done. APPENDIX II. 259 12. — The error in this adjustment. — After the first approximation, denote by «', e", &a., the reading of the east end of the level; by w', w", &c., the same of the west end, and let the parenthesis denote the end of the axis marked by some peculiarity, such as the clamp, or illumination ; then mounting the level in its place, and writing its readings in any one position upon the same horizontal line, we may have First position of level . . . . e' (w') Level reversed e" (w") „,, . e'+e" (w')+{w"y Half sums of . . . ^^ — '——^ — '- 2 2 These half sums are the readings which the level would have indicated in both positions had it been in perfect adjustment, and (y,') + (w")-e' +e" ~ 4 = * (5) ,the error, or inclination of the axis to the horizon, expressed in the level's unit, provided its pivots be of the same size. But lest there may be a difference! in the pivots, reverse the axis, and apply the level as before, and we may have For first position of level . . («'") . . . w'" Level reversed (^"") • • • w"" Half sums ^ — - — ^ , . 2 2 whence w"'+w""-{e"')+{e"") ^^ and s-s' {w')+{w")+{e"')+{e"")-w"'+w""+e'+e" ^ ... ~T~~ ^ 16 -^ •^'> will be the angle which the axis makes with the line whose inclination is given in equation (5), whence, denoting the inclination of the axis to the horizon, or the angle which a plane perpendicular to the axis makes with a vertical plane at right angles to the projection of the axis, on the hori- zon, hy I', we shall have This value is expressed in terras of the level's unit ; if «' denote the 260 SPHERICAL ASTRONOMY. value of this unit in seconds, -we shall have, representing tlje angle in seconds of arc by I, l=n'l'=^'{s±t) ' (8) The value of t for the same axis is constant, and must be determined by talring a mean of a great many careful observations. If it be positive, the pivot at the clamp end of the axis is the larger, but if negative, it is the smaller. "When the half sum of the readings on the west end is greater than that on the east, the inclination is counted positive, and the plane perpen- dicular to the axis will fall to the east of the zenith ; and as it is obvious that the axis will be depressed on the side of the greater pivot, when the level indicates perfect adjustment, the upper sign, in equation (8), must be taken when that pivot is to the east, and lower when to the west. Example. — Performing the operations indicated, let the following be the record, viz. : First position of level Level reversed . . . Axis reversed. First position of level . . Level reversed (81.30) 4)7.50(1.875=s' Adding the indications of the level diagonally, we have (322.95) — 312.75 71.40 78.60 (87.60) (80.10) 150.00 167.70 4)17.70(4.425= (73.95) (81.30) 167.70 s 84.90 77.85 155.25 162.75 162.75 16 : 0.6375 = «. Applying the level to the face of some vertical graduated circle, § 95, let 23.5 of 1*3 units correspond to 30" then will „'=i^ = l".276. 23.5 Whence for Clamp end west Z=(4.425-0.037)Xl".276=4".83348 Clamp end east Z=(1.875-f 0.637)Xl .276 = 3".205312 APPENDIX II. 261 13. — Azimuth adjustment. — It is now supposed tliat tLe errors of aoUimation aud of level are destroyed. By a reference to a map of the stars it will be seen that a straight line drawn from the Pole star to a point midway between the fifth and sixth stars, called s and ^ respectively, in the constellation of the Great Bear, will pass sensibly over the pole. About the time when this line assumes a vertical position, direct the tel- escope to the Pole star, and keep its image on the middle normal wire by a motion of the horizontal adjusting screws of the Y, or by the mo- tioQ of the Ys themselves, if the requisite range be beyond that of the screws, and at the instant when it is inferred from a suspended plum- met, that the line referred to is exactly vertical, arrest the motion and secure the Ys. The adjustment will be sufficient for the first approx- imation. Next find the amount of azimuth error. The axis being horizontal, and the line of collimation perpendicular to the axis, it is plain that in the motion of the telescope the line of collimation will describe the plane of a vertical circle, and that the angle made by this plane with the me- ridian is the error in question. LetffOIi be the horizon, SZH the meridian, P the pole, Z the ze- nith, and S the star when on the line of collimation. Make, X ^latitude of plaee=90°— ZP; S =declinationofstar=:90°— PaS, positive when north, nega- tive when south ; PzzzZ P S—liom: angle of star ; Z=^Z(S=azimuth of star's position, and equal to the eiTor sought when east. s =Z5=:zenith distance of star. Then, in the triangle Z P S, sin P= sin Z .sin z cos S ' and because the sines of P and Z are very small, cos (9) in which P and Z are expressed in seconds of arc 262 SPHERICAL ASTRONOMY. Divide both members by 15 and make. ■ sin (\—S) J — =«; cos and equation (9) becomes P h .Z 15-^ ^''^ P . in which — is the time required for the star to pass from the vertical de- scribed by the line of coUimation to the meridian, and if i denote the time indicated by a timepiece at the instant the star is on the central normal wire, the time of meridian passage will be '+^'^4 = ^ (") Let e be the error of the timepiece at the time t referred to the vernal equinox ; m the rale or quantity by which this error is increased or di- minished in one day or twenty-four hours ; then, if R denote the right ascension of the star, supposed known, will t-\-e^-'k. — = R (12) V 15 and for a second star 1 o in which *' — < is reduced to the decimal part of a day. Subtracting the first from the second, we get in which the upper sign is used when the timepiece runs too slow, and the lower wheq too fast ; whence, Z R'—R-(t'—t)±{f-t)m, , - = ^--- . .... (13) Z is hence known, and for which ^he instrument may be corrected, if desired. This value jn equation (11), gives the time of meridian pass- age, and in equation (12), which may be written ez=R-t-h. — = R-T, ....... (13') 15 ^ gives the error of the timepiece. APPENDIX II. 263 The sign of the quantity Jc changes when +e'+c.C' +I.L' +Z.Z/ =R' t" +e'+c.C" \l.L" +Z.Z/' =R" t'" +e'+c . C" +1 . L'" +s . Z/" =B"' t""+e'+c . G""+l . L""+z . Z,""=B"" which are sufficient. But as there are always slight errors in the obser- vations themselves, it would be well, where great accuracy is required, to increase the number of these equations, and treat them after the method of least squares. 15. — The finding circles. — These may indicate zenith distances, altitudes, or polar distances. The rule for adjusting is the same for all. Direct the telescope to the distant horizon, and move it till the image of some small object appear midway between the double axial wires : clamp the axis, move the index-arm till its level indicates the same read- ing at both ends of the bubble, and note the reading of the vernier. Unclamp and reverse the axis ; bring the image of the same object again between the same wires, and clamp the axis ; move the index-arm till the bubble has the same reading at each end, and again note the reading of the vernier. If the vernier reading be the same as before, the circles 'are in adjustment ; if not, add the readings together, take the half sum, move the" index-arm till the vernier is brought to the reading indicated by this half sum, clamp the index-arm, and bring the air-bubble so as to have the same reading at each end by the adjusting screws of the level. It would be well to verify by repeating the process. It may be, that the finders are gradu- ated from 0° to 360°, in which case, if the first reading were a°, the second ought to be 360°— a°. IC. — The adjustments in azimuth, collimation, and level being per- fected, the middle normal wire will be a visible representation of that portion of the celestial meridian to which the telescope is pointed ; and when a star is seen to cross this wire in the telescope, it is in the act of culminating. The precise instant of this event being noted by the clock or chronometer, the time of meridian passage is known, and any eiTor in noting this precise time is lessened by the use of the lateral wires of the reticle, as already explained, 17. — Besides, these lateral wires increase the chances of secunng an observation that might, without them, be lost. It frequently happens that efforts to obtain the time of a body's passing the middle or other wire are defeated by the- presence of clouds, or other accidental circumstances, 266 SPHERICAL ASTEOIsrOMY. Ill which, if the time of passing any one be obtained, that of passing the middle or mean place of the wires, when not equally distant, may bij leduced thus. Let i„ <2, ifj, &c., be the times of crossing the several wires in order, hen will ti + t,+h+...t. --L (18) n which t„ denotes the time of the body's crossing the mean position of the wires, and n the number of wires. And {L—ti)-oosS=ij, ■ (<„— <2).cos5=4, {L—h).cos5—i3, (19) in which 5 denotes the declination of the body observed, and I'l, ij, ig.. . i„ the constant intervals of time required for a body in the equatcff to pass over the distances which separate the several wires from their mean position. Adding equations (19) together, we obtain '' (20) «»=— + • ^ n n cos in which 2 denotes the algebraic sum of the quantities expressed by the letter written after it. ' By carefully observing a star whose declination is known, we obtain the values of Ji, 4, &c. ; and these being tabulated with their proper signs, equation (20) will give the time of a body's passing the mean position from the time of passing one or more of the threads. The Collimating Telescope. 1. — In some situations it would not be possible to obtain a distant mark by which to coUimate, and a near one could not be used in conse- quence of its image falling too far behind the reticle. In such cases recourse must be had to what is called the collimating telescope. Fig. 28. This is a telescope whose eye-piece is removed, and upon its tube is mounted a small swing-frame, supporting a reflector, by means of which APPENDIX II. 267 Fig. 24. sufficient light may be thrown through the tejpscope to illuminate a pair of cross wires, situated at the solar focus of the object-glass. In this position of the wires, we have, from the principles of optics, these facts, viz. : the rays composing the pencil of light proceeding from any point of the cross, vrill emerge from the collimator parallel to a line drawn through that point and the optical centre of the lens ; and if the telescope of the transit be directed towards the collimator so as '.o receive these rays, an image of the point in question will appear in its solar focus, and on a line drawn through the optical centre of its object-glass, par- allel to these same. rays. The Vertical Collimator. 1- — This instrument is used for the double purpose of collimating, and for finding the zenith or horizontal point of circles, used in the meas- urement of .vertical angles. It consists of a collimating telescope T mounted in a vertical position upon an annular plate' B, of cast-iron, float- ing upon the free surface of mercury, contained in an annular trough 'S, also of cast-iron. The annular plate is called the float. The telescope is mounted upon the float in a manner similar to .the transit, except that the axis is near- er to the object end. One of the Ys may be elevated or depressed by an ad- justing screw A, while the telescope is turned about its axis by another A\ thus affording the means of giving the line joining the cross wires and the optical centre of the lens a vertical position. L is the lamp, and G the reflector, to catch its light and throw it upon the cross wires at the lower end of the tube. 2. — The collimatinij process. — Take the transit for instance. Level the axis carefully ; turn the telescope in a vertical position ; place the collimator below, and bring the image of the intersection of its cross wires, seen upon the bright ground O, accurately on the intersection of the middle wires in the transit, by means of the adjusting screws of the collimator; next turn the float in azimuth through 180°. If the emer- gent rays from the collimator be vertical, the image of the intersection of the collimator's wires will remairf stationary, but if not, the image will move in the circumference of a circle ; because, the plane of floatati(>n 268 SPHERICAL ASTRONOMY. remaining the same, tlie emergent rays from the collimator will preserve their inclination to the horizon unchanged, thus causing the line through the optical centre of the transit's lens, and parallel to these rays, to de- scribe a conical surface. The axis of this cone, which is a vertical line, is the position for the line of coUimation. Supposing, then, the image to have changed its position during the semi-rotation of the float, renew the contact of the image and wires ; one half by the adjusting screws of the collimator, and the other half by a motion of the transit and the adjusting screws of the diaphragm of its wires. This process being re- peated once or twice, the adjustment is iijade. 3. — The zenith or horizontal points. — Direct the telescope of anv circle to the collimator, and bring the image of the intersection of the cross wires in the collimator to the line of collimation ; read the circle, and revolve the float through an azimuth. of 180° ; renew the contact of the image line of collimation by moving the circle, if necessary, and read, again; denote the first reading by a, the second by a', and that of the zenith point by s, and we have g=180°-f^; and denoting the reading of the horizontal point by A, h= -^± 90°. The Collimating Eye-piece. 4. — If now the swing-frame and its reflector be transferred from the collimating telescope to the eye-piece of the telescope of the instrument sup- posed to be vertical over a basin of mercury, this latter telescope becomes its own vertical collimator by reflection, on applying the lamp to the swing reflector. By perforating the' swing reflector, and applying the eye behind it, two sets of wires will be seen in the solar focus of the telescope, and the collimating process consists in making the wires of one of these sets coincident with those of the other, by the joint motion of the telescope and its reticle. The little swing reflector, with a single microscope as an eye- piece, just behind its perforation, to magnify the wires and their images, constitutes the collimating eye-piece. This beautiful little instrument, which has done so much to facilitate the process of collimating and the measure- ment of zenith or nadir distances, is due to Professor Bohnenberger of Tubingen. APPENDIX II. 269 The Mural Circle. 1 . — By means of the transit and a time-keeper, distances are meas- ured on the equinoctial in titne ; and by an easy reduction tliis time is converted into arc. The object of the Mural Circle is to measure dis- tances on the meridian. This instrument consists of a metallic circle A A, varying in diameter from four to eight feet, strongly framed together or cast in one entire piece, and a telescope, of considerable optical power, having a focal length about equal to the diameter of the circle. The circle is firmly attached Fig. 36. to the larger end of a hollow conical-shaped axis at right angles to its plane, which axis is mounted on Ys, placed in an opening through a heavy wall, whose front face is in the plane of the meridian. The gradu- ation is usually, though not always, upon the outer rim, and the readings are made by a pointer and six ,or more reading microscopes F, mounted upon the face of the wall, at equal distances from each other, around the circle. The't61escope is mounted upon the front face -of the circle, so as 270 SPHERICAL ASTRONOMY. to move parallel to the plane of the latter by means of a second axis, ■which turns freely and concentrically within ^hat of the circle. The axis of the telescope is also conical, and is kept in place and proper contact "vvith that of the circle, by means of a strong nut, which receives a screw cut upon its smaller end, the head of the nut bringing up against the end of the circle's axis. By turning this screw in the direction of its thread, the two axes are brought as closely in contact as may be found desirable. Permanently connected with each end of the telescope is a clamping , arrangement, for the purpose of seizing the rim of the circle, and when these are in bearing, the telescope can only move with the circle, and when loose, it may move independently, thus affording the means of meas uring the same angular distance on different parts of the circle. Five clamping and tangent screw arrangements are permanently at- tached to the face of the wall, for the purpose of restricting the motion <>f the circle t© the minute adjustments necessary to complete the contact (if the objects observed with the reticle of the telescope, and to secure the instrument till the readings are made andTecorded. They are made thus numerous, that one may always be at hand, in the various positions of the observer about the circle ; one of them is shown at E. The proportions of the whole instrument are so adjusted as to throw its centre of gravity on the axis just behind the circle, and between it and the wall, where the axis is received by a stirrup with friction-rollers C C, the stirrup being connected by rods D D with levers and counter- poising weights, which take the bearing from the Ys. The front Y, or that nearest the circle, is movable in azimuth about a vertical pintle, and that at the smaller end admits of both a vertical and hoiizontal motion, by means of two sets of antagonist screws. The tube of the telescope is perforated on the .side opposite that of the axis to admit the light from a lamp at a short distance in front of the circle ; this light is received upon a perforated reflector within, after the manner of the transit, and thrown to the eye to illuminate the field of view in nocturnal observations. The intensity of the illumination is reg- ulated by square perforations in two sliding plates, placed over the aper- ture in the tube, and so connected with rack and pinion work as to move in opposite directions, on turning a large milled-headed screw near the eye-glass ; one of the diagonals of each square being placed in the direc- tion of the motion of the plates, the figure of the opening will be un- changed, while its size may be varied at pleasure. At P and P are two small tu^es, permanently fixed to that of the telescope, and at riglit angles to its length. They are cut away on ona APPENDIX II. 271 side at the' middle, and each is closed at one end by a small disk of molher-of-pearl, movable about an axis perperidiculai- to its plane, and concentric with the tube. Between the disk and middle of the tube is a convex lens, which admits of a motion in the direction of the tube, and by which an image of a small eccentric perforation in the disk is formed about the ,middle xif the cut, and of course on one side of the axjs. A motion of the pearl causes this image to describe the circumference of a circle, of which the centre is on the axis of the tube. In the opposite end of the tube is a small microscope to view this image. The image is technically called the ghost, being a visible but unsubstantial representa- tion of the perforation. A small metallic style projects from the face of the wall at S, from the end of which may be suspended a plumb-line of fine silver wire, with its hob immersed in a vessel of water or other liquid at the bottom of the wall. The style is so arranged by an adjusting screw as to bring the plumb-line to intersect the axes of' the small tubes in the cuts, or to throw it clear of the instrument, at pleasure. In the tail end of the telescope, and at the solar focus of the object- glass, is a reticle, of which the axial wires are parallel to the axis of the circle. An additional wire is driven by a micrometer screw in the direc- tion, peipendicular to the axial wires, while it is also kept constantly par- allel to them. The telescope has a collimating eye-piece, which is used for the same puipose and in the same manner as in the transit. Adjustments. 2. — The adjustments are, first, to make the line of collimation per- pendicular to the axis, and, second, to make the axis perpendicular to the meridian. The plane of the circle and tube of the telescope are placed at right angles to the axis by the manufacturer ; the face of the wall is built as nearly in the meridian as possible by the aid of meridian marks ; and the Ys are so placed as to bring the axis, when mounted, nearly per- pendicular to the face, so that the adjustments are approximately made when the instrument is put up. To complete them, begin with 3'. — The line of collimation. — Turn the circle till the telescope is vertical, suspend the plumb-line and bring it by its adjusting screw to co- inci ie with the upper ghost as seen thyough the microscope : examine the position of the lower ghost ; if it be not on the line, turn the pearl about its axis till it is : clear the line from the instrument, and invert the telescope by revolving the circle through 180°; bring the line tc the 272 SPHERICAL ASTRONOMY. ■upper ghost as before, and again examine the lower ghost ; if it be on the line, the axis of the circle is horizontal, but if not, bring it to the line, one-half by the vertical adjusting screws of the circle's axis and half, by a revolution of the pearl. When by repeating this process once or twice the axis is made horizontal, put on the collimating eye-piece, and directing the telescope to the trough of mercury at the foot of the pier, and immediately below, move the diaphragm of the cross wires till the wire, which is perpendicular to the axis, coincides with its image — the line of collimation will be in a vertical plane, and of course perpendicular to the axis, which is horizontal. Should the telescope have no collimating eye-piece, recourse may be had to the vertical collimator, which is to be used exactly as in the transit. Since reflection takes place in a plane normal to the reflecting surface, the axis may be made horizontal by observing the' same star directly, and by reflection from the free surface of mercury. If the time of the star's appealing on the line of colHmatio'n in both views be the same, the two positions of the line of colhmation will lie in the same vertical plane, and being equally inclined to the horizon, the axis with which they make a constant angle must be horizontal. 4. — Axis perpendicular to the meridian. — This adjustment may be made by the method pointed out for the same adjustment in the transit ; and when not perfected, the amount of error may be found by the process explained for that instrument. » Polar and horizontal points. — On the circumference of the circle is a scale of equal parts, each part having an angular value of five minutes. Every twelfth division is numbered, the numbers varying from 1 to 360° inclusive ; these indicate the degrees of the scale; and to facil- itate the reading, the intermediate divisions are also numbered, but in smaller characters. If the reading be known when the line of coUimation is either hori- zontal or directed to the pole ^f the heavens, and the reading be taken when directed upon the centre of any body as it passes the meridian, the dificrence of the readings will in the first case be the observed meridian altitude of the body, and in the second its observed polar distance. 5. — Tlie horizontal point. — This is found by means of the cdhma- ting eye-piece, or vertical collimator, by the process indicated at page 268, or as follows, viz. : having carefully ascertained the value of a revolution of tiie micrometer in the eye-pifece of the telescope, and the reading of its divided head when the movable wire is coincident with that parallel to he axis, set the telescope nearly in the position at which a star would APPENDIX II. 273 appear by reflection on the stationary wire ; clamp the circle and record the reading of the index and microscopes ; when the star is at a conve- nient distance from the meridian wire, bisect it by the movable wire with- out moving the circle, and note the time accurately. Uuclamp the circle, and biing the star by direct view accui;ately on the stationary wire, by turning the whole circle about its axis ; again note the time, and record the reading by the index and microscopes. Denote by JR the first lead- ing, by D the second, and by m the angular value of the distance between the fixed and movable wire, as indicated by the micrometer ; then, if tlie star had been observed accurately on the meridian, would the reading of the horizontal point be 2 ' since the star must appear as far below the horizon by reflection as it actually is above it. But as the star cannot be taken at the same instant in both positions of the instrument, the readings S and D, taken as above indicated, must be reduced to what they would have been if taken on the meridian. 