A TREATISE ON AST RONOA SPHERICAL AND PHYSICAL; WITH ASTRONOMICAL PROBLE MS, AND SOLAR, LUNAR, AND OTIIER ASTRONOMIICAL TABIES. FOR THIE USE OF COLLEG1ES, AND SCIENTIFIC SCHOOLS. BY WILLIAM A. NORTON, M3.A., PROFESSOR OF CIVIL ENGINEERING IN YALE COLLEGE. FOURTH EDITION. REVISED, REMODELLED, AND ENLARGED. NEW YORK: JOHN WILEY & SON' 15 ASTOR PLACE. 1874. Entered according to Act of Congress, in the year 1867, BY WILLIAM A. NORTON, In the Clerks Office of the District Court of the United States for the Southern District cf.New York. PREFACE TO THE REVISED EDITION. IN the preparation of the present edition the work has been entirely remodelled. The chapters which treat of Astronomical Instruments, Comets, the Fixed Stars, and the Tides, and the portion of the chapter on the Sun, that treats of the Sun's Spots and Physical Constitution, and the Zodiacal Light, have been wholly, or mostly, rewritten. Several changes of plan and arrangement have been made with the view of facilitating study and class instruction. The more difficult investigations of astronomical formulas, occnrring in the text of the former editions, have been transferred to the Appendix. On the other hand, the text has been enlarged by giving a more extended description of astronomical facts and appearances, and a more complete discussion of physical phenomena, including a detail of the important results of recent investigations concerning the physical constitution of the different classes of heavenly bodies, and a succinct exposition of the physical theories that have been generally received, or explain the phenomena most satisfactorily. Such theoretical discussions are kept distinct from the universally recognized truths of the science. The results of the author's own investigations on the physical constitution and phenomena of Comets, and on the physical constitution of the Sun, and the origin of the Sun's Spots, are briefly given in the same connection. New theoretical views are offered, in a note in the Appendix, on the possible development of sidereal systems from primordial nebulous masses; under the operation of recognized material forces, originated and sustained by the Creator, which 1V PREFACE. unceasingly execute His will. Some prominence is given to the author's theory of the variable intensity of the repulsive force of the Sun, acting on different portions of cometic matter, as the operative cause of the lateral dispersion of the nebulous matter that makes up the train of a comet. This is believed to have been substantiated by a detailed discussion and comparison with observations; and as recent astronomical treatises, published in this country and in Europe, have advocated it without making mention of its previous publication and mathematical discussion, it is but just and proper that it should be distinctly set forth in the present work. The Astronomlical Problems in Part III. remain substantially the same as in the last edition. The table of Latitudes and Longitudes of Places, the tables of the Planetary Elements, and the table of the Mean Places of Fixed Stars, have been replaced by others that are more accurate and more extended. The tables of the Sun's and Moon's Epochs have been extended to 1884. Many new illustrative figures, and several plates of telescopic appearances, have been added. One of the most important of the special improvements introduced consists in the adoption of the new and more accurate determination of the Sun's parallax, and mean distance from the earth. This is now generally adopted, as one evidence of which may be mentioned its introduction into the computations of the English Nautical Almanac for 1870. It brings with it a more accurate determination of the distances of all the planets from the Sun, and of the satellites from their primaries, and the dimensions and densities of the planets. The present is the first American treatise in which this important advance in exact astronomical science has been incorporated. Another improvement is the insertion of a brief description of the mnethods used in the United States Coast Survey in determining from astronomical observations the latitude and longitude of a place. These may be characterized as the AmericaLn PREFACE. V ilethocls, as they were devised and perfected by American astronomers and engineers; and are superior to all others that hlave yet been tried. Without further specification of alterations and supposed improvements, it is hoped that the work will be found, in all its features, a true exposition, within the limits necessarily prescribed, of the present condition of the sublime science of Astronomy; from both the theoretical and practical point of view. A large number of astronomical treatises and scientific periodicals have been consulted. Professor Chauvenet's admirable work on Spherical and Practical Astronomy, should be particularly mentioned as having been especially consulted in preparing the chapter on Instruments. In the mnention of new discoveries and theoretical views, as well as of the signal advances vhTllich modern Astronomy has made, the name of the discoverer, or author, is generally given. The history of Astronomy cannot properly be wholly omitted from a text-book on the science. although it may be simpler to present the science as a body of admitted truths, without making mention of their discovery. The author takes occasion here to acknowledge his obligations to Professor C. S. Lyman, of Yale College, for important advice, and valuable assistance frequently rendered. TABLE OF CONTENTS. PART I. SPHERICAL ASTRONOM.Y. CHAPTER I. PAGa Definitions and Fundamental Conceptions.-General Phenomena of the Heavens......1 CHAPTER II. Celestial and Terrestial Spheres..11 CHAPTER III. Astronomical Instruments.-Astronomical Observation 24 The Transit Instrument 30 Astronomical Clock. 38 Meridian Circle. 39 Altitude and Azimuth Instrument...... 43 Equatorial 45 Sextant. 47 Errors of Instrumental Admeasurement. 51 The Telescope.. 52 CHAPTER IV. Corrections of Measured Angles... 54 Refraction. 54 Parallax..60 A berration. a66 Ch:APTER V. Ffigure and Dimensions of the Earth.-Latitude and Longitude of a Place. 72 Determination of the Latitude and Longitude of a Place. 75 Viii CONTENTS. CHAPTER VI. PAGE Apparent 5Motion of the Sun in the Heavens. 81 CHAPTER VII. Precession of the Equinoxes.-Nutatioll.. SG Nutation.......... 8 CHAPTER VIII. Measurement of Time. 91 Different kinds of Time ib.,Conversion of One Species of Time into Another.. 92 Determination of Time, and Regulation of Clocks by Astronomical Observations.... 94 The Calendar.......... 98 CHAPTER IX. Motions of the Sun, Moon and Planets in their Orbits... 102 Kepler's Laws.......... ib. Definitions of Terms.106 Elements of the Orbit of a Planet...... 10S Determination of the Sun's Apparent Orbit, or the Earth's Real Orbit.109 Mean Motion.ib. Semi-MAajor Axis. ib. Eccentricity. 110 Longitude and Epoch of the Perigee. 112 Determination of the Elements of the Moon's Orbit 113 Longitude of the Node...i. b. Inclination of the Orbit........ 114 _Mean Motion ib. Longitude of the Perigee, Eccentricity, and Semi-MaLjor Axis. 115 Mean Longitude at an Assigned Epoch.. 16 Determination of the Elements of a Planet's Orbit. ib. Heliocentric Longitude of the Ascending Node 117 Inclination of the Orbit..... 118 Periodic Time. 119 To find the Heliocentric Lonoitude and Latitude, and the Radius Vector, for a given time.. i Longitude of the Perihelion. Eccentricity, and Semi-Major Axis 121 Epoch of the Perihelion Passage.122 True and Mean Elements. 123 CHAPTER X. Determination of the Place of a Planet, or of the Sun or Moon, for a given time, by the Elliptic Theory.-Verification of Kepler's Laws......... 126 CONTENTS. ix PAGi Place oi a Planet in its Orbit... e; Heliocentric Place of a Planet.. 127 Geocentric Place of a Planet.. 12 Places of the Sun and Moon....... b. Verification of Kepler's Laws.129 CHAPTER XI. Inequalities of the Motions of the Planets and of the MIoon.-Tables for Finding the Places of these Bodies.. 130 Tables of the Sun, Moon, and Planets.. 135 CIT-A1PTER: XII. Motions of the Comets.130 Halley's Comet.. 140 Encke's Comet.... 14 Biela's Comet. 142 Fayes' Comet. ib. Lexell's Comet of 1770... 143 The Great Comet of 1843..... 184 Donati's Comet.. 1'4 Conspicuous Comets of the Present Century.. 1 O CHAPTER XIII. Motions of the Satellites... 1 CHAPTER XIV. The Sun and the Phenomena Attending its Apparent iMotionls. 15 Inequality of Days......... Twilight 1.; The Seasons........ 3 Form and Dimensions of the Sun.1... ( Sun's Spots, and Rotation on its Axis. Physical Constitution of the Sun. 164 Zodiacal Light. 175 CHAPTER XV. The Moon and its Phenomena.. 1 Phases of the Moon..'b. Moon's Rising, Setting, and Passage Over the Meridian.. 1 Rotation and Librations of the MIoon.. Dimensions and Physical Constitution of the Mloon... l Description of the Moon's Surface..... 188 CHAPTER XVI. Eclipses ef the Sun and!Moon.-Occultations of the Fixed Stars 190 :x CONTENTS. PAGE Eclipses of the Moon.190 Calculation of an Eclipse of the Moon.194 Construction of an Eclipse of the Moon. 198 Eclipses of the Sun. 199 Calculation of an Eclipse of the Sun.206 Occultations.i. b. CHAPTER XVII. The Planets and the Phenomena Occasioned by their Motions in Space. 208 Apparent Motions of the Planets -with Respect to the Sun. ib. Phases of the Inferior Planets.. 214 Transits of the Inferior Planets. 215 Appearance, Dimensions, Rotation, and Physical Constitution of the Planets......... 216 Mercury...... 217 Venus.. 218 ifars........... 220 Jupiter and its Satellites..... 221 Saturn, with its Satellites and Ring.... 223 Uranus and its Satellites. 227 Neptune. ib. The Planetoilds. 228 CHAPTER XVIII. Comets.229 Their General Appearance. -Varieties of Appearance ib. Form, Structure and Dimensions of Comets 233 Physical Constitution of Comets.... 236 Constitution and Mode of Formation of Tails of Comets.. 237 Condition and Origin of the Nebulous Envelopes... 241 CHAPTER XIX. The Fixed Stars......... 245 Constellations.-Division into AMagnitudes.... ib. RNumber and Distribution over the Heavens.... 247 Annual Parallax, and Distance of the Stars.... 250 NRature and Magnitude of the Stars..... 252 XvTariable Stars.. 253 Double Stars. 256 Proper Motions of the Stars..... 259 Clusters of Stars..... 261 ebule..... ib. Distance and Magnitude of Nebul.e. 2 67 Number, Mutual Distance, and Comparative Brightness of the Component Stars of Clusters.269 CONTENTS. xi PAGH Structure of the Sidereal Universe. 270 General Dynamical Condition of Sidereal Systems... 273 CHAPTER XX. Theories of the Evolution of Sidereal and Planetary Systems. 275 Nebular Hypothesis. ib. Development of the Solar System ib. PART II. PHYSICAL ASTRONOMY. CHAPTER XXI. Principle of Universal Gravitation... 278 CHAPTER XXII. Theory of the Elliptic Motion of the Planets.. 282 CHAPTER XXIII. Theory of the Perturbations of the Elliptic Motions of the Planets and the hIoon..... 287 CHAPTER XXIV. Relative Masses and Densities of the Sun, ZMIoon, and Planets.Relative Intensity of the Force of Gravity at their Surface.. 297 CHAPTER XXV. Form and Density of the Earth.-Changes of its Period of Rotation.-Precession of the Equinoxes, and Nutation... 299 CHAPTER XXVI. The Tides. 302 Comparison of the Theory of the Tides with the Results of Observation. 307 Tides of the Atlantic Coast of the United States 309 Tides of the Pacific Coast.. 311 Tides of the Gulf of Mexico.. 312 Tides of the Mediterranean... 313 Tides of Inland Seas and Lakes...... ib. Tides of the Coast of Europe... ib. Establishment of the Port.-Tide Tables. ib. Sl1l CONTENTS. PART III. ASTRONOMICAL PROBLEMS. PAGH EXPLANATIONS OF THE TABLES. 315 PROB. I. To work by logistical logarithms a proportion, the terms of which are degrees and minutes, or minutes and seconds, of arc; or hours and minutes, or minutes and seconds, of time 320 PROB. II. To take from a table the quantity corresponding to a given value of the argument, or to given values of the arguments of tlhe table.321 PROS. III. To convert Degrees, Minutes, and Seconds of the Equator into Hours, Minutes, &c., of Time.327 PROB. IV. To convert Time into Degrees, Minutes, and Seconds. ib. PROB. V. The Longitudes of two Places, and the Time at one of them being given, to find the corresponding time at the other 328 PROB. VI. The Apparent Time being given, to find the corresponding MIean Time; or, the Mlean Time being given, to find the Apparent... 329 PROB. VII. To correct the Observed Altitude of a Heavenly Body for Refraction.332 PROB. VIII. The Apparent Altitude of a Heavenly Body being given, to find its True Altitude. 333 PROB. IX. To find the Sun's Longitude, Hourly Motion, and Semidiameter, for a given Time, from the Tables. 335 PROB. X. To find the Apparent Obliquity of the Ecliptic, for a given time, from the Tables.337 nROB. XI. Given the Sun's Longitude and the Obliquity of the Ecliptic, to find his Right Ascension and Declination.. 338 PROB. XII. Given the Sun's Right Ascension and the Obliquity of the Ecliptic, to find his Longitude and Declination... 339 PROB. XIII. The Sun's Longitude and the Obliquity of the Ecliptic being given, to find the Angle of Position ib... PROS. XIV. To find from the Tables, the Moon's Longitude, Latitude, Equatorial Parallax, Semi-diameter, and Hourly Motions in Longitude and Latitude, for a given Time.... 340 PROB. XV. The Moon's Equatorial Parallax, and the Latitude of a Place, being given, to find the Reduced Parallax and Latitude 349 PRoB. XVI. To find the Longitude and Altitude of the Nonagesimal Degree of the Ecliptic, for a given Time and Place.. ib. PROB. XVII. To find the Apparent Longitude and Latitude, as affected by Parallax, and the Augmented Semi-diameter of the Moon; the Moon's True Longitude, Latitude, Horizontal Semidiameter, and Equatorial Parallax, and the Longitude and Altitude of the Nonagesimal Degree of the Ecliptic, being given. 352 CONTENTS. xi. PAGE PROB. XVIII. To find the Mean Right Ascension and Declination, or Longitude and Latitude of a Star, for a given Time, from the Tables.356 PROB. XIX. To find the Aberrations of a Star in Right Ascension and Declination for a given Day. 357 PROB. XX. To find the Nutations of a Star in Right Ascension and Declination, for a given Day.358 PROB. XXI. To find the Apparent Right Ascension and Declination of a Star, for a given Day... 36C PROB. XXII. To find the Aberrations of a Star in Longitude and Latitude, for a given Day. 361 PROB. XXIII. To find the Apparent Longitude and Latitude of a Star, for a given Day. ib. PROB. XXIV. To Compute the Longitude and Latitude of a Heavenly Body from its Right Ascension and Declination, the Obliquity of the Ecliptic being given. 362 PROB. XXV. To compute the Right Ascension and Declination of a Heavenly Body from its Longitude and Latitude, the Obliquity of the Ecliptic being given.363 PROB. XXVI. The Longitude and Declination of a Body being given, and also the Obliquity of the Ecliptic, to find the Angle of Position.......... 364 PROB. XXVII. To find from the Tables the Time of New or Full Moon, for a given Year and Month..... 365 PROB. XXVIII. To determine the number of Eclipses of the Sun and Moon that may be expected to occur in any given Year, and the Times nearly at which they will take place.. 368 PROB. XXIX. To calculate an Eclipse of the Moon.. 371 PROB. XXX. To calculate an Eclipse of the Sun, for a given Place 375 PROB. XXXI. To find the Moon's Longitude, &c., from the Nautical Almanac. 392 APPENDIX. TRIGONO0METRICAL FORMULAE. 395 I. Relative to a Single Arc or angle a.... ib. II. Relative to Two Arcs a and b, of which a is supposed to be the greater.. ib. III. Trigonometrical Series..... 397 IV. Differences of Trigonometrical Lines.... ib. V. Resolution of Right-angled Spherical Trianoles... ib. VI. Resolution of Oblique-angled Spherical Triangles.. 399 xiv CONTENTS. PAGH INVESTIGATION OF ASTRONOMICAL FORMULE.. 402 Formulae for the Parallax in Right Ascension and Declination, and in Longitude and Latitude.ib. Formulae for the Aberration in Longitude and Latitude, and in Right Ascension and Declination... 409 Formulae for the Nutation in Right Ascension and Declination 413 Formulae for computing the effects of the Oblateness of the Earth's Surface,- upon the Apparent Zenith Distance and Azimuth of a Star. 417 Solution of Kepler's Problem, by which a Body's Place is found in an Elliptical Orbit.418 Formulae for calculating the Parallax in Altitude of a Heavenly Body, from its True Zenith Distance.421 Formulae for computing the Annual Variations in the Right Ascension and Declination of a Heavenly Body.. 422 Formulae for computing the Heliocentric Longitude and Latitude and Radius Vector of a Planet, from its Geocentric Longitude and Latitude. 423 Formulae for computing the Geocentric Longitude and Latitude of a Planet, from its Heliocentric Longitude and Latitude and Radius Vector...... 424 Calculation of an Eclipse of the Sun. 426 Calculation of an Occultation.431 NOTE I. Construction of Tables 432 NOTE II. Relative to Sun's Spots. 434 NOTE III. Kirkwood's Law...... 436 NOTE IV. Relative to Origin of Comets. 437 NOTE V. Origin of Sidereal Systems...... 439 ASTRONOMY. PART I. SPHERICAiL ASTRONOMY. CHAPTER I. DEFINITIONS AND FUNDAMENTAL CONCEPTIONS; GENERALPHENOMENA OF THE HEAVENS. 1. The sun, moon, and stars-the luminous bodies disseminated through the heavens, or indefinite space surrounding the, earth-are called _Heavenly Bodies. The heavenly bodies, considered collectively are often termed the Heavens. The science which treats of the heavenly bodies is called Astronomy. It is divided into Theoretical and Practical Astronomy. Theoretical; Astronomy is divided into Spherical and Physical Astronomy. 2. Spherical Astronomy treats of the positions, motions, and distances of the heavenly bodies; and of their appearance, magnitude, form, and structure. It comprises the theory of the methods of observation and calculation by which the positions, motions, etc., of the heavenly bodies have been determined; and the whole body of exact knowledge thus acquired, which is often termed Descriptive Astronomy. Physical Astronomy investigates the general physical cause of the motions and constitution of the bodies of the material universe, and deduces from this general cause, called the force of universal gravitation, all the details of the celestial mechanism. Practical Astronomy treats of astronomical instruments, and astronomical observation; practical determinations, as of the latitude or longitude of a place, from instrumental observation; and the solution of astronomical problems with the aid of' tables. 2 FUNDAMENTAL CONCEPTIONS. 3. Formn of the Earth. We learn from the following cir. cumstances that the earth is a body of a globular form, insulated in space. (1.) When a vessel is receding from the land, an observer, from a point on the coast, first loses sight of the hull, then of the lower parts of the sails, and lastly of the topsails. It will be readily perceived, on glancing at Fig. 1, that no part of the earth could become interposed between the hull, and then the lower portions of the sails of a distant vessel, and the eye of the observer, if the sea were really what it appears to be, an indefinitely extended plane; also that if the earth be round, a receding ship should disappear in the manner it is actually observed to do, as the hull, mainsail, and topsails pass in succession below the line of sight tangent to the surface of the sea. If the observer take a more elevated position the ship should begin to sink out of sight at a greater distance, because the line of sight will touch the sea at a more distant point. FIG. 1. (2.) At sea the visible horizon, or the line bounding the visible portion of the earth's surface, is everywhere a circle, of a greater or less extent according to the altitude of the point of observation, and is on all sides equally depressed. To illustrate this proof, let BOA (Fig. 2) represent a portion of the earth's surG FIG. 2. face supposed to be spherical, P the position of the eye of the observer, and IDPC a horizontal line. If we conceive lines, such VISIBLE PORTION OF THE HEAVENS. 3 as PA and PB3, to be drawn through the point of observation P, tangent to the earth in every direction, it is plain that these lines will all touch the earth at the same distance from the observer, and therefore that the line AGB, conceived to be traced through all the points of contact, A, B1, etc., which would be the visible horizon, is a circle. It is also manifest that the angles of depression CPA, DPB, etc., of the horizon in different directions, will be equal; and that a greater portion of the earth's surface will be seen, and thus that the horizon will increase in extent, ill proportion as the altitude of the point of observation, P, increases. (3.) Navigators, as it is well known, have sailed entirely around the earth. These facts prove the surface of the sea to be convex, and the surface of the land conforms very nearly to that of the sea; for the elevations of the highest mountains bear an exceedingly small proportion to the dimensions of the whole earth. 4. Visible and Invisible Portions of the Heavens. If an indefinite number of lines, PA, PB, etc., be conceived to be drawn through the point of observation P, (Fig. 2,) touching the earth on all sides, a conical surface will be formed, having its vertex at P, and extending indefinitely into space. All heavenly bodies, which at any time are situated below this surface, have the earth interposed between them and the eye of the observer, and therefore cannot be seen. All bodies that are above tlhis surface, which send sufficient light to the eye, are visible. That portion of the heavens which is above this surface, presents tlhe appearance of a solid vault or canopy, resting upon the earth a-tt tlhe visible horizon, (see Fig. 2;) and to this vault the sun, moon, ncl stars seem to be.attached. It is hardly necessary to renlmarlk itli,:t this is an optical illusion. It will be seen in the sequel that thle heavenly bodies are distributed through space at vawrious distances from the earth, and that the distances of all of then are very great in comparison with the dimensions of the earth. It will suffice, in the conception of phenomena, to suppose the eye of the observer to be near the earth's surface, and that the imaginary conical surface above mentioned, which separates the visible from the invisible portion of the heavens, is a horizontal plane, confounded for a certain distance with the visible part of the earth. This is called the plane of the horizon, and sometimes the horizon simply. 5. Up and down, at any place on the earth's surface, are from and towards the surface; and thus at different places have evelry variety of absolute direction in space. 6.'The Sky. The earth is surrounded with a transparent gaseous medium, called the earth's atmosphere, estimated to be so)me fifty miles in height; through which all the heavenly bodies are seen. The atmosphere is not perfectly transparent, 4 GENERAL PHENOMENA. but shines throughout with light received from the heavenly bodies, and reflected from its particles; and thus forms a luminous canopy over our heads by day and by night. This is called the Skcy. It appears blue because this is the color of the atmosphere; that is, because the particles of the atmosphere reflect the blue rays more abundantly than any other color. By day the portion of the atmosphere which lies above the horizon is highly illuminated by the sun, and shines with so strong a light as to efface the stars. 7. Diurnal Motion of tlhe Heavens. The most conspicuous of the celestial phenomena, is a continual motion common to all the heavenly bodies, by which they are carried around the earth in regular succession. The daily circulation of the sun and moon about the earth is a fact recognised by all persons. If the heavens be attentively watched on any clear evening, it will soon be seen that the stars have a motion precisely similar to that of the sun and moon. To describe the phenomenon in detail, as witnessed at night: —if, on a clear night, we observe the heavens, we shall find that the stars, while they retain the same situations with respect to each other, undergo a continual change of position with respect to the earth. Some will be seen to ascend from a quarter called the East, being replaced by others that come into view, or rise; others, to descend towards the opposite quarter, the lIest, and to go out of view, or set: and if our observations be continued throughout the night, with the east on our left, and the west on our right, the stars which rise in the east will be seen to move in parallel circles, entirely across the visible heavens, and finally to set in the west. Each star will ascend in the heavens during the first half of its course, and descend during the remaining half. The greatest heights of the several stars will be different, but they will all be attained towards that part of the heavens which lies directly in front, called the South. If we now turn our backs to the south, and direct our attention to the opposite quarter, the XNorth, new phenomena will present themselves. Some stars will appear, as before, ascending, reaching their greatest heights, and descending; but other stars will be seen, further to the north, that never set, and which appear to revolve in circles, from east to west, about a certain star that seems to remain stationary. This seemingly stationary star is called the Pole Star; and the stars which revolve about it, and never set, are called C(ircumpolar Stars. It should be remarked, however, that the pole star, when accurately observed by means of instruments, is found not to be strictly stationary, but to describe a small circle about a point at a little distance from it as a fixed centre. This point is called the North Pole. It is, in reality, about the north pole, as thus defined, and not the pole star, that the apparent revolutions of the stars at the north are performed. At the corresponding hours of the following night the aspect of THE PLANETS. 5 the heavens will be the same, from which it appears that the stars return to the same position once in about 24 hours. It would seem, then, that the stars all appear to move from east to west exactly as if attached to the concave surface of a hollow sphere, which rotates in this direction about an axis passing through the station of the observer and the north pole of the heavens, in a space of time nearly equal to 24 hours. For the sake of simplicity this conception is generally adopted. This motion, common to all the heavenly bodies, is called their DiurnaGl iotion. It is ascertained, by certain accurate methods of observation and computation, that the diurnal motion of the stars is strictly untbforn and circular. S. Rotating Sphere of the Heavens. It is important to observe, that the conception of a single sphere to which the stars are supposed to be attached, will not represent their diurnal motion, as seen from every part of the earth's su7fcace, unless the sphere be supposed to be of such vast dimensions that the earth is comparatively but a mere point at its centre. A circle cut out of the heavens conceived to be a rotating sphere, by a plane passing through the axis of rotation, has a north and south direction. 9. Fixed Stars and Planets. The greater number of the stars constantly preserve the same relative positions, and are therefore called Tilxed Stars. But there are also many stars which are perpetually changing their places in the heavens. These are called Planets, or wandering stars. Each planet has received a distinctive name. For convenience of desirgnation they are divided into the two classes of Planets, and Planetoids or Minor Planets. The former class comaprises the planets Mercury, Venus, Mars, Jupiter, Saturn, UraniUs, and Neptune. The first five of these are visible to the naked eye; but Uranus and Neptune, and the planetoids, cannot be seen without the aid of a telescope; and have all been discovered since the year 17S0. Table II. (a), p. 5, &c. contains a list of the planetoids at present known, with the date and place of discovery of each, and the name of the discoverer. The number of planetoids hitherto discovered is ninety-one. Every year adds one or more to the list. 10. Distinctive Peculiarities of Different Planets. The planets are distinguishable from each other, either by a difference of aspect, or by a difference of' apparent motion with respect to the sun. Venus and Jupiter are the two most brilliant planets. They are quite similar in appearance, but their apparent motions with respect to the Sun are very different. Thus Venus never recedes beyond 40~ or 50~ from the Sun, while Jupiter is seen at every variety of angular distance from him. Mars is known by the ruddy color of his light. Saturn has a pale, dull aspect. 11. Apparent Mllotionls of the Planets. The apparent mo 6 GENERAL PHENOMENA. tion of each of the planets, is generally directed towards the east; but they are occasionally seen moving towards the west. As their easterly prevails over their westerly motion, they all, in process of time, accomplish a revolution around the earth. The periods of revolution are different for each planet. 12. Apparent Motions of the Sun and Mloon. The sun and moon, are also continually changing their places among the fixed stars. From repeated observations of its position among the stars, it is found that the moon has a progressive circular motion in the heavens from west to east, and completes a revolution around the earth in about 27 days. The mrotion of the sun, is also constantly progressive, and directed from west to east. This will appear on observing for a number of successive evenings, the stars which first become visible in that part of the heavens where the sun sets. It will be found that the stars, which in the first instance were observed to set just after the sun, soon cease to be visible, and are replaced by others that were seen immediately to the east of them; and that these in their turn, give place to others situated still further to the east. The sun must then be continually approaching the stars that lie on the eastern side of him. To make this more evident, let us suppose that the small circle aon (Fig. 3) repro / Xt Fig. 3. sents a section of the earth perpendicular to the axis of rotation of the imaginary sphere of the heavens, (8,*) conceived to pass through the earth's centre; the large circle H Z S a section of * Numbers thus inclosed in a parenthesis refer to a previous article. APPARENT MOTION OF THE SUN. 7 the heavens perpendicular to the same line,*and passing through the sun; and the right line H o r the plane of the horizon at thi station o. The direction of the diurnal motion is from H towards Z and S. Suppose that an hour or so after sunset the sun is at S, and that the star r is seen in the western horizon; also that the stars t, u, v, &c., are so distributed that the distances rt, ta, Utv, &c., are each equal to S r. Then, at the end of two or three weeks, an hour after sunset the star t will be in the horizon; at the end of another interval of two or three weeks the star u will be in the same situation at the same hour; at the end of another interval, the star v, &c. It is plain, then, that the sun must at the ends of these successive intervals be in the successive positions in the heavens, r, t, u, &c.; otherwise, when it is brought by its diurnal motion to the point S, below the horizon, the stars t, u, v, &c., could not be successively in the plane of the horizon at r. Whence it appears that the sun has a motion in the heavens in the direction S r t u v, opposite to that of the diurnal motion; that is, towards the east. Another proof of the progressive motion of the sun among the. stars from west to east, is found in the fact that the same stars rise and set earlier each successive night, and weel, and month, during the year. At the end of six months the same stars risc and set 12 hours earlier; which shows that the sun accomplishes half a revolution in this interval. At the end of a year, or of 365 days, the stars rise and set again at the same hours, from which it appears that the sun completes an entire revolution in the heavens in this period of time. It is to be observed that the sun does not advance directly towards the east. It has also some motion from south to north, and north to south. It is a matter of common observation that the sun is moving towards the north from winter to summer, and towards the south from summer to winter. When the place of the sun in the heavens is accurately found fiom day to day by certain methods of observation, hereafter to be explained, it appears that his path is an exact circle, inclined about 23~ to a circle running due east and west (8). 13. The Zodiac. The motions of the sun, moon, and planets, are for the most part confined to a certain zone, of about 185 in breadth, extending around the heavens obliquely from west to east, which has received the name of the Zodiac. 14. Counets. There is yet another class of bodies, called Comets, or hairy Stars, that have a motion among the fixed stars. They appear only occasionally in the heavens, and continue visible only for a few weeks or months. They shine with a diffusive nebulous light, and are commonly accompanied by a fainter divergent stream of similar light, called a tail. The motions of the comets are not restricted to the zodiac, 8 GENERAL PHENO MENA. These bodies are seet in all parts of the heavens, and moving in every variety of direction. 15. Satellites. By inspecting the planets with telescopes, it has been discovered that some of them are constantly attended by a greater or less number of small stars, whose positions are continually varying. These attendant stars are called Siztellites. The planets which have satellites are Jupiter, Saturn, Uranus, and Neptune. The satellites are sometimes called Secondary Planets; the planets upon which they attend being denominated Primary Planets. 16. Thle Solar Systesm. The sun and moon, the planets, (including the earth,) together with their satellites, and the comets, compose the Solacr System. From the consideration of the apparent motions and other phenomena of the solar system, several theories have been formed in relation to the arrangement and actual motions in space of the bodies that compose it. The theory, or system, now universally received, is, in its most prominent features, that which was taught by Copernicus in the sixteenth century, and which is known by the name of the Copernican System. It is as follows: The sun occupies a fixed centre, about which the planets (including the earth) revolve from west to east, in planes that are but slightly inclined to each other, and in the following order: Mercury, Venus, the Earth, Mars, the Planetoids, Jupiter, Saturn, Uranus, and Neptune. The earth rotates from west to east, about an axis inclined to the plane of its orbit about 66-0~, and which remains continually parallel to itself as the earth revolves around the sun. The moon revolves from west to east around the earth as a centre; and in like manner the satellites circulate from west to east around their primaries. Without the solar system, and at immense distances from it are the fixed stars. A motion in space from west to east, is a motion from right to left, as observed from a station within the orbit described, and on the north side of its plane. To obtain a clear conception of the motions of the planets, the reader should place himnself in imagination at or near the centre of the system, and on the north side of the plane of the earth's orbit within which the planets may all, for the present, be conceived to revolve. 17. ~yiubols. The principal planets, and the sun and moon, are often designated by the iollowing conventional characters or symbols. The Sun,... Jupiter,... Mercury,.... Saturn,. Venus,.... Uranus,. The Earth,... ~ Neptune,... MAars,.... The Moon,... 1S. aagerior, and Superior Planets. The two planet,% EFFECTS OF THE EARTH'S ROTATION. 9 Mlercury and Venus, whose orbits lie within the earth's orbit, are called Inferior Planets. The others are called S1perior Planets. The terms inferior and superior as here used, have merely the signification of lower and higher in place, or in position with respect to the sun, as compared with the earth. 19. Vast Distaance of thle Fixed Stars. The angular distance between any two fixed stars, is found to be the same from whatever point of the earth's surface it is measured. It follows, therefore; that the diameter of the earth is insensible, when compared with the distance of the fixed stars; and that with respect to the region of space which separates us from those bodies, the whole earth is a mere point. Moreover, the angular distance between any two fixed stars, is the same at whatever period of the year it is measured. Hence, if the earth revolves around the sun, its entire orbit must be insensible in comparison with the distance of the stars. 20. Explanation of the lDiurnaal otlioa oi tohe Heavens. On the hypothesis of the earth's rotation, the diurnal motion of the heavens is a mere illusion occasioned by the rotation of the earth. To explain this, suppose the axis of the earth to be prolonged till it intersects the heavens considered as concentric with the earth. Conceive a great circle to be traced through the two points of intersection, and the point directly overhead, and let the position of the stars be referred to this circle. It will be readily perceived that the relative motion of this circle and the stars will be the same, whether the circle rotates with the earth from west to east, or, the earth being stationary, the whole heavens rotate about the same axis and at the same rate in the opposite direction. Now, as the motion of the earth is perfectly equable, we are insensible of it, and therefore attribute the changes in the situations of the stars with respect to the earth to an actual motion of these bodies. It follows, then, that we must conceive the heavens to rotate as above menhioned, since, as we have seen, such a motion would give rise to the same changes of situation as the supposed rotation of the earth. It was stated (7) that the sphere of the heavens appears to rotate about a line passing through the north pole and the station of the observer; but, as the radius of the earth is insensible in comparison with the distance of the stars, an axis passing through the centre of the earth will apparently pass through the station of the observer, wherever this may be upon the earth's surface. 21. Explanation of the Sial's appareznt M!otion. Ve in like manner infer that the observed motion of the sun in the heavens is only an apparent motion, occasioned by the orbital motion of the earth. Let E, E' (Fig. 4) represent two positions of the earth in its orbit EE'E" about the sun S. When the earth it at E, the observer will refer the sun to that part of the 10 GENERAL PHENOMENA. heavens marked s; but when the earth is arrived at E', he will refer it to the part marked s'; and being in the mean time insensible of his own motion, the sun will appear to him to have described in the heavens the arc s s', just the same as if it had' E Fig. 4. actually passed over the arc SS' in space, and the earth had, during that time, remained quiescent at E. The motion of the sun from s towards s' will be from west to east, since the motion of the earth from E towards E' is in this direction. Moreover, as the axis of the earth is inclined to the plane of its orbit under an angle of 661-~ (16), the plane of the sun's apparent path, which is the siame as that of the earth's orbit, will be inclined 238- to a circle perpendicular to the earth's axis, or to a circle directed due east and west. CELESTIAL AND TERRESTIAL SPHERES. 11 CHAPTER II. CELESTIAL AND TERRESTRIAL SPHERES. 22. Celestial Sphere. In determining from observation the apparent positions and motions of the heavenly bodies, and, in general, in all investigations that have relation to their apparent positions and motions, astronomers conceive all these bodies, whatever may be their actual distance from the earth, to be referred to a spherical surface of an indefinitely great radius, having the station of the observer, or what comes to the very same thing, the centre of the earth, for its centre. This imaginary spherical surface is called the Sphere of the Heavens, or the Celestial ASphere. It is important to observe, that by reason of the great dimensions of this sphere, if two lines be drawn through any two points of the earth, and parallel to each other, they will, when indefinitely prolonged, meet it sensibly in the same point; and that, if two parallel planes be passed through any two points of the earth, they will intersect it sensibly in the same great circle. This amounts to saying that the earth, as compared to this slhere, is to be considered as a mere point at its centre. Not only is the size of the earth to be neglected in comparison with the celestial sphere, but also the size of the earth's orbit. Thus the supposed annual motion of the earth around the sun, does not change the point in which a line conceived to pass from any station upon the earth in any fixed direction into space, pierces the sphere of the heavens; nor the circle in which a plane cuts the same sphere. The fixed stars are so remote from the earth that observers, wherever situated upon the earth, and in the different positions of the earth in its orbit, refer them to the same points of the celestial sphere, (19.) The other heavenly bodies are referred by observers at different stations to points somewhat different. Definitions. For the purposes of observation and computation, certain imaginary points, lines, and circles, appertaining to the celestial sphere, are employed, which we shall now proceed to define and explain. (1.) The Vertical Line, at any place on the earth's surface, is the line of descent of a falling body, or the position assumed by a plumb-line when the plummet is freely suspended and at rest. Every plane that passes through the vertical line is called a 12 CELESTIAL SPHERE. Vertical Plane. Every plane that is perpendicular to the vertical line, is called a forirzontal Plane. (2.) The Sensible Horizon of a place on the earth's surface, is the circle in which a horizontal plane drawn through the place, cuts the celestial sphere. As its plane is tangent to the earth, it separates the visible fiom the invisible portion of the heavens, (4.) (8.) The Rational Horizon is a circle parallel to the former, the plane of which passes through the centre of the earth. The zone of the heavens comprehended between the sensible and rational horizon is imperceptible, or the two circles appear as one and the same at the distance of the earth. (4.) The Zenith of a place is the point in which the vertical line prolonged upwards pierces the celestial sphere. The point in which the vertical line, when produced downwards, intersects the celestial sphere, is called the Nadir. The zenith and nadir are the geometrical poles of the horizon. (5.) The Axis of the Hleavens is an imaginary right line passing through the north pole (7) and the centre of the earth. It is the line about which the apparent rotation of the heavens is performed. It is, also, on the hypothesis of the earth's rotation, the axis of rotation of the earth prolonged on to the heavens. (6.) The South Pole of the heavens is the point in which the axis of the heavens meets the southern part of the celestial sphere. ~~H AdRZi N FIG. 5. To illustrate the preceding definitions, let the inner circle nOs (Fig. 5) represent the earth, and the outer circle HZRN the sphere of the heavens; also let O be a point on the earth's sur CELESTIAL SPHERE. 13 face, and OZ the vertical line at the station O.-Then IIOR will be the plane of the sensible horizon, HCR the plane of the rational horizon, Z the zenith, and N the nadir; and if P be the north pole of the heavens, OP, or CP its parallel, will be the axis of the heavens, and P' the south pole. The lines CP and OP intersect the heavens in the same point, P; and the planes HIOR, and HCR, in the same circle, passing through the points H and R. Unless we are seeking for the exact apparent place in the heavens of some other heavenly body than a fixed star, we may conceive the observer to be stationed at the earth's centre, in which case OP will become the same as z CP, and HOR the same as 11CR; as represented in Fig. 6. In this / diagram, the circle of the horizon P/ being supposed to be viewed from a point above its plane, is represented by the ellipse HARa, Z A. and N are its geometrical poles. In the construction of Fig. 5, the / eye is supposed to be in the plane / / / of the horizon, and HARa is projected into its diameter 11CR. Every different place on the F surface of the earth has a different FIG. 6. zenith, and except in the case of diametrically opposite places, a P. E FIG. 7. different horizon. To illustrate this, let nesf (Fig. 7) represent the earth, and HZRP' the sphere of the heavens; then considering fue earth, and IIZRP' the sphere of the heavens; then considiering 14 CELESTIAL SPHERE. the foulr stations, e, O, n, and q, the zenith and horizon of the first will be respectively E and PeP'; of the second Z and HOR; of the third P and QnE; of the fourth Q and P'qP. The diametrically opposite places O and O' have the same rational horizon, viz. H1CR. The same is true of the places n and s, and e and q. Their rational horizons are respectively QCE and PCP'. (7.) Vertical circles are great circles passing through the zenith and nadir. They cut the horizon at right angles, and their planes. are vertical. Thus ZSMM (Fig. 6) represents a vertical circle passing through the star S, called the Vertical Circle of the Star. (8,) The iferidian of a place is that vertical circle which contains the north and south poles of the heavens. The plane of the meridian is called the Mleridian Plane. Thus, PZRP' is the meridian of the station C. The half HZR, above the horizon, is termed the Superior lerfidiatn, and the other half RNH, below the horizon, is termed the Inferior Mferidian. The two points, as H and R, in which the meridian cuts the horizon, are called the North and South Points of the horizon; and the line of intersection, as HCR, of the meridian plane with the plane of the horizon, is called the ilferidian Line, or the North anrd South Line. (9.) The Prime Vertical is the vertical circle whicli crosses the meridian at right angles. It cuts the horizon in two points, as e, w, called the East and WVest Points of the Horizon. (10.) The Altitude of any heavenly body is the arce of a vertical circle, intercepted between the centre of the body and the horizon, or the angle at the centre of the sphere, measured by this are. Thus, SM or MICS is the altitude of the star S. (11.) The Zenith Distance of a heavenly body is the are of a vertical circle, intercepted between its centre and the zenith; or the distance of the centre of the body from the zenith as meamured by the are of a great circle. Thus, ZS, or ZCS, is the zenith distance of the star S. It is obvious that the zenith distance and altitude of a body are com2plements of each other, and therefore when either one is known that the other may be found. (12.) The Azimuth of a heavenly body is the are of the horizon, intercepted between the meridian and the vertical circle passingl throulgh the centre of the body; or the angle comprehended between the meridian plane and the vertical plane containing the centre of the body. It is reckoned either from the north or from the south point, and each way from the meridian. HAI or 11CM represents the azimuth of the star S. The Azimuth and Altitude, or azimuntth and zenith clistance of a heavenly body, ascertain its position with respect to the horizon and meridian, and therefore its place in, the visible hemispqhere. Thus, the azimuth HUIt determines the position of the vertical cir DEFINITIONS. 15 cle ZSM of the star S with respect to the meridian ZPI, and the altitude MS, or the zenith distance ZS, the position of the star in this circle. (13.) The Amgnpl7tde of a heavenly body at its rising, is the arc of the horizon intercepted between the point where the body rises and the east point. Its amplitude at setting, is the arc of the horizon intercepted between the point where the body sets and the west point. It is reckoned towards the north, or towards the south, according as the point of rising or setting is north or. south of the east or west point. Thus, if aBSA represents the circle described by the star S in its diurnal motion, ea will be its amplitude at rising, and vwA its amplitude at setting. (14.) The Celestial Eqlator, or the Equinoctial, is a great circle of the celestial sphere, the plane of which is perpendicular to the axis of the heavens. The north and south poles of the heavens are therefore its geometrical poles. The celestial equator is represented in Fig. 6 by EwcQe. This circle is also frequently called the Eq'uator, simply. (15.) Parallels of Declination are small circles parallel to the celestial equator. aBSA represents the parallel of declination of the star S. The parallels of declination passing through the stars, are the circles described by the stars in their apparent diurnal motion. These, by way of abbreviation, we shall call Dirnctal Circles. (16.) Celestialct Mfericlianls, flour Circles, and Declination Circles, are different names given to all great circles which pass through the poles of the heavens, cutting the equator at right angles. PSP' is a celestial meridian. The angles comprehended between the hour circles and the meridian, reckoning firom the meridian towards the west, are called TIozur Angles, or HIorary Angles. (17.) The Ecliptic is that great circle of the heavens which the sun appears to describe in the course of the year. (18.) The Obliquity of the Ecliptic is the angle under which the ecliptic is inclined to the equator. Its amount is 23l-. (19.) The Eqztinoctial Points are the two points in which the ecliptic intersects the equator. That one of these points which the sun passes in the spring is called the Vernal Equinox, and the other, which is passed in the autumn, is called the Autumnal Equinox. These terms are also applied to the epochs when the sun is at the one or the other of these points. These epochs are, for the vernal equinox the 21st of March, and for the autumnal equinox the 23d of September, or thereabouts. (20.) The Solstitial Points are the two points of the ecliptic 90~ distant from the vernal and autumnal equinox. The one that lies to the north of the equator is called the Summer Solstice, and the other the WTVinter Solstice. The epochs of the sun's arrival at these points are also designated by the same terms. The summer 16 CELESTIAL SPHERE. solstice happens about the'21st of June, and the winter solstice about the 22d of December. (21.) The Equzinoctial Colulre is the celestial meridian passing through the equinoctial points; and the Solstitical Colure is the celestial mericlian passing through the solstitial points. (22.) The Polar Circles are parallels of declination at a distance fiom the poles equal to the obliquity of the ecliptic. The one about the north pole is called the Arctic Circle; the other, about the south pole, is called the Antcarctic Circle. The polar circles contain the geometrical poles of the ecliptic. (23.) The Tropics are parallels of declination at a distance from the equator equal to the obliquity of the ecliptic. That which is on the north side of the equator is called the Tropic of Cancer, and the other the Tropic oj Capricorn. The troIpics touch the ecliptic at the solstitial points. / T FIG. 8. Let C (Fig. 8) represent the centre of the earth and sphere, PCP' the axis of the heavens, EVQ\ the equator, WVTA the ecliptic, and K, K', its poles. Then will V be the vernal and A the autumnal equinox; W the winter, and T the summer solstice; PVP'A the equinoctial colure; PKW~K'T the solstiticllt colure; the angle TCQ, or its measure the arc TQ, the obliquity of the ecliptic; KmU, K'nm'U', the polar circles; -and TnZ, Wn'7', the ecliptic, a nd K, K', its poles. Then will V be the venal and A tropi7cs. DEFINITIONS. 17 It is important to observe that, agreeably to what has been stated (p. 11), the directions of the equator and ecliptic, of the equinoctial points, and of the other points and circles just defined and illustrated, are the same at any station upon the surface of the earth as at its centre. Thus, the equator lies always in the plane passing through the place of observation, wherever this may be, and parallel to the plane which, passing through the earth's centre, cuts the heavens in this circle. In like manner the ecliptic lies, everywhere, in a plane parallel to that which is conceived to pass through the centre of the earth and cut the heavens in this circle, and so for the other circles. (24.) The Zodiac (13) extends about 9~ on each side of the ecliptic. (25.) The ecliptic and zodiac are divided into twelve equal parts, called Szqns. Each sign contains 30~. The division commences at the vernal equinox. Setting out from this point, and following around from west to east, the Signs of the Zodiac, with the respective characters by which they are designated, are as follows: Aries r, Taurus 6, Gemini u, Cancer ~s, Leo St, Virgo -ay, Libra -, Scorpio 1l, Sagittarius t, Capricornus V3. Aquarius I, Pisces Ne. The first sik are called northern signs,8 being north of the equinoctial. The others are called southern signs. The vernal equinox corresponds to the first point of Aries, and the autumnal equinox to the first point of Libra. The summer solstice corresponds to the first point of Cancer, and the winter solstice to the first point of Capricornus. The mode of reckoning arcs on the ecliptic is by signs, degrees, minutes, &c. A motion in the heavens in the order of the signs, or from west to east, is called a direct motion, and a motion contrary to the order of the signs, or from east to west, is called a retrograde motion. (26.) The Right Ascension of a heavenly body is the are of the equator intercepted between the vernal equinox and the declination circle which passes through the centre of the body, as reckoned from the vernal equinox towards the east. It measures the inclination of the declination circle of the body to the equinoctial colure. Thus, PSR being the declination circle of' the star S, and V the place of the vernal equinox, VR is the right ascension of the star. It is the measure of the angle VPS. If' PS'R' be the declination circle of another star S', SPS', or RR', will be their difference of right ascension. (27.) The JDeclination of a heavenly body is the arc of a circle of declination, intercepted between the centre of the body a(nd the equator. It therefore expresses the distance (f the body from the equator. Thus, RS is the declination of the star S. Declination is North or South, according as the body is north or south of the equator. 2 18 CELESTIAL SPHERE. In the use of formulae, a south declination is regarded as negative. ]Tze rigiht ascension and declinattion of a heavenly body are two co-ordinates, which, talcen together, fix its position in the sphere of t/ie heavens: for they make known its situation with respect to two circles, the equinoctial colure, and the equator. Thus, VR and RS ascertain the position of the star S with respect to the circles PVP'A and VQAE. (28.) The Polar Distance of a heavenly body is the are of a declination circle, intercepted between the centre of the body andl the elevated pole. The polar distance is the complement of the declination, and, therefore, when either is known the other may be found. (29.) Circles of Latitude are great circles of the celestial sphere, which pass through the poles of the ecliptic, and therefore cut this circle at right angles. Thus, KSL represents a part of the circle of latitude of the star S. (30.) The Longitude of a heavenly body is the are of the ecliptic, intercepted between the vernal equinox and the circle of latitude which passes through the centre of the body, as reckoned firom the vernal equinox towards the east, or in the order of the signs. It measures the inclination of the circle of latitude of the body to the circle of latitude passing through the vernal equinox. Thus, VL is the longitude of the star S. It is the measure of the angle VKS. (31.) The Latitude of a heavenly body is the arc of a circle of latitude, intercepted between the centre of the body and the ecliptic. It therefore expresses the distance of the body from the ecliptic. Thus, LS is the latitude of the star S. Latitude is North or iSouth, according as the body is north or south of the ecliptic. In the use of formulae, a south latitude is affected with the minus sign. The longitude and latitude of a heavenly body are another set of co-ordinates, which serve to fix its position in the heavens. They ascertain its situation with respect to the circle of latitude passing through the vernal equinox, and the ecliptic. Thus, VL and LS fix the position of the star S, making known its situation with respect to the circles KVK'A and VTAW. (32.) The Angle of Position of a star is the angle included at the star between the circles of latitude and declination passing through it. PSK is the angle of position of the star S. (33.) The Astronomical Latitude, or the Latitude of a place, is the arc of the meridian intercepted between the zenith and the celestial equator. It is North or South, according as the zenith is north or south of the equator. ZE (Fig. 7) represents the latitude of the station O; QOE or QCE being the equator. 23. Terrestrial Sphere. The earth's surface, considered as DEFIN'iTIONS. 19 spherical (which accurate admeasurement, upon principles that will be explained in the sequel, shows it to be, very nearly), is called the Terrestrial Sphere. The following geometrical constructions appertain to the terrestrial sphere, as it is emnploved for the purposes of astronomy. It will be observed that they correspond to those of the celestial sphere above described, and are used for similar objects. Definiitions. (1.) The North and South Poles of the Earth are the two points in which the axis of the heavens intersects tfhe terrestrial sphere. They are also the extremities of the earth's axis of rotation. (2.) The Terrestrial Equator is the great circle in which a plane passing through the centre of the earth, and perpendicular to tile axis of the heavens and earth, cuts the terrestrial sphere. The terrestrial and the celestial equator, then, lie in the same plane. The poles of the earth are the geometrical poles of the terrestrial equator. The two hemispheres into which the terrestrial equator divides the earth are called, respectively, the Northern Hfemvi-,sphere and the Southern Hemisphere. (3.) Terrestrial Meridians are great circles of the terrestrial sphere, passing through the north and south poles of the earth, and cutting the equator at right angles. Every plane that passes through the axis of the heavens cuts the celestial sphere in a celestial meridian, and the terrestrial sphere in a terrestrial meridian. Let PP' (Fig. 9) represent the axis of the heavens, O the centre of the earth, and p and p' its poles. Then, elq will represent the terrestrial equator (ELQ representing the celestial equator); and pep' and psp' terrestrial nmeridians (PEP' and PSP' representing celestial meridians). (4.) The Reduced Latitude of a place on the earth's surface is the are of a terrestrial meridian, intercepted between the place and the equator, or the angle at the centre of the earth measured by this are. Thus, oe, or the angle oOe, is the reduced latitude of the place o. Latitude is Norlth or ~South, according as the place is north or south of the equator. The reduced latitude differs somewhat from the astronomical latitude, by reason of the slight deviation of the earth fiorn a spherical form. Their difference is called the Red(ltction of Latitude. (5.) Parallels of Latitude are small circles of the terrestrial sphere parallel to the equator. Every point of a parallel of latitude has the same latitude. The parallels of latitude which correspond in situation widtl the polar circles and tropics in the heavens, have receivedl the same appellations as these circles. (See defs. 22, 23, p. 16.) (6.) The Longitude of a place on the earth's surface is the inclination of its meridian to that of some particular station, fixed 20 TERRESTRIAL SPHERE. upon as a circle to reckon from, and called the First llfericdian. It is measured by the arc of the equator, intercepted between the first meridian and the meridian passing through the place, and is called East or West, according as the latter meridian is to the east FIG. 9. or to the west of the first meridian. Thus, if pqp' be supposed to represent the first meridian, the angle spq, or the are ql, will be the longitude of the place s. Different nations have, for the most part, adopted different first meridians. The English use the meridian which passes through the Royal Observatory at Greenwich, near London; and the French, the meridian of the Imperial Observatory at Paris. In the United States the longitude is, for astronomical purposes, reckoned from the meridian of Washington, or of Greenwich. The longitude and latitude of a place designate its situation on the earth's surface. They are precisely analogous to the right ascension and declination of a sitar in the heavens. 