6. — This correction will now be explained. Let S' S S" be the small diurnal circle of the star ; PMS' an arc of the meridian ; S the position of the star when observed on the intersection of the axial and one of the side normal wires ; MSV the arc of a great circle, of which the axial wire is a portion. The point M will be that to which the line of collimation is actually directed, and S' is that in which the star will reach the meridian ; the are MS' is, therefore, the reduction to the'nieridian. Make P = MP S = hour angle of star ; d =1 P S = polar distance of star ; y zzz P M = polar distance of line of collimation ; a; = MS' = reduction to meridian. Then in the triangle MP S, right-angled at M, sin y cos d Fig. 2T. P =z tan y .cot d = and subtracting this from cos y ' sin d ' 1 = 1, 18 274 SPHEEICAL ASTRONOMY. we have, after reducing, and replacing 1— cos P by 2 sin' ^P, 2sin'ii>= ^"('^-t - " cos 3/ . sin a The observation being made very near the meridian, P and ^= 2 ' and the true reading in the second position becomes a — a'— 180° ' «'+. Again, denote by y the small angle which the line of collimation mates with the plane passing through the axis of the vertical circle and that zero of this latter circle nearest the hne of collimation, and suppose the line of collimation to lie above this plane when the telescope is directed to the same object, as before. Let b denote the apparent altitude, supposing the circle in the position to mark altitudes ; the true altitude is sensibly equal to 6 + y; turn the instrument in azimuth 180°, and bring the telescope again on the object ; the hne of collimation will now be below the plane of the axis and zero, but the circle now indicates a zenith distance 6', whence the true zenith distance is b'+y, adding these measures together, ve have APPENDIX II. 2S1 6+ 6'+ 2y=;90° 90°-{b + b') y= ^ , and the true zenith distance becomes 90°-(6+ 6') 2 Whence to adjust the line of collimation we have this rule, viz. : Direct the telescope to some vpell-defined and distant object, not far from the horizon, and bring its image to the intersection of the middle wires ; record the reading of the azimuth and vertical circles ; turn the instru- ment in azimuth 180°, bring the line of collimation again on the object, and record the new readings of the circles ; subtract from the difference of the azimuthal readings 180°, divide by 2, and add (algebraically) the quotient to the last azimuthal reading for a new reading in azimuth. Add the two readings of the vertical circle together, subtract the sum from 90°, and add half the difference to the last reading for a new reading on the vertical circle. Set the circles to these new readings, clamp, and by the adjusting screws of the reticle bring the line of collimation to the object, and the adjustment is made ; it should be verified, however, by repetition. 7. — To make the normal wires perpendicular to the transit axis, proceed as in the case of the transit instrument, viz. : Move the diaphragm about in its own plane, till the image of some object appears to run accu- rately along some one of the wires, say the middle one, while the tele- scope is turned about its axis. 8. — The altitude and azimuth instrument is regarded by nlany as the most Universally useful of all astronomical instruments. It is portable and accurate. When used in the meridian, it may perform the work of the transit and mural circle, though with somewhat diminished accuracy. But its principal merit consists in the ease with which it may be moved in azimuth without impairing its measurement of altitudes and zenith dis- tances. 9. — The instrumental hearing of an object is the angle indicated ■ by the reading of the azimuth circle when the centre of the object is ap- parently on the line of collimation. From the instrumental, the true bearing, or true meridian, is found by a process to be explained hereafter. 10. — To find the altitude and instnimental bearing of an object at any instant, it is only necessary to make the object pass the line of colli- mation by turning both tangent^screws as it moves through the field of view, and to note the time of passage, ana read the circles. 282 SPHERICAL ASTKONOMT. 11. — The altitude and time, or the instrumental bearing and time, are the elements more commonly observed in the case of celestial objects. 12. — To obtain the altitude and time.- — With the circles undamped, direct the telescope, which it will be remembered inverts, so as to bring the image of the object in the lower or upper part of the field of view, as the body may be rising or setting ; clamp the circles, and by the tangent screw of the azimuth motion, bring the image to the middle normal wire, and keep it there tfll it passes all the axial wires, carefully noting the time of its passing each, and also notin'g the indications of the level be- fore it passes the first and after it passes the last one. Now read the vertical limb, unclamp, and, by an azimuthal motion, reverse the face of the vertical circle without unnecessary loss of time, and go through the same operation as before. Reduce the vertical readings to the same de- nomination of altitude or zenith distance, correct them by applying the level readings, and take half the sum for the altitude or zenith distance, us the case may be. Add the times together, and divide the sum by the number of recorded times for the corresponding time. 13.— To find the instrumental hearing and time, direct the "telescope as before, and clamp; with the tangent-screw of the vertical motion, bring the image of the object to the middle axial wire, and keep it there till it passes all the normal wires, on each of which record the time. The read- ing of the azimuth circle will give the instiiimental bearing, and a mean of all the times will give the corresponding time. All of this supposes that the object's change in altitude and azimuth is uniform ; and although this is not strictly true, it is nevertheless so nearly so for the short time its image is in the field of view, that the error will be inappreciable during the interval required for a single set of ob- servations. The Equatorial. 1. — The object of the equatorial or parallactic, as it is frequently called, is to support a telescope, generally of great size and optical power, in su jh manner as to give to the observer the means of directing it with ease to any part of the heavens, and to measure at once the apparent hoilr angle and polar distance of a heavenly body. In the principles of its con- struction, it is like the altitude and azimuth instrument, but differs from it in the position of its axes, which, instead of being vertical and horizontal, are, when in position, respectively perpendicular, and' parallel to the plane of the equinoctial. The first is called the polar, the second the declina- tion axis. It has two graduated circles, one securely attached to each APPENDIX II. 28i axis ; the plane of one, viz., tLat attached to the polar axis, is parallel, and the other perpendicular to the equinoctial. The firet is called the liour, and the second the dedinatioA circle. By a motion of the polar axis, to which the supports of the declination axis are attached, the declination circle may be made parallel to any assumed declination circle of the celes- tial sphere. The polar axis, always much loaded, is, in low latitudes, con- siderably inclined to the horizon, and the practical difficulty of supporting it has given rise to a variety in the form of the instrument. That repre- sented in the figure is the one now most generally used, and it is intro- duced here on that account. The principle is the same in all. The supporting-stand is shown at IT, H, H. It is made either of a strong frame of wood-work, or is cut from a solid block of stone. ^ is a 'plate of metal, firmly secured to the stand, the surface of contact being pai'allel to the axis of the heavens. Upon this plate the instrument is mounted. The polar axis is seen at /. It is of steel, and revolves in two cylindrical collars near the extremities, and the lower end, being rounded off and highly polished, rests upon a >steel plate attached to a bearing- ■ piece K. To the lower end of this axis is attached the hour-circle R, which is either graduated into hours, minutes, and seconds, or into degrees and the usual subdivisions, at the option of the person ordering the instrument. The verniers, or reading microscopes, and tangent-screw arrangement, are supported by pieces connected with the plate B. The declination axis revolves in a metallic tube M, which forms a part of the frame-work se- cured to the top end of the polar axis. To one end of the declination axis is attached the declination circle P, which is graduated so, as to I'ead polar distances or declinations — suppose the former, its micrometers and tangent- screw being mounted upon pieces- projecting from the extremity of the tube M, and to the other end, which projects slightly beyond the frame- work, is attached the telescope at a point nearer the eye-end than the middle. The excess of weight towards the object-end is, in the mounting by Mr. Henry Fitz, of New York, compensated by a counterpoise cylin- drical lever within the tube of the telescope, and so arranged in bearing %is to counteract all tendency in the tube to bend. Attached to the end of the declination axis, is a counterpoise weight 0, the office of which is to throw the centre of gravity of the entire movable part of the instrument in the polar axis near its upper end, where it is received by a pair of fric- tion-rollers. At C is a box containing a system of wheel-wort, feo connected with the polar axis as, by the aid of weights and a centrifugal governor, to give 284 SPHERICAL ASTRONOMY Fig. 29. it a uniform motion of rotation. The velocity of rotation is regulated bj a vertical motion of the axis of the governor, whose balls in their retro- cession and increasing velocity, force a pair of rubbing-surfaces against tlia interioi* of an inverted conical box : the moment of the friction thence arising equilibrates that of a descending weight, and the motion become" APPENDIX II. 285 nniform. By elevating the axis of the governor, the motion is aooeler- ated ; by depressing the axis, it is retarded, and thus the velocity of rota- tion may be made equal to that of the earth about its axis, in which case a star in the field of view will be kept there ly the instrument itself, the effect being the same, abating refraction, as though the earth were at rest. 2. — With a divided pbject-glass for the telescope, to be explained presently, or with the position micrometer, the equatorial is mostly used as a differential instrument, and particularly when the observer is pro- vided with a very full and accurate catalogue and rnap of the stars, which serve as points of reference. Whenever it is possible to bring a known object into the field of view with one that is not known, the j)lace of the latter is found by measuring its bearing and distance from the known object. 3. — To measure directly the hour angle and polar distance of an object with the equatorial, requires the parts of the instrument to be iu perfect adjustment among one another, and its polar axis to be parallel to the axis of the earth. For these adjustments and a full analysis of the equatorial. Analysis of the Equatorial. , The true instrumental position of an object is that indicated by an in- strument in perfect adjustment within itself. The apparent instrumental position is that actually indicated by an iijgtrument whether in adjustment or not. When the several parts of an instrument are adjusted with respect to each other, these two positions are the same. The instrumental hov,r angle of an object, is its angular distance from a vertical plane passing through the polar axis, estimated' upon the hour circle. Its instrumental declination is its angular distance from a plane.peipen- dicnlar to the polar axis, estimated upon the declination circle ; and its in- strumental polar distance, its angular distance from the polar axis. The line of collimation should be perpendicular to the declination axis, and the latter perpendicular to the polar axis. The index of the hour cir- cle should stand at the zero of the scale when the Une of collimation is parallel to the vertical plane of the polar a4^, and, supposing the instru- ment to read polar distances, the index of the declination circle should be at the zero of its scale, when the line of collimation is parallel to the polar 286 SPHERKJAL ASTEONOMY. Supposing none of the conditions to be fulfilled, the apparent instru- mental position of an object will differ from the true, and the first thing to be done is to find the latter from the former, when the error in each of the above particulars is known. To do this, we will premise that the equatorial may be regarded as an universal transit instrument, whose liorizon is the equinoctial, and zenith the pole. The formulae of reduc- tion applicable to the transit will apply at once to the equatorial by making therein the symbol for the latitude 90° ; in which case we sliall have for the difference between the true and apparent instrumental hour angle in arc, the sum of the last three terms of Eq. (IS), viz., _ c cos (X — ^ , sin (X — 5) _ cos ,cos cos which reduces, by making X = 90°, and replacing S by 90° - ■ *, to c . cosec * -f- Z . cot * + s ;. in which c is the error in the line of coUimation, I that of the declination axis, and a a constant correction to be applied to every reading of the hour circle arising'from the improper position of its index, and therefore the in- dex eiTor of the hour circle, and if the instrumental polar distance of an object whose image is on the line of coUimation. Denoting by tf' the true, and by rf the apparent instmmental hour angle, and writing ^ d for g, we have a' = v,sln(p^^-^ — ^'. The first member is the projeotipfl pf the aro X, on the declination circle at right angles to the meridiap, apd is, therefore, the deviation of the pole of the instrument fi:om this latter plane, This error being treated in a man- ner similar to the preceding, by means of adjusting screws which act at right angles to the meridian, the polar axis is brought to this latter plane, and the instrument will be so pearly in adjustment as to bring the eiTors within the limitations required to render equations (d) and (e) exact. The approximation may be continued, if desirable, or the value of each error found by recourse to celestial objects properly selected, and these errors employed as elements of reduction. To find e. — Observe an equatorial star about the time of its meridian passage, and again after reversing the declination circle ; the readings of Jhe hour circle in Eq (d) give s = ring, and making, after eliminating s' and s by their equals t' — a, t — a, 2 = («' — t) — (tf' — tf — 12'') — (n' — n), n = {p'-p)-{if,-if), we obtain X . cot * [sin (a" — 12'' — (p) — sin (tf — (p)] = Z > X . [cos (o" — 12" - ip) - cos (a- - ip)] = n i " " ' ' but ' • — ), (I A _ loh \ 292 SPHERICA.L ASTEONOM-?. substituting these above, and dividing the second equation by the first, we have, using p for *, tanJ-^ ip\ = — cotp . . . . {i) whence o-' + tf— 12'" ^ , n 9 = tan'.— cotp_. . . . (k) and from equations (h) we have . in I- 2 • smi(ir— ir— lafc). Biul $1 smi(i7— o— 12"). oosi ■ ^1. -w cot^ To find ^ d. — ObseiTe a star before its culmination in the hour angle 360° — tf, and at an interval after its culmination in the hour angle tf', such, that 360° — rf and fl" shall be equal, or very nearly so, without re- versing the declination circle ; Eq. (d) will then give ^ 24'' — s = 24'' — tf + A rf + X cot * sin (860° — rf + ip) — «, s' = tf' + A tf + X cot ir . sin (rf' — (p) + ». Adding and reducing, «' — «=:fl"— i)' + 2A(r + Xcotir. [sin {i' — ) If the law which connects the varying density I) with the height x b« given, one of these variables may be eliminated and the integration per- formed. But in a practical point of view this is not necessary*; for X is known to be a very small fraction, as is also the greatest value of ar, the latter not exceeding 0.01931, being the height of the first stratum of air that has sensible action upon light, divided by the radius of the earth,or W miles divided by 4000 miles. Developing the factors e~ ^' ~ ■'and e~ ' \ neglecting the second and higher powers of X and x, and also the teim of which X sin' Z ia a factor, which may be done without sensible error when Z does not exceed 80°, wo find 308 SPHEEICAL ASTEOITOMT. X . sin ZdD X sin Z . dD dr ■■ ■Vl + %x — sai^ Z Vcos^ Z + 2x^ whence X. tan Z .dD , „ ,^ , ^, , dr = . = X tan Z . (1 — a; sec" Z) dj) ; Vl + ^xseo" Z r = X tan zf(dD — sec' ZxdB) ; imd performing the integration, that of the last term by parts, r = X tan Z [Z* - sec' Z {Dx —fl> .dx)\; but if A denote the height of the mercurial column at any stratum of air above the observer, J5^, the density of the mercury, and g the force of gravity regarded as constant, then will g .f I)dx = gD„h\ and r = X tan Z [Z> - sec' Z {Dx - I),,h) + C] ; and from the limit a; = 0, where 2> = i)' and h = h^, to the limit x = height of4he entire atmosphere, where I> = 0, r = 0, and A = 0, we find r = \ta.aZ .D' (l-h.^' sec' z\. Taking the density of Mercury as unity, we have the mean valud of ■'^ — To 4T5' The mean value of h is found from the proportion, milM iiichei 4000 : 29.6 : : 1 : A: which will give for the coefiScient of sec' Z, h.^ = 0.0012517. J) Also, if D, be the density of air when the thermometer is 50, and the ba- rometer 30 inches; and we take a = 0.00208, and j3 = 0.0001001, the coefficients of expansion for air and mercury respectively, then. Analytical Mechanics, § 245, ' ■ 30 ■ 1 + (< — 50) . a ■ in which t denotes the actual temperature of the air and mercury supposed the same, and A the height of the barometer. Hence APPENDIX III. 309 Had the second power of x been retained in Eq. (7), then would the last term df which, within the limits supposed, is insignificant. Make . u = — . ^ti!°~.^!^ . tan 2 . (1 - 0.0012517 se6= Z) . . (9) 30 1 + (< — 50) a ^ ' ^ ' and we have r = 'KJ},u . (lb) Denote by s and s' the greatest and least observed zenith distances of a circumpolar star, r and r' the coiresponding refractions, and c the zenith distance of the pole ; then will. g + '* + s' + »■' c = ^ . In like manner, if e^ and 2/ be the greatest and least zenith distancbs of another circumpolar star, r, and r/ the corresponding re&actions, _ e, + r,+ g/ + r/ fe[uating these values, replacing the refractions by the values given in Eq, (10), we find •» KJJ = -. jr. ' u + u — (u, — u/) The indications of the barometer and thermometer being substituted in Eq. (9), give H, m', u^, and u/, and therefore the value of XD^. Numerous and careful observations make \D^=. Si ".82, which substituted in equa- tions (8) and (8)', give the refraction for every observed zenith distance, temperature of the air, and height of the barometer. 310 SPHERICAL ASTKOITOMT. APPENDIX IV. SHAPE AND DIMENSIONS OF THE EAKTH. Let A MP I A' represent a meridional section of the terrestrial ellipsoid, Jl/'th.e place of the spectator, B the north pole (6) APPENIJiX IX. 321 In which c'otaii^i = cos a, . taii ^, ....... (V) Also, if I denote the geocentric colatitude of the place of observation, p the corresponding ra;dius of the earth, and 2*^ the sidereal time of observation reduced to degrees, then will 5^, = p . sin Z . sin yA (8) h = p . cos I . ) mi sun^s horizontal parallax at the place of observation ' number of seconds in an arc equal in length to radius ' ' ^ ' Multiplying- the first of' Eqs. (5) by tan a, and subtracting the product from the second, then by tan tf, and subtraqtingthe product from the third, and reducing by the relations of Eqs, (6), we have y,-ai,tan«, = (X.-/.)tan<.,-r.+i7. + vp,[^ + ^-tan«,g,+ -^)] in like manner ,-a- j.>. ^ , ,- rdVt , dTi , (dx, , dX.A-\ y -il!3tanaa=(Xa— /a)tanad— ra+Va + v()8|^-^ + -^J-— tlVUa2(^-^+-^jJ 8. - - ici tan 9j= (Xj — /j) tan fli — ^ + A + V pa[-^ + •^'' — tan flj (^-^^ + -^^ j J and -ii'^-dr-^^'"K-di^-dr)\ in which, as in equations (3) and (4), •pj.= (gj + Zj)sec^j) ,j^, P3 = (2, + ^,) sec iS, j ^ ' or, if the body be near the equator, Pj = (3^;+ Xs) sec ttj . cosec ^j ) .^ 2> Pj =: {x^-y X)) sec a, . cosec ^3 ' ' * Now make «,'= t, — f, ^a t3 = ti + r'; 21 (10) 322 SPHERICAL ASTKONOMT. then, because s, and Xi are functions of t, whicli become Xt when r and r' become zero, we have by Taylor's formula, dxt Xi — X} — -j— , T -)- d'aTj ■■(is; 73 in equations . . . . (19) , coseo ^1 . coseo ^j (20). Vfo obtain dvs, dx, . , yi — X, tan a, — -^ t + tan aj -^ t = ^, + o, s, — »s tan^i ^ -:^- r+ tasB, -y- t = Bi+ 6i at'- o*> y, — Xj tan a, = ^, + Oj Zi — aJj tan ^j = 5, -f* ^i ji — jii,tan«, .+ —ft' — taiLft', rrr r = Jj ;+ Oi ;,i; Of at' ' d Zt I dx» .. _. • \ z, — ar, tan «, + -jj r' — tan 0, ^ r? = .B, + 6, (21) AFPENDIXL IX ^ 325 K^rding »„ y„ gj, — , — , and — as unknown quantities, we ob- tain by elimination, in which, by making rt _ y 1 ^ T^y sin.(qa— ,iiC|^)jeos'a3 : \ T / ' sin .(«! — aj) cos Wj S=(l+ -\ - s'^ (^g — '^f) "OS '^s • \ T / ■ sin (4, — ^s) cos ^2 , m m we have p_ ■ cos a, cos aj f i< '' , j j A , '^\l i (M — S) sm (ui — a,) L T \ «" /J cos (i(, cos a, r t' / t'\t ^= (j;- ^) sin («,-«,) •r7+''' - "r(^ + 7JJ cos ^, c os ^a |- T' . /, , t'Xt ^ = (ig-.y)sin(^.-^3) • l^'T + *'-*« (i + 7) J K24) or by making we have and making we have jp _ /'OS "■ cos gj «'_7)'^' (^ - ;S) sin (a, - aj) ' ^--^7' cos ^1 cos ^, a—w"' ~(B-S) sin (6, -6,)' ^-^r' p = D {03 — as) + E{ay — a^ (26) 326 SPHERICAL ASTRONOMY. j/j = aj tan ai + Ai + a^ gj = x, tan 6i + Bi + b. dx, 1 J = l(i?-5)(P+i,)-p-(l+i2) (26) dv, dxo 1 , , , J , X -fi = -ji tan Ui (Xt tan a, + ^, + o, — y,) at at r ■ -Ji - -^ tan 6i - - (x, tan «, + ^, + 6, - g,) at at 7 or instead of the last two, ' ' dy, dx, . . 1 /• . , J . X •^ = ^ tan a, + p (a:s tan a, + ^3 + o, — y,) -rr = -r; tan «3 + — (a:, tan 6, + £, + b, — z,) dt dt r Now although the Eqs. (26) express the values of the co-ordmat«8 and components of the velocity of the body at the time of the second observa- tion, they involve the geocentric distances p,, pj, pj, and the radius vector rj, which are unknown, and the solution of the problem can only be ac- complished by successive approximations. J'irst Approximation. Let us first neglect the terms involving aberration, and those containing as factors powers of r and r' higher than the second. This will give, Eqs. (1*?), Z7= and Eqs. (18), 2r7 Tr=0; ^' = &' TF-O; V.'f 6, — — -^ (gj — Xi tan «,) ir. 60 = a, = (27) APPENDIX IX. 327 And as all terms involving powers of t and r' higher than the second ar« to be neglected we obtain from Eqs. (21), for the values of the severa. &ctors above, y, — Xs tan a, = A, ' gj — Xa tan Si = £i yj — Xi tan aj = A^ gj — X, tan 6^ = Bt yt — «s tan a^ = A, Zj — aJj tan 6} == £^ , which substituted in Eqs. (27) give (28) Aiii.i' a, = J- ; ' 2r,' ' a, = 0; A,ii.t'^ 6,= ^,(JH-' 03 = 2ra' ' 6, = 0; 2»-s' ' b,= 2rj' (29) and these in the first of Eqs. (26) give u,t' / r" t'2 ' \ 1 and making JT = !^ (i) ^ -5 + ^^. - i?'^, ;^ - 6'^.) . . (30) we have x,= C + (31) and this value of x„ and the foregoing values of a^ and 6j in the second and third of Eqs. (26), give _ , N tan a, . yi= C tan aj + ^3 H 5 — ; 2, = (7 tan ^s + ^2 H , — - ; or making C^' = C tan ttj + A ; JV = iVtanaj) C"= C7tan«, + ^,; i7" = iV^ tan «a J " * ' '^ ^ 32S SPHERICA.L ASTRONOMY. we have _ y, = C" + -j ........ {83) ^.= ^" + ^ (34) and hence This equation must be solved tentatively. If the body be considerably more distant than the earth from the sun, (7, C", C" are respectively ap- proximations to the values of x^, y^, % ; and if considerably less distant, N- N' N" then — , — 3 , and — j- are approximations to the same quantitie3. It is evident that a value for rj must be selected whioh will make the greatest N N' N" values of C -1 — 5, C" -| 5, and C" H 5-, without regard to signs less than r. and greater than — I; . And if C -\ — ; , for instance, be the ■y/s «^ greatest, the value of r^ which satisfies the equation will diflfer from the true value by a quantity less than -Ml — j , that is, less than 0.31 r^. But the solution of Eq. (35) may be greatly -facilitated by means of an elegant geometrical construction, due to J. J. Waterston, Esq. Thijs : Squaring the terms as indicated in the second member, recalling the re- lation in Eq..(7), which will give 1 + tan' ttj 4" tan^ 6^ = tan' 6^ sec' ^j ; substituting the values of C", C", JV, N" in Eqs,,(32), ^nd makjng r = N . tan {i sec ^s ; £" = -iij tan ttj . cot ^2 cos ^j + B^ cos ^, ; /3 = ^ + C tan ^j sec ySj ; equation (35) may be written , ' r,' = a',+ (^+^y (36) APPtEKDIX IX. or Make then will and Eq. (37), 5"— (^i-$)' («) - = cosec^, a -\ — 1=00^ S, and -; = sin'i cot6 = — I- -7 . sin' I a a »nd piajang CQt 6 — y, aud siu' ^ = a:, |8 r y=^+;?* m Also 1 + cot' 6 = coseo' 4 = ■ sinVd' or 1 whence , i+y'=^; «= 3 (39) (i + s^)^ If the curve of which this is the equation be described graphically, and the straight li^e, of which (38) is the equation, be drawn, the abscissa of their a' point of intersection will give the value of sin' d, or — ; and r^ becomes known. Its value may be verified by substitution in Eq. (35). This value of rj in Eq. (31) gives x^, and this in, the .second.and third ot Eqs. (26) gives yj and gj; also, r^ in Eqs. (29) gives Oi, 6i, O3, 63, ,and these in the third of (25) will give p, which with o, and 61 iurfourth, fifth, and sixth, or fourth, ^yenth, and eighth of (26), give -rr, -77 > and -j^, ^(b t at (It and the values off,, pj, ps in Eqs. (1 9) or (20). Second JLpproicimation. By differentiating the equation ■r^^^ + .»s' + «=S mi diwdipg by .ra.dt. we liave dr, dt ' X, dXi ~rt'dt "^ ^2 dyt Zi r^' dt r,' dt, ' dt ' ' ' ' ' (40, 330 SPHERICAL ASTRONOMY. The first term of this equation becoming thus known, the values of U, W, XT, and W, Eqs. (lY), may be computed to include the third powers of (Aaj — Aa,) + J" (Affli — Aa,) ) A g = J?" (A 6j - A 6s) + ff (A 6, - A 6,) j am ben from Eqs. (26), A a!j = Ap — Ag A y, = A a:, tan aj + A Oj A gj = A a;, tan 63 46' .334 SPHERICAL ASTRONOMY. For the doubtful case9> an eclipse will result when B<^{P + x-c) + s+ 16" in -vrliich F, s denote the equatorial horizontal parallax and semi-diameter of the moon, and t, e those of the sun. 2. At the time of new moon an eclipse of the sun will be ««''"'". iwhen^ <1°23'15' impossible > ( > 1 S4 52 and doubtful between these limits. For the doubtful cases, an eclipse will happen when /J <(P — t)+ff+! + 25' PARALLAX. If a straight line be drawn from the centre of the earth to any assumed place, it will be the radius of the earth for that place, and this radius we shall designate by the letter p. This radius p, produced upward towards the heavens, will determine what we shall call the central zenith, being that point which spherically deter- mines our true position in relation to the centre of the earth. The apparent ze- nith, however, is naturally determined by a line which is vertical to the observer, and therefore a normal to the spheroidal surface of the earth. The small angular deviation of this normal from the radius of the earth, or the angular distance be- tween the central and apparent zeniths, is what astronomers call " the angle of the vertical ;" and, the earth being an oblate spheroid, it is evident that the cen- tral zenith will be nearer to the equator than the apparent, and also that the hor- izontal parallax will always ,be less than that at the equator, in consequence of the diminution of the earth's radius in proceeding towards the poles. The effect of parallax on the position of a body above the horizon is to augment its zenith dis- tance, and for this we have the well-known relation, "sin par. in zen. dist, = sin hor. par. X sin app. zen. dist." This relation will hold strictly for the spheroidal figure of the earth, provided we adopt the central zenith, and that horizontal parallax which appertains to the ra- dius p of the place of observation. Consider the equatorial semi-diameter of the earth as unity, and let y denote the polar semi-diameter, which, adopting the mean between La Lande and Delam- 304 bre, will be . Let also / be the latitude of the central zenith, or what is usu- 305 ally called the " geocentric latitude," and I' that of the apparent zenith, which may be termed the spheroidal or geographical latitude. Then the co-ordinates of this place, referred, in the plane of its meridian, to the polar axis, will be z = p sin /, y = P cos I. By the generating ellipse 7 + 3'' = !. and therefore for the angle I', which the normal makes with y or the tangent with f, we have „ dy 1 X tan 2 tan 2' =- -J- = -f . - = — — , ax y' y y" .-. tan /=y taal> . . . . . (1) APPENDIX XI. 335 Again, the values of x and y, Bubstituted in the above equation of the ellipse, give , /sin' I , . , \ '>'(-pr-4- r, a total contact first commences with A ' = s — it ; when s < IT, a total eclipse first begins on the earth, when A=P' + s-a. If s < r, an annular eclipse first begins on the earth, when A=P' -s + T. , (3) When contact of centres first takes place on the earth, A' = and A=i". For the time of true conjunction in right ascension, assume P, the true declination of the moon ; a, the true difference of right ascension, or 5 's right ascension minus ©'s riglit ascension, in space ; Di, the relative motion in declination, or the motion of the moon in declina- tion, minus that of the sun, at that time ; a„ the relative motion in right ascension at the same time ; I, the inclination of the relative orbit S, with a parallel of declination through the point C, or the angle OSn; u, the angle under the distance and the line of nearest approach, or the angl« MS n. This angle is always measured on the northern side of the dis- tance, so that when R falls below /S, or when diflf. dec. C Sis negative, it will exceed 90°. ' Then the relations of the figure will give these equations : tan I = -; re ^ (diff. dec.) cos i (1) ■Ji cos i) Hourly motion in the orbit = -; — , ■^ sin I arc nC = n tan i. For the time of desci'ibing the arc n C, or the interval between the middle of the general eclipse and the time of conjunction, it must be divided by the hourly motion in the orbit. Therefore, t denoting this interval, (re sin I \ , -^)tan.. Assume resini r« .,.„„n " sin 1 "1 ,= 3600" X--pp = [3.65630]-^ 1 ^^^ and ' I t in seconds = e tan i J The sign of t will be determined by combining the signs of diiE dec and Di ; and then ' time of middle = time of Mm -8m, or A P' — A', that is, A must be between the values J" — A' and /"+ A' : this leads to two spe- cies of curves. 1. When the nearest approach is greater than P' —r A'.' Here the formation of the triangles SmM, Sm' M, will always be possible du- ring the appearance of the phase on the earth. At the first appearance and final departure of the phase, 8 M= Mm -{■ Sm, the triangle SmM will be simply the line SM, and only one place Z will result. By taking positions of 31 on both sides of the middle point n, it will also appear that the relative positions of the places ZZ' become inverted, and that the curves described by them must intersect each other at some intermediate place. Hence it appears that the curve of risings and settings commences with a single point, which immediately after divides itself into two points moving in opposite directions on the earth, and which describe two curves intersecting each other, and finally meeting again in a single point, the whole forming one continued curve, returning into itself, and assuming the figure of an 8 much distorted. At the place where they intersect, the phase will begin at sun rise and end at sunset, or it will begin at sunset and end at sunrise. 2. When the nearest approach is less than P' — A '. In this case the triangles SmM, Sm' M, will resolve into the line 5 J/ when A = P' + A ' and also when A = -P' — A ', each of which positions will give only one place Z. Thus it appears thdt the points Z will form two distinct, oval, and isolated curves, the former curve being generated between the decreasing values A =P'-f- a' and A =P' — A', and the latter between the increasing values A =P' — A 'and A =P' + A'. Tlie leading point of the first oval and the terminating point of the second oval are the places where the phase begins and ends on the earth. The terminating point of the first oval and the leading point of the second oval are simply determined by using a =P' — A', and computing the same as for the beginning and ending of a phase on the earth. •Let us now turn our attention to the determination of the two places ZZ', at anytime, or for any position of M. Join ZS and draw Md perpendicular tti We shall, throughout our investigation, usually denote Sdhj (x), dMhy (y), and the Z dSM\>j S, this angle being estimated from 5iV towards the'east. To determine these quantities, let the declination of the point d=:{D), which will a little exceed that of M, and which is distinguished from it by being placed within a parenthesis ; then, supposing JVJf to be joined, the right-angled spherical triangle Nd Mwill give tan (D) = . As a is always small, the difference of the declinations (D) — i> = tan-' -D may be arranged in a small table ^ ' cos a as annexed. 342 SPHERICAL ASTEONOMY. Difference bet^reen {B} andi>, or o corr. Arguments : D and a . D 1 10 / 20 1 3o 1 4o 5o / 6o 70 8o 90 100 o » 11 „ // // n It 11 II // O o o D o o o I O o I I I I 2 2 I I I 2 2 3 3 o o I I 2 2 3 4 5 4 o I 2 2 3 4 5 6 5 o o I 2 3 4 5 6 8 6 o o I 2 3 4 6 7 9 7 o 2 3 4 5 7 9 II 8 o o 2 3 4 6 8 10 ij • 9 2 3 5 7 9 11 i3 lO 2 4 5 7 lo 12 i5 lit 3 4 6 8 lO i3 16 12 o 2 3 4 6 9 II i4 ,'8 i3 2 3 5 7 9 12 i5 '9 i4 o 2 3 5 7 10 i3 17 20 i5 2 3 5 8 II i4 18 22 i6 o 2 4 6 8 II i5 '9 23 17 2 4 6 9 12 i6 20 =4 i8 2 4 6 9 i3 i6 21 26 '9 o 2 4 7 lO i3 17 22 =7 20 o 3 4 7 lO i4 18 23 28 21 3 5 7 TI i4 '9 24 29 22 3 5 8 II i5 ■9 25 3o 23 3 5 8 II i5 20 25 3i 24 u 3 5 8 12 i6 21 26 32 25 3 5 8 12 i6 21 27 33 26 3 6 9 12 17 22 28 34 27 3 6 9 l3 17 23 29 35 28 o 3 6 9 l3 i8 23 29 36 29 3 6 9 i3 i8 24. 3o 37 The number of seeondB given by thjs table, which we hav^ denoted by the term a corr, is to be applied so as to increase D, whether it be north or south. The value of (D) being found by so correcting D with this table, we shall evi- dently have {x) = {D)-i, (y) = acos(i)) "(»)' \- (A) sin 8 cos 8' the quadrant in which 8 is to be taken being determined by (as) and (y) as co- ordinates. tanS: APPENDIX XI. 343 We shall afterwards have frequent occasion to use these quantities. If t denote the time from the middle of the general eclipse, they may be deter- mined more easily, though less accurately, by means of the following formulae, vhich may readily be inferred from -what has preceded. tan w = t (x) ^ A cos S, ') (B) cos 0) (y) = A sin S, _ the upper sign being for the time t before the middle, and the under sign for the same time after the middle. Denote the / mMS by m. In the triangle mMS, which may, on account of its smallneBs, be considered as a plane one, we also have Mm=:P', Sm.= a', andSM=A. Assume P'— A' P'+ A- and then if-') (>-f) P' . A (1^ As ZS,Zm maybe considered as quadrantal arcs, they will be parallel at the extremities S, m ; and thus the /_ Z S M= /, m MS = m. Therefore the / MSZ:= S ± m ; and the sun being supposed in the horizon, the spherical tri- angle NSZ will have ZS=: 90°, and lience the places Z, Z', will depend on the following formulse, in which Z is called the place advancing, and Z' the place fol- lowing. Place following, sin / = cos (S — m) cos S, tan A = - Place advancing, sin I = cos {8 + m) cos i, tan A = - tan {S — m) sin i ' tan {S + m) sin I ' (2) In these expressions the symbol S represents the declination of the sun at the time for which we calculate ; but for common purposes the value of i at the time of conjunction may be used in all cases. IIL NoaiHERN AND SoCTUEEN LrMTTS TOE ANY PhASE. The determination of the extreme latitudinal limits of a phase, or of the terres trial lines whereon that phase will appear as the middle of the local eclipse, is tlic most complex and unmanageable of all operations which relate to a general eclipse. For any given phase, at different places on the earth, the moon must be so rediieeJ by parallax as to touch a given concentric circle on the solar disk ; and if we con- sider this circle, by way of illustration, to represent, instead of the sun, the disk of the luminous body, the places on the earth which severally see the given phase must be situated in the surface of the penumbral or umbral cone, according as t)ie interfering limb of the moon only approaches or projects over the centre of the sun ; tjiat is, the places must all be found in the intersection of this cone with the surface of the earth. This intersection will assume a complete or partial oval SM SPHERICAL ASTRONOMY. Fig. T. lorm, according as the cone falls wholly or partially on the earth's illuminated disk. When it falls only partially on the earth, the extreme points will evidently see the sun in the horizon, and be therefore two points belonging to the horizon limits ; but in the other case the phase cannot at that instant be seen in the hori- zon. It is evident then, that these two cases have been already characterized in the discussion of the rising and setting limits. Let us now suppose the bodies to assume consecutive positions, answering to very small intervals of time, the earth also turning round its axis, and we shall have a series of these ovals. It is obvious that the extreme goographioal limits of the phase will be represented by curves which envelope all these ovals; — that at each instant the place of limit, by reason (/f the compound of the motions, will be proceeding relatively in the direction oi the tangent to the oval ; — that there will be two of these limits when the oval becomes entire during the eclipse, but only one when it is always partial. This is tlie most, popular and natural idea that can be furmed of the nature of these limits ; and we may here remark, as an inference fi'om what has been said, that if the rising and setting limits of any phase do not extend throughout the general partial eclipse, there will be both a northern and southern limit to that phase; but that, on the contrary, when the rising and setting limits continue throughout the eclipse, there will be only one of these limits to tiie phase, viz. : a southern limit when the difference of declination at conjunction is positive, and a northern one when that difference is negative. As before, let the system be referred to a sphere concentric with the earth, and let M be the place of the moon ; Z, Z\ the zeniths of the places which are respectively in the northern and southern limits; and m, m', the corres- ponding apparent places of the moon. Draw the meridians iV )w', N 8, N m, N Z, NZ',; also m r, m' r', and M k d h' perpendicular to NS; and assume ;S'd:=(a;),dJlf=(y),m A=:3-, hM^=:-y, Sr:=u, mr^=v, Sm.= a', Zm=Z, Z. N mZ^M, I. m NS=a', declination of f«=i)', and the latitude of Z=;. Then the /.mNZ=.h — o', w 1/= Z' sitt JI5, a = m J/" cos M=P' sin Z cos M and ^ = m .Jf sin If = P sin ZsmM; these by spherics resolve thus: * a; = P' sin Z cos M = P' [sin I cos D' — cos I sin D' cos (A — o') ] y = P' sin 2 sin M = P' cos I sin (A — a!) From these we deduce « = a! — (a;) — P'sinZoos JIf— (a;) = P' [sin I cos D' — cos Z sin i)' cos ( v = (y)—y = (j/) — P'waZua.M = (y) — P' cos I sin (A — a') L«t us now keep our attention to the same place Z on the earth, and suppose the system to he in motion as in nature. The hour angle A will increase ^t tbt I s(A_a')]-wl (1) APPENDIX XI. 34.5 rate of 16° per hour, and the latitude I will by hypothesis remain unchanged; m that the following equations will ensue : j^ = — F< sin l"-jr [sin I sin D' + cos I cos Z»' cos (A — a') ] + P' sin l'Yl5° — ^) cos I sin B' sin (A — „') — ^ = — P'sin l"'^'cosZ+P'sinl"Cl5° — ^')sini)'6in^sinjf— ^ at \ dtf at, dv d(y) _, . ,„/,_o da'\ -—-■=1-^ — P' sin 1" (15° — -=— I cos I cos (h — o') , at at \ all = ^ — P' sin 1" ^1 6° — ~\ (cos Z cos B' — sin Z sin D' cos J^T), Now, in order that m may be the apparent place of the moon at the middle of the eclipse, and consequently her nearest apparent contiguity with the sun, we must have -^— = 0; or since m''4-«'°= a", «-r- + «^- =:0, which is the condition of at at at limit Before we substitute the preceding values of 3— , -;— , it may be observed, to at dt avoid complexity, .that the quantities P' sin 1" -j — , P' sin 1" -j— may be neg- lected as being very small compared with P'. 15° sin 1", -~ and -5-^; also that i may be substituted for D', which will equally serve the purpose of both northern and southern limits. With these modifications we have ^ = P: 15° sin 1" sin S sin Z sin if — ^ 1 '^^ ^'^ \ (2) «_^iy) _ pi ;^go Bini"(cy8Zcosi — sinZsinicos Jf) I dt dt J . and, for the condition of limit, u Ip'. 15° sin 1" sin I sin Z sin Jf |-J + V \-^ — P'- 15° sin 1" (cos .?f cos 5 — sin Z sin i cos if ) J = 0. Instead of P' sin Z cos M put (x) -f- ?«. and for P' sin Z sin if put (y) — v, and it becomes « ri5° sin 1" (y) sin i — -^^1 + v [15° sin 1" (a;) sin S + ^] — P'vl5° sin 1" cos Z cos 6 = ; dix) ) „ ( rf(y) ^)(^)"°^-i5 ^rr^'J (-)"°^ + l^^' .'. COB Z ^ ^^^ — ^ r- V cos j 3i6 SPHERICAL ASTRONOMY. But, if oi denote the true relative motion in right ascension, and Di the true relative motion in declination, and D the declination of the moon, at the time of true conjunction, dt dt ■■ a, cos D ; cos ^ = V cos i Make now the following assumptions : ai cos D (^) = 15° sin 1" [0.58204] ai «03 D Xsin. = (^)-(-^)^™* X cos 1 (.0) (3) /" cos h _ (^) + (») sin i P'cosi in which (.4), (5) may be used as constant quantities throughout the eclipse, and we get cos Z-^ — ( — u sin V + « cos v\ V The angle r 8m is equal to the inclination of the apparent relative orbit with the parallel of declination ; denote it by i', and then « = A' cos •', jj =: a' sin i', and cos Z^:X sin (i' — v) (4) which is a concise form of the condition to be fulfilled by Z and i', in order that the place Z may be situated in the limit of a phase. Since the Z MSd = 8, and the Z MSm = 180° — (S + •'). / MSm' = S + i', we have for the triangle MS m Mni' = A" + A'" ± 2 A A' cos (8 + •'). Divide this by P"" and we get sin'X _ A'+ A" , 2 A A' F" pii C03(S + .') (6) for the geometrical relation between 8 and i', the upper sign applying to the northern, and the under sign to the southern limit. Add this to the square of the preceding equation (4), and there results A'?^=^±24#:cos(.+ 0+^^.^ = l sm" i' — P' for the determination of the angle J3I2 (6) The solution of this equation is by no means very practicable ; but as a small error in the value of Z will not sensibly affect the angle i', we may have recourse to the following indirect process, in which we first consider the- angle i' to be equal to i, which in most instances is very nearly so. The letter M designates the the angle Mmh, APPENDIX XI. 347 U ^ A 203 i V ^ /^ ' Bin I {D):=J) + {a. y = (t — a') COS (D) y a'= ± tan Jf = siu 2 =: • COS D' a') corr. x = {J)) — D' X y (7) X JP' cos M P' sin M the upper signs being for the northern, and the under signs for the southern limit. Or, if t be the time from the middle of the general eclipse, and u' the angle under Mm and the line of nearest approach, we shall have ifm sin u' = n tan u :=n — , and 3fm cos u' =r » ± a ', c which, observing that Mm = P' sin Z, give the following equations, wherein S and i''are constant for all the compntatious. S= F = n ± A tan ( «(« ± A') ( upper ) . - ( northern ) ,. ., I under [^"g" ^""^ ] southern [ 1™'*- F J M={—,)T^' ■t.E sia Z=- ■ (8) 1 "^^ { under [ ^'^n for the interval t j )^l^^ [ the middle. The sign of the constants E, F, are the same as that of « ± A ' ; and when this is negative, the angle u' will be in the second quadrant. ' The value of Z determined in this manner will be sufficiently approximate for the purposes of a general map ; and where greater minuteness is wanted, it will serve very well to get the angle i from the equation (4). For this we have , , cos Z cot 1 = cot V which may be resolved thus : sin ^ = y ; A sin V cos Z tan I = - tan 1 (9) 2 X cos V - - - ^^^ 2 ^ After i' is so found, which is only wanted roughly, the accuracy of the calculation may be tested by the equation (4) ; and then we may proceed to a correct compu- tation of MZ, by the equations (7), only using i' instead of i. "We shall thus have in the spherical triangle ZmN, ZM=Z, ^■m. = 90° — D', and the angle ZmN=- M; and I v spherics the following formulie : tan 6 = tan Z cos M tan (h — a<) = ■■| 'Vn,> tan -^; tan i = tan (9 + J)') cos (/* — «') ",os (9 + JJ ) sin e sin Z cos M 'cob{B + J)') check . . . cos (e + Z>') cos (A — a') cos / (10 For a map the equations (8) and (10) will alone be amply sufficient. In fact, where a very acciu-ate calculation is wanted, the most satisfactory method, will consist in &-st computing the places roughly ; then to reduce the horizontal paral- lax to the latitude by means of the radius p, from the table at page 337, and with 348 SPHERICAL ASTKONOMY. the use of the value of Z, to find the augmented semi-diameter of the moon by means of the table at page 360, and thenee the proper value of A', and then to follow the equations (3), (9), (4), (1), (10). The first and last points of these limits will have Z = 90°. For these places we have therefore by (5) i"' = A° + a" ± 2 A A' cos (S + <'). If we assume t' = i, we shall obviously have S -jr •' =: /Sf + i = u, and A cos (S + i')=n, o being the angle under the distance A and the nearest ap- proach n, as before used. .•. P«= A"+ a" ±2 A'n, = A' — 7<.' + (A-±nf. Consequently A=sin»/l"'' — in± a'Y. n Therefore by taking the constant caused in the computation of the beginning and ending of a phase on the earth, we shall have semi-duration = c tan u ^ - V P'" — (« ± A')', n which may be arranged for calculation as follows : , « ± A' ■ A r P' ■ I cos u ^ — ..r; — , semi-duration = c — sin id , P' n Time of j j"'"!^!!" ^ [ = t™e of middle j T [ semi-duration, (11) The places of entrance and departure of the limits, by continuing the assump- tion •' = ', may be hence calculated as for the beginning and ending of a phase only using i ^u instead of i, thus : a = (— — ". i = ( — ') + "i For place of entrance. . , ♦ »^ 1 tan a sin J =; cos a cos B', tan rt = — _. , sin Pot place of departure. (12 . , , T,, , tan i Sin t = cos 6 cos D\ tan A = : — =-, sin D Having assumed i' = i, the times and places so computed will only be approxi- mate, though BufBciejotly near for general purposes. For an accurate calculation, we must first determine the true value of i'. Since Z= 90", the equations (9) give t' = V, which is also shown by (4). We may, therefore, with the quantities taken out for the respective times of entrance and departure, proceed with the equations (C), (3), use v instead of i in (7), and then the final results will be deter- mined by (10). It ought, however, to be observed, that jt will be advisable to take the time of entrance in excess to the next higher integral minute, and to re- ject fractions of a minute in the time of departure ; since by fixing on a time a trifle without the actual limits, the value of sin Z would come out greater liian APPENDIX X 349 unity, and the calculation rendered useless in consequence. The places so compu- ted will be accurately situated in the limiting lines, and though not strictly the first and last points of these Unes, they will be very nearly so. IV. Deteemination of the Place where a givex Phase will appear both ai Sunrise and Sunset. We have seen (page 341) that when the rising and setting lines of a phase ex- tend throughout the eclipse, they will compose the figure of an 8 much distorted. The point of intersection or nodus is a place where the phase will be seen to begin and end in the horizon ; that is, it will either commence at sunrise and end at sunset, or commence at sunset and end at sunrise. At the time of the middle of the eclipse, the sun will therefore be very nearly on the meridian : if diff. dec. and 6 are of the same sign, it will be midnight, because the pole of the earth will have the zenith and sun on opposite sides of it ; but when those values are of difi'erent signs, it will be noon at the place, for then the zenith and sun will be both on the same side of the pole. If t denote the semi-duration of the eclipse, which begins and ends with the given phase, r -y- will express the semi-diurnal arc of the sun; and,'. — tan i tan i = cos It —I = cos (r . 15°), which being nearly unity, wc must have I '^ S or Z nearly = 90°. Consequently for the values of du dv dt' dt' or midnight, we may assume sin Z=i unity, and M=0'' or 180°. So we get, from the equations (1) and (2), page 344-5, « = — (a;) ± i", V = (y), dt dt ' dt dt Let /t denote the hourly motion on the apparent relative orbit, and i' the incli nation with a parallel of declination ; then , d V . , du ,cos.' = -, Msm.=-^; or, in sin i'= J>i ) /j> M cos i' — 90° when diff. dec. is negative. These serai-durations will give two times of beginning and ending; the one an- swering to tlie point jif and the other to the point M" The middle of an eclipse in the horizon will take place from the first beginning to the second beginning, and from the second ending to the first ending. Tlie places will be determined by producing m M to a distance of 90° from m. If a great circle be drawn through S, so as to be at this point parallel to mM, it will evidently intersect the former at a distance of 90° and determine the same place. We shall therefore, in supposing the places to be determined in this man- ner, have the following formuloa : First place of beginning, ui = 90°, tan /i := — cot I (2) sin I = — r sin * cos i. v«.. .~ , h must be taken in the 2d semicircle, or between 0° and — ISO' First place of ending. Change the name of the latitude of the place of beginning, and to the hour angle A apply ± 180°. The results will determine the place of ending. Second place of beginning, 6 = — 1 -I- «>a tan a a = — I — (J J, sin I = cos a cos S, Second place of ending, sin I = cos b cos I, tan h^^- tan h = ■ sin i tan b aa i 18) S62 SPHERICAL ASTRONOMY. The second places of beginning and ending will be two of the extreme points of the lines traced on the earth. The other two extremes may be determined by computing coa u = gj , and proceeding as before, observing that n mast be considered positive, and u > 90° when diff. dec. is positive. These four extreme points are the same as those of the northern and southern limits, the phase being simply external contact. 