24. Altitude of the Pole. The diagram (see Fig. 6) that was made use of in Art. 22, in illustrating the description of the circles of the celestial sphere, represents the aspect of this ALTITUDE OF THE POLE. 21 sphere at a place at which the north pole of the heavens is somewhere between the zenith and horizon. Such is the position of the north pole at all places situated between the equator and the r north pole of the earth. For, let 0 (Fig. 10) represent a place on the earth's surface, HOR the horizon, a / 1" OZ the vertical, UHZR the me- x ridian, and ZE the latitude. QOE will then represent the \S equinoctial, and P, 90~ distant from E and on the su- Q perior meridian, the elevated FIG. 10. pole. Now we have HP ZH ZP - 90~-ZP; ZE - PE -ZP - 90 -ZP. Whence, HP ZE. Thus, the altitude of the pole is everywahere equal to the latitude of the place. It follows, therefore, that in proceeding from the equator to the north pole, the altitude of the north pole of the heavens will gradually increase from 0~ to 90~. By inspecting Fig. 7, it will be seen that this increase of the altitude of the pole in going north, is owing to the fact that in following the curved surface of the earth the horizon, which is continually tangent to the earth, is constantly more and more depressed towards the north, while the absolute direction of the pole remains unaltered. If the spectator is in the southern hemisphere, the elevated pole, as it is always on the opposite side of the zenith from the equator, will be the south pole. At corresponding situations of the spectator, it will obviously have the same altitude as the north pole in the northern hemisphere. 25. Oblique Sphere. Let us now inquire minutely into the principal circumstances of the diurnal motion of the stars, as it is seen by a spectator situated somewhere between the equator and the north pole. And in the first place, it is a simple corollary from the proposition just established, that the parallel of declination to the north, whose polar distance is equal to the latitude of the place, will lie entirely above the horizon, and just touch it at the north point. This circle is called the circle of perpetual apl2parition; the line aH (Fig. 11) represents its projection on the meridian plane. The stars comprehended between it and the north pole will never set. As the depression of the south pole is equal to the altitude of the north pole, the parallel of declination oR, at a distance from the south pole equal to the latitude of the 22 CELESTIAL SPHERE. place, will lie entirely below the horizon, and just touch it at the south point. The parallel thus situated is called the circle of perpetual occultation. The stars comprehended between it Z', and the south pole will never rise. PI,,/ / a/ /sv The celestial equator (which 7i/> H~ a// \passes through the east and west points) will intersect the /n, 1 meridian at a point E, whose zenith distance ZE is equal to the latitude of the place (deft 33, Art. 22), and consequently, /S, /whose altitude RE is eqeal to the co latitude oftheplace. There_-~ _~ o fore, in the situation of the observer above supposed, the FIG. 11. equator QOE, passings to the south of the zenith, will, together with the diurnal circles nr, st, etc., which are all parallel to it, be obliquIely inclined to the horizon, making with it an angle equal to the co-latitude of the place. As the centres c, c', etc., of' the diurnal circles lie on the axis of the heavens, which is inclined to the horizon, all diurnal circles situated between the two circles of perpetual apparition and occultation, allI and oR, with the exception of the equator, will be divided unequally by the horizon. The greater parts of the circles ur, n'r', etc., to the north of the equator, will be above the horizon; and the greater parts of the circles st, s't', etc., to the south of the equator, will be below the horizon. Therefore, while the stars situated in the equator will remain an equal length of time above and below the horizon, those to the north of the equator will remain a longer time above the horizon than below it; and those to the south of the equator, on the contrary, a longer time below the horizon than above it. It is also obvious, from the manner in which the horizon cuts the different diurnal circles, that the disparity between the intervals of time that a star remains above and below the horizon will be the greater the more distant it is from the equator. Again, the stars will all culmrinate, or attain to their greatest altitude, in the meridian: for, since the meridian crosses the diurnal circles at right angles, they will have the least zenith distance when in this circle. Moreover, as the meridian bisects the portions of the diurnal circles which lie above the horizon, the stars will all employ the same length of time in passing from the eastern horizon to the meridian, as in passing from the meridian to the western horizon. The circumpolar stars will pass the meridian twice in 24 hours; once above, anld once beioi the pole. These nmeridian passages are called, respectively, ASPECTS OF THE CELESTIAL SPHERE. 23 Upper and Lower Caulrniations, or Inferior and Superior Transits. It will be observed, that in travelling towards the north the circles of perpetual apparition and occultation, together with those portions of the heavens about the poles which are constantly visible and invisible, are continually on the increase. It is evident, from what is stated in Art. 24, that the circumstances of the diurnal motion will be the same at any place in the southern hemisphere, as at the place which has the same latitude in the northern. The celestial sphere in the position relative to the horizon which we have now been considering, which obtains at all places situated between the equator and either pole, is called an Oblique Sphere, because all bodies rise and set obliquely to the horizon. 26. Right Sphere. When the spectator is situated on the equator, both the celestial poles will be in his horizon (24), and therefore the celestial equator and the diurnal circles in general will be perpendicular to the horizon. This situation of the sphere is called a Right Sphere, for the reason that all bodies rise and set at right angles with the horizon. It is represented in Fig. 12. As the diurnal circles are bisected by the horizon, the stars will all remain the same length of time above as below the horizon. 12'1) t!t FIG. 12. FIG. 13. 27. Parallel Sphere. If the observer be at either of the poles, the elevated pole of the heavens will be in his zenith (24), and consequently the celestial equator will be in his horizon. The stars will move in circles parallel to the horizon, and the whole hemisphere, on the side of the elevated pole, will be continually visible, while the other hemisphere will be continually invisible. This is called a Parallel Sphere. It is represented in Fig. 13. 24 ASTRONOMICAL INSTRUMENTS. CHAPTER III. ASTRONOMICAL INSTRUMENTS.-ASTRONOMICAL OBSERVATION. 28. Astronomical instruments are used to measure arcs of the celestial sphere, or their corresponding angles at a station on the earth. They consist, essentially, of a refracting telescope turning upon an axis, and a graduated limb, or two graduated limbs at right angles to each other, to indicate the angle passed over by the telescope. When designed to mea-,sure angles in the meridian plane, the axis of rotation is horizontal, and a single vertical limb is used. 29. The Reticie. At the common focus of the object-glass and eye-glass of the telescope, is a piece of apparatus called a reticle, the design of which is to furnish a definite line of sight. In its simplest form it consists of a flat circular ring, attached to which are two very fine wires, or spider lines, crossing each other at right angles in its centre (Fig. 14). The line passing through the point of intersection of the cross wires and the centre of the object-glass, indefinitely prolonged, is the line of sight, or Line of Cotlimzation of the telescope. The reticle can be moved up or down, or to the right or left, by adjusting screws; and the line of collimation thus made perpendicular to ______ These screws are shown at aa and bb, Fig. 14. They pass through narrow,g,;,a slits in the tube of' the telescope, so that they can be turned from without, and each pair of screws, aa or bb, gives a motion to the wire-plate. The line passb ing through the centre of the eye-glass FIG. 14. and the centre of the object-glass, or the oltical axis of the telescope, is perpendicular, or nearly so, to the axis of rotation. When it is in that precise position and the line of collimation accurately adjusted, the two lines will coincide. But it is not important in the use of instruments that this coincidence should be perfect. It is suffi. cient if the line of collimation is perpendicular to the axis of rotation. MOVABLE MICROMETER WIRE. 25?eticle lTube. The reticle is placed in a tube, which slides in the lower end of the principal tube of the telescope. The eye-piece is inserted in the outer end of this tube; and can be pressed in or drawn out until the wires of the reticle are distinctly seen. In making an observation, the reticle tube with the eye-piece is moved out or in if necessary, by means of a milled head screw that works a pinion in a rack connected with the tube, until the image of the star, formed by the object-glass, falls upon the wires, when both the wires and the star will be distinctly seen. The reticle tube can also be turned around until the wires have the right direction in the field of view. A star is known to be on the line of collimation when it is bisected by each of the two cross wires. 30. hnlproved Formn of Reticle. The form of reticle just described is now attached only to portable instruments. That which is adapted to the larger instruments of an observatory, differs from this in the number of the wires, the form of the wire-plate, and the mode of attaching the plate to its tube and of adjusting the wires. It has several parallel and equidistant wires, crossed at right angles by a, single wire, or more commonly by two very close parallel wires (Fig. 15). In meridian instruments, and those for measuring altitudes, the single wire, or the equivalent pair of close parallel wires, is made horizontal. The middle wire of the others is brought into the meridian plane; these __ are called transit wires. The star is made to pass through the field between the b, 1 Il d two horizontal wires. The point of the middle transit wire that lies midway FIG. 15. between the two horizontal ones, corresponds to the point of intersection of the two cross wires in Fig 14. The wire-plate lies within a frame fastened across the outer end of the reticle tube (see Fig. 19, p. 31), and is adjusted by screws that act upon pieces projecting from its outer rim. The eye-piece is screwed into a plate that slides within the same transverse fralne, and is moved by means of a screw. By turning this screw the eye-piece may be brought into such positions that that the star observed is kept in the middle of the field of view. 31. liovable Iriecrometer Wire. In the focus of the eyeglass there is often fastened to a transverse sliding plate. and movable with it, a wire at right angles to tile direction in whiich the plate is moved by a screw. The screw has the form of the micrometer-screw with graduated head, soon to be described. This wire is called the movable micromneter-wire, and the whole apparatus, being especially designed for the measurement of small angular distances, is called a Micrometer. The same name 26 ASTRONOMICAL INSTRUMENTS. is sometimes, though improperly, given to the reticle alone, when the movable wire is not employed. 32. Reading off the Angle. The telescope, and the graduated limb which is perpendicular to the axis of rotation, are, in most instruments, firmly attached to each other, and turn together about this axis. The limb glides past a fixed index. The angle read off is that which is pointed out by the index. The limbs of even the largest instruments, are not divided into smaller parts than 2'; but by means of certain subsidiary contrivances, the angle may, with some instruments, be read off to within a fraction of a second. The principal contrivances in use for increasing the accuracy of the reading off of angles, are the Vernier, and the Reading Aficroscope. a3. The Vernlier is simply the index-plate so graduated that a certain number of its divisions occupy the same space as a number one less on the limb. A division, or space on the vernier, will therefore be less, by a certain amount, say 1', than a division on the limb. The index will, therefore, have moved 1', or 2', or 3', etc., beyond the last line of division on the limb, passed before it became stationary, according as the first, second, third, etc., line of division of the vernier beyond the index coincides with a line of division on the limb. In Fig. 16, MN represents a portion of the limb of an instrument, divided into degrees and 10' spaces; V the Vernier, ten 1A FIG. 16. equal divisions of which have the same extent as nine of the 10' spaces on the limb; and A the index-arm, which is here supposed to revolve with the telescope. The index-point, or zero of the vernier, is seen to be just beyond the point 30~ 10' on the limb; and on looking along the vernier, we perceive that the fourth line of division from the zero coincides with one of the lines of division of the limb. The zero of the vernier is, therefore, 4' beyond the point 300 10' on the limb; and the whole reading is 30~ 14'. It is here implied that one of the divisions of the vernier is less by 1' than the 10' space on the limb. Tc THE READING MICROSCOPE. 27 show this, let x = the number of minutes in a division of the vernier, then by what is stated above, 10 x = 9 x 10'; whence x = 9', and 10' - x = 1'. By increasing the number of divisions of the vernier that corresponds to a number one less on the limb, the angle may be read off more accurately. For example, if sixty divisions of the vernier were made equal to fifty-nine of the limb, a division of the vernier would be 10" less in value than a division of the limb; and the reading would be within 10" of the exact position of the zero of the vernier on the limb. But when the highest degree of accuracy is sought for, as in the large, fixed instruments of an observatory, the angle is read off by means of the Reading Microscope, instead of the vernier. 34. The Reading Mlieroscope is a compound microscope, firmly fixed opposite to the limb, and furnished with cross wires in its focus, which are movable by a fine-threaded Micrometer Screw. This is a screw to the head of which is attached a graduated cylindrical head, that moves past a fixed index, to measure, by means of the turns and parts of a turn of the screw, the exact distance through which it is moved in the direction of its axis. In A Fig. 17, AC is the microscope, and MN a portion of the limb seen edgewise. At D, on the optical axis, is the conjugate focus of the object-glass C; when the S T -T > microscope is set at the proper distance from the limb, it is coincident with the E. focus of the eye-glass A. An image of a portion of the limb below C is formed at this point, and is seen distinctly through the eye-glass. ST is a box containing the sliding frame to which the crosswires are attached; G, the milled head of the screw; EF, the graduated cylin- C drical head, called the graduated head M AL of the screw; and i the fixed index. IG. 17. The cross-wires, with the connected apparatus for giving motion to them, and measuring the distance through which they are moved, is called a Micrometer. Fig. 18 shows, upon an enlarged scale, the whole of the micrometer, as it would appear if viewed from A in Fig. 17. aa is the sliding frame to which the cross-wires are attached; c is the end of the screw working into this frame; and bb, spiral springs between the end of the frame and the end of the box, to prevent dead motion of the screw, and give more steadiness and regularity to the movements of the frame under the action of the screw. The divisions of the limb are shown as short, heavy, equidis 28 ASTRONOMICAL INSTRUMENTS. tant lines. The cross-wires are the fine lines intersecting under an acute angle. A wire-pointer, not shown in the figure, in a position such that its prolongation would bisect this acute angle, FIG. 18. is generally used. On one side of the field is shown a notched scale of teeth, called a comb-scale; the distance from the middle of one notch to the middle of the next being the samne as that between the threads of the screw. The wire-pointer is moved over this scale along with the cross-wires. This scale is attached to the micrometer-box, and does not move with the cross-wires. The number of teeth passed by the intersection of the wires, therefore, shows the number of turns made by the screw; and the fractional part of a turn is indicated by the number of divisions of the graduated head that move past the index (i, Fig. 17), from the zero. If one revolution of the screw answers to a space of 1' on the limb of the instrument, the number of teeth passed by the intersection of the wires will be the number of minutes of arc through which it is moved; and if the head of the screw is divided into sixty equal parts, the line of division opposite the fixed index will give the number of seconds to be added to the minutes, to determine the additional space moved over. In reading off the angle the observer looks through the microscope at the limb. The point of intersection of the cross-wires of the microscope, when brought against the central notch of the scale, is a fixed point of reference, like the zero of a fixed vernierplate. When the angle is to be read, this point will not, in general, fall upon one of the lines of division of the limb. By turning the micrometer-screw, the intersection of the wires is moved over the space which separates it from the line of division beyond which it falls; the number of teeth passed on the notched scale, will then be the number of minutes, and the number of the division of the screw-head opposite the index, will be the number of seconds, to be added to the angle taken from the limb. To increase the accuracy of the reading, and determination of an angle, several microscopes are used, set opposite the limb at equally distant points. The fraction of a division in the reading ASTRONOMICAL OBSERVATIONS. 29 is thus measured at different points of the circle, and the mean of tile different measures is taken. Four reading microscopes, sometimes six, or even a greater number, are thus used. The whole degrees and minutes are read at only one of the microscopes. 35. Accuracy of Instruments. It is obvious that, other things being the same, instruments are accurate in proportion to the power of the telescope and the size of the limb. The large instruments now in use in astronomical observatories, are relied upon as furnishing angles to within a fraction of 1". 36. Time is an essential element in astronomical observations. Three different kinds of time are employed by astronomers; Sidereal, Apparent or True Solar, and Mlean Solar Time. Sidereal T'ime is time as measured by the diurnal motion of the stars; or, as it is now considered, of the vernal equinox. A Sidereal Day is the interval between two successive meridian transits of a star; or, as now defined, the interval between two successive transits of the vernal equinox. It commences at the instant when the vernal equinox is on the superior meridian, and is divided into 24 Sidereal Hours. Apparent, or True Solar Time, is deduced from observations upon the sun. An Apparent Solar Day is the interval between two successive meridian passages of the sun's centre, commencing when the sun is on the superior meridian. It appears from observation that it is a little longer than a sidereal day, and that its length is variable during the year. It is divided into 24 Apparent Solar Hours. Aiean Solar Time is measured by the diurnal motion of an imaginary sun, called the MAean Sun, conceived to move uniformly from west to east in the equator, with the real sun's mean motion in the ecliptic, and to have at all times a right ascension equal to the sun's mean longitude. AMean Solar Day commences when the mean sun is on the superior meridian, and is divided into 24 Mean Solar tTours. Since the mean sun moves uniformly and directly towards the east, the length of the mean solar day must be invariable. The Astronomical lDay commences at noon, and is divided into 24 hours; but the Calendar Day begins at midnight, and is divided into two portions of 12 hours each. 37. Astronomical Observations are, for the most part, made in the plane of the meridian. But some of minor importance are made out of this plane. The chief instruments employed for meridian observations, are the Meridian Circle, and the Transit Instrument, used in connection with the Astronomical Clock. These are the capital instruments of an observatory, inasmuch as they serve, as will soon be explained, for the determination of the places of the heavenly bodies, which are the fundamental data of astronomical science. The principal instru 30 ASTRONOMICAL INSTRUMENTS. ments used for making observations out of the meridian plane, are the Altitude and Azimuth] Instrument, the Eqaatorial, and the Sextant. THE TRANSIT INSTRUMENT. 3S. The Transit Instrument, or Transit, is an instrument em. ployed, in connection with a clock, for observing the passage of celestial objects across the meridian; either for the purpose of determining their right ascension, or obtaining the correct time. It is constructed of various dimensions, from a focal length of 20 inches, to one of 10 feet. The larger and more perfect instruments are permanently fixed in the meridian plane, and rest upon stone piers. The smaller ones are mounted on portable stands. Fig. 19 represents a fixed transit instrument in its most approved form. It is a sketch of the meridian transit instrument of the Washington Observatory, made by Ertel & Sons, Munich. The telescope has a focal length of 85 inches, with a clear aperture of 5.3 inches. TT is the telescope, firmly fixed to an inflexible axis, AA, at right angles to its length. The axis consists of two hollow cones, AA, proceeding from the opposite sides of a hollow cube, AI; the whole being cast in one piece. The tube of the telescope is composed of two tubes, which are fastened by screws to the other two faces of the cube, M. The axis terminates in two steel pivots, V, accurately turned to the cylindrical shape, and of equal size. These pivots rest on two angular bearings, in form like the upper part of a Y, and called Y's. The Y's are notches cut in two blocks of metal, set in metallic boxes; the latter being imbedded in the tops of the stone piers PP. Sufficient play is given to the blocks in their boxes to allow one of the pivots to be raised or lowered, and the other to be moved to the right or left by means of adjusting screws, that give a motion to the blocks. To relieve the pivots of a portion of the weight of the telescope, a brass pillar, S, is firmly set upon the top of each pier, and furnishes a fulcrum to a lever, R., firom one end of which depends a strong brass hook that supports the friction rollers X, under the end of the axis. A counterpoise, XV, is adapted to the other end of the lever, whiclh serves to sustain the greater part of the weight of the telescope, and leaves only a sufficient pressure at the pivots to secure a perfect contact with the Y's. This not only saves the pivots from wear, but gives the greatest possible freedom of motion to the telescope —tle lightest touch of the finger being sufficient to rotate the instrument upon the friction rollers on which the axis chiefly rests. Illumination of the Reticle- Wires. The pivots are perforated, to admit the light of a lamp placed on the top of either pier. TIIE TRANSIT INSTRUMENT.;1 Zq...............................:I VY~~~~~~FG M 32 ASTRONOMICAL INSTRUMENTS. The light is received upon a plane metallic speculum, set within the hollow cube, M, at an angle of 45~ to the axis of the telescope, and is reflected to the eye-glass; thus illuminating the field of view, and exhibiting the wires of the reticle, at m, as dark lines on a comparatively bright ground. The reflector has an elliptical opening at its centre, to permit the light that enters the telescope from a star, to pass on to the eye-glass. In observing small stars the wires are illuminated from the side of the eyeglass, by two small lamps (omitted in the drawing) suspended upon the telescope, near the eye-piece, which throw their light obliquely upon the wires, through openings in the eye-tube, without illuminating the field. The wires are thus made to appear as bright lines on a dark ground. The reticle has seven transit wires, placed at equal intervals, and two horizontal ones, between which the star is made to pass (30). Finding Circles. On each side of the eye-end of the telescope, is fastened a small vertical graduated circle, F, about the centre of which turns freely an index-arm which carries a spirit-level and a vernier. This piece of apparatus is called a Finding Circle, or a Finder. An outline sketch of the finding circle, in one of its forms, is shown in Fig. 20; a is the index-arm, and I FIG. 20. the level fastened at right angles to this, at the centre of the divided circle. Both turn freely about this centre. At the lower end of a is a vernier, and also a clamp and tangent-screw (not shown in the figure). The finding circles attached to the present instrument have a vernier at each end of a horizontal arm that carries the level; and the vertical arm serves only for clamping, and the tangent-screw motion. By means of the finder, the telescope can be set to any given altitude or zenith distance, preparatory to an observation of the meridian passage of a star. This is done by setting the vernier of the finder to the given angle, and then depressing the eye-end of the telescope until the spirit-level is horizontal. In accomplishing this, the handles, BB and D, are used. The handle D acts upon a clamp that fastens the rotation axis. When the telescope has been depressed nearly to the required position, it is ADJUSTMENTS. 33 clamped by this handle, and the handles BB, which are connected with tangent-screws, serve to give the telescope a slow motion in altitude. By the same means, when the star to be observed enters the field of view of the telescope, it can be made to pass through the middle of the field. A Reversing Apparatus, or Car, with which the instrument may be lifted from the Y's, and the rotation axis reversed, is shown at Rt. It is mounted on grooved wheels that run upon two rails laid in the observatory floor, between the piers PP. The telescope having been placed in a horizontal position, the car is brought directly beneath the axis. By turning the crank h, acting upon two bevelled wheels, e and f, the latter of which has an internal screw engaging in an external screw upon the lower end of the vertical shaft t, two forked arms, aa, are lifted and brought into contact with the axis at AA; then, continuing the motion, the telescope is lifted sufficiently for the axis to clear the Y's and the fiiction rollers at XX. The car is then rolled out from between the piers, hearing the telescope with it; the instrument is turned half around upon the vertical shaft, the car rolled back to its former position, and the axis lowered into the Y's. The exact semi-revolution is determined by the stop, d. An observing couch, C, runs on the rails between the piers. It is so arranged that the observer, reclining upon it, may give his head any required elevation; and thus promotes facility and. accuracy of observation, by giving greater steadiness to the head,. and relieving the observer of the fatigue of a constrained position when the telescope is directed upon stars at high altitudes. L is a striding level, which is used in levelling the rotation axis.. 39. Adjustnents of the Transit. To secure accurate observations with the transit, three adjustments of the instrument are necessary: 1. The axis of rotation is to be brought into a horizontal position. 2. The line of collimation is to be made perpendicular to the axis of rotation. 3. The line of collimation is to be brought accurately into the meridian plane. When these adjustments have been effected, the line of sight:: will lie in the plane of the meridian in every position given to, the telescope. 49. First Adjustment. The first adjustment is efetbcted by means of the striding level, L, which is applied to the pivots, VV; - the feet of the level having the form of an inverted V for this purpose. By alternately working the screws that raise or depress one of the pivots, and the adjusting screws of the spirit-level, until the level is horizontal, whichever leg rests upon the eastern end of the axis, the axis may be made truly horizontal. Instead of attempting to secure in this way a perfect adjustment of the 3 34 ASTRONOMICAL INSTRUMIENTS. axis, it is found more convenient to determine the inclination of the axis to the horizon, by means of the scale marked off upon the tube of the spirit level, and calculate the error that is entailed from this cause, upon the observation. 41. Second Adjtistiealt. The secon(d, or collimation adjustmnent, is now generally made by means of special contrivances for the purpose, but it may also be accomplished in the following manner. Bring the telescope into a horizontal position, and direct it upon a well-define(l point of a distant terrestrial object. Then, by means of the reversing apparatus raise the telescope from the Y's, and replace it with the ends of the axis reversed. Bring the telescope again into a horizontal position, and note whether it is directed upon the same point as before. If not, bring it half-way back to this point by the adjusting screws of the reticle, ond the remaining distance by the screws that give a lateral motior:o one end of the rotation axis. By one or more repetitions.' this process, the desired adjustment may be effected. The better plan, and the one ordinarily adopted by astronomical observers, is, after the error of collimation has been reduced to a small amount, to determine its value, and allow for it. This can readily be done when the reticle is provided with a movalble microrneter-wire (31). It is only necessary to measure, with the micrometer, the distance of the point observed from the middle wire of the reticle in both positions of the telescope, convert each of the measured distances, expressed in revolutions of the screw-head, into their equivalent angular measures, and take the half difference of the two results. This will be the error of.col i mation. The opportunity of reversing the instrument also enables the observer to determine the correction for inecuality of the pivots; that is, the inclination of the mathematical axis of rotation to the horizon that may result from any such inequality. This correction is equal to one quarter of the difference between the inclinations of the line on which the feet of the level virtually rest, as determined by the level, in both positions of the telescope. Collimating Eye-Piece. The most convenient method of determining the error of collimation is by making a certain observation with what is x... called the collimating eye-piece, substituted for the ordinary eye-piece of the telescope (Fig. lt 1). This differs from the common eye-piece in having an opening in one side of the tube, and a metallic reflector, of the form of an ellipSUN - E Xtical ring, set obliquely within the tube, to reflect the light of a lamp upon the wires of the FIG. 21. micrometer. rl'he observation to be made with it consists simply in looking vertically downward through the telescope at the image of the micrometer THIRD ADJUSTMENT. 35 wires, reflected from a basin of mercury placed on an immovable stone slab under the telescope. If the axis has been truly levelled, the error of collimation will be half the distance between the middle wire, as seen directly, and its reflected image. This distance can be measured by means of the movable wire of the micrometer. By working the adjusting screws of the reticle, and the vertical adjusting screws of the axis of rotation, the interval between the wire and its image may be made to disappear entirely; when the axis will be truly level, and the line of collimation in perfect adjustment. 42. Third Adjiastmnent. The piers must first be established in such positions that the telescope, when the pivot ends of' the axis have been placed in the Y's, and the axis levelled, will lie nearly in the meridian plane. This may be accomplished by bringing the telescope, after repeated trials, into such a position that it will be directed upon the pole-star when it is on the meridian. By referring to a map of' the stars, it may be seen that the pole-star will be nearly on the meridian when a straight line from it to a point midway between the fifth and sixth stars, designated as E and C, in the constellation of the Great Bear, is in a vertical position. The pole-star is also known to be on the ineridian when it attains to its greatest, or least altitude. When the instrument has thus been approximantely established, it may be more accurately adjusted to the meridian, with the aid of the screws that give a horizontal motion to one end of the axis. For this purpose observations may be made upon the pole-star at its upper and lower meridian transits, and the telesco)pe moved in azirnuth, until the interval between the upper and lower transit is made equal to that between the lower and upper transit. The more convenient method is to ascertain fi-om existing tables the time of the meridian passage of some known star, and bring the middle wire of the telescope upon the star at the instant of the transit. In order to effect this, the error of the timepiece must be known. If it indicates sidereal time, its error may be approximately determined with the instrument that is being established, by selecting a star that passes the meridian near the zenith, and noting the time of its transit across the middle wire of the telescope. This time should differ very little from the instant of the true meridian passage, as determined from astronomical tables; the difference will then be the error of the timepiece, nearly. The subsequent observations for adjustment to the meridian plane should be made upon stalrs remote from the zenith (the pole-star in preference). This process may be many times repeated, until the line of collimation of the transit telescope is brought, with all attainable accuracy, intothe meridian plane. Or, the error of the adjustment may be calculated from the results of the observations upon the star near the zenith and the pole-star, and allowed for in subsequent obser c3 dt- ASTRONOMICAL INSTRUMENTS. vations. This method of adjustment is called the method of high and low stars. The final result obtained by it may be tested by the method of circumpolar stars already alluded to; which has the advantage of being independent of the error of the clock. If the timepiece used in setting up the transit keeps mean solar time, its error may be determined by measuring an altitude of the sun with the transit or sextant, as will hereafter be explained. 4.. The Time -of the Meridiant Passage of a Star is ascertained as follows: the telescope is first set by means of the finding circle, to the meridian altitude, or zenith distance of the star to be observed, and the instants of its crossingo each of the parallel wires of the reticle noted. The sum of these observed times, divided by the number of the wires, will be the time of the star's crossing the middle wire; provided the wires are equidistant. The distances between the wires, in time, are called the wire-intervals. They can be determined, and their equality tested, by noting the intervals of time employed by a star situated on the celestial equator, in passing over them successively; these equatorial intervals, divided by the cosine of the declination of any star, will be the wire-intervals for that star. By means of these intervals the time of the star's passing either wire can be reduced to the middle wire. The mean of such reduced times obtained for all the wires, will be the time of the meridian transit of the star. The utility of having several wires, instead of one only, will be readily understood, from the consideration that a mean result of several observations is deserving of more confidence than a single one; since the chances are that an error which may have been made at one observation will be compensated by an opposite error at another. If the body observed has a disc of perceptible magnitude, as in the cases of the sun, moon, and planets, the time of the passage of both the western and eastern limb across each of the parallel wires is noted, and reduced to the middle wire; the mean of all the results is then taken, which will be the instant of the meridian transit of the centre. We may, at the present day, obtain the time of the meridian passage of the centre of the sun, moon, or any planet, from an observation upon the western limb only, by adding "the sidereal time of the semi-diameter passing the meridian," taken from the Nautical Almanac, to the observed time. Or, the observation may be made upon the eastern limb, and the same quantity subtracted. aL. Electro-Chronograph. The accuracy of transit observations has recently been greatly increased, by the introduction of the electro-chronograph. This valuable contrivance consists of an electro-magnetic recording apparatus, put into communication with the pendulum of an astronomical clock, in such a mani-ner that the circuit is broken at a certain point of each oscillation THE RIGHT ASCENSION OF A STAR. 37 and, as a consequence, the seconds beat by the pendulum are designated by a series of equally distant breaks in a continuous line, upon a roll of paper to which an equable motion is given by machinery. The observer holds in his hand a break-circuit key, by means of which he interrupts the circuit at the instant that the star is bisected by one of the wires in the field of the telescope, and thus makes a break in one of the short lines that answer to the successive seconds; as shown between 44s. and 45s., in Fig. 22. 4a0. 41s. 42s. 43s. 44s. 45s. 46s. 47s. 48s. FIG. 22. In this way, the instant of the transit across a single wire can be noted to within a much smaller fraction of a second than by the common method. Besides, the number of bisections in a single culmination of a star, by increasing the number of wires, may be augmented fivefold. This method of observation was adopted at the Washington Observatory, in 1849, and soon after at the Observatory of Harvard College. It has since been introduced at the Greenwich and other principal observatories. 45. To determoine the Right Ascensio n of a Star. VWhen a star is on the meridian, its declination circle (def. 1G, p. 15) coincides with the meridian; moreover, the arc of the equator which lies between the declination circles of tiweo stars, measures their difference of right ascension. Thus, RR' (Fig. 8) is thel difference of right ascension of the stars S and S'; their absolute right ascensions being VR and VR'. In the interval between the transits of the two stars, the are RR', which is equal to their difference of right ascension, passes across the meridian at the rate of 15~ to a sidereal hour. If, therefore, the times of their meridian transits be determined with the transit instrument and sidereal clock, the difference between these times, converted into degrees by allowing 15~ to the hour, will be the difference of right ascension of the two stars. In this way, the difference between the right ascension of any standard star, S, fixed upon as a point of reference, and other stars, may be successively determined. This having been done, the absolute right ascensions of these stars will become known as soon as the position of the vernal equinox with respect to the standard star has been found. For, it is plain that RR' being known, if VR be also determined, VR' may be found by adding VR and RR'. The manner of determining the position of the vernal equinox, or the value of VR, will be explained in the chapter on the Apparent Motion of the Sun. Right ascensions are commonly expressed in time. 38 ASTRONOMICAL INSTRUMENTS. ASTRONOMICAL CLOCK. 46. The Astronomical Clock is provided with a pendulum so constructed that its length is unaffected by changes of temperature. The mercurial compensation pendulum, in which the ordinary brass bob is replaced by a glass jar containing a certain quantity of mercury, is generally employed. The clock is secured to a stone pier resting upon a firm foundation, which is disconnected from the floor of the observatory. It keeps sidereal time. 47. To Regulate a Sidereal Clock. When a clock is used for determining differences of right ascension (45), it is adjusted to sidereal time if it goes equably and marks out twenty-four hours in a sidereal day; it being altogether immaterial at what time it indicates Oh. Om. Os. To ascertain its daily rate, note by the clock the times of two successive meridian transits of the same star: the difference between the interval of the transits and twenty-four hours will be the daily gain, or loss (as the case may be), of the clock with respect to a perfectly accurate sidereal clock. If the gain or loss, when found in this manner, proves to be the same each day, then the mean rate of going is the same each day. Error.-The sidereal clock now in use in astronomical observatories, is made to indicate Oh. Om. Os. when the vernal equinox is on the superior meridian; and it is necessary to know not only its rate but also its error. This may be found from day to day by noting the time of the transit of some known star, whose place has been accurately determined, and comparing this with its right ascension expressed in time. If the two are equal the clock is right; otherwise their difference will be its error. For greater accuracy in the determination of the error and rate, the successive transits of several standard stars should be noted. To facilitate these and other determinations, the apparent places of a large number of stars are given in nautical almanacs, and other similar works. Clock Stars. The stars most favorably situated for determining the clock correction are those which pass the meridian near the zenith; or, next to these, the stars which cross the meridian between the zenith and equator. Stars considerably to the north of the zenith pass too slowly through the field of the telescope; and if the transit instrument has not been accurately adjusted to the meridian, the error in the time of the transit will be greater in proportion as the star observed is further from the zenith. 4~. A,1,ean Solar Clock is usually regulated by observations upon the sun. The methods by which its error and rate are determined will be explained in the chapter on the Measuremnent of Time. MERIDIAN CIRCLE. 39 MERIDIAN CIRCLE. 49. The Meridian Circle is an instrument used to measure the zenith distance, or altitude of a heavenly body, at the instant of its arrival on the meridian. It is, in its general construction, a combination of the transit instrument and a graduated vertical circle; and is hence sometimes called the Transit Circle. In the larger observatories, it is mounted on two piers, like the transit. Tile graduated circle is firmly attached at right angles to the horizontal axis of rotation, and turns with it. The angle is read frori the circle by a reading miscroscope, attached to the adjacent pier; or in some instances, to a frame which rests upon the axis itself. For greater accuracy four or six reading microscopes are used, at equally distant points of the limb. The degrees, minutes, and seconds, are read fiom one of the microscopes, and the seconds only from the others. If the seconds read from either microscope be added to the degrees and minutes obtained from the first, the result will be the reading of that microscope reduced to the first. By taking the mean of all the results, for the different microscopes, the errors fiom imperfect graduation, inaccurate centring, and unequal expansion of the limb, may be materially lessened. 50. Fig. 23 represents a meridian circle manufactured by Repsold, a celebrated German instrument-maker, and mounted in 1852 in the observatory of the United States Naval Academy. It has two graduated circles, CC and C'C', of the same size, but only one of these, CC, is graduated finely; this is read by four microscopes, two of which are seen at tRR. The microscopes are attached to the four corners of a square frame which is centred upon the rotation axis; but does not turn with it, being held in a fixed position by screws connected with the piers. Each horizontal side of the frame carries a spirit level, by which any change of inclination of the frame with respect to the horizon may be detected. The second circle, constructed of the same size as the first, for the sake of symmetry, is graduated more coarsely, and is used only as a finder. The counterpoises WW act at XX, to support the greater part of the weight of the instrument upon friction rollers, as in the case of the transit instrument. The inclination of the rotation axis is measured with a hanging level, LL. A horizontal arm, vFG, seen to the right of the telescope in the figure, extends out froln the pier, and receives a vertical arm which is connected with a collar upon the rotation axis. By turning a screw, the head of which is at G, the telescope is clamped in the collar; and then a screw (not seen in the drawing), connected with the arm FG, and acting horizontally upon the 40 ASTRONOMICAL INSTRUMENT$i It Cf Cy~~~~~~~~,i. 3_ I~~ a: ~i &w F PI P FExc. 23. MfERIDIAN CIRCLE. 41 vertical arm, gives a fine motion to the telescope. FG turns upon a joint at F; and to the left of the telescope is shown in the position it takes when detached from the vertical arm, preparatory to a reversal of the instrument. Another arm, fg, similar in its form and arrangement to FG, receives a vertical arm attached to the microscope frame. Screws connected with fg, and acting horizontally at g upon the vertical arm, serve to adjust the frame. The field is illuminated by light thrown into the interior of the telescope through tubes at AA, and reflected towards the reticle by a mirror in the central cube. The quantity of light is regulated by revolving discs with eccentric apertures, a.t the extremities of the tubes nearest the Y's. These discs are revolved by means of a cord to which hangs a small weight, S. The micrometer at m contains seven fixed transit threads, and three equally distant horizontal threads movable by a micrometerscrew. The more common form consists of a reticle with several stationary transit wires, or threads, and one stationary horizontal wire; in connection with one or more movable horizontal wires (31). The movable micrometer-wires serve for the measurement of small differences of declination. 51. 1n1ral Circle. This is another form of the meridian circle that has been much used in large observatories. The graduated limb of the mural circle is secured to one end of a horizontal axis, which is let into a massive pier or wall of stone. Its axis, therefore, is not symmetrically supported, and it cannot be reversed. On these accounts it is inferior, for nice determinations, to the form of meridian circle just described. Mural circles have been constructed as large as eight feet in diameter. 52. Adjistiments. The same adjustments have to be effected with the meridian circle as with the transit; and the same methods may be adopted. But it is also necessary to determine with great accuracy what is called the horizontal point of the limb. This is the place of the index, or zero of the reading microscope, answering to a horizontal position of the line of collimation of the telescope. 53. I'o S8eterminaie tiae [Horizointal Point of the Lfnimlb. Direct the telescope upon any known star, at the time of its pass. ing the meridian, and read off the angle on the limb. On the next night, when the star comes to the meridian, direct the telescope upon the image of the same star reflected from a basin of mercury, and note the angle as before. By a fundamental law of reflection the angle of' depression of th'is image will be equal to the angle of elevation of the star. Accordingly the are on the limb, which passes before the reading microscope, in moving the telescope from the star to its image, will be double the altitude of the star, and its point of bisection the horizontal point. 42 ASTRONOMICAL INSTRUMENTS. This method will not give an exact result unless a correction is applied for the difference in the values of the atmospheric refraction at the times of observation (81). The necessity of making this correction may be avoided, and a more reliable result obtained, whenever the instrument is provided with a micrometer having a movable horizontal wire. By a rapid manipulation an observation may then be made upon the star at the time it is crossing the first transit wire, and another observation taken upon its image, as it is crossing the last transit wire. The instrument is first set to the altitude of the star, as nearly known, and the correction to this altitude measured by bringing the movable horizontal wire upon the star at the instant it is crossing the first transit wire. In observing the image of the star, it is brought near the fixed horizontal wire, the limb clamped, and the observation completed by the tangent screw of the limb. The observer may then read, at his leisure, the microscopes for the last measured angle! and the micrometer correction to the first angle. To each of the angles measured, a small correction must be applied to reduce it to the meridian. 54. To nneasure the Altitu6de of a Heavenly Body. (1). Of a fixed star. Direct the telescope of the meridian circle upon the star, bring it on the horizontal wire of the reticle, and clamp the limb; then by means of the tangent screw that gives a small motion to the limb and telescope, bisect the star with the horizontal wire at the instant of its crossing the middle transit wire. Then read off the angle from the different microscopes, as already explained (49), and take the mean of the several results. This must be corrected for the deviation of the horizontal point from the zero of the limb, and all the detected errors that result fromn imperfect adjustments, or defects of construction. If an observation be made upon the star at the time of its crossing any other than the central wire, it can readily be reduced to the meridian. (2). Of the sun, moon, or any planet. Measure the altitudes of the upper and lower limbs, and take their half sum for the altitude of the centre, or measure the altitude of the upper or lower limb, and add or subtract the apparent semi-diameter of the body, taken from the Nautical Almanac. The observations are facilitated by using the movable micrometer wire in establishling the contact with the limb; then, by turning the micrometer screw, measuring the interval between the position of the movable and that of the parallel stationary wire, and adding this measured interval to the mean of the microscope readings. 55. To determine the Deehlinaation of a Heavernly odly. The meridian altitude, or zenith distance of a heavenly body, having been measured at a place the latitude of which is known, its declination may easily be found. For let s (Fig. 10, p. 21) represent the point of meridian passage of a star which crosses to the north of the zenith (Z), Es will be its declination (def. 27, p. 17), Zs its meridian zenith distance, and ZE the latitude of the place of observation (O), (def. 33, p. 18); and we obviously have Es=ZE+Zs.... (a) If the star cross the meridian at some point s' between the zenith (Z) and the equator (E), we shall have E.s'=ZE-Zs', (b); and if its point of transit be some point s" to the south of the ALTITUDE AND AZIMUTH INSTRUMIENT. 43 equator (E), we shall have Es" =Zs"-ZE, and-Es"=ZE-Zs", (c). The three formula (a), (b), and (c), may all be comprehended in one, viz.: Declination =latitude+ meridian zenith distance,... (1) If we adopt the following conventional rules: (1) north lati. tude is +, south latitude -; (2) the zenith distance is north, or south, according as the star passes to the north or south of the zenith; and it has the same sign as the latitude when it has the same name, the contrary sign when it is of a contrary name; (3) north declination is +, south declination-. The latitude which is here supposed to be known, may be found by measuring the meridian altitudes of a circumpolar star with the meridian circle, and taking their half sum. For, as the pole lies midway between the points at which the transits take place, its altitude will be the arithmetical mean, or the half sum of the altitudes of these points; and the altitude of the pole is equal to the latitude of the place (24). It will be seen in the next Chapter, that certain corrections must be applied to all measured altitudes. 56. To determuiane the Lonsgitaude an-d Latitnde of a Body. When the right ascension and declination of a heavenly body have been obtained from observation, with a transit instrument and circle (45, 55), its longitude and latitude may be computed. For, let S (Fig. 8) represent the place of the body, VRQE the equator, VLTW the ecliptic, and P, K, the north poles of the equator and ecliptic. In the spherical triangle PKS we shall know PS the complement of SR the declination, and the angle KPS = ER=EV+VR= 90~ + right ascension; and if we suppose the obliquity of the ecliptic to be known, we shall know PK. We may therefore compute KS, and the angle PKS. But KS is the complement of SL, which is the latitude of the body S; and PKS = 1800-WKS = 180 —(WV+ VL)=-180~ -(90~ + longitude) = 90~ -longitude. The obliquity of the ecliptic, which we have here supposed to be known, is, in practice, easily found; for it is equal to TQ, the sun's greatest declination. ALTITUDE AND AZIMUTH INSTRUMENT. 57. This instrument consists essentially of a te escope mounted upon either a fixed or portable stand, and provided with both a vertical and a horizontal graduated limb. The telescope turns with the vertical limb about a horizontal axis, and the whole turns about the vertical axis of the horizontal limb. The instrument is so adjusted, that when the line of sight of the telescope is in the meridian plane, the zero of the reading microscope of the horizontal limb will answer to the zero of the limb, or nearly so. If they do not correspond, the distance between them will 44 ASTRONOMICAL INSTRUMENTS. be the index error. This having been determined, if the telescope be directed upon a star out of the meridian, the reading of the horizontal limb, corrected for the index error, will be the azimuth of the star at the instant of the observation. The vertical circle serves to measure the altitude. The altitude and azimuth instrument is sometimes called the A2tazimuath also the Astronomical Theodolite. 58. The Mleridiatn Line (def. 8, p. 14) at a place may easily be determined with the altitude and azimuth instrument, by a method called the Method z of Equal A lttuades. Let 0 (Fig. 24) represent the place of observation, NPZ the meridian, and S, S' two positions of the same star, at which the altitude N \ \' / \ \ is the same. Now, the spheriR cal triangles ZPS and ZPS' have the side ZP common, ZS= ZS', and (allowing the stars to:movein circles) PS=PS'. Hence Fri. 24. they are equal, and consequently the angle PZS-PZS'; that is, equal atitadcles of a star correspond to equal azfimuths. Therefore, by bisecting the arc of the horizontal limb, comprehended between two positions of the vertical limb for which the observed altitude of a star is the same, we shall obtain the meridian line. The meridian line may be approximately determined, by this method, with the common theodolite; the observations being made upon the sun. The result will be more accurate if they be made towards the summer or winter solstice, when the sun will have but a slight motion towards the north or south in the interval of the observations. It is, however, easy to determine and allow for the effect of the sun's change of place in the heavens. When the time is accurately known, the north and south line may be found very easily by directing the telescope of any instrument that has a motion in azimuth, upon a star in the vicinity of the pole, at the instant of its arrival on the meridian. 59. Zenith Telescope. This may be regarded as a modified form of the portable altitude and azimuth instrument. It is of great value for the convenient and accurate determination of the latitude of a place; and has been used for this purpose with great success in the United States Coast Survey. Its chief peculiarities consist in the substitution of a finding circle with a delicate spirit level, similar to the finding circle of the transit instrument (38), for the ordinary vertical limb of the altitude and azimuth instrument, and the adaptation to the telescope of a micrometer with a movable horizontal wire. If such a micrometer be adapted to a transit instrument, that EQUATORIAL. 45 instrument mnay be successfully used as a substitute for the zenith telescope, for the accurate determination of the latitude of a station.* EQUATORIAL. 60. The equatorial consists of a telescope mounted with two axes of motion, at right angles to each other, one of which is parallel to the axis of the earth, and of the celestial sphere. The angular movement about this axis is measured by a graduated circular limb at right angles to the axis, and therefore parallel to the plane of the equator; fiom which the instrument takes its name. This limb is called the hour circle. There is also a graduated circle, called the declination circle, adapted to the other axis; which lies, in every one of its positions, in the plane of a celestial meridian. The telescope turns in the plane of a celestial meridian about this axis; and can at the same time be made to rotate, in connection with it, about the other, or polar axis. It can thus be readily set upon any star, whose hour angle and declination are known; and when once directed towards it, can be made to follow the star in its diurnal motion, by simply producing a continuous movement about the polar axis. This motion is generally communicated by clock-work, without the use of the hand. Plate I. represents the large equatorial telescope mounted under the dome of the observatory of Harvard College. It is connected with a bed-plate which is fastened by screw-bolts to the top of a granite block, in a position parallel to the axis of the heavens. This block is ten feet in height, and rests upon a granite pier forty-two feet high. The clock-work is on the further side of the stone support, and does not appear in the figure. The instrument is so nicely counterpoised that it can be moved with the greatest ease by the pressure of the hand upon the end of one of the balance rods. 61. Uses of the Equatorial. A telescope thus eqzatoriaTly mounted, and provided with a movable micrometer-wire, is especially adapted to the measurement of the apparent diameter of a heavenly body, the angular distance between stars in close proximnity, and in general to all observations that require the telescope to be directed upon a body for a considerable interval of time. Accordingly the large telescope of every prominent observatory is mounted in this manner. * This has been satisfactorily shown by Professor C. S. Lyman, of Yale College (see American Journal of Science, Vol. XXX., p. 52). The zenith telescope is essentially the invention of Capt. Andrew Talcott, of the United States Corps of Engineers, who also devised a method of determining the latitude by this instrument which surpasses all others, both in simplicity and accuracy. This is now known as Talcott's method (Chauvenet's Spherical and Practical Astronomy). 46 ASTRONOMICAL INSTRUMENTS. The equatorial can also be advantageously used'for determin ing the unknown place of a fixed star, or planet, in the heavens, by measuring the angular distance and direction of the star from some known star seen with it in the field of the telescope; or by noting the interval of the transits, and measuring directly the difference of declination of the two stars. For this purpose the telescope is furnished with a certain form of micrometer, called the Position, Filar Micrometer; with which the measurements in question can be made with great accuracy. I)ifferences of right ascension and declination can also be measured with the equatorial, by means of the hour and declination circles, but with much less accuracy than with the transit instrument and meridian circle. 62. Position Filar Micrometer. This piece of apparatus serves at the same time to measure small angular distances, and the angle included between the line connecting two stars in close proximity and the celestial meridian. This angle is called the angle of position of one of the stars with respect to the other. It is estimated from the S. round by the W. to 360~. The Filar Micrometer, designed for the measurement of small angles, is shown in Fig. 25. It is the same in principle as the micrometer employed in the reading microscope (34). kLi a' ht f FIG. 26. FTIT. 25. It consists of two forks of brass, bb'b, cc'c, sliding within a rectangular brass box, aa'a, and one within the other. Each of these forks carries a very fine wire, or spider line, stretched perpendicularly across from one prong to the other; they are mov-ablbe, and the parallel wires which they carry, by micrometer screws passing' through the ends of the box, and attached to the forks. A third an'J stationary wire, 1, perpendicular to the other two, is attached to a diaphragm disconnected from the forks. The heads of the screws are not shown in the figure, but they may be seen in Fig. 26, in which b is the micrometer-box. The eyepiece is screwed into the micrometer-box, as shown in Fig. 26. The graduated screw-heads are connected with nuts which turn, without advancing, upon the screws that are fastened to the forks. Accordincrly by turning the nuts, the forks may be moved either forwards or backwards. A stationary comb-scale on one side of the box, indicates the number of revolu EQUATORIAL. 