2) When n > P' — (« + tr) and < « + », ) The places will be determinable throughout the whole of the first > . . (4) duration found as above. ) (3) ■When«> «+ a, n — (s + f) /c-P'\ . (fi) F' ' n must here be considered a positive quantity, and u will be > 90° when diff. dec. is negative. The phase will continue throughout the whole duration, and the ex- treme places may be computed from this value of u according to the equations (3). Having found the limits between which the phase is possible, the places for any intermediate times may be determined thus, t denoting the time from the middle, ''" " = (tt) '• u > 90° when diff. dec. is negative, and the places by the equations (3). If n < s + "i suppose n to be positive, and compute n~{s + „) /cP' \ . cos u := =- ; r = I I sm u. F \ n f Then for times, without the limits of this duration, we may determine four places ; two with o < 90° and two with u > 90°, which will all fulfil the neeesaary conditions. Tlie preceding results have been derived on the assumption of i' =: t. They will be suifieiently approximate for a general drawing of the lines on a map, and more particularly as these phenomena cannot be subject to minute observation. When, however, from local circumstances or otherwise, greater accuracy is wanted, we must use the proper value of i' and the relative horizontal parallax reduced to the latitude thus determined. Since Z = 90°, the condition for the middle of the eclipse, according to the equation (4) page 346, is i' — » = or i' = i'. Let the figure at page 344 represent the positions which answer to the particulars of the present case. Then as Mm = Mm! ^ P\ the / Mmm' = / Mm' m.» Denote this angle by e ; the angles Nm M, NirCM hy M,M ; and we shall have lNm8 = y, I Nm' S=\m° — V, M=0—v, if = 180° — V — fl, lMSm = m'' — S—v, ZM8m> = 8 + v, /.SMm = S + v — 9, I S Mm' =160''— {S + v + B). APPENDIX XI. 358 "With the triangles MSm, MSm', we hence find sin 8 = -p; Bin (S + v); sm {8 + v) ' ^•"-^ sin(« + v) = vhich, for computation, may be thus arranged sin {8 + v) 9— —J,-' — ; sin e = J' . A i 9 to be + or — but less than 90' o-._ °''^(^ + " — ») ■ 8m': 9 sin (8 + + «). 9 M=e — i J/'=(180° — 9) — V (6) The points m, m', may in some cases be both on the same side of 8, and the value of iS»i is only necessary to indicate whether any portion of the sun is ecUpsed or not. To have an eclipse, S m, taken as a positive quantity, must be less than s + v, and we must only determine a place from the angle M when the corresponding value of Sm is within this limit. 1i Sm, 8 m', taken as positive quantities, are both greater than s + ", the middle of an eclipse cannot be seen OD the earth under the assumed conditions ; on the contrary, ii 8 m, 8 m' so taken are both less than « + », the angles M, Zf may both be used, and consequently two places will be determined. In each case, similarly to (3), we adopt the formulee tan M) sin / = oos if cos i, tauA=: : — -y . . . (1) sin i ) VL CzNiBAi. lass. The places which in succession see a central eclipse are evidently determined by producing SMio a. distance Z from 8, so that A sin 2 = - 0) for then the relative parallax P' will bring the centres to a coincidence. To de- termine the position of the place on the earth for any given time, we have in the triangle iV£f 2, thus formed, N8 = 90° — S, ZN8Z = 8, 82 — Z, and hence the following formnlse : tan 8 = tan Z cos 8, 9 to be + or — and less than 90° sin 9 tan h = -tan jS; tan / = tan (9 + i) cos h. cos (9 + i) A to be in the same semicircle with jS; sin 9 sin Z cos S check . cos (9 + i) cos h cos / l (2) In the course of the general central eclipse, one of the places on the earth will have the central eclipse at noon. At this instant the bbdiea will obviously have 23 354 SPHERICAL ASTRONOMY. true as well as apparent conjunction in right ascension, and .'. A = diff. dec, and £ = 0. Tliis place is hence determined thus : . diff. dec. ;_j_i_7 sinZ = — ^, — , l=S + Z, Z to have the same sign as diff. dec. App. time of true 6 ^ ■west long, of place, _ (8) These equations (1), (2), (3), involve the horizontal parallax P', answering to a mean latitude of 45°, which will be suflSciently near for ordinary purposes. Where an accurate result is wanted, the calculation must be repeated with the use of the equatorial relative parallax properly reduced to the latitude thus determined. The first and last places on the earth which see a central eclipse, are to be found by the formulse at pages 338-40. The preceding discussions comprise all that is necessary for the calculation of the lines which are shown in the maps now inserted in the Nautical Almanac, and which are quite sufficient to indicate the general character of the eclipse that may be expected for any particular place. We might now proceed to show the appli- cation of these equations in the resolution of innumerable other curious and in- teresting problems; but such a field of speculation would not conform with the object of this paper, and may the more willingly be abandoned on the considera- tion that the means of solution may, in most cases, be readily elicited from the equations already established. The following classification of these equations will be found to exhibit, in a comprehensive form, all that will be requisite to direct and facilitate the operations of the calculator, and relieve the mind from any un- necessary reference or consideration. IfOTATION. D ^ the II 's true declination ; i = the O's true declination; a = the true difference of right ascension in are, or 5 's right ascension — o's right ascension; Di = the B 's relative motion in declination, or P 's motion in declination — 0's motion in declination, ai = the 9 's relative motion in right ascension, or the motion of the i — that of the O ; Diff. dec. = the true difference of declination at 6 in right ascension, viz., D 's declination — Q'b declination, at that time ; P = the 5 's equatorial horizontal parallax ; r = the O's equatorial horizontal parallax ; /" = [9.99929] (P-ir); a = the ]) 's true semi-diameter ; V = the o's true semi-diameter; A ^ the true distance of the centres ; D', «', «', a', the apparent values of D, o, », A; H = the angle under A and n: in all cases thb angle is to be taken poi- itively, and between 0° and 180°. APPENDIX XI. 355 I. — SEQmsma and ending of a phase on the earth. (Z), Di and ai at c5 ) ; tani= =; n = diff. dec. X cos «; ai cos jO ' I of the same sign as Di ; n of the same sign as diff. dec. n sin I [3.65630] « 2) ; t = e tan i ; sin ( to be found by combining the preceding values of cos i and tan i ; sign of t to be determined by diflf. dec. x A. Time of middle = time of ci — i ; For ■ ' partial I r p' 4. , ^. „_ central ,. I", total r"''p=^'^ = ip'+.-., annular J [ j" _ , + , ; n cos w ^ — : T =r c tan u, A Ti«« of j enXf ^ 1 = t-oof -<^din + ['' "=(-')-"; b = (-t) + u. 4 Place of beginning, {S at d); • J , i , tan a sm < = cos a cos d : tan A ^ : — - ■ an I Zr= apparent Greenwich time of beginning; longitude east = h —:II; A to be in the same semicircle with a. 6. Place of ending, ( j at d); • 7 r 1 . . tan J sm I = cos cos : tan A = ; — r ; smd £^^ apparent Greenwich time of ending; longitude east =h—H; A to be in the same semicircle with b. 6. For more accurate calculations, reduce the true relative horizontal parallai^ Dj means of the table at p. 387, to the latitudes so determined, and recompute. n. BISING AND SETTING LINES. » For partial eclipse, a' = * + a. 7. ■Whenn> J"- A'. These limits will extend throughout the entire duration of the general eclipse, and form the distorted figure of an 8, the first and last points being the places of beginning and ending on the earth. 35G SPHKEICAL ASTRONOMY; 8. Whenn ^1. . . ' -P'- A' P'+ A' Prepare the constants, p ^ , j ^ ! , and let t be the time from the middle of the general eclipse ; . t n tan (I) = — ; A = ; c cos u c > 90° when n is — . 10. S = {-i)^,^. 11. sm — 2 ~V P'.A ■ to be leas than 90° and positive 12. Place following. • I /o % . II tan(iS — »») sm (= cos (P'_A'. • 16 Find P' ^f {P — ir), for a latitude equal to the complement of j at c5. fi sin i' = Bi, H COS j' =ai cos D ± [9.41796] P' sin I, — diff. deciP' ro i=. <.„«-! ifc sin, k = y~, r , t m seconds = 13.656801 — . cos (• ~ ■^ II U-|uXl-S--''-^'ff'i-HneS ;iye, igatiye. 16. At the place, When diffi dec. and a have | ^^^^^^^ \ signs, app. time of true rf = i l^" ; _ ^^ which, compared with the Greenwich apparent time of the true d , will determine the longitude of the place. 17. k cos ( A' sin u cos « A' • ■- ^ ' cob(t. 15°) tani=±^ — J— i; tau 2 to be of the same name as di£F. dec. IT. PLACES WHEKE THE MTDDLB OF THE ECLIPSE IS SEEN WITH THK SUN IN THE HOKIZON. eP' Ti = . cos I 18. When n P' — (s + ") and < s+ "> compute cP' The phenomenon will continue throughout the whole of the duration so found. The two extreme points will be determined as above with the angle uj. The places of first and last appearance also as above. 20. When n > » + », compute uj, tj, as above. The phenomenon will continue throughout the whole duration, and the extreme places will be determined by proceeding with this value of u as for the beginning and ending of a phase. These places will in this case be also those ot first and last appearance. 21. Places for any time within the limits : Let t be the time from the middle, and compute '■=(^) If re < s + ff, this (D may be taken both gi'tater and less than 90° when t is greater than r^ before found ; and then four places will be determined. In all other cases whatever u must be > 90° when diff. dec. is negative. The places to be determined by proceeding with u as for the beginning and end- ing of a phase. 22. For a more accurate determination at any time : Find P' = f(P — tt) for the latitude before found. Find (a), (y), /S, and A, as in No. 14. For the time of cS form the constants (.4) = [0.58204] ai cos D, (jB) = [0.68204] i),. Compute V from the equations, X cos » = „, ' , X am » = ■ ''„, ^' . — . P' cos i P' cos i APPENDIX XI 359 23. Then sinf^f+v) 9 = — _pr—'' s'° " =l>- A, S to be + or — but less than 90°. sin J = cos M cos 3, tan A = : . Bin d If K, k', be both less than » + tr, the angles M, Jlf, may be both used in these equations, and two places determined. If one of the quantities k, k', be greater than « + (T, the corresponding ifwill be excluded, and only one place determined with the other yalue. If k, k', be both greater than a + e, both computations will be excluded, and the assumed time will be without the limits of the appear- ance on the earth. NOETHEEN AND SOUTHEEN LIMITS FOE ANY PHASE. C Partial ) f (« + 6") + '• f Partial ^ f (s + 6") + ir, For -J Total > appearance, A' = P' — A ' only one limit will have place, vit : A ;K:rn}'-*-''-»HT: 26, First and last points or places of entrance and departure : n±A' /c-P'N •=(^) sm w; 3:r}^'g"^-Sro°;lh:rn[l-"- Ti- of j grt^ure I =^-« °f-d^« {~\^- Places of entrance and departure determined as in Nos. 4 and 6, for the begin- ning and ending of a phase, using a = ( — i) — w and h = ( — <) + w. For the appearance of external contact these determinations are included in No. 18, tod therefore need not be repeated for these limits. jaV. Places for any times within the limits: Prepare the following constants, using a at c5i a' sin c u=A'co3i, -jy = i^ii, a=±- cos D' n n ± a' , E'= r. cos w := — T,- — "8 above ; c (» ± A') P 3:;}»Snfor{»-tr^^-*- 360 SPHERICAL ASTRONOMY. 28. Let t be the time from the middle of the general eclipse, tan a' = i . M, M= (— T m' ; . „ /COS TV sm Z = ; , Srf-S-^-laftrf*''-"''^^- tan 6 = tan Z cos M, tan (A — a') = sin B ^ „-r tan M, tan I = tan (fl + D') cos (A — a'), cos (9 + jD') / \ /• sin 8 sin Z cos Jtf' cos (8 + J)') cos (A — o') cos i' < 90°, and same sign as cos M; and A — o' to be in the same semicircle ■with M. 80. For a more accurate determination at any time. Find S" = i> {P — i) for the latitude before found. Also, with Z find the augmented semi-diameter s' =s + augmentation, from the table annexed. Augmentation of the ]> 's ^mi-diameter. Argument : True Zenith Distance Z. z ¥otP = 54' Var. for 10' ini>. Z SorP = 54' Var. for 10' in P. Z ForP = 54' Var. for 10' in P. o „ II „ „ /, „ o i4'0 5.7 3o_ 1 2 'I 4.9 60 6.9 2.9 I 74-0 5.7 3i I2-0 4.8 61 6.7 2-8 2 i4-o 5.7 32 II. 9 4-8 62 6.5 2.7 3 i4-6 5.7 33 11.7 4.7 63 6.2 • 2.6 4 i4-o 5.7 34 11-6 4.7 64 6-0 2.5 5 l3'Q 5.7 35- 1 1.5 4.7 65 5.8 2-4 6 .3.9 5.7 36 II. 3 4-6 66 5-6 2.3 7 13.9 5.7 37 II.2 4.6 67 5.4 2-2 8 i3.8 5.7 38 II -O 4-5 68 5.2 2.1 9 i3.8 5.7 39 10-8 A-4 69 4.9 2-0 lO i3.8 5-6 40 10-7 4.4 70 4.7 1.9 u 13.7 5-6 4i 10.5 4.3 71 4-5 1.8 12 •13.7 5.6 42 10.3 4.3 72 4-2 1-7 i3 i3.6 5-6 43 10-2 4-2 73 4.0 1-6 i4 i3.6 5.5 44 lO-O 4.1 -74 3-8 1.5 i5 i3.5 5-5 45 9-8 4-1 75 3^5 1-4 t6 i3.4 5-5 46 9.7 4-0 76 3-3 1.3 17. i3.4 5-4 47 9.5 3.9 77 3-1 1.2 i8 i3.3 5-4 48 9.3 3.9 78 3-8 I.I '9 l3.2 5.4 49 9-2 3.8 79 2.6 I.I 20 i3.i 5-4 5o 9.0 3.7 80 2.4 I-O 21 i3.o 5-4 5i 8-8 3.6 81 2-1 0.9 22 12.9 5.3 52 8-6 3.5 82 1.9 0-8 23 12.8 5-3 53 8.4 3.4 83 1-7 0.7 24 12.7 5-3 54 8-2 3.3 84 1.4 0.6 25 12.6 5-2 55 8-0 3.2 85 1.2 0-5 26 12.5 5.1 56 7.8 3.2 86 I-O 0.4 =7 12.4 5.1 57 7-5 3.1 87 0.7 0.3 28 12.3 5-0 58 7-3 3.1 88 0.5 0.2 29 12-2 4.9 59 7.1 3.0 89 0-3 Q.I So I2-I 4.9 60 6.9 2.9 90 CO 0-0 APPENDIX XI. Thjn, f Partial ^ («'+», For -j Total > phase, A' = J «'- », (Annular) (_a — ^ 81. For the time of rf form the constants, (.4) = [0.58204] <■, COS p, (S) = [0.58204] J5,. Find the values of J), i, a, for the given time. {D) = jD + (o corr. from table, page 342). {x) = {D)-i, (2/) = acos(Z)), 1 „„„ _ (^) + («) sin ^ , . (-B) — (») sin i X cos » = > D, — ^, A Bin V = V_i-_«!1^ — , P' cos 4 ' P' COS 4 32. {Z fi-om the first computation), sin 361 tan V . / cos 2 sin # = 4/ -r , tan .' = - r 2 A cos * cos 2 # «=A'cost', J)' = iTw, »=A'8in.', a'=:±— !^-, cos D (D) = D + (a — a') corr. y = (a — a')cos(i)), z = (D) — D', V tanM=-, sin 2: ' a' P' cos Jf P' sin Jf ' lSr}»'snBfor|-^-(limit. Bemaining coigputation the same as in Ko. 29. VI. CENTEAL LINE. S3. The compntation of the limiting times and places is comprehended under the head, " Beginning and Ending of a Phase o& the Earth." 34. Places for any times within the limits : t = the time from the middle. t n tan w = — , A c cos u u > 90° when n is negative. 36 5 = (— i)T«; I uX } ^^ f» { aft'r } *« «- °f -'ddle. 86 (i at d). sin 2 := -^, tan 9 = tan Z cos iS, tan A = — ^-; — r tan 5, tan I = tan (9 + i) cos A, cos (e -<- f ) 862 SPHEKICAL ASTEOITOMT. , , sin 9 sin Z cos S check ^~^^^^~^^^^~' " " * ^^^— — — — oos-(9 + i) C08 A C08 I ' 0, same sign as cos S, and less than 90°: A; same semicircle with S. 37. For a more accurate determination at any time, find P', S, A, as in No. 14, and proceed again witli these as in No. 36. 88. Place where the eclipse will be central at noon : («atc5)- . „ diE dec. , , „ sin2 = — pj— , l = S + Z. Apparent Greenwich time of true c5 = longitude W. Z < 90° and same sign as diS dec. 39. For a more accurate determination, find the horizontal parallax for the lati- tude, and with it repeat the operation. [All latitudes in the preceding formulae are to be recognized as geocentric, and will therefore need reducing by the table at page 336.] Examples. For an elucidation of the practical application of the preceding formula;, we shall take the solar eclipse of May 15, 1836. At the time of new moon, viz. 2>' 7"'.o, the moon's latitude |3 is 25' 43'', which being less than i° 23' 17'', the eclipse is certain. (See the limits at page 333.) The elements of this eclipse, as related to the equator, are d. b. m. 8. Greenwich mean time of (j in B. A. . . . May i5 2 21 22.9 ]) 's declination K 19 25 9-8 ©'a declination N. 18 5? 58-8 5 's hourly motion in R A. 3o 8-3 ©'s hourly motion in B. A 2 28 • a )) 's hourly motion in declination .... If. 9 58-7 ©'s hourly motion in declination .... N. 35 •! D 's equatorial horizontal parallax ... 54 23 • 9 ©'s equatorial horizontal parallax ... 8-5 D 's true semi-diameter 1 4 49-5 ©'3 true semi-diameter 1 5 49-9 from which we prepare the following values : Oil, I „ D'sdec. . . + 19 25 10 5 's H. M. in R. A. . So 8 ©'s dee. . . + 18 57 59 ©'a H. M. in R. A. . 2 28 Diff. dec. . . + 27 1 1 o, . . . . 27 4o I II I II 5 's H. M. in dec. . -1- 9 59 J> 'a eq. hor. par. . 54 24 ©'a H. M. in dec. . + 35 ©'s eq. hor. par. . 9 J>i . . 4- 9 24 Eel. eq. hor. par. . 54 1 5 log. 3 -51 255 const 9-99929 P' . . . .54 10 log, 3-51 184 APPENDIX XI. S63 I. BEGINNING AND ENDING ON THE EAETH. A + 9' 24" . . . Ol 27 4o . . . . 2.75128 (i) . 3. 2201 I D + 19° 25'. 2 COS . 9.53117 . 9.97456 ( tan . . « + 19 49 \ ( cos . diff. dec. + 27' n" . . . . 9-55661 (2) • 9-97349(3) . 3-21245 n + 25 34 ... sin 1 const. . 3-18594 . 9-530IO (2) + (3) . 3.55630 6.27234 (4) « . . + 19™ 56' . e tan t . 3-52106 (4) — (I) . 3-07767 6 • i5 2 21 23 _ i5 2 I 27 middle of general eclipse -P' . . . 54' 10" = » + ff . . . 3o 39 A for central pbase 84 49 = A for partial phase Partial. n . + 3-18594 A . 3.70663 Central, n . + 3-18594 A . 3-5ri84 (Iog.i 72' 27' I tan cos -J- 9.47931 0.49999 3.52106 4-02105 11° 49' I tan d. h. m. 8. 2 54 57 i5 2 r 27 , 1 4 23 6 3o beginning 1 5 4 56 24 ending cos -}- 9-67410 0-27109 3.52106 3-79215 d. b. In. 8. I 43 17 i5 2 I 27 1 5 o 18 10 beginning i5 3 AA AA ending (-0 — 19 49 72 27 (-0 a b — 92 16 + 52 38 o . — 19 46 61 /i<) — 8i 38 + 42 o Place of Partial Beginning. cos a COS i sin I I — 8-59715 tan o + 9-97576 sin i — 8-67291 tan h , a 2° 9' h . Beduclion i H . Latitude S. a 10 Longitude W. 76 53 + 1.40251 + 9.51191 — 1.89060 — 89° 16' ■347 37 h. in 8. Greenwich time 23 6 3o Equation . . 3 56 time °^inj space 23 10 26 347° af 364 SPHEEICAL ASTRONOMY. In the same manner may the places of partial ending and central beginning and ending be calculated, -irbich will come out Partial ending Central beginning Central ending Long. E. 28 5i Long. W. 98 16 Long. E. 52 4i Lat.N. 35 i3 Lat. N. 7 58 Lat, N. 44 5o n. KISOTG AND SETTING LIMITS. P' . . » + ff=A' . . J"— A' 54 10 So 39 23 3i p^ii 46 P'+ A' 84 49 g = 43 25 Since « > P' — a', these limits will extend throughout the whole duration of the eclipse ; and we may therefore calculate the position of a, place for any time between the Greenwich times i4'^ 23'> 6"" 3o', and iS*" 4'" 56™ 24'. As an ex- ample, take the time iS*" o'' So". S m B — m S + m Assumed time Time of middle . — 1949 58 5i — 78 40 34 2 — 112 42 — 44 38 A iA . i A— i> ? — 4 A d. h. m. H. i5 o 3o i5 2 I 27 I 3i 27 . .' . . . . 3.73933 e. - - . 3.52106 58 5i 49 24 24 42 tan cos 0.21827 9.71403 3.18594 log A comp. 3.47191 6.52809 12 56 2. 17 43 3.02653 Comp. log P' . 6.48816 2)18.93264 17 0.9 34 2 einim . 9.46632 Place Followihq. cos^(S — m) cos i . , sin I I .' . Beduction Latitude . .9.58648 9 '97576 — 9.56224 S. 21° 24' 8 R. 21 32 tan {8- sin S , -to) + 0.37850 + 9.51191 tan h . , — 0-86659 h . . — 82° i5' S . , 8 29 Longitude . W 90 44 b. m. s. Greenwich time o 3o o b. m. ime Equation . + 3 5G (time o 33 56 space ( tin; s (sps 8° 29 APPENDIX XI. 365 COS (S + m) cos I . sin 2 . I Reduction Latitude . Plaob Advanoimg. + 9-85225 + 9-97576 + 9.82801 N. 42 18 II N. 42 29 tan (S + m) sin S tan h . h . H . Longitude — 9-99444 + 9.51191 + 0-48253 — 108 i3 8 20 "W. 116 42 B7 taking 5=:( — t) + u instead of ( — ») — u, similar computations -will give the places following and advancing for the interval t ^ i' 3i"' 27' after the time of middle, or for the Greenwich time iS'^ 3'' 32"' 54". Much time will be saved by taking the computations two and two in this manner. in. PLACK WHEKE THE KISING AND SETTING LINES INTEKSEOT. i 90 c . . 18 58 * I . . 71 2 P P — » 9.99872 3.51255 i* . . . . 54' 5" + 4 36 sin i . const cos J) 3.51127 . +9-51191 9.41796 + 2-44114 3-220II 9-97456 + 26 6 = n cos l' . /I sin 1' . 3-19467 3o 42 = + 3-26529 . + 2.75128 (A) .' . . 17" 1' tan . . cos . . + 9.48599 + 9.98056 c - • 19 49 (1 ... + 3.28473 ,' ~ . . 2 48 — diff. dec. . — 27' n" . +54 5 + 26 54 . + 3.20790 cos(.'~i) . . 9-99948 k. . . . cos ( . . . + 3-20842 + 9-97349 + 3-18191 , 3-26458 sin 1 , . , + 3.20842 + 9.53010 3.55630 a' . . . + 6.29482 COS 10 . . . sin u , 9.91733 9.75036 log < . . . + 3.01009 366 SPHERICAL ASTRONOMY. a' Bin u C • • • 3.01494 3-28473 9-73021 2-95424 2-68445 t . . . . App. time true 6 h. m. s. + 17 4 12 it the i5' . . . 1 1 42 56 I place 8° 4' . . h. m. s. 2 21 23 COS . . 9-99568 9-5361 5 Equation . App. time true cS 3 55 tGre tanil 2 25 19 c enwioh tan I I . Reduction N 0-45953 ' 70 5i 7 Long, in ' time space 9*' 1 7"' 37' , 139'' 24' h Latitade . K 70 58 Tlius we find the required place to be in longitude E. 139° 24' and latitude N. 70° 58', where simple contact will have place at sunset and again at sunrise ; also the middle of the eclipse would be seen at midnight if it were not iiitercepted by the opacity of the earth. The duration of the eclipse will correspond with the duration of the night, and therefore no portion of it will be visible. rv. PLACES WHERE THE MIDDLE OF THE ECLIPSE HAS THE SUN IN THE HOKIZON. In the present case n is > P' — (» + ») and < s + c. We must therefore pro- ceed as in No. 19. 1. J^or the extreme pointt, e . P' . . 3-52io6 . 3.51184 1 it n + 25 34 i^. 03290 3-18594 _(, + ,) _3o 39 5 5 n . 3-84696 (i) 2-48430 ' „, . . 95 23 . ' (_,)_ 19 49 r, ua 95 23 a — Ii5 12 4 + 5 34 h. . I 2 3 P' . J cos ( sin m. t. 56 39 . I 27 time 4 48 time 58 6 time . 3-5ii84 — 8.97246 . 9-99808 (2) . . 3-845o4 (0 of middle of beginning of ending + (»). APPENDIX XI. 367 Place of Bzginnino, ok First Extekmb Piaok. b. m. B. Greenwich time b 4 48 cos a . . cos j . — 9-62918 + 9-97576 tan sin a i . . + 0-32738 . + 9-51191 sin 2 . — 9-60494 tin h . —0-81547 I . . Reduction S. 23 45 8 S. 23 53 Loi h H Jgitudi ' . —81 18 . + 2 n Latitude e W. 83 29 Equation time space 3 56 8 AA 2° II' Place op EwDiNa, or Last Extreme Place. b. m. B. cos 6 . cos j . + 9.39664 + 9-97576 + 9-37240 ' N. i3 38 5 tan b • Bin S . tan A . h . H . Longitude + 0-58943 + 9-51191 — 1-07752 ' + 94 47 60 3i Greenwich time 3 58 6 Equation . . 3 56 ival . . I . . Beduction time 422 jffin < space 60° 3r' Latitude N. i3 43 E. 34 16 2. For the extretne times, eP the value of t^ taken out from the preceding logarithm of is i"" 57" 10'. n h. m. s. 2 I 27 time of middle I 57 10 . . T, o 417 first appearance 3 58 37 last appearance PtACB OF FiasT Appeaeanoe. Bin 1 . cos j . . + 9-53oio + 9-97576 — 9-5o586 S. 18 42 7 cot < . sin S . tan A . h . R . + 0-44339 . + 9.51191 . —0-93148 . —83 19 .+23 h. m. 8. Greenwich time 4 17 Equation . 3 56 sin 2 . . time 8 i3 Reduction space 2° 3' Latitude S. 18 49 Longitude W. 85 22 Latitude K. 18 49 Place of Last Appkaeakob. • — 83 19 180 Greenwich time Equation . . time space li. m. s. 3 58 37 3 56 h a . . + 96 4i , + 60 38 4 2 33 60° 38 Longitude E. 36 3 388 SPHEKIOAL ASTRONOMY. For the computation of places in this line, we hare therefore the whole range between the Greenwich mean times oi" 4" 17' and 3'' 58"" 3j'. As an example, take the time i'' 3o™, h. m. s. Time of middle 2 i 27 I 3o (— ) 3i 27 / — 19 49 i5 34 — 35 23 — 4 i5 . . . 3.27577 — . 3.84696 sin , 9.42881 cos a cos S . sin / , Redaction Latitude + 9.91132 tan a + 9' 97576 sin S , + 9.88708 tan h . N. 5o 27 II N. 5o 38 — 9.85i4o + 9.51191 + 0.33949 — ii4 35 + 23 29 Longitude W. i38 4 b. m, e. Greenwich time i 3o , o 3 56 Equation jti time space I 33 56 23° 29' By similarly using the angle 6, we shall find the position for the interval 3i™ 27' after the time of middle, or for the time 2'' 32™ 54' ; , thus. cos b cos i sin I I . . Reduction Latitude + 9.99880 + 9.97576 + 9-97456 N. 70 35 7 N. 70 42 tan b . sin I , tan h . h . — 8.87106 + 9.51191 + 9.35915 — 167 7 + 39 i3 Longitude^ 5- '°^f ^ ( E. i53 4o h, nt B, Greenwich time 2 32 54 Equatior Hi „ 3 56 time 2 36 5o space 39° 1 3' The places may be computed by two together in this way ; and it will perhaps be a little more couTenient to assume a value of t in the first instance. We may take any value which does not exceed tj or i* 57" 10'. In the present example we should take i = Si"" 27', and begin as under: ■o- (II . . — 19 49 i5 34 a . b . . —35 2J . — 4 i5 log* cP' 3.27577 3.84696 9.42881 and then proceed for the places as above. Time of middle t . . h. m. t. 2 I 27 , 3l 27 Time before middle i 3o o Time after middle 3 32 54 lyitMii APPENDIX XI. 369 V. NOETHEEN AND SOUTHEEN LIMITS. 1, Fob the partial phase, we have only southern line of simple contact. Constants H, cos w, D', a'. V ■ . i5 5o a' . n . «— A' . 3o 46 + 25 34 — 5 12 C , + 3.18594 — 2.49415 — 0.69179 3.52106 T" . — 2.49415 3.51184 E . A' cost . — 7.17073 3-26623 + 9-97349 cos vr sin t . _ 8.98231 3.26623 + 9.53010 log u . u + 3-23972 28' 57" COS D' + 2-79633 + 9-97448 i + 18 57 59 + 19 26 56 logo' a' — 2-82185 — 11' 4" The extreme places will be the same as those which have the middle of tlie eclipse with the sun in the horizon, page 366 ; and we may compute for any tiiiie between the corresponding times of beginning and ending, viz. : oh 4'" 48' and 3'' bS'" 6'; or we may take any value of t less than 1'' 56™ 39'. For an example, take t = oh 58"' 33'. h. m. s. Time of middle . 2 i 27 (- t . . 58 33 Before middle . i 2 54 . . After middle . 3 . . -')- • — 19 m' . 100 M . —lie M . + &i Z . + 3o 311 0" o'. + 9-77167 + 9-19113 + 8-96280 + 8-96098 + 9-95835 + 9-00263 . + 0-80357 + 9-80620 + 9-92544 ) + 9-66258 + 9-58802 , TS. 21 10-2 7.6 , t . 3.54568 49 E . —7.17073 53 tan a . — 0.71641 42 cos m' . — 9-27571 4 cos w . — 8-98231 Eemaining calculation for the time tauZ . cos M . fl + 5 j4-7 tan 9 . . D' + 19 26.9 sin fl . 6 + D' + 24 4i-6 cos . . . tan M . ° ' ( tan . . . h-a' +32 37.2 . \ a' — ii-i ( cos . . . ooc, oi^in^ . + 9-70660 "^ ■''(tan2 . + 9-77167 sin Z . . + 9 - 70660 cos M . +9-19113 + 8-89773 Comp, cos {h — a') + o- 07456 Comp. cos I . . + o-o3o35 check . . . + 9-00264 li. m. b. Greeuwich time 3 A .. 32 26.1 ^ tan Z . . Equation . . 3 56 ( time . 3 3 56 I . . Reduction . 1 space +45° 59' h . . . + 32 26 Latitude . N. 21 18 — Longitude . "W.i3 33 24 370 SPHEKICAL ASTEONOMT. The calculation for the time i^ i" 54' is to be performed in this manner, with the same values of tan Z, sin Z, only taking the value of M= — 120° 4^', A MORE ACCtJRATK CALCULATION lOR IHK ToiB S"" O" O'. Constants (A), (£). + 3.22011 9.97456 2»i . . 11 . cos D const. o . 58204 + 2.7512.8 0.58204 U) 3.77671 + i°39'4o" (B) 3.33332 These- constants may serve for the computations at example the following is the process employed : «... I II + 17 49.0 . +o°35'54" all times. For the present D . a corr. I . (') • log {x) sin i (x) sin I (A). . I log • P' cos i \ 00a y • 2 • • + 19 3i 34) (J) a . + 18 58 21 « . . + 3.02898 + o 33 i4 cos (D) + 9-97428 + 3.29973 log (y) + 3.00326 + 9.51204 + 9.51204 + 2.81177 + 0° 10' 48" + 1 39 4o +1 5o 28 + 3.82138 3-48811 + 0.33327 o.3oio3 2 A COB r + 0.6343O + 2.5i53o (1/) sin I '+ 0° 5' 28'' -f o 35 54 -t- o 3o 26 { log X sin V . X cos » • tan V , + 3 .26150 3.4881 [ + 9-77339 + 0.33327 + 9.44012 -I- 24° 38'. 9 P — p . . . -P' . . . COS S . . P' coal . cos Z . . 2 X cos r sin' . . sin . . . 2S> . . . COS 2 # . . tan •' . . COS r . . sin t' . . 3.51255 9-9998' 3.51287 9-97574 3.488II 9.93493 o -63430 9.30063 9.65o3a 26° 33'. I 53 6.2 9.77843 + 9.44012 + 9.66169 + 9-95851 + 9-62020 t 14 5o aug. . 12 »' . . . i5 2 « . . • i5 5o A'. . . 3o52 i If 3.26764 . -f 9.95851 • • • sin i' . 3.26764 . + 9.62020 + 3.22615 cosi)' 2.88784 9.97451' 2 •91333 + o''28' 3" -f 18 58 21 -f IQ 26 34 APPENDIX XI. 371 «' . . -o°i3'39" a . + 17 49 D . . . + 19 3i 34 (a — a' . (log . + 3i 28 {« — a') corr. 3 + 3.27600 (D) . D' . . . + 19 3i 37 . + 19 26 24 . COS y • ■ + 9.97428 + 3 . 25028 M . . . + 5 i3 . +80° i' 4 ( tan + 2.49554 + 0.75474 ( sin + 9.99338 Z . . . + 33° 43 -9 J" j sin ( cos + 3.5I23-7 + 3.50675 + 9-74453 + 9.91994 tan 2 . . +9.82459 sin .? . . + 9.74453 O ' cos M . tan e . +9.23864 . . + 9.06323 ■ + 9.23864 « . . + 6 35-9 + 8.983.7 D' . + 19 26.4 sin 9 . + 9-06034 comp. cos (A — a')+ 0.09214 e + i)' + 26 2-3 . cos . . . +9.95352 + 9.10682 . comp. cos I check . + o.o3i5i + 9.10682 tanJf . . +0.75474 ' j tan . . ( COS . . . +9.86156 /j_a' + 36 I.I . 1). m. a. a . — 13.7 . +9-90786 Greenwich time 3 A . + 35 47-4 tan{9 + i)' ) +9.68892 Equation . 3 56 tan! . . +9.59678 ( time . ^in i + 3 3 56 I . . . N. 21° 33'. 7 ( space + 45°59'.o BeductioQ 7 -7 A . . . + 35 47-4 Latitude . N. 21 4i -4 Longitude "W. 10 II .6 This result differs materially frotn the former one ; but we are not to infer that the former position is so far wide of the truth. In general the second determination may be considered as an almost accurate point in the limit, and though the first result be some distance apart, yet it will always be very near to the limiting line, sufficiently near indeed for the mapping of the lines. By direct calculations of the eclipse for these places, the former will have an eclipse of about ^L of the sun's diam- eter, and the latter about jif Jn;- of the diameter, which is too small to be pereeptibla 2. For the anmulab phase, toe have both northern and soutJiern limits. Constants M, cos w, J)', a', for northern limit. « + 6" . ff ■ > • i4' 56" i5 5o A' . . n . . «+A' . 54 . + 25 34 . . . + 26 28 . . . +3.18594 . +3.20085 . + 3. 20085 c + 9-98509 3.52106 P' . +3.51184 £ . . + 6 464o3 COB W . + 9-68901 3T2 SPHERICAL ASTRONOMY. + 60" 44' -8 A' C03 t log « U . I . D' . h. m. 8. I 42 l4 Bin w eP" n 4- 9-94075 4- 3-84696 4- 3.78771 + 1-73239 9.97349 sin I 1-73239 4- 9-53010 + 1.70588 4- 1-26249 + 0° o'5i" oosi)' + 9-97579 + 18 57 59 4- 1-28670 4- 18 57 8 4- 0° o' 19" The semi-duration of the northern limit on the earth is therefore i"" 42" i4', and ■we may calculate for any Talue of t not exceeding this. A calculation of the extreme places on the earth is to be performed the same as for the beginning and ending of a phase on the earth, and will bo unnecessary here. As an example, for a time within the limits, we shall take i =: i"" lo"" p'. b. m. 8. Time of middle . 2 i 27 ( — 1) . / — 19 49 5o 43 . — 70 32 . 4- 3o 54 t . E . tan u' . cos u' . cos w . 3-62325 4- 6-464o3 t . . I ID ' 4- 65 7-3 /j_a' +45 44-6 «' . -f o^ h 4-45 44-9 tan 9 sin 9 . cos . tan M ( tan . ( cos . tan (9 4- tan I I 4- 0-01775 4- 9-82106 4- 9-85817 comp. cos (A — a') 4- o..i5622 4-9-62397 comp. COS Z . . 4-0-20693 check . : 4- 0-23420 - + 9-77706 4- 0-01126 . 4- 9-84378 D') 4- 0-33374 4- 0-17752 4- 0-23421 h. m. 9, Greenwich time 3 11 27 Equation . . 3 56 ST. 56° 23'. 8 10 -4 Reduction . Latitude . N. 56 34 • ffin] h. . Longitude time , 3 i5 2i space 4-48° 5i' • • + ^S 45 W. 3 6 The calculation for o'' Si™ 27' is to be performed in the same manner, with APPENDIX XI. 373 A MORE ACCCBATE CAtCOLATION FOR THB TiMB 3'' II™ 27'. a COIT. . i . . + '9 33 27K^j + i8 58 28 U . a + 23 6 + 3.14176 P — T, . . 3.51255 (x) . . + o 35 o COS (2)) log (y) + 9-97419 + 3-11595 + 9-5i2o8 P ■ ■ -P' - . . ; COS 5 9.99901 logW . ein S + 3-32222 + 9.51208 . . 3.5Ti5ft 9-97574 + 2.83430 + 0" 11' 23" +1 39 4o (y) sia S (5) . + 2.62803 J" COS S coaZ . 2 X cos v 3-48730 (x) sin S (A). . + 0" i 5'' -f 35 54 9-8o33i 0-53740 + I 5i 3 log. -f 28 49 -f 3-23779 3-48730 sin' . sin i>

• . + 23 6 D . . . •f 19 33 27 ( « — a' . (log - . cos . -f 22 47 (a — 0') corr. I . 4-3-13577 (D) . . + 19 33 28 - -t- 9.97419 D' . . . + 18 57.40 y . • . + 3-10996 X . . . + 35 48 .' . . + 3-33203 M . . . + Be" 57'. I tan cos . -f 9-77793 . + 9-93328 J" + 3-39875 . +3.5ti56 Z . . . + 5o° 27' -9 ( sin J cos . + 9.88719 . + 9-80383 374 SPHERICAL ASTKOKOMT. 9. . +46 5-8 B' . + i8 57-7 9 + B' + 65 3-5 h—a + 45 4i-9 a.' . + 0-4' h + 45 42-3 tan 2 003 JW tan sin S C03 . tan M itan . cos . tan (9 + tan / I Seduction + o -08336 + 9-93328 sin Z cos M .+ 9-88719 + 9-93328 . . + 6 01664 • ■ +9 - . +_9^ + - . +9 85764 62499 -23265 .77793 . . + -01058 • • +9 -D') +0 844i2 -33249 . . +0 1766 c . . K 56° 20'. 5 10 .4 + 9.82047 comp. cos {h^-a) + 0.1 5588 comp. cos / check + p.2563o + 0-23265 h. m, a. 3 II 27 3 56 + 3 i5 23 Greenwich time Equation . . ( time ZTin i { space + 48° 5o'.8 h . . . . + 45 42 .3 Latitude . N. 56 3o .9 Longitude . W. 3 8-5 TI. CENTEAL LINE. We have, at page 363, found the semi-duration of the central appearance on the earth to be i'" 43" 17', which is therefore the greatest value of t for this phase. As an example for a time within the limits, take the same value of t as 'in the two preceding examples. t e Time of middle . h. m. 8. 2 I 27 (-0 • . — i°9 49 t . . I 10 0) . . . s . . + 5i 4i . — 71 3o tai Before middle , 5l 27 CO After middle 3 II 27 . . s . . + 3i 52 n A Eemaining computation for the time 3'' 1 1" 27". + 0-06994 . +9.92905 S + AA 56. o i + 18 58.0 d + i + 63 54.0 Jr 44 56.6. tanZ cos 8 tan B sin e . cos . tan ;S ( tan h ^ cos h tan (9 + tan / Reduction Latitude 49 35.6 I sin Z. cos S sin Z tanZ ' + 9-99899 + 9.84898 + 9.64339 + 0.20559 + 9.79354 + 9-999I3 + 9.84991 + 0.30990 + 0.15981 N.55° 18'. 7 I0.6 comp. cos /» , comp, cos I . check . 3-02323 , 3.52106 . O.IO219 . 9.79246 . 3.18594 . 3.39348 . 3.5ti84 . 9.88164 . 0.06994 + 9.88164 + 9.92905 + 9.81069 + o-i5oo9 + 0.24480 + o.2o558 h. m. Greenwich time 3 11 Equation . . 3 ( time . 3 i5 Sin \ — — I space + 48 • N.55 h : . . Longitude + 48° W. 3 54 APPENDIX XL 375 A MORE ACCCBATE OaLCHLATIOK. . f eorr , i . . + 19 33 27 I + 18 58. 28 (') • • + 35 P — » . ; 3.5ia55 . 9-99903 P' . . . 3.5ii58 l\i^) + 23 6 cos JD + 3i° 52'. 7 iv) . i tan 8 ( cosS Z . +49° 35'. 6 1 « . i . e + s + 44 55.8 + 18 58.5 + 63 54-3 + 44 57-5. tan 2 cosS tan e ein e cos . tan ,9 tan h cos h tan(d tan I + 0.06994 + 9.92899 sin Z taaZ sin Z cos S S) Beduction Latitude + 9-99893 + 9-84895 + 9.64331 •f 0.20564 + 9-79373 + 9-99937 -f 9.84980 + o.3iooo + 0. 15980 N. 55° 18'. 7 ro .6 K. 55 29 .3 Centrai Eclipse at IfooN. conip. cos h comp. cos / check . Greenwich time Equation . ( time . ( space , A . . . . Longitude . . + 3.14176 + 9-97419 + 3.ir595 + 3-32222 + 9-79373 + 9.92899 + 3.39323 3.5ii58 ■f 9.88165 + 0.06994 + 9.88r65 + 9.92899 + 9-81064 4- o.i5o2o 4- 0.24480 + o.2o564 m. B. II 27 3 56 + 3 i5 23 + 48°5o'.8 + 44 57 .5 W. 3 53 .3 Diff. dec. P . . sin Z Z . . 6 . . I . . Reduction Latitude 21245 5ii84 9.70061 3o° 8' r8 58 49 6 II Time of d Equation h. m, 8. 2 21 23 + 3 56 Long, in ( time . ( space 2 25 36 -if) °20') W, N. 49 17 By assuming a series of times, and so computing, in conformity with the preced- ing examples, a series of points on each of the several limits will be determined ; and these points being laid down in a geographical map, with respect to latitude and longitude, it will be easy to trace the lines through them. In this manner has the following map been executed, the assumed law of projection being that the parallels of latitude are concentric and equidistant circles. This projection will be found very suitable when an eclipse, as in the present instance, extends com- pletely round one of the poles of the earth. In other cases, any hypothesis what- ever may be assumed, with respect to the law of projection, provided the geo- graphical sketching and eclipse-lines be both laid down on the same principle, (See Fig. 11.) PRINCIPAL LINE'S FOR THE SOLAR ECLIPSE OF MAY 14-15, 1836 Fig. 11. — r— *■■'■'■■ T r- APPENDIX XI. , 3Y7 PHENOMENA FOK A PAETICULAE PLACE. I. — ^Eclipses op the Sun. The chief objeeta of determination for any particular place are — 1. For a partial eclipse, its magnitude, and the times of beginning, greatest phase, and ending. 2. For a total eclipse, the times of external and internal contact of limbs, or the times of partial and total beginning and ending. 3. For an annular eclipse, the times of exterior and interior contact of limbs, or the times of partial and annular beginning and ending. Also, to secure certainty in the observation, it is necessary to determine, in each case, the particular points on the limb of the sun, as related either to the vertical or a circle of declination, where these contacts take place ; and hence the general configuration of the eclipse. We first proceed to find expressions for calculating, at any time, the apparent relative position of the two bodies, and the augmentation of the semi-diameter of the moon. The parallax in altitude depends on the Eq. (8) or (9), page 336. It will here be necessary to investigate the effects which this parallax will produce in the right ascension and declination of the moon. These might be accurately determined by the theory of the small variations of sphfirical triangles, but not quite so simply as in the following manner: — Assume, as before, I, the geocentric latitude of the place ; JR. .4., the true right ascension of the moon; D, the true declination of the moon, + north, — south ; h, the true hour angle of the moon, + west^ — east ; r, the distance of the centres of the earth and moon. Then if, from the earth's centre, we take X, on the intersection of the planes of the meridian and equator, + towards upper meridian ; y, in the plane of the equator, + west, — east; z, parallel to the earth's axis, + north, — south ; we shall haVe, for the position of the moon, a; = r cos Z> cos h, y = »■ cos D einh, z = r sin JD ; and, for the position of the observer, (x) = p cos I, (y) = 0, (2) = p sin I. Thus the position of the moon, in relation to the observer as an origin, wiU be ■n' = x — (a) = r cos D cos h — p cos I ; y' =y — (}') = »■ cos D sin It ; z' = 2 — (z) = r ein D — p sin / ; and hence, D', h' denoting the apparent declination and hour angle, and r' th« distauce of the moon from the observer, we shall have a' = / cos D' cos h' =^r cos I) cos h — p cos I ; y' = r' cos D' sin K = r cos D sin A ; i' = r' sin D' = r sin Z? — p sin /. 378 SPHERICAL. ASTEONOMY 3! «' 1 Therefore, as cot A' =: -; , tan i)" = -; sin A', - = sin P, we find y y ■f cot A' ^ cot A - p sin P cos 2 cos D sin A „, /. p sin P sin Z\ sin A' , . „ tan i)' = ( 1 — ^ ^-F, — I -^-r tan -^ \ sin Z* / sin A / p sin P \ , cot A — cot A^ ^ I rv ■ — i I«08 ' \cos j> sin A/ tan i) t ani?' _ / p sin P \ . lin A' Vcos Z> sin A/ (1) sin A which present a direct method of calculating the apparent position of the moon, at any time, from that of the true. The former of these equations is evidently subservient to the other, and must necessarily be computed first. As the calcula- tion of these expressions will, in general, require seven places of figures, it will be more convenient to determine the simple effects of the parallax, or the small dif- ferences A.R. — A.R.', D — JD', for which other expressions may be derived from them. Let A.R. — A.R.' =.h' — h= A A, and D — S' = aD; then by mnlti plying the equation P sin P cos / cos J) sin A cot A — eot A' by sin A sin A', the left-hand member will become sin (A' — A) or sin A A. p sin P cos I , , , . • . sin A A ■= iT sin A . cos D Again we have tan J> tan D' p sin P sin I sin A sin A' cos 2) sin A' But tanD sin A tan jD' sin A' _ tan J) — tan i)' /I 1 > sin A Vsin A sin A'> sin (D — J)') sin A' — sin A sin A cos D cos J)' sin A sin A' tanD'; sin A Z> 2 sin i A A cos (A -)- 4 A A) , ^ ■ ^ s fv + ■ . ■ , -7 : — - tan J/. sin h cos JJ cos 1) sin A sin A' . , p sin P sin I Equate this with =-—. — =-, and we nnd ^ cos 1) Bin A sin A jD p sin P sip I 2 sin ^ A A cos (A -f i A A) sin ID cos J) cos B' cos D sin A' sin A A p sin P cos I sin A' ' coalf But 2 sin 4 /, A = cos -} A A cos 2) ' cos it A V APPENDIX XI. 379 Substitute this value and multiply by cos D cos D', and we deduce sin A D = p sin P fsin I coa jy — cos / sin i)' °°« (^ + i ^ ^0"| L cos i A A J We shall therefore have, for the parallax of the hour angle, and that of the decli- nation. , (f cos T)im P . , , sm A A = — ~ sin h' cos D Sin A D = sm P I (p sm /) cos D — (p cos I) sin i) ^ — -- i— ^ — -' I L vr / cos i A A J .(2) These are still however not adapted for direct calculation, since they involve the apparent quantities h', D' , which it is our object to determine. The only use that can be made of them is, first to use the true quantities, in order to get the parallaxes and apparent values approximately, and then to repeat the operation. To avoid this difficulty, substitute in the former h + A h instead of li', and in the latter put J) — A Z> instead of J)', and we get, by expansion, , » cos / sin P , . _ , , , . ,. Bin A A = i, — (sm A cos A A + cos A sin A A) : cos i) . . „ ^ r . , T^ ■ • ,1 cos (A -f ■} A A)l sin A Z>^f sm P cos A i) I sm I cos Jj — cos I sm D ^ — ; — - I ; L cos 4 A A J + P sin P sin A i> I sin IsinD + cos I cos D ^ — ; — - 1. L cos i A A J Divide these by cos A h, cos A -D, respectively, and solve for tan A A and tan A 1>, and we find tan A A = (p cos Z sin P\ . , •- -— 1 sin A cos J) / /p COS / sin P\ . 1 - l"- ~ — 1 cos A \ eoa D / (3) tan a2>: ;• sin P I sin Z cos jD — cos I sin D cos (h + i A A)"| • cos i A AA)-| A J 1 — P sin P I sin ^ sin D -\- coa I cos J) (4) [. , • T. . , _ COS (A + i A A)l sin I sin Jj + cos { cos J) ^ ; ; — - I ' COS i A A J , . , . p, n r, UnD cosCA + iAAn (p sin I sm P) cos Z* I 1 — . ^— ^; j — I ^•^ ' L tan I cos -J A A J ^l-(psin;sinP)sini>p+— L-^. ''°-(" + ;pn ^•^ ' L tan I tan J) cos 4 A A J ■> These expressions are all of them perfectly rigorous, and better suited to calcu- lation than they would appear at first sight. The process of the calculation, in which five places of figures will be sufficient, is more detailed in the following equations : (p cos /) sin P cos JJ tan A A = n sin' A 1 — h cos A • -(5) 380 SPHERICAL ASTEONOMY. c=:{p sin I) sin P ; Di = k tan J) ; tan A I> = COS -J A A "" tanD c cos J){1 — ni) 1—c sin D{1 + Dj) (6) rhe expression (4) for tan A D may, however, be neatly resolved by means spherical triangle as follows : Assume ^ cos (h + i A h) cos -J A A of i cos (A) : («) (7i) being v«ry nearly equal to A + ^ A A. And let JV be the nor(h pole, Z the central zenith, and M the moon ; then N'M = 90° - J), NZ = 90° - I, and the /_ h' = h. 'Without changing these values of NM, NZ, let U3 suppose the hour aogle iVto become increased to the value of (A); and with the triangle so constituted suppose the altitude of the moon to be t, so that ZM'=- 90° — t; then the spherical relations sin ZMcos M= cos NZ sin NM— sin iV^cos iV^fcos N, cos ZM= cos NZ cos NM+ sin NZ sin iV^cos N, will give cos t cos Jf = sin J cos D — cos I sin D cos (A) . , „ , . _ COS (A 4- i A A) = sm i cos iJ — cos I sin Z) ^ — - ' ; , cos^ A A sin £ = sin i sin J) + cos Z cos J) cos (A) • I • n 1 . I, ''08 (/' + i A A) := sm < sin Z* + cos I cos i> ■ ^^ — -— =. cos ■} A A Comparing these with the former expression of (4), we have therefore _ (p sin P) cos t tan A i) = — ^H — : „, . . cos Jf . . . . 1 — (p sin P) sin c Before this can be used the angles M and c must be determined. Draw ZD perpendicular to MN, and by spherics. tan ND = tan NZ cos iV . . . . . sin MD tan Jf = tan Zi) = sin iVi) tan A''; . ,, sin iVD . • . tan Jf = -; — -— — tan JV sin MB Also by (c) tan MZ = — -, or cot MZ = cot ^i) cos M cos Jh sin ND tan M cos N sin M cos JV sin NZ sin MJ) tan N cos Jf ' sin 7'" cos ^sin ifZ (/) (6) APPENDIX XI. 381 Let now JVD = 9, and MD = MN— fl = 90° — (9 + i)) ; and the equations (o), (J), (c), [d), (e), (/), will give the following: cos (A) cos (/» + ^ A A) coa -J A A tan 6 = cot I cos (A) . . . , ,, sin 9 ^ ,j^ tan JK = — ; — — tan (A) cos {B-\-J)) ^ ' tan t = tan (9 -|- 2)) cos M . sin 9 cos (A) cos Z cos (9 4"^) cos Jf cost (p sin P) cos c tan A i) : 1 — (p sin P) sin e cos Jf. .(b) ■ w ■ (<«) • (^) • (/) ^ -a) in which the equation (e) is used as a check on the preceding computations. This check affords a good secuiity to tlie accuracy of the work, and gives to these equa- tions a decided preference over those of (6), although a trifle more perhaps in point of calculation. They have also another advantage, inasmuch as Mmaj be consid- ered as the parallactic angle, and £ the altitude of the moon ; the former of these is useful in determining the position of the line joining tlie centres of the two bod- ies in relation to the vertical, and the other is useful in finding the augmentation of the moon's semi-diameter, which we shall now consider. If s' denote the moon's apparent semi-diameter, and s her true semi-diameter as seen from the centre of the eartli, the actual semi-diameter of the moon will be represented by both r sin s, and r' sin «' ; also, if a perpendicular be drawn from the centre of the moon upon the rijdius p produced, this perpendicular will be rep- resented by both r sin Z, and r' sin Z'. We must therefore have — =: -: . sm s sm Z Let M be the true position of the moon, in the preceding figure, and sin ZM sin / NZM=. sin JVJ/siii N will be sin 2 sin/ NZM=: cos D smh; for the apparent position of tlie moon the angle N ZM will remain the same, and sin Z sin / NZM-=. cos B' sin A'. sin Z' cos D' sin A' " mn Z cos Z) ' sin A ' . Also, by means of tlie equations (8) and (9), page 336, sin Z' f sin P sin Z' sin z cos z siaZ f sin P sin if p sin F bin Z p sin P sin Z tan z : 1 — (1 sin P cos Z sm s sin » sin Z' cos D' sin A' sin Z cos D sin A 1 — p sin P cos Z (8) All the preceding formulas are strict in theory. It now remains to consider what allowances may be made and what fiicilities given in their actual calculation. In the first place the value of cos i A A may be safely assumed equal to unity, and may therefore be rejected in tlie equations (2), (4), (6), and (7), so that (A) •■= h-\-i A h; it may be shown that this supposition cannot involve an error of more than 0".03 in the value of A I>. 382 SPHERICAL ASTRONOMY Also, as the area P, A h, A D, are emaU, we must have very nearly tan A -Z> -'i^ = sin 1" = [468657], *J^' = aJ) = tan 1" = [4.68657], ■wher^ P, A Ji, A P, denote respectively the numbers of seconds they contaia These equations may be made more exact, for the limits between which the angles are always comprised, by adopting numbers differing a little from sin 1" and tan 1"; thus, by assuming 'i^ = [468566], tan A h A A = [468561]. The first supposition will not in any case involve an error exceeding that of 0''.05 in the value of P, nor the second an error of more than 0".l in the value of A h, and these are much too small to merit attention ; the latter assumption aj}- plies equally the same to A iJ. Thus we shall have (A) = A + | A h, sin P = [468555] P, A A = [6.81489] tan A A, A D = [5.31439] tan A D ; also, A A = A o, the parallax in right as- cension. The equations (3) and (7) may therefore be commodiously arranged as follows : *" A = cP; n,^=k cos A ; c =: [468556] p ; m=^ A cos I ; r,„, .„„T k sin A A a = [5.31439] ; 1 — n k = cos D (9) By taking A less than 180°, positively or negatively, A o will have the same sign as A, (A) = A + i A..; tan fl = cos (A) cot I; sin P sin t 1 — «i ' fti being the number which enters into the computation of A D. Hence , _ t _ [9.4353'7] P 1- •»i 1 — fti (11) SSi SPHERICAL ASTRONOMY. This and the last formulee for A <«, A -D, entirely pi'celude the necessity of having recourse to a table of the sines and tangents of small arcs, and possess much uni- formity and simplicity in their application. To get the relative parallax of the moon with respect to the sun, we must use JP — TT, instead of P. If, therefore, JP' denote the value oi p(P — ir), or the rela- tive horizontal parallax reduced to the latitude of the place, we must use sin P', instead of p sin P, in the preceding formul33. The determination of the apparent relative positions of the centres of the two bodies, as well as the augmentation of thfe semi-diameter of the moon, at any time, has now been reduced to a practical and expeditious set of formulse. A series of these apparent positions of the moon, with respect to that of the sun, will trace out her apparent relative orbit; and the contact of limbs will evidently take place when the apparent distance of the centres becomes equal to the sum or differeece of the semi-diameter of the sun and the augmented semi-diameter of the moon. For a distance equal to the sum of these semi-diameters we shall have partial be- ginning or ending ; for a distance equal to their difference we shall have annular y ^^S'nning or ending, when »' | < [<'• * Since the hour angle of the bodites is subject to the rapid variation of nearly 15° per hour, the effect produced by parallax will be of so irregular a nature as to give a decided curvature to the apparent relative orbit of the moon. Tliis curva- ture will be more strongly characterized when the eclipse takes place at some distance from the meridian or near to the horizon ; and the apparent relative liourly motion of the moon, even during the short interval of the duration of the eclipse, will, through the same irregular influence, experience considerable varia- tion. These circumstances will, in some measure, vitiate any results deduced in the usual manner, by supposing the portion of the orbit described during the eclipse to be a straight line, and using the relative motion at the time of apparent conjunction as a uniform quantity. The method we are about to pursue is very simple, and consists in assuming any time within tlie eclipse, and computing for this time tlie relative positions and motion of the bodies, and thence finding, with- out any reference whatever, either to the time of the middle of tlie eclipse or to the time of conjunction, the times of beginning, greatest phase, and ending, and the relative positions pf the bodies at these times. The nearer the assumed time is to the time of the greatest phase, the more accurately will the time of that phase be determined ; and, similarly, the nearer that time is to the time of begin- ning or ending, the more certainty will attach to the determination. To find the apparent relative motion of the moon, we must first determine the variation which takes place in the parallax. For this, take the equations (2), p. 379, viz.: , sin P' cos / . , , sm A o = sin A A = k— sm '* > cos I) sin A -D = sin P' I sin I cos D' — cos I sin J)' or, substituting small arcs instead of their sines. Til cos Z . , , A a=P irSin A', cos U , cos (h-}- i A h) cos -J A A ]■ i)=pT ■ 1 n 7 ■ n, COS (A + J^aA)-] sm I COS J) — ^ cos I sin JJ' ■ ; ; — COS i a A J APPENDIX XI. 385 Since a portion of the apparent disk of the moon is projected on that of the sun, the apparent declination D' can differ Tery little from i. As the hourly variations of these small quantities are only required approximately, ve may therefore use i instead of 2)' and neglect A h, so as to have Aa = P =- sin A, cos J) A i) = P' (sin I COS S — cos Z sin i cos h) ; 'vhich expressions, though rough values of A a, A D, will give their hourly varia- tions pretty accurately. For these, observing that h is the only quantity which, by its rapid variation, has any sensible influence on these values, we have by differentiation, — i-; — - = 1 1" -T- sin 1 I K cos h, dt \ dt / cosi) ^ -^ — • = IP'-— sin 1' I COS I sin o sin A. dt \ dt / But by the equations (9), m = [4.68556] F' cos I, Substitute, therefore. _ , „, COS I , n = [4.68565] F =- cos A. cos /> pi ^^iL cog A = [6.31445] n, cos jD and we get P' cos/ = [6.3 1446] m; 1^) = [5.31445] (If sin !")