47 tions of either screw, answering to any distance that the wires may be separated from each other; and the fractional part of' a revolution is shown by the graduated head of the screw. The value of one revolution of the micrometer-screw may be ftund by bringing the two parallel wires into a position perpendicular to the celestial equator, separating them by a certain number of revolutions, and then noting the time taken by an equatorial star to traverse the interval between them. The interval of tiile thus obtained, converted into the equivalent angular space by allowing 15" to 1', will be the number of seconds of are answering to the assumed number of revolutions of the screw. To adapt the filar micrometer to the measurement of angles of position, the micrometer-box, with its attached eye-piece, is so mounted as to admit of a rotation around the centre of a graduated circle (Fig. 26). The circle is fastened at the end of the reticle-tube, and in a plane perpendicular to the optical axis of the telescope. The revolving motion is produced by a milled-head screw s, which works on an interior toothed wheel; and the angle is read off upon the stationary graduated circle, by aid of the vernier movable with the plate a. SEXTANT. 63. The instruments which have now been described are observatory instruments, the chief design of whose construction is to furnish the places of tlhe heavenly bodies with all attainable exactness. That of -which we are now to treat is much less exact, though still of great utility in effecting certain important astronomical determinations; as of the latitude or longitude of a place, and the time of day. It is chiefly used by navigators, and astronomical observers on land, who are precluded by their situation, or other circumstances, from using the more accurate instruments of an observatory. It is much more conveniently porrtable than ally of these, and has not to be set up and adjusted at every new place of observation. Besides, as it is held in the hand, it can be used at sea, where by reason of the agitations of the vessel, no instrument supported in the ordinary way is of any service. 64. Coiastruction: —Prinaciple of Conistraictioim. The sextant may be defined, in general terms, to be an instrument which serves for the direct admeasurement of the angular distance between any two visible points. The particular quantities that may be measured with it, are; 1st, the altitude of a heavenly body; 2d, the angular distance between any two visible objects in the heavens or on the earth. Its essential parts are a graduated limb BC (Fig. 27), comprising about 60 degrees of the entire circle, which is attached to a triangulalr frame BAC; two mirrors, of which one (A) called the ilcdlex Glass, is movable in connection with an index, G, about A, the centre of the limb, and the other (D) called the Horizon Glass, is permanently fixed parallel to the radius AC (lrawn to the zero point of the limb, and is only half silvered (the upper half being transparent); and a small immovable telescope at E, directed towards the horizon-glass. The principl-e of the construction and use of the sextant may be understood from 48 ASTRONOMICAL INSTRUMENTS. what follows: A ray of light SA from a celestial object S, which impinges against the index-glass, is reflected off at an equal angle, and striking the horizon-glass (D) is again reflected to E, where the eye likewise receives through the transparent S' uA. FIG. 27. part of that glass a direct ray from another point or object S'. Now, if AS' be drawn, directed to the object S', SAS', the angular distance between the two objects S and S', is equal to double the angle CAG measured upon the limb of the instrument (AC being parallel to the horizon-glass). For, when the index-glass is parallel to the horizon-glass, and the angle on the limb is zero, AD, the course of the first reflected ray, will make equal angles with the two glasses, and therefore the angle SAD will become the angle S'AD, (=ADE;) and the observer, looking through the telescope, will see the same object S' both by direct and reflected light. Now, itf the indexglass be moved from this position through any angle, CAG, the angle made by the reflected ray which follows the direction AD, with this glass, will be diminished by an amount equal to this angle; for, we have DAG=DAC —AG. Therefore the angle made with the index-glass by the new incident ray SA, which after reflection now pursues the same course ADE, and reaches the eye at E, as it is always equal to that made by the reflected ray, will be diminished by this amount. Consequently, the incident ray in question will on the whole, that is, by the diminution of its inclination to the mirror by the angle CAG, and by the motion of the mirror through the same angle, be displaced towards the right, or upwards, an angle S'AS equal to 2GAC. Thus, the angular distance SAS' of two objects S, S', seen in contact, the one (S') directly, and the other (S) by reflec THE SEXTANT. 49 Lton from the two mirrors, is equal to twice the angle CAG that the index-glass is moved from the position (AC)j of parallelism to the horizon-glass. IIence the limb is divided into 120 equal parts, which are called degrees; and to obtain the angular distance between two points, it is only necessary to sight directly at one of them, and then move the index until the reflected image of the other is brought into contact with it; the angle read off on the limb will be the angle sought. To obtain the angular distance between two bodies which have a sensible diameter, bring the nearest limbs into contact, and to the angle read off on the limb add the sum of the apparent semi-diameters of the two bodies, or bring the farthest limbs into contact, and subtract this sum. 65. The Detail of the Construction of the Sextant is shown in Fig. 28. The limb, and the triangular frame to which it is FIG. 28. attached, are of hammered brass, and strengthened by cross-plates. The graduation is upon silver inlaid in the brass. Each degree is divided into six equal parts, of 10'. N is the horizon-glass, fastened to the frame in the position before stated; I the index-glass, in a brass frame, attached to the index-bar CD, by the screws s ss, and movable with it about the centre C of the graduated arc. These two mirrors are of plate-glass silvered. The upper half 4 50 ASTRONOMICAL INSTRUMENTS. of the horizon-glass is left unsilvered, that the direct rays from the object towards which the small telescope, T, is directed may not be intercepted. The telescope is supported in a ring, IK, attached to a stem underneath, which can be raised or lowered by a screw. By this means the relative brightness of the direct and reflected images can be regulated. M is a microscope, movable about a centre on the index-bar, used in reading the angle from the vernier at D. The vernier is so divided as to give the angle to within 10". At B13, under the index-bar, is a screw for clamlping it to the limb; and G is a tangent screw for giving the bar, witih the index-gloass, a small motion, in securing the accurate conttact or coincidence of the images. HI is a wooden handle at the back of the sextant, by which it is held when an observation is taken. At E and F are colored glasses of different sh1iacdes, to diminish the intensity of the light when the sun is observed. Those at F are interposed between the index-glass and the horizon-glass when the sun is seen by reflection from the index-glass. The others are used when the telescope is directed upon the sun. 66. Adjustments. The adjustments of the sextant consist in setting the index-glass and the horizon-glass, and bringing the line of 3ight of the telescope parallel to the plane of the graduated arc, and in determining tha index error. The index-glass may be adjusted by setting the index near the middle of the arc, placing the eye nearly in the plane of the sextant, and near the index-glass, and observing whether the arc seen directly and its reflected image form one continuous arc. if the reflected image does not appear to form a true continuation of the are, the index-glass is not perpendicular to the plane of the sextant. It may be corrected by loosening the screws s s s, and inserting a piece of paper under the plate through which they pass. The horizon-glass is adjusted by sighting through the telescope at a star, and moving the index until the direct and reflected image of the star pass each other. If, in passinlg, the two images can be made to coincide, the horizon-glass is perpendicular to the plane of the instrument. If any correction is necessary, it can be made by turning a small screw at the top or bottom of the horizon-glass. To test the position of the line of sight of the telescope, select two objects, as two stars, 1000 to 1200 apart, and bring the reflected image of the one in contact with the direct image of the other, on the wire within the telescope that is nearest the plane of the sextant: if then, on moving the instrument, the contact remains when the images are thrown upon the other parallel wire of the telescope (although a separation occurs in the interval between them), no adjustment is required. It can be made, when necessary, by means of two small screws in the ring which supports the telescope. To find the itdex error. Bring the direct and reflected images of the same point of a distant terrestrial object, or of the same star, into coincidence, and read off the are. This reading will be the index error, and may be either positive or negative. 67. "Taking atn Angie. When observing with the sextant, it is held in the right hand by the handle, and the telescope directed upon one of the two objects whose angular distance is to be measured, generally the fainter one. It is then turned about the line of sight until the other object lies in its plane and the index moved with the left hand until the reflected imnage of this object is brought, at the centre of the field of the telescope, into apparent contact with the object seen directly; THE SEXTANT. 51 the contact being finally effected by the use of the tangent screw. Tile angle is then read from the limb and vernier, with the microscop~e. When the sextant is employed to take the altitude of a heavenly body, a horizontal reflector, called an Artlficial Jiorizon, is placed inT front of the observer. The angle between the body and its reflected image is then measured as if this image were a real object: the half of which will be the altitude of the body. A small quantity of mercury, poured into a shallow vessel of tinned iron or copper, forms a very good artificial horizon. In obtaining the altitude of a body at sea, its altitude above the visible horizon is measured, by bringing the lower limb into contact with the horizon. To this angle is added the apparent semi-diameter of the bodvy, and from the result is subtracted thea depression of the visible horizon below the horizontal line, called the Dip of the Horizon. 6~. Hladley's Quadu'abut. HIadley's Quadrant differs from the sextant in having a graduated limb of 45~, instead of 60~, in real extent, and a sight-vane instead of a small telescope. It is not capable, then, of measuring any angle greater than about 90~, while the sextant will measure an angle as great as 1200; or even 140~ (for the graduation generally extends to 140~). The quadrant is also inferior to the sextant in respect to materials and workmanship, and its measurements are less accurate. 69. Reflecting Circle. The Reflecting Circle is but an einlarged sextant. Its limb is a full circle, and the index-arm is prolonged in the other direction, and carries a vernier on each end. The angle is read from each vernier, and the mean of the two readings taken, to eliminate the error of eccentricity. 70. Prismatic Sextant. This is an improved form of sextant, recently introduced. It takes its name from the fiict that a refiectillng prism is used in place of the ordinary horizon-glass. This prism als) occupies a different position with respect to the index-glass. The graduated linmb extends 120~. The prismatic sextant can be used to measure an angular distance of 180~, and an altitude of 90~. It is also superior to the ordinary sextant in certain other peculiarities of construction. ]Prismatic Reflecting Circles are also constructed which possess similar advantages over the ordinary reflecting circle. ERRORS OF INSTRUMENTAL ADMEASUREMENT. 71. \Whatever precautions may be taken, the results of instrumental admeasurement will never be wholly free from errors. Errors that arise from inaccuracy in the workmanship or adr justment of the instrument, may be detected and allowed for. But errors of observation are, obviously, undiscoverable. Since, ho(wever, the chances are, that an error committed at one obser 52 ASTRONOMICAL INSTRUMENTS. vation, will be compensated by an opposite error at another, it is to be expected that a more accurate result will be obtained if a great number of observations, under varied circumstances, be made, instead of one, and the mean of the whole taken for the element sought. And accordingly, it is the uniform practice of astrorlornical observers to multiply observations as much as is practicable. 72. Instrumental Errors may be divided into three classes; viz. errors of construction, errors of adjustment, and incidental errors. Errors of construction, in the best instruments, result chiefly from imperfect graduation, an eccentricity of the limb, an inequality or an ellipticity of thepivots, and an imperfect rigidity of the telescope or axis. The effect of eccentricity and of the ellipticity of the pivot, may be eliminated by taking the mean of the readings of two microscopes, at opposite points of the limb. The error of graduation may be greatly reduced, by reading the angle from several equidistant points of the limb, and taking the mean of all the readings. When the construction of the instrument is such that the principle of erepetition may be adopted-that is, the angle read off from all parts of the limb -the error of graduation may, theoretically speaking, be removed entirely. It is not the practice of astronomical observers to strive to bring instruments into the nicest possible adjustment, but instead, after a good adjustment has been effected, to deduce, by a systematic series of observations, the several errors that remain, and derive from these the corrections to be applied to the quantity to be determined. Incidental errors may arise from diverse effects produced by changes of temperature, especially an unequal expansion of different parts of the limb, and a derangement of the microscopes; from flexure produced by weight; and also from vibrations produced by passing vehicles, and other derangements from extraneous mechanical causes. All such errors may be mostly neutralized by making numerous measurements, under a great variety of circumstances. THE TELESCOPE. 73. An observatory is not completely furnished unless it is supplied with a large telescope for examining the various classes of objects in the heavens; and one or more smaller ones for exploring the heavens and searching for particular objects invisible to the naked eye, as faint comets, and making observations upon occasional celestial phenomena, as eclipses of the sun and moon, occultations of the stars, etc. Telescopes are divided into the two classes of Reflecting and Refracting Telescopes. In the former class, the image of the object is formed by a concave speculum, and in the latter by a converging achromatic lens. This image is viewed and magnified by an eye-glass; or rather by an achromatic eye-piece consisting of two glasses. In the simplest form of the reflecting telescope, the Herschelian, the image formed by the concave speculum is thrown a little to one side, and near the open mouth of the tube, where the observer views it through the eye-glass, with his back turned towards the object. 74. Magnifyingpower-illtuninating power —spac-penetrating power. The magnifying power of a telescope is to be carefully listinguished from its illuminating, and space-penetrating power. A telescope magnifies by increasing the angle under which the object is viewed; it increases the light received from objects, and reveals to the sight remote stars, nebule, etc., by intercepting and converging to a point a much larger beam of rays. The magnifying power is measured by the ratio of the focal length of the object-glass, or speculum, to that of the eye-piece. The illuminating power, by which it reveals stars invisible to the naked eye, if we leave out of view the amount of light lost by reflection and absorption, is measured by the proportion which the area of the object-glass, or speculum, bears to that of the pupil of the eye. Since the quantity of light received from any luminous point, viewed at different distances by the naked eye, decreases in the same proportion that the square of the distance increases, and the quantity of light from tho same point, conveyed to the eye by a telescope, is augmented in the ratio of the Ti'E TELESCOPE. 53 square of the diameter of its aperture to the square of the diameter of the pupil of the eye, it follows that the diminution of the light from an increase of distance, will be just suppliel if the aperture of the telescope exceed in its diameter that of the pupil of the etye in the same ratio that the distance is augmented. The power of a telescope to penetrate into space, and discern stars, therefore, exceeds that of the naked eye in the same ratio that the diameter of its aperture exceeds that of the pupil of the 6ye (0.2 in.). In the larger reflecting telescopes, the space-penetrat ing power, calculated by this rule, requires to be diminished about one-fifth, in consequence of the loss of light incident to the use of the telescope. Telescopes are provided with several eye-glasses, of various powers. The power to be used varies with the object to be viewed, and the purity and degree of tranquillity of the atmosphere. Of two telescopes of the same focal length, that which has the largest aperture will form the brightest image in the focus, and therefore, other things being equal, admit of the use of the most powerful eyepiece. In this way, it happens that the available magnifying power indirectly depends materially upon the size of the aperture. In all telescopes, there is a certain fixed ratio between the aperture and focal length, or at least limit to this ratio. In reflecting telescopes, it is one linear inch of aperture for every foot of focal length, and in refracting telescopes one inch of aperture for from one to two feet of focal length. Reflectors and refractors of the same focal length, have about the same actual magnifying and illuminating power. The highest theoretical magnifying power that has yet been obtained is about 7,000. But the highest actually available power, in observing any celestial object, does not exceed 2,500. The higher powers can be used only upon double stars, and clusters of stars. EWith the best telescopes, a magnifying power of four or five hundred is the highest that can be applied to the moon and planets; owing to the great diminution of brightness that results from the enlargement of the image. 7 5. Defining power. Telescopes of equal size may differ materially in their defining power: that is, in their capability to show the planets, and other celestial objects which have a sensible disc, with a sharp outline, and all their peculiarities of appearance with distinctness, and to separate close double stars and clusters of stars. The excellence of telescopes in this respect, depends upon the precision of form and perfection of polish of the lenses, their freedom from chromatic and spherical aberrations, and other niceties of construction. 76. The field ofview of telescopes diminishes in proportion as the magnifying power increases. It is stated that with a magnifying power of between 100 and 200 it is a circle not as large as the full moon; and with a power of 600 or 1,000 is nearly filled by one of the planets, while a star will pass across it in from two to three seconds. The diminution of the field of view, and the trepidations of the image occasioned by the varying density of the atmosphere, and the unavoidable tremors of the instrument, must ever affix a practical limit to the magnifying power of telescopes. This limit, it is probable, is already nearly attained; for the highest powers of the best telescopes can now be used only in the most favorable states of the weather. The illuminating and space-penetrating power of telescopes may, however, yet be materially increased, and a greater distinctness and definiteness in the outline of objects obtained. 77. Large Telescopes. The largest reflecting telescope that has yet been constructed and directed to the heavens, is the great P2osse Telescope, devised and constructed by Lord Rosse, of Ireland. It has a focal length of 53 feet, and an aperture of 6 feet. Its illuminating power is about 78,000; and its space-penetrating power, for single stars, about 280 times the distance of the most remote star visible to the naked eye. The most powerful refractor yet constructed, is the great Clark Telescope, made by Clark & Sons, Cambridgeport, Mass., and recently set up in the Chicago Observatory. It has a clear. aperture of 181 inches, and a focal length of 23 feet. It has, by the adaptation of different eye-pieces, different magnifying powers, varying from 70 to about 2,000. The great telescope of the Observatory of Harvard College has an aperture of 15 inches, and a focal length of 22~ feet. Its highest magnifying power is 2,000. The refractor of the observatory at Pulkova, in Russia, is but slightly inferior to this in its dimensions and capabilities Refracting telescopes of large dimensions and great excellence. are mounted equa. torially in all the prominent observatories m the United States and in Europe. C-I CORRECTIONS OF MEASURED ANGLES CHAPTER IV. CORRECTIONS OF MEASURED ANGLES. 87. Angles measured at the earth's surface with astronomical instruments answer to the Apparent Place of a heavenly body, and are termed Apparent elements. In astronomical language the True Place of a heavenly body is its real place in the heavens, as it would be seen from the centre of the earth. Angles which relate to the true place are denominated True elements. The co-ordinates of the apparent place of a body are termed its acpparent co-ordinates, and those of its true place its true co-ordinales. 79. Corrections. The apparent co-ordinates are reduced to the true, by the application of certain corrections, called Refrac. tion, Parallax, and Aberration. Refraction and aberration are corrections for errors committed in the estimation of a star's place, while parallax serves to transfer the co-ordinates from the earth's surface to its centre. The object of the reduction of observations from the surface to the centre of the earth, is to render observations made at different places on the earth's surface directly comelparable with each other. Observers occupying different stations upon the earth refer the same body, unless it be a fixed star, to different points of the celestial sphere. Their observations cannot, therlefore, be compared together, unless they be reduced to the same point, and the centre of the earth is the most convenient point of reference that can be chosen. REFRACTION. SO. Atmospheric Refraction. We learn from the principles of Pneumatics, as well as by experiments with the ba, rometer, that the atmosphere gradually decreases in density from the earth's surface upwards. We learn also firom the same sources, that it may be conceived to be ma(le up of an infinite number of strata of decreasing density, concentric with the earth's surface. From the known pressure and density of the atmosphere at the surface of the earth, it is computed, that by the laws of the equilibrium of fluids, if its density were throughout the same as immediately in contact with the earth, its altitude would be about 5 miles. Certain facts, hereafter to be mentioned, show REFRACTION. 55 that its actual altitude is not far from 50 miles. Now, it is an established principle of Optics, that light in passing from a vacuum into a transparent medium, or from a rarer into a denser medium, is bent or refracted towards the perpendicular to tlb surface at the point of incidence. It follows, therefore, that ti'~ light which comes from a star, in passing into the earth's atmt sphere, or in passing from one stratum of atmosphere into anothe, is refracted towards the radius drawn from the centre of ti \ earth to the point of incidence. Path of a ray of light. Let MmnnN, NnoO, OoqQ, (Fig. 29, R / Tz / q FIG. 29. represent successive strata of the atmosphere. Any ray, Sp, wily then, instead of pursuing a straight course, Spx, follow the broken line _pabc; being bent downwards at the points 1, q, b, c, &c., where it enters the different strata. But, since the number of strata is infinite, and the density increases by infinitely small degrees, the deflections apx, bay, &c., as well as the lengths of the lines pa, ab, &c., are infinitely small; and therefore pabc, the path of the ray, is a broken line of an infinite number of parts or a curved line concave towards the earth's surface, as it is represented in Fig. 30. Moreover, it lies in the vertical plane con. taining the original direction of the ray; for this plane is perpendicular to all the strata of the atmosphere, and therefore the ray will continue in it in passing from one to the other. 81. Astroiuomical lefraction. The lineOS' (Fig. 30) drawn tangent to paO, the curvilinear path of the light, at its lowest point, will represent the direction in which the light enters the eye, and therefore the apparent line of direction of the star. If, then, OS be the true direction of the star, the angle SOS' will be the displacement of the star produced by Atmospheric Refrac. tion. This angle is called the Astronomical Refraction, or simply the Refraction of the star. Since paO is concave towards the earth, OS' will lie above ~56 CORRECTIONS OF OBSERVATIONS. OS, consequently, refraction makes the cpparent altitude of a star greater than its true attitude, and the apparent zenith distance of a star less than its true zenith distance. (We here speak of the true altitude and true zenith distance, as estimated from the station -tA FIG. 30. Ol the observer upon the earth's surface.) Thus, to obtain the true altilude from the apparent, we must suzbtract the refraction; and to obtain the true zenith distance from the apparent, we must add the refraction. As refraction takes effect wholly in a vertical plane (80), it does not alter the azimuth of a star. Tzhe amount of the refraction varies with the apparent zenith distance. In the zenith it is zero, since the light passes perpendicularly through all the strata of the atmosphere: and it is the greater, the greater is the zenith distance; for, the greater the zenith distance of a star, the more obliquely does the light which comes from it to the eye penetrate the earth's atmosphere, and enter its different strata, and therefore, according to a well-known principle of optics, the greater is the refraction, s2. To finald the Amount of the Refraction for a given Zenith Distance or Altitude. Let us first show a method of resolving this problem by the general theory of refraction. According to this theory, the amount of the refiraction, except so far as the convexity of the strata of the atmosphere may have an effect, depends wholly upon the absolute density of the air immediately in contact with the earth, and not at all upon the law of variation of the density of the different strata; that is, the actual refraction is the same that would take place if the light passed from a vacuum immediately into a stratum of air of tlhe density which obtains at the earlth's surface. Let us suppose, then, that the whole atmosphere is brought to the same density as that portion of it which is in contact with the earth, and let bah (Fig.;1) represent its surface; also let 0 represent the station of the observer upon the earth's surfhce, and Sa a ray incident upon the atmnosphllere at a. Denote the angle of refraction OaC by p, and tile refraction Oax by r. The angle of incidence REFRACTION. 57 Z'aS = Z'aS'+ S'aS = OaC + Oax =p + r. Now if we represent the index of refraction of the atmo. sphere by m, we have, by a law of refraction, sin Z'aS =n sin OaC, or sin (p + r)= m sin p; z FIG. 31. developing, (App. For. 15,) sin p cos r + cosp sin r =m sin p; or, dividing by sin p, cos r + cot _p sin r = m. But, as r is small, we may take cos r = 1, and sin r - r r" sin 1", (App. 47.) rn - 1 Whence, 1 +cot p.r" sin 1"= m, or r" -si,,A tangp; -sin 1" Cot p putting A sinl'' Let ZCa C; and ZOa- Z. OaC = ZOa - ZCa, or p= Z - C. Substituting, we have r" A tang (Z - C); or, omitting the double accent, and considering r as expressed in seconds, r = A tang (Z- C)..... (2) When the zenith distance is not great, C is quite small compared with Z. If we neglect it, we have r = A tang Z..... (3); which is the expression for the refraction, answering to the supposition that the surface of the earth is a plane, and that the 58 CORRECTIONS OF OBSERVATIONS. light is transmitted through a stratum of uniformly dense air, parallel to its surface. We perceive, therefore, that the refraction, except in the vicinity of the horizon, varies nearly as the tangent of the apparent zenith distance. It has been ascertained by experiment that mn, the index of refraction (the barometer being 29.6 inches, and the thermometer = 50),- 1.0002803. Substituting in equation (3), after having restored the value of A, and reducing, there results r 57".8 tang Z..... (4). S3. Formulea of Refraction. With the aid of this formula, or of others purely theoretical, astronomers have sought to determine the precise amount of the refraction at various zenith distances from observation, and by collating the results of their observations to obtain empirical formulae that are more exact. One of the simplest methods of accomplishing this is the following": When the latitude or co-latitude of a place, and the polar distance of a star which passes the meridian near the zenith, have been determined, the refraction may be found for all altitudes from observation simply. Z For, let P (Fig. 32) be the elevated pole, Z the zenith, PZE the meridian, HOR the POX I \ horizon, S the true place of a star, and S' its apparent place. Suppose the apparent zenith distance ZS' to have been measured. Now, in the triangle ZPS, ZP V alS\ / \ the co-latitude and PS the polar distance are known by hypothesis, and the angle X P is the sidereal time which has elapsed since the star's last meridian transit, (or, if the star be to the east of the meridian, the diffbrence between this interval and 24 sidereal hours,) converted into degrees by allowingo 15"1 to the hour. Therefore we may compute the true zenith distance ZS, and subtracting from it the apparent N zenith distance ZS', we shall have the reFIG. 32. fraction. For the solution of this problem, the polar distance may be found by taking the complement of the declination computed from al observed meridian zenith distance (55); and, since the upper and lower transits of a circumpolar star take place at equal distances from the pole, the co-latitude may be found by taking the half sum of the greatest and least zenith distances of the pole-star. But it is obvious that neither of these quantities can be accurately determined, unless the measured zenith distances be corrected for refraction. When, however. the zenith distances in question differ considerably from 900, the corresponding refrao. tions may be at first ascertained with considerable accuracy by means of equation (4). When more correct formulae have been obtained by this or any other process, the latitude and polar distance, and therefore the refraction answering to to the measured zenith distance, will become more accurately known. The various formulae of refraction having been tested by numerous observations, it is found that they are all, though in different degrees, liable to material errors when the zenith distance exceeds 80~, or thereabouts. At greater zenith distances than REFRACTION. 59 this the refraction is irregular, or is frequently different in amount when the circumstances on which it is supposed to depend are the same. 84. Mean Refractions.- Corrections for the varaying density of the Air. The refractive power of the air varies with its density, and hence the refractions must vary with the height of the barometer and thermometer. The refractions which have place when the barometer stands at 30 inches and the thermometer at 50~, are called mean refractionzs. The refractions corresponding to any other height of the barometer or thermometer, are obtained by seeking the requisite corrections to be applied to the mnean refractions in consequence of the difference between the actual density of the air and its assumed mean density. Tables of Refraction. To save astronomical observers the trouble of calculating the refraction whenever it is needed, the mean refractions corresponding to various zenithl distances, or altitudes, are computed from the formula, as also the corrections for various heights of the barometer and thermometer, and inserted in tables. (See Tables VIII. and IX.) On inspecting Table VIII., it will be seen that the refraction amounts to about 34' when a body is in the apparent horizon, and to about 58" when it has an altitude of 45~. S5. Other Effects of Atumosphleric Refractiona. Atmospheric refraction makes the apparent distance of any two heavenly bodies less than the true; for it elevates them in vertical circles which continually approach each other from the horizon till they meet in the zenith. Refraction also gives to the discs of the sun and moon an eliptical form when near the horizon, As it increases with an increase of zenith distance, the lower limb of the sun or moon is more refracted than the upper, and thus the vertical diameter is shortened, while the horizontal diameter remains the same, or very nearly so. This effect is greatest near the horizon, for the reason that the increase of the refraction is there the most rapid; and it is most observable at sea, as the sun and moon, at their rising or setting, can there be seen in closer proximity to the horizon than at most stations on land. The difference between the vertical and horizontal diameters may amount to 8 part of the whole diameter. When a star appears to be in the horizon, it is actually 34' below it (84): refraction, then, retards the setting and accelerates the rising of the heavenly bodies. Having this effect upon the rising and setting of the sun, it must increase the length of the day. The apparent diameter of the sun is about 32'; as this is less than the refraction in the horizon, it follows, that when the sun appears to touch the horizon it is actually entirely below it. The (30 CORRECTIONS OF OBSERVATIONS. same is true of the moon, as its apparent diameter is nearly the same with that of the sun. PARALLAX. The correction for atmospheric refraction having been applied, the zenith distance of a body is reduced from the surface of the earth to its centre, by means of a correction called Parallax. S6. Definitions. Parallax is, in its most general sense, the angle made by the lines of direction, or't the are of the celestial sphere comprised between the places of an object, as viewed from two different stations. It may also be defined to be the angle subtended at an object by a line joining two different places of observation. Let S (Fig. 33) represent a celestial object, and A B two places from which it is viewed. At A it will be referred to the point s of the celestial sphere, and at B to the point s'; the angle BSA, or the are ss', is the parallax. The are ss' is taken as the measure of the angle BSA, on the principle that the -A.~ celestial sphere is a sphere of an indefiFIG. 33. nitely great radius, so that the point S is not sensibly removed from its centre. The term parallax is, however, generally used in Astronomy in a limited sense only, namely, to denote the angle included between the lines of direction of a heavenly body, as seen from a p)oint on the earth's surfhce and from its centre; or the angle subtended at a heavenly body by a radius of the earth. If C (Fig. 34) is the centre of the earth, 0 a point on its surface, and S a heavenly body, OSC is the parallax of the body. When there is occasion to distinguish this angle from other angles of parallax, it is termed the Geocentric Parallax. The parallax of a heavenly body above the horizon is called Parallax in Altitude. The parallax of a body at the time its apparent altitude is zero, or when it is in the plane of the horizon, is called the Horizontal Parallax of the body. Thus, if the body S (Fig. 34) be supposed to cross the plane of the horizon at S', OS'C will be its horizontal parallax. OSC is a parallax in altitude of this body. It is to be observed, that the definition just given of the horizontal parallax, answers to the supposition that the earth is of a spherical form. In point of fact, the earth (as will be shown in PARALLAX. 61 the sequel) is a spheroid, and accordingly the vertical and the radius at any point of its surfice are inclined to each other; as zS C fi FIG. 34. represented in Fig. 35, where OC is the radius, and OC' the vertical. The points Z and z, in which the vertical and radius z FIG. 35. pierce the celestial sphere, are called, respectively, the Apparent Zenith and the True, or Central Zenith. In perfect strictness, the horizontal parallax is the parallax at the time zOS, the apparent 62 CORRECTIONS OF OBSERVATIONS. distance from the true zenith, is 90~. But no material error will be committed in supposing the earth to be spherical, except when the question relates to the parallax of the moon. S7. Triae Zenaith Dlist-ance. Let the apparent zenith distance ZOS _ Z, (Fig. 34,) the true zenith distance ZCS =z, and the parallax OSC -p. Since the angle ZOS is the exterior angle of the triangle OSC, we have ZOS ZCS + OSC, and hence also ZCS - ZOS - OSC; or, Z =z + p, and z = Z -p.... (5). Thus, to obtain the true zenith distance from the apparent, we have to subtract the parallax; and to obtain the apparent zenith distance from the true, to add the parallax. Parallax, then, takes effect wholly in a vertical plane, like the refraction, but in the inverse manner; depressing the star, while the refraction elevates it. Thus, the refraction is added to Z, but the parallax is subtracted from it. ~S. To find an Expression for tihe Parallax in Altitude, in terms of the apparent zenith distance. In the triangle SOC (Fig. 34) the angle OSC parallax in altitude =p, OC radius of the earth- R, CS = distance of the body S -D, and COS - 1800 — ZOS == 180 — apparent zenith distance = 180~ - Z; and we have by Trigonometry the proportion sin OSC: sin COS:: CO: CS; whence, sin p: sin (180~- Z)::: D; and D sinp R sin Z; or, sin p = sin Z (6). This equation shows that the parallax p depends for any given zenith distance Z upon the distance of the body, and is less in proportion as this distance is greater: also, that for any given distance of the body it increases with an increase in the zenith distance. When Z - 90~, p has its maximum value, and then horizontal parallax — H; and equa. (6) gives sin H R -..... (7); substituting, we have sinp = sin IH sin Z... (8). This equation may be somewhat simplified. The distances of the heavenly bodies are so great, that p and HI are always very small angles; even for the moon, which is much the nearest, the value of H does not at any time exceed 62'. We may, PARALLAX. 63 therefore, without material error, replace sin p and sin H by ii and Il. This being done, there results, 2p- -1 sin Z.... (9).'Wherefore, the parallax in altitude equals the product of the horizontalparallax by the sine of the apparent zenith distance. If we take notice of the deviation of the earth's form from that of a sphere, Z, in equation (8), will represent the apparent distance from the true zenith, (86,) and II the horizontal parallax as it is defined in Art. 86. In order to be able to compute the parallax in altitude by means of formula (9) it is necessary to know IIH, the horizontal parallax. S9. To find al ]Expression for the Horizontal Parallax, in terms of measurable quantities. Let O, O' (Fig. 35) represent two stations upon the same terrestrial meridian OEO', and remote from each other, Z, Z' their apparent zeniths, and z, z' their true zeniths, QCE the equator, and.S the body (supposed to be in the melidian) the parallax of which is to be found. Let the ancle OSO'- A, zOS - Z, z'O'S - Z'; also let CO - R, CO' 1R', CS D, the parallax in altitude OSC p, and the parallax in altitude O'SC p'. Now, by equation (6), replacing the sine of the parallax by the parallax itself; (88,) p= sin Z, nd' sin andsin D 1) whence R R' R sin Z + R' sin Z' ~p + p':j-ysin Z + -sin Z'- D (10); and, (equ. 7,) R R IIT D) or D - Substituting this value of D, and deducinl the value of H, we have _H _((p'A) RxA (11). R sin Z + R' siln Z' - sin Z -- R sin Z It remains now to find an expression for A in terms of measurable quantities. Let Os and O's (Fig. 35) be the directions, at O and O', of a fixed star which crosses the meridian nearly at the same time with the body. Owing to the immense~distance of the star, these lines will be sensibly parallel to each other (19). Let the angle SOs, the difference between the meridian Zeenith distances of the body and star, as observed at O, be represented by d, and let the same difference SO's for the station 0', be represented by d'. Now, OSO' - OLO' - SO's = SOs-SO's, or A = d-d'. 64 CORRECTIONS OF OBSERVATIONS. If the body be seen on different sides of the star by the two observers, we shall have A d+ d'. Substituting in equation (11), there results, = RI (ddc')... (12). R sin Z+R' sin Z' If we regard the earth as a sphere, R-R', and dividing by R, we have:-_did'....(13). sin Z+ sin Z' 90. To Determilne tlhe Horizontal Parallax of a body, from Observationi; by n-eans of this formula. Let each of the two observers measure the meridian zenith distance of the body, and also of a star which crosses the meridian nearly at the same time with the body, and correct the measured distances for refraction. The difference of the two will be, respectively, the value of d and d'; and the corrected zenith distances of the body will be the values of Z and Z'. If formula (12) be used, the measured zenith distances of the body must still be corrected for the reduction of latitude, (Art. 23, def. 4.) It is not necessary that the two stations should be on precisely the same meridian; for if the meridian zenith distance of the body be observed from day to day, its daily variation will become known; then, knowing also the difference of longitude of the two places, the following simple proportion will give the change of zenith distance during the interval of time employed by the body in moving from the meridian of the most easterly to that of the most westerly station, viz.: as interval (T) of two suecessive transits: diff. of long., expressed in time, (t):: variation of zenith dist. in interval T: its variation in interval t. This result, applied to the zenith distance observed at one of the stations, will reduce it to what it would have been if the observation had been made in the same latitude on the meridian of the other station. The horizontal parallax of the moon has been determined by this process with sufficient accuracy. The parallaxes of the sun and planets, which are very small, have been determined by much more accurate methods. The importance of havilg recourse to methods of the greatest possible accuracy, in the case of the sun and planets, will appear in the sequel. 91. Horizontal Parallax ial Differeiat Latituades. In consequence of the spheroidal form of the earth, the horizontAl parallax of a body is somewhat different in different latitudes. Let H and H' denote the horizontal parallaxes of the same body, at the distance D, and R and R' the radii of the earth at two different latitudes; then, by equ. 7, PARALLAX. sin H R-, and sin HI' D' -- R R' whence, sin H: sin H':: D: D:: R: R'. Also, as H and 1' are small, we have very nearly, H: H'::R: R'. Thus the horizontal parallax is greatest at the equator, and decreases nearly in the same ratio with the radius of the earth from the equator to the poles. The horizontal parallax of the moon is about 11" greater at the equator thlan at the poles. In the case of the sun, or of any planet, the difference is in every instance less than /". 92. Equatorial Parallax. The horizontal parallax of a body, for a station on the equator, is called its equatorial horizontal parallax, or simply its equatorial parallax. The equatorial parallax of the moon varies from 52' 50" to 61' 32", according to the distance of the moon from the earth. At the mean distance its value is 57' 3". The equatorial horizontal parallax of the sun, at the earth's. mean distance, is 8".95. The sun's horizontal parallax varies with the earth's distance less than ~". The horizontal parallaxes of the planets, at their varying distances from the earth, are comprised between the limits 34" and 0."3. The greater limit is the parallax of Venus when nearest the earth, and the smaller limit is the parallax of Neptune when farthest from the earth. The fixed stars have no geocentric parallax. Tables of Parallax. In the present condition of astronomical science, the horizontal parallax of the sun, moon, or any planet, may be calculated for any particular time from the results of astronomical observations, or may more readily be obtained by the aid of tables that have been computed for the purpose of facilitating its determination. It may also be obtained by simple inspection, from the Nautical Almanac. The American, or English Nautical Almanac, is a collection of data to be used in nautical and astronomical calculations, published annually, two or three years in advance of the year firl which it is calculated. 93. Parallax in Right AseelsiioDa acid inl Deelihantion. Since the parallax of a body displaces it in its vertical circle, which is generally oblique to the equator, it will alter its right ascension and declination. The consequent corrections to be applied to the right ascension and declination are called, respectively, parallax in right ascension, and parallax in declination. For a similar reason the parallax of a body, generally alters both its longitude and latitude; and the requisite corrections are termed parallax in longitude, and parallax inc latitude. 5 66 CORRECTIONS OF OBSERVATIONS. Formulae for calculating the parallax in right ascension, and in declination, as well as in longitude and latitude, are investigated in the Appendix. ABERRATION. 94,. The celebrated English astronomer, Dr. Bradley, commenced in the year 1725 a series of accurate observations, with the view of ascertaining whether the apparent places of the fixed stars were subject to any direct alteration in consequence of the continual change occurring. in the earth's position in space. The observations showed that there had been in reality, during the period of observation, small changes in the apparent places of each of the stars observed, which, when greatest, amounted to about 40"; but they were not such as should have resulted from the orbital motion of the earth. These phenomena Dr. Bradley undertook to examine and reduce to a general law. After repeated trials, he at last succeeded in discovering their true explanation. HEis theory is, that they are different effects of one general cause, a progressive motion of light in conjunction with the orbital motion of the earth. 95. Aberration of Lig,ht. Let us conceive the observer to be stationed at the earth's centre; and let ACB (Fig. 36) be a' A A' c/ GB FIG. 36. portion of the earth's orbit, so small that it may be considered a right line, CS the true direction of a fixed star as seen from the point C, AC the distance through which the earth nioves in some small portion of time, and aC the distance traversed by a wave of light, in the same time. Then, a ray of light, which, coming from the star in the direction SC, is at a at the same time that the earth is at A, will arrive at C at the same time that the earth does. Suppose that Aa is the position of the axis or central line of a tele ABERRATION. 67 scope, when the earth is at A, and that, continuing parallel to itself, it takes up, by virtue of the earth's motion, the successive positions A'a', A"a"...... CS'. A ray of light which follows the line SC in space will descend along this axis: for aa' is to AA' and aa" is to AA", as aC is to AC, that is, as the velocity of light is to the velocity of the earth; consequently, when the earthl is at A' the ray of light is on the axis at ca', and when tile earth is at A" the ray is on the axis at a", and so on for all the other positions of the axis, until the earth arrives at C. The apparent direction of the star S, as far, at least, as it depends upon the cause under consideration, will therefore be CS'. The angle SCS', which expresses the chancge in the apparent place of a star S, produced by the motion of light combined with the motion of the spectator, is called the Aberration of the star; and the phenomenon of the change of the apparent course of the light coming from a star, thus produced, is called Aberration of L2ght, or simply Aberration. The phenomenon of the aberration of light may be familiarly illustrated by taking falling drops of rain instead of supposed particles of light, and a vessel in motion at sea instead of the earth moving through space; and considering what direction must be given to a small tube by a person standing upon the deck of the vessel, so as to permit the drops falling perpendicularly to pass through the tube. It is plain that if the tube had a precisely vertical position, its forward motion would bring the back part of the tube against the drop; and that the only way to prevent this is to incline the upper end of the tube forward, or draw the lower end backward, whereby the back part of it would be made to pass through a greater distance before it comes up to the line of (lescent of the drop. The quantity that it is made to deviate in direction from this line, must depend upon the relative velocities of the falling drop and moving tube. To the observer, unconscious of his own motion, the drop will appear to fall in the oblique direction of the tube. 96. Angle of Aberration. If through the point a (Fig. 37) FIG. 37. a line, as', be drawn parallel to AC, and terminating in CS', the figure Aas'C will be a parallelogram, and therefore as' will be equal to AC. Hence it appears, that if on CS, the line of direc 68 CORRECTIONS OF OBSERVATIONS. tion of a star S, a line Ca be laid off, representing the velocity of light, and through a a line, as', be drawn, having the same direction as the earth's motion and equal to its velocity, the line joining s' and C will be the apparent line of direction of the star, the point S' its apparent place in the heavens, and the angle aCs' its aberration. WVe conclude, therefore, that by virtue of aberration a star is seen in advance of its true place, in the plane passing through the line of direction of the star and the line of the earth's motion. The amount of the aberration of a star is always very small (never greater than about 20"), because of the very great disproportion between the velocity of light and the velocity of the earth. It is very much exaggerated in Figs. 36 and 37. The aberration is the same wchen a star is viewed with the naked eye as when it is seen through a telescope. For, let aC, the velocity of the light, be decomposed into two velocities, of which one, AC, is equal and parallel to the velocity of the earth, the other will be represented by s'C. Now, since the velocity AC is equal and parallel to the velocity of the earth, it will produce no change in the relative position of a supposed particle of light and the eye, and therefore the relative motion of the light and the eye will be the same that it would be if the earth were stationary and the light had only the velocity s'C; accordingly, the light entering the eye just as it would do if it actually came in the direction s'C, and the eye were at rest, Cs' will be the apparent direction of the star from which it proceeds. If we regard the observer as situated upon the earth's surface, instead of being at its centre, the aberration resulting from the earth's motion of revolution will be still the same, for all points of the earthl advance at the same rate and in the same direction with the centre. The motion of rotation will produce an aberration proper to itself, but it is so small that there is no occasion to take it into account. 97. To find a General Expression for the Aberratiort. We have by Trigonometry (Fig. 37), sin AaC: sin CAa:: CA: Ca:: vel. of earth: vel. of light; whence, sin AaC = sin CAac A or, since AaC = SCS', Ca sin aberr. - sin CA l. of earth (14). vel. of light When CAa is 90~, the aberration has its maximum value, and this has been found by observation to be 20".445; whence, sin 20". 445 el of earth (5) vel. of light ABERRATION. 69 substituting, and.taking sin BCa for sin CAa, to which it is very nearly equal, we have sin aberr. = sin BCa sin 20".445....(16). We may conclude from this equation, that the aberration increases with the angle 13Ca made by the direction of the star with the direction of the earth's motion; that it is equal to zero when this angle is zero, and has its maximum value of 20".445 when this angle is 90~. 98. Annual Curve of Aberration. Let us now inquire into the entire effect of aberration in the course of a year. Let S (Fig. 38) be the sun; E the earth; Efg its orbit; ZTV that orbit FiG. 38. extended to the fixed stars, or the ecliptic (p. 15, def. 17); ET a tangent to the earth's orbit at E; o the place of S among the fixed stars or in the ecliptic, as seen from the earth; s a fixed star; siT the arc of a great circle passing through s and T. Then, by what has preceded (96), the earth moving in the direction Efy, the apparent place of the star may be represented by s' and the aberration by sEs'. Thus, the effect of aberration at any one time is to displace the star by a small amount, directly towards the point T of the ecliptic, which is 90~ behind the sun. As the earth moves, the position of the point T will vary; and in the course of a year, while the earth describes its entire orbit in the direction Efg, this point will move in the same direction entirely around the ecliptic. In this period of time, therefore, ss', the small arc of aberration, will revolve entirely around s, the true position of the star; from which we conclude, that in consequence of aberration a star appears to describe a closed curve in the heavens around its true place. As the inclination of the direction of the star to the direction of the earth's motion will vary during a revolution of the earth, the aberration will also vary during this period (97), and hence the curve in question will not be a circle. It appears upon investigation that it is an ellipse, having the true place of the star for l70 CORRECTIONS OF OBSERVATIONS. its centre, and of which the semi-major axis is constant and equal to 20".445, and the semi-minor axis variable and expressed by 20".445 sin 2, (;~ denoting the latitude of the star). Each star, then, describes an ellipse which is the more eccentric in proportion as the star is nearer to the ecliptic; for, the expression for the minor axis shows that the smaller the latitude the less will be this axis. For a star situated in the ecliptic the minor axis will be zero, and the ellipse will be reduced to a right line. For a star in the pole of the ecliptic the minor axis will be equal to the major, and the ellipse therefore becomes a circle. In following the motion of the star in its ellipse, it is to be observed that the orbit of the earth is a mere point at the centre of the celestial sphere, and the angle sET as the earth moves forward, decreases from 90~ at E to its minimum value at f and then increases to 90~ at r; and that similar changes occur while the earth is describing the other half of its orbit. When the earth is at E, the star is at one extremity of the major axis of its ellipse, and when the earth is arrived at r, the star is at the opposite extremity of the major axis. The points j and g, where the angle sET has its minimum value, answer to the extremities of the minor axis. 99. Aberration of the sun.-Displacement of the moon and planets. Since the motion of the earth is at all times in a direction perpendicular, or nearly so, to the line followed by the light which comes from the sun to the earth, the aberration of the sun, which takes place only in longitude, is continually equal to about 20"'.44. Thus the sun's apparent place is always about 20".44 behind its true place. The apparent displacement of a planet, resulting from the progressive motion of light, differs from that of a fixed star in a similar position. As a planet changes its place during the interval of time that a ray of light is passing from it to the earth, it would, if the earth were stationary, appear to be as far behind its true place as it has moved during this interval. This angular displacement, dependent upon the motion of the planet, combined with the aberration proper due to the earth's motion, constitutes the actual angular displacement of the planet from the cause under consideration. The apparent change of place caused by the motion of the moon around the earth is very small. 100. Aberration inil Right Asceension antd ial Declination. Since aberration causes the apparent place of a star, that has been (orrected for refraction, to differ slightlv from its true place, the true and apparent co-ordinates will differ somewhat from each other. The effects of the aberration of light upon the right ascension and declination of a star are called, respectively, the aberration ilz right ascension and the aberration in declinattion. These are to be determined and applied as corrections to the apparent right ascension and declination; the result will be the true co-ordinates, which will define the actual place in the heavens of the body observed. Formulae for computing the aberrations of a star in right as. cension and declination, are investigated in the Appendix. 101. Proof of the Progressive Mlotion of Light. If the apparent places of a star, found at various times, be corrected for aberration, the same result for the true place of the star is obtained. Again, the deductions of Art. 98 agree in every par. ABERRATION. 71 ticular with the observed phenomena of the apparent displacement of the stars, first discovered by Dr. Bradley. These facts show that the aberration of light is the true cause of these phenomena, and consequently establish at the same time the fact of the progressive motion of light, and that of the orbital motion of the earth. Although Bradley derived from the phenomena of aberration decisive proof of the progressive motion of light, it was first discovered by Roemer, a Danish astronomer, in 1675, from a comparison of observations upon the eclipses of Jupiter's satellites. Velocity of Light. We have by equation (15), vel. of earth: vel. of light:: sin 20".445: 1:: 1: 10,088.8; and taking the velocity of the earth in its orbit at 65,460 miles per hour, or 18.1833 miles per second, we obtain for the velocity of light 183,448 miles per second. The orbital velocity of the earth here used is that which answers to the recent more accurate determination of the earth's distance from the sun (viz. 91,328,100 miles). The result obtained for the velocity of light is nearly 8,000 miles per second less than the former determination, in which the mean distarne of the earth from the sun was taken a little over 95,000,000 miles. Light traverses the distance from the sun to the earth in 8m. 18s. 72 FIGURE AND DIMENSIONS OF THE EARTH. CHAPTER V. FIGURE AND DIMENSIONS OF THE EARTH.-LATITUDE AND LONGITUDE OF A PLACE. io02. ALTHOUGH it is in general sufficient for astronomical purposes to regard the earth as a sphere, still it is necessary in some cases of astronomical observation and computation, when accurate results are desired, to take notice of its deviation from the spherical form. No account need, however, be taken of the irregularities of its surface, occasioned by mountains and valleys, as they are exceedingly minute when compared with the whole extent of the earth. It is to be understood, then, that by the figure of the earth is meant the general form of its surface, supposing it to be smooth, or that the surface of the land corresponds with that of the sea. 103. lMethod of determining the Fortii of a Terres. trial Mleridian. The figure of the earth is ascertained from an examination of the form of the terrestrial meridians. A Degree of a terrestrial meridian is an arc of it corresponding to an inclination of 1~ of the vertical lines at the extremities of the are. It is also called a Degree of Latilude. Thus, if QNE (Fig. 39) represent a terrestrial meridian, ab will be a degree of it if' it be of such length that the angle aCb between the vertical lines Z'aC, ZbC, is 10. z Q FIG. 39. The length of a degree at any place will serve as a measure of the curvature of the meridian at that place; four it is LENGTH OF A DEGREE. 