„, 1^—^= [5.31445] (^ Bin 1") m sin i sin h. If we adopt 14° 29' as a mean value of -7-, we shall have -j- sin 1" = [9.40274], and [5.31445] (-^ sin 1") — [4.71719] or [4.7172]. Therefore, if (S), the value of the sun's declination at the time of the middle of the eclipse, be adopted in tne value of -^-j — -, we may form the constants, dt <3. = [4.7172], ) , C, = [4.7172] m sin (a) J ^ ' and then, using A oi, A Di in place of , , , we shall have A ai = Qi «, ) /jj,\ A 2>i = Qj sin /i J which offer a simple calculation. 25 386 SPHERICAL ASTRONOMY. Let now, at any assumed time within the Kg. 10. duration of the eclipse, S and M be the ap- parent positions of the centres of the sun and moon ; and S M E an are of a great cir- cle coinciding with the relative direction of the moon's motion at that time, which arc we shall first adopt in place of the curvilin- ear orbit actually described. On the circle of declination 8 N, demit the great circle perpendicular Md, and suppose B and E to be the positions of the moon at the respective times of partial beginning and ending of the eclipse, and n the middle point. Assume/S.B = jS .£•=«' + IT = a', 8 d= x, d M = y, S M = W 8n=:n, /.N8M=z8, LSMd— I ; then tan {(D) — i)] = [(-D) — D] sin 1" ; sin- = -sin 1' = (30 sin I")". 2 2 Therefore, by substitution, we find (B) — D = (900 sin 1" sin 2 D) a'. Consequently, assuming F= 90000 sin 1" sin 2 i) = [9.63982] sin 2 2), we shall have a corr. = (!)) — B-F. (^ . rhe Tnlue of J!*, argument B, is contained in the following small tkU* APPEN^DIX yi. 387 Factor F for o correction. D Jf i) F D F o O •GOO 10 •149 20 .280 I • oi5 II • 164 21 .292 3 • o3o 12 .178 22 .3o3 3 •o46 i3 ■ 191 23 .3i4 4 • o6i i4 •205 24 .324 5 • 076 i5 .218 25 .334 6 .091 16 •231 26 •344 7 • io6 17 •244 27 .353 8 . • 120 18 • 256 28 • 362 9 • i35 •9 .268 29 .370 lO .149 20 • 280 a corr. in seconds — F . (— ) a denoting the number of minutes it contains. From what has preceded, it is evident that a -^a — A 1, is the apparent dif- ference of the right ascensions of the bodies, and that D' ^=D — L. D is the appa- rent declination of the moon ; and that a; = [D' -f (a — A «) corr.] —i\ y ^ [a — A a] COS D' ) and consequently also 1 i)' J • • (14) (15) y\ = (ai — A ai) COS . Moreover, the figure occupying so small a portion of the sphere, and being com- posed of arcs of great circles, we may, without any appreciable error, treat these arcs as straight lines ; thence we shall obviously have tan 18= -, X W- sin S cos 8 cot. = ?^. Hourly motion in the orbit = -=^ n=JFcos(>S+0. Again, in the triangles B SM, E S M, I B8M=<^ + iS+,), and consequently, by plane trigonometry, W .B jr = -^^ sin [« -f (5 + .) ], cos u cos w = ' (16) Z£!SM = u, — {B+i); w coa u nM=WBin(S+i), 388 SPHERIJAL ASTRONOMY. With the above hourly motion in the orbit ive shall therefore hare Time of describing - 5 jj/ = Jl£2ii siu [» + (,S + .) ], «i cos ui ■• "■ ' ' J' IF COS I . ,™ , . n M = sm (S+i), EM= y^J^LL Bin r„ _ (SJf ]. yi COS w ■■ > ' ' ■' Let, now, U, ti, be corrections to be applied to the time assumed to get the times of beginning and ending, and (f) the correction for the time of the greatest phase. Then we have evidently { ti ) (SM\ fnegative^ .< (() V = the time of describing ■< « Jf t with a < negative V ( ^ ' and then we shall derive h = c&m [— (S+i) — "], <3 = c8in [— (S + O + u], (<) = c cos u sin [ — (;Si + 1) ], and hence ( beginning \ f c sin [ — (S + ■) — "] ) The time of .J greatest phase > = assumed time + ■< e cos u sin [ — (S + <) ] V (IS) ( ending ) Ic sin [ — (/S + + "] 3 It has been observed, that any one of these values will be the more to be de- pended on the more nearly it approximates to the assumed time. Thus, if the assumed time be within ten minutes or so of the end of the eclipse, the point M will approximate so closely to the point E, that no sensible error can arise by supposing the small portion M M oi the orbit to be a straight line, and to be passed over by the moon with a uniform motion. This circumstance renders it advisable, in the first instance, to take the ' assumed time near to the time of the middle of the eclipse, so as to give a good result for the time of the greatest phase, and results for the times of beginning and ending, which may be nearly equally relied on. Such a computation will be sufficiently exact for the usual purposes of prediction. When the time of beginning or ending is wanted to great minute- ness to compare with observation, it will only be necessary to repeat the operation for a time assumed as near as convenient to the first determination, which will mostly give within a fractional part of a second of the true theoretical result ; a degree of accuracy, however, seldom wished for, and quite unsupported by the present state of the lunar theory. To fix on a time near.to the middle of the eclipse for the radical computation, one of the most simple expedients wiU be to determine roughly the time of the apparent conjunction. APPENDIX XI. 389 We shall now briofly consider the apparent positions of the moon, as related to '.be sun's centre. It is clear that S is the angle of position of the moon's centre from the north towards the east, at the time assumed ; also that the angle N'S £ = a ■{- t is the similar angle of position from the north towards the west at the time of begin- ning; and that the angle JV^/S jF =; o — (is the angle of position from the north towards the east at the time of ending ; and that the angle SfSn = iia the same angle towards the west at the time of the greatest phase. Therefore, by estima- ting all these angles towards the east we shall have f beginning 1 r (_ ,) _ , At ■< greatest phase > Z of ]>'s centre from N. towards E. = <(— i) V (19) .( ending C(-')-") In the computation of the parallax in declination, we find an angle M, which in practice may be supposed to be the angle N 8 Z for the assumed time, the zenith Z being reckoned towards the east ; conseqijently, at this ti(ne we shall have S — M for the angle of position of the moon's centre from the zenith towards the east. At any other time the parallactic angle .Jf for the latitude of Greenwich maybe taken from the following table, arguments the corresponding apparent time and the sun's declination. This table, for any other place, may be computed by for- mulae, such as at page 881, viz. : , !■ tan 9 = cot I cos /*, tan M = ——. — rr tan A, cos (9 + ") A being the angle answering to the apparent time. Those who may be engaged in the computation of eclipses, for any particular places, will find considerable facility in the formation of similar tables. For an occultation of a star by the moon, the argument, instead of the apparent time, will be the star's hour angle, or the sidereal time minus the star's right as- cension. In this case the required positions will be those o£ the star with respect to the moon's centre, which will therefore be different from the angles of position for a eolar eclipse, in which the moon's centre is referred to that of the sun. The angular positions of the contacts at immersion and emersion will consequently be determined in the same way as for an eclipse of the sun, and will be estimated in the opposite directions. Thus, for an oocultatioh, Ati i'""«':"''° I Z of * from N. towards E. = ] [J^O"-.) - .. ) (emersion J ^ j (180 — i) + ui) And so must 180° be applied to the other angles of position, as expressed for a solar eclipse : this will make the expressions for the direct images of occultationg the same as those for the inverted images of eclipses of the sun, in estimating the contacts either from the north point or from the vertex. 390 SPHERIUAL ASmOlfOMT. Parallactic Angles for the Latitude of Greenwich, {mme sign as h) Arguments : Apparent Sour Angle and Declination. Hour Angle h. Dec North. 10 20 3o 40 5o bo 70 80 90 100 110 120 i3o 140 o D O 8 i5 22 27 3i 35 37 38 39 38 37 35 3i 27 I 8 i5 22 27 32 35 37 38 39 38 37 34 3i 27 2 8 16 22 28 32 35 37 38 39 38 37 34 3i 27 3 8 16 22 28 32 35 37 38 39 38 36 34 3i 26 4 8 16 23 28 32 35 37 38 39 38 36 34 3i 26 5 9 16 23 28 33 36 38 39 39 38 36 34 3o 26 6 9 '7 23 29 33 36 38 39 39 38 36 34 3o 26 7 9 '7 24 29 33 36 38 39 39 ?s 36 34 3o 26 8 9 17 24 29 34 36 38 39 39 38 36 33 3o 25 9 9 •7 24 3o 34 37 38 39 39 38 36 33 3o 25 10 9 18 25 3o 34 37 39 39 39 38 36 33 3o 25^ II 9 18 25 3i 35 37 39 39 39 38 36 33 29 25 12 10 18 25 3i 35 38 39 40 39 38 36 33 29 25 i3 10 '9 26 3i 35 38 39 40 39 38 36 33 29 25 i4 10 "9 26 32 36 38 40 40 39 38 36 33 29 25 i5 10 '9 27 32 36 39 40 4o 39 38 36 33 29 24 i6 11 20 27 32 37 39 4o 40 4o 38 36 33 29 24 17 II 20 28 33 - 37 39 40 4i 4o 38 36 33 29 24' i8 II 21 28 34 38 40 4i 4i 4o 38 36 33 29 24 '9 11 21 29 34 38 40 4i 4i 4o 38 36 33 29 24 20 12 22 29 35 39 41 4i 41 4o 38 36 33 29 24 21 12 22 3o 36 39 4i 42 42 4o 39 36 33 29 24 22 12 ?3 3o 36 40 42 42 42 4i 39 36 33 29 24 23 i3 23 3i 37 40 42 43 42 4i 39 36 33 29 24 24 i3 24 32 38 41 4^ 43 42 4i 39 36 33 29 24 25 i4 25 33 38 42 43 43 43 4i 39 36 33 29 24 26 i4 26 34 .39 42 44 44 43 42 39 36 33 29 24 27 i4 26 35 40 43 44 44 43 42 39 36 33 29 24 28 i5 27 35 4i 43 45 45 44 42 40 37 33 29 24 29 if. 28 36 4i 44 45 45 44 42 40 37 33 29 24 By subtracting the parallactic angle, for the respective times of beginning, greatest phase, and ending, from the foregoing angles of position of the moon's centre from the north towards the eas';, we shall evidently obtain the same angles from the zenith or vertex towards the east. li, however, the operation be. repeated for the accurate determination of the times of beginning and ending, we shall have in the calculations the angle M also at these times. Let ii, ui. Mi be the angles appertaining to the beginning, and <2, U2, M2 those for the ending, and we shall evidently have the following values, which will be more accurate than the preceding : APPENDIX XI. 391 Parallactic Angles for the Latitude of Greenwich. {lame sign as h) A rguments: Apparent Hour Angle and Declination. Hour Angle h. Dec. South. 10 20 3o 40 So 60 70 Bo 90 100 110 120 l3o 1 40 o O 8 i5 22 27 3i 35 37 38 39 38 37 35 3i 27 I 8 i5 21 27 3i 34 37 38 39 38 ■57 35 32 27 2 8 i5 21 27 3i 34 37 38 39 38 37 35 32 28 3 8 i5 21 26 3i 34 36 38 39 38 37 35 32 28 4 7 i5 21 26 3i 34 36 38 39 38 37 35 32 28 5 7 i5 21 26 3o 34 36 38 39 39 38 36 33 28 6 7 M 20 26 3o U 36 38 39 39 38 36 33 29 7 7 14 20 26 3o 34 36 38 39 39 38 36 33 29 8 7 14 20 25 3o 33 36 38 39 39 38 36 34 29 9 7 14 20 25 3o 33 36 38 39 39 38 37 34 3o lO 7. 14 20 25 3o 33 36 38 39 39 39 37 34 3o II 7 14 20 25 29 33 36 38 39 39 39 37 35 3i 12 7 14 20 25 29 33 36 38 39 40 39 38 35 3i i3 7 14 19 25 29 33 36 38 39 40 39 38 35 3r i4 7 i3 19 25 29 33 36 38 39 40 40 38 36 32 i5 7 i3 '9 24 29 33 36 38 39 40 40 39 36 32 i6 7 i3 '9 24 29 33 36 38 4o 40 40 39 37 32 17 7 i3 '9 24 29 33 36 38 4o 4i 40 39 37 33 i8 7 i3 •9 24 29 33 36 38 4o 4i 4i 40 38 34 •9 7 i3 19 24 29 33 36 38 4o 4i 4i 40 38 34 20 7 i3 '9 24 29 33 36 38 4o 41 41 41 39 35 21 (1 6 i3 19 24 29 33 36 39 4o 42 42 41 39 36 22 6 i3 19 24 29 33 36 39 4i 42 42 42 40 36 23 6 i3 18 24 29 33 36 39 4i 42 43 42 40 37 24 6 i3 18 24 29 33 36 39 4i 42 43 43 4r 38 25 6 i3 18 24 29 33 36 39 4i Ai 43 43 42 38 26 6 i3 18 24 29 33 36 39 42 43 44 44 42 39 27 6 i3 18 24 29 33 36 39 42 43 44 44 43 4o 28 6 12 18 24 29 33 37 40 42 44 45 45 43 41 29 6 12 18 24 29 33 37 40 42 44 45 45 44 4i ( beginning For -J greatest phase ( ending [ Z of 5 's centre from N. towards E. Z of J> 's centre from vertex towards E, C ( — ") — "1 — ^') = i{-,)-M > (20) '( — '2) + "a — -*^!* These angles relate to the natural appearance or direct images of the bodies. For the same angles, as they will appear through an inverting telescope, ± 180° must be applied : this may be simply done by using (180° — 1) instead of ( — 1). 392 SPHERICAL ASTEONOMY. To find the time when the apparent conjunction takes place, let t denote the - interval, in units of an hour, to be applied to the time of the true conjunction, and h the common hour angle of the bodies at the true conjunction. Then the position of the sun, not being supposed to be influenced by parallax, the common apparent hour angle of the bodies, at the time of the apparent conjunction, -will be A' = A + 15° . < ; and therefore at this time, a-n,t, A a = Cp' ^^) sin (A + 15° . t), \ cos JJ/ ' 80 that the conditi n for apparent conjunction, viz. a' = a — A a = 0, gives for the determination of the interval t, which from this equation will be best found, perhaps, by the usual method of double position. We only want, however, an ap- proximate value, and may therefore avoid much unnecessary labor in estimating this time. Thus, at the time of true conjunction, the same approximate formulae may be adopted as used at page 385, viz. ; _, cos I . , i^ a := F — sm h, cos JJ T^, / .+ a' corr.) — I o cos Z)' . D = i ■ ; — cos 1 ; — em c ^ a' A k k g=: — ; [(i' -f- "' corr.) — i] sin i-{ — -a cos D' cos I A D A a COS D' . A » = — r co^ ' n ^"' ' A A k k A y = — ; A i> sin 1 -| ; A a cos i)' COB 1 (*) (5) and, observing the above values of x and y, the equations (2), (3), will beeome u =zp — Ap, t = T--^k Bin a cos w =:p — Ap, I . . (6* 394 SPHERICAL ASTRONOMY. Let ^, tp be determined by the equations Y COS 1^ — (^ + °' " O" ) — ' a COS D' y sm li = ; and j>, g will take the following values : p= yoo8(i;' + ) q = k Y Bin {if, -{■ i) J ' . (?) (8) It y^et remains to determine the values of Ap, A g, which depend on the po- sition of me place of observation. Adopting the notation used in the equations (3), (4), (9), (10), pages 379 and 382, we shall have [5.31439]^ coal . , A a = i^- ^— . =5 sm A, 1 — » cos p „ [5.31439]^ r. , _ , . „cos(A+iAfln Al> = ^ --■ — I sm i cos i) — cos / sin i) ^ ' '- I. 1 — »i L cos -J A « J To simplify the expressions, let [5.31439]^ cos J' ~ (I— ») V" ■ eoaj)' [5.31439] A „ c = J- -^ . cos D, (1 — «i)a' [5.31439] -4 . „ (1— «,)A' ' and A"" = - b A ' cos 1 sin A >D' A J) = c A' Bin I — a A ' cos i cos (A + i A a) cos i A o Aa ^e a' sin Z — a a' cos I cos h-\-a A' tan —- cos I sin h. J* These substituted in (5) give A p =: c cos 1 sin Z — cos l\ a cos i cos h — ( a cos i tan — i sin i) sin h I A ; = £ c sin I sin / — cos M ^ a sin i cos k — (£ a sin i tan -— -{-kh qob i) sin h I cos D' The Value of b contains the factor =■, for which we have cos D cos D'. =- = cos A i> (1 + tan J) tan A D). cos D ^ ' Substitute the first value of tan A D, p. 379, and cos D' cos D = cos A J5 . . „ cos Z 1 — p sm P Tjcos (A) cos jD ' 1 — p sin P [sin I B\n J) ■\- cos /'cos I> cos (A)]" Or, putting h instead of (A) in the numerator, which cannot sensibly affect the value of the fraction, cos D' _ 1 — n = cos aB . cos J) 1 «! APPENDIX XI. 395 This, eupnosing cos aJ) = 1, reduces the values of the constants a, b, c, to the following: J _ [5.31439] A • 1 ~{1— ni) A' I (9) e = bco3D; a = b Bin D ) If « be a small arc determined by ^ cos e = &, ^ sin « = a tan — -, we shall have A o a cos I tan — 6 sin < = ^ sin ( — • + e) = ^ cos (90° + i — e) ; A a ia sm e tan — + kb cos i = kg cos (< — «) = kr; sin (90° + t — e). ' However, as e must always be a very small arc, we may suppose cos e = 1 also g = b, and, e being expressed in minutes, ' = ^-F--T^ = r2f6-^'- = [''-»208]Aasin,i). . . .(10) If therefore X = (90"' + i) — « (11) the values of Ap, A g, will be m Ap ^ c cos J sin i — cos I (a cos i cos h — b cos x sin h) ) A g ^kc sini Bin I — coa I (ka sin i cos A — fc 6 sin x sin A) ) ' Assume now A := the longitude of the place, + east, — west. H= the true hour angle of the moon, for the meridian of Greenwich. L' ^c cos I T y' cos (>/.' — H) = a cos I V (18) / sin ( and we shall have A ^ ^ i' sin / — y' cos / cos (t//' + a — H) = L' sin I — y' cos / cos {y\i' + A), Ag = L" sini — y"co3 I aoa {^" -\- h — H) = L" aia. l — y" coal cob { — Z" Bin I + y" cos I cos (i//' + >), will take the forms (' [sin I' sin i + "os '' "os i cos (lA' + X)J, f " [sin I" sin Z + eos I" cos Z cos (i//'' -j- A)] ; and, without the factors (', (", will represent the cosines of the distances of the proposed place from two other places whose latitudes are I', I", and west longi- tudes 1^', il/". The former of these two places will be near to the southern pole of the true relative orbit, and the latter will be near to the orbit itself, and will pre- cede the moon by a distance nearly equal to 90°. For purposes which do not require great minuteness, the preceding equations will admit of some simplification by neglecting the small angle c Add the squares of the equations (13) and (14), observing that ff' + a' = b\ and Z" -1- y'' = i" (cos" 1 -f cos" x), X"= -f y'" = P f (sin" c + sin" %) ; ' which give the general relation Tin yll :2 6' By neglecting e, x = 90" -J- i, cos x^ — sin i, sin x ^= c°s ' > *°d then Z" + y" = i", r"+Y"' = *"*'; which united with the equations (16) give (' = b, (" =z kb, and hence sin V = — Z' sin (i//' - H) = i' b (' cos P = b cos I' ; b cos X 6 sin I y' 6 eos I' = — cos I> cos I ; cos I' . „, Z" i c sm t „ . sm f = 177 = 7-, — ■ = — COB J) Bin i : f kb Bm{'^J' — S) = y" = i" COS I" ; fc i sin X ■.kb COS I"; k b cos « COS I Or. y" iioosT sin ? = — COS D cos i i' = — 6 sin V ; y' t= 4 cos V ^^ ' cos / sin V — — cos i) sin i COS I (11) (18) (19) APPENDIX XI. 397 These may be employed instead of the equations (18) and (14) ; or the equations (18) and (14) may be adopted in their reduced form, viz. : (20) -— = cos 2) cos 1 — cos (t//' — -ff ) = sin J) cos « v' — . sin {\^' — S) = — sin 1 Til — — r= cos D sin ( kb ■^ cos {xp" — H) = sin D sin , ^sin (.//" — J?) = cos 1 kb (21) Id which the coefficients c, u, will not be required. III. — ^TKANsrrs of Mkrockt and Venus oves thb Disk of thb Sun. These phenomena are, in many respects, analogous to tljat of an annular eclipse of the sun, and admit of a similar calculation ; the principal distinction consists in the negative sign of the relative motion of the planet in right asceneiou, which will make the inclination of the orbit always obtuse, and therefore render some modifications necessary in tlie determination of the particular species of the other angles which enter into tlie computation. To avoid any confusion that miglit thus arise, we shall adopt the sun as the movable body, and refer his positions to that of the planet which we now suppose to be stationary. Thus, i =: the O's declination; D = the planet's declination ; T = the G's equatorial horizontal parallax; P = the planet's equatorial horizontal parallax ; o = 's right ascension minus that of the planet ; X =(i' + al C0TT.)—D; y = o' cos I' ; Xi == the O's motion in declination minus that of the planet ; yi = ( O 's motion in right ascension minus that of planet) . cos i' ; and so we might proceed as with an eclipse of the sun, only observing that the relative parallax p (t — jP) is a negative quantity, and that the positions of the contacts on tlie limb of the sun, as in the case of an occultation, will be at points opposite to those which come out in the calculation. However, as the relative parallax is always very small, the ingress and egress of the planet will be seen at all places on the earth at nearly the same absolute time ; it will, for this reason, be best to compute first the circumstances for the centre of the earth, and then to ascertain the small variations produced by parallax for any assumed place on the surface, which may be readily deduced from the preceding equations for the reduc- tion of an eclipse of the sun. Let w, (t), be the values of u, t, for the centre of the earth, and, by separating the effects of parallax from the equations (6), 398 SPHERICAL ASTKONOMT. cos vr =p, (t) = (T— g)7k sin yr, A cos-w= Ap, At = — A J T i A sin w. But, as the quantities A cos w, A sin w are very small, Asinwr^— Acosw- that IS, A Bin w = -^ A p -: . Therefore, • -^ sm w COS w At — — Aq ±K Ap -. : (c lAp- 6 ■T A? )• In this ezpi-ession substitute the yalues of A p, A g, according to the equation;. (12), and we find A < = r oos[— iTw] . / C08[— iTw] eos[— xTw] . \T ±1 *«- — -. -Bial—aoBlllea =; ^cosA— «o =; = sm A) J, L sin, w \ sm w sin w /J in wJiich 6 = ^-^ = — ~ , c = J ccs i and o = 6 sin J. A A Because of the smallness of the parallax, the angle ewill not be appreciable, an! consequently x = 90° + 'i coa [ — ^ -f w] = sin [ — i T w]. We shall thcrtfurc have for the time of ingress or egress the following general expression, in wliioh the terms within the brackets depend on the position of the place of observation ; also the upper signs apply to the ingress, and the under signs to the egress. t = T — ; =F ^ sin w sin[— iTw] . ^TiT iC08[— jTw] . /. cos[— iTw] s Tkbl cobS -. ^ sin Z— I Bin a- — ■^ ^cosA— - I_ sin w \ sin w i/tlcos{ I ■kb Assuming k" = — : , this expression will resolve into the following : tan I : S' k = [3.56630] A cos I y cos l// : y sin ip = {S-\-a corr.) — D I cos S cos w = Y cos (li + i) J = fc y sin (i^ + ') (0 = T— y T * sin w A sm w 1 (») (6) W L" — ^ cos [ (— «) T w] cos i y" p; cos (i//" — fi) = cos [ {— i) T w] sin S ■ r^»m( >'-S =Bin[(-.)Tw] t = (<) T [y" l> cos / cos (;//' + A) — Z" p sin ] (d) («) APPENDIX XI; 399 In these equations, ff r= the O's true hour angle from the meridian of Greenwich, at the time (t). For \ «^'«r "■ I contact of limbs, A=\' + H { mtenor ) }c — si For contact of centre of planet with 0's limb, A = tr ; s denoting the true semi-diameter of the planet, and = (#(!" cos S — ,pO) sin I cos /i) . /", A -D. = 0(') . P sin ^ sin A. If, in the values of A o, A a,, we use cos i instead of cos D, the values of x, y, z„ ji, p. 38'7, will become X =(i> — i) — (#!=). P cos a — ^(').P sin i cos /t) ' ^ = a cos i — ^C) . P sin A Ki = J9i — #(') . P sin a sin A yi = a, cos i — 0'') , P cos A in which we have disregarded the a correction. With the values of x, y, xi, yi, so found, we may then proceed with the equa- tions (IG) and (IS), pages 387 arid 388, as in the case of a solar eclipse. This method is similar, and, as far as accuracy goes, the same as the recent method of Professor Bessel, who divides all the quantities by the equatorial hori- zontal parallax of the moon. lie assumes a cos i D — l P' = ' , cos S 1 =- .A sin h. \ (3) P ' " P « = ^C) sin A, «' ^ 0W cos h V ^ ^f^' cos i — ^C) sin i cos A, v = ^O sin S sin i so that if we change the signification of the symbols x, y, Xi, yi, and suppose them now to represent the preceding values divided by P, we shall have x = q — V, yz=p — u, These values being adopted, in proceeding with the equations (16) and (18) we must use A' = -p-, the value of which, according to Burckhardt's Tables de la Jimnt (Paris, 1812), p. 73, is [9.48687]. Much facility is thus given to the calculation pf occultations, for different places, if the values of y, y, p', q', which are indepeii- yi —p' — «' r (5) APPENDIX XI. 401 dettt of goograpliieal posilion, are published; but if these quantities are to be pre- pared by the computer, the equations (2) -will be more simple and advantageous. The chief difficulty in the calculation of oceulta- tions, for any particular place, rests in the selection of the list of stars : in the course of any year a great number will be liable to oocultation on the earth generally, though the majority of them ■will not be occulted at the particular place for which the special calculations are to be made. It will therefore be expedient to reject such stars as may at different stages of the calculation be shown to violate any conditions necessary for the existence of the occulta- tion, its appearance above the horizon, or its exemp- tion from the glare of sun-light For the general list yre may observe, that the difference of declina- tion at the time of conjunction must be within the limit of about 1° 30', and that all stars, whose con- junctions with the moon occur within two days of new moon, may be omitted. In the process of exclu- sion for the particular place, the first and most pal- pable condition is, that at the time of conjunction the sun must be below, or near to, the horizon ; if niDre than half an hour above the horizon, the oocul- tation will surely be useless ; another condition is, that the star must be above the horizon ; and, to satisfy this, the hour angles at the times of immer- sion and emersion must be less than its semi-diurnal arc. The value of the hour angle at the time of apparent conjunction may be determined by increas- ing that at the time of true conjunction by the quan- tity - SO -, according to the' .tables on pages 401 and 402 ; and it may be ojiserved that this hour angle must not exceed the ?emi-diurnal' arc by more than half an hour. For the latitude of Greenwich, the semi-diurnal arcs, allowing 33' for refraction in tile horizon, are shown in the annexed table. As a final test for the exclusion of unnecessary stars, it is useful to calculate the extreme limits of latitude between which the star will be visibly oc- culted on the earth. These -will evidently appertain to the extreme northern and southern points of the northern and southern limits of contact,' determined as for a solar eclipse, a point in the northern or southern limit will depend on the formula Nos. 27, 28, pages 359-60. Thus, ^ _^ ^, cos W =: =r: . ^ Deo. Semi-diurnal Arcs, for ' the Latitude or of Greenwich. Star. J?ec. NoHh. Dec. South. a h. m. h, in. 6 4 , 6 4 , I 6 9+ 5 5 559- 5 54 ^ 2 6 i4 3 6 19 5 5 5 5 5 5 6 5 5 549 I 543 I 5 38 ^ 5 33 ^ 4 6 24 5 6 29 6 6 34 7 6 39 5 28 5 8 6 A4, 5 23 I 9 ID 6 5o 6 55 5 18 ^ 5 i3 5 5 V I II 7 12 7 6 6 5 2 5 l3 i4 i5 7 " 7 "7 723 5 6 6 5 6 6 7 6 7 4 56 % 4 5. g 445 4 40 . 434 4 22 i6 728 17 734 i8 19 20 21 7 4o 7/7 7' 53 8 22 8 6 6 ii'^ 23 8 i3 7 24 8 21 8 3 49 I 341 ^ 25 8 28 7 26 8 36 8 334 I 326 I 3 18 ^ 27 8 AA 8 28 8 53 9 29 9 = , 9 3 9 9 3o 9 12 ' 10 3 0-9 P' e\n Z = ■ M= — I ± J; and thence, sin I = sin i)' cos Z + cos D' sin Z cos M. 26 402 SPHERICAL ASTRONOMT. It is now oar object to aecertain what Talue of u' will render the value of I, so deduced, a maximum or a minimum, and what will be the corresponding value of /. Let ^ be an arc determined by the equation, cos 2 = cos sin w (6) Then by uniting with it the equation cos m' sin 2 ^ cos w . . . . . . (V) we infer that sin u' sin 2 = sin ^ sin w , (8) because the squares of these three equations added together will give um cos w 7 sin i sin ^ sin w ; and, consequently, sin / ^ cos jy cos i cos w + sin w (sin D' cos ^ T cos 2)' sin i sin ^), which now involves only one variable 0. Again, assume two arcs, 8, iji, which will fulfil the equations, cos 8 cos \l/ = Bia S' (9) COB sm\p ^ ± cos i>' sin I (10) A third equation will follow from these, viz. : sin fl = cos i)' COS « (11) because, as before, the squares of these three equations will together make ttnity. The value of sin I will now become sin 2 = cos w sin e 4" sin w cos fl cos (^ + \f/). The angle jS + i/- being the only variable in this expression, it is evident that the greatest value of I will have ^ -|- 1//^ 0, and the least (p-\-\(/^ 180°. Therefore, feTt" * } -lue ofl=\l + Z\, nsing w for { -f-„ } limit Thes^ would be the extreme latitudes for the appearance of the occultation if the earth were a transparent body ; as this, however, is not the case, it will be neces- sary that the star should be above the horizon, a condition not included in the preceding equations. The zenith distance Z must not exceed 90°, and therefore cos Z must necessarily be a positive quantity. By the equation (6) cos Z must have the same sign as cos ^, and this must be the same as -)- cos for southern limit, because in the former case "•' "^ ^^ ^^^ same sign as j ^ n,' It is evidept, therefore, that the extreme northern limit will have the star below the horizon, and be excluded when J)' is negative, and that for the same reason the soutliern limit will be excluded when D' is positive. Thus the only admissi- ble extreme limit will be determined by the equations cos w = " ^,^ . 7, = fl±w . ... (12) usmg upper signs when D' is positive, and under signs when J)' is negative. The other limit for the actual appearance of the occultation will evidently; be on» APPENDIX XI. 403 of the two places where the other liniiting line meets the rising and setting limits, and will be determined by n T A' sin h = cos D' cos [ (— ') T w] (18) using, as before, upper signs when D' is positive, and under signs when Df is neg- ative. The equations (11), (12), (13), for convenience in determining the species of the angles^ may be put in the following form : cos wi : T n— A cos Wa = T «+ A' :i (14) I" • '~ J" sin fl =: cos D' cos i ii = w, — fl sin Za = T cos J)' cos (Wj — i) observing that Wi, Wa, 0, and i, must here take the same sign as JO' ; also, 3:jfaignswheni,'is|Pj[[v^ These formulae arc applicable to a solar eclipse. For an occultation of a star by the moon, P' will be the moon's horizontal parallax, and a' her eemi- diameter, which, as these limits are not wanted very accurately, may be regard- ed as true quantities ; also, we may neglect u and so take i instead of D'. Sinca t ■■ [9.4358'?] = .2725, the formulae for an occultation will hence be tan I = - ai cos i T;^-.2725, n = (diff, dec.) cos i cos Wa= i: -- -I-.2726 (16) sin 9 = cos contact with penumbra, A ' = semi-diam. penumbra ± ». THe angular positions of the points where the contacts take place will be esti- mated on the circumference of the shadow or penumbra the same as they were before on the limb of the sun. These angles will therefore be in a reversed posi- tion on the disk of the moon, and consequently as they come out from the compu; . t^tion will have reference in the first instance to the inverted appearance of the p^lise. ■ The relative orbit of the moon, not being affected with parallax, will not sensibly deviate from a great circle in the course of the eclipse ; and hence the assumption of ,the particular time, on which to found the calculation, Avill be but of little im- portance: any convenient time may be assumed near the time of opposition. It will be unnecessary to add any further remarks. We shall conclude this pa- per with a tabular recapitulation of the formuliE which relate to the phenomena for a particular place, in which eclipses of the moon, for the sake of clearness, are given separately. The object of this ta')le, like the former one for the general eclipse, is to simplify and expedite, by an sasy reference, the actual operations of the computer. APPKKDIX XI. 405 I. ECLIPSE OF THE SUN FOE A PAETICULAE PLACE. I. h = apparent time of tcue cS in K A. to nearest minute. j With this as an argument^ take ont the numbers 6, £">, from the following table : i Table for reducing the true to the app. d in B. A. S «(1) 6 / ff, the eclipse ■wiU be total if «" > », or annular if »' < » : in this ease these last equations No. 7 must be repeated for this phase with A' = s* .~ », the results of which ought to give the same time for the greatest phase. Take A' for partial phase, and Portion of sun's disk eclipsed := A ' — n. a' — n Magnitude of eclipse = — — , the sun's diameter being unity. it V 8. For the positions of the points of contact on the limb of the sun, ^t \ egJDi'Dg ( _ angle from north towards east = l ; 'i jT " [ ''"■ <^'''««' image. At \ ^^'f^^^ \ , angle from north towards east = j i^^°! " '^ 7 " I H '""'"""^ I ending ) ((180° — 0+»S image. For the position of the moon's centre at greatest phase, Angle from \ °°' i towards east = ■} , '/ ,, Mor direct image. < vertex ) ( ( — •) — M } Angle from ■< °' f- towards east =■) ;. . V ,^ > for inverted image, ^ I vertex J ( (180° — i) — if J ^ 408 SPHERICA'L ASTEONOMT. 9. For a more accurate calculation of the time, &c, of beginning of the partial phase, assume a convenient time near to the preceding determination. For this time, talce out the quantities D, Di, I, a, a^, from the Ephemeris ; and proceed as in Nos. 3, 4, 6, 6, 7, omitting b, ti, and the times of greatest phase and ending. Let Ml, (i, ui, be the values of the angles in this computation ; then, for the po- sition of the point of contact on the limb of the sun, Angle from \ I towards the east = \) ''; "' ,, Kor direct image, (vertex) (( — i,) — un — Mi) inverted (180" — ii) — Ml — J/i ) image. Angle from \ '""^*^ I towards the east = \ f'f 0° " '■) " "' ,, I ^°'. *""«•' ^ (vertex) I (ISO" — i,) — ui — Mi ) image. 10. For a more accurate calculation of the time, iSec., of ending of the partial phase, assume a convenient time near to the first determination. For this time, take out the values of D, iJ,, i, a, a, ; and proceed as in Nos. 3, 4, 5; 6, 7, omitting a, ti, and the times of beginning and greatest phase. ^ Let M„ 12, U2, be the angles in this computation ; then, for the position of the point of contact on the limb of the sun. Angle from \ "^^ , t towards the east => ■} > , , a cos J y cos i// = ^ , y sm i/- = — — -, cos w = y cos(i// + i), q = kysi.n{;/" — Xf) = sinicos[( — <) T w]; ^ sin W - fi-) = sin [(- .) T w]. lY. Then, for the centre of the earth, (<) = (y— y) T Ajsin w; and, for any place whose latitude is I and east longitude A, ( = (<) T [y" p cos I cos (A + i/-") — X" e sin /], using the upper signs for the ingress, and the under signs for the egress. The positions of the points of ingress and egress, estimated from the north point of the sun's limb towards the east, as the transit would be seen from the centre of the earth, will be determined in the same manner as for the immersion and 410 SPHERICAL ASTRONOMY. emersion of an ocoultation, No. 19, using -w^ for u. These angles may be assumed to be the same for any place on the surface, the effect of parallax being so Tery .minute. IV. OCOULTATION OF A STAK BY THE MOON. Oenebil Limits of Lathude. 18. (a, and Di at true c5)- tan c ^ — , n = (diff. dec) cos i, i uj cos i cos Wi = ^ -p — .212S, cos Wa = T -p- + .2125, ■ sin B = cos i cos i, ii = Wi — e, sin 1, = ^ coai cos (ws — i), Vi, Ws, I, 9, same sign as I, nKh'sn.-hen^isjP^. When Wi is impossible, li = 90°, with the same name as i. When W3 is impossible, h ^ complement of i, with different name from t, CALCCtATIOH FOR FASTICTJLAR FlACE. 19. For the latitude of the place prepare the constants ^0)=,co8;, fi)=piml = ~-, #1^1 = [9.41916] «('), which will serve for all bccultations at that place. For the time of true i find h = sidereal time at place — right ascension of star ; and thence determine the. time T, as in No. 1. For this time talce out the quanti- ties jP, s, D, Diy a, oi ; and compute ^ X =(]> — l) — (0(2) . P cos a — 0(1) .Paint cos h); y = or cos I — 0(') P sin A ; -c, = i), — 0(3) . P sin i sin A ; yi ^ o, cos I — 0(') . P cos A. ~ With these proceed as in Nos. 6 and 1, using A ' = » ^ [9.4858T] P. 20. For the positions of the points of immersion and emersion on the limb of the moon, ^jj immersion | angle from north towards east= \ i]^°°~'}~" lfor 's dee. ; yi = oi cos > 's dec i" = [9.99929] P. Semid. shadow ^— -(P'-J-ir — a), 60 fil Semid. penumbra = — (P' + " — tf) + 2 ». •^'"' 1 i*ntern*l f '^°°'^'' '^'^^^ shadow, A' = semid. shadow ■]_[<• For .j . . , [ contact with penumbra, A ' = semid. penumbra ■)_(■•■ The remaining computation as in Kos. 6 and 7. 24. For the positions of the points of contact on the limb of the moon. At \ '""version ) ,^ ^^^ j^ ^^^^^^^ ^ ^ ( (180° — ')-") for rfir«< imago. < emersion ) ((180° — ■) + ") At \ '«""«';"<'° I angle from H. towards E. = i f~ '| ~ " I for inverted image. ( emersion ) l ( — <) + " ' At the middle of the eclipse, Z cent, shadow from N. towards E. = | (|^*|° ~ '^ I for | ^^J^^^j I image. To get tho same angles from the vertex, the parallactic angle must be deducted for the respective times. 413 SPHERICAL ASTROITOMY. Examples. 1. ECLIPSE OF THE SUN. Let it be required to calculate the circumstances of the solar eclipse of May IS 1836, as it will be seen at the observatory of Edinburgh. The elements of this eclipse are stated at page 362. h. va, 6. Greenwich sidereal time at Greenwich mean noon Longitude 12 43-6 W. Edinburgh sidereal time at Greenwich 1 „ „ i-3 20 i4-4 "1 mean noon J t t 1 Sun's right ascension at c5 . . . . 3 29 25-2 f Hour angle h at Greenwich mean noon — o 910-8 ( Greenwich mean time of (J . . 2 21 22-9 (Acceleration 23-2 ni -/ 'I3 32 58-0 3 .o3 83 -I • 8 + 2 i3 6 + 21 fi(i) + 55 (+ .87 ./— 6+ 63 Greenwich mean time of true c5 «<"-^(«.-/— fi) . . T . . . h. m. 2 21 + 52 3T3 5o-4 4-6 Constants. 54 23-4 8-5 54 i4-9 logP . const. 3-5i367 9-43537 logs i + 2-94904 i5'49<'-9 i8='58'.5 const. A . cos I m sin i 9-99902 4- 68555 4-68457 3-51254 8-19711 9-75001 7.94712 4-7172 9-5i2i Qj + 2.1764 Computation for 3f' 1 3"", Greenwich time. Z) + 19 33 43 J -f 18 58 29 o + 23 49 i), H- 9 19 oi + 27 43 b. m. s. Edinburgh sidereal time at Greenwich mean noon . . . - 3 20 i4-4 is"" o" 3 o 29-6 i3 i3 2-1 6"33 46-1 Moon's right ascension 3 3i 9-0 {time ... 3 2 37-1 arc . . . . +^y^^i APPENDIX XI. 413 ooai> k cos h log . Aai 7-94712 9-97418 const. 5-3i439 • 7-97294 7-97294 + 9-84446 sin A . + 9-85440 . . + -7-81740 oorr. for n . 286 . 4-7172 ( log . - +3-14459 Qa + 2-5346 + 5' 42" + 23' i5" flog . 1 A A + 9-8544 + 2-1764 + 2-o3o8 + i' 47" h . i A • + 45 39-3 + II. 6 + 45 5o-9 9 + 25 20-9 J) + 19 33-7 + D + 44 54-6 M+3i 54-5 ■ , + 4o i4-3 logs {7: v ■ Partial A' . Annular A' 2.94964 444 + 9-84295 . + 9-83256 cos 2 . . + 9-84296 + 9-75001 + 9-67552 + 9-63i56 . . + 9-59297 + 9-85017 B . . + 9-8ii58 + 9-78139 + 0-01286 . check . + 9-78139 + 9-79425 + 9-92885 . + 9-99864 cos < . + 9-92885 + 9-88273 + 9-92749 + 9-88273 B . . const. , + 9-8ii58 + 5-31439 + 9-81022 5-12597 8-19711 . + 8-00733 corr. for ni 8-19711 444 (log 1 A D . 2-95348 . i4' 58"-4 . i5 49 -9 . 3o 48-3 . o 5i -5 A. A A . 3-32752 + 35' 26" + 9 19 + I 47 + 7 32 D , AJD jy. a', corr. i . + 19 33 43 + 35 26 + 18 58 17 ( o (log + 18 58 29 — o 12 cos D' y • „ + 23 49 A a + 23 1 5 0'+ o 34 . +i-53i48 - + 9-97574 ilog- + 1.50722 y, a, + 27 43 A ai + 5 42 + 22 I + 3-12090 . + 9-97574 . + 3-09664 (1) 414 SPHKRICAL ASTROlfOMY. ' ;S. . -f no 28.0. « . . + 19 53-5 — {S + ,) — i3o 21.5 X . taaS cos S W. . COB . n . - log A' COS u c sio a . + 1-50722 . — I -07918 . — 0-42804 . —9-54364 . + 1-53554 . — 9-81129 . — I -34683 3-26677 . —8-08006 y, . + 3-09664 (l, X, . + 2-655i4 cot « + o-44i5o coe 1 + 9-97328 . . -t- 1-53554 const. 3-55630 Partial . u + 90 4i-i + 5-o65i2 (2) ff+ 1.96848 (2)- . . — 8-o8oo0 -(') a — 221 2-6 6 — 39 4o-4 . — 3-88842 . + 9-81732 e —3-88842 sin 6 — 9.8o5io — 3-70574 -f 3-69352 h. m. B. b. m. fl. i, — I 24 39 (i + I 22 18 Assumed time , 3 i3 . . . 3 i3 Beginning . . . . I 48 21 Ending4 35 18^ '^™"|r«='' ^ ( mean times Longitude . . T2 44 W. . . 12 44 W. Partial, Beginning . . . . '35 37 Ending4 22 34{-f-'i-£° n . — 1-34683 Annular . log A' . 1-71181 S+ 1-96848 w + ii5°33'-9 cos u . — 9-635o2 . . — 9-635o2 - (S + .) — i3o 21 -5 c . — 2-33346' c — 2-33346 a — 245 55.4 sin a . + 9-96047 sin b — 9-40711 b — i4 47 -6 — 2-29393 + i-74o57 h. m. B. h. m. s. '. — 3 17 (3 + 55 Assumed time . 3 i3 . . . 3 i3 Beginning . . . . 3 9 43 Ending 3 i3 55 j tt™enw,eh ' " ( mean times. Longitude . . 12 44 W. . . 12 44 W. Anndlar. Beginning . . . . 2 56 59 Ending 3 x „ j -^-^-^ Positions of ConTiicTS fok dibect Image. Partial contact at [ beginning ! ending . ( — —•9-9 u + 90-7 II0.6 ) (west, o rfrom north towards! 70 [ east Annular contact ( beginning * ( ending . (_,) — 19.9 ui + ii5-6 f from : orth towards! 95 ! eaet APPENDIX XI. 415 For the samo angles from vertex we must estimate them towards the east, and deduct the angle M, thus o o Beginning — i35.5 Ending -f-95-7 M + 31.9 M +31-9 167.4 towards west. 63.8 towards east. COMPDTATION FOR l"" 48°", FOR AN AOOURATK DETEKMlIfATIOIf OF PaETIAL BeGISNINQ. D + 19 19 35.9 A + 9 26 I + 18 57 39.3 Edinburgh Sid. Time at Greenwich Mean Noon Sidereal Equivalent for ) ' ° tn . cos D k . cos h n . e. ■ (log • (All h . i Aa (A) . 9 . + 2> Ml 7.94712 9.97481 7.97231 + 9^9571)7 7.92938 4.7172 + 2.6466 + 7' 23" / + 25 3.5 + 6.9 4" 25 10.4 O ' + 3i 36.7 + 19 '9-6 + 5o 56-3 -f- 21 21. 1 + 48 55.9 J) 's R. A. , ( time . A in ■< a — l5 23-2 Hi + 27 38 h. Di. e. 3 20 14.4 I o 9.9 48 7-9 5 8 32.2 3 28 18.2 + I 4° 14.0 + 25° 3'.? const. sin h corr. for n (log. I Ao cos . cot I 5.31439 , 7.97231 + 9-62690 + 9.6269 370 + 2.91730 Qa . . . 2.1764 + i3'46".6 (log. . . + I -8033 1 A A . . + ~'~7' + 9.95666 +9-95666 + 9-83256 tan 9 . sin 9 . cos . + 9-78922 + 9.71946 + 9.79945 tan (It) . tan Ml + 9. 9200 [ . +9.67209 + 9.59210 cos Ml . tan (9 + Z* + 9. 9691 1 1 +0.09068 tan cos . . + 0.05979 + 9-81754 sin , . A . . . +9-87733 + 8.19711 cos I . • +9-75oo[ a . . . +9.70667 B . . . +9.78665 check . +9.92002 const. ■8.07444 corr. for ni log . . aZ> + 9-81754 + 9.78665 5.31439 5.10104 8 . 1 97 1 1 5i8 ' 3-3o333 . +33'3o"-6 416 SPHERICAL ASTEQNOMT log 1 ir. . . 2.94904 5i8 2.95422 . . l5 Q.O . . i5 49-9 A . . + 9' 26" ADi . . + I 4 ai . . . + 8 22 A' . , 3o 49.9 s . . + Al) . + 19 19 35-9 33 3o.6 a Aa . — l5 23.2 + i3 46-6 "1 A a, . + 27 38 . + 7 23 J)' . . + a! corr. . i . . + 18 46 5.3 2-2 18 57 39-3 ■ log . cos ly — 29 9.8 3.24299 + 9-97627 log. + 20 i5 . +3.08458 • +9-97627 X . . — II 3i.8 y ■ ■ — '3.21926 — 2.83998 Xl . . +3 -06085(1) . +2.70070 S . . — «■ . . + ' 112 39'8i 23 34-49 89 ,5.32 89 6-93 t tan S . BinS . W. . cos n . log A'. cos ui . + 0.37928 — 9 ■ 965 1 + 3.25416 -f 8.20168 + 1 .4-5584 +-3. 26715 + 8.18869 cot li COS 1, const. H . . +o.36oi5 - +9-9^2i5 . +3.25416 3-55630 -, . . + 6.77261 (2) . + 3.71 176 (2) -(1) . +8-18869 a . . — I '61 sin a c . (sin I . +5-523o7 6-46373 61 _o. 20683 — 2.19363 ■ ti . . — o'' 2"'36' Assumed time i 48 . Beginning Long. I 45 24 Green' MT. 12 44 W. Partial. Beginning I 32 4oEdin.M.T If the oakulation be repeated for tlie Greenwich time i"" 45"", it will lead to ex actly the same result, which is therefore to the accurate second, according to tha data employed. Position of Contact fok Direct Imaqb. ( — '1) — "I ( — ii) — m — Ml o — 23.6 + 89-1 — 112. 7 + 21.4 ■i34.i The point of contact is therefore ■! ,3 / [■ from \ °_j.gy \ towards west. APPENDIX XI. 417 n. EQTJAllONS FOR EEDTTCTIOIT OF PARTIAL BKGINNINQ. The data for this compatation are taken from the preceding one. 3-55630 5-3i439 7.9208 a' 3'267i5 a . 8.I97II Ao . +2-9173 cos 1 9.962I5 corr. forMj 5i8 sin i) -J- 9-5198 .^— ^^ :_ o ' 6.78660 3-51668 -f 0-3579 . . e + 02-3 y, 3.06085 A' 3-26715 90° + 1 . ii3 34-5 h +3.72475' J . +0.24953 X ■ "3 32a k . +3.72475 kb. +3.97428 o ' " D +191935.9) a —2.96530 o' corr. 2. 2 J COB iJ' +9.97627 * + 18 57 3o.3 , . 7-7- ^ A y sin t^ — 2.94157 + 21 58-8 A'ycosii +3.i2oi8 J, _ 33 32.2 (tan./, -9.82139 . +2334.5 1"°^"^ +9^2092 ^^ ^ ' ' . A' . . 3.26715 Y 9.93211 y . . 9.93211 c03(i/' + i) + 9-99341 sin(ip + «) — 9-23802 1 77" i .. . 3.72475 I + 9.92552 ' \p + ^41^0 i -!:!E^ A + 25 d .5 r . . + 1 48 Long. 3 10 .9 W. ^+^r^- z^. . +2 1 5 COB J) +9.97481 efts i> +9- 97484' cos 1 + 9.96215 sin 1 +9-60200 b . +0-24953 kb . +3.97428 L' . +0.18649 • L" . +3.55109 sin J? + 9'5i977 sin Z* +9-51977 cos I +9-96215 sin £ +9-60200 + 9.48192 +9-12177 COS X — 9-60134 sin x +9-96228' {tan . — 0-11042 ,„ „, „ „ , , (tan - + o-84o5i sin . -9^9^ -^+ S'^^^'-iJ^i^ . 19^955, £^+28 14.4 +9.70025 if+ 28 i4 -4 +9-96676 ^ . —24 32-4 J . 0-24953; 1//' +110 I .5 kb . 3-97428 V • +9*94978 r" • +3.94104 27 418 SPHERICAL ASTRONOMY. We have hence, for the Greenwich time t of beginning, at any place whose lat- itude is I, = north, — south, and longitude X, + east, — west, the two following equations, which may be safely depended on for any place in Scotland or the North of England. cos u :i=o- 84240— [o- 18649] sin/ + [9 -94978] cos/ cos (X— 24° 32' .4) ^ — jh im5»_[3. 72475] sin u + [3 -55109] sin/ — [3.94104] cos / cos (X + 110° i'-5) Contact on 0's limb, u -f- 28° 34' -5 from the north towards the west. As a check on this calculation take the assumed radical place, Edinburgh, and / = + 55°46'-9, X = — 3°io'-9, giving u = 89° 6' • 9 and < = 1 1- 45'" 24', which perfectly coincide with the results of the original calculation. Similar calculations for the ending of the eclipse give the equations, , cos "=0-93848— [0-20291] sin / + [9-88677] cos/ cos (X+ 27°6'-7) (=!'■ 38™ 33' + [3.66890] sin a>+ [3-35544] sin/— [3-90073] cos/ cos{x+i53''3'-8) Contact on O's limb, u — 16° 56' -2 from the north towards the east. Also by calculating with T= 3*' i3™ for the annular phase there will result cosm=29-666oo— [1-75169] sin/ +[1-46950] cos /cos (*+ [ i''42'.4) (= i"- 43°" 7"? [2 -14475] sin 0) + [3 ■45484'] 8iii/-[3 -92550] cos/ cos (X + r3i°55'-9) Contact on 0's limb, — 19° 53' -5 T u from the north towards the east, the upper sign appertaining to the beginning and the under sign to the ending. If cos u > I, the place will be without the limits, and the eclipse will not be annular. By taking / = + 55° 46' -9, X = — 3° 10' -9, the results will exactly correspond with the special calculation. Jfbte. — ^The expression of cos u for the annular phase, as the appearance of this phase is comprised within narrow limits on the surface of the earth, will afford a very oonvenient and simple determination of the places which range in those lim- its as well as those which range in the central line ; and we may expect very ac- curate results throughout the portion of country originally taken into considera- tion. Thus for the southern limit we must obviously have cos u = -j- i, for the central line cos u := o, and for the northern limit cos u = — i ; and hence the following conditions : f + I J C southern limit. p — i' sin / + / cos / cos (X + i^') = ■< o >■ for < central eclipse. ( — I ) ' northern limit. By making the assumptions n' cos iV' = / cos (X 4- ;/,')) n' sin N'=L' ) ^''> ♦hey will give f — p + I'J f southern limit J n' cos (N" ■{■1)= < — p > for ■< central eclipse > ....(») ( — p — I } t northern liinit ) If we therefore take any meridian whose east longitude is X, these two equa- tions (r), («) will serve to determine the extreme latitudes /, on this meridian, be- tween which the eclipse will he annular as well as that where it will be centraL For the preceding eclipse, these equations will be n' cos iP' = [1 .46950] cos (X 4- 1" 42'.4), »' sin JV' = [1.75159]; ( — [i .45737] "i C southern limit n' cos {y -\-l) = <— [1 .47226]> for ^central eclipac. (— [i .48665] ) (northern limit. APPENDIX Xr. 419 If we take, for example, the meridian of Edinburgh, and use A = — 3° lo'-gv there will result, O ' Extreme southern point of annular appearance, N. 54 19-7 Point of central appearance, N. 55 20-4 Extreme northern point of annular appearance, N. 56 21-7 which are geocentric latitudes. in. CALCULATION OF THE TRANSIT OF MEECUEY, November 1, 1835. The conjunction in right ascension takes place about 7'' 38°"; take therefore T= y^ 4o", and we readily find from the cphemeris the following data: 6—16" i5' 58" -2 D - A- I P ■ 16 22 4-2 2 32.6 4-8 12-66 a + O 10-95 a, + 5 32-7 ff 16 IO-4 I 8-66 With these quantities, the calculation, for external contact of limbs, is as follows : P c 16 10-4 s 4-8 A 16 i5-2 a + I -03941 COS S + 9-98226 a COS i + I -02167 O I II 12-66 8-66 4-00 o- 60206 2 - 98909 6 + 7-61297 ai + 2-52205 + 9-98226 . . + 9-79523 const. . 9.43537 #(') + 9.21439 (—3.27124 (— 3i' 7" A' . 2.96223 f + 2.40108 \ + 4' 12" y • • — I' 38" l. . + 25 29 — 1.99123 + 3-18441 X . . . + 2.74741 - Xl . —2.75967 tanS . — 9.24382 8 . . — 9 56.6 cot 1 . —0.42474 cos/S . . + 9-99343 ( . — 20 36.6 cos t . +9.97128 w .. . + 2.75398 ■ > . . . . W . +2.75398 cos— (-S + .) + 9-935o8- -('S+O + So 33.2 3.55630 n . . . + 2.68906 + 6.28i56 A ' . . / 2.96223 1 H . +3.09715 COSu . . + 9.72683 . u + 57 47.0 . cos B + 9.72683 c . . + 3-37032 a — 27 i3.8 c . . +3.37032 sin a . -9.6604') — 3.03077 b . + 88 20.2 sini . +9-99982 + 3.37014 <, . . . — o''i7"'.9 h . . . . . + o''39'°.i r . . . II 6 10 48 . 1 T. . . . , Emersion II 6 Immersion II 45 • I mean times. Acceleration i .8 Acceleration 2 .0 S.T. mean noon 19 4-4 S. T. mean noon 19 4-4 ■ Immersion 5 54-3 , . Emersion 6 5i .5 . Bid. times. Star's R. A 10 23 .4 Star's R. A . 10 23 .4 lm.h . —4 29.1: = -67° f Em. h = . ( Parallactic l . —331.9=— 53* Parallactic^ —39° .7 — 36°. 9 (-.) . . . + 20 .6 (-.). . . . + 20 .6 u . . . +57 8 . . . . + 57 .8 From i ''°''*'' 7^7 • 2 Uo the east. From \ I vertex +2.5) I north + 78 -4)t„thee»t vertex + no -3 ] APPENDIX XI. 423 These angles are for the inverted image ; and, being estimated towards the east, the negative values must be considered as towards the west. The declination of the star gives for the latitude of Greenwich a semi-diurnal arc of 7'' 23'° ; as this exceeds the value of A both at immersion and emersion, the immersion and emer- sion will both occur above the horizon. V.^CALCULATION OF THE ECLIPSE OF THE MOOK, "April 30, 1836. The opposition or full mocn takes place at 19'' 58'°. For the computation assume the time 20'' o™. J'sRA. . . ©•3 E. A. -f 12'' time 19K h. m. 8. i4 32 5i-35 . i4 33 52-38 . — I I -03 20'' li. m. B. i4 35 11-19 - . i4 34 I -91 . + I 9-28 + 17' 19" b. m. 8. . 143731-43 . i4 34 11-45 -f 3 19-98 space - i5' i5" + 5o' 0" - i5' i5" a= + :7 i<> I 3, 4: .. = + 32' 38" + 5o 19I' 20'' 2lk • " ' " ' " >'sdeo...