73 obvious, from considerations already presented (3), that the earth, if not strictly spherical, must be nearly so, and therefore that a degree ab (Fig. 39) may, with but little if any error, be considered as an arc of 1~ of a circle which has its centre at C, the point of intersection of the verticals Ca, Cb, at the extremities of the arc. The curvature will then decrease in the same proportion as the radius of this circle increases, and therefore in the same proportion as the length of a degree increases. Wherefore, the form of a meridian may be determined by measuring the length of a degree at various latitudes. 104. To determinme the Lemngth of a Degree of a Terrestrial iHieridian. To accomplish this, we have, (1.) To run a meridian line; an operation which is performed in the following manner. An altitude and azimuth instrument (or some other instrument adapted to meridian observations) is first placed at the point of departure, and accurately adjusted to the meridian. A new station is then established by sighting forward with the telescope. To this station the instrument is removed, and is there adjusted to the meridian by sighting back to the first station. A third station is then established by sighting forward with the telescope as before, to which the instrument is removed. By thus continually establishing new stations, and carrying the instrument forward, the meridian line may be marked out for any required distance. The meridian adjustments may be corrected from time to time by astronomical observations (42, 58). (2.) To find the length of the arc passed over. When the ground is level, the length of the are may be directly measured. In case the nature of the ground is such as not to allow of a direct measurement, it may be determined with great precision by means of a base line and a chain of triangles, the angles of which are measured. (3.) To find the inclination of the verticals at the extreme stations. This angle may be obtained by measuring the meridian zenith distances of the same fixed star at the two stations, correcting them for refraction, and taking their difference. For, let O, O' (Fig. 89) be the two stations in question, Z, Z' their zeniths, and OS, O'S, the directions of' a fixed star, and we shall have OcO' = ZOI - OIc = ZOS - Z'IS ZOS- Z'O'S; that is, the angle comprised between the verticals equal to the difference of the meridian zenith distances of the same star. (4.) The length of an arc of the meridian, either somewhat greater or less than a clegree, having been found by the foregoing operations, thence to co?mrpute the length of a degree. Let N denote the number of degrees and parts of a degree in the measured arc, A its length, and x the length of a degree. Then, allowing that the earth for an extent of several degrees does not differ sensibly from a sphere, we may state the proportion 74: FIGURE AND DIMENSIONS OF THE EARTH. l~ x A N: A:: 1~: x; whence x =- A —....(17). 105. Results of the Measurenients of Degrees. Degrees have been measured with the greatest possible care, at various latitudes and on various meridians. Upon a comparison of the measured degrees, it appears that the length of a degree increases as we proceedJfrom the equator towards either pole. It follows, therefore (103), that the curvature of a meridian is greatest at the equator, and diminishes as the latitude increases; and consequently, that the earth is flattened at the poloes. The fact of the decrease of the curvature of a terrestrial meridian from the equator to the poles, leads to the supposition that it is an ellipse, having its major axis in the plane of the equator and its minor axis coincident with the axis of the earth. Analytical investigations, founded on the lengths of a degree in different latitudes, and on different meridians, have established that a meridian is, in fact, very nearly an ellipse, and that the earth has very nearly the form of an oblate slpheroid. The same investigations have also made known the dimensions of the earth. The amount of the oblateness at the poles is measured by the ratio of the difference of the equatorial and polar diameters to the equatorial diameter, which is technically termed the Oblatenzess of the earth. R E/ z FIG. 40. The form of the earth has also been determined by other methods, which cannot here be explained. All the results of measurements, taken together, indicate an oblateness of 1 7 0 7 ~~~299' The following are the dimensions of the earth in miles: Radius at the equator............... 3,962.80 miles. Radius at the pole.................. 3,949.55 " Difference of equatorial and polar radii. 13.25 " Radius at 450 latitude............... 3,956.20" Mlean length of a degree of meridian... 69.048 " The fourth part of a meridian......... 6,2 14.33'" LATITUDE OF A PLACE. 75 106. 1nclinationi of Radius to Vertical Line. Owing to the elliptical form of a terrestrial meridian, the radius and vertical line at a place do not coincide. Let ENQS (Fig. 40) represent a terrestrial meridian. For any point 0 situated on this meridian, CO will be the radius, and the normal line ZOB the vertical. The position of the vertical line will always be such that the apparent zenith Z will lie between the true zenith z and the elevated pole P. The inclination of the radius to the vertical line, or the angle COB, called the reduction of latitude, is greatest at the latitude 45~, and is there equal to about 11~'. DETERMINATION OF THE LATITUDE AND LONGITUDE OF A PLACE. 107. The latitude and longitude of a place ascertain its situation upon the earth's surface, and are essential elements in many astronomical in vestigations. o10. To fimnd 1the Latitude of a Place. (1.) By the zenith distances or altitudes of a circumpolar star, at its upper and lower transits. The principle of this method has already been stated (55), and represented to be a particular case of a well-known principle of arithmetical proportions; the following is a detailed proof of it. Let Z (Fig. 41) repre-?R FIG. 41. sent the zenith, HOR the horizon, P the pole, and S, S' the points at which the upper and lower transits of a circumpolar star take place; HP will be equal to the latitude (24), and ZP will be equal to the co-latitude. Now, we have HP i HS + PS, and HP = HS'- PS' HS'- PS; whence, 2HP = IHS+HS', or, HP _ HS HS.(18) I'n like manner we obtain, ZP = ZS+ZS' (19) z =zszs' 2 Wherefore, let the altitudes of a circumpolar star at its upper and lower transits be measured and corrected for refraction, and their half sum will be the latitude; or, let the zenith distances be measured, and corrected for refraction, and their half sum sub 76 LATITUDE AND LONGITUDE OF A PLACE. tracted from 90~ will be the latitude. Stars should be selected that have a considerable altitude at their inferior transit, for, the greater is the altitude the less is the uncertainty as to the amount of the refraction. On this principle the pole-star is to be preferred to all others. (2.) By a single meridian altitude or zenith distance. Let s, s', s" (Fig. 10, p. 21) be the points of meridian passage of three different stars, the first to the north of the zenith, the second between the zenith and equator, and the third to the south of the equator ZE - the latitude, and we have for the three stars, ZE sE- Zs, ZE- s'E t Zs', ZE = Zs" —s"E. Thus, if the zenith distance be called north or south, according as the zenith is north or south of the star when on the meridian, in case the zenith distance and( declination are of the same name their sum will be equal to the latitude; but if they are of different names their difference will be the latitude, of the same name with the greater. This method supposes the declination of the body observed to be known. The declination of a star or of the sun at any time is, in practice, obtained for the solution of this and other problems, by the aid of tables, or is taken by inspection from the American Nautical Almanac, or other similar work. If the time of the meridian transit be known, the altitude may be measured by a sextant (67). The observed altitude must be corrected for refraction, and also for parallax if the body observed be the sun, or moon, or either one of the planets. This method of finding the latitude is the one most generally employed at sea, the sun being the object observed. As the time of noon is not known with accuracy, several altitudes about the time of noon are taken, and the meridian altitude is deduced from these. (3.) By the difference of the meridian zenith distances of two stars that cross the meridian near the zenith, on opposite sides. This is'Talcott's lfethod alluded to in connection with the subject of the zenith telescope (59). It is to be preferred to all other methods of determining the latitude, when the observer is provided with a zenith telescope. Let z be the true zenith distance of the star that passes to the south of the zenith, and if its declination; z' and Y' the true zenith distance and declination of the other star; and I the latitude of the station: we then have l=- — z, and l = — 6 z', and thefoere, I - 8+')+ (z - z').... (a). Alao, let Z denote the apparent zenith distance of the star that passes to the south of the zenith, r its refraction, and Z', r' the coeaponding quantities for the other star; then, LATITUDE OF A PLACE. 77 z = Z+r, and z' Z' r'; and substituting in equation (a) we obtain, As we may'suppose the declinations of the two stars to be known, it is then only necessary to determine the values of Z-Z', and r- r'. Now, if two stars be selected whose zenith distances are nearly equal, their difference, Z - Z', can be directly measured by the micrometer of the zenith telescope, and thus a result obtained for the latitude free from the instrumental errors that attend all methods in which the absolute zenith distances are measured. Also, if the selected stars pass the meridian near the zenith, their refractions will be small, and the amount of their difference, r - r', very minute, and liable to no appreciable uncertainty. If m and m' denote the micrometer readings in observing the two stars, converted into their equivalent angular values, equation (b) becomes, Z-(8 I 8') + 7W(m - m') + 2 (r 4')... (C). It is here tacitly supposed that the micrometer reading increases with an increase of zenith distance. If the reverse be true, the second term should be affected with the negative sign. The only instrumental correction that is to be applied to the result given by this formula, is for any error that may occur in the position of the vertical axis of the zenith telescope, when either star is observed. This is determined by means of a horizontal level, attached to the instrument in a position perpendicular to the horizontal axis of rotation of the telescope; and therefore turning with the instrument around the vertical axis. The method of making the observations is briefly as follows: the instrument having been previously adjusted to the meridian, the observer, by means of the finding circle (p. 82), sets the telescope to the mean of the zenith distances of the selected pair of stars, and when the precedinc star has entered the field follows it with the movable micrometer wire, and bisects it as it reaches the meridian. He then reads the micrometer, and also the level; and turns the instrument around its vertical axis, 180~ in azimuth. When the second star enters the field of the telescope, it is bisected, like the first, with the micrometer-wire as it reaches the meridian. The micrometer and level are then read as before. The micrometer readinogs multiplied by the angular value of one revolution of the micrometer-screw, are the values of m and m' in equation (c). Both the north and south ends of the bubble of the level are read in each observation, and the south end reading subtracted from the north end reading,. Half the difference multiplied by the value of one division of the level in seconds of arc, will be the inclination of the level to a horizontal line, in each observation. The half algebraic sum of these inclinations for the two observa 78 LATITUDE AND LONGITUDE OF A PLACE. tions, will be the correction to be applied, according to its sign, to the result obtained by equation (c), for the deviation of the vertical axis from the truly vertical position. It is found that the probable error, from all causes, of a single determination, by a practised observer, does not exceed 1"; and that by continuing the observations upon a series of pairs of suitably selected stars, for a number of nights, the latitude of a station can be determined with a probable error of only 0".1, which answers to a distance on the meridian of only ten feet. Reduced Latitude. The astronomical latitude being known, the reduced latitude (p. 19, def. 4) may be obtained by subtracting from it the reduction of latitude. For if OC (Fig. 40) represents the radius, and OB the vertical, at any place 0, and ECQ represents the terrestrial equator, OBQ will be the astronomical latitude, OCQ the reduced latitude, and COB the reduction of latitude; and we have, OB3Q OCQ + COB, and OCQ- OBQ -COB.... (20). (For the practical method of resolving this problem, see Problem XV.) 109. Longitude of a Place: General Principle. There are various methods of finding the longitude of a place, nearly all of which rest upon the following principle: The difference at any instant between the local timnes (whether sidereal or solar), at any place and on, the first meridian, is the longitude of the place expressed in timne; and consequently, also, the diference between the local times at any two places is their difference of longitude in time. The truth of this principle is easily established. In the first place, we remark that the longitude of a place contains the same number of degrees and parts of a degree as the are of the celestial equator comprised between the meridian of Greenwich and the meridian of the place. Now, it is Oh. Om. Os. of mean solar time, or mean noon, at any place, when the mean sun (36) is on the meridian of that particular place. Therefore, as the mean sun, moving in the equator, recedes from the meridian towards the west at the rate of 15~ per mean solar hour, when it is mean noon at a place to the west of Greenwich, it will be as many hours and parts of an hour past mean noon at Greenwich, as is expressed by the quotient of the division of the arc of the celestial equator, or its equal the longitude, by 15. If the place be to the east instead of to the west of Greenwich, when it is mean noon there, it will be as much before mean noon at Greenwich as is expressed by the longitude of the place converted into time (as above). In either situation of the place, then, the principle just stated will be true. It is plain that the equality between the difference of the times and of the longitudes will subsist equally if sidereal instead of solar time be used. LOXGITUDE OF A PLACE. 79 110. To finsd tlae Lo-ngitude of a Place. (1.) Let two observers, stationed one at Greenwich and the other at the given place, note the times of the occurrence of some phenomenon which is seen at the same instant at both places; the difference of the observed times will be the longitude in time. The same observations made at any two places will make known their diference of longitude. If the stations are not distant from each other, a signal, as the flashing of gunpowder, or the firing of' a rocket, may be observed. When they are remote from each other, celestial phenomena must be taken. Eclipses of the satellites of Jupiter tand of the moon, are phenomena adapted to the purpose in question. But as in these eclipses the diminution of the light of the body is not sudden, but gradual, the longitude cannot be obtained with very great accuracy from observations made upon them. (2.) tiancsport a chronometer which has been carefuZy actdjsted to the local time at Greenwic.h, to the place whose longitude is souight, and compare the time given by the chronometer withl the local time of the place. In the same way, by transporting a chronometer from any one place to another, their difference of longitude may be obtained. The error find rate of the chronometer must be determined at the outset, and as often afterwards as circumstances will admit, that the error at the moment of the observation may be known as accurately as possible. To insure greater certainty and precision in the knowledge of the time, a number of chronometers are often taken, instead of one only. This method is much used at sea; the local time being obtained from an observation upon the sun or some other heavenly body, in a manner to be hereafter explained. (3.) Let the Greenwich time of the occurrence of some cele.stial phenomenon be computed, and note the time of its occurrence at the given place. Eclipses of the sun and moon, and of Jupiter's satellites, occultations of the stars by the moon, and the angular distance of the moon from some one of the heavenly bodies, are the phenomena employed. The Greenwich times of the beginning and end of the eclipses of Jupiter's satellites, are published for the solution of the problem of the longitude in the English Nautical Almanac. When the'lngitude is estimated from Washington, the Washington times of the occurrence of the same phenomena may be taken from the American Nautical Almanac. Eclipses of the sun, and occultations of the stars, furnish the most exact determinations of the longitude, but they cannot be used for this purpose unless the longitude is already approximately known. The method of lunar distances is chiefly used at sea, and is given in detail in treatises on navigation and nautical astronomy. (4.) Another and more accurate method of determining the dif 80 LATITUDE AND LONGITUDE OF A PLACE. ference of longitude of two places, has recently been introducedand perfected by American astronomers. It consists in the use of the electric telegraph fo)r the transmission of signals from one station to the other, and the introduiction of the electro-chronograph into tlhe circuit, to measure off and record, at each station, the beats of a sidereal clock. The clock may be at either station, or at some other astronomical station in the circuit. Its beats are electrically transmitted, and recorded upon a moving roll of paper, adapted to the registers at each station, in a series of equally distant dots, or in a succession of equally distant breaks in a continuous line (see Fig. 22, p. 37). The signals adopted are the passages of a star across the wires of a transit instrument. The observer at the most easterly station strikes his breakcircuit key as the star passes each of the wires in succession. As the result, the instants of these successive transits are shown upon the roll of paper at each station, by breaks in the line of seconds, falling between those which indicate the seconds. When the star reaches the meridian of the other station, a similar set of observations are made by the other observer; and the instants of the successive transits are recorded as before, upon the roll of paper at each station. It then only remains for each observer to remove the roll upon which the instants of the passage of the star across the wires of the transit instrument at each station are noted, and carefully measure the distance between each break in the time-line, obtained by the one set of observations, fromn the corresponding break obtained by the other set; then convert this into the equivalent interval of time, and take the mean of all the intervals. This will be his determination of the difference of longitude of the two stations, in time.'The mean of the results thus obtained by the two observers, is then to be taken as more reliable than either of the single determinations. For greater accuracy a number of selected stars should be observed. The observations should also be many times repeated; the clocks at the two stations being alternately thrown into the circuit. The result obtained is free from the errors that may exist in the tabular places of the stars observed, and from the clock error; since neither of these errors will affect the intervals of time employed by the stars in passing from the meridian of the one station to that of the other. But each observer should carefully determine and allow for the errors of adjustment of his transit instrument. The longitudes of the principal observatories in the United States, and of several important stations of the United States Coast Survey, have been very accurately determined by this method. OBLIQUITY OF THE ECLIPTIC. 81 CHAPTER VI. APPARENT MOTION OF THE SUN IN THE HEAVENS. ill. The sun's declination and the difference between the right ascension of the sun and that of some fixed star, found from day to day (45 and 55) throughout a revolution, are the elements from which the circumstances of the sun's apparent motion are derived. The curve on the sphere of the heavens, passing through all the successive positions thus determined fiom day to day, is the Ecliptic. If we suppose it to be a circle, as it appears to be, its position will result from the position of the equinoctial points and its obliquity to the equator. 112. To find the Obliquity of the Ecliptic. Let EQA (Fig. 42) represent the equator; ECA the ecliptic; and OC, OQ, nes. 42. lines drawn through 0, the centre of the earth, and perpendicular to the line of the equinoxes, AOGE: then the angle COQ will be the obliquity of the ecliptic. This angle has for its measure the arc CQ, and therefore tke ohliquity of the ecliptic is equal to the greatest declination of the sun. It can but rarely happen that the time of the greatest declination will coincide with the instant of noon at the place where the observations are made, but it must fall within at least twelve hours of the noon for which the observed declination is the greatest. In this interval the change of declination cannot exceed 4", and therefore the greatest observed declination cannot differ more than 4" from the obliquity. A formula has been investigated, which gives in terms 6 82 APPARENT MOTION OF THE SUN. of determinable quantities the difference between any of the greater declinations and the maximum declination. By reduci/rq, by means of this formula, a number of the greater declinations to the maximum declination, and taking the mean of the individual results, a very accurate value of the obliquity may be ibund. r'The obliquity of the ecliptic changes slightly from year to year. It is also subject to a slight diminution from century to century. Its mean value at tile present date (Jan., 1867) is 230 27' 24". 1i3. To find the Position of the Vernal or Autumnal Equinaox. (1.) On inspecting the observed declinations of the sun, it is seen tlihat about the 21st of March the declination changes in the interval of two successive noons from south to north. The vernal equinox occurs at some moment of this interval. Let RS, IR'S' (Fig. 43) represent the declinations at the noons between FIG. 43. which the equinox occurs: as one is north and the other south, their sum (S) will be the daily change of declination at the time of the equinox. Denote the time from noon to noon by T. Now, to find the interval (x) between the noon preceding the equinox and the instant of the equinox, state the proportion Tx RS S: RS:: T: x= T S SI on the principle that the declination changes, for a day or more, proportionally to the time. Next, take the daily change in right ascension (RR') on the day of the equinox and compute the value of RE, by the proportion T x RS T: x, or [LIRS: RR': RE; add RE to MR, the observed difference of right ascension (111) on the day preceding the equinox, and the sum ME will be the POSITION OF TIIE EQUINOX. 83 distance of the equinox from the meridian of the star observed in connection with the sun; if the star be to the west of the sun, as in the figure. The position of the autumnal equinox mnay be found by a similar process, the only difference in the circumstances beino that the declination cllanges fromn north to south instead of from south to north. if the value of x which results from the first proportion be added to the time of noon on tile day preceding the equinox, the result will be the time of the equinox. (2.) In the triangle RES (Fig. 42) we have the angle RES - the obliquity of the ecliptic, and RS D the declination of the sun, both of which we may suppose to be known, and we have by Napier's first rule (Appendix), sin ER -— tan (co. RES) tan RS - cot cs tan D.... (21), whence we can: find ER. And by taking the sum or difference of ElR and MR, according as the star observed is on the opposite side of the sun from the equinox or the same side, we obtain MAE as before. If this calculation be effected for a number of positions, S, S', S", etc., of the sun on different days, and a mean of all the individual restlts be taken, a more exact value of ME will be obtained. MAE being accurately known, the precise time of the equinox may readily be deduced from the observed daily variation of right ascension on the day of the equinox. The calculations just mentioned rest upon the hypothesis that the ecliptic is a great circle. The close agreement which is found to subsist between the values of ME deduced from observations upon the sun in different positions, S, S', S", etc., establishes the truth of this hypothesis. It is also confirmed by the fact that the right ascerisio'ns of the vernal and autumnal equinox differ by 180~, since we may infer from this that the line of the equinoxes passes through the centre of the earth. 114. Longitude of the Suai. The longitude of the sun may be expressed in terms of the obliquity of the ecliptic and the right ascension or declination. In the t'riangle ERS (Fig. 42), ES ( = L) represents the longitude of the sun supposed to be at S, ER (- R) its right ascension, and RS ( =D) its declination. Now, by Napier's first rule, cos RES cos RES= tan ER cot ES, or cot ES cos RES=cosRES cot ER; tan ER thus, tan. cot L - cos o cot R, or tan L = __-.... (22), cos co Also (Napier's second rule, Appendix), 84 APPARENT MOTION OF THE SUN. sin RS - cos (co. RES) cos (co. ES); whence, sin ES - sin RS sin RES' or, sin L - sin.... (23). sin o With these formulae the longitude of the sun may be computed from either its right ascension or declination. (See Prob. XII., Part III.) Formulae (22) and (23) may be written thus, tan R = tan L cos a; sin D = sin L sin o.... (24). These formulae will make known the right ascension and declination of the sun, when its longitude is given. (See Prob. XI.) It will be seen in the sequel that in the present condition of astronomical science, the longitude of the sun at any assumed time may be computed from the ascertained laws and rate of the sun's motion. 115. Tropical Year. The interval between two successive returns of the sun to the same equinox, or to the same longitude, is called a Tropical Year. The interval between two successive returns of the sun to the same position with respect to the fixed stars, is called a Sidereal Year. It appears from observation that the length of the tropical year is subject to slight periodical variations. The period from which it deviates periodically and equally on both sides, is called the Jlean Tropical Year. As the changes in the length of the true tropical year are very minute, the length of the mean tropical year is obviously very nearly equal to the mean length of the true tropical year, in an interval during which this passes one or more times through all its different values. In point of fact, it may be found with a very close approximation to the truth by comparing two equinoxes observed at an interval of 60 or 100 years. According to the most accurate determinations, the length of the mean tropical year, expressed in mean solar time, is 365d. 5h. 48m. 46.1s. 116. Sun's Daily Motion in Longitude. In a mean tropical year the sun's mean motion in longitude is 360~; hence, to find his mean daily motion in longitude we have only to state the proportion 365d. 5h. 48m. 46s.: ld.:: 360: x = 59' 8".33. If from the right ascension or declination of the sun, found on two successive days, the corresponding longitudes be deduced (equs. 22, 23), and their difference taken, the result will be the sun's daily motion in longitude at the date of the observations. SUN S DAILY MOTION IN LONGITUDE. 85 The sun's daily motion in longitude is not the sace dthoughout the year, but, on the contrary, is continually varying. It gradually increases during one-half of a revolution, and gradually decreases during the other half, and at the end of the year has recovered its original value. Thus, the greatest and least daily motions occur at opposite points of the ecliptic. They are, respectively, 61' 10" and 57' 12". Tlie exact law Elf the sun's uneqguable motion, can only be obtained by taking into account the variation of his distance from the earth; for the two are essentiallv connected by the physical law of gravitation, which determines the nature of the earth's motion of revolution around the sun. That the distance of the sun from the earth is in fact subject to a variation, may be inferred from the observed fact that his apparent diameter varies. On measuring with the micrometer the apparent diameter of the sun from day to day throughout the year, it is found to be the greatest when the daily angular motion, or in longitude, is the greatest, and the least when the daily motion is the least; and to vary gradually between these two limits. Accordingly the sun is nearest to us when its daily angular motion is the most rapid, and farthest from us when its daily motion is the slowest. The greatest apparent diameter of the sun is 32' 36"; and the least apparent diameter 31' 32". 86 PRECESSION OF THE EQUINOXES. CHAPTER VII. PRECESSION OF THE EQUINOXES.-NUTATION. 117. Proof of an Aununal Precession of the Equinoxes. The determination of the position of the vernal equinox, consists in deducing from the results of certain observations the difference between the right ascension of the equinox and that of one of the fixed stars (113). This difference is represented by ME, in Fig. 42, and by VR in Fig. 8. We have seen (45) that when this has become known, the absolute right ascensions of all the stars mlay be determined. We have seen also (56), that when the right ascension and declination of a star are known, its longitude and latitude may be computed. Now, if the position of the vernal equinox be determined at two epochs separated by a nurnber of years, it is found that the value of ME has materially increased, if the star s, observed with the sun, is to the east of the equinox; and decreased if the star lies to the west of the equinox. From this fact we may conclude that the equinox has a retrograde motion, or towards the west, from year to year. Again, if the longitudes and latitudes of the same fixed stars, obtained as above, at different periods, be compared, it is found that their latitudes continue very nearly the same, but that their longitudes all increase at the same mean rate of about 50" per year. Thus, EL (Fig. 42) represents the longitude of the star s, and sL its latitude, and it is found thtat st remains the same, but that EL increases at the mean rate of 50" per year. It follows, therefore, that the vernal equinox must have an annual motion of about 50" along the ecliptic, in a direction contrary to the order of the signs, or from east to west. As it has been ascertained that the autumnal is always at the distance of 180~ from the vernal equinox, it must have the same motion. This retrograde motion of the equinoctial points is called the Precession of the EEquinoxes. 11S. Ecliptic Stationmary. As the latitude of a star is its angular distance from the ecliptic, it follows from the circumstance of the latitudes of all the stars continuing very nearly the same, that the ecliptic remains fixed, or very nearly so, with respect to the situations of the fixed stars. The ecliptic being stationary, it is plain that the precession of the equinoxes must result from a continual slow motion of the equator in one direction. It appears from observation that the obliquity of the ecliptic, or the inclination of the equator to the MOTION OF THE POLE OF THE HEAVENS. 87 ecliptic, remains in the course of this motion very nearly the same. 119. Progressive Motioni of the Pole of the Heavens. Since the equator is in motion, its pole must change its place in the heavens. Let VLA (Fig. 44) represent the ecliptic; K its, I{ FIG. 44. stationary pole; P the position of the north pole of the equator, or of the heavens, at any given time, and VEA the corresponding position of the line of the equinoxes: KPL represents the circle of latitude passing through P, or the solstitial colure. Now, the point V being at the same time in the ecliptic and equator, it is 90~ distant from the two points K arid P, the poles of these circles; therefore, it is the pole of the circle KPL passing through these points, and hence VL- 90~. It follows from this, that when the vernal equinox has retrograded to any point V', the pole of the equator, originally at P, will be found in the circle of latitude KP'L' for which V'L' equals 90~: it will also be at the distance KP' from the pole of the ecliptic, equal to KP. Whence it appears that the pole of the equator has a retrograde motion in a small circle about the pole of the ecliptic, and at a distance from it equal to the obliquity of the ecliptic. As the motion of the equator which produces the precession of the equinoxes is uniform, the motion of the pole must be uniform also; and as the pole will accomplish a revolution in the same time with the equinox, its rate of' motion must be the same as that of the equinox, that is, 50" of its circle in a year. The period of revolution of the equinox and the pole of the equator, is about 24,500 years. It is an interesting consequence of this motion of the pole of the equator and heavens, that the pole-star, so called, will not always be nearer to the pole than any other star. The pole is at the present time approaching it, and it will continue to approach it until the present distance of 11-~ becomes reduced to less than io, which will happen about the year 2100: after which it will begin to recede from it, and continue to recede, until about the 88 PRECESSION OF THE EQUINOXES. year 3200 another star will come to have the rank of a pole-star. The motion of the pole still continuing, it will, in the lapse of centuries, pass in the vicinity of several pretty distinct stars in succession, and in about 12,000 years will be within a few degrees of the star Vega, in the constellation of the Lyre, the brightest star in the northern hemisphere. The present pole-star has held that rank since the time of the celebrated astronomer IHipparchus, who flourished about 120 B. C. In very ancient times, a pretty bright star in the constellation of the Dragon (a Draconis) was the pole-star. The motion of the equator which produces the precession of the equinoxes, must also produce changes in the right ascensionzs and declinations of the stars. These changes will be different according to the situations of the stars with respect to the equator and equinoctial points. 120. Effect of Precession on the Length of the Year. The precession of the equinoxes makes the tropical year shorter than the sidereal year. For, since the precession is a retrograde movement of each equinox of 50".24 per year, when the sun has returned to the same equinox, it will not have accomplished a sidereal revolution into 50".24. The excess of the sidereal over the tropical year results from the proportion 59' S".33:: 50".24:: ld: x = 20m. 23.3s. Thus the length of the mean sidereal year, expressed in mean solar time, is 365d. 6h. 9m. 9.4s. 121. Secular Diminution of the Obliquity of the Ecliptic. The ecliptic, although very nearly stationary, as stated in Art. 118, is not strictly so. By comparing the values o(f the obliquity of the ecliptic, found at distant periods, it is:ascertained that it is subject to a gradual diminution of 46" from century to century. It appears front observation that there are minute secular changes in the latitudes of the stars, which establish that the progressive diminution of the obliquity of the ecliptic arises from a slow displacement of the plane of the ecliptic, or of the earth's orbit, in space. It remains for us now to take notice of a minute inequality in the motion of the equator and its pole, which we have thus far overlooked. NUTATION. 122. Discovery of Nutatioi. Dr. Bradley, in obserNvilngthe polar distance of a certain star (y Draconis) with the view of verifying his theory of aberration, discovered that the observed polar distance did not agree with the polar distance as computed fiiom the results of previous observations, by allowing for the clhange due to the precession in the interval; the proper correetiuns for refraction and aberration having been applied in both ELLIPSE OF NUTATION. 89 cases. On continuing his observations he found that the polar distance alternately increased and diminished, and that it returned to the same value in about 19 years. These phenomena led him to suppose that the pole, instead of moving uniformly in a circle around the pole of the ecliptic, oscillated from the one side to the other of a point conceived to move in this manner. 123. Ellipse of iNutation. If the pole has such a motion it is plain that, allowing the fact of the earth's rotation, it must result from a vibratory motion of the earth's axis. To this supnosed vibration of the axis of the earth, and consequently of that of the heavens, Dr. Bradley gave the name of Nutation. Upon a detailed examination of all his observations, it appeared that the oscillation of the pole did not take place in a right line, but in a minute ellipse. The motion may accordingly be regarded as a motion of revolution in an ellipse around its centre. This central point, about which the pole revolves, is the mean position of the pole, and is called the Mean Pole. The direction of the motion of revolution is retrograde, or from east to west, and the period is about 19 years. In Fig. 45, pgf'g' represents the ell'ipse of mutation, and P the K i/ T. FIG. 45. mean pole; the direction of the motion of revolution being from p towardsf. The major axis gg' lies in the solstitial colure KPL, and is equal to 19"; and the minor axis if' is equal to 14". While the true pole revolves in its ellipse about the mean pole P, the mean pole has a uniform retrograde movement in a circle NPP', around the pole of the ecliptic K. Accordingly the pole has two cotem poraneous motions; one in a minute ellipse, and about its centre, and another in a circle of 23~~ radius, about the pole of the ecliptic. Its actual motion must therefore be in a slightly waving curve, passing alterFIG. 46. nately from one side to the other of this 90 PRECESSION AND NUTATION. circle, as shown in Fig. 46; in which, however, the deviations from the circle are greatly exaggerated. The ellipse of nutation is also greatly exaggerated in Fig. 45. 124,. Effeets of iNutatioan. As the equator must move with the axis of the earth or heavens, nutation nmust change the position of the equinox and the obliquity of the ecliptlic. It is plain that its effect upon the position of the equinox will be to make it oscillate periodically, and by equal degrees, from one side to the other of the position which corresponds to the mean pole; and that its effect upon the obliquity of the ecliptic will be to make it alternately greater and less than the obliquity corresponding to the mean pole. The position of the equinox which corresponds to the mean pole is called the Mlean Equinox; and the obliquity correspondiug to the mean pole is called the lfean Obliquity..Mean Equator has a like signification. The real equinox and the real equator are called, respectively, the True Equinox and the True Equator. The actual obliquity of the ecliptic is termed the Apparent Obliquity. In like manner, the right ascension, declination, etc., of a star, referred to the mean equator and mean equinox, are designated the mean right ascension, mean declination, etc.; to distinguish them from the corresponding elements referred to the true equator and true equinox. The distance of the true from the mean eql-d nox in longitude, is called the equation of the equinoxes in longitude. DIFFERENT KINDS OF TIME. 91 CHAPTER VIII. MEASUREMENT OF TIME. DIFFERENT KINDS OF TIAME. 125. IN Astronomy, as we have already stated, three kinds of time are used -Sicdereal, T'ue or Apparent Solar, and lean Solar Time; sidereal time being measured by the diurnal motion of the vernal equinox, true or apparent solar time by that of the sun, and mean solar time by that of an imaginary sun called the Jfean sun, conceived to move uniformly in the equator with the real sun's mean motion in right ascension or longitude. 126. True Solar Day. The sidereal day and the mean solar day are each of uniform duration, but the flength of the tlrue soear clday is variable, as we will now proceed to show. The sun's daily motion in right ascension, expressed in time, is equal to the excess of the solar over the sidereal day. Now this are, and therefore the true solar day, varies from two causes, ViZ.: (1.) ]ie inequality qf the San's daily mnotion in longitucde. (2.) The obliquity of the ecliptic to the equator. If the ecliptic were coincident with the equator, the daily arc of right ascension would be equal to the daily arc of longitude, and therefore would vary between the limits 57' 12" and 61' 10", which would answer, respectively, to the apogee and perigee. But, owing to the obliquity of the ecliptic, the inclination of the daily are of longitude to the equator is subject to a variation; and this, it is plain (see Fig. 42), will be attended with a variation in the daily arc of right ascension. The tendency of this cause is obviously to make the daily arc of right ascension least at the equinoxes, where the obliquity of the arc of longitude is greatest, and greatest at the solstices, where the obliquity is least. 127. MIean Solar Time. As the length of the apparent solar clay is variable, it cannot conveniently be employed for the expression of intervals of time; moreover, a clock, to keep apparent solar time, requires to be frequently adjusted. These inconveniences attending the use of apparent solar time, led astronomers to devise a new method of measuring time, to which they gave the name of mean solar time. By conceiving an imaginary sun to move uniformly in the equator with the real 92 MEASUREMENT OF TIME. sun's mean motion, a day was obtained of which the length is inva. riable, and equal to the mean length of all the apparent solar days in a tropical year. The point and time of departure of this fictitious sun, were also so chosen that its distance from the mean equinox would always be equal to the sun's mean longitude; the time deduced from its position with respect to the meridian, was thus made to correspond very nearly with apparent solar time. To find the excess of the mean solar day over the sidereal day, we have the proportion 360~: 24 sid. hours:: 59' 8".33: x - 3m. 56.555s. A mean solar day, comprising 24 mean solar hours, is therefore 24h. 3m. 56.555s. of sidereal time. Hence, a clock regulated to sidereal time will gain 3m. 56.555s. in a mean solar day. TSo find the expression for the sidereal day in'mean solar time, we must usa the proportion 24h. 3m. 56.555s.: 24h.:: 24h.: x - 23h. 56m. 4.092s. The difference between this and 24 hours is 3m. 55.90ss.; and therefore, a mean solar clock will lose with respect to a sidereal clock, or with respect to the fixed stars, 3m. 55.908s. in a sidereal day, and proportionally in other intervals. This is called the daily acceleration of the fixed stars. To express any given period of sidereal time in mean solar tine, we must subtract for each hour 3m 5591s. = 9.83s., and for minutes and seconds in the same 24 proportion. And, on the other hand, to express any given period of mean solar time in sidereal time, we must add for each hour 3m. 56.55s. - 9.86s., and for miI24 utes and seconds in the same proportion. It is the practice of astronomers to adjust the sidereal clock to the motions of the true instead of the mean equinox. The inequality of the diurnal motion of this point is too small to occasion any practical inconvenience. Sidereal time, as determined by the position of the true equinox, will not deviate from the same as indicated by the position of the mean equinox, more than 2.3s. in 19 years. CONVERSION OF ONE SPECIES OF TIME INTO ANOTHER. 12s. The difference between the apparent and mean time is called the Equation of Time. The equation of time, when known, serves for the conversion of mean time into apparent, and the reverse. 129. To find the Equation of Tiine. The hour angle of the sun (p. 15, def. 16) varies at the rate of 360~ in a solar day, or 15~ per solar hour. If, therefore, its value at any moment be divided by 15, the quotient will be the apparent time at that moment. In like manner, the hour angle of the mean sun, divided by 15, gives the mean time. Now, let the circle EQUATION OF TIME. 93 VSD (Fig. 47) represent the equator, V the vernal equinox, M the point of the equator which is on the meridian, and VS the right ascension of the sun; and we shall have, M FIG. 47. MS VM- VS appar. time - - 15 Again, if we suppose S' to be the position of the mean sun (VS' being equal to the mean longitude of the sun), we shall have MS' TA -VS' mean time- thus, equa. of time = mean time — ap. time = -_-S~.. (25); or, the equation of time is equal to the difference between the sun's true right ascension and mean longitude, converted into time. This rule will require some modification if very great accuracy is desired; for, in seeking an expression for the mean time, the circle VSD ought properly to be considered as the mean equator, answering to the mean pole (124), and the mean longitude of the sun is really estimated from the mean equinox V', and ought therefore to be corrected by the arc VV', or the equation of the equinoxes in right ascension. The value of the equation of time, determined from formula (25), is to be applied with its sign to the apparent time to obtain the mean, and with the opposite sign to the mean time to obtain the apparent. A formula has been investigated, and reduced to a table, which makes known the equation of time by means of the sun's mean longitude. (See Table XII.; also Art. 158.) The value of the equation of time at noon, on any day of the year, is also to be found in the tables of calculations for the sun, published in the Nautical Almanac. If its value for any other time than noon be desired, it may be obtained by simple proportion. Tlhe equation of time is zero, or mean and true time are the same four times in the year, viz. about the 15th of April, the 15th of June, the 1st of September, and the 24th of December. 94 MEASUREMENT OF TIM E. Its greatest additive value (to apparent time) is about 141: minutes, and occurs about the 11 th of February; and its greatest subtractive value is about 16}- minutes, and occurs about the 3d of November. 130. To convert Sidereal Time into Mean Timne, and vice versa.-Making use of Fig. 47, already employed, the arc ATM, called the Right Ascension of Mid-feaven, expressed in time, is the sidereal time; VS' is the right ascension of the mean sun, estimated from the true equinox, or the mean longitude of the sun corrected for the equation of the equinoxes in right ascension (124); and MS' expressed in time, is the mean time. Let the arcs VM, MS', and VS', converted into time, be denoted respectively by S, M, and L. Now, VM = MS' +VS'; or, S = M+L..(26); and M= -- S-L..(27). If M +L in equation (26) exceeds 24 hours, 24 hours must be subtracted; and if L exceeds S in equation (27), 24 hours must be added to S, to render the subtraction possible. This problem may in practice be solved most easily by means of an ephemeris of the sun (220), which gives the value of S, or the sidereal time, at the instant of mean noon of each day, together with a table of the acceleration of sidereal on mean solar time, and the corresponding table of the retardation of mean on sidereal time. The conversion of apparent into sidereal time, or sidereal into apparent time, may be effected by first obtaining the mean time, and then converting this into sidereal or apparent time, as the case may be. DETERMINATION OF THE TIME AND REGUTLATION OF CLOCKS BY ASTRONOMICAL OBSERVATIONS. 131. The regulation of a clock consists in finding its error and its rate. 132. MIean Solar Clock. The error of a mean solar clock is most conveniently determined from observations with a transit instrument of the time, as given by the clock, of the meridian passage of the sun's centre. The time noted will be the clocktime at apparent noon, and the exact mean time at apparent noon may be obtained by applying to the apparent time (24h., or Oh. Om. Os.) the equation of time with its proper sign, which may for this purpose be taken from the Nautical Almanac by simple inspection. A comparison of the clock time with the exact mean time, will give the error of the clock. The daily rate of a mean solar clock may be ascertained by finding as above the error at two successive apparent noons. If the two errors are the same and lie the same way, the clock goes accurately to mean solar time; if they are different, their differ ence or sumrn,: according as they lie the same or opposite ways, will be the daily gain or loss, as the case may be. 133. Sidereal, Clock. The methods of determining the error and rate of a sidereal clock have already been explained (47). In practice, the apparent right ascension of the clock star to be observed, is taken from the table of the apparent places of stars, in the Nautical Almanac, as already intimated. The TIME BY OBSERVATIONS OUT OF THE MERIDIAN. 95 method of calculating such apparent places is given in Prob. XXI. 131. Tinme by 1Observations out of thie M1eridian. In default of a transit instrument, the time mav be obtained and time-keepers regulated by observations made out of the meridian. There are two methods by which this may be accomplished, called, respectively, the method of Aingle Altitudes, and the method of Doblde Altitudes, or of Equal Altitudes. These we will now explain. (1.) To determine the time from a measured altitude of the sun, or of a star, its declinatio, and also the latitude of the place being given. Let us first suppose that the altitude of the sun is taken; correct the measured altitude for refraction and parallax, and also, if the sextant is the instrument used, for the semi-diameter of the sun. Then, if Z (Fig. 48) represents the zenith, P the z elevated pole, and S the sun; in the triangle ZPS we shall know ZP- co-latitude, PS - co-declin;ltion, and ZS co-alti- / tude, from which we nmay v colpute the angle ZPS ( p), which is the angular dist:ance of the sun fromn the meridian or, if expressed in time, the FIG. 48. time of the observation fromn apparent noon; by the following equations (App., Resolution of oblique-angled spherical triangles, Case 1), 2 lc Z P + PS + ZS - co-lat. + co-dec. + co-alt.... (28); sin P sin(- - ZP) sin(k- PS).... (29) sin ZP sin PS or, sin" ~p __ sin (k - co-lat.) sin (k - co-dec.)....(30). sin (co-lat.) sin (co-dec.) The value of P being derived from these equations an.d converted into time (see Prob. III.), the result will be the apparent time at the instant of the observation, if it was made in the afternoon; if not, what remains after subtracting it from 24 hours will be the apparent time. The apparent time being found, the mean time may be deduced from it by applying the equation of time. Amore accurate result will be obtained if several altitudes be measured, the time of each measurement noted, and the mean of all the altitudes taken and re. garded as corresponding to the mean of the times. The correspondence will be sufficiently exact if the measurements be all made within the space of 10 or 12 minutes, and when the sun is near the prime veertical. If an even number of altitudes be taken, and alternately of the upper and lower limb, the mean of the whole will give the altitude of the sun's centre, without it being necessary to know his ap 96 MEASUREMENT OF TIME. parent semi-diameter. In practice, the declination of the sun may be taken for the solutio - of this problem from an ephemeris of the sun. For this purpose, the time of the observation and the longitude of the place must be approximately known. Evample. On March 20, 1867, the following double altitudes of tile stn were taken with a prismatic sextant, at New Haven; upper limb, 64~ 12' 0", 6i40 21' 35", 64~ 33' 0", -lower limb, 63~ 3b' 50", 630 51' 0", 63~ 5~' 5"; the corresponding times of observation, noted by a watch, were 9h. 6m. 49s. A.M., 9h. 7m. 20.5s., 9h. 7m. 56s., 9h. Sin. 29.5s., 9h. 9m. 7.7s., 9h. 9m. 31s.; the barometer stood at 30.47in., and the thermometer at 340. What was the mean time answering to the mean of the times of observation? Mean of times of observation....... 9h. 8m. 12.3s. A.M. Long. of station of observer, west of Greenwich,..................4 51 42 Corresponding Greenwich time.......1 59 54.3. P.M. Sun's dec. at that time, Am. Naut. Aim... 0~ 11' 37"S Sun's co-dec., or N. P. dist.............90 11 37 Mean of measured double altitudes......64:~ 5' 45" Index error................. -- 1 3 2)61 4 42 Appar. alt. of sun's centre.............320 2' 21" Refraction (Tables VIII., and IX.)...... 1 37.3 True alt. of sun's centre...............32 0 43.7 Lat. of station..410 18' 37"' Co-lat.........48 41 23...a..r. co. sin. 0.124276 Co-dec...... 90 11 37......a. co. sin. 0.000003 Co-alt.........57 59 16.3 2) 196 52 16.3..........98 26 8 k-co-dec..... 8 14 31............sin. 9.156408 k -co-lat.....49 44 45...........sin. 9.882630 2 ) 19.163317 ~P - 22~ 26' 7".8......9.581658 P =44 52 15.6.4 179m. 29s. 2"' 2b. 59m. 29.03s. 12 TIMIE BY OBSERVATIONS OUT OF TIHE MERIDIAN. 9~ 9h. Om. 30.97s. A. M. Equa.oftime. + 7 41.37 BI. time sought 9 8 12.34 A. M. Time by watch 9 8 12.3 Error of watch - 0.04s. The error of the watch, as estimated from transit observations, was less than is. On the same date, the following measurements were made: Double Altitudes of Sun. Times of Observation. 640 11' 45"............ 9h. 0ln. 15s. A. M. L. L. 64 18 35............9 10 86 64 25 20............9 10 58 65 41 410............9 11 37 U.L. 4950............9 12 4.5 65 59 50...........9 12 36 Barometer, thermometer, and index error, same as above. The error of the watch, as determined from these data, was -- 0.08s. In case the altitude of a star is taken, the value of P derived from formula (30), when converted into time, will express the distance in time of the star from the meridian; and being added to the right ascension of the star, if the observation be made to the westward of the meridian, or subtracted from the right ascension (increased by 24h., if necessary) if the observation be made to the eastward, will give the sidereal time of the observation. (2.) To determine the time of noon from equal altitudes of the sun, the times of the observations being given. If the sun's declination did not change while he is above the horizon, he would have equal altitudes at equal times before and after apparent noon. Hence, if to the time of the first observation one-half the interval of time between the two observations should be added, the result would be the time of noon, as shown by the clock or watch employed to note the times of the observations. The deviation from 12 o'clock would be the error of the clock with respect to apparent time. The difference between this error and the equation of time would be the error of the clock with respect to mean time. But, as in point of fact the sun's declination is continually changing, equal altitudes will not have place precisely at equal times before and after noon, and it is therefore necessary, in order to obtain an exact result, to apply a correction to the time thus obtained. This correction is called the Equation of Equal Altitudes. Tables have been-constructed by the aid of which the equation is easily obtained. This is at the same time a very simple and quite accurate method of finding the time, and the error of a clock. 93 MEASUREMENT OF TIME. If equal altitudes of a star should be observed, it is evident that half the interval of time elapsed would give the time when the star passed the meridlian, without any correction. From this the error of the clock (if keeping sidereal time) may be found, as explained in Art. 133. THE CALENDAR. 135. Natural Periods oi Time. The apparent motions of the sun, which bring about the regular succession of day and night and the vicissitude of the seasons, and the motion of the moon to and from the sun in the heavens, attended with conspicuous and regularly recurring changes in her disc, furnish three natural periods -for the measurement of the lapse of time: viz., 1, the period of the apparent revolution of the sun with respect to the meridian, comnrising the two natural periods of day and night, which is called the solar day; 2, the period of the apparent revolution of the sun with respect to the equator, comprehending the tfour seasons, which is called the tropical year; 3, the period of time in which the moon passes through all its phases and returns to the same position relative to the sun, called the hlnar month. The day is arbitrarily divided into twenty-four equal parts, called hours; the hours into sixty equal parts, called minutes; and the minutes into sixty equal parts, called seconds. The tropical year contains 365d. 5h. 48rn. 46s. The lunar month consists of about 29~ days. The week, consisting of seven days, has its origin in Divine appointment alone. A Calendar is a scheme for taking note of the lapse of time, and fixing the dates of occurrences, by means of the four periods just specified, viz., the day, the week, the month, and the year, or periods taken as nearly equal to these as circumstances will adlnit. Different nations have, in general, had calendars more or less different: and the proper adjustment or reglulation of the calendar by astronomical observation has in all ages, and with all nations, been an object of the highest importance. We propose, in what follows, to explain only the Julian and Gregorian Calendars. 136. The Julian Calendar divides the year into 12 months, containing in all 365 days. Now, it is desirable that the calendar should always denote the same parts of the same season by the same days of' the same months: that, for instance, the summer and winter solstices, if once happening on the 21st of June and 21st of December, should ever after be reckoned to happen on the same days; that the date of the sun's entering the equinox, the natural commencement of spring, should, if once, be always on the 2Oth of March. For thus the labors of agriculture, which really depend on the situation of the sun in the heavens, would be simply and truly regulated by the calendar. THE CALENDAR. 99 This would happen if the civil year of 365 days were equal to the astronomical; but the latter is greater; therefore, if the calendar should invariably distribute the year into 365 days, it would fall into this kind of confusion, that in process of time, and successively, the vernal equinox would happen on every day of the civil year. Let us examine this more nearly. Suppose the excess of the astronomical year above the civil to be exactly 6 hours, and on the noon of March 20th of a certain year, the sun to be in the equinoctial point; then, after the lapse of a civil year of 365 days, the sun would be on the meridian, but not in the equinoctial point; it would be to the west of that point, and would have to move 6 hours in order to reach it, and to complete the astronomical or tropical year. At the completions of a second and a third civil year. the sun would be still more and more remote from the equinoctial point, and would be obliged to move for 12, and 18 hours, respectively, before he could rejoin it and complete the astronomicall year. At the completion of a fourth civil year the sun would be more distant than on the two preceding ones from the equinoctial point. In order to rejoin it, and to complete the astronomical year, he must move for 24 hours; that is, for one whole day. In other words, the astronomical year would not be completed till the beginning of the next astronomical day; till, in civil reckoning, the noon of lfarch 21st. At the end of four more common civil years, the sun would be in the equinox on the noon of March 22d. At the end of S and 64 years, on March 23d and April 6th, respectively; at the end of 736 years, the sun would be in the vernal equinox on September 20th; and in a period of 1460 years, the sun would have been in every sign of the zodiac on the same day of the calendar, and in the same sign on every day. If the excess of the astronomical above the civil year were really what we have supposed it to be, 6 hours, this confusion of the calendar might be very easily avoided. It would be necessary merely to make every fourth civil year to consist of 366 days; and for that purpose to interpose, or to intercalate, a day in a month previous to March. Bv this intercalation, what would have been March 21st is called'MarcL 20th, and accordingly the sun would be still in the equinox on the same day of the month. This mode of correcting the calendar was adopted by Julius Caesar. The fourth year into which the intercalary day is introduced was called Bissextile; it is now frequently called Leap year. The correction is called the Julian correction, and the length of a mean Julian year is 365d. 6h. By the Julian Calendar, every year that is divisible by 4 is a leap year, and the rest common years. 100 MEASUREMENT OF TI~ME. 137. Reformation of the Caleindar.-Gregorian Calendar. The astronomical year being equal to 365d. 5h. 48m. 46.1s, it is less than the mean Julian by rlm. 13.9s., or 0.007800d. The Julian correction, therefore, itself needs correction. The calendar regulated by it would, in process of time, become erroneous, and would require reformation. The intercalation of the Julian correction being too great, its effect would be to antedate the happening of the equinox. Thus (to return to the old illustration) the sun, at the completion of the fourth civil year, now the Bissextile, would have passed the equinoctial point by a time equal to four times 0.007800d.; at the end of the next Bissextile, by eight times 0.007800d.; at the end of 130 years, by about one day. In other words, the sun would have been in the equinoctial point 24 hours previously, or on the noon of lfatrch 19th. In the lapse of ages this error would continue and be increased. Its accumulation in 1300 years would amount to 10 days, and then the vernal equinox would be reckoned to happen on March 10th. The error into which the calendar had fallen, and would continue to fall, was noticed by Pope Gregory XIII., in 1582. At his time the length of the year was known to greater precision than at the time of Julius Caesar. It was supposed equal to 365d. 5h. 49m. 16.23s. Gregory, desirous that the vernal equinox should be reckoned on or near March 21st (on which day it happened in the year 325, when the Council of Nice was held), ordered that the day succeeding the 4th of October, 1582, instead of being called the 5th, should be called the 15th: thus suppressing 10 days, which, in the interval between the years 325 and 1582, represented nearly the accumulation of error arising from the excessive intercalation of the Julian correction. This act reformed the calendar. In order to correct it in future ages, it was prescribed that, at certain convenient periods, the intercalary day of the Julian correction should be omitted. Thus the centurial years 1700, 1800, 1900. are, according to the Julian Calendar, Bissextiles, but on these it was ordered that the intercalary day should not be inserted; inserted again in 2000, but not inserted in 2100, 2200, 2300; and so on for succeeding centuries. By the Gregorian Calendar, then, every cenlurial year that is divisible by 400 is a Bissextile or Leuap year, and the others common years. For other than centurial years, the rule is the same as with the Julian Calendar. This is a most simple method of regulating the calendar. It corrects the insufficiency of the Julian correction, by omitting in the space of 400 years 3 intercalary days. It is easy to estimate the degree of its inaccuracy; for the real error is 0.007800d. in one year, and 400 x 0.007800d., or 3.1200d. in 400 years. Consequently 0.1200d., or 2h. 52m. 4Ss. in 400 years, or 1 day in 3333 TE CALENDAR. 101 years, is the measure of the degree of inaccuracy of the Gregorian correction. Th/e Greyorian Calendar was adopted immediately on its promulgation, in all Catholic countries, but in those where the Protestant religion prevailed it did not obtain a place till some time after. In England, "the change of style," as it was called, took.place after the 2d of September, 1752, eleven nominal days being then struck out; so that the last day of Old Style being the 2d, the first of SNew Style (the next day) was called the 14th, instead of the 3d. The same legislative enactment which established the Gregoriarn Calendar in England, changed the time of the beginning of the year from the 25th of MIarch to the 1st of January. Thus the year 1752, which by the old reckoning would have commenced with the 25th of March, was made to begin with the 1st of January; so that the number of the year is, for dates falling between the 1st of January and the 25th of March, one greater by the new than by the old style. In consequence of the intercalary day omitted in the year 1800, there is now, for all dates, 12 days difference between the old and new style. Russia is at present the only Christian country in which the Gregorian Calendar is not used. The calendar months consist, each of them, of 30 or 31 days, except the second month, February, which, in a common year, contains 28 days, and in a Bissextile, 29 days; the intercalary day being added to the last of this month. To find the number of days comprised in any number of civil years, multiply 365 by the number of years, and add to the product as many days as there are Bissextile years in the period. 102 MOTIONS OF THE SUN, MOON, AND PLANETS. CHAPTER IX. MOTIONS OF THE SUN, MOON, AND PLANETS, IN THEIR ORBITS. KEPLER'S LAWS. 138. The celebrated astronomer, Kepler, by examining the observations upon the planets that had been made by the renowned Danish observer, Tycho Brahe, discovered, early in the seventeenth century, that the motions of these bodies were in conformity with the following laws: (1.) The areas described by the radcius-vector of a planet (or a line from the sun to the planet) are proportional to the times. (2.) The orbit of a planet is ant ellipse, of which the sun occupies one of the foci. (3.) The squares of the periods of revolution of the planets are proportional to the cabes of their mean distcances from the suan, or of the semi-major axes of their orbits. These laws are known by the denomination of KIepZer's Laws. They were announced by Kepler as the fundamental laws of the planetary motions, after a partial examination only of these motions. They have since been completely verified, and shown to hold good for all the planets, including the earth. We shall adopt the first two laws for the present, as hypotheses, and show in the sequel that they are verified by the results deducible fromn them. These laws being established, the third is obtained by simply comparing the known major axes and periods of revolution. 139. Iotion of the Sen iin its Apparenat Orbit. The apparent motion of the sun in space must be subject to Kepler's first two laws; for the apparent orbit of the sun is of the same form and dimensions as the actual orbit of the earth, and the law and rate of the sun's motion in its apparent orbit are the same as the law and rate of the earth's motion. To establish these two principles, let EE'A (Fig. 49) represent the elliptic orbit of the earth, and S the position of the sun in space. If the earth move from E to E', as it seems to remain stationary at E, it is plain that the sun will appear to move fiom S to S', on the line ES' drawn parallel to E'S the actual direction of the sun from the earth; and at a distance ES' equal to E'S the actual distance of the sun from the earth. Thus, for every KEPLER'S LAWS. 103 position of the earth in its orbit, the corresponding apparent position of the sun is obtained by drawing a line parallel to the radius-vector of the earth, and equal to it. It follows, therefore, that the area SES' apparently described by the radius-vector of FLO. 49. the sun (or a line drawn from the sun to the earth) in any interval of time, is equal to the area ESE' actually described by the radius-vector of the earth in the same time; and consequently that the arc SS' apparently described by the sun in space, is equal to the are EE' actually described in the same time by the earth. Whence we conclude, that the apparent motion of the sun in space, and the actual motion of the earth, are the same in every particular. 140. It has been ascertained that the motion of the moon in its revolution around the earth, is subject to the same laws as the motion of a planet in its revolution around the sun. We shall assume this to be a fact, and show that the hypothesis is verified by the results to which it leads. 141. Perihelion.-Aphelion. That point of the orbit of a planet, which is nearest to the sun, is called the Perihelion, and that point which is most distant from the sun, the Aphelion. The corresponding points of the moon's orbit, or of the sun's apparent orbit, are called, respectively, the Perigee and the Apogee. These points are also called Apsides; the former being termed the Lower Apsis, and the latter the Iigher Apsis. The line joining them is denominated the Line of Apsides. The orbits of the sun, moon, and planets, being regarded as ellipses, the perigee and apogee, or the perihelion and aphelion, are the extremities of the major axis of the orbit. 14'2. Latyw of the Angular MIotiosa of a Platct. The law of the angular motion of a planet about the sun may be deduced from Kepler's first law. Let PpAp" (Fig. 50) represent the orbit of a planet, considered as an ellipse, and p, p' two positions of the planet at two instants separated by a short inter 104 MOTIONS OF THE SUN, MOON, AND PLANETS. val of time; and let n be the middle point of the arc pp'. With the radius Sn describe the small circular arc mnl', and with the radius Sa, equal to unity, describe the arc ab. It is plain that the FIG. 50. two positions p, p' may be taken so near to each other, that the area Slop' will be sensibly equal to the circular sector SlI'. If we suppose this to be the case, as the measure of the sector is lnyz1' X Sn- ~ab x Sn2 (substituting for lnZ' its value, ab x Sn), we shall have area Spp'- = ab x Sn2. When the planet is at any other part of its orbit, as n', if Sp,"p."' be an area described in the same interval of time as before, we shall have area Sp"' = -a'b' X S n'. But these areas are equal according to Kepler's first law: hence, lab x Sn2 = ~a'b' x Sn'....(31); and ab: a'b':: Sn': Snn; that is, the angular motion of a planet about the sun for a short interval of time, is inversey proportional to the square of the radius-vector. It results from this that the angular motion is greatest at the perihelion, and least at the aphelion, and the same at corresponding points on either side of the major axis: also. that it decreases progressively from the perihelion to the aphelion, and increases progressively from the aphelion to the perihelion. 143. Milean Place.-Trnie Place. Now to compare the true with the mean angular motion, suppose a body to revolve in a circle around the sun, with the mean angtular motion of a planet, and to set out at the same instant with it from the per[helion. Let PAMAM' (Fig. 51) represent the elliptic orbit of the planet, and PBaB' the circle described by the bod(y. Thle position B of this fictitious body at any time, will be the mean l)t1ce of the planet as seen from the sun. The two bodies will accoin. MEAN AND TRUE PLACE. 105 plish a semi-revolution in the same period of time, and therefore be, respectively, at A and a at the same instant; for it is obvious that the fictitious body will accomplish a semi-revolution in half the period of a whole revolution, and by Kepler's law of areas, the planet will describe a semi-ellipse in half the time of a revolution. At the outset, the motion of the planet is the most rapid (142), M but it continually decreases until the planet reaches the aphelion, \ while the motion of the body remains constantly equal to the r mean motion. The planet will A therefore take the lead, and its angular distance pSB from the X body will increase until its motion becomes reduced to an equality with the mean motion; FIG. 51. after which it will decrease until the planet has reached the aphelion A, where it will be zero. In the motion from the aphelion to the perihelion, the angular velocity of the planet will at first be less than that of the body (142), but it will continually increase, while that of the body will remain unaltered: thus, the body will now get in advance of the planet, and their angular distance p'SB' will increase, as before, until the motion of the planet again attains to an equality with the mean motion, after which it will decrease as before, until it again becomes zero at the perihelion. It appears, then, that from the perihelion to the cbphelion the true place is in aclvanee of the mean place; and that from the aphelion to the perihelion, on the contrary, the mean place is in aclvance of the true place. The angular distance of the true place of a planet from its mean place, as it would be observed from the sun, is called the Equation of the Ceztre. Thus, pSB is the equation of the centre corresponding to the particular positionp of the planet. It is evident, from the foregoing remarks, that the equation of the centre is zero at the perihelion and aphelion, and greatest at the two points, as M and M', where the planet has its mean motion. The greatest value of the equation of the centre is called the Greatest Equat~ion of the Centre. As the laws of the motion of the moon (140), and of the apparent motion of the sun (139), are the same as those of a planet, the principles established in the two preceding articles are as applicable to these bodies in their revolution around the earth, as to a planet in its revolution around the sun. 106 MTOTIONS OF THE SUN, MOON, AND PLANETS. DEFINITIONS OF TERMS. 144. (1.) The Geocentric Place of a body is its place as seen from the earth. (2.) The Heliocentric Place of a body is its place as it would be seen from the sun. (3.) Geocentric LongiMtude and Latitude appertain to the geocentric place, and KIelioeentric Longitude and latitucde to the heliocentric place. (4.) Two heavenly bodies are said to be in Conjunction when their longitudes are the same, and to be in Opposition when their longitudes differ by 180~. When any one heavenly body is in conjunction with the sun, it is, for the sake of brevity, said to be in Conjunction; and when it is in opposition to the sun, to be in Opposition. The planets Mercury and Venus, allowing that their distances from the sun are each less than the earth's distance (18), can never be in opposition. But they may be in conjunction, either by being between the sun and earth, or by being on the opposite side of the sun. In the former situation they are said to be in I9feriori Conjzunction, and in the latter in Superior Cojunction. (5.) A Synodic Revolution of a body is the interval between two consecutive conjunctions or oppositions. For the planets Mercury and Venus a synodic revolution is the interval between two consecutive inferior or superior conjunctions. (6.) The Periodcic Time of a planet is the period of time in which it accomplishes a revolution around the sun. (7.) The Nocles of a planet's orbit, or of the moon's orbit, are the points in which the orbit cuts the plane of the ecliptic. The node at which the planet passes from the south to the north side of the ecliptic is called the Ascending Node, and is designated by the character A. The other is called the Descending Node, and is marked U. (S.) The Eccentricity of an elliptic orbit is the ratio which the distance between the centre of the orbit and either focus bears to the semi-major axis. 145. To illustrate these Deffivltions, let EE'E" (Fig. 52) represent the orbit of the earth; C'DC the orbit of Venus, or Mercury, which we will suppose, for the sake of simplicityr, to lie in the plane of the ecliptic or of the earth's orbit; LNP a part of the orbit of Mars, Qr of any other planet more distant from the sun S than the earth is; and ANB a part of the projection of this orbit on the plane of the ecliptic. N or fg will represent the ascending node of the orbit; alnd the descending node will be diametrically opposite to this ia the direction Sn'. DEFINITIONS OF TERMS. 107 Also let SV' be the direction of the vernal equinox, as seen from the sun, and EV, E'V the parallel directions of the same point, as seen from the earth in the two positions E and E'; and P being supposed to be one position of Mars in his orbit, let p be V FIG. 52. the projection of that position on the plane of the ecliptic. The heliocentric lon~yitude and latitucde of Mars, in the position P, are respectively VSp and PSp; and if the earth be at E, his geocentric longitude and latitude are respectively VEp and PEp. If we suppose that when Mars is at P the earth is at E', he will be in conjunction; and if we suppose the earth to be at E"' he will be in o92position. Again, if we suppose the earth to be at E, and Venus at C, she will be in suvperior comjunction; but if we suppose that Venus is at C' at the time that the earth is at E, she will be in irfelrior conjunction. The term ir)ferior is used here in the sense of lower in place, or nearer the earth; and superior in the sense of higher in place, orfart/herfiorm, the earth. Since the earth and planets are continually in motion, it is manifest that the positions of conjunction and opposition will recur at different parts of the orbit, and in process of time in every variety of position. The time employed by a planet in passing around from one position of conjunction or opposition to another, called the synodic revolution, is, for the same reason, longer than the periodic timne, or time of passing around from one point of the orbit to the same again. 108 MOTIONS OF THE SUN, MOON, AND PLANETS. ELEMENTS OF TiE ORBIT OF A PLANET. 146. To have a complete knowledge of the motions of the planets, so as to be able to calculate the place of any one of them at any assumed time, it is necessary to know for each planet, in addition to the laws of its motion discovered by Kepler, the position and dimensions of its orbit, its mean motion, and its place at a specified epoch. These necessary particulars of information are subdivided into seven distinct elements, called the Eements of Ahe Orbi of a Plcanet, which are as follows: (1.) The longitude of the ascending node. (2.) The inclination of the plane of the orbit to the plane of the ecliptic, called the inclination of the orbit. (3.) The mean distance of the planet from the sun, or the semimajor axis of its orbit. (4.) The eccentricity of the orbit. (5.) The heliocentric longitude of the perihelion. (6.) The epoch of the perihelion passage of the planet, or instead, the mean longitude of the planet at a given epoch. (7.) The periodic time of the planet. The first two ascertain the position of the plane of the planet's orbit; the third and fourth, the dimensions of the orbit; the fifth, the position of the orbit in, its plane; the sixth, the place of tihe pclanet at a given, epoch; and the seventh, its mean, rate of notion. The elements of the earth's orbit, or of the sun's apparent orbit, are but five in number; the first two of the above-mentioned elements being wanting, as the plane of the orbit is coincident with the plane of the ecliptic. The elerments of the moon's orbit are the same with those of a planet's orbit, it being understood that the perigee of the moon's orbit answers to the perihelion of a planet's orbit, and that the geocentzric longitude of the perigee and the geocentric longitude of the node of the moon's orbit answer, respectively, to the heliocentric longitude of the perihelion and the heliocentric longitude of the node of a planet's orbit. 147. The Linear Unit adopted, in terms of which the semi-major axes and radius-vectors of the planetary orbits are expressed, is the mean distance of the sun from the earth, or the semi-major axis of the earth's orbit. When thus expressed, these lines are readily obtained in known measures whenever the mean distance of the sun becomes known. The lines of the moon's orbit are found in terms of the moon's mean distance from the earth, as unity. MEAN DISTANCE OF THE SUN. 109 DETERMINATION OF THE ELEMENTS OF THE SUN'S APPARENT ORBIT, OR OF THE EARTH'S REAL ORBIT. MEAN MOTION. 14s. The sun's mean daily motion in longitude results from the length of the mean tropical year obtained from observation (115). SEMI-MAJOR AXIS. 149. As we have just stated, the semi-major axis of the sun's apparent orbit, is the linear unit in terms of which the dimensions of the planetary orbits are expressed. Its absolute length is computed from the mean horizontal parallax of the sun. The Horizointal Parallax of a body being givent, to find its Distance from the Earth. We have (equation 7, Art. 88) D -; sin i; where II represents the horizontal parallax of the body, D its distance from the centre of the' earth, and R the radius of the earth. The parallax of all the heavenly bodies, with the exception of the moon, is so small, that it may, without material error, be taken in this equation in place of its sine. Thus, D - -Rx1...(32). sin l. Again, since 6.2831853 is the length of the circumference of a circle of which the radius is 1, and 1296000 is the number of seconds in the circumference, we have 6.2831853: 1:: 1296000": x = 206264."806 the length of the radius (1) expressed in seconds. Hence, if the value of H be expressed in seconds,,2062 64."806 H 150. Deteri niation of the Sum's Ieant Horizontal Parallax. In the determination of the sun's parallax, by the process of Art. 90. an error of 2" or 3", equal to about onefourth of the whole parallax, may be committed, so that the distance of the sun, as deduced by equation (33) from his parallax found in that manner, may be in error by an amount equal to one-fourth or more of the true distance. There are more accurate methods of obtaining the sun's parallax. By one method, which will be noticed in another connection, the equatorial parallax of the sun (92) was deduced from certain observations made upon Venus, when seen to pass between the sun and earth, in 1761 and 1769, and the value 8".58 obtained. This is the value of the sun's equatorial horizontal parallax which has been uni 110 APPARENT MOTION OF THE SUN, versally adopted until within a very few years. Quite recently, several different determinations have been made of this important element, by independent astronomical methods. The different values obtained fall between 8".93 and 8".97, the mean of which is 8".95. One of these has been the deduction of the solar parallax, by the process of Art. 90, from the parallax of Mars determined by direct observations at the opposition of this planet, in 1862, when its distance from the earth attained its minimum value. This deduction was easily effected, since, as will appear in the next Chapter, the theory of the orbital motions of the planets would give the distance of Miars from the earth at the epoch of the observations, in terms of the mean distance of the sun from the earth as the linear unit (147). The mean of two results obtained from the observations made by two sets of observers, at localities remote from each other, is 8".95. This value, which is the mean of all the results, has been definitively fixed upon in the most approved Solar Tables (Leverrier's); and has since been adopted in the English Nautical Almanac for 1870. It may be relied upon as exact to within a small fraction of a second. 1i1. Calculation of Sun's Mean Distance. We have, then, for the sun's mean distcance fromn the earth, or the semimajor axis of its orbit, D -= R 2062 806 23046.347 R - 91,328,064 miles; H. taking for R the equatorial radius of the earth, 3962.80 miles. ECCENTRICITY. 152. First MIethod. By the greatest and least daily motions in longitude. We have already explained (116) the mode of deriving from observation the sun's motion in longitude from day to day. Now, let v - the greatest daily motion in longitude; v'- the least daily motion in longitude; r = the least or perigean distance of the sun; and r' the greatest or apogean distance; and we shall have, by the principle of Art. 142, r:r'::'v' v; whence, r' r r' -r::/ v + V v': -/ v- V v', r' + r VV~V'' —r" or, r +:r' r v + v v- v: 7 2 2 but, r +r= semi-major axis =_ 1 and r'- r =_ 2(eccentricity) 2 e; ECCENTRICIrY OF SUN'S APPARENT ORBIT. 111 thus, 1: 2e:: + v~ - -v, and e V -/....(34). and e - -. (34). V v + v' v The greatest and least daily motions are, respectively (at a mean), 61'.167 and 57'.200. Substituting, we have e - 0.016761. The eccentricity may also be obtained from the greatest a'nd least apparent diameters, by a process similar to the foregoing, on the principle that the distances of the sun at different times are inversely proportional to its corresponding apparent diameters (116). 153. Seoeemadt Mlethod. B2y the greatest equation of the centre. (1.) To find the greatest equation of the centre. Let L =the true longitude, and M - the mean longitude, at the time the true and mean motions are equal between the perigee and apogee (143); L' = the true longitude and M' = the mean longitude, when the motions are equal between the apogee and perigee; and E = the greatest equation of the centre. Then (143) L = MA + E, and L' = Mi' -E; whence, L'- L -= AM'- M- 2E, and E = (' -M) - (LL)....(35). About the time of the greatest equation the sun's true motion, and consequently the equation of the centre, continues very nearly the same for two or three days; we may therefore, with but slight error, take the noon, when the sun is on either side of the line of apsdcles, that separates the two days on which the motions in longitude are most nearly equal to 59' 8", as the epoch of the greatest equation. The longitude L or L' at either epoch thus ascertained, results from the observed right ascension and declination. M' - l -- the mean motion in longitude in the interval of the epochs, and is found by multiplying the number of mean solar days and fractions of a day comprised in the interval by 59' 8".330, the mean daily motion in longitude. For example: from observations upon the sun, made by Dr. Miaskelyne, in the year 1775, it is ascertained in the mannr just explained, that the sun was near its greatest equation at noon, or at Oh. 3m. 35s. mean solar time, on the 2di April, and at noon on the 31st, or at 23h. 49m. 35s. mean solar time, on the 30th of September. The observed longitudes were, at the first period 12~ 33' 39".06, and at the second 188~ 5' 44'.45. The interval of time oetween the two epochs is 182d.14m. Mean motion in 182d. - 14m.................179Q 22' 41".56 Difference of two longitudes............... 17..175 32 5.39 Difference................2) 3 50 36.17 Greatest equation of centre...................... 1 55 18.08 More accurate results are obtained by reducing observations made during seve, ral days before and after the epoch of the greatest equation, and taking the mean of the different values of the greatest equation thus obtained. According to M. Delamrbre, the greatest equation was in 1775, 1o 55' 31".66. (2.) The eccentricity of an orbit may be derived from the greatest equation of the centre by means of the following formula: 112 APPARENT MOTION OF THE SUN. K 11 K 587 ]_5 e = K- - -&c....(36), 2 3.28 3.5.21 in which IK stands for the expression __(E being the greatest equation 57~.2957795 of the centre). In the case of the snn's orbit, K being a small fraction, all its powers beyond the first mlay be omitted. Thus, retaining only the first term of the series, and taking E = 10 55' 31".66 the greatest equation in 1775, we have K 10 55' 31".66 e - -- =-.016803. 2 2 x 57~.2957795 154. Equatioan of Ceatre depcaids on Eccentricity. It appears from the law of the angular velocity of a revolving body, investigated in Art. 142, that the amount of the proportional variation of this velocity, which obtains in the course of a revolution, depends altogether upon the amount of the proportional variation of distance, or, in other words, upon the eccentricity of the orbit (def. S, p. 106). It follows, therefore, that the amount of the greatest deviation of the true place from the mean place, that is, of the greatest equation of the centre (143), must depend upon the value of the eccentricity. If the eccentricity be great, the greatest equation of the centre will have a large value; and if the eccentricity be equal to zero, that is, if the orbit be a circle, the equation of the centre will also be equal to zero, or the true and mean place will continually coincide. If either of the two quantities, the greatest equation and the eccentricity, be known, the other will then become determinate; and formul.e have been investigated which make known either one'when the other is given. Equation 36 is the formula for the eccentricity. From observations made at distant periods it is discovered that the equation of the centre, and consequently the eccentricity, is subject to a continual slow diminution. The amount of the diminution of the greatest equation, in a century, is 17".6. LONGITUDE AND EPOCH OF THE PERIGEE. 155. Methods of Determkination. As the sun's angular velocity is the greatest at the perigee, the longitude of the sun at the time its angular velocity is greatest will be the longitude of the perigee. The time of the greatest angular velocity may be easily obtained, within a few hours, by means of the daily motions in longitude derived from observation (116). T/he mor'e accurate Tmethodl of determining' the longitude and epoch of the perigee, rests upon the principle that the apogee and perigee are the only two points of the oribit whose longitudles differ by 180~, in passing from one to the other of which the sun employs half a year. This principle may be intfrred from Kepler's law of areas, for it is a well known properly of the ellipse, that the major axis is the only line drawn through the focus that LONGITUDE AND EPOCH OF THE PERIGEE. 113 divides the ellipse into equal parts, and, by the law in question, equal areas correspond to equal times. 156. Progressive Motion of the Perigee. By a comparison of the results of observations made at distant epochs, it is discovered that the longitude of the perigee is continually increas ing at a mean rate of 61.".7 per year. As the equinox retrogrades 50".2 in a year, the perigee must then have a direct angular motion of 11".5 per year. It will be seen that as a consequence the interval between the times of the sun's passage through the apogee and perigee, is not, strictly speaking, half a sidereal year, but exceeds this period by the interval of timte employed by the sun in moving through an are of 5".7, the sidereal motion of the apogee and perigee in half a year. According to the most exact determinations, the mean lolngitude of the perigee of the sun's orbit at the beginning of the year 1800, was 279~ 29' 56". It is now 280-~. 157. The Heliocentric Loangitnae of thle Perihelion of the Earth's Orbit, is equal to the geocentric longitude of the perigee of the sun's apparent orbit minus 180~. For, let AEP (Fig. 49, p. 103) be the earth's orbit, and PV the direction of the vernal equinox. When the earth-is in its perihelion, P, the sun is in its perigee, S, and we have the heliocentric longitude of the perihelion, VSP - VPL angle abce- 1S0~ = geocentric longitude of the sun's perigee - 180~. It is plain that the same relation subsists between the heliocentric longitude of the earth and the geocentric longitude of the sun in every other position of the earth in its orbit. 158. The Mean Longitude of the Suin, at any assumed epoch, may be obtained by means of the mean motion in longitude (116), the epoch and mean longitude of the perigee of the sun's orbit having once been found. DETERMINATION OF THE ELEMENTS OF THE MOON'S ORBIT. LONGITUDE OF THE NODE. 159. In order to obtain the longitude of the moon's ascending node, we have only to find the longitude of the moon at the time its latitude is zero, and the moon is passing from the south to the north side of the ecliptic. This may be deduced from the longitudes and latitudes of the moon, derived from observed right ascensions and declinations (56); by methods precisely analogous to those by which the right ascension of the sun, at the time its declination is zero, and it is passing from the south to the north side of the equator, or the position of the vernal equinox, is ascertained (113). 8 114 MOTION OF THE MO0ON IN SPACE. INCLINATION OF THE ORBIT. 160. Among the latitudes computed from the moon's observed right ascensions and declinations, the greatest measures the inclination of the orbit. It is fiund to be about 5~; sometimes a little greater, and at other times a little less. MIEAN MOTION. 161. Tropical Revolution. -With the longitudes of the moon, found from day to day, it is easy to obtain the interval from thle time at which the moon has any given longitude till it returns to the same longitude again. This interval is called a Tr5opica~l Revolution of' the moon. It is found to be subject to considerable periodical variations, and thus one observed tropical revolution may differ materially from the mean period. In order to obtain the mean tropical revolution, we must compare two longitudes found at distant epochs. Their difference augmented by the product of 360~ by the number of revolutions performed in the interval of the epochs, will be the mean motion in longitude in the interval; from which the mean motion in 100 years, or 36,525 days, called the Secular motion, may be obtained by simple proportion. The secular motion being once known, it is easy to deduce from it the period in which the motion is 360~, which is the mean tropical revolution. It should be observed, however, that to find the precise mean secular motion in longitude, it is necessary to compare the mean longitudes instead of the true. Now, the true longitude of the moon at any time having been found, the mean lonlliude at the same time is derived from it by correcting for the equation of the centrle and certain other periodical inequalities of longitude hereafter to be noticed. But this cannot be done, even approximately, until the theory of the moon's motions is known with more or less accuracy. The longitude of the moon, at certain epochs, may be very conveniently deduced from observations upon lunar eclipses. For, the time of the middle of the eclipse is very near the time of opposition, when the longitude of the moon differs 180~ from that of the sun, and the longitude of the sun results from the known theory of its motion. The recorded observations of the ancients upon the times of the occurrence of eclipses, are the only observations that can now be made use of for the direct determination of the longitude of the moon at an ancient-epoch. 162. Mean Daily Motion in Longitude. The mean tropical revolution of the moon is found to be 27.321582d. or 27d. 7h. 43m. 4.7s. (5s. nearly). Hence, 27.321582d.: id.:: 360: 130.17639 = 130 10' 35".0 = moon's mean daily motion in longitude. LONGITUDE OF THE PERIGEE. 115 163. Sidereal Revolution. Since the equinox has a retrograde motion, the sidereal revolution of the moon must exceed the tropical revolution, as the sidereal year exceeds the tropical year. The excess will be equal to the time employed by the moon in describing the arc of precession answering to a revolution of the moon. Thus, 365.25d.: 50/.2:: 27.3d.: 3".752 = the arc of precession, and 13~.176: id.:: 3".752: 6.8s. - excess. Wherefore, the mean sidereal revolution of the moon is 27d. 7h. 48rm. 11.5s. 164. Secular Acceleration of Mloon's Ttlotion. It has been found, by determining the moon's mean rate of motion for periods of various lengths, that it is subject to a continual slow acceleration. This acceleration will not, however, be indefinitely progressive; Laplace investigated its physical cause, and showed, from the principles of Physical Astronomy, that it is really a periodical inequality in the moon's mean motion, which requires an immense length of time to go through its different values. The mean motion given in Art. 162 answers to the commencement of the present century. LONGITUDE OF THE PERIGEE, ECCENTRICITY, AND SEMI-MAJOR AXIS. 165. The methods of determining these elements of the. moon's orbit are similar to those by which the corresponding elements of the sun's orbit are found. 166. Orbit Longituades. The only essential difference in the methods adopted, is that in place of the longitudes of the sun, which are laid off in the plane of the ecliptic. in the case of the moon corresponding an- _M gles are laid of il thle plane of its orbit. These angles are reckoned from a line drawn making r an angle with the line of nodes equal to the C longitude of the ascending node, and are called Orbit Longitudes. The orbit longitude is equal to the moon's angular distance from the ascending node plus the longitude of the ascending IN node. Thus, let VNC (Fig. 53) represent the plane of the ecliptic, andV'NM a portion of the moon's orbit; N being the ascending node; also let EV be the direction of the vernal equinox, and let EV' be drawn in the plane of the moon's orbit, making an angle WEN with the line of the nodes equal to VEN, the longitude B of the ascending node N. The orbit longitudes / lie in the plane of the moon's orbit, and are estimated from this line, while the ecliptic longitudes lie in the plane of the ecliptic, and are estimated from the line EV. Thus, VEM, or its measure V'NM, is the orbit longitude of the FIG. 53. moon in the position M; and VEim is the ecliptic longitude; that is, the longitude as it has been hitherto considered. V'NM = V'N + NM = VN +NM; that is, orbit long. = long. of 2 + 3)'s distance from i. 1-6l oMOTIONS OF THE PLANETS IN SPACE. The orbit longitudes are calculated from the ecliptic longitudes; these being derived from observed right ascensions and declinations. 167. The ecliptic longitude of the moon at any time being given, to find the orbzl longitude. As we may suppose the longitude of the node to be given (159), the equation of the preceding article will make known the orbit longitude so sooll as lIN, the moon's distance from the node, becomes known: now, by Napier's first rule we have cos MNm = cot NM tan Nm; or, cot NM - cos MNm cot Nm. Nm = ecliptic long. -long. of node; and MNm = inclination of orbit. 16S. The Horizontal Parallax of the llIoon, like almost every other astronomical element, is subject to periodical changes of value. It varies not only during one revolution, but also from one revolution to another. The fixed and mean parallax, about which the true parallax may be conceived to oscillate, answers to the mean distance, that is, the distance about which the true distance varies periodically, and is called the Constant of the Parallax. It is, for the equatorial radius of the earth, 57' 2".7, from which we find by equation (32) the mean distance of the moon from the earth to be 238,824 miles. 169. Thle Eccentricity of the moon's orbit is more than three times as great as that of' the sun's apparent orbit. Its greatest equation exceeds 6~ (154). MEAN LONGITUDE AT AN ASSIGNED EPOCH.'eo. We have already explained (161) the principle of the determination of the mean longitude of the moon from an observed true longitude. Now, when the mean longitude at any one epoch whatever becomes known, the mean longitude at any assigned epoch is easily deduced from it by means of the mean motion in longitude. DETERMINATION OF THE ELEMENTS OF A PLANET'S ORBIT. 171. Heliocentric Longitude and Radius-Vector of the Earth. The methods of determining the elements of the planetary orbits, suppose the possibility of finding the heliocentric longitude and radius-vector of the earth for any given time. Now, the elements of the earth's orbit having been found by the processes heretofore detailed, the longitude may be computed by means of Kepler's first law; and the radius-vector from the polar equation of the orbit, as given in treatises on Analytical Geometry. The manner of effecting such computation will be considered hereafter; at present the possibility of effecting it will be taken for granted. LONGITUDE OF THE ASCENDING NODE. 117 HELIOCENTRIC LONGITUDE OF THE ASCENDING NODE. 172. ]Fiarst Method. When the planet is in either of its nodes, its lati. tude is zero. It follows, therefore, that the longitude of the planet at the time its latitude is zero, is the geocentric longitude of the node at the time the planet is passing through it. Now, if the right ascension and declination of the planet be observed from day to day, about the time it is passing from one side of the ecliptic to the other, and converted into longitude and latitude, the time at which the latitude is zero, and the longitude at that time, may be obtained by a proportion. When the planet is C again in the same node, the geocentric longitude of the node may again be found in the same manner as before. On account of the different position of the earth in its orbit, this longitude will differ from the former. Now, if two geocentric longitudes of the same node be found, its heliocentric longitude may be conjputed. Let S (Fig. 54) be the sun, N the node, and E one of the positions of the A earth for which the geocentric longitude of........ the node (VEN) is known. Denote this angle by G, the sun's longitude VES by S, and the radius-vector SE by r. Also, let E' be the other position of the earth, and de- - note the corresponding quantities for this position, VE'N, VE'S, and SE', respectively, FIG. 54. by G', S', and r'. Let the radius-vector of the planet when in its node, or SN = V; and the heliocentric longitude of the node, or VSN = X. The triangle SNE gives sin SNE: sin SEN:: SE: SN; but SEN Y VES- VEN = S-G, and SNE = VAN - VSN = VEN- VSN = G -X; hence, sin (G-X): sin (S-G):: r: V, or, r sin (S - G) = V sin (G- X)....(37). In like manner, r' sin (S' - G') = V sin (G' - X). Dividing, r sin (S- G) _ sin (G- X) r' sin (S' - G') sin (G'-X)' r sin (S - G) _ sin G cos X - sin X cos G _ sin G — cos G tan X r' sin (S'- G') sin G' cos X- sin X cos G' sin G' -cos G' tan X whence, tan x - r sin (S — G) sin G' —r' sin (S'- G') sin G 38) r sin (S - G) cos G'-r' sin (S' - G') cos G Equation (37) gives - r sin (S —G) G (39). sin (G - X) 173. Seconid i.Method. The longitude of the node may also be found approximately from observations made upon the planet at the time of conjunction or opposition. It will happen in process of time that some of the conjunctions and oppositions will occur when the planet is near one of its nodes; the observed longitude of the sun at this conjunction or opposition, will either be approximately the heliocentric longitude of the node in question, or will differ 1S0~ from it. This will be seen on inspecting 1 13 MTOTIONS OF THE PLANETS IN SPACE. Fig. 55. If at a certain time the earth should be at E, crossing the line of nodes, and the planet in conjunction, it will be in the node N, and VES, the longitude of the sun, will be equal to VSN, the heliocentric longitude of the node. If the earth should be P C p N FIG. 55. at E" and the planet in opposition, the longitude of the sun would be VE"S = VE"N + 1800~ VSN + 1800 - hel. long. of node + 180~. If the daily variations of the latitude of the planet should be observed about the time of the supposed conjunction or opposition near the node, the time when the latitude becomes zero, or the planet is in its node, could approximately be calculated by simple proportion; and then so soon as the rate of the angular motion about the sun becomes known (176) the longitude of the node could be more accurately determined. INCLINATION OF THE ORBIT. 174. The longitude of the node having been found by the preceding, or some other method, compute the day on which the sun's longitude will be the same or nearly the same: the earth will then be on the line of the nodes. Observe on that day the planet's right ascension and declination, and deduce the geocentric longitude and latitude. LIt ENp (Fig. 55) be the plane of the ecliptic, V the vernal equinox, S the sun, N the node, E the earth on the line of the nodes, and P the planet as referred to the celestial sphere, from the earth. Let i, denote the geocentric latitude PEp; E the arc rNp Vp - VN - geo. long. of planet - long. of node; and I the inclination PXNp. The right- angled triangle PNp gives PERIODIC TIME. 119 sin Np -= tan Pp cot PNp - tan 2, cot I; nence, cot I = sin E and tan I - tan. tan X- sin E tan lat. or, tan inclination.... (40). sin (long. - long. of node) It will be understood, that to obtain an exact result, we must compute the precise time of day at which the longitude of the sun is the same as that of the node, and then, by means of their observed daily variations, correct the longitude and latitude of the planet for the variations in the interval between the time thus ascertained and the time of the observation above mentioned. PERIODIC TIME. i75. The interval from the time the planet is in one of its nodes till its return to the same, gives the periodic time or sidereal revolution. Another and more accurate method is to observe the length of a synodic revolution and compute the periodic time from this. If we compare the time of a conjunction which has been observed in modern times, with that of a conjunction observed by the earlier astronomers, and divide the interval between them by the number of synodic revolutions contained in it, we shall have the mean synodic revolution with great exactness; from which the mean periodic time may be deduced, as will be shown hereafter. 176. llean Daily MNotion. The periodic time being known, the mean daily motion around the sun may be found by dividing 360~ by the periodic time expressed in days and parts of a day. TO FIND THE HELIOCENTRIC LONGITUDE AND LATITUDE, AND THE RADIUS-VECTOR, FOR A GIVEN TIME. 177. General Problelm. The earth being in constant motion in its orbit, and being thus at different times very differently situated with regard to the other planets, as well in respect to distance as direction, it is necessary for the purpose of comparing the observations made upon these bodies with each other, to refer them all to one common point of observation. As the sun is the fixed centre about which the revolutions of the planets are performed, it is the point best suited to this purpose, and accordingly it is to the sun that the observations are in reality referred. The reduction of observations from the earth to the sun, as it is actually performed, consists in the deduction of the heliocentric longitude and latitude from the geocentric longitude and latitude; these being calculated from the observed right ascension and declination. The requisite fo)rmulre for effecting this reduction are investigated in the Appendix. 120 MOTIONS OF THE PLANETS IN SPACE.'17. Special Cases. The heliocentric longitude, or radiusvector of a planet, may be more readily obtained if the observations be made upon it when it is in certain favorable positions. case 1. WVhen the planet is in cornjunction or opposition, its heliocentric longitude will then. either be r equal to the geocentric longitude, or differ 180~ from it. is When the heliocentric longitude cow\ is thus found, the latitdcle for the N same time may be obtained by solving the triangle PNp (Fig. 56).'For, by Napier's first rule, sin Np - cot PNp tan Pp, or tan Pp sin Np tan PNp; where Pp is the latitude sought, PNp the known inclination of the orbit, and Np VNpp- VN= long. of planet- long. of node, both FIG. 56. of which may be considered as known. TDhe radtcius-vector may be computed for the same time from the triangle ESP; for the side SE, the radius-vector of the earth, is known, as well as the angle SEP, the geocentric latitude of the planet, and the angle ESP - 180~ - PSp - 180~ - heliocentric latitude. Case ]IF. When an inferior planet is at its maximum elongation from the sun. The radius-vector of either of the inferior planets at the time of maximum elongation, or greatest angular distance from the sun, may be approximately deduced from the amount of the greatest elongation determined from observation. The elongation which obtains at any time, may be found by ascertaining from instrumental observations the places of the planet and sun in the heavens, and connecting these by an are of a great circle, and with the pole by other arcs. In the triangle PSp (Fig. 57) thus formed, there will be known the two polar disP FIG. 57. tances PS and Pp, which are the complements of the observed declinations, and the angle SPp the'difference of their observed right ascensions, from which the angular distance Sp between the two bodies may be calculated. The maximum elongation being LONGITUDE OF THE PERIHELION. 121 then supposed to be known, let NPP' (Fig. 5S) represent the orbit of the inferior planet. The line EP drawn from the earth to the planet will, at the time of maximum elongation, be perpendicular to SP, the radiusvector of the planet; and thus we shall have in the right-angled triangle EPS, the line ES, and the anole SEP, from which the radiusvector SP may be computed. / As the earth and planet are in motion, the greatest elongation will occur at different points of the planet's orbit, and therefore we may find by the foregoing process different radi us-vectors. 179. Thle orbit Loagitude of a Planet may be derived fiom the ecliptic longitude in the same manner that the orbit longitude of the moon is calculated from its ecliptic longitude (166j). The orbit longitude and radius-vector, when found for a given time, ascertain the position of the planet in the plane of its orbit at that time. LONGITUDE OF THE PERIHELION, ECCENTRICITY, AND SEMI-MAJOR AXIS. 1~O. These elements may be calculated from the heliocentric orbit longitude and radius-vector, found for three different times. Let SP, SP', SP" (Fig. 59), be the three given radius-vectors; V'SP, V'SP', V'SP", the p three given longitudes; and AB the line of apsides of the planet's orbit. Let the angles PSP', PSP", which P are klnown, be represented A by m and n, and the angle BSP, which is unknown, by x; and let the radius-vectors SP, SP', SP", be denoted by v, v', v", the semli-major FIG. 59. axis AC by a, and the eccentricity by e. Then the three unknown quantities which are to be determined are a, e, and the angle x; and the general polar equation of the ellipse furnishes for their determination the three equations, 122 MOTIONS OF THE PLANETS IN SPACE. v a( -e') v - a(l -e2),, a( -- e') 1 +ecosx' e cos (x -r -)' + e cos (x + n)' The process of solution is given in the Appendix. When x has been found, by subtracting it from V'SP we obtain V'S1B, the longitude of the perihelion. 1S1. Other:vgethods of Determiziatiang the Sen!i-naior Axis. The semi-major axis, or mean distance from the sun, may also be had by taking the mean of a great number of values of the radius-vector found for every variety of position of the planet in its orbit (178). Now that KIepler's third law has been established by investigations in Physical Astronomy, it furnishes the most accurate method of findingo the mean distance of a planet from the sun. Thus let P denote the periodic time of a planet. and a its mean distance from the sun; then the length of the sidereal year being 365.256359 days (120), (365.256359d.)2: P2: 1: a'; whence, a (36-.2 Aid.) *.. (41). 1~2. Loangitude of Perihe~ions, -taaad Eceenitricity, by Approximate Methods. If a great number of values of the radius-vector, in a great variety of positions of the planet in its orbit, be found by the method explained in Art. 178, the longitude of the planet at the time when its calculated radius-vector is the least, will be approximately the longitude of the perihelion; or, if it chances that among the calculated radius-vectors there are two equal to each other, the position of the line of apsides may be found by bisecting the angle included between these. The ratio of the ditference between the greatest and least calculated radii to the mean of the whole, will be the approximate value of the eccentricity. EPOCH OF THE PERIHELION PASSAGE. 1I3. From several observations upon the planet about the time it has the same longitude as the perihelion, the correct time of its being at the perihelion may be easily determined by proportion. The liean Longitude at amz assigzzed epoch is obtained on the same principles as the mean longitude of the sun or moon (158, 170). REMARKS. 184. The foregoing methods of determining the elements of a planet's orbit suppose observations to be made at two or more successive returns of the planet to its node; but it is not necessary to wait for the passage of a planet through its node. Soon after ~ — ~0 (3 TRUE AND MEAN ELEMENTS. 128 the planet Uranus was discovered by Sir William Herschel, Laplace contrived methods by which the elements of its elliptic orbit were determined from four observations within little more than a year from its first discovery by Herschel. After the dis. covery of Ceres, Gauss invented another general method of calculating the orbit of a planet from three observations, and applied it to the determination of the orbit of Ceres, and subsequently to the determination of the orbits of Pallas, Juno, and Vesta. This method can be more readily employed in practice than that of Laplace, or than any of the solutions which other mathematicians have given of the same problem, and is now generally used in computing the orbit of a newly discovered planet. TRUE AND MEAN ELEMENTS. 185. True elements and their variations. The elements of the planetary orbits, obtained by the foregoing processes, are the true elements at the periods when the observations are made. Upon determining them at different periods, it appears that they are subject to minute variations. A comparison of the values found at various distant epochs shows that they are slowly changing from century to century, and that the changes experienced during equal long periods of time are very nearly the same. The amount of the variation of an element in a period of 100 years is called its Secular Variation. Upon reducing the elements, found at different times, to the same epoch, by allowing for the proportional parts of' the secular variations, the different results for each element are found to differ slightly from each other, which shows that the elements are also subject to slight periodical variations. These variations being very minute, the true elements can never differ much from the mean, or those from which they deviate periodically and equally on both sides. 186. Miean Elements and their Secula'r Variations. The mean elements at an assigned epoch may be had by finding the true elements at various times, and reducing them to the given epoch, by making allowance for the proportional parts of the secular variations, and then taking for each element the mean of all the particular values obtained for it. A comparison of the mean values of the same element, found at distant epochs, makes known the variation of its mean value in the interval between them, from which the secular variation may be deduced by simple proportion. 18 7. Variations of Elements of Moon's Orbit. The elements of the moon's orbit are also subject to continual variations. These are, for the most part. periodic, and are far greater than the variations of the corresponding elements of a planet's orbit. It will be seen, then, that in determining the mean elements, a much greater number of observations will be required than in the case of a planetary orbit. The mean node and perigee have a rapid and nearly uniform motion. Their motions, in connection with the mean motion of revolution of the moon, are subject to minute secular variations. The mean eccentricity, and inclination of the orbit, are constant. 188. Verifications. The mean elements which have been derived as above, directly from observation, have subsequently been verified and corrected by coin-paring the computed with the observed places of the planet; and for this purpose many thousands of observations have been made. 189. Tables II. and Ill. contain the elements of the orbits of the principal planets, and of the moon's orbit, together with their secular variations for the beginning of the year 1850. Table II. (a) contains the mean distances, sidereal revolutions, and eccentricities of the orbits of the planetoids. 121 MIOTIONS OF THE PLANETS IN SPACE. If an element be desired for any time different from the epoch of the table, we have only to allow for the proportional part of the secular variation, in the interval between the given time and the epoch of the table. 190. Secanar Variations. It will be seen on inspecting Table II., that the mean distances of the planets from the sun, or the semi-major axes of their orbits, are the only elements that are invariable. The rest are subject to minute secular variations. The nodes have all retrograde motions. The perihelia, on the contrary, have direct motions, with the single exception of the perihelion of the orbit of Venus, which has a retrograde motion. The eccentricities of some of the orbits are increasing; of others, diminishing. That of the earth's orbit is diminishing. The node of the moon's orbit has a retrograde motion, and the perihelion a direct motion. The former accomplishes a tropical revolution in about 18 years and 224 days, and the latter in about 8 years and 309 days. The mean motion of the node and the mean motion of the perigee are both subject to a slow secular diminution. 191. Ecceentricities and Inclinations. It will be seen also, that the orbits of the principal planets are ellipses of small eccentricity, or which differ but slightly from circles; and that they are inclined under small angles to the plane of the ecliptic. The eccentricity is in almost every instance so small that, if a representation of the orbit were accurately delineated, it would not differ perceptibly from a circle. The most eccentric orbits are those of Mercury and Mars; and the least eccentric those of Venus, Neptune, and the earth. The eccentricity of Mercury's orbit is 12 times that of the earth's, of Mars 6 times, of Venus less than -. The eccentricities of the orbits of Jupiter, Saturn, and Uranus, are each about three times that of the earth's orbit. Tlhe orbit of Mercury is more inclined to the ecliptic than the orbit of any other of the eight principal planets; and the orbit of Uranus is less inclined than that of any other planet. The inclination of the latter is a, of the former 70. The orbits of the planetoids are in general more eccentric, and more inclined to the plane of the ecliptic than those of the other planets. The inclination of the orbit of Pallas is nearly 35~. 192. The Meaui Distances of tahe Planets farom the Snn, expressed in miles, are in round numbers as follows: Mercury 35 millions, Venus 66 millions, the earth 91 millions, M/iars 139 millions, Juno 244 millions, Jupiter 475 millions, Saturn 871 millions, Uranus 1752 millions, Neptune 2,743 millions. The range of distance is from 1 to 77 —. The distance of Neptune is 30 times the earth's distance. 193. Tihe Approximnate Periods of Htevoltltion of the planets are: of Mercury 3 months, Venus 7- months, Mars 1I DIMENSIONS OF THE SOLAR SYSTEM. 125 years, Juno 4- years, Jupiter a little less than 12 years, Saturn 29-1- years, Uranus 84 years, Neptune 161-y vears. The periods and mean distances are more exactly given in Table II. (For the planetoids, see Table II. (a) ). 194. Bode's Law. A remarkable empirical law, called Bode's Law of the Distances, was announced in 1772 by Professor Bode, of Berlin, as connecting the distances of the planets from the sun. It is as follows: If we take the numbers 0, 3, 6, 12, 24, 48, 96, 192, and add 4 to each one of them, so as to obtain 4, 7, 10, 16, 28, 52, 100, 196, this series of numbers will express the order of distance of the planets from the sun. This law embodies the following curious relation between the distances of the orbits from one another, viz.: setting out from Venus, the distance between two contiguous orbits increases nearly in a duplicate ratio as we recede from the sun; that is, the distance from the orbit of the earth to the orbit of Mars, is twice the distance from the orbit of Venus to the orbit of the earth, and one-half the distance from the orbit of lMars to the orbits of the planetoids. Previous to the discovery of the planetoids, to complete the above law a planet was wanting between Mars and Jupiter. It was, on this account, surmised by Bode that another planet might exist between these two. Instead of one such planet, however, no less than ninety-one have since been discovered, revolving at pretty nearly the distance from the sun that Bode had derived from his law for the distance of the supposed planet; some at a little greater, and others at a little less distance. Bode's law, though it holds good for the planets in general, fails in the case of the planet Neptune; the error for this planet being more than one-fburth the whole distance. The error is one-twentieth of the distance for Mars, and also for the planetoids. For Mercury, Venus, and Saturn it is about one-thirtieth. For Uranus and Jupiter it is a still smaller fraction. 195. Dimensions of the Solar System. A better idea of the dimnensions of the solar system than is conveyed by the statement of distances above given, may be gained by reducing its scale sufficiently to bring it within the range of familiar distances. Thus, if we suppose the earth to be represented by a ball only one inch in diameter, the distance of IMvercury fronm the sun will be represented, on the same scale, by 370 feet, the distance of Venus by 700 feet, that of the earth by 960 feet, that of Mars by 1,500 feet, that of Juno by half a mile, that of Jupiter by 1 mile, that of Saturn by 1- miles, that of Uranus by 31 miles, and that of Neptune by 5 miles. On the same scale, the distance of the moon from the earth would be only 2j feet. 126 DETERMINATION OF THE PLACE OF A PLANET. CHAPTER X. DETERMINATION OF THE PLACE OF A PLANET, OR OF THE SUN OR MOON, FOR A GIVEN TIME, BY THE ELLIPTIC THEORY.VERIFICATION OF KEPLER'S LAWS. PLACE OF A PLANET IN ITS ORBIT. 196. True and lMean Anomaly. The angle contained between the line of apsides of a planet's orbit and the radiusvector, as reckoned from the perihelion towards the east, is called the Trute Anomaly. Thus, let BPAP' (Fig. 60) represent the!) C0 R FIG. 60. orbit, B the perihelion, and P the position of the planet; then BSP is its true anomaly. The angle contained between the line of apsides and the mean place of the planet, also reckoned from the perihelion towards the east, is called the lfean Anomaly. Thus, let M be the mean place of a planet at the time P is its true place, and BSM will be its mean anomaly. The diffierence between the true anomaly BSP and the mean anomaly BSM, is the angular distance MSP between the true and mean place of the planet, or the equation of the centre (143). Describe a circle BpA on the line of apsides as a diameter; through P draw pPD perpendicular to the line of apsides, and join p and C; the angle BCp, which the line thus determined makes with the line of apsides, is called the Eccentric Anomaly. The corresponding angles appertaining to the sun's apparent orbit, and to the moon's orbit, have received the same appellations. 197. Anomalistic Revolution. The interval between two HELIOCENTRIC PLACE OF A PLANET. 127 consecutive returns of a body to either apsis of its orbit, is called the Anornralbstic Revolution. The anomalistic revolution of the earth, or of the sun in its apparent orbit, is termed, also, the Anomalistic Year. The periodic time, or the mean motion of a body, and the motion of the apsis of its orbit, being known, the anomalistic revolution may be easily computed. Let m = the sidereal motion of the apsis answering to the periodic time, and M - the mean daily motion of the planet; then, ATM: id.:: m: x cdiff. of anomalistic rev. and periodic time. When the epoch of any one passage of a planet through its perihelion, or of the sun or moon through its perigee, has been found, we may, by means of the anomalistic revolution, deduce from it the epoch of every other passage. The length of the anomalistic year exceeds that of the sidereal year by 4m. 39s. 19~. Calctlulation of 1feaz A oinaly. From the anomalistic revolution, and the epoch oft the last passage through the perihelion or perigee (as the case may be), we may derive the mean anomaly for any given time. Let T - the anomalistic revolution t- the time that has elapsed since the last passage through the perihelion or perigee, and A -- the mean anomaly; then, T: 360:: t: A- 360~ t....(42). 199. The Place of a Body in its Elliptic Orbit is ascertained by findingc its true anormaly. The problem which has for its object the determination of the true anomaly from the mean, was first resolved by Kepler, and is called Kepler's Problem. Another and more convenient method of obtaining the true anomaly, is to compute the equation of the centre fiom the mean anomaly, and add it to the mean anomaly, or subtract it from it, according to the position of the body in its orbit (143). (See Appendix, Solution of Kepler's Problem.) HELIOCENTRIC PLACE OF A PLANET. 200. Tlhe Place of a Plainet in the Plaine of its Orbit is designated by its orbit longitude (166) and radius-vector. To find the orbit longitude we have the equation V'SP - V'SB + BSP (see Fig. 60); or, long. long. of perihelion + true anomaly. The orbit longitude may also be deduced from the mean Iongitude, by adding or subtracting the equation of the centre; for, V'SP = V'SM t- MSP, 128 DETERMINATION OF THE PLACE OF A PLANET. or, true long. = mean long. + equa. of centre: also, V'SP' V'SM' - 1M'SP', or, true long. = mean long. - equa. of centre. The radius-vector results from the polar equation of the elliptic orbit, viz.: a (1_-e2) V 1 + e cos x in which x denotes the true anomaly, e the eccentricity, and a the semi-major axis. 201. To find the IHeliocevitric Lotngitude and Latitude, which ascertain the position of the planet with respect to the ecliptic, the triangle NPp (Fig. 56, p. 120) gives sin Pp sil NP sin PN2p; or, sin lat. - sin (orbit long.-long. of node) x sin (inclin.)...(44); and cos PNp — = tan Np cot NP, or tan Np = tan NP cos PNp, 01, tan (long. - long. of node)- tan (orbit long. - long. of node) x cos (inclination).... (45). GEOCENTRIC PLACE OF A PLANET. 202. The theoretical determination of the place of a planet, as it would be seen from the centre of the earth, consists inr deducing its geocentric longitude and latitude, and its distance from the earth, from its heliocentric longitude and latitude and radiusvector; the latter having been calculated by the methods just explained. (For the detail of the solution of this problem see Appendix.) PLACES OF THE SUN AND MO00N. 203. The place of the sun, as seen from the earth, may be easily deduced from the heliocentric place of the earth; for the longitude of the sun is equal to the heliocentric longitude of the earth plus 180~ (157), and the radius-vector of the earth's orbit is the same as the distance of the sun from the earth. But it is more convenient to regard the sun as describing an orbit around the earth, and compute its true anomaly (199); and thence the longitude and radius-vector, by the equation long. - true anomaly -t long. of perigee, and the polar equation of the orbit. 204. The Orbit Longitude and the alh adaVlus-vector of the Moon, are found by the same process as the longitude and radius-vector of the snn. The orbit longitude being known VERIFICATION OF KEPLER'S LAWS. 129 the ecliptic longitude and the latitude may be determined by a process precisely similar to that by which the heliocentric longitude and latitude of a planet are found (201). VERIFICATION OF KEPLER'S LAWS. 205. If Kepler's first two laws be true, then the geocentric places of the planets, computed by the process that we have described (202), which is founded upon them, ought to agree with the true geocentric places as obtained for the same times by direct observation; or, the heliocentric places computed from the observed geocentric places (177), ought to agree with the same as computed by the elliptic theory (200, 201). Now, a great number of comparisons have been made between the observed and computed places, and in every instance a close agreement between the two has been found to subsist. We infer, therefore, that the motions of the planets must be very nearly in conformity with these laws. The truth of the third law has been established by a direct comparison of the mean distances of' the different planets with their periodic times. Kepler's laws have been verified for the sun and moon, in a similar manner. 206. The Relative Distances of the Sunt or Moon, at different times, result for this purpose, from measurements of the apparent diameter, upon the principle that any two distances are inversely proportional to the corresponding apparent diameters. Let A semi-diameter corresponding to the mean distance, and = semi-diameter corresponding to any distance D: then:::: D; whence, D....(46); an equation which, when a has been found, will make known the distance corresponding to any observed semi-diameter S, in terms of the mean distance as a unit. Now, to find a, denote the greatest and least semi-diameters, respectively, by S', 3", and the corresponding distances by D' and D", and we have D' — ~ D " =~; and thence, (D'+ D/") ( -+), or, 1 =; whence, A... (47). -6.... 130 INEQUALITIES OF PLANETARY MOTIONS. CHAPTER XI. INEQUALITIES OF THE MOTIONS OF THE PLANETS AND OF THE AlooN; TABLES FOR FINDING THE PLACES OF THESE BODIES. 207. Gravitation. It is a general law of nature, discovered by Sir Isaac Newton, that bodies tend or gravitate towards each other, with a force directly proportional to their mass, and inversely proportional to the square of their distance. The force which causes one body to gravitate towards another, is supposed to arise from a mutual attraction existing between the particles of the two bodies, and is hence called the Attraction of Gravitation. This force of attraction, common to all the bodies of the Solar System, is the general physical cause of their motions. Tile sun's attraction retains the planets in their orbits, and the planets, by their mutual attractions, slightly alter each other's motions. The reasoning by which Newton's Theory of Univers.al Gravitation is established, appertains to Physical Astronomy, and will be presented in Part II. 20S. Perturbations;-Inequalities. If a planet were acted on by no other force than the attraction of the sun, it is proved that its orbit would be accurately an ellipse, and the areas described by its radius-vector, in equal times, would be precisely equal. But it is in reality attracted by the other planets, as well as the sun, and therefore its actual motions cannot be in strict conformity with the laws of Kepler. In fact, if we descend to great accuracy, the agreement between the observed and computed places, noticed in Art. 205, is found not to be exact. The deviations from the elliptic motion, which are produced by the attractions of the planets, are called Perturbations, or, in Spherical Astronomy, Inequalities. Although, as we have just seen, the fact of the existence of inequalities in the motions of the planets is discoverable from observation, their laws cannot be determined without the aid of theory. 209. Disturbing Force. In treating of the perturbations in the motions of one planet, resulting fiorn the attractions of another, the attracting planet is called the Disturbing Body, and the force which produces the perturbations the Disturbing Force. To find the disturbing force, let P (Fig. 61) be the planet, S the sun, and M the disturbing body; and let PD represent the attraction of MI for the planet. Decompose PD into two forces, COMPONENTS OF DISTURBING FORCE. 131 PE and PF, one of which, PE, is equal and parallel to SG, the attraction of M for the sun; the other, PF, will be known in position and intensity. The two forces, PE and SG, being equal and parallel, they cannot alter the relative motion of the sun ir' p FIG. 61. and planet, and accordingly may be left out of account: there remains, therefore, the component PF, which will be wholly effective in disturbing this motion. This, then, is the' disturbing force. It happens in the case of each planet, that the distances of some of the other planets are so great that their disturbing forces are insensible. The attractions of these bbdies for the sun and planet, when thev are exterior to the planet, are sensibly equal and parallel. Owing to the great distance of the planets from each other, and the smallness of their mass compared with that of the sun, the disturbing force is in every instance very minute in comparison with the sun's attraction. 210. Cornponenlts of Disturbing Force; —their Effects. It is plain that the disturbing fbrce will, in general, be obliquely inclined to the perpendicular to the plane of the orbit, PK, the tangent to the orbit, PT, and the radius-vector, PS; and may, therefore, be decomposed into forces acting along these lines. The component along the perpendicular will alter the latitude, and the two others both the longitude and radius-vector; that along the tangent by changing the velocity of the planet, and that along the radius-vector by changing the gravity towards the sun. It appears, therefore, that the disturbing force produces at 132 INEQUALITIES OF PLANETARY MOTIONS. the same time perturbations or inequalities of longitude, of latitude, i-nd of rcadius-vector. 211. Deternmitation of Inequalities. Let us now (-)nsider how these inequalities may be determined. In the frst place, the inequalities produced by each disturbing body may be separately investigated upon mechanical principles, as if the other bodies did not exist; for the reason that the effect of each disturbing body is sensibly the same that it would be if the other bodies did not act. That this is very nearly, if not quite true, may be at once inferred from the minuteness of the whole disturbance produced by the joint action of all the disturbing forces of the system. The problem which has for its object the determination of the inequalities in the motions of one body, in its revolution around a second, produced by the attraction of a third, is called the Problem of the Three Bodies. If, in the case of any one planet, this problem be solved for each of the other bodies of the system which occasion sensible perturbations, all the inequalities to which the motion of the planet is subject will become known. The general solution of tAe problem of the three bodies, that is, for any mass and distance of the disturbing body, or any intensity of the disturbing force, cannot be effected in the existing state of the mathematical sciences. But the problem has been solved for the case that presents itself in nature, in which the disturbing force is very minute in comparison with the central attraction. The results obtained by the analysis are certain analytical expressions for the perturbations in longitude, latitude, and radiusvector, involving variables and constants. 212. Equations of Specific lnequalities of Loigai. tude. The general expression for the whole perturbation in longitude, due to the action of any one disturbing body, is of the form C sin A + C' sin A' + C" sin A", etc., in which C, C', C", etc., are constants, and A. A', A", etc., angles depending upon the positions of the disturbing and disturbed planets, with respect to each other and the sun, and also, in some cases, with respect to the nodes and perihelia of their orbits. Each of the terms, C sin A, C' sin A', etc., is technically called an Equation, and is considered as representing a specific ine quality. The variable angle whose sine enters into the term is called the Argument of the inequality, and the constant is called the Coefficient of the inequality. As the greatest value of the sine of the argument is unity, the coefficient is equal to the greatest value of the inequality. 213. Calculationl of In.equalities. The value of each argument may be derived for any assumed time, from the elliptic theory of the planetary motions; and the coefficients of all the CALCULATION OF INEQUALITIES. 133 inequalities may be calculated by malking repeated determinae tions of the difference between the observed and computed longitude of' the disturbed planet. By putting the entire expression, C sin A + C' sin A', etc., equal to each one of the differences of longitude so determined, we may form as many equations as there are unknown quantities, C, C', etc., from which their values may be deduced. The coefficient of any inequality being known, the value of the inequality, at any particular time, will become known if that of the argument be found. This value will be the correction for that inequality, to be applied to the elliptic place of the planet computed for the assumed time. 214. Inequalities of Latitude anad Radiaus-vector. The theory of' these inequalities, and of their computations, is similar to that of the inequalities of longitude just explained. 215. Inequalities are Periodic. WVe have seen that the arguments of the inequalities are angles depending on the configurations of the disturbing and disturbed planets with respect to each other and the sun, or with respect to the nodes ori perihelia of their orbits. Whenever these configurations become the same, as they will periodically, the arguments, and therefore the inequalities themselves, will have the same value. It follows, therefore, that the inequalities in question are periodic. The interval of time in which an inequality passes throughl all its gradations of positive and negative value, is called the Peroiod of the inequality. It is manifestly equal to the interval of time employed by the argument in increasing -from zero to 360~; for, in this interval sin A or cos A takes all its values, both positive and negative, and at the expiration of it recovers the same value again. 216. [nequalities of Elliptic Elemeants. It has been stated that the elements of the elliptic orbits of the planets are, for the most part, subject to a slow variation from century to) century. Investigations in Physical Astronomy have established that the variations of the elements are due to the action of the disturbing forces of the planets, and that they are not progressive (except in the cases of the longitude of the node and the longitude of the perihelion), but are really periodic inequalities whose periods comprise many centuries. From the great lengths of their periods these inequalities are termed Secular Inequalities, in order to distinguish them from the inequalities of the elliptic motion, denominated Periodic Inequalities, the periods of which are comparatively short. Physical Astronomy firnishes expressions called Secular Equations, which give the value of an element at any assumed time. 217. The Inequalities of the Ml7loon's Motion arise from the disturbing action of the sun. The attractions of each of the planets for the moon and earth are sensibly equal and parallel. 13 94 INEQUALITIES OF THE MOON'S AIMOTION. The lunar inequalities are investigated upon the same principle as the planetary, and are represented by equations of the same general form, that is, consisting of a constant coefficient and the sine or cosine of a variable argument. They far exceed in number and magnitude those of any single planet. There are three lunar inequalities of longitude which are prominent above the rest. and were early discovered by observation. The most considerable is called the Evection, and was discovered by Ptolemy in the first century of the Christian era. It has for its argument double the angular distance of the moon from the sun minus the mean anomaly of the moon, and amounts when greatest to 1~ 20' 80". The second is called the Variation, and was discovered in the sixteenth century by Tycho Brahe. Its argument is double the angular distance of the moon from the sun, and its maximum value is 35' 42". The third is denominated the Annutal Equation, from the circurnstance of its period being an anomalistic year. Its argument is the mean anomaly of the sun. The discovery of the other lunar inequalities (with the exception of one inequality of latitude), is due to Physical Astronomy. 21 8. Calculation of Exact Heliocentric Place of a Planet. To present now at one view the entire process of calculating the co-ordinates of the exact heliocentric place of a planet, or of the geocentric place of the moon, at any assumed time,(1). Seek the elements of the elliptic orbit from a table of elements, such as Table II. or III., allowing for the proportional part of the secular variation; or (more exactly) obtain them from their secular equations (216). (2). Compute the longitude, latitude, and radius-vector, by the elliptic theory (200, 201). (3). Compute the values of the inequalities in longitude, latitude, and radius-vector, by means of their equations (212, 213, 214), and apply them individually, with their proper signs, as corrections to the elliptic values of the longitude, latitude, and radius-vector. When the exact heliocentric place of a planet has been found, its geocentric place may be determined by the process referred to in Art. 202. Geocentric Place of the Sun. The elements of the sun's apparent orbit are the same as those of the earth's actual orbit, except that the geocentric longitude of the perigee of the one exceeds the heliocentric longitude of the perihelion of the other by 180~. From these elements the longitude and radius-vector are obtained as in Art. 203. The values of the inequalities resulting from the earth's motion are then to be applied to these as corrections. ASTRONOMICAL TABLES. 135 TABLES OF TEE SUN, MOON, AND PLANETS. 219. The calculation of the co-ordinates of the place of the sun, moon, or any planet, for any assumed time, may be greatly facilitated by the use of tables. The principle and mode of construction of tables adapted to this purpose are explained in Part III. We will only remark here that the tables save the necessity of calculating the equations of the inequalities (218); since they make known their values corresponding to the values of the arguments at the time supposed. These values of the arguments are also readily obtained from tables especially designed for this purpose. Tables of the sun, moon, and of each of the principal planets, have been calculated by different astronomers, and are now in general use. 220. Ephenmeris. With the aid of these tables an ephemeris of each body is computed, and published for each year in advance, in the American and English Nautical Almanacs. Aln E.phemeris of a heavenly body is a collection of tables exhibitmng the longitude, latitude, right ascension, declination, paralklcr, semi-diameter, etc., of the body, at stated periods of time, a' at noon of each day throughout the year. 136 MOTIONS OF THE COMETS. CHAPTER XII. MOTIONS OF THE COMETS. 221. Apparent ioitions. VWhen first seen, a comet is ordinarily at some distance from the sun in the heavens, and moving towards it. After this, it continues to approach the sun, for a certain time, and then recedes to a greater or less distance, and finally disappears. In many instances comets have come so near the sun, as to be for a time lost in its beams. It has sometimes happened that a comet has not made its appearance in the firmament until after the time of its nearest apparent approach to the sun, and when it is receding from him in the heavens. This was the case with the great comet of 1843. It was first seen, in this country, in open day, on the 28th of February, in the immediate vicinity of the sun; and after this moved away from it, and, gradually diminishing in brightness, in about a month became invisible. Comets resemble the planets in their changes of apparent place among the fixed stars, but they differ from them in never having been observed to perform an entire circuit of the heavens. Their apparent motions are also more irregular than those of the planets, and they are confined to no particular region of the heavens, but traverse indifferently every part. 222. Orbits of ComlIets. Sir Isaac Newton, from observations that had been made upon the remarkable comet of 1680, ascertained that this comet described a parabolic orbit, having the sun at its focus, or an elliptic orbit of so great an eccentricity as to be undistinguishable from a parabola, and that its radiusvector described equal areas in equal times. Since then, the orbits of 240 comets have been computed, and found to be, the majority of them, of a parabolic form, or sensibly so. It was demonstrated by Newton, on the theory of gravitation, that a body projected into space may describe about the sun as a focus either one of the conic sections, and that the formn of the orbit will depend upon the projectile velocity alone. WVith one particular velocity the orbit will be a parabola; with any less velocity it will be an ellipse or circle; and with any greater velocity it will be an hyperbola. Now, as there is but one velocity from which a parabolic orbit will result, and as any come-'t. lwhich may have originally moved in a hyperbola, must have COMETS OF KNOWN PERIOD. 137 passed its perihelion, and receded beyond the limits of the solar system, it may be inferred, with great probability, that the orbits of the comets whose observed courses are not distinguishable from parabolic arcs, are in fact ellipses of great eccentricity. This is the theory of the cometary motions proposed by Newton. The orbits of some of the comets are known from observation to be very eccentric ellipses. 223. Eleaents of Parabolic Orbit. The elements of the parabolic path conceived to be traced by a comet during the period in which it remains visible, are: the longitude of' the ascending node, the inclination of the orbit, the longitude of the perihelion, and the epoch of the perihelion passage. Assuming that the radius-vector describes areas proportional to the times, these elements may be computed from three observed geocentric places. But the problem is one of considerable difficulty. 224. SEntire ]Elliptic Orbit..-Perzo~ s of Revohlutioat. Astronomers do not in general seek to deduce, from the observations made during one appearance of a comet, its entire elliptic orbit. It is impossible, from such observations, to compute the major-axis of its orbit and its period with any accuracy, inasmuch as in the interval during which they are made, the comet descrbibes but a small portion of its entire orbit. As examples of the uncertainty of such determinations, four periods have been found by Bessel for the comet of 1807, of which the least is 1,483 years and the greatest 1,952 years; and for the great comet of 1811 the two periods, 2,301 years and 3,056 years, have been computed. The uncertainty becomes much less when the period of revolution is short. The only mode of obtaining the period of a comet's revolution with certainty is by directly comparing the times of its successive perihelion passages. A comet cannot be recognized at a second appearance by its aspect; for this is liable to great alterations. But it may be identified by means of the elements of its parabolic orbit (223), as it is extremely improbable that the elements of the orbits of two different conmets will agree throughout. This method of identifying a comet may sometimes fail of application, inasmuch as the orbit of a comet may experience great alterations from the attractions of the planets. 22a. Comets of Rianowia Period. Owing to the great lengths of the periods of revolution of most of the comets, and the comparatively short intervals of time during which their motions have been carefully observed, there are but eight comets whose periods and entire orbits have been determined with certainty. These have all reappeared, and in some instances repeatedly, and verified the determinations of their paths through space, and the predictions of their return to their perihelia. A comet usually receives the name of the astronomer who first determines its orbit and period of revolution. The comets just alluded to 138 MOTIONS OF THE COMETS. are designated as ITalley's, Encke's, Biela's, Faye's, De Vico's, BJrorson's, D'Arrest's, and WVinnecke's. The last seven are known as Comets of Short Period; their periodic times being comprised within the limits of 3.3 years and 71- years. Their mean distances fiom the sun are less than that of Jupiter, and they revolve within the orbit of Saturn. Halley's comet, in its recess firom the sun, passes beyond the limits of the solar system, and its period approximates to that of Uranus. Fig. 62 shows the relative dimensions and positions of the orbits of Halley's, Encke's, and Biela's comets. \ / FIG. 62. 226. Counets whose Per iods lhave been Approximately Calculated. There are a number of cometary bodies whose periods of revolution and elliptic orbits have been approximately deduced, by calculation, from observations made at the periods of their first discovery, but which have not since been seen. Five of these belong to the class of comets of comparatively short period, and small mean distance from the sun; their computed periods being from five to seven years. Two have periods of 10 years and 16 years, respectively. Five form, with Halley's comet, a distine.t class; their periodic times are all about 75 years, and their mean distances from the sun nearly equal to that of Uranus. There are also more than twenty comets whose entire elliptic NUMBER OF COMETS. 139 orbits are believed to have been ascertained with a certain decree of approximation to the truth. Their mean distances exceed the limits of the solar system, and their periods are much longer than that of the most distant planet. The same is known to be true of the mean distances and periods of all the remaining comets that have been carefully observed. 227. All the Comraels of Comnparatively Short Period (viz., from 3.3 years to 16 years) revolve around the sun in the same direction as the planets, and like the planetoids, in planes inclined less than 35~ to the plane of the ecliptic. But their orbits are much more eccentric than the orbits of the minor planets. They form a group of bodies whose orbits bear a striking resemblance to each other, and occupy a position, in respect to their orbital motions, intermediate between the planetoids and the comets of long period (75 years and more). They are comparatively faint objects, and have generally been visible only with the aid of a telescope. All the other comets, whose mean distance from the sun does not exceed that of the most distant planet, with the exception of Halley's, also have a direct motion. Some of these, on their return to their perihelia, have become visible to the naked eye; Halley's comet conspicuously so. 228. Comets of Long Period. Of 220 observed comets, whose mean distances from the sun exceed that of Neptune, about an equal number have a direct and a retrograde motion. The perihelia of more than two-thirds of the orbits fall within the orbit of the earth. The aphelia lie far beyond the orbit of Neptune. There is little reason to doubt that many comets recede tens of thousands of millions of miles before they begin to return to the sun again; and that the periods of most of them include a number of centuries, and of many of them even tens of centuries. The planes of their orbits are inclined under every variety of angle to the plane of the ecliptic. 229. Conets of Small Perihelios Distance. Some comets come into close proximity to the sun. The great comet of 1680, according to the computation of Newton, came 166 times nearer the sun than the earth is. The no less remarkable comet of 1843 approached still nearer; when at its perihelion, it was less than 70,000 miles from the sun's surface. Its orbital velocity at that time was 350 miles per second; and it accomplished a semi-revolution around the sun (from n to n', Fig. 63) in the astonishingly short interval of 2 hours. 230. lnumber of Comets. The number of recorded appearances of comets is about 800, but the actual number of cometary bodies connected with the solar system is undoubtedly far greater than this. This list of recorded appearances comprises, for the great number of years which precede the date of the invention of the telescope, only those comets which were very conspicuous to the naked eye; giving, for example, only three in 140 MOTIONS OF THE COMETS. the thirteenth, and three in the fourteenth century; and, since the heavens have been attentively examined with telescopes, from two to three comets, on an average, have made their appearance every year, of which the great majority were telescopic. The periods of these, as well as of the others, are in general of such vast length that probably not more than half the whole number of comets have returned twice to their perihelia during the last two thousand years. From these considerations it appears, that, had the heavens been attentively surveyed with the telescope during the last two thousand years, as many as 2,500 different cometary bodies would have been seen. But, as there are various causes which may tend to prevent a comet from being seen when present in our firmament,-as continued proximity to the sun in the heavens, too great distance from the sun and earth, want of intrinsic lustre, etc., —it is highly probable that there are, in fact, many thousands of these bodies. HALLEY'S COMET. 231. Halley's comet is so called from Sir Edmund lHalley, Second Astronomer Royal of England, who ascertained its period, and correctly predicted its return. From a comparison of the elements of the orbits described by the comets of 1531, 1607, and 1682, he concluded that the same cornet had made its appearance in these several years, and predicted that it wonuld again return to its perihelion towards the end of 1758 or the beginning of 1759. Previous to its appearance, Clairaut, a distinguished French astronomer, undertook the arduous task of calculating its perturbations from the disturbing actions of the planets during this and the preceding revolution. He found, that, from this cause, it would be retarded about 618 days, 100 days from the effect of Saturn, and 518 days from the action of Jupiter,-and predicted that it would reach its perihelion within a month, one way or the other, of the middle of April, 1759. It actually passed its perihelion on the 12th of March, 1759. Assuming the earth's mean distance from the sun to be unity, the perihelion distance of this comet is 0.6, and aphelion distance 35.4. Accordingly it approaches the sun to within about one-half the distance of the earth, and recedes from him to nearly twice the distance of Uranus. (See Fig. 62.) Its period is about 76 years, but is liable to a variation of a year or more from the effect of the attractions of the planets. The inclination of its orbit is 18~, and its motion is retrograde. The last perihelion passage took place on the 16th of November, 1835, within a few days of the predicted time. The next will occur in the year 1911. It is to be expected that the perturbations will now be determined with ENCKE'S COMET. 141 such increased accuracy that the error in the prediction of its next perihelion passage will be less than one day. Probable repeated appearances of this comet have been traced as far back as the year 11 B. C. It seems to have been particularly conspicuous in the years 1066 and 1456. ENKCE'S COMET. 232. This comet is remarkable for its short period of revolution, which is only 3.3 years. It moves in an orbit inclined only 13~ to the plane of the ecliptic, and whose perihelion is at the distance from the sun of the planet Mercury, and aphelion at a distance somewhat less than that of Jupiter (see Fig. 62). Its period and elliptic orbit were determined on the occasion of its fourth recorded appearance, by Professor Encke, of Berlin. Since then it has returned a number of times to its perihelion, and in every instance very nearly as predicted. At some of its returns it has become visible to the naked eye. Its last return took place in 1865; the next will be in September, 1868. 233. Disturbing Effects of a Resisting Nl[ediutm. The motions of this comet present the anomalous fact in the solar system of a period continually diminishing, and an orbit slowly contracting, from the operation of some other cause than the disturbing actions of the other bodies of the system. Professor Encke found that after allowance had been made for all the perturbations produced by the planets, the actual time of each perihelion passage anticipated the time calculated from the duration of the previous revolution about 22 hours; and that the comet now arrives at its perihelion about 22 days sooner than it would itf the period had remained unaltered since the comet was first seen in 1786. This continual acceleration of the time of the perihelion passage, discovered by Encke, could not be attributed to the disturbing attraction of some unknown body, because this attraction would produce other effects, which have not been noticed. He conceived that it could arise from no other cause than the action of a resisting medium, or ether in space. The immediate effect of such a medium subsisting in the regions of space traversed by the comet, would be to diminish the velocity in the orbit, which it would at first seem should delay the time of the perihelion passage; but the velocity being diminished, the centrifugal force is weakened, and consequently the comet is drawn nearer to the sun, and moves in an orbit lying within the orbit due to the sun's attraction alone; its mean distance is therefore diminished, and its period shortened. A similar phenomenon to this is presented in the oscillations of a pendulum freely suspended. It is well known that the are of vibration of 142 iMOTIONS OF THE COMETS. the pendulum shortens, and consequently its rapidity of oscillation increases, under the influence of the resistance of the air. BIELA'S COMET. 234. In February, 1826, M. Biela, of Josephstadt, in Bohemia, detected a telescopic comet in the constellation Aries; and subsequently made repeated observations upon its varying position in the heavens. From the results of his observations, he calculated the elements of its supposed parabolic orbit, and found on inspecting a catalogue of comets that the computed elements bore a striking resemblance to those of' the comets of 1772 and 1805. He also ascertained that the entire observed path of the comet could not be accurately represented by a parabolic orbit, and proceeded to compute from his observations the elements of an elliptic orbit. He found the period of revolution to be 6.7 years, and that it accorded with the supposition that the same comet had been previously seen in 1772 and 1805. The period, as since more accurately determined, is 6.6 years. Its orbit is inclined 12-1~ to the plane of the ecliptic; and the perihelion lies just within the orbit of the earth, while the aphelion falls beyond the orbit of Jupiter (Fig. 62). By a remarkable coincidence, the orbit of this comet very nearly intersects the orbit of the earth. At the return of the comet in 1832, Dr. Olbers found that in going through its descending node it would pass within 20,000 miles of the earth's orbit, on the inside, and that a portion of the orbit would fall within the filmy mass of the comet. The earth was more than 60,000,000 miles distant from the comet at the time of the nodal passage, and did not reach the point of nearest approach of the two orbits until one month after the comet had passed by it. In 1805 the same comet passed within 6,000,00o miles of the earth. According to calculation, the last return of Biela's comet to its perihelion took place in February, 1866; but the comet escaped detection. The next return will be in September, 1872. FAYE'S COMET. 2~s. This comet was discovered and its orbit determined by M. Faye, of the Paris Observatory. Its period of revolution is 7{ years. The eccentricity of its orbit (0.556) is less than that of any other known cometary body, although nearly twice as great as that of the most eccentric planetary orbit. The return of this comet to its perihelion appears to be accelerated, like that of Encke's comet, and in a much greater degree, by the operation of a resisting medium in space. As the perihe LEXELL'S COMET OF 1770. 143 lion distance of this comet is much greater than that of Encke's, it seems probable that the resistance encountered by these comets is due to a collision with meteoric bodies, or some other form of cosmical matter. The remaining comets of short period need not be specially noticed. LEXELL'S COMET OF 1770. 236. It has already been intimated that the motions of the comets are liable to great derangements, from the operation of the attractive forces of the planets. This results from the elongated form of the cometary orbits, in consequence of which the comets, while pursuing their course within the limits of the planetary system, may come into proximity to the planets, and be strongly attracted by them. Halley's comet has already furnished an illustration of this general fiact. Lexell's comet offers a still more striking example of the disturbances to which the cometary motions are exposed. From observations made upon this comet in the year 1770, Lexell made out that its period was 5, years; still, though a very bright comet, it has not since been seen. Burckhardt, an eminent French calculator, undertook to investigate the cause of this phenomenon, and found that on its return to the perihelion in 1776, the comet was so situated with regard to the earth and sun as to be continually hid by the sun's rays; and that in 1779, before its next return, it passed so near the planet Jupiter, that his attraction was very many times greater than the attraction of the sun. The consequence was that its orbit was greatly enlarged, so that it no longer comes near enough to the earth to be visible. Another fact to be accounted for was, that the comet had not been seen previous to the year 1770. In seeking for its explanation it was discovered, by tracing back the orbit of the comet, that in 1767 it must have passed near Jupiter, and that the action of his attractive force must have altered its orbit from one of large dimensions to the comparatively small orbit, with short period, of the comet as seen in 1770. While describing, previous to 1767, an orbit with a large perihelion distance, it could not have come near enough to the earth and sun to be visible. This comet is also remarkable as having made a nearer ap. proach to the earth than any other on record. On July 1, 1770, its distance from the earth was less than 1,500,000 miles. 144 MOTIONS OF THE COMETS. THE GREAT COM[ET OF 1843. 237. This comet has already been allnded to as remarkable for having made a nearer approach to the sun than any other comet. Its parabolic path is represented in Fig. 63. The positions of the -a 8 Mareb 6'ti ua/rch, 9' A/.L as \eb N-ahrc~i~Fe Fin. 63. comet at several different dates, with the corresponding positions of the earth, are also indicated; n is the ascending and,'' the descending node. The perihelion is within 500,000 miles of the sun's centre, and nearly midway between n and n'. The inclination of the orbit is 36~. The comet passedl its perihelion on February 27, at about 5 P.r. (Philadelphia time). On the DONATI'S COMET. 145 28th it was observed in full daylight in various parts of New England, in Mexico, at several places in Italy, and off the Cape of Good Hope. It was then about 3~ distant from the sun, and of a dazzling brightness. Its great lustre at that time doubtless resulted in part firom its tail being foreshortened by the obliquity under which it was seen. After the 28th it showed itself with great distinctness early in the evening, over the western horizon; and though growing fainter from night to night, as it receded from the sun, continued visible to the naked eye until about the 3d of April. This comet is believed to move in an elliptic orbit answering to a period of 175 years. DONATI'S COMET. 238. This is the great comet that made its appearance in 1858. It was first seen by Donati at Florence, on the 2d of June, 1858. It was then but a faint nebulosity, discernible only with a telescope. Although becoming more distinct in the field of the telescope from week to week, it did not become visible to the naked eye until near the 1st of September. It attained to its greatest size and splendor after the perihelion passage on September 30, after which it decreased in brightness as it receded from the sun and earth, moved off rapidly towards the south, and finally disappeared from view in March, 1859, in /a'LjLIN OF NODES OCT 15 OCT 30 FIG. 64. the southern heavens. Fig. 64 represents a portion of the orbit of the comet, as projected on the plane of the earth's orbit, and several corresponding positions of the comet and earth. The plane of the orbit is inclined to that of the earth's orbit under an angle of 63~, the portion of the orbit containing the perihelion 10 146 MOTIONS OF THE COMETS. lying on the north side of the plane of the earth's orbit. When first seen, otl June 2, the cornet wfas about 240,000,000 miles from the earth. At the perihelion (September 30) the distance was less than 70,000,000 miles. It was at its least distance from the earth (nearly 52,000,000 miles) on October 10, but attained its greatest brilliancy five days earlier. The period of revolution of Donati's comet has not been determnined; but it is estimated to, exceed 1,600 years. CONSPICUOUS COMETS OF THE PRESENT CENTURY. 239. These are, in addition to Donati's comet, and the great comet of 1843, the great comet of 1811, the bright comets of 1819, 1825, and 1835 (lalley's comet), and the great comet of 1861. The comet of 1811 affords an instance of a large and bright comet, with a perihelion distance exceeding the earth's distance from the sun. REVOLUTION OF THE SATELLITES. 14i CHAPTER XIII. MOTIONS OF THE SATELLITES. 240. As before stated, the planets which have satellites are Jupiter, Saturn, Uranus, and Neptune. The number of Jupiter's satellites is four, of Saturn's eight, of Uranus' eight, of Neptune's one. 241. The Satellites of Jupiter are perceptible with a telescope of very low power. It is found, by repeated observations, that they are continually changing their positions with respect to one another and the planet; being sometimes all to the right of the planet, and sometimes all to the left of it, but more frequently some on each side. They are distinguished from each other by the distance to which they recede from the planet; that which recedes to the least distance being called the ]F'irst Satellite, that which recedes to the next greater distance the Second, and so on. The satellites of Jupiter were discovered by Galileo, in the year 1610. The Satellites of Saturn, Uranus, and NAeptune cannot be seen, except through excellent telescopes. They experience changes of apparent position, similar to those of Jupiter's satellites. 242. The Satellites Revolve arouin d the Planet. The apparent motionl of Jupiter's satellites alternately from one side to the other of the planet, leads to the supposition that they actually revolve around the planet. This inference is confirmed by other phenomena. While a satellite is passing from the eastern to the western side of the planet, a small dark spot is frequently seen crossing the disc of the planet in the same direc-. tion; and again, while the satellite is passing from the western to the eastern side, it often disappears, and, after remaining for a time invisible, reappears at another place. These phenomena are easily explained, if we suppose that the planet and its satellites are opake bodies illuminated by the sun, and that the satellites revolve around the planet from west to east. On this hypothesis, the dark spot seen traversing the disc of the planet is the shadow cast upon it by the satellite on passing between the planet and the sun; and the disappearance of the satellite is an eclipse, occasioned by its entering the shadow of the planet. As the transit of the shadow occurs during the passage of the satellite from the eastern to the western side of the planet, and 3 [ J MOTIONS OF THE SATELLITES. the eclipse of the satellite during its passage from the western to the eastern side, the direction of the motion must be from west to east. Analogous conclusions may be drawn from similar phenomena exhibited by the satellites of Saturn. The satellites of Uranus also revolve around their primary; but the direction of their motion, as referred to the ecliptic, is from east to west. The satellite of Neptune revolves around the planet from west to east. 243. Eclipses. —Tralnsits of Shadows. Let us now examine into the principal circumstances of the eclipses of Jupiter's satellites, and of the transits of their shadows across the disc of the primary. Let EE'E" (Fig. 65) represent the orbit of the / f5 F FIG. 65. earth, PP'P" the orbit of Jupiter, and ss's" that of one of its satellites, supposed to lie in the plane of. Jupiter's orbit. Suppose that E is the position of the earth, and P that of the planet, and conceive two lines, aa', bb', to be drawn tangent to the sun and planet: then, while the satellite is moving from s to s' it Mwill be eclipsed; and, while it is moving from f tof,' its shadow will PERIODS, MEAN MOTIONS, MIEAN DISTANCES. 149 fall upon the planet. Again, if Ee, Ee' represent two lines drawn from the earth tangent to the planet on either side, the satellite will, while moving from g to g', traverse the disc of the planet, and, while moving from h to h', be behind the planet, and thus concealed fromn view. It will be seen on an inspection of the figure, that, during the motion of the earth from E", the position of heliocentric opposition, to E' that of' conjunction, the disappearances or Immersions of the satellite will take place on the western side of the planet; and that the enmersions, if visible at all, can be so only when the earth is so far from opposition and conjunction that the line Es', drawn from the earth to the point of emersion, will lie to the west of Ee. It will also be seen, that, during the passage of the earth from E' to E" the emersions will take place on the eastern side of the planet, and that the immersions cannot be visible, unless the line Fs, drawn from the earth to the point of immersion, passes to the east of the planet. It appears from observation that the immersion and emersion are never both visible at the same period, except in the case of the third and fourth satellites. If the orbits of the satellites lay in the plane of Jupiter's orbit an eclipse of each satellite would occur every revolution, but, in point of fact, they are somewhat inclined to this plane, fromi which cause the fourth satellite sometimes escapes an eclipse. 244. Periods.-Mean I!Motions. —Rcealt Distances. The periods and other particulars of the motions of the satellites, result from observations upon their eclipses. The middle point of time between the instants when the satellite enters and emerges f'oro the shadow of the primary, is the time when the satellite is in the direction, or nearly so, of a line joining the centres of the sun and primary. If the latter continued stationary, then the interval between this and the succeeding central eclipse would be the periodic time of the satellite. But, the primary planet moving in its orbit, the interval between two successive eclipses is a synodic revolution. The synodic revolution, however, being observed, and the period of the primary being known, the periodic time of the satellite may be computed. The mean motions of the satellites differ but little from their true motions; and hence the forms of their orbits must be nearly circular. The orbit, however, of the third satellite of Jupiter has a small eccentricity; that of the fourth, a larger. The distances of the satellites from their primary, are determined from micrometrical measurements of their apparent distances at the times of their greatest elongations. A comparison of the mean distances of Jupiter's satellites with their periodic times proves that Kepler's third law with respect to the planets applies also to these bodies; or, that the squares of their sidereal revolutions are as the cubes of their mean distances from the primary. 150 MOTIONS OF THE SATELLITES. The same law also has place with the satellites of Saturn and UTranus. 245. The Computation of the Place of a Satellite for a given time, is effected upon similar principles with that of the place of a planet. The mutual attractions of Jupiter's satellites occasion sensible perturbations of their motions, of which account must be taken when it is desired to determine their places with accuracy. 246. Relations of Mlleanl Motion and Position. Laplace has shown from the theory of gravitation, that, by reason of the mutual attractions of the first three of Jupiter's satellites, their mean motions and mean longitudes are permanently connected by the following remarkable relations. (1.) The mean motion of the first satellite, plus twice that of the third, is equal to three times that of the second. (2.) The mean longitude of the first satellite, plus twice that of the third, minus three times that of the second, is equal to 180~. It follows, from this last relation, that the longitudes of the three satellites can never be the same at the same time, and consequently that they can never be all eclipsed at once. INEQUALITY OF DAYS. 151 CHAPTER XIV. THE SUN, AND THE PHENOMENA ATTENDING ITS APPARENT MOTIONS. INEQUALITY OF DAYS.* 247. Sun's MIotion relative to the Equtator. WVe will first give a detailed description of the sun's apparent motion with respect to the equator, the phenomenon upon which the inequality of days (as well as the change of seasons, soon to be treated of) immediately depends. Let VEAQ (Fig. 66) represent the equator; VTAW (inclined to VEAQ, under the angle TOE, measured by the are TE, equal to 23~~), the ecliptic; TnX and Wn'X', the two tropics; POP', z the axis of the heavens; and - PEP'Q the meridian, and HVRA the horizon, in one of their vari- ous positions with respect to the / x other circles. About the 21st D. of March the sun is in the ver- X nabl equinox V, crossing the equator in the oblique direction VS, towards the north and east. At this time its diurnal circle is identical with the equator; and w it crosses the meridian at the FIG. 66. point E, south of the zenith a distance ZE equal to the latitude of the place. Advancing towards the east and north, it takes up the successive positions S, S', S", etc., and from day to day crosses the meridian at r, r', etc., farther and farther to the north. Its diurnal circles will be, respectively, the northern parallels of declination passing through S, S', S", etc., and continually more and more distant from the equator. The distance of the sun, and of its diurnal circle from the equator, continues to increase until about the 21st of June, when he reaches the summer solstice T. At this point he moves for a short time parallel to the equator; his declination changes but slightly for several days, and he crosses the meridian from day * The day here considered is the interval between sunrise and sunset. 102 THE SUN AND ATTENDANT PHENOMIENA. to day at nearly the same place. It is on this account, -viz., because the sun seems to stand still for a time with respect to the equator, when at the point 90~ distant from the equinox,-that this point has received the name of solstice.* The diurnal circle described by the sun is now identical with the tropic of Cancer, Tn[X; which circle is so called because it passes through T the beginning of the sign Cancer, and when the sun reaches it he is at his northern goal, and ltrons about and goes towards the south.t The sun is, also, when at the summer solstice, at its point of nearest approach to the zenith of every place whose latitude ZE exceeds the obliquity of the ecliptic TE, equal to 23'~. The distance ZT = ZE - ET - latitude - obliquity of ecliptic. Dunring the three months following the 21st of June, the sun moves over the arc TA, crossing the meridian from day to day at the successive points r", r', etc., farther and farther to the south, and arrives at the autumnal equinox A about the 23d of September, when its diurnal circle again becomes identical with the equator. It crosses the equator obliquely towards the east and south, and during' the next six months has the same motion on the south of the equator, that it has had during the previous six months on the north of the equator. It employs three months in passing over the arc AWT, during which period it crosses the meridian each day at a point farther to the south than on the preceding day. At the winter solstice, which occurs about the 22d of December, it is again moving parallel to the equator, and its diurnal circle is the same circle as the tropic of Capricorn. In three months more it passes over the arc WVV, crossing the meridian at the points " s', etc.; so that on the 21st of March it is again at the vernal equinox. 241. IExplaiaaloait oxf Ieqtuality of Days. The phenomenlon of the inequality of days obtains at all places on the earth situa'ted north or south of the equator. At all such places, the observer is in an oblique sphere; that is, the celestial equator and tile parallels of declination are oblique to the horizon. This position of the sphere is represented in Fig'. 11, p. 22, where HOR is the horizon, QOE the equator, and ncr, set, etc., parallels of declination; WOT is the ecliptic. It is also represented in Fig. 66, from which Fig. 11 differs chiefly in this, that the horizon, equator, ecliptic, and parallels of declination, which are represented as ellipses in Fig. 66, are in Fig. 11 p)rojected into right lines upon the plane of the meridian. Since the celtres of the parallels of declination are situated upon the axis of' tile heavens, which is inclined to the horizon, it is plain that these parallels, as it is represented in the Figs., and as we have before seen (25), will be divided into unequal parts, and that the disparity between the parts will be greater in proportion as tlhe parallel is more distant froom the equator; also, that to the iior'th * From Sol, the sun, and 8to, to stand. + Fhm Tp.iw, to tur:. INEQUALITY OF DAYS. 153 of the equator the greater parts will lie above the horizon, and to the south of the equator below the horizon. Now, the length of the day is measured by the portion of the parallel to the equator, described by the sun, which lies above the horizon; and it is evident, from what has just been stated, that (as it is shown by the Fig.) this increases continually from the winter solstice W to the summer solstice T, and diminishes continually from the summer solstice T to the winter solstice NV; whence it appel-ars that the daly will increase in length from the winter to the summer solstice, and diminish in length fromn the summer to the winter solstice. 249. Length of Day. As the equator is bisected by the horizon at the equinoxes, the clay and night must be each twelve hours long. But, when the sun is north of the equator, the greater part of its diurnal circle lies above the horizon, in northern latitudes; and therefore, from the vernal to the autumnal equinox, the day is, in the northern hemisphere, more than 12 hours in length. On the other hand, when the sun is south of the equator, the greater part of its circle lies below the horizon, and hence from the autumnal to the vernal equinox the day is less than 12 hours in length. In the latter interval, the nights will obviously, at corresponding periods, be of the same length as the days in the former..2a@. IE iectsa of Increase of Latitlude. The variation in the length of the day, in the course of the year, will increase with the latitude of the place; for the greater is the latitude the more oblique are the circles described by the sun to the horizon, and the greater is the disparity between the parts into which they are divided by the horizon. This will be obvious, on referring to Fig. 11, p. 22, where 11O0R, H'OR', represent the positions of the horizonls of two different places with respect to these circles; Il'OR' being the horizon for which the latitude, or the altitude of the pole, is the least. For the same reason, the days will be the longer as we proceed from the equator xnorthward, during the period that the sun is north of the equinoctial, and the shorter, during the period that he is south of this circle. 251t. Lolagest Day. At the equator, the horizon bisects all the diurnal circles (26); and, consequently, the day and night are there each 12 hours in length throughout the year. At te arcltic cir'cle the day will be 24 hours long at the time of the surnmmer solstice; for the polar distance of the sun will then be 661-o, which is the same as the latitude of the arctic circle; whence it -follows, that the diurnal circle of the sun, at this epoch, will correspond to the circle of perpetual apparition for the parallel in question. On the other anld, when the sun is at the winter solstice, the night will be 24 hours long on the arctic circle. 134 THE SUN AND ATTENDANT PHENOMENA. T/o the north of the aretic circle, the sun will remain continually above the horizon during the period, before and after the summer solstice, that his north polar distance is less than the latitude of the place, and continually below the horizon during the period, about the winter solstice, that his south polar distance is less than the latitude of the place. At the north pole, as the horizon is coincident with the equator (27), the sun will be above the horizon while passing from the vernal to the autumnal equinox, and below it while passing from the autumnal to the vernal equinox. Accordingly, at this locality there will be but one day and one night in the course of a year, and each will be of six months' duration. 252. In the.ouPtheern Heaisplhere, the circumstances of the duration of light and darkness are obviously the same as in the northern, for corresponding latitudes and corresponding declinations of the sun. 253. Problen I. Th]e latitude of the place and the declination of the sun being given, to find the times of the sun's rising and setting and the length of the day. Let HPR (Fig. 67) be the meridian, HMR the horizon, and pt/ / A\ BsD the diurnal circle described by the sun. The hour angle EPt, or its measure Et, which, converted into time, expresses the interval between the rising or setting of the sun and his passage over the meridian, is called the Semidiurnal Arc. Now, Q"- /Et = EM + Mt 90~ + Mt, FIG. 67. which gives cos Et - sin it; and we have, by Napier's first rule, sin Mt = cot tMs tan ts - tan PMIH tan EB = tan PH tan EB: whence, cos Et = -tan PH tan EAB, or, cos (semi-diurnal arc) = - tan lat. x tan dec.... (48). The semi-diurnal arc (in time) expresses the apparent time of the sun's setting, and, subtracted from 12 hours, gives the apparent time of its rising. The double of it will be the length of the day. In resolving this problem it will, in practice, generally answer to make use of the declination of the sun at noon of the given day, which may be taken from an ephemeris. Exam. 1. Let it be required to find the apparent times of the sun's rising and setting, and the length of the day at New York, at the summer solstice. TIME OF SUN'S RISING OR SETTING. 155 Log. tan lat. (40~ 42' 40").............. 9.93474 — Log. tan dec. (23~ 27' 24")............. 9.63740 Log. cos (semi-diurnal arc).............. 9.57214 - Semi-diurnal arc................... 111~ 55' 26"' Time of sun's setting................ 7h. 27m. 42s. Time of sun's rising................. 4 32 18 Length of day....................14 55 24 Exam. 2. WVhat are the lengths of the longest and shortest days at Boston; the latitude of that place being 42~ 21' 15" N? Ans. 15h. 6m. 25s., and 8h. 53m. 35s. Exanm. 3. At what hours (apparent time) did the sun rise and set on May 1, 1866, at Charleston; the latitude of Charleston being 32~ 47', and the declination of the sun being 15~ 9' 30" N? Ans. Time of rising, 5h. 19m. 48s.; time of setting, 6h. 40m. 12s. 254. Problemn nIl. To find the time of the sun's cpparent rising or setting, the latitude of the place and the declination of the sun being given. At the time of apparent rising or setting, the sun, as seen from the centre ot the earth, will be below the horizon a distance sS (Fig. 67) equal to the refraction minus the parallax. The mean difference of these quantities is 34' 45" (according to Bessel). Let it be denoted by R. Now, to find the hour angle ZPS (= P), the triangle ZPS gives (see Appendix), ZP + PS + ZS co-lat. + co-dec. + (90~ + R) 49 2 -2...(49) 2 2 and sin" P - sin (k-ZP) sin (k- PS) and sin2~P -- sin ZP sin PS' siln (k — co-lat.) sin (k - co-dec.) o sin (co-lat.) sin (co-dec.) The value of P, in time, will be the interval between apparent noon and the time of the apparent rising or setting of the centre of the sun's disc; from which the apparent times of the apparent rising and setting are readily obtained. To obtain the mean times, these results must be corrected for the equation of time. If the time of the rising or setting of the upper limb of the sun, instead of its centre, be required, we must take for R 34' 45" + sun's semi-diameter, or 50' 47". Unless very accurate results are desired, it will be sufficient to take the declinations of the sun at 6 o'clock in the morning and evening. A more accurate calculation may be made by first computing the times of true rising and setting from equation (48), and making use of the declinations answering to these times. .156t THE SUN AND ATTENDANT PHENO.IENA. TWILIGHT. 255. Explanation. When the sun has descended below the horizon, its rays still continue to fall upon a certain portion of the body of air that lies above it, and are thence radiantly reflected down to the earth, so as to occasion a certain degree of light; which gradually diminishes as the sun descends fnlrther below the horizon, and the portion of air posited above the horizon, that is directly illuminated, becomes less. The same effect, though in a reverse order, takes place in the morning, previous to the sun's rising. The light thus produced is called the Cre2pusculttz or Twilight. The explanation of twilight will be better understood on examining Fig. 6S, where AON represents a pork FIG. 68. tion of the earth's surface, IHR the surface of the atmosphere above it, and kmS a line drawn touching the earth and passing tlhrough the sun. The unshaded portion, keR, of the body of air which lies above the plane of the horizon, HIOR, is still illuminated by the sun, and shines down, by reflection, upon the station of the observer at O. As the sun descends, this will decrease, until finally, when the sun is in the direction RNS', it will illuminate directly none of that part of the atmosphere which lies above the horizon, and twilight will be theoretically at an end. It is assumed that, when the sun has reached this position, in whichl no portion of air that lies above the horizon is directly illuminated, faint stars will become visible over the western horizon; and thus that the end of evening twilight is definitely nlarked by the appearance of such stars. In like manner, morn ing twilight is astronomically defined as beginning when faint stars situated in the vicinity of the eastern horizon begin to disappear. It has been ascertained from numerous observations that, at the beginning of the morning and end of the evening twilight, as thus defined, the sun is about 18~ below the horizon. TWILIGHT. 157 256. Ajpproixinate Determniautiou of Meiglat of Atuosphere. As we have just seen, at the end of evening twilight, the angle TRS' (Fig. 68) is equal to 18~; H/kR being the limnit of that portion of the atmosphere which is capable of reflecting a sensible amount of light to the eye, in the direction RO. Now, if the vertical lines at O, m, and N, be produced to the centre of the earth, C, we shall have the angle OCN equal to TRS', or 18~, and therefore OCR equal to 9~. If, then, we denote the radius of the earth Cmn by R, we shall have, height of atmos. — mR -CCR-Cm -R sec 9~- R - R (sec 9~ 1). Making the calculation, we obtain for the height of the atmosphere, 49.3 miles. It is plain that the actual height of the atmosphere must be greater than this, since a stratum of air of considerable thickness may lie above kR, and yet not have sufficient density to send a sensible amount of reflected light to the eye at 0. through the body of air lying on the line RO. 257. Problemn. The latitude of the place and the sun's declinanation being giuen, to find the time of the beginning or end of twilight. The zenith distance of the sun, at the beginning of morning or end of evening twilight, is 90~ + 180; we may therefore solve this problem by means of equations (49) and (50), taking R- 18~. If the time of the commencement of morning twilight be subtracted from the time of sunrise, the remainder will be the duration of twilight. #5S. Variable Duration of Twilight. The duration of twilight varies with the latitude of the place, and with the time of the year. In the northern hemisphere, the summer are longer than the winter twilights, and the longest twilights take place at the summer solstice; while the shortest occur when the sun has a small southern declination, different for each latitude. The summer twilights increase in length from the equator northward. In the southern hemisphere, the phenomena are similar for corresponding declinations of the sun. These facts are consequences of the different situations with respect to the horizon of the centres of the diurnal circles described by the sun in the course of the year, and of the different sizes of these circles. To make this evident, let us conceive a circle to be traced in the heavens parallel to the horizon, and at the distance of 180 below it; this is called the Crepusculum Circle. The duration of twilight will depend upon the number of degrees in the arc of the diurnal circle of the sun, comprised between the horizon and the crepusculum circle, which, for the sake of brevity, we will call the arc of twilight: and this will vary from the two causes just mentioned. For, let hkr (Fig. 69) represent the equator, and h'k'r' a diurnal circle described by the sun when north of the equator; and let hr, st, and h'r', s't', be the intersections of the equator and diurnal circle, respectively, with the planes of the horizon and the crepusculum circle. When the sun is in the equator, the arc of twilight is hs, and when he is on the parallel of declination h';'r' it is h's'. Draw the chords hs, h's', san, and the radii, cs, cs', cr', cn, cp. The angle r"h's' is the half of r'cs', and the angle jmn is the half of pen; but r'cs' is less than pcn, and therefore r'h's' is less than pmnn. Again, chs is the half of rcs, and 158 THE SUN AND ATTENDANT PHENOMENA. therefore greater than pmn, the half of the less angle pcn. Whence it appears that the chord h's' is more oblique to the horizon, and therefore greater than the chord man, and this more oblique and greater than the chord hs. It follows, therefore, that the are h's' is greater, and contains a greater number of degrees than the arc inn, and that this arc is greater than As. Thus, as the sun receder from the equator towards the nort',, the arc of twilight, and therefore the duration of twilight, increases from two causes, viz.: 1st. The increase rt~~~~~~ { c 1 \ in the distance of the line of intersec-...... -;:......... | tion of the horizon with the diurnal t circle from the centre of the circle; and, 2d. The diminution in the size of the circle. The change will manifestly rn FIG. 76. rounding regions that are less elevated are involved in darkness. The disc is also diversified with spots of different shapes and different degrees of brightness. The brighter parts are supposed to be elevated land, and the dark to be plains, and valleys, or cavities. 316X. Liunar nouatains. The number of the lunar mountains is very great. Many of them, by their form and grouping, furnish decided indications of a volcanic origin. From measurements nmade with the micrometer of the lengths of their shadows, or of the distance of their summits when first illuminated, from the adjacent boundary of the disc, the heights of a number of' the lunar mountains have been computed. According to Herschel, the altitude of the highest is only about 13 English miles. But Schroeter, of Lilienthal, a distinguished Selenographist, makes the elevation of soame of the lunar mountains to exceed 5 miles; and the more recent measurements of MM. Baer and Middler, of Berlin, lead to similar results. 313. There are e no Seas, nor other bodies of water, upon the surface of the moon. Certain dark and apparently level parts of the moon were for some time supposed to be extended sheets of water, and, under this idea, were named by Ilevelius fi/fare Imbriurn, Mare Crisia.m, etc.: but it appears that when the boundary of light and darkness falls upon these supposed seas, it is still more or less indented at some points and salient at others, instead of being, as it should be, one continuous regular curve; 188 THE MOON AND ITS PHENOMENA. besides, when these dark spots are viewed with good telescopes, they are found to contain a number of cavities, whose shadows are distinctly perceived falling within themn. The spots in question are therefore to be regarded as extensive plains diversified by moderate elevations and depressions. The entire absence of water also from the farther hemisphere of the moon may be inferred from the fact that the moon.'s face is never obscured by clouds or mists. 31 1. Lumaar AtnlosplIere. It has longbeen aquestion amongo Astronomers, whether the moon has an atmosphere. it is asserted, that, if it has any, it must be exceedingly rare, or very limited in its extent, since it does not sensibly diminish or refract the light of a star seen in contact with the moon's limb; for when a star experiences an occultation by reason of the interposition of the moon between it and the eye of the observer, it does not disappear or undergo any diminution of lustre until the body of the moon reaches it; and the duration of the occultation is as it is computed, without making any allowance for the refraction of a lunar atmosphere. But it is maintained, on the other hand, that these facts, if allowed, are not opposed to the supposition of the existence of an atmosphere of a few miles only in height; and that certain phenomena which have been observed afford indubitable evidence of the presence of a certain lima ited body of air upon the moon's surface. Thus, the celebrated Schroeter, in the course of some delicate observations made upon the crescent moon, perceived a faint grayish light extending from the horns of the crescent a certain distance into the dark part of the moon's face. This he conceived to be the moon's twilight, and hence inferred the existence of a lunar atmosphere. From. the measurements which lie made of the extent of this light he calculated the height of that portion of the atmosphere which was capable of affecting the light of a star to be about one mile. Again. in total eclipses of the sun, occasioned by the interposition of the moon, the dark body of the moon has been seen terminated by a luminous ring, which was at first most distinct at the part where the sun was last seen, and afterwards at the part where the first ray darted from the sun. This is supposed to have been a lunar twilight. A similar phenomenon was observed in the annular eclipse of 1836, just before the conmpletion of the ring, at the point where the junction took place. DESCRIPTION OF THE MOON'S SURFACE. 315. Getueral Topographical Features. The surface of the moon, like that of the earth, presents the two general varieties of level and mountainous districts; but it differs from the earth's surface in having no seas or other bodies of water upon it, and in being more rugged and mountainous. The comparatively level regions occupy somewhat more than one-third of the nearer half of the moon's surface. These are, in general, the darker parts of the disc. The lunar plains vary in extent from 40 or 50 miles to 700 miles in diameter. 3 6. lThe iaIoHnt;l iaous Fornaiattionas of the other parts of the surface offer three marked varieties, viz.: (1.) Insulated cMfouetains, which rise from plains nearly level, and which may be supposed to present an appearance somewhat similar to Mlount Etna or the Peak of Teneriffe. The shadows of these mountains, in certain phases of the moon, are as distinctly perceived as the shadow of an upright staff when placed opposite to the sun.* The perpendicular altitudes of some of them, as determined from the lengths of their shadows, are between four and five miles. Insulated mountains frequently occur in the centres of circular plains. They are then called Central Mountains. (2.) Ranges of Mountains, extending in length two or three hundred miles. These ranges bear a distinct resemblance to our Alps, Apennines, and Andes; but they are much less in extent, and do not form a very prominent feature of the lunar * Dick's Celestial Scenery, p. 256. DESCRIPTION OF THE MOON'S SURFACE. 1 9 surface. Some of them appear very rugged and precipitous, and the highest ranges are, in some places, above four miles in perpendicular altitude. In some instances they run nearly in a straight line from northeast to southwest, as in the range called the Apennines; in other cases they assume the form of a semicircle or a crescent.*: (3.) Circular Formations. The general prevalence of this remarkable class of mountainous formations is the great characteristic feature of the topography of the moon's surface. It is subdivided by late selenographists into three orders, viz.: WValled Plains, whose diameter varies from one hundred and twenty to forty or fifty miles, IKing Mountains, the diameter of which descends to ten miles; and Craters, which are still smaller. The term crater is sometimes extended to all the varieties of circular formations. They are also sometimes called Caverns, because their enclosed plains or bottoms are sunk considerably below the general level of the moon's surface. The different orders of the circular formations differ essentially from each other only in size. The principal features of their constitution are, for the most part, the same, and they present similar varieties. Sometimes terraces are seen going round the whole ring. At other times ranges of concentric mountains encircle the inner foot of the wall, leaving intermediate valleys. Again, we have a few ridges of low mountains stretching through the circle contained by the wall, but oftener isolated conical peaks start up, and very frequently small craters having on an inferior scale every attribute of the large one.t The smaller craters, however, offer some characteristic peculiarities. Most of them are without a flat bottom, and have the appearance of a hollow inverted cone with the sides tapering towards the centre. Some have no perceptible outer edge, their margin being on a level with the surrounding regions: these are called Pits. The bounding ridge of the lunar craters or caverns is much more precipitous within than without; and the internal depth of the crater is always much lower than the general surface of the moon. The depth varies from one-third of a mile to three miles and a half. These curious circular formations oocur at almost every part of the surface, but are most abundant in the southwestern regions. It is the strong reflection of their mountainous ridges which gives to that part of the moon's surface its superior lustre. The smaller craters occupy nearly two-fifths of the moon's visible surface. * Dick's Celestial Scenery, p. 257. j Nichol's Phenomena of the Solar System, p. 167. 190 ECLIPSES OF TIIE SUN AND MOON. CH.APTER XVI. ECLIPSES OF THE SUN AND MOON.-OCCULTATIONS OF THE FIXED STARS. 317. AN eclipse of a heavenly body is a deprivation of its light, occasioned by the interposition of;_- - some opake body between it and the eye, or between it and the sun. -__ -______ Eclipses are divided, with respect ______- - to the objects eclipsed, into eclise. of the sun, of the moon, and of the satellites; and, with respect to circumstanc's, into totztl, partial, an_'' - zular, and central. A total eclipse is one in whiclh the whole disc of the luminary is darkened; a par-w < tial one is when only a part of the disc is darkene(l. In an annular eclipse the wlhole is darkened, except a ring or annlulus, which appears round the dark part like I \, an illuminated border; the definition of a central eclipse will be given in another place. ECLIPSES OF TIlE MOON. 31a. An eclipse of the moon is o(casioned by tin intcrpositioll of the bo(dy of the ea:rtih directly between tile sun and moon, alnd thus intercepting the light of tlhe sun; or the moon is eclipsed when. "// 1it passes through part of the sha.dow of the earth, as projected fronthe sun. Hence it is obvious that lunar eclipses can happen only at FIG. 77. the time of full moon, for it is then only that the earth can be between the moon and the sun. 319. Earth's Shadlow. Since the sun is much larger than CIRCUMSTANCES UNDER WHICH AN ECLIPSE OCCURS. 191 thie earth, the slhadow of the earth must have the form of a cone, the length of wlhich will depend on the relative magnitudes of the two bodies and their distance from each other. Let the circles AGB, agy (Fig. 77), be sections of the sun and earth by a plane passing through their centres S and E; Aca, Ib, tangents to these circles on the sarme side, and Ad, 1Bc, tangents on different sides. The trianrgular space aCb will be a section of the earth's shadow or Umbra, as it is sometimes called. The line EC is called the Ax;s ot /he Shadow. If we suppose the line cp to revolve about EC, and f)rm the surface of the frustum of la cone, of which pcdq/ is a section, the space included within that surface and exterior to the umbra, is called the Penumnnbra. It is plain that points situated within the umbra will receive no light from the sun; and that points situated within the penumbra will receive light frorn a portion of the sun's disc, and from a greater portion the more distant they are fromIl the umbra. 32@o. To field lhe Lenlth iof the Earth'.s Ilaadowv. Let L — the length of the shadow R - the radius of the earth; - the sun's apparent selni-dinameter, and p - sun's parallax.''he right-angled triangle EcLa (Fig. 7 7) gives - XEC s ECa silln ECa Ea -R; and ECc SEA -EAC -p; whence, L - 11_.. (54). Si 1 (1 -12') As the angle (( —p) is only about 16', it will differ but little from its sine, and therefore, L = R _ (nearly); oS-P or, if S and p be expressed in seconds, 206i264".8 L -64".8 (nearly).. (.(55). o 319 The shadow will obviously be the shortest when the sun is nearest to the earth. -We then have S - 16' 18", and p - 9"', which gives L -213 R. The grezatest (listance of the moon is 65R. It appears, then, that the earth's shadow ralways extends to snore than three times the distrance of the moon. 321. Careumau.taauces un~der xl;hieah 3a~ EcDi-pse occmn'a. Let khMh be a circular arc, describedl ahout E thle centre of t!he earth, and with a radius equal to thle distance between the centres of the earth ald moon at the time of opposiition. The angle IMEfm, the apparent semi-diameter of a section of the earth's shadow, made at the distance of the moon's centre, is called the Semi-diacmeter of the ~Earth's S/hadow. And the angle MEh, the apparent semi-diameter of a section of the penumbra, at the same distance, is called the Semi di'ameter of the Penumbra. 192 ECLIPSES OF THE SUN AND TMOO'N. Were the plane of the moon's orbit coincident -with the plane of the ecliptic, there would be a lunar eclipse at every full moon; but, as it is inclined to it, an eclipse can happen only when the full moon takes place either in one of the nodes of the moon's 7'" NX — -M orbit, or so near it that the iroon's latitude does not exceed the sum of:i! \ "',1 c the apparent semi-diameters of the moon and of the earth's shadow. This will be better understood on \\~ \referring to Fig. 78, in which N'C represents a portion of the ecliptic, and N'M a portion of the moon's orbit, N' the descending node, E the earth, ES", three different directions of the sun, s, s', ", sections of the earth's shadow in the three several positions corresponding to these directions of the sun, E and m, n', mn", the moon in opposition. It will be seen that the moon FIG. s8. will not pass into the earth's shadow unless at the time of opposition it is nearer to the node than the point m', where the latitude m's' is equal to the sum of the semi-diameters of the moon and shadow. s'22. Calcnlation of $ci- n=diamaeier of Slhadow. To determine the distance from the node, beyond which there can be no eclipse, we must ascertain the semi-diameter of the earth's shadow. Let this be denoted by A, arid let P- the moon's parallax. parallax. MEm = Ema - ECrn (Fig. 77); but Ema = P and E =m —p - (320); therefore, MEm = A P + jp -..... (56). The semi-diameter of the shadow is the least when the moon is at its greatest and the sun is at its least distance, or when P has its minimum and 8 its maximum value. In these positions of the moon and sun, P = 52' 40", - 16' 18", and 1p = 9". Substituting, we obtain for the least semi-diameter of the earth's shadow 36' 31", and for its least diameter 1~ 13' 2". The greatest apparent diameter of the moon is 33' 32". Whence it (Rppears that the diameter of the earlths shcdclow is always 7more tlhtne twice the diameter of the moon. The means of the greatest and least values of P and c are, respectively, 57' 11" and 16' 2"; which gives for the mean semiidiameter of the earth's shadow, 41' 1S". 323. Lunar Ecliptic Limits. If to P +p -,, the semidiameter of the earth's shadow, we add d, the semi-diameter of PARTICULAR FACTS. 193 the moon, the sum P +p + d — will give the greatest latitude of the moon in opposition, at which an eclipse can happen. It is easy for a given value of P + p + d - A, and a given inclination of the moon's orbit, to determine within what distance from the node the moon must be in order that an eclipse may take place. By taking the least and greatest inclinations of the orbit, the greatest and least values of P +p + d —-, and also taking into view the inequalities in the motions of the sun and moon, it has been found, that when at the time of mean full moon the difference of the mean longitudes of the moon and node exceeds 13~ 21', there cannot be an eclipse; but when this difference is less than 7~ 47' there must be one. Between 7~ 47' and 130 21' the happening of the eclipse is doubtful. These numbers are called the Lunar Eclioptic Limits. To determine att what fitll moons in the course of any one year there will be an eclipse, find the time of each mean full moon (301); and for each of the times obtained find the mean longitude of the sun, and also of the moon's node, and compare the difference of these with the lunar ecliptic limits. Should, however, the difference in any instance fall between the two limits, farther calculation will be necessary. This problem may be solved more expeditiously by means of tables of the sun's mean motion with respect to the moon's node. (See Prob. XXVIII.) 324. Central Eclipse. The magnitude and duration of an eclipse depend upon the proximity of the moon to the node at the time of opposition. In order that the centre of the moon may be on the same right line with the centres of the sun and earth, or, in technical language, that a central eclipse may happen, the opposition must take place precisely in the node. A strictly central eclipse, therefore, seldom, if ever, occurs. As the mean semi-diameter of the earth's shadow is 41' 18", the mean semidiameter of the moon 15' 35", and the mean hourly motion of the moon with respect to the sun 30' 29", the mean duration of a central eclipse would be about 31h. 325. Particular Facts. Since the moon moves -from west to east, an eclipse of the moon must commence on the eastern limb, and end on the western. In the preceding investigations, we have supposed the cone of the earth's shadow to be formed by lines drawn from the edge of the sun, and touching the earth's surface. This, probably, is not the exact case of nature; for the duration of the eclipse, a:nd thus the apparent diameter of the earth's shadow, is fulnd by observation to be somewhat greater than would result from this supposition. This circumstance is accounted for by supposin,(r those solar rays that, from their direction, would glance by and raze the earth's surface, to be stopped and absorbed by the lower strata of the atmosphere. In such a case the conical boundary 13 194 ECLIPSES OF THE SUN AND MOON. of the earth's shadow would be formed by certain rays exterior to the former, and would be larnger. The moon in approaching and receding from the earth's total shadow, or umbra, passes through the penumbra, and thus its light, instead( of being extinguished and recovered suddenly, experiences at the beginning, of the eclipse a gradual diminution, and at the end a gradual increase. On this account the times of the beginning and end of the eclipse cannot be noted with precision, and in consequence astronomers differ as to the amount -)f the increase in the size of the earth's shadow from the cause above menltioned. It is the practice, however, in computing an eclipse of the moon, to increase the semi-diameter of the shadow by at 4- part; or, which amounts to the same, to add as many secondls as the semi-d(iameter contains minutes It is remarked in total eclipses of the moon, that the moon is not wholly invisible, but appears with a dull reddish light. This phenomenon is doubtless another effect of the earth's atmosphllere, though of a totally different nature from the preceding. Certain of the sun's rays, instead of being stopped and absorbed, are bent from their rectilinear course by the refracting power of the atmosphere, so as to form a cone of faint light, interior to that cone which has been mathematically described as the earth's shadow, which falling upon the moon renders it visible. As an eclipse of the moon is occasioned by a real loss of its light, it must begin and end at the same instant, and present precisely the same appearance to every spectator who sees the moon above his horizon during the eclipse. It will be shown that the case is different with eclipses of the sun. CALCULATION OF AN ECLIPSE OF THE MOON. 326. The apparent distance of the centre of the moon from the axis of the earth's shadow, and the arcs passed over by the centre of the moon and the axis of the shadow during an eclipse of the moon, being necessarily small, they may, without material error, be considered as right lines.;Ve may also consider the apparent motion of the sun in longitude, and the motions of the moon in longitude and latitude, as unitbrm during the eclipse. These suppositions being made, the calculation of the circum-,stances of an eclipse of the moon is very simple. 327. PRelative Orbit. Let NF (Fig. 79) be a part of the zecliptic, N the moon's ascending node, NL a part of the moon's orbit, C the centre of a section of the earth's shadow at the moon, CK perpendicular to NF a circle of latitude, and C' the centre of the moon at the instant of opposition: then CC', which is the latitude of the moon in opposition, is the distance of the centres CALCULATION OF AN ECLIPSE OF THE MOON. 195 of the shadow and moon at that time. The moon and shadow both have a motion, and in the same direction, as from N towards F and L. It is the practice, however, to regard the shadow as stationary, and to attribute to the moon a motion equal FiG. 7 9. to the relative motion of the moon and shadow. The orbit that would be described by.the moon's centre if it had such a motion, is called the Relative Orbit of the moon. Inasmuch as the circumstances of the eclipse depend altogether upon the relative motion of the moon and shadow, this mode of proceeding is obviously allowable. As the shadow has no motion in latitude, the relative motion of the moon and shadow in latitude will be equal to the moon's actual motion in latitude: and since the centre of the earth's shadow moves in the plane of the ecliptic at the same rate as the sun, the relative motion of the moon and shadow in longitude will be equal to the difference between the motions of the sun and moon in longitude. We obtain, therefore, the relative position of the centres of the moon and shadow at any interval t, following opposition, by ly1ing off Cm equal to the difference of the motions of the sun and moon in longitude in this interval, through m drawing mM perpendicular to NF, and cutting off mM equal to the latitude at opposition plus the motion in latitude in the interval t: M will be the position of the moon's centre in the relative orbit, the centre of the shadow being supposed to be stationary at C. As the motion of the sun in longitude, and of the moon in longitude and latitude, are considered uniform, the ratio of C'm' (- Cm, the difference between the motions of the sun and moon in longitude) to Mm' the moon's motion in latitude, is the same, whatever may be the length of the interval considered. It follows, therefore, that the relative orbit of the moon N'C'AM is a right line. The relative orbit passes through C', the place of the moon's centre at opposition: its position will therefore be known, if its inclination to the ecliptic be found. Now we have Mme' moon's motion in latitude tan inclina. = C'm' moon's mot. in long.- sun's mot. in long. ~196 ECLIPSES OF THE SUN AND AMOON. 32S. Requisite Data. The following data are requisite in the calculation of the circumstances of a lunar eclipse: T = time of opposition. M = moon's hourly motion in longitude. n = moon's hourly motion in latitude. m = sun's hourly motion in longitude. A = moon's latitude at opposition. d = moon's semi-diameter. -- sun's semi-diameter. P = moon's horizontal parallax. p = sun's horizontal parallax. s = semi-diameter of the earth's shadow. I = inclination of relative orbit. h = moon's hourly motion on relative orbit. T, M, n, Mn, A, d, J, P, and p, are derived from Tables of the sun and moon. (See Problems IX and XIV.) The quantities s, I, and h, may be determined from these: s = + p - -(P + p —) (322 and 325)....(57); tang I n (327)....(58). M - n The triangle C'Mm' gives M- = C'm' or, h M-....(59). cos Md:C'm' cos I 3'29. Process of Calculation. The above quantities being supposed to be known, let N'CF (Fig. 80) represent the ecliptic, and C the stationary centre of the earth's shadow. Let CC' -, and let N'C'L' represent the relative orbit of the FIG. 80. moon. We here suppose the moon to be north of the ecliptic at the time of opposition and near its ascending node; when it is south of the ecliptic A is to be laid off below NOCF, and when it is approaching either node, the relative orbit is inclined to the right. Let the circle KEFK'R, described about the centre C, represent the section of the earth's shadow at the moon; and let f, f', and g, g', be the respective places of the moon's centre, at the beginning and end of the eclipse, and at the beginning and end of the total eclipse. Cf= Cf' = s + d, and Cg = Cg'= s -- d. Draw CM perpendicular to N'C'L', and M will represent the place of the moon's centre when nearest the centre of the shadow: it will also be its place at the middle of the eclipse; for since Cf= Cf', and CM is perpendicular to N'Cf' Mf= Mf'. 3Middle of the eclipse. The time of opposition being known, that of the middle of the eclipse will become known when we have found the interval (x) employed by the moon in passing from M to C'. Now CALCULATION OF AN ECLIPSE OF THE MOON. 197 MC' (expressed in parts of an hour) x =MC; and in the right-angled triangle CC'M we have CC' =, and < C'CM = < C'ON' = I, and therefore MC' = A sin I; whence, by substitution, X sin I X sin I X sin I cos I X- h = -[m(equ. 59)=- M —m cos I 3600.9. cos I or (expressed in seconds), x = X3600s. c sin I.. (60). Hence, if M = time of middle, we have 3600s. cos I M T:F T:F.Xsin I....(61). M-m It is obvious that the upper sign is to be used when the latitude is increasing, and the lower sign when it is decreasing. The distance of the centre of the moon from the centre of the shadow at the middle of the eclipse, = CMI = CC' cos C'CM = A cos I....(62). Beginning and end of the eclipse. Let any point I of the relative orbit be the place of the moon's centre at the time of any given phase of the eclipse. Let t = the interval of time between the given phase and the middle; and k = Cl, the distance between the centres of the moon and shadow. In the interval t the moon's centre will pass over the distance Ml; hence Ml MI cos I h i -- but, MI = 4 C, 2 - = -4/ A2 COS2 I (equa. 62), and therefore t M — COS I M[ — q6 z 1// __ 2 cos I or (in seconds), t- 3600s.I c o I) (D-A cos I)..(63). Let T' denote the time of the supposed phase of the eclipse, and M3 the time of the middle; and we shall have T'= M + t, or, T' = — M —, according as the phase follows or precedes the middle. Now, at the beginning and end of the eclipse, we have, kI =.Cf or Cf' = s + d: substituting in equation (63) we obtain - 3600s. cos I M-rm 4/(s + d + A cos I) (s + d - A cos I))....(64). t' being found, the time of the beginning (B), and the time of the end (E), result from the equations B=M — t', E - M + t'. Beginning and end of the total eclipse. At the beginning and end of the total eclipse, k = Cg = Cg' = s- d; whence, by equation (63),,, _ 3600s. cos I - (5): -m 6/(s- d + A coS I) (s — d-A cos I)...(65 and, denoting the time of the beginning by B' and the time of the end by E', we have B' = M - t", E'= M +t". Quantity of the eclipse. In a partial eclipse of the moon the magnitude or quantity of the eclipse is measured by the relative portion of that diameter of the moon, which, if produced, would pass through the centre of the earth's shadow, that is involved in the shadow. The whole diameter is divided into twelve equal parts, called Digits, and the quantity is expressed by the number of digits and fractions 198 ECLIPSES OF THE SUN AND MOON. of a digit in the part immersed. When the moon passes entirely within the sha. dow, as in a total eclipse, the quantity of the eclipse is expressed by the numbE r of digits contained in the part of the same diameter prolonged outward, which is comprised between the edge of the shadow and the inner edge of the moon. Thus the number of digits contained in SN (Fig. 80) expresses the quantity of the eclipse represented in the figure. Hence, if Q = the quantity of the eclipse, we shall have NS _12NS 12 (NM + MS)_ 12 (NM + CS - CGM) _ 12-NV NV NV NV 12 (d + s- X cos I) 2d 6 (s + d-X cos I).... or, dQ - — _ — —....(66). If X cos I exceeds (s + d) there will be no eclipse. If it is intermediate between (s + d) and (s - d) there will be a partial eclipse; and if it is less than (s- d) the eclipse will be total. CONSTRUCTION OF AN ECLIPSE OF THE MOON. 330. The times of the different phases of an eclipse of the moon may easily be determined by a geometrical construction, within a minute or two of the truth. Draw a. right line N'F (Fig. 81) to represent the ecliptic; and assume upon it any point FIG. 81. C, for the position of the centre of the earth's shadow, at the time of opposition. Then, having fixed upon a' scale of equal parts, lay off CR - M - m, the difference of the hourly motions of the sun and moon in longitude; and draw the perpendiculars CC' - - the moon's latitude in opposition, and RL' i i n the moon's latitude an hour after opposition. The right line C'L', drawn through C' and L', will represent the moon's relative orbit. It should be observed, that if the latitudes are south they must be laid off below N'F, and that N'C'L' will be inclined to the righlt when the latitude is decreasing. With a radius CE = ECLIPSES OF THE SUN. 199 s (equation 56) describe the circle EKFK', which will represent the section of the earth's shadow. With a radius s + d, and another radius s -d, describe about the centre C arcs intersecting N'L' in f f', and g, g'; f andf' will be the places of the moon's centre at the beginning and end of the eclipse, and g and g' the places at the beginning and end of the total eclipse. From the point C let fall upon N'C'L' the perpendicular CM; and NM will be the place of the moon's centre at the middle of the eclipse. To render the construction explicit, let us suppose the time of opposition to be 7h. 23m. 15s. At this time the moon's centre will be at C'. To find its place at 7h., state the proportion, 60m.: 23m. 15s.:: moon's hourly motion on the relative orbit: a fourth term. This fourth term will be the distance of the moon's centre from the point C' at 7 o'clock; and if it be taken in the dividers and laid off on the relative orbit from C' backward to the point 7, it will give the moon's place at that hour. This being found, take in the dividers the mooon's hourly motion on the relative orbit, and lay it off repeatedly, both forward and backward, from the point 7, and the points marked off,, 9, 10, 6, 5, will be the moon's places at those hours respectively. Now, the object being to find the times at which the moon's centre is at the points/ff, g, g', and M, let the hour spaces thus found be divided into quarters, and these subdivided into 5-minute or minute spaces, and the times answering to the points of division that fall nearest to these points, will be within a minute or so of the times in question. For example, the pointf falls between 9 and 10, and thus the end of the eclipse will occur somewhere between 9 and 10 o'clock. To find the number of minutes after 9 at which it takes place, we have only to divide the space from 9 to 10 into four equal parts, or 15-minute spaces, subdivide the part which containsf' into three equal parts, or 5-minute spaces, and again that one of these smaller parts within which f' lies, into five equal parts or minute spaces. ECLIPSES OF THE SUN. 331. Lutminons Frustum and C(one. An eclipse of the sun is caused by the interposition of the moon between the sun and earth; whereby the whole, or part of the sun's light, is pre-, vented from falling upon certain parts of the earth's surface. Let AGB and agb (Fig. 82) be sections of the sun an(l earth by a plane passing through their centres S and E; Aa, Bb, tangents to the circles AGB and agb on the same side; and Ad, Be, tangents to the same on opposite sides. The figure AabB will be a section through the axis, of a frustum of a cone torrned by rays tangent to the sun and earth on the same side, and the triangular space Fcd will be a section of a cone formed by rays 290 ECLIPSES OF THE SUN AND MOON. tangent on opposite sides. An eclipse of the sun will take place somewhere upon the earth's surface, whenever the moon comes within the frustum AabB, and a total or an annular eclipse whenever it comes within the cone Fed. FIG. 82. 332. ~emi-dianaeters of ]Fruwituln arid Cone. Let;mmn'MS/ (Fig 82) be a circular arc described about the centre E, and with a radius equal to the distance between the centres of the moon and earth at the time of conjunction. The angle mES is the apparent semi-diameter of a section of the frustum, and m/ES the apparent semi-diameter of a section of the cone, at the distance of the moon. To find expressions for these semi-diameters in terms of determinrate quantities, let the first be denoted by A, and the second by A'; and let P the parallax of the moon, p = the parallax of the sun, and A = the semi-diameter of the sun. Then we have mES - A - mEA + AES - Ema -EAm + AES; or, A r -p - S....A-(67): and rn'ES - m'EB - BES Em'c -EBrn' - BES; or, A' - P -p -- 5....(68S). Taking the mean values of P, p, and S (322), we find for the mean value of A, 1~ 13' 3"; and for the mean value of A', 41' 1 ".,333. (Circ iumstlanuces of ilooDil's Position ina Solar lcegilmies. As the plane of the moon's orbit is not coincident with the plane of the ecliptic, an eclipse of the sun can happen only when conjunction or new moon takes place in one of the inocdes of the moon's orbit, or so near it that the moon's latitude does not exceed the sumr of the semi-diamLneters of the moon allnd luminous firustum at the moon's orbit. This may be illustrated by means of Fig. 78, already used for a lunar eclipse, by supposing the sun to be in the directions Es, Es', Es", nnd tihat s, s', s", are sections of the luminous frustum corresponding to these directions of the sun; also that n, m', mn", represent the m31-oon in the correspondilng positions of conjunction. Thus, de NUMIBER OF ECLIPSES IN A YEAR. 201 noting the moon's semi-diameter by d, and the greatest latitude of the moon in conjunction, at which an eclipse can take place, by L, we have L -P-p + + d.... (69). For a total ecl}pse, the greatest latitude will be equal to the sum of the semi-diameters of the moon and the luminous cone. Hence, denoting it by L', L' -P -p- +d....(70). In order that an annular eclipse may take place, the apparent semi-diameter of the moon must be less than that of the sun, and the moon must come at conjunction entirely within the luminous frustum. Whence, if L"= the maximum latitude at which an annular eclipse is possible, we have -" = P-p + 6 d....(71). In the same manner as in the case of an eclipse of the moon, it has been found that when at the time of mean new moon the difference between the mean longitude of the sun or moon and that of the node, exceeds 19~ 44', there cannot be an eclipse of the sun; but when the difference is less than 13~ 33', there must be one. These numbers are called the Solar Ecliptic Limits. 334. Predictioia of Eclipses: —Period. In order to discover at what new moons in the course of a year an eclipse of the sun will happen, with its approximate time, we have only to find the mean longitudes of the sun and node at each mean new moon throughout the year (301), and take the difference of the longitudes and compare it with the solar ecliptic limits. (For a more direct method of solving this problem, see Prob. XXVIII.) Eclipses both of the sun and moon recur in nearly the same order and at the same intervals at the expiration of a period of 223 lunations, or 18 years of 365 days, and 15 days;* which for this reason is called the Period of the Eclipses. For, the time of a revolution of the sun with respect to the moon's node is;346.619851d., and the time of a synodic revolution of the moon is 29.53058S7d. These numbers are very nearly in the ratio of 223 to 19. Thus, in a period of 223 lunations, the sun will have returned 19 times to the same position with respect to the moon's node, and at the expiration of the period will be in the same position with respect to the moon and node as at its commencement. The eclipses which occur during one such period being noted, subsequent eclipses are easily predicted. This period was known to the Chaldeans and Egyptians, by whom it was called Saros. 335. /Miin a er of Eclipses igt a Year. As the solar ecliptic limits are more extended than the lunar, eclipses of the sun must occur more frequently than eclipses of the moon. l More exactly, 18 years (of 365 days) plus 15d. 7h. 42m. 29s. 202 ECLIPSES OF THE SUN AND MOON. As to the number of eclipses of both luminaries, there cannot be fewer than two nor more than seven in one year. The most usual number is four, and it is rare to have more than six. When there are seven eclipses in a year, five are of the sun and two of the moon; and when but two, both are of the sun. The reason is obvious. The sun passes by both nodes of the moon's orbit but once in a year, unless it passes by one of them in the beginning of the vear, in which case it will pass by the same again a little before the end of the year, as it returns to the same node in a period of 346 days. Now, if the sun be at a little less distance than 19~ 44' fronm either node at the time of mean new moon, he may be eclipsed (333), and at the subsequent opposition the moon will be eclipsed near the other node, and come round to the next conjunction before the sun is 13~ 33' from the former node; and when three eclipses happen about either node, the like number commonly happens about the opposite one; as the sun comes to it in 173 da-ys afterwards, and six lunations contain only four days more. Thus there may be two eclipses of the sun and one of the moon about each of the nodes; and the twelfth lunation from the eclipse in the beginning of the year may give a new moon before the year is ended, which, in consequence of the retrogradation of the nodes, may be within the solar ecliptic limit; and hence there may be seven eclipses in a. year, five of the sun and two of the moon. But when the moon changes in either of the nodes, it cannot be near enough to the other node, at the next full moon, to be eclipsed; as in the interval the sun will move over an arc of 140 32', whereas the greatest lunar ecliptic limit is but 13~ 21', and in six lunar months afterwards it will change near the other node. In this case there cannot be more than two eclipses in a year, both of which will be of the sun. If the moon changes at tile distance of a few degrees from either node, then an eclipse both of the sun and moon will probably occur in the passage of that node and also of the other. Although solar eclipses are more frequent than lunar, when considered with respect to the whole earth, yet at any given place more lunar than solar eclipses are seen. The reason of this circumstance is, that an eclipse of the sun (unlike an eclipse of the moon) is visible only over a part of a hemisphllere of the earth. To show this, suppose two lines to be drawn froln the centre of the moon tangent to the earth at opposite points: they will make an angle with each other equal to double the moon's horizontal parallax, or of 1~ 54'. Therefore, should an observer situated at one of the points of tangency, refer the centre of the moon to the centre of the sun, an observer at the other would see the centres of these bodies distant from each other an angle of 1~ 54', and their nearest limbs separated by an are of more than 10. MOON S SHADOW CAST UPON THE EARTH. 203 336. Moon's Shadow Cast upon the Earth. Instead of regarding an eclipse of the sun as produced by an interposition of the moon between the sun and earth, as we have hitherto considered it, we may regard it as occasioned by the moonh's shadow falling upon the earth. Fig. 83 represents the moon's shadow, as projrcted from the sun and cnvnring a portion of the earth's surface. Wherever the umbrnl falls, there is total eclipse; and wherever the penumbra falls, a partial eclipse. IN — /],I FIG. 83. In order to discover the extent of the portion of the earth's surface over which the eclipse is visible at any particular time, we have only to find the breadth of the portion of the earth covered by the penumbral shadow of the moon; but we will first ascertain the length of the moon's shadow. As seen at the vertex of the moon's shadow, the apparent diameters of the moon and sun are equal. Now, as seen at the centre of the earth, they are nearly equal, sometimes the one being a little greater and sometimes the other. It follows, therefore, that the length of the moon's shacdow is about equal to the distance of the earth, being sometimes a little greater and at other times a little less. When the apparent diameter of the moon is the greater, the shadow will extend beyond the earth's centre; and when the apparent diameter of the sun is the greater, it will fall short of it. If we increase the mean apparent diameter of the moon as seen from the earth's centre, viz. 31/ 7/", by 61, the ratio of the radius of the earth to the distance of the moon, we shall have 31' 38" for the mean apparent diameter of the moon as seen from the nearest point of the earth's surface. Comparing this with the mean apparent diameter of the sun as viewed from the same point, which is sensibly the same as at the centre of the earth, or 82' 3", we perceive that it is less; from which we conclude, that when the sun and moon are each at their mean distance from the earth, the shadow of the moon does not extend as far as the earth's surface. 204 ECLIPSES OF THE SUN AND MOON. 337. To ffi.d a General Expression for the Length of thie Mloon's Shadow. Let AGB, ag'b', and agb (Fig. 84) be sections of the sun, moon, and earth, by a plane passing through A z sHa\-rc, -— t=< FIG. 84. their centres S, I, and E, supposed to be in the same right line, and Aa', B13b, tangents to the circles AGB, a'g'b': then a'Kb' will represent the moon's shadow. Let L - the length of the shadow; D - the distance of the moon; D' - the distance of the sun; d -= the apparent semi-diameter of the moon; and J = the apparent semi-diameter of the sun. At K the vertex of the shadow, MlKa' the apparent semi-diameter of the moon, will be equal to SKIA the apparent semi-diameter of the sun; and as the distance of this point from the centre of the earth, even when it is the greatest, is small in comparison with the distance of the sun (336), the apparent semi-diameter of the sun will always be very nearly the same to an observer situated at K as to one situated at the centre of the earth. Now, since the apparent semidiameter of the moon is inversely proportional to its distance, angle MKa': d:: IE: MK; and thus,: d:: ME:MK:: D: L (nearly): whence, L=D. D. (72). If a more accurate result be desired, we have only to repeat the calculation, after having diminished ~ in the ratio of D' to (D' + L -D). 335. To finid the Breadth of the Penumbnral Shadow cast upon the earth, let the lines Ad', B3c' (Fig. 84) be drawn tangent to the circles AGB, a'g'b', on opposite sides, and prolonged to the earth. The space hc'd'k will represent the penumbra of the moon's shadow, and the arc gh one-half the breadth of the portion of the earth's surface covered by it. Let this arc or the angle gEh =- S, and denote the semi-diameter of the sun and the semi-diameter and parallax of the moon by the same letters as in previous articles. The triangle MEh gives angle MEh = S = MhZ - hME. The angle hME is the moon's parallax in altitude at the station h, and MhZ is its zenith distance at the same station. Denote the former by P' and the latter by Z. Thus, S = z- P'....(73). The triangle hlMS gives hME = P' = MSh + MhS; LENGTH AND BREADTH OF MOON'S SHADOW. 205 MhS = d + 1; and AMSh is the sun's parallax in altitude at the station h: let it be denoted by p'. We have, then, P' = d + j +' = d + 3 (nearly)....(74); and to find Z we have (equa. 9, p. 63), P' = P sin Z, or sin Z = p....(75). P' and Z being found by these equations, equa. (73) will then make known the value of S. If great accuracy be required, the calculation must be repeated, giving now to p' in equa. (74) the value furnished by equa. (9) which expresses the relation between the parallax in altitude of a body and its horizontal parallax, instead of neglecting it as before; and Z must be computed from the following equation: sin PI sin z = sin.... (76). sin P The penumbral shadow will obviously attain to its greatest breadth when the sun is at its least and the moon is at its greatest distance. The values of d, 6, and P under these circumstances are resplectively 14' 24", 16' 18", and 52' 50". Performing the calculations, we find that the breadth of the greatest portion of the earth's satface ever covered by the penumbral shadow is 70~ 17', or about 4,850 miles. 339. The Breadth of the Umbra may be found in a similar manner. The arc gh' (Fig. 84) represents one half of it: denote this arc or the equal angle gEh' by S'. MEh' = S' = Mh'Z'- h'ME; or, S' =- P'....(77). h'ME = P' = M Sh' + Mh'S; but AMh'S = d- -, and MSh' = p', the sun's parallax in altitude at h'; whence, P' = d- d + p' = d- a (nearly)....(78): and we have, as before, P P' = P sin Z. or sin Z =f....(79). The greatest breadth will obtain when the sun is at its greatest and the moon is at its least distance. We shall then have - = 15' 45", d = 16' 46", P = 61' 32". Making use of these numbers, we deduce for the maximum breadth of the portion of the earth's surface covered by the moon's shadow, 10 54'; or 130 miles. It should be observed that the deductions of the last two articles answer to the supposition that the moon is in the node, and that the axis of the shadow and penumbra passes through the centre of the earth. In every other case, both the shadow and penumbra will be cut obliquely by the earth's surface, and the sections will be ovals, and very nearly true ellipses, the lengths of which may materially exceed the above determinations. 3410. Phases of Eclipse Different at each Place. Parallax not only causes the eclipse to be visible at some places and invisible at others, as shown in Art. 335, but, by making the distance 20 6 ECLIPSES OF TIIE SUN AND MOON. between the centres of the sun and moon unequal, renders the circumstances of the eclipse at those places where it is visible, differ. ent at each place. This may also be inferred from the circumstance that the different places, covered at any time by the shadow of the moon, will be differently situated within this shadow. It will be seen, therefore, that an eclipse of the sun has to be considered in two points of view: 1st. W'ith respect to the whole earth, or as a general eclipse; and, 2d. With respect to a particular place. 341. Particular Facts. The following are the principal facts relative to eclipses of the sun that remain to be noticed: 1st. The duration of a general eclipse of the sun cannot exceed about 6 hours. 2d. A solar eclipse does not happen at the same time at all places where it is seen: as the motion of the moon toward the sun, and consequently of its shadow, is from west to east, the eclipse must begin earlier at the western parts and later at the eastern. 3d. The moon's shadow being tangent to the earth at the commencement and end of the eclipse, the sun will be just rising at the place where the eclipse is first seen, and just setting at the place where it is last seen. At intermediate places, the sun will at the time of the beginning and end of the eclipse have various altitudes. 4th. An eclipse of the sun begins on the western side and ends on the castern. 5th. When the straight line passing through the centres of the sun and moon passes also through the place of the spectator, the eclipse is said to be central: a central eclipse may be either annular or total, according as the apparent diameter of the sun is greater than that of the moon, or the reverse. 6th. A total eclipse of the sun cannot last at any one place more than eight minutes; and an annular eclipse more than twelve agd a half minutes. 7th. In most solar eclipses the moon's disc is covered with a faint light, a phenomenon which is attributed to the reflection of the light from the illuminated part of the earth. CA.LCULA TION OF AN ECLIPSE OF THE SUN. 342. The complete calculation of a solar eclipse involves the solution of two distinct problems, viz.: (1), the determination of all the circumstances of the eclipse for the earth as a whole; (2), the determination of the times of all the phases, and the corresponding apparent relative positions of the moon and sun for a )particular place. Different methods of solving these problems have been devised. Processes of calculation, comparatively simple and direct, are given in the Appendix. OCCULTATIONS. 343. An occultation is an eclipse, or deprivation of the light of a star, resulting fiom the interposition of the moon between the star and the eye of the observer. At all places on the earth which at a given time have the moon in the horizon, its apparent place will differ from its true place (78), by the amnount of the horizontal parallax. It follows, therefore, that a star will be eclipsed by the moon, somewhere upon the earth, in case its true distance from the moon's centre is less than the sum of the moon's semi-diameter and horizontal parallax. OCCULTATIONS. 207 341. Lmits of Positiosa of Stars Liable to Ocfutltation. The greatest value of the apparent semi-diameter of the moon is 16' 46"', and that of its horizontal parallax is 61' 32". If we add the sum of these quantities to 50 20' 6", the greatest possible latitude of the moon, we obtain as the result 6~ 38' 24". It is then only the stars which have a latitude, either north or south, less than 6~ 33' 24" that can experience an occultation from the moon. In order that any of the stars situated within this distance from the ecliptic may suffer occultation at some point on the earth, it is necessary that, at the time of the true conjunction (144) of the moon and star, the actual difference of latitude of the two should not exceed the sum of the actual apparent semi-diameter and horizontal parallax of the moon. 208 THE PLANETS. CHAPTER XVII. THE PLANETS, AND THE PHENOMENA OCCASIONED BY THEIR MOTIONS IN SPACE. APPARENT M10TIONS OF TIHE PLANETS WITH RESPECT TO THE SUN 345. THE apparent motion of an inferior planet with reference to the sun, is materially different from that of a superior planet. The infeirior planets always accompany the sun, being seen alternatelyv on the east and west side of it, and never receding from it beyond a certain moderate distance, while the superior planets are seen, at different times, at every variety of angular distance. This difference of apparent motion arises firom the difference of situation of the orbits of an inferior and superior planet, with respect to the orbit of the earth; the one lying within, and the other without the earth's orbit. C FIGS. 85. 316. Apparent![otioln of an lanferior Planet. Let CAC'B (Fig. 85) represent the orbit of either one of the inferior MOTION OF A SUPERIOR PLANET. 209 planets, Venus for example, and PKT the orbit of the eartl — which we will suppose to be circles, and to lie in the same plane, -and let MLN represent the circle of intersection of this plane with the sphere of the heavens, to some point of which the planet will be referred by an observer on the earth. Suppose, for the present, that the earth is stationary in the position P, and through P draw the lines PA, PB. tangent to the orbit of Venus, and prolong them until they intersect the heavens at a and b. The earth beino at P, Venus will be in superior conjunction at C, and in inferior conjunction at C'. Now, by inspecting the figure, it will be seen that in passing from C to C' she will be seen in the heavens on the east side of the sun, and in passing from C' to C on the west side of the sun; also, that in passing from C to A she will recede from the sun in the heavens, from A to C' approach it, from C' to B recede from it again, and from B to C approach it again. a and b will be the positions of the planet in the heavens at the times of the greatest eastern and western elongations. When to the east of the sun, Venus is seen in the evening, and called the Evening Star; and when to the west of the sun, is seen in the morning, and called the Morning Star. We have in the foregoing investigation supposed the earth to be stationary, a supposition which is contrary to the fact; but it is plain that the only effect of the earth's motion in the case under consideration, as it is slower than that of the planet, is to cause the points A, C', B, to advance in the orbit, without altering the nature of the apparent motion of the planet with respect to the sun. The orbits of the earth and planet are also ellipses of small eccentricity, and are slightly inclined to each other, instead of being circles and lying in the same plane: on this account. as the greatest elongations will occur in various parts of the orbits, they will differ in value. The greatest elongation of Venus varies from 45~ to 470 15'. Its mean value is about 46~. Owing to the circumstance of the orbit of Mercury being within the orbit of Venus, the greatest elongation of this planet is less than that of Venus. It is never so great as 30~. 347. Apparent Motion of a Superior Planet. Suppose PKT (Fig. 85) to be the orbit of a superior planet, and CAC'B: that of the earth; and as the velocity of the earth is much greater than that of the planet, let us. for the present, regard the planet as stationary in the position P, while the earth describes the circle CAC'. When the earth is at C, the planet being at P, is in conjunction with the sun. When the earth is at A, SAP, the elongation of the planet, is 90~. When it arrives at C', the planet isin opposition, or 180~ distant from the sun: and when it reaches B, the elongation is again 90~. At intermediate points, the elongation will have intermediate values. If, now, we restore to the 14 210 iTHE PLANETS. planet its orbital motion, we shall manifestly be conducted to the same results relative to the chance of elongation, as the only effect of such motion will be to throw the points A, C', B, forward in the orbit. It appears, then, that in the course of a synodic revolution a superior planet will be seen at all angular distances from the sun, both on the east and west side of him. From conjunction to opposition, that is, while the earth is passing from C to C', the planet will be to the right, or to the west of the sun; and will therefore be below the horizon at sunset, and rise some time in the course of the night. But, from opposition to conjunction, or while the earth is moving from C' to C, it will be to the east of the sun, and therefore above the horizon at sunset. 34S. wo fijed the Length of the Sytnodic Revolution of a Planet. Let us first take an inferior planet, Venus for instance. Suppose we assumne, at a given instant, the sun, Venlus, and the earth to be in the same right line; then, after any elapsed time (a day for instance), Venus will have described an angle m., and the earth an angle M around the sun. Now, the value of m is greater than that of M; therefore, at the end of a day, the separation of the planet from the earth (measuring the separation by an angle formed by two lines drawn from the planet and the earth to the sun), will be m — M; at the end of two days (the mean daily motions continuing the same) the angle of separation wil be 2 (m - M); at the end of three days, 3 (m - MI); at the end of s days, s (m — M). When the angle of separation amounts to 860~, that is, when s (m- MI)= 360~, the sun, planet, and earth must be again in the same right line, and in that case 3(00-...(80) In which expression s denotes the mean duration of a synodic revolution, m and M being taken to denote the mean daily -: motions. We may obtain from equat;Jon (80) another equation, in which -the synodic revolution is expressed in terms of the sidereal periods of the earth and planet. Let P and p denote the sidereal periods in question, then, since id.: M:: P: 360~, and 1: m::p:360; 3600 3600 M 360, and m- 0; substituting, P oP 360~ PP (81). 360(, P- ) Equnations (80), (81), although investigated for an inferior STATIONS AND RETROGRADATIONS OF THE PLANETS. 211 pllanet, will answer equally well for a, superior planet, provided we regard m as standing for the mean daily motion of the earth, M for that of the planet, p for the sidereal period of the earth, and P for that of the planet. For the earth holds towards a superior planet the place of an inferior planet, and a synodic revolution of the earth, to an observer on the planet, will obviously be a synodic revolution of the planet to an observer on the earth. 349. Lengths of Synodic Revolutions of Planets. Equa. (80) shows that the length of a mean synodic revolution depends altogether upon the amount of the difference of the mean daily motions of the earth and planet, and is the greater the less is this difference. It follows, therefore, that the synodic revolution is the longest for the planets nearest the earth. It appears by equa. (81) that the length of a synodic revolution is, for an interior planet, greater than the sidereal period of the planet, and, for a superior planet, greater than the sidereal period of the earth. The actual lengths of the synodic revolutions of the different planets are given in Table V. The mean synodic revolution of a planet being known, and also the time of one conjunction or opposition, we may easily ascertain its mean elongation at any given time, and thus approximately the time of its rising, setting, and meridian passage. 350. Planets as E venling or MYorning Stars. A planet will rise and set at the same hours at the end of a synodic revolution; and will be an evening star, that is, above the horizon at sunset, during half of a synodic revolution, and a morning star, that is, above the horizon at sunrise, during an equal interval of time. The inferior planets will be evening stars from superior to inferior conjunction; and the superior planets from opposition to conjunction. Mercury is an evening star for a period of 2 months; Venus during an interval of 94 months; Mars for 1 year and 1 month; Jupiter for 61 months; Saturn and Uranus each a few days more than 6 months. STATIONS AND RETROGRADATIONS OF THE PLANETS. 351. The apparent motions of the planets in the heavens, as has already been stated (11), are not, like those of the sun and moon, continually from west to east, or direct, but are sometimes also from east to west, or retrograde. The retrograde motion takes place over arcs of but a small number of degrees; and in changing the direction of their motions, the planets are for several days stationary in the heavens. These phenomena are called the Stations and Retrogradations of the planets. We now propose to 212 THE PLANETS. inquire theoretically into the particulars of the motions in question, and to show how the phenomena just mentioned result from the motions of the planets in connection with the motion of the earth. 352. Case of an Inferior Planet. Let CAC'B (Fig. 85) represent the orbit of an inferior planet, and PKT the orbit of the earth; both considered as circles, and as situated in the same plane. If the earth were continually stationary in some point P of its orbit, it is plain that while the planet was moving from B the position of greatest western elongation to A the position of greatest eastern elongation, it would advance in the heavens from b to a; that, while it was moving from A to B, that is, from greatest eastern to greatest western elongation, it would retrograde in the heavens from a to b; and that, in passing the points A and B, as it would be moving directly towards or from the earth, it would for a time appear stationary in the heavens, in the positions a and b. But the earth is in fact in motion, and the actual apparent motion of the planet is in consequence materially different from this. Let A, A' (Fig. 86) be the positions of the planet and earth, at the time of the greatest eastern elongation, C', P their positions at inferior conjunction, and B, B' their positions at the! ta FIG. 86. greatest western elongation. At the time of the greatest eastern elongation, while the planet describes a certain distance AD on the line of the centres of the earth and planet, the earth moves for STATIONS AND RETROGRADATIONS OF THE PLANETS. 213 ward in its orbit a certain distance A'D'; so that, instead of appearing stationary at a in the interval, the planet will advance in the heavens from a to d. From the same cause it will have a direct motion about the time of the greatest western elongation. As it advances from A towards C', the direct motion will continue; but, as the daily arc described by the planet will make a less and less angle with the daily arc described by the earth, the rate of motion will continually decrease, and finally, when the planet has come into a position with respect to the earth, such that the lines of direction of the planet, mp. myp', at the beginning and end of the day are parallel, it will be stationary in the heavens. As the daily arc of the planet is greater than that of the earth, and becomes parallel to it in inferior conjunction, the planet will be in the position in question before it comes into inferior conjunction. Subsequent to this, the inclination of the daily arcs still dirninishing, the lines of direction of the planet at the beginning and end of the day will diverge, and therefore the motion will be retrograde. After inferior conjunction, the inclination of the arcs will, at corresponding positions of the earth and planet, obviously be the same as before. It follows, therefore, that the planet will be at its western station when it is at the same angular distance firom the sun as at its eastern station; that its motion will be retrograde until it has passed inferior conjunction and arrived at its western station; and that after this it will be direct. q and n rel)resent the positions of the planet and the earth at the time of the western station; C'q = C=p, and Pn-Pm. The diminution of the elongation of the planet at its two stations is not the only effect of the earth's motion in the case under consideration; it also accelerates the direct, and retards the retrograde motion of the planet, and gives to the planet along with the sun an apparent motion of revolution around the earth. 353. Case of a Superior Planet. Suppose AC'B (Fig. 86) to be the orbit of the earth, and A'PB' that of the planet. Since the earth is an inferior planet to an observer stationed upon a superior planet, it appears by the foregoing article that it will, to an observer so situated, have a retrograde motion while it is passing over a certain arc pC'q in the inferior part of its orbit, fand a direct motion during the remainder of the synodic revolution. Now, it is plain that the direction of the planet's motion, as seen from the earth, will always be the same as the direction of the earth's motion as seen from the planet. When the earth is at C', the middle of the arc pC'q, the planet is in opposition. It follows, therefore, that a superior planet has a, retrograde motion during a small portion of its synodic revolution, about the time of opposition. (See Table V.) 214 THE PLANETS. PHASES OF THE INFERIOR PLANETS. 354. To the naked sight the disc of the planet Venus appears circular, like that of each of the other planets, but the telescope shows this to be an optical illusion. When Venus is repeatedly observed with a telescope, it is seen to present in its various positionlS with respect to the sun the same variety of phases as the moon; being a full circle at superior conjunction, a half circle at the greatest eastern and western elongations, and a crescent, with the horns turned from the sun, before and after inferior conjunction. Mercury exhibits precisely similar phases, but being smaller, at a greater distance from the earth, and much nearer the sun, its phases are not so easily observed as those of Venus. 355. Explanation. The phases of Venus are easily accounted for, by supposing it to be an opake spherical body, and to shine by reflecting the sun's light, and by taking into consideration its motion with respect to the sun and earth. The hernisphere turned towards the sun 0 is illuminated and the other is in the dark, and as the planet revolves around the sun, various portions of the enlightened half It~~~. S tare turned towards the earth; in superior conjunction, the whole of it; at the greatest elongations, one half; and near inferior cnllju lction, but a small part. This will be abundantly evident on inspecting Fig. 87. The phases corres\/ \\, ponding to the positions represented are delineated in the figure. The phases of Mercury are obFIG. 87. viously suceptible of a similar expl anation. 3.6. Changes of Form of the Disc of Mars. The disc of the planet Mars also undergoes changes of form, but they are of comparatively moderate extent. It is sometimes gibbous. but never has the form of a crescent. Indeed, on the supposition that Mars is an opake body illuminated by the sun, we wou;d not see the whole of the enlightened hemisphere, except in conjunction and opposition, but there would always be more than half of it turned towards the earth, and therefore the disc should always be larger than a half circle. The discs of the other superior planets do not experience any perceptible variation of form, for the reason, doubtless, that their TRANSITS OF THE INFERIOR PLANETS. 215 orbits are so large with respect to the orbit of the earth, that all, or very nearly all of their illuminated hemispheres, is constantly visible from the earth. Jupiter offers the only exception to this general truth; it is slightly gibbous in quadratures. TRANSITS OF THE INFERIOR PLANETS. 357. The two inferior planets Venus and Mercury, at inferior conjunction, sometimes, though rarely, pass between the sun and earth, and are seen as a dark spot crossing the sun's disc. This phenomenon is called a Transit. It will take place, in the case of either planet, whenever, at the time of inferior conjunction, it is so near either node that its geocentric latitude is less than the apparent semi-diameter of the sun. 35S. Epochls of Tranisits:-Periods of Recurrence. The transits of Venus take place alternately at intervals of 8 and 105~ or 1211 years. The last were in the years 1761 and 1769. The next will be in 1874 and 1882; of which the latter will be visible in this country. In consequence of the greater distance of Mercury from the earth, a greater portion of its orbit is directly interposed between the sun and earth than of the orbit of Venus; moreover, the synodic revolution of Mercury is shorter than that of Venlus. On these accounts it happens that the transits of Mercury are much more frequent than those of Venus. The last transit of Mercury was on November 11, 1861. The next two will take place in 1868 and 1878, in November and May. The first, which will occur on the 4th, will be visible in this count'ry. 359. A Transit is Calculated in a precisely similar manner with a solar eclipse; the planet in the one calculation answering to the moon in the other. 360. A Transit is an Important Phenomenoni in a practical point of view, as it furnishes an indirect but accurate method of ascertaining the sun's parallax. In order to understand how this phenomenon can be used for this purpose, we have only to consider that, in consequence of the difference of A FrG. 88. the distances or parallaxes of the sun and Venus, observers at different stations upon the earth will refer the planet to different points upon the sun's disc, and that therefore, to such observers, 216 THE PLANETS. the transit will take place along different chords, and be accomplished in unequal portions of time. This fact is represented to the eye in Fig. 88. It is then to be expected, that, if the durations of the transit at two different places should be noted, the difference of the parallaxes of the sun and Venus, upon which alone the difference of the durations depends, could be computed. This computation is in fact possible. Also, the ratio of the parallaxes being inversely as that of the distances, could be found by the elliptic theory of the planetary motions, and thus the parallax both of the sun and Venus would become known. 3:61. The Parallax of the Sun was quite accurately deduced from observations upon the transits of Venus in 1769 and 1761. Expeditions were fitted out on the most efficient scale, by the British, French, Russian, and other governments, and sent to various parts of the earth, remote from each other, to observe the transit of 1769, that the parallax of the sun might be computed from the results of the observations. The sun's parallax, as determined by Professor Encke from the observations made upon the transit in question, and that of 1761, is 8".5776. We have already seen that the sun's parallax has recently been more accurately determined (150). APPEARANCE, DIMENSIONS, ROTATION, AND PHYSICAL CONSTITUTION OF THE PLANETS. 362. Variations of Apparent Diameter. It appears from admeasurement with the telescope and micrometer, that the apparent diameter of a planet is subject to sensible variations. The apparent diameter of Venus, as well as of Mercury, is greatest in inferior conjunction, and least in superior conjunction; while the apparent diameter of each of the other planets is greatest in opposition and least in conjunction. These variations of the apparent diameters of the planets are necessary consequences of the changes that take place in the distances of the planets from the earth. (See Fig. 85.) 363. Absolute and Relative Mlagnitudes. The real diameter of a planet is deduced from its apparent diameter and horizontal parallax. (See Art. 310.) When the diameters of the planets have been found, their relative surfaces and volumes are easily obtained; for the surfaces are as the squares of the diameters, and the volumes as the cubes. The order of magnitude of the planets is as follows: 1 Jupiter, 2 Saturn, 3 Neptune, 4 Uranus, 5 the Earth, 6 Venus, 7 Mars, 8 Mercury, 9 Pallas, 10 Vesta, 11 Ceres, 12 Juno, 13 the other planetoids. The range of magnitude, for the principal planets, is from 1 to about 25,000. The relative magnitudes of the principal planets are given in Table IV. 364. Rotation of Planets. Spots more or less dark have MERCURY. 217 been seen upon the discs of most of the principal planets; and by passing across them from east to west and reappearing at the eastern limbs, have established that the planets upon which they a;'e observed rotate upon axes from west to east. From repeated careful observations upon the situations of these spots, the periods of rotation, and the positions of the axes, have been determined. The periods of rotation of Mercury, Venus, the Earth, and lMars, are all about 24 hours, and of Jupiter and Saturn about 10 hours. Those of the other planets are not known. The axes of rotation remain continually parallel to themselves, as the planets revolve in their orbits. 365. The Amount of Light and Heat, which the sun bestows upon the planets decreases in the same ratio that the square of the distance increases. (See Table IV.) It will be seen in the sequel that the planets are all opake bodies, like the earth, and shine wholly by the reflected light of the sun; and that most, if not all of them, are surrounded with atmlospheres. MIERCURY. 366. Tn consequence of its proximity to the sun, Mercury is rarely visible to the naked eye. When seen under the most favorable circumstances, about the time of greatest elongation, and at periods of the year when twilight is of short duration, it presents the appearance of a star of the third or fourth magnitude. Its phases indicate that it is opake and illuminated by the sun. Its apparent diameter varies with its distance from the earth, from 5" to 13". Its real diameter is a little less than 3,000 miles, or A- of that of the earth; and its volume is about -i of the earth's volume.* Mercury performs a complete rotation on its axis in 24h. 5Em., and according to Schroter, its axis is inclined to the plane of the ecliptic under as small an angle as 20~. 367. Telescopic Appearances. Owing to the dazzling splendor of its light, and the tremulous motion induced by the ever-varying density of the air and vapors near the earth's surface, through which it is seen, the telescope does not present a well-defined image of the disc of this planet. SchrSter is the only observer who has supposed that he discerned distinct spots upon it. Later observers have only noticed on rare occasions, slight inequalities of brightness on its disc. From the fact that such appearances are only occasionally seen, it has been inferred that the planet is surrounded with a dense atmosphere loaded with clouds, that reflect a strong light, and, except when the atmosphere clears up in an unusual degree, prevent the darker body of the planet from being seen. But the evidence in support of this conclusion needs confirmation. Schrdter, in making observations upon Mercury at the time the disc had the form of a crescent, discovered that one of the horns of the crescent became blunt at * The exact diameters, volumes, times of rotation, &c., of the different planets, as tar as known, may be found in Table IV. 218 THE PLANETS. the end of every 24 hours; from which he inferred that the planet turned upon an axis, and had mountains upon its surface, which were brought at the end of every rotation into the same position with respect to his eye and the sun. VENUS. 368. Venus is the brightest of all the planets, and generally appears larger and brighter than any of the fixed stars. But it is much more conspicuous at some times than at others, during a synodic revolution. It is found by calcuLlation, that the epochs in the course of a synodic revolution at which Venus gives most light to the earth, are those at which, being in the inferior part of its orbit, it has an elongation from the sun of a little less than 40~ The disc is then less than one-quarter of' a circle, but the increased proximity to the earth more than compensates for the diminished size of tile disc. Venus attains to greater splendor in some revolutions than in others, in consequence of being nearer the earth when in the favorable position just noticed A. coma bination of the most favorable circumstances recurs every eight years, when Venus becomes visible in full daylight, and casts a sensible shadow at night. This last happened in February, 1862. As seen through a telescope, Venus presents a disc of nearly unitbrm brightness, and spots have very rarely been seen upon it. From the regular succession of phases through which the dise passes, as the planet changes its position with respect to the earth and sun, we infer that it is an opake spherical body, shinillu by the reflected light of the sun. Its apparent diameter varies with its distance from the earth, from 10" to 66". Its real diameter is 7,600) miles; and its volume I- less than that of the earth. The period of its rotation is 23h. 21m. The inclination of its axis to the plane of its orbit is not exactly known, but is supposed to be not far from 18~o 369. Evidences of an Atmosphere surrounding Venus. From the remarkable vivacity of the light of this planet, which far exceeds that of the light reflected from the moon's surface, as well as the transitory nature of the few darkish spots that have been seen upon its disc, it is inferred that it is surrounded by a dense and highly reflective atmosphere, which in general screens the whole of the darker body of the planet from view. The truth of this inference is confirmed by certain delicate observations made by Schrdter. This Astronomer distinctly discerned, when the disc was seen as a narrow crescent, a faint light stretching beyond the proper termination of one of the horns of the crescent into the dark part of the face of the planet, as is represented in Fig. 89, where the left extremity of the dotted line represents FIG. 89. the natural terminating point of one of the horns. The same appearance has since been repeatedly noticed by other observers It VENUS. 219 was distinctly perceptible before and after the last inferior conjunction of the planet, December 11th, 1866. The planet was watched from day to day by Professor Lyman, with the nine-inch equatorial of the Sheffield Observatory, Yale College, until, on the day before conjunction, its distance from the nearest limb of the sun was only 1~ 8'. The very slender crescent which it exhibited, was each day seen more and more extended beyond a semicircle; until at favorable moments, when the sun, but not the planet, was covered by a passing cloud, it was distinctly observed as an entire ring of light, thinnest on the side furthest from the sun. The entire ring was seen also, by the same observer, with a fiveafoot telescope, so placed as to have the sun covered by a distant chimney. The maximum extent of the crescent observed at Dorpat, at the conjunction in 1849, was 240~; the planet being 3~ 26' from the sun's centre. This remarkable prolongation of the cusps must be attributed mainly to the refraction of the sun's rays by the atmosphere of the planet. On this assumption, Mddler, from the Dorpat observations of the extent of the cusps, made the horizontal atmospheric refraction of Venus 43'.7. The observations of Professor Lyman, at the late conjunction, give 45'.3. This is about 4 greater than the horizontal refraction produced by the earth's atmosphere; and indicates that the density of the atmosphere of Venus is decidedly greater than that of the earth's atmosphere. 370. C(ouds in the Atmosphere. Since the transparency of Venus's atmosphere is variable, becoming occasionally such as to admit of the body of the planet's being seen through it, we must suppose that it contains aqueous vapor and clouds, and therefore that there are bodies of water upon the surface of the planet. It is in fact supposed that isolated clouds have actually been seen. The most natural explanation of the bright spots which have sometimes been noticed on the disc, is, that they are clouds, more highly reflective than the atmosphere, or than the clouds in general. 371. Irequalities on the Surface. There are great inequalities on the surface of Venus, and, it would seem, mountains, much higher than any upon our globe. Schrdter detected these masses by several infallible marks. In the first place, the edge of the enlightened part of the planet is shaded, as seen in Figs. 89, 90, 91, and as the moon appears when in crescent even to the naked eve. FIG. 90. FIG. 91. This appearance is doubtless caused by shadows cast by mountains; which are naturally best seen on that part of the planet to which the sun is rising or setting, where they are longest. In the next place, the edge of the disc shows marked irregularities. Thus it sometimes appears rounded at the corners, as in Fig. 90, owing undoubtedly to part of the disc being rendered invisible there by the shadow or interposition of some line of eminences, and at other times, as in Fig. 91, a single bright point point appears detached from the disc-the top of a high mountain, illuminated across a dark valley. Schrdter found that these appearances recurred regularly, at equal intervals of about 231 hours; the same period as that which Cassini had previously found fol the completion of a rotation, by observations upon the spots. 220 THE PLArNETS. MARSo 372. Mars is of the apparent size of a star of the first or second magnitude, and is distinguished from the other planets by the ruddy color of its light. The observed variation in the firm of its disc (356) shows that it derives its light from the sun. Its greatest and least apparent diameters are respectively 4" and 30/". Its real diameter is somewhat less than 5,000 miles, and its bulk is about ~ of that of the earth. Mars revolves on its axis in 24h. 37m.; and its axis is inclined to the plane of the ecliptic in an acngle of about 600~ It appears, from measurements rmade with the micrometer, that its polar diameter is less than the equatorial, and thus, that like the earth, it is flattened at its poles. According to the latest deters minations its oblateness (105) is ~!. 37 3. Telescopic Appeaancenee: —nferker cies. When the disc of Mars is examined with telescopes of great power, it is generally seen to be diversified with large spots of different shades, which, with occasional variations, retain constantly the same size and form. These are conjectured to be continents and seas. In fact, Sir J. F. W. Herschel has on several occasions, in examining this planet with a good telescope, noticed that some of its spots are of a dull red color, while others have a greenish hue. The former he supposes to be land, and the latter water. Fig. 92 represents Mars in its gibbous state, as seen by Herschel in his twenty-feet reflector, on the 16th of August, 1830. The darker parts are the supposed seas. The ~_ y_ _ bright spot at the top is at one of the poles of Mars. At other'_ g _ _ times a similar bright spot is seen at the other pole. These brilliant white spots have been conjectured, with a great deal of probability, to be snow; as they FIG. 92 are reduced in size, and sometimes disappear when they have been long exposed to the sun, and are greatest when just emerging from the long night of their polar winter. The great divisions of the surface of AMars are seen with different degrees of distinctness at different times, and sometimes disappear, either partially or entirely; parts of the disc also appear at times particularly dark or bright. From these facts it is to be inferred that this planet is environed with an atmosphere, and that this contains aqueous vapor, which, by varying in quantity and density, renders its transparency variable. 374. No rountains have been detected qupon JAfars. But this is no good reason for supposing that they are really wanting there; for, if the surface of Mars be actually diversified with mountains and valleys, since its disc never differs much from a full circle, we have no reason to expect that its edge would present that shaded appearance and those irregularities which have been noticed on Venus JUPITER AN D ITS SATELLITES. 221 and Mercury, when of the form of a crescent. The same remarks will apply, with still greater force, to the other superior planets. 375. The ruddy color of the liydt of Jiars' has generally been attributed to its atmosphere, but Sir John Herschel finds a sufficient cause for this phenollenor in the ochrey tinge of the general soil of the planet. JUPITER AND ITS SATELLITES. 376. Jupiter is the most brilliant of the planets, except Venus, and sometimes even surpasses Venus in brightness. The general fact and special circumstances of the eclipses of its satellites, and of the transits of the shadows across the disc of their primary (243), indicate that Jupiter, as well as its satellites, are opake bodies, and shine by the reflected light of the sun. Its apparent diameter, when greatest, is 51/", and when least, 31." Jupiter is the largest of all the planets; its equatorial diameter is 11 times that of the earth, or 88,000 miles, and its bulk is very nearly 1,300 times that of the earth. It turns on an axis nearly perpendicular to the ecliptic, and completes a rotation in 9h. 55 m. The polar diameter is ~7- less than the equatorial. 377. Belts of Jupiter. Whllen Jupiter is examined with a good telescope, its disc is always observed to be crossed by several obscure spaces, which are nearly parallel to each other and to the equator. These are called the Belts of Jupiter (see Fig. 93; which represents the appearance of Jupiter as seen by Sir John Herschel, in his twenty-feet reflector, on the 23d of September, 1832.) They vary somewhat in number, breadth, and situation on the disc, but never in direction. Sometimes _ only one or two are visible; on other occasions as many as eight have been seen at the same time. Sir WVilliam Herschel even saw them, on one or two occasions, broken up and distributed over the aG. 903 whole face of the planet; but this phenomenon is extremely rare. Branches running out from the belts, and subdivisions, as represented in the figure, are by no means uncommon. Darc7 S2pots of invariable form and size have also been occasionally seen upon them. These have been observed to have a rapid motion across the disc, and to return at equal intervals to the same position on the disc, af'ter the same manner as the sun's spots; which leaves no room to 209 THE PLANETS. dcubl)t that they are on the body of the planet, an th that this turns ul)on an axis, Brighlt Sjpols lhave also recently been detected uj pon the belts by two observers; Dawes and Lassell. The belts generally retain pretty nearly the same appearance for several months together, but occasionally marked chances of form and size take place in the course of an hour or two. They are even said to change sometimes very sensibly in the course of a few minutes. Explanation of the Belts. The occasional variations of Jupiter's belts, and the occurrence of spots upon them, which are undoubtedly permanent portions of the mass of the planet, render it extremely probable that they are the body of the planet seen through an atmosphere of variable transparency, but, in general, having extensive tracts of comparatively clear sky in a direction parallel to the equator. These are supposed to be determined by currents analogous to our tradewinds, but of a much more steady and decided character, as would be the necessary consequence of the superior velocity of rotation of this planet. As remarked. by Herschel, that it is the comparatively dark body of the planet which appears in the belts, is evident from this, that they do not come up in all their strength to the edge of the disc, but fade away gradually before they reach it. The bright belts, intermediate between the dark ones, are believed to be bands of clouds, or tracts of less pure air. It is possible that these bright belts may be of the nature of auroral rather than aqueous clouds, and that the dark belts may result from their dispersion along certain tracts, the process being controlled by the varying operation of the sun and planets: after the manner that the planets operate upon the photosphere of the sun, to develop spots upon the sun's disc. Such clouds may have a certain degree of luminosity, and yet at the distance of the earth may shine by the reflected light of the sun. The general prevalence of dark belts on either side of the equator, separated by a bright band at the equator, is analogous to the two spotbelts of the sun, withl an intervening region from which the spots are absent. If, as has been maintained by the author in other publications, the collision of the particles of the earth and planets with the ether of space develops heat, not only directly, but by the origination of electric currents which subsequently pass off in the form of heat, then since a point on the equator of Jupiter has a rotatory velocity 28 times greater than that of a point on the equator of the earth, the temperature at the surface of Jupiter may be much greater than that of the earth, notwithstanding its greater distance from the sun. Upon this idea, it is natural to expect a certain degree of similarity in the photospheric condition of this planet and the sun. 37S. The Satellites of Jupiter, as it has been already remarked, are visible with telescopes of very moderate power. With the exception of the second, which is a little smaller, they are a little larger than the moono The orbits of the satellites lie very nearly in the plane of Jupiter's equator. Thley are, therefore, viewed nearly edgewise from the earth, and in consequence the satellites always appear nearly in a line with each other. Sir W. Herschel, in examining the satellites of Jupiter with a telescope, perceived that they underwent periodical variations of brightness. These variations he supposed to proceed from a rotation of the satellites upon axes which caused them to turn different faces towards the earth; and from repeated and careful observations made upon them, he discovered that each satellite made one turn upon its axis in the same time that it accomplished SATURN, WITH ITS SATELLITES AND RING. 223 a revolution around the primary, and therefore, like the moon, presented continually the same face to the primary. SATURN, WITH ITS SATELLITES AND RINGo 379. Saturn shines with a pale dull light. Its apparent diameter varies less than 6", by reason of the change of distance, and is 16" at the mean distance. The eclipses of its satellites indicate that it is opake, and illuminated by the sun. Saturn is the largest of the planets, next to Jupiter. Its equatorial diametter is 9 times that of the earth, or 72.000 miles; and its volume is 670 times that of the earth. The rotation on its axis is performed in 10ho 29m. The inclination of its axis to the ecliptic is about 62~o Its oblateness is A-d-. 380. Belts of Saturn. The disc of Saturn, like that of Jupiter, is frequently crossed with dark bands, or belts, in a direction parallel to its equator. But Saturn's belts are far more indistinct than those of Jupiter. Extensive dusky spots are also occasionally seen upon its surface. ( See Fig. 94.) The cause of Saturn's belts is doubtless the same as that of Jupiter's. They accordingly establish the existence of an atmosphere upon the surface of Saturn. The results of Herschel's observations on the occultations of the satellites by the planet, indicate the existence of a dense atmosphere. 3~1. Saturn's Ring. The planet Saturn is disinuunished from all the other planets in being surrounded by a broad, thin, luminous ring, situated in the plane of its equator, and entirely detached from the body of the planet. (See Fig. 94.) This ring sometimes casts a shadow upon the planet, and is, in turn, at times partially obscured by the shadow of the planet; from which we conclude that it is opake, and receives its light from the sun. It is inclined to the plane of the ecliptic in an angle of about 28~, an:rd during the motion of Saturn in its orbit remains continually parallel to itself. The face of the ring is, therefore, never viewed perpendicularly from the earth, and for this reason never appears circular, al though such is its actual fbrm. Its apparent form is that of an ellipse, more or less eccentric, accord- FIr. 94. ing to the obliquity under which it is viewed, which varies with the position of Saturn 224 THE PLANETS, in its orbit. When it is seen under the larger angles of obliquity, it appears as a luminous band nearly encircling the planet, and is visible in telescopes of small power. Stars can also be seen between it and the planet in these positions. At other times, when viewed very obliquely, it can be seen only with telescopes of high power~. When it is approaching the latter state, it has the appearance of two handles or ansce, one on each side of the planet. It is also at times invisible. This is the case whenever the earth and sun are on different sides of the plane of the ring, for the reason that the illuminated face is then turned from the earth. WVhen the plane of the ring passes through the centre of the sun, the illuminated edge can be seen only in telescopes of extraordinary power, and appears as a thread of light cutting the disc of the planet. 3S -,. Circulstatnceg of Disappearaince of Ring. Since the orbit of Saturn is very large in comparison with the orbit of the earth, the plane of the ring, during the greater part of the revolution of Saturn, will pass without the orbit of the earth; and when this is the case the ring will be visible, as the earth an(l sun will be on the same side of its plane. During the period, which is about a year, that the plane of the ring is passing by the orbit of the earth, the earth will sometimes be on the same side of it as the sun, and sometimes on opposite sides. In the latter case the ring will be invisible, and in the former will be seen so obliquely as to be visible only in telescopes of considerable or great power. All this will perhaps be better understood on consulting Fig. 95, where efg represents the orbit of the earth. FIrT 95, The appearances of the ring in the different positions of the planer in its orbit are delineated in the figure. The plane of the ring will pass through the sun every semi revolution of Saturn: or, at a mean, about every fifteen years; and at the epochs at which the longitude of the planet is re. spectively 170~ and 350~o The ring will then disappear once in SATURN'S RING. 22.5 about fifteen years; but, owing to the different situations of the earth in its orbit, under varied circumstances': and the disappearance will occur when the longitude of the planet is about 170~ or 350~. The ring will be seen to the greatest advantage when the longitude of the planet is not far from 800, or 2600. The last disappearance took place in 1861; the next will be in 1877. At the present time (1867) the north face of the ring is visible. 383. Rotation of Ring.-Dinenslions. From observations made upon bright spots seen on the face of the ring, IHerschel discovered that it rotated from west to east about an axis perpendicular to its plane, and passing through the centre of the planet (or very nearly). The period of its rotation is 1Oh. 32m. It is remarkable that this is almost the exact period in which a satellite assumed to be at a mean distance equal to the mean distance of the particles of the ring, would revolve around the primary, according to the third law of Kepler. The breadth of the ring is 28,400 miles, which is a little more than one-half greater than its distance from the surface of the planet, and exceeds one-third the equatorial diameter of the planet. 394. Divisions of the Ring. What we have called Saturn's Ring consists in fact of two principal concentric rings; which turn together, although entirely detached from each other. The void space between them is perceived in telescopes of high power, under the form of a black oval line. Calculations from the micrometric measures of Professor Struve give for the breadth of the inner ring 16,500 miles, and of the outer, 10,150 miles. The interval between the rings is 1,700 miles, and the distance from the planet to the inside of the interior ring is 18,300 miles. The thickness of the rings is not well known; the edge subtends an angle less than w", which at the distance of the planet, answers to 210 miles. The division of the ring was discovered as early as the year 1665. The improved telescopes in the hands of modern observers, have revealed the existence of a dark line on the exterior ring, indicative of a subdivision of this ring. It is outside of the middle of the ring, and its breadth is estimated by Dawes at about one-third of that of the principal division of the whole ring. 385. A new Riing of Saturn, interior to the other two, was discovered by G. P. Bond, then assistant at the Observatory at Harvard College, on the 11th of November, 1850. It was subsequently observed by the Messrs. Bond on repeated occasions: from that date to the 7th of January, 1851. It shone with a pale dusky light. Its inner edge was distinctly defined, but the side next the old ring was not so definite; so that it was impossible to make out with certainty whether the new was connected with the old ring or not. The same appearances were noticed by Dawes, at his observatory near Maidstone, England, on the 25th and 29th of November, and subsequently by Lassell, with his large reflector, at Starfield, near Liverpool. According to Dawes, the breadth of the new ring is 1".7, or 7,200 miles; and its distance from the inner edge of the bright ring 0".3, or 1,270 miles. 386. Form of Cross Section of the Ring. Bessel has shown that the double ring is not bounded by parallel plane surfaces. He infers this to be the case from 15 226 THE PLANETS. the fact that at almost every disappearance or reappearance of the ring, the two ansme have not disappeared or reappeared at the same time. He has also found, from a discussion of the observations which have been made upon the disappearances and reappearances of the ring, that they cannot be satisfied by supposing the two faces of the ring to be parallel planes. In view of all the facts, it seems most probable that the cross section of each ring has the approximate form of a very eccentric ellipse instead of a rectangle, and that it varies somewhat in size from one part of the ring to another. It may have irregularities on its surface, as great or greater than those which diversify the surface of the earth. 387. Centre of Gravity of each Ring-Stability of the Rings. Whatever may be the form of the rings, their matter is not uniformly distributed; for micrometric measurements of great delicacy made by Struve, have made known the fact, that the rings are not concentric with the planet, but that their centre of gravity revolves in a minute orbit about the centre of the planet. Laplace had previously inferred, from the principle of gravitation, that this circumstance was essential to the stability of the rings. He demonstrated that if the centre of gravity of either ring were once strictly coincident with the centre of gravity of the planet, the slightest disturbing force, such as the attraction of a satellite, would destroy the equilibrium of the ring, and eventually cause the ring to precipitate itself upon the planet. 388. Physica7 Constitution of the Ring. G. P. Bond has propounded a bold and ingenious theory, relative to the physical constitution of Saturn's rings; which is, that "' they are in a fluid state, and within certain limits change their form and position in obedience to the laws of equilibrium of rotating bodies." He conceives also, that under peculiar circumstances of disturbance several subdivisions of the two fluid rings may take place, and continue for a short time until the sources of disturbance are removed, when the parts thrown off would again reunite. Profesfor Pierce has followed up the speculations of Bond, by undertaking to demonstrate from purely mechanical considerations, that Saturn's ring cannot be solid. He maintains that there is no conceivable form of irregularity, and no combination of irregularities, consistent with an actual ring, which would serve to retain it permanently about the primary if it were solid. He is led by his investigations to the curious result, that Saturn's ring is sustained in a position of stable equilibrium about the planet, solely by the attractive power of his satellites; and that no planet can have a ring unless it is surrounded by a sufficient number of properly arranged satellites. Upon the theory of the development of heat by collision with the ether of space (377), the temperature of the mass of Saturn's rings should be much higher than that of the body of the planet, as its actual velocity of rotation is nearly twice as great, and the possibility of a liquid condition of its mass may be admitted. 389. Origin. In respect to the origin of Saturn's ring, Sir John Herschel has offered the interesting suggestion, that as the smallest difference of velocity in space between the planet and ring must infallibly precipitate the, latter on the former, never more to separate, it follows either that their motions in their common orbit around the sun must have been adjusted by an external power with the minutest precision, or that the ring must have been formed about the planet while subject to their common orbital motion, and under the full influence of all the acting forces. The latter supposition accords with Laplace's theory of the progressive development of the planetary system. 390. The Satellites of Saturn were discovered, the 6th, in the order of distance, by Huyghens, in 1655, with a telescope of 12 feet focus; the 3d, 4th, 5th and 8th, by Dominique Cassini, between the years 1670 and 1685, with refracting telescopes of 100 and 136 feet in length; and the 1st and 2d, by Sir William Herschel, in 1789, with his great reflecting telescope of 40 feet focus. All these but the 1st and 2d, are visible in a telescope of large aperture, with a magnitfying power of 200. The 7th satellite, in the order of distance from the primary, was discovered by the Messrs. Bond, with the great refractor of NEPTUNE. 227 the Cambridge Observatory, on the 16th of September, 1848; and observed two days afterwards by Lassell. It has received the name of Hyperion. The periods of revolution and the mean distances of the satellites of Saturn from their primary, together with the mythological names proposed for them by Sir John Herschel, are given in Table VI. All of Saturn's satellites, with the exception of the Sth, revolve very nearly in the plane of the ring, and of the equator of the primary. The orbit of the 8th is inclined under a considerable angle to this plane. The 6th satellite is much the ]argest, and is estimated to be not much inferior to Mars in size. The others interior to this, diminish in size, towards the ring. The 1st and 2d are so small, and so near the ring, that they have never been discerned but with the most powerful telescopes which have yet been constructed, and with these only at the time of the disappearance of the ring (to ordinary telescopes), when they have been seen as minute points of light skirting the narrow line of the luminous edge of the ring. The new satellite (the 7th) is described as fainter than either of these two interior satellites, discovered by Sir William Herschel. The 8th satellite is subject to periodical variations of lustre, which indicate a rotation about an axis in the period of a sidereal revolution of Saturn. URANUS AND ITS SATELLITES. 391. The planet now known by the name of Uranus, was discovered by Sir William Herschel. It is not visible to the naked eye, except in opposition, when it becomes barely discernible. In a telescope it appears as a small, round, uniformly illuminated disc. Its apparent diameter is about 4", from which it never varies much, owing to the small size of the earth's orbit in comparison with its own. Its real diameter is 33,000 miles, and its bulk 73 times that of the earth. Analogy leads us to believe that this planet is opake and turns on an axis, but there is no positive evidence that this is the case. Of the eight satellites of Uranus, six were discovered by Herschel, one by Lassell, and one by 0. Struve. NEPTUNE. 392. It is a remarkable fact that the existence of this planet was first detected from the disturbances it produced in the motions of Uranus. It having been ascertained that there were outstanding inequalities in the motion of this planet, which could not be referred to the action of the other planets, Le Verrier, the 2 a2 THE PLANETS. eminent French astronomer, undertook in 1845 the problem of determining the orbit and mass of a planet capable of producing such inequalities. The same problem was independently under. taken and successfully solved by Mr. Adams, of Cambridge, Eng]and. Le Verrier, as the final result of his computations, indicated the probable place of the theoretical planet in the heavens; and Dr. Galle, of Berlin, upon directing the great telescope of the Royal Observatory on the region indicated, on the evening of the 23d of September, 1846, descried the new planet within 1~ of its most probable place, as assigned by Le Verrier. The apparent diameter of Neptune is a little less than 3". Its real diameter is 36,00() miles; and its volume 93 times that of the earth. Neptune, like Uranus, is destitute of visible spots and belts, and the period of its axial rotation is unknown. Neptune's satellite was discovered by Lassell in 1846. The same observer has since obtained traces of the existence of a second satellite. THE PLANETOIDS. 393. Vesta is the brightest of the minor planets. In the telescope, it appears as a star of 6th or 7th magnitude. Pallas, Ceres, and Juno appear of the 7th or 8th magnitude. The great majority of the other planetoids are of the 10th or 11th magnitude. Pallas is the largest of this class of bodies. According to Lamont, Director of the Royal Observatory, Munich, its diameter is 670 miles. The diameter of Vesta is believed not to exceed 300 miles; and that of Ceres to be somewhat smaller. Juno is the smallest of the four planetoids first discovered. All of the other minor planets are supposed to be less than 100 miles in diameter. COMIETS. 229 CHAPTER XVIIL CoMETS. THEIR GENERAL APPEARANCE:-VARIIETIES OF AiPPEARAlNCE. 394. The general appearance of comets is that of a mass of some luminous nebulous substance, to which the name Comaz has been given, condensed towards its centre around a brilliant nrucleus that is in most cases not very distinctly defined; from which proceeds, in a direction opposite to the sun, a stream of FIG. 9 6 similar but less luminous matter, called the Tail or Train of the comet (Fig. 96). The nucleus, with the surrounding coma, forms the Head of the comet. The tail gradually increases in width, and at the same time diminishes in distinctness from the head to its extremity, where it is generally many times wider than at the head, and fades away until it is lost in the general light of the sky. It is, in 23,0 COMETS. general, less bright along its middle than at the borders. From this cause the tail sometimes seems to be divided, along a greater or less portion of its length, into two separate tails or streams of light, with a comparative dlark space between them. Ordinarily it is not straight, that is, coincident with a great circle of the heavens, but concave towards that part of the heavens which the cornet has just left. This curvature of the tail is most observable near its extremity. The most remarkable example is that of the cornet of 1744, which was bent so as to form nearly a quarter of a circle. Nor does the general direction of the tail usually coincide exactly with the grce:t circle ptassing through the sun and the head of the comet, but deviates more or less fromr this, the position of exact opposition to the sun in the hceavens, on the side towards the quarter of the heavens just traversed by the comet. This deviation is quite different for different comets, and varies materially for the same comet while it continues visible. It has even amounted in some instances to a right angle. 395. VYriations of Le@tgat of Tail. The apparent length of the tail varies, from one comet to another, from zero to 1000 and more; and ordinarily the tail of the same comet increases and diminishes very much in length during the period of its visibility. WVhen a comet first appears, in general no tail is perceptible, and its light is very faint. As it approaches the sun, it becomes brighter; the tail also, after a time, shoots out from the coma, and increases from day to day in extent and distinctness. As the comet recedes from the sun, the train precedes the head, being still on the opposite side from the sun, and groows less and less at the same time that, along' with the head, it decreases in brightness, till at length the comet resumes nearly its first appearance, and finally disappears. (See Fig. 98.) It sometimes happens that, owing to FIG. 97. peculiar circumstances, a comet does not make (Cveea~ C onet of1843. its cappea'rance in the firmament until after it has lpassecd the sun in the heavens, and not unt;l it has attaiined to more or less distinctness, and is furnished with a train of considerable or even great length. This was remarkably the case with the great comet of 1843. (See Art. 237; also Fig. 97.) THEIR GENERAL APPEARANTCE 231 396. Effects of the Position of tile Earthl, on the apparent size and brightness of a comet. The tail of a cornet is the longest, and the whole comet is intrinsically the mnost luminous, not long after it has passed its perihelion. Its apparent size and lustre will not, however, necessarily be the greatest at this time, as they will depend upon the distance and position of the earth, as well as the actual size and intrinsic brightness of the comet. To illustrate FIG. 98. this, let abcd (Fig. 98) represent the orbit of the earth, and MPN the orbit of a comet, having its perihelion at P. Now, if the earth should chance to be at a when the comet, moving towards its perihelion, is at r, it might very well happen that the comet would appear larger and more distinct than when it had reached the more remote point s, although when at the latter point it would in reality be larger and brighter than when at r. It would be the most conspicuous possible if the earth should be in the vicinity of e or b soon after the perihelion passage; and it would be the least conspicuous possible if the comet be supposed to be moving in the direction NPM, and to pass from N around to Mf, while the earth is moving around from a to b or c; so as to be continually comparatively remote from the comet, and so that the comet will be in conjunction with the sun at the time after the perihelion passage when its actual size and intrinsic lustre are the greatest. It is to be observed that the apparent lustre of A comet is sometimes very much enhanced by the great obliquity of the tail, in some of its positions, to the line of sight. This seems to have been the case with the comet of 1843, on February 28 (see Fig. 63), and, it has been already intimated, 232 COMETS~ was one reason of its being so very bright as to be seen in open day in the immediate vicinity of the sun. Since the earth may hlave every variety of position in its orbit at the successive returns of the same comet to its perihelion, it will be seen, on examining Fig. 98, that the circumstances of the appearance and disappearance of the comet, as well as its size and distinctness, may be very diffirent at its different returns. This has been strikingly true in the case of Halley's Comet. Biela's Comet was also invisible on its return to its perihelion in 1839, by reason of its continual proximity to the line of direction of the sun as seen from the earth, and its great distance from the earth. 397. Varieties of Aspect. Individual comets offer considerable varieties of aspect. Some comets have been seen which were wholly destitute of a tail: such, among others, was the conlet of 1 682, which Cassini describes as being as round and as bright as Jupiter. Others have had more than one luminous train. The cornet of 1744 was provided with six, which were spread out like an immense fan, through an angle of 117~; and that of 1823 with two, one directed from the sun in thie heavens, and, what is very remarkable, another smaller and fainter one directed towards the sun. Others still have had no perceptible nucleus, as the comets of 1795 and 180-4. The comets that are visible only in telescopes, which are very numerous, have generally no distinct nucleus, and are often entirely destitute of every vestige of a tail. They have the appearance of round masses of luminous vapor, somewhat more dense towards the centre. Such are Encke's and Biela's comets. (Fig. 99.) The point of greatest condensation is often more or less ZFIG. 99.-Ecke's Cornmet. removed from the centre of figure on the side towards the sun; and sometimes also on the opposite side. 39~. The Comnets which have had the Longest Trains are those of 1680, 1769, and 1618. The tail of the great comet FORM AND STRUCTURE OF COMETS. 233 of 1680, when apparently the longest, extended to a distance of 70~ friom the head; that of the comet of 1769, a distance of 97~: and that of the cornet of 1618, 104". These are the apparent lengths as seen at certain places. By reason of the different degrees of purity and density of the air through which it is seen, the tail of the same comet often appears of a very different length to observers at different places. Thus, the comet of' 1769, which at the Isle of Bourbon seemed to have a tail of 97~ in length, at Paris was seen with a tail of only 60~. From this general fact we may infer that the actual train extends an unknown distance beyond the extremity of the apparent train. FORM. STRUCTURE, AND DIMENSIONS OF COMETS. 399. The general form and structure of comets, so far as they can be ascertained from the study of the details of their appearance, mnay be described as follows: The head of a cornet conlsists of a central nucleus, or mass of matter brighter and denser than the other portions of the comr-et, enveloped on the side towards the sun, and ordinarily at a great distance from its surface in comparison with its own dimensions, by a globular nebulous mass of great thickness, called the Aebulosity, or nebulous Envelope. This, it is said, never completely surrounds the nucleus, except in the case of comets which have no tails. It forms a sort of hemispherical cap to the nucleus on the side towards the sun. Its form, however, is not truly spherical, but varies between this and that of a paraboloid having the nucleus in its focus and its vertex turned towards the sun. The tail begins where the nebulosity terminates, and seems, in general, to be merely the continuation of' this in nearly a straight line beyond the nucleus. There is ordinarily, as has been already intimated, a distinct space containing less luminous matter between the nucleus and the nebulosity, but this is not always the case. The tail of a comet has the shape of a hollow truncated cone, with its smaller base in the nebulosity of the head; with this difference, however, that the sides are usually more or less curved. That the tail is hollow is evident from the fact, already noticed, that on whichever side it is viewed it appears less bright along the middle than at the borders. There can be less luminous matter on a line of sight passing through the middle, than on one passing near one of the edges, only on the supposition that the tail is hollow. The whole tail is generally bent so as to be concave towards the regions of space which the comet has just left. 400. tlnltiple Tails. In some instances the nucleus is furnished with several envelopes concentric with it: which are formed in succession as the comet approaches the sun, and then 234 COMETS. recedes from it again. For example, the comet of 1744, eight days after the perihelion passage, had three envelopes. Sometimes each of them is provided with a tail. Each of these several tails being hollow, may in consequence appear so faint along its middle as to have the aspect of two distinct tails. A comet which has in reality three separate trains, might thus appear to be supplied with six, as was the comet of 1744. If the different envelopes were not distinctly separate from each other, then all the trains would appear to proceed from the same nebulous mass. Supernumerary tails, shorter and less distinct than the principal one, are by no means uncommon; but they generally appear quite suddenly, and as suddenly disappear in a few days, as if the stock of materials from which they were supplied had become exhausted. 401. The general Position of the Tail of a Comet is nearly but not exactly in the prolongation of the line of the centres of the. sun and head, or of the radius-vector of the comet. (See Fig. 98.) It deviates from this line on the side of the regions of space which the comet has just left; and the angle of deviation. which, when the comet is first seen at a distance from the sun, is very small or not at all perceptible, increases as the comet approaches the sun, and attains to its maximum value soon after the perihelion passage; after which it decreases, and finally, at a distance from the sun, becomes insensible. For example, the angle of deviation of the tail of the great comet of 1811 attained to its maximum about ten days after the perihelion passage, and was then about 110. In the case of the comet of 1664, the same angle about two weeks after the perihelion passage was 43~, and was then decreasing at the rate of 8~ per day. The comet of 1823 might seem to present an exception to the general fact that the tail of a comet is nearly opposite to the sun; but Arago has suggested that the probable cause of the singular phenomenon of a secondary tail, apparently directed towards the sun in the heavens, was that the earth was in such a position that the two tails, although in fact inclined to each other under a small angle, were directed towards different sides of the earth, and thus were referred to the heavens so as to appear nearly opposite. The same principle will serve to show that the deviation of the train of a comet from the position of exact opposition to the sun may appear to be much greater than it actually is, by