-i4 519). . — f4i958). . a cor. . . J I ) — i4 34 32 ) O'sdec. . -fi5 6 35 . . -j- 15 720 - - + i5 8 6 a; . . . + I I 16 + 47 21 + 33 29 + 61' 16' _ 13, 55„ ie = + 47 21 _ ,3 52 «i=-i3'54" -f-33 29 a -{-3-01662 01 + 3-29181 P = 6o'l9" cos J? + 9-98627 . . +9-98627 P 3-55859 jf + 3.oo289 yj + 3-27808 0.00929 a; + 3-45347 1,-2-92117 ZJZLZ 3-55788 ' tan S + 9-54942 cote — 0-35691 8 +i9°3o'.7 ^ „, ^, , -— P'.Mi3"l ( cos /S + 9-9743i COS1 + 9-90163 ^ j- , —23 43-9 ir+3-479i6 . . +3-47916 ' .15 53 I +9.99882 3-5563o 44 29 JL- . 44 « + 3-47798 +6-99709 ^^ — A' 3-56808 „ ,'Z 45 l3 SHADOW £-+3-71901 , ,6 26 -(a+i)+ 4 l3.2 . External . +35 38-7 cos M + 9-90990 . . +9-90990 — —~~ I < 61 39 external a -3i 25.5 + 3-809.1 « + 3-8o9.i A -j ^8 47 internal i +39 5i -9 sina — 9-71716 sm6 + 9-8o685 — 3.52627 +3-61596 <,_ 56-.0 t,+ iM^.S Assnmed time 20'' o ... 20 o Beginning 19 4 -o Ending 21 8 - 8 Green w' mean times. 4M SPHERICAL AST'EOKOMY. ]?or the timea «t anj other place,, it will only be neoeasaiy to take into laoconnt '^e difference of longitude. The positions of the points of contact on the limb of the moon may be deter- mined tin the same manner as those of an occultation, and will here be unnecessary. As A ' for internal contact with shadow is less than n, no internal contact can take place, and therefore the eclipse is only partial The contacts with the penumbra are to be determined in a similar manner from the same values of n, ff, and will also be unnecessary here. a' for external contact with shadow 6i' 89" n 5o 6 II 33 Eclipsed . . . which divided by 2 s = 32' S2", gives o-35i for the magnitude of the eclipse, the mooit!s diameter being utiity. APPENDIX XII. EQUATION OF EQUAL ALTITUDES. Let P he the pole, Z the zenith, S' «» li- the place of the sun in the afternoon, S the place he would have occupied had his declination or polar distance F S remain- ed unchanged. Make I = latitude of place = 90° — P Z X = declination of sun = 90° — PS a = altitude of sun = Q0° — Z S P t= hour angle Z P S ; then in the triaBgle Z P S, sin a = sin / sin x + cos 7 cos x . cos P Differentiating, supposing x and P alone to vary, we have dx .coax, ami = cos I. cos x .sin P .d P + oosl .cos P .tin xdx, («) d X . (cos X .sinl — cost .COS P .sin x) =dP . sin P cos ? cos « ; whence , ^ , /tan I tan x \ dp = a r. I — \si] \sinjP tanP^ (i) APPENDIX XIII. 425 Denote by 6 the ohange in declination from the next preceding to the next following noon or change in 48 hours, and by t the interval in hours between the epochs of equal altitudes in the morning and afternoon. Then •whence also is : 5 :: t S.t dx = 48 which substituted in Eq. (b) give , p _ S , tan I .t S . tan x. t ^ ~ 48 . sin 7^ t ~ 48 . tan 7^ t ' converting both members into time and taking one- half, we have, after writing d for x, and making t^ ^ ^ x jt ■ d P = -^g d P, S . tan I . t S . tan d , t t,= 1440 .sin 7i t 1440 . tan 7^ < as in the text, page 187. w APPENDIX XIII. CORRECTION FOR DIFFERENCES OF REFRACTION. Let P be the pole, Z the zenith, and S the place of the sun had the air undergone no change, and S' the place as determined by a change of atmospheric refraction. Then, employing the same notation as in the pre- ceding appendix and resuming its equation (a), regarding the altitude a as referring to the place 5 and P to the hour angle ZP S,yre have, writing d for x, sin o = sin Z . sin cZ + cos I . cos d . cos P, and denoting the altitude of 5' by a' and the hour angle Z P S' hj P' , sin o' = sin / . sin d + coa l.coad . cos P' ; and by subtraction, sin a' — sin a = cos l.coad (cos i" — cos P) ; 426 SPHERICAL ASTEOlfOMY. but sin o' — sin a =2 sin i (a' — a) . cos ^ (a' + a), cos P' - cos P = 2 sini (P - P') . sin^ (P' + P) ; whence by substitution, sin I (ffl'— a) . cos ^ (a'+ a) = cosl. cos ti . sin ^ (^— -P') • sin ^ (P'+ P), and because a and o' as also P' and P differ by very small quantities, the above becomes, by transposing and dividing, (a' — a) . cos a F-P' = cos I . cos d . sin P" But denoting the refraction in the afternoon by r' and that in the momipg by r, we have a' — a = r' — r ; substituting and converting both members into time, and writing. 3 241-9 287-4 336-9 390.2 7 129.3 163-2 200-9 242-6 288-2 337.7 391-1 8 129.9 i63-8 201 -6 243-3 289-0 338-6 392-1 9 i3o.4 164-4 202-2 244-1 289-8 339.4 393-0 lO i3i -0 i65-o 202-9 244-8 290-6 340.3 393-9 II i3i-5 i65-6 2o3-,6 245-5 291-4 341-2 394-8 12 l32-0 166.2 204-2 246-3 292-2 342 -b 395-8 i3 132-6 166-8 204-9 247-0 293-0 342.9 396.7 i4 I33-I 167-4 2o5-6 247-7 293-8 343.7 397.6 i5 i33.6 168-0 206-3 .248-5 294-6 344-6 398.6 i6 i34-2 168.6 206-9 249-2 295-4 345.5 399.5 17 i34-7 169-2 207-6 249-9 296-2 346.4 400.5 i8 135-3 169-8 208-3 25o-7 297-0 347-2 401-4 19 i35.8 170-4 208-9 25i-4 297-8 348-1 402-3 20 i36-3 171-0 209-6 252-2 298-6 349.0 4o3-3 21 136-9 171-6 210-3 253-0 299-4 349.8 4o4-a 32 137.4 172-2 211.0 253-6 3oo-2 35o-7 4o5-i 23 i38-o 172-9 211 -7 254-4 3oi-o 351.6 4o6-o 24 i38.5 173-5 212-3 255-1 301-8 352.5 407-0 25 .39-1 174.1 2l3-0 255.9 3o2-6 353.3 408-0 26 139-6 174 7 213-7 256-6 3o3-5 354-2 408.9 27 l40-2 175-3 214-4 257-4 3o4-3 355.1 409-9 38 . l4o-7 175-9 215-1 258-1 3o5-i 356-0 410-8 29 i4i-3 176-6 2i5-8 s58-9 3o5-9 356.9 411.7 TABLES. 443 Table V. — (Continued.) Fw the Reduction to the Meindian : showing the value of 2 sin'i i P A = sin 1' Sec. 8™ gm 10™ 11™ 12'" IS" 14". 1/ " II ^, ,^ ^^ II 3o i4i-8 177-2 216.4 259-6 3o6-7 357-7 412-7 3i 142-4 177-8 217-1 260-4 307-5 358-6 4i3.6 32 143.0 178-4 017-8 261 -1 308.4 359-5 4i4<6 33 143.5 179-0 218-5 261 -9 309-2 36o-4 4i5-5 34 144-1 179.7 219-2 262-6 3io-o 361.3 4.16-5 35 144-6 180-3 219.9 263-4 3io-8 362-2 417-5 36 145.2 180-9 220-6 264-1 311-6 363-1 418-4 , 37 145-8 i8i-6 221-3 264-9 3i2.5 364 -0 419-4 38 146.3 182-2 222.0 265-7 3i3.3 364-8 420.3 39 i46-9 182.8 222-7 266-4 3i4-i 365-7 421-3 4o 147-5 183-5 223-4 267-2 3i5-o 366-6 422-2 4i i48-o i84-i 224-1 267-9 3i5-8 367-5 423-2 42 i48-6 184-7 224-8 268-7 3i6-6 368-4 424-2 < 43 i49-2 185-4 225-5 269-5 , 317-4 369-3 425-1 44 149-7 186-0 226-2 270-3 3i8-3 370-2 426-1 45 i5o-3 186-6 226-9 271-0 319-1 371-1 427-0 46 i5o-9 187.3 227-6 271-8 319-9 372-0 428-0 47 i5i-5 187-9 228-3 272-6 320. 8 372.9 429.0 48 l52-0 188-5 229-0 273-3 321-6 373.8 429-9 49 i52-6 189-2 229.7 274-1 322-4 374-7 430-9 5o i53.2 189-8 23o-4 274-9 323-3 375-6 431.9 5i i53-8 190-5 23l.l 275-6 324-1 376.5 432-8 52 i54-4 191-1 23i-8 276-4^ 325-0 377-4 433-8 53 154-9 191-8 232-5 277:2 325-8 3^8-3 434-8 54 i55-5 192-4 233-2 278-0 326-7 379-3 435-8 55 i56-i 193 -I 234 -0 278-8 327-5 38o-2 436-7 56 i56-7 193-7 234 1 7 279.5 328-4 381-1 437-7 57 157-3 194-4 235-4 280-3 329-2 382-0 438-7 58 i57-8 195-0 236-1 '281 -I 33o-o 382-9 439-7 59 i58-4 195-7 236-8 281-9 33o-9 383-8 440-6 Ui SPHERICAL ASTRONOMY. Table V. — (Continued.) For the Reduction to the Meridian : showing the value of _ 2sin'^ P ~ sin 1" ■ Sec. 15"" IB"" lY™ IS" ID"" 20™ 21™ o 441-6 5p2-5 567.2 635.9 708-4 11 784-9 - 865.3 I 442-6 5o3.5 568-3 637.0 709-7 786-2 ' 866.6 2 443-6 5o4-6 569.4 638.2 ' 710-9 787-5 868.0 3 444-6 5o5.6 570.5 639.4 712. 1 788.8 869.4 4 445-6 506.7 571-6 640.6 713-4 790.1 870-8 5 446-5 507.7 572-8 64t-7 714.6 791.4 872-1 6 447-5 5o8.8 573.9 642-9 715.9 792.7 .873.5 7 448-5 509.8 575.0 644-1 717. 1 794.0 874.9 8 449-5 510.9 576.1 645-3 718.4 795.4 876.3 9 450-5 5ii-9 < 577-2 646.5 719.6 796.7 877.6 lO 451-5 5i3.o 578.4 647-7 720.9 798.0 879-0 II 452.5 5i4-o 579-5 648-9 722.1 799.3 880.4 12 453-5 5i5.i 580-6 650-0 723.4 . 800.7 881.8 i3 454-5 5i6-i 581.7 65i.2 724.6 Bo2.o 883.2 i4 455-5 5i7-2 582.9 652-4 725.9 803.3 884.6 i5 456-5 5i8.3 584-0 653.6 •ji-i-i 804.6 886.0 i6 457-5 519.3 585-1 654-8 728.4 80b. 887.4 17 458-5 520.4 586-2 656.0 Tit)--] 807.3 888.8 i8 459-5 521-5 587-4 667.2 730.9 808.6 890.2 '9 460,. 5 522-5 588-5 658.4 732.2 809.9 891.6 20 461-5 523.6 589.6 659-6 733.5 811. 3 893.0 21 462-5 524-6 590.8 660.8 734.7 8i2.6 894.4 22 463-5 525-7 591.9 662.0 736.0 813.9 895.8 23 464-5 526-8 593.0 663.2 737.3 8:5.2 897.2 24 465-5 '527.9 594-2 664-4 738.5 816.6 89S;6 25 466-5 528.9 595.3 665.6 739.8 817.9 900.0 26 467-5 53o.o 596.5 666.8 741-1 819.2 901.4 27 .468-5 53i-i 597.6 668.0 742-3 820.5 902-8 28 469.5 532-2 598.7 669 -2 743-6 821.9 904-2 29 470-5 533-2 599.9 670.4, 744-9 823.2 905 -fi TABLKS 445. Table V. — (Continued.) For the Reduction to the Meridian : showing the value of 2 sin« I P Sec 15" le-" 11" 18"> 19™ 20" 21" i 3o 471 -5 II 534-3 601-0 671-6 746-2 824-6 II 907-0 3i 472-6 535-4 602 -2 672.8 747-4 825.9 908 4 32 473-6 536-5 6o3-3 674-1 748-7 827.3 909 8 33 474-6 537-6 604 -5 675-3 750.0 828-6 911 2 34 475-6 538-7 6o5-6 676-5 751.3 829-9 912 6 35 ■476-6 539-7 606-8 677.7 752-6 831.2 914 36 . 477-6 540-8 607-9 678-9 753-8, 832.6 9.5 5 37 478-7 54t-9 609-1 680-1 755-1 833-9 916 9 38 479-7 543-0 610-2 681-3 756-4 835-3 918 3 39 480.7 544-1 611. 4 682-6 757-7 836-6 919 7 40 481-7 545-2 612-5 683-8 759-0 838-0 921 1 41 482.8 546-3 613.7 685-0 760-2 839.3 922 5 42 483-8 547-4 6i4-8 686-2 761-5 840.7 923 9 43 484-8 548-4 616-0 687-4 762.8 842-0 925 3 44 485-8 549-5 617-2 688-7 764-; 843.4 926 8 45 486-9 55o-6 618.-3 689-9 765-4 844.7 928 2 46 487-9 55i-7 619-5 691-1 766.7 846.1 929 6 47 488-9 552-8 620-6 692-4 768-0 847-5 93. 48 490.0 553-9 621.8 693-6 769-3 848-9 932 4 49 491.0 555-0 623-0 694-8 770-6 85o-2 933 8 5o 492-0 .556-1 624-1 696-0 771-9 861.6 935 2 5i 493.1 557-2 625.3 697-3 773-1 852-9 936 6 52 494-1 558-3 626-5 698.5 774-5 854-3 938 I 53 495-2 559-4 627.6 699-7' 775-8 855-7 939 5 54 496-2 560-5 628-8 701-0 777.1 857.1 940 9 55 497-2 56i-6 63o.o 702-2 778.4 858.4 942 3 56 498-3 562.7 63r-2 703.5 779.7 ^ 859.8 943 8 57 499.3 563-9 632-3 704.7 781.0 * 861. 1 945 2 58 500-3 , 565-0 633-5 705.9 782.3 862.5 946 6 59 501-4 566-1 634-7 707.1 783.6 863.9 948.1 U6 SPHERICAL ASTRONOMY. Tabm V. — (Continued.) For the Seduction to the Meridian : showing the value of 2 sin" i P A = sin 1" ■ Sec. 22™ 23"- 24™ 25™ 26" 27™ 28" o 949.6 1037.8 1129.9 n 1225-9 11 1325.9 If 1429-7 1537.5 I 951 -o 1039-3 ii3(.4 1227-5 1327.6 i43i-4 1539.3 2 952.4 _^i 040-8 1I33.0 1229-2 1329.3 1433.2 i54i.r 3 953-8 1042.3 1134-6 1230-8 i33i.o 1434-9 1542-9 4 955.3 1043 -8 II36-2 1232-5 1332-7 1436-7 1544-8 5 956.7' 1045-3 II37-8 1234. I i334'-4 1438-5 1546-6 6 958-2 1046-8 II39-3 1235.7 1336-1- i44o-3 1548-4 7 959-6 1048-3 1140-9 1237.3 1337-8 1442-1 i55o-2 8 961 -I 1049-8 1142.5 1239-0 1339-5 1443-9 1552-1 9 962.5 io5i-3 1 1 44 ■ 1240-6 i34i-2 1445-6 1553-9 lO 963-9 io52-8 1145-6 1242-3 i342-9 1447-4 1555-8 II 965.4 1054-3 1147-2 1243-9 1344-6 1449.2 1557-6 12 966.9 io55-9 1148.8 1245-6 1346-3 i45i-o 1559-5 i3 968-3 io57-4 ii5o-4 1247-2 i348-o 1452-8 i56i-3 i4 969.8 io58-9 ll52-0 1248-9 1349-7 1454-5 1563-2 i5 971-2 1060-4 1153-6 i25o-5 i35i-4 1456-3 I565-0 i6 972-7 1062-0 1155-2 ia52-2 1353-2 1458-1 1566-9 «7 974-1 I063-5 r 156-8 1253-8 1354-9 t459-9 1568-7 i8 975-5 io65-o 1158-3 1255-5 1356-6 1461-6 1570-5 '9 977-6 1066-5 1159-9 I257-I 1358-3 1463-4 1572-4 20 978.5 1068 -I 1161-5 1258-8 i36o-i i465 - 2 1574.3 21 979.9 1069-6 1163-1 12O0-4 i36i-8 1466-9 1576.1 22 981.4 1071.1 1164-7 1 262 - 1 1363-5 1468-7 1578.0 23 982.9 1072-6 1166-3 1 263 . 7 1365.2 1470.5 1579.8 24 984.4 1074.2 ■1167-9 1265.4 i367-o 1472.3 1581.7 25 985.8 1075.7 1169-5 1267.0 1368-7 1474.0 1583.5 26 987.3 1077.2 1171-1 1268.7 1370-4 1475.9 1585.3 2? 988.8 1078.7 1172-7 1 270 ■ 3 1372-1 ,477.7 1587.2 28 990.3 1080.3 1174-3 1272. 1373.9 1479-5 1589.1 29 991-8 1081.8 1175-9 1273.7 1375-6 r48i.3 1590.9 TABLES. 447 Table V. — (Continued.) For the Eeditction to the Meridian : showing the value of 2 sin« I- P ~ sin \~' Sec. 22"» 23" 24°' 25"" 26"" 27™ 28"° ' 3o 993.2 1083.3 1177.5 1275.4 1377.4 1483-1 • It , 1592.7 ■ 3i 994-7 1084.8 1179-1 1277. I 1379.0 1484-9 1594-6 32 ' 996.2 1086.4 1180.7 1278-8 i38o.8 1486-7 1596.5 33 997.6 1087.9 1182.3 1280-4 1382.5 i/(88-5 1598-3 34 999.1 1089-5 ii83.9 I282-I 1384.2 1490-3 1600-2' 35 1000.6 1091 .0 1185-5 1283-8 1385.9 1492-1 1602-1 36 1002. I 1092.6 1187.1 1285-5 1387.7 1493.9 1604.0 37 ioo3.5 1094.1 1188-7 1287-1 1389-4 1495.7 1605.9 38 ioo5.o 1095.7 1190-3 1288-8 1391 .2 1497-5 1607.7 39 1006-5 1097.2 1191.9 1290-5 1392.9 1499.3 1609.6 4o 1008.0 1098.8 1193.5 1 292 - 2 1394-7 i5oi.i 1611.5 4i 1009.4 1100.3 1195.1 1293.8 1396-4 1502.9 1613.3 42 1010.9 1101.9 1196-7 1295.5 1398-2 1 504-7^ i6i5.2 43 1012.4 iio3-4 1198-3 1297.2 1399.9 i5o6-5 1617.1 AA 1013.9 iioS-o 1199.9 1298-9 1401.7 i5o8-4 1619.0 45 1015.4 I io6-5 1201.5 i3oo.5 1403.4 l5[0-2 1620.8 46 1016.9 iio8.i 1 2o3 . 1 l302-2 i4o5.2 l5l2-0 1622.7 47 1018.4 1109-6 1204.7 i3o3-9 1406.9 i5i3-8 1624-6 48 1019.9 III1-2 1206.4 i3o5-6 1408.7 i5i5-6 1626-5 49 1021.4 1112-7 1208.0 1 307 - 3 i4io.4 i5i7-4 1628-3 5o 1022.6 1114.3 1209.6 1309.0 l4l2.2 1519.2 i63o.2 5i 1024 -3 iii5.8 1211.2 i3io.7 1413.9 l52I .0 i632.i 52 1025.8 1117-4 I212.9 i3i2.4 1415.7 1522.9 1634.0 53 1027.3 1118.9 1214.5 i3i4-i 1417.4 1524.7 1635-9 54 1028.8 1120.5 1216-1 i3i5-7 '1419-2 i5^6.5 1637-7 55 io3o.3 1122-0 1217-7 1317-4 [420-9 1528.3 1639.6 56 io3i.8 1123-6 1219.4 i3i9-i 1422.7 i53o.2 i64i.5 57 1033.3 1I25-I 1221 -0 i32o-8 1424.4 i532-o 1643.3 58 1034.8 1126.7 1222-6 1322-5 1426.2 1533-8 1645-2 59 io36.3 1128.3 1224.2 1324.2 1427.9 1535-6 1647-1 u^ SPHERICAL ASTRONOMY. Tablk V.>— (Continued.) ^or the Reduction to the Meridian : showing the value of 2 sin' I P A = - sin 1' Sec 29°' 30™ 31"" 32" 33- 34™ 35"" o It i649'0 1764-6 1884-0 2007-4 2134-6 2265-6 2400-6 I 1650.9 1766-6 1886-0 2009 4 2i36.8 2267-8 2402-9 3 1652-8 1768-5 1888-0 2011 5 9138.9 2270-0 24o5-2 3 1654-7 1770.5 1890-0 20l3 6 2l4l .1 2272-2 2407 - 5 4 1656-6 1772-4 1892.1 20(5 7 2143.2 2274-5 2409:8 5 1658-5 1774.4 1894-1 2017 8 2145.3 2276-7 2412.0 6 1660.4 1776-3 1896- I 2019 9 2147.5 2278-9 2414-3 7 1662.3 1778-3 1898. I 2022 2149-7 2281 -2 2416-6 8 1664-2 1780-3 1900.2 2024 I 2i5i-8 2283-4 2418-9 9 1666- 1 1782-3 1902.2 2026 2 2i53-9 2285-6 2421-2 10 1668-0 1784-2 1904.3 2028 3 2i56-i 2287-8 2423-5 II 1669-9 1786-2 1906-3 2o3o 5 2158-3 2290-0 2425-8 12 1671-9 1788.2 1908-4 2032 5 2160-5 2292-3 2428-1 i3 1673-8 1790. I 1910-4 2034 6 2162-6 2294-5 2430.4 i4 1675-7 1792 -I 1912-4 2o36 7 2164-8 2296.8 2432.7 i5 1677-6' 1794- I i9i4-'4 2o38 8 2166-9 2299-0 2435-0 i6 i679-'5 1796-1^ 1916-5 2o4o 9 2169-1 2301-3 2437-3 '7 1681-4 1798-1 1918-5 2043 2171 -2 23o3-6 2439-6 '8 1683-3 1800-0 1920-6 2045 I 2173.4 23o5-8 2441-9 '9 1685-2 1802 -0 1922-6 2047 2 2175-6 23o8-o 2444-2 20 1687-2 i8o4-o 1924-7 2049 3 2177-8 23l0.2 2446-5 21 1689- I i8o5-9 1926-7 2o5i 4 2179-9 2312.4 2448-8 22 1691 -0 1807.9 1928-8 2o53 5 2182-1 2314-7 2451-1 23 1692-9 1809-9 1930-8 ao55 7 2184-3 2316-9 2453.4 24 1694-8 1811-9 1932-^9 2o57 8 2i86'-5 23(9-2 2455.7 25 . 1696.7 1698-6 i8i3-9 1935-0 2059 9 2188-6 2321-5 2458.0 26 i8i5-8 1937-0 2062 2190-8 2323.7 2460.3 27 , 1700-5 1817-8 1939-0 2064 I 2193-0 2325-9 2462.6 28 1702-5 1819-8 1941-1 2066 2 2195-2 2328-2 2464-9 29 1704.4 1821-8 1943-1 2068 3 2197-3 2330-4 2467-2 TABLES. U9 Table V. — (Continued.) Jihr the Seduction to the Meridian : showing the value of _ 2 sin" ^ P sin 1" ■ • Sec. 29"" 30" 31" 32" 33" 34" 35" 1 3o 1706.3 1823-8 1945.2 2070.4 II 2199-5 2332-7 2469.5 3i I708'2 1825.8 1947.2 2072-6 2201-7 2334-9 2471-8 3i 1710-2 1827-8 1949-3 2074-7 22o3-9 2337-2 2474-2 33 I7I2-I 1829-8 1951.3 2076-8 3206-1 2339-4 2476-5 34 I7i4'0 i83i-8 1953.4 2078.9 2208-3 2341-7 2478-8 35 1715.9 1833-8 1955.5 2o8r-o 22IO-5 2343-9 2481 -I 36 1717-9 1835-8 1957-6 2083-2 2212-7 2346-2 2483-5 3? 1719.8 1837-8 1959-6 2085-3 2214-9 2348-5 2485-8 38 1721.7 1839-8 1961-7 2087.4 2217-I 2350-7 2488-1 39 1723.6 i84i-8 1963.7 2089.6 2219-3 2353 -0 2490-4 4o 1725-6 1843-8 1965-8 2091-7 2221 -5 2355-2 2492-8 4i. 1727-5 1845-8 1967.8 2093-8 2223-7 2357-5 2495-1 42 1729.5 1847-8 1969-9 2095-9 2225-9 2359-7 2497-4 43 1731.5 1849.8 1972-0 2098 - 2228-1 2361-9 2499-7 44 1733.4 i85i.8 1974-1 2100-2 2230-3 2364-2 25o2-l 45 1735.3 1853.8 1976- I 2I02-3 2232.5 2366-4 25o4-4 46 1737. i 1855-8 1978-2 2I04-5 2234-7 2368-7 25o6-7 47 1739.2 1857-8 1980-3 2106-6 2236-9 2371.0 2509-0 48 1741-2 1859-8 1982-4 2108-8 2239-1 2373.3 25II-4 49 1743.1 1861-8 1984-4 2IIO-9 2241-3 2375-5 25i3-7 5o 1745. I 1863-8 1986-5 2H3-I 2243-5 2377-8 25i6-i 5i 1747-0 1865-8 1988-6 2Il5-2 2245-7 2380-1 25i8-4 5i 1749-0 1867-8 1990.7 2II7-4 2247-9 2382-4 2520-8 53 1750-9 1869-8 1992-7 2119-6 225o-l 2384 6 2523-1 54 1752.9 1871-8 ,994.8 2I2I-7 2252-3 2386-9 2525-4 55 1754.8 1873-8 1996-9 2123-8 2254-5 2389-2 2527-7 56 1756.8 1875.9 1999-0 2126-0 2256-7 2391-5 2530-1 57 1758-7 1877-9 2001 -0 2 [28 -I 2258-9 2393.7 2532-4 58 1760-7 1879.9 2003-l 2i3o-3 226I-I 2396-0 2534-8 59 1762-6 1882.0 2005-3 2132-4 2263-4 2398-3 2537.1 450 SPHERICAL ASTKONOMY. TABLE VI. For the second part of the Beduction to the Meridian: showing the value cf B: 2 sin^ \ P sin 1' Miautes 0" 10' 20- 30* 40' 50' ^j // „ II // It 5 O-OI O-OI O-OI o.oi O'OI o-oi 6 O'OI O-OI O-OI 0.02 0.02 0-02 7 0.02 0-02 o.o3 o.o3 o.o3 o.o4 .8 o.o4 o.o4 o.o5 o.oS o.o5 o.oG 9 o.o6 0.07 0.08 0.08 0-08 0-09 lO o>o9 O.IO o.ii o.ii 0.12 o.i3 II o.i4 o.i5 o.i5 0.16 0-17 0.18 12 0M9 0.20 0.22 0.23 0.24 0.25 l3 0-27 0.28 o.3o o.3i 0.33 0.34 i4 0.36 0.38 0.39 0.41 0.43 0.45 i5 0.47 0.49 0-52 0.54 0.56 0.59 i6 o-6i 0.64 0.67 0.69 0.72 0.75 17 0-78 o.8i 0.84 0.88 0.91 0-95 i8 0.98 1.02 1.06 1.09 I.i3 I. .8 '9 1.22 1.26 i.3o 1.35 1.40 1-44 20 1-49 1.54 1.60 1.65 1-70 1.76 21 1.82 1.87 1.93 1.99 a. 06 2*12 22 2.19 2.25 2.32 2.39 2-46 2.54 23 2-6l 2.69 2.77 2.85 2.93 3.01 a4 3.10 3.18 3.27 3.36 3.45 3.55 25 3-64 3.74 3.84 3.94 4.o5 4.15 26 4-26 4-37 4-48 4.60 4.72 4.83 27 4.96 5.08 5 .20 5.33 5-46 5-60 28 5.73 5.87 6.01 6.i5 .6.30 6.44 ^9 6.59 6.75 5.90 7-06 7.22 7.38 3o 7.55 7.72 7.89 8.06 8.24 8.42 3i 8 -61 8.79 8.98 9.17 9.37 9.57 32 9.77 9.97 10.18 10.39 10.61 10.82 33 11 -04 11.27 II. 5o 11.73 11.96 12-20 34 12.44 12.69 12.94 i3i9 13.45 13.71 35 13.97 14.24 i4.5i 14.78 i5.o5 15.35 TKIGONOMETRICAL FORMUI.^. I. Equivalent expressions forsin :». 1. cos X .taax. cos X 2. . cot X 3. VI — cos' X. 4 1 Vl + cot' X 5, tan cc Vl +tatf « 6. 2 sin ^x . cos ^2. 1. ,/l — cos 2 « Y '2 8. 2 tan ^ 2: 1 +tan»^a;' o 2 cot •!■ a + tan -J^ a; ' sin (30° + g) — sin (30° — x) 10. ^ ■ ■'. 11. 2 sin' (46° + ^ !t) - 1. 12. 1 — 28iB'(45° — I*). 1 — tan' (45°- I g) 1 +tan'(45°-i«)' tan (45° +\x)- tan (45° — ^a) tan (45° +\x)+ tan (45° i-* \ x)' 15. sin (60° + a;) — sin ((SO' — *). 1 13 14, 16. cowcant x 452 SFHEIIIOAL ASTRONOMY. '- n. EqtiiTalent expreesioos ibr cos x. sin X ' Stan X 2. tmx , cot X. 3. Vl — sin' !»• VI + tan' « cot a! V 1 + cot' » 6. cos' ^x — siu' J «. ■7. 1 — 2 sin' ^ «. 8. 2co9*Ja!— 1. 9, ^ /l+co8 2g 2 "' 10 ^-^"'i'' * 14twi'^a;" cot ^ » — tan I « cot -^a; + tan ^a;' 12, 1 + tan a; . tan ^ a; o 13. tan (45° + ^ a;) + cot (45° + J x)' 14. 2 cos (45° + J ar) cos (45» — J x). 15. cos (eo" + a;) + cos (60° - at). secant a; TRIGONOMETRICAL FORMULA 453 m. Equivalent expressions for tan x. -I* ~~~~~" • cos X 2. ' cot a; 3. 4/^V^ cos' X sm X 5. 6. 1. Vl — sin' X Vl — cos" k cos a 2 Unix 1 — tan'^«' 2 cot ^x cot'i«-l" 2 g cot J « — tan ^ «■ g. cot a; — 2 cot 2 a;. 1 — cos 2 a; 10. ■11. sin 2x sin 2x 1 + cos 2 a: 12. /-. 1 — cos 2 « 13. 1 + cos 2 « tan (45° + ^ a) - tan (45° -jx) 454 - SPHERICAL A8TR0K0MT. TV. Eelative to two arcs A and J?. 1. a\a(A + B) = sin ^ . cos J5 + cos ^ . sin £. 2. sin (A — £) = sin ^ . cos ^5 — cos ^ . sin £. 3. cos (^ + ^) = cos ^ . cos ^ — sin ^ . sin A 4. cos (^ — ^) = cos ^ . cos J& + sin A . sin B. f.,jj. tan ^ + tan 5 5. tan (A + B) = 3—- =. ^ ' 1 — tan ^ . tan B_ A . , J T,\ tan j^ — tan 5 6. tm{A-B) =r-p- J— — 5. '^ ' 1 -f- tan j4 . tan ^ 1. sin (4,5° ± ^) I _ cos .g ± sin B 8. cos(45°=F-B)) V2 0. tan(46»±^) =\^^- 10. tan'(46<>±i5^=l±^. * ' 1 qp sin £ . . « . , ™ 1 ± sin jB cos 5 11. tan {45" ± 4 5) = ^- = r-^. '^ * ' cos 5 1 qp sin B sin (^ + ^) tan ^ + tan jB cot JS + cot A sin (A —'B) ~ t&n A— tan 5 ~ cot 5 — cot A cos (2I + ^) _ cot 5 — tan ^ _ cot ^ — tan 5 cos (^ — .6) ~ cot J? + tan A "~ cot ^ + tan jB ' gin A + sin B _ tan ^ (^ + 5) sin j1 — sin 5 ~~ tan |(^ — ^) ' CO6 .g + cos A _ cot ^ (.^ + jB) cos £ — cos .^ ~ taa^ 'A — B)' \eontinwd. TRIGONOMETRICAL FORMULAE. 455 IV. continued. Relative to two area A and JB. 16. sin A.co&B = i sin (^ + 5) + + sin {A — B). 17. cos ^ . sin .B = ^ sin (^ + .B) — J sin {A — B). 18. sin -4. sin 5 = \ co& {A — B) — \ cos. {A + B). . 19. cos A. cos, B = ^ cos {A + B) + ^ cos (A — B). 20. sin ^ + sin J5 =2 sin \ {A + B) . cos ^ {A — B). 21. cos ^ + cos 5 =2 cos \{A + B) . cos ^ (^ — jB). sin (A + jB) 22. tan ^ + tan 5 = \ ^. cos A . cos ^ sin (A + jB) 23. cot ^ + cot 5 =-^— Y— , — i. sm .<1 . sin B 24. sin J — sin 5 =2 sin i (^ — 5) . cos ^ (.4 + B). 25. cos 5 — cos ^ = 2 sin ^ (^ — .B) . sin \{A + B). sin (^ — ^) 26. tan ^ — fan 5 = cos ^ . cos B sin (^ - B) 27. cot 5 - cot ^ = ^-^. — r—j.. sin ^ . sin B 28. sin" ^ — sin» 5 ) V = sin (^ — B) .sin {A + 5). 29. cos' 5 — cos' A ) 30. coa' ^ — sin' B = cos {A — B) . cos (yl + ^). sin (A-B). sin (^ + B) 31. tan'^-tan'^=_A___i__i__^. 32. sin {A-B). sin (^ + J?) cofi?-c^f^ sin'i.sin'^ ^' 456 SPHERICAL ASTRONOMY. V. Differences of trigonometrical lines. 1. A sin a; = + 2 sin ^ Ax . cos (« +' J A *), 2. A cos X = — 2 sin ^ A a; , sin (« + J A x). sin A X 3, A tan a; = + 4. A cot a; = — cos X . cos (a; + A ar) ' sin A X sin a; . sin (a; + A x) 5. A sin' X = + sin A a; . sin (2 a; + A x). 6. A cos' X = — sin A a; . sin (2 a: -f- ^ *)• .. , , sin A a; . sin (2 a; + A ai) 1. A tan' a; = H -, ^-^ — , ' / . cos' X . cos' (a; + A a!) sin A a; . sin (2 a; 4- A a;) '1? — -. ^ c sin' X . sin' (a; + A a;) 8. A cot' X = — VI. Differentials of trigonometrical lines. 1, d sin a; = + d a; . cos x. 2. d cos a; = — ,d x . sin «. da; d tan a; = + 4. d cot a; = — cos" X dx 5. d sin' X = -)- 2 d a; . sin a; . cos r. 6. d cos' x = — 2 d a; . sin a; , cos ar. , . 2 d a; , tan a; 7. d tan' X = ■] 5 . „ , ,, 'idx. cot,a! 8. d cot' a; = — :-; . TRIGONOMETEICAL FORMULA. 457 VII. General analytical expressions for the sides and angles of any spherical triangle. 1. 008 5 = cos ^ . sin 5' .sin S" + cos S' . cos S" 2. cos S' = cos ^' . sin S" .sinS + cos S" . cos S. 3. cos S" =coaA". sin S .sinS'+cosS . cos S'. 4. cos ^ = cos /S . sin ^' . sin A" — cos A' . cos A". 5. cos A' = cos S' . sin A" . sin ^ — cos A" . cos A. 6. cos A" = cos )Si" . sin -4 . sin ^' — cos ^ . cos A', 1. cos 8 . cos ^' = cot Given, Two sides and the included angle. Btquired, 1°. One of the other angles. ten a' = cos given angle X tan ffiven side, a" = the base - -a' t&n X = tan given angle sin a' Sin a" In this formula, the given side is assumed to be the side opposite the angle sought : the other knoirn ^ide is called the base. Required, 8°. The third side. ten a' =z cos given angle X tan given side, a" = the base ~ a', cos a" cos « = cos given side X cos a In this formula, either of the given sides may be as- sumed as the bade; and the other as the given side. In these formulae, x denotes the quantity sought : a' and a" are (mxUiary angles introduced for the purpose of facilitating the compu- tations. If the side sought in formula 8 he small, the formula maj not give the value to a sufiScient degree of accuracy ; and some other mode must be adopted for obteining the correct value. [continued. SPHERICAL ASTRONOMY. IX. continued. Solutions of the cases of SMgtie-ax^\ef\< spherical triangles. GiTEN', 'A gide qnd the two ad;0cent angles, Bequired, 9°. One of the other sides. cot o' = tan given angle X cos given side, o" = tte vertical angle ~ a', tan X = tan given side X cos a' In this formula, the angle, opposite the side 6ough<| is asBumcd As the ^en angle: the other kno'wn angle is called the vertical angle. Required, lO". The third angle. cot a' = tan given angle X cos given side, a" = the vertical angle — a', sin a" cos « = cos given angle X -; — -• In this formula, either of the given angles may be assumed as the vertical' angle ; and the other as the given angle. In these formulae, x denotes the quantity sought: a' .and d" are auxiliary angles introduced for :th<;) purposii of facilitating the compu- tatiC'D.t. If the aogle sought' in formula 10 be small, the formula may not giye the value to a sufficient 'degree of accuracy ; and some .other mode must be adopted for obtaining the correct value. \eontinHid. TRIGONOMETRICAL FORMULAE. 46^ IX. contimted. Solutions of the cases of obUg^ue-mgled spherical triangles. GreEN, The three tides. Required. 11°, An angle. « 1 * = Bin* i X . (A + B+a \ . {A + B + C „\ «n ( Bjx sm ( Cj sia B . sin C . /A + B+C\ /A + B + C .\ cos'^:r = — ^ ' n r sia B . sm 6 In these formulis, A, B, C are the three sides of the triangle ; and A is assumed aa the bide opposite to tfas angle required. GiVEK, The three angles. I Required, 12°. A side. (a + h + c\ la + b + c \ ___ j X cos [—^— ~ a) Cos lin' 4 a; = — Bin* ^ X ^ sin 6 . sin c cos (a + h + c \ ia + h + c \ sin 6 . sin c In these wrmuIiB, a, o, c are the three angles of the triangle ; and a is assumed as the angle opposite to the side required. In these fonnula, x denotes the quantity sought. The formulae, which are resolved by the co«ne, are used only when the angle or side x is imall. 461 SPHERICAL ASTEONOMT. X. Trigonometrical series. \ 1, .sin X = X 1 &c. 2.3^2.3.4.5 3? a/' X 2. COS.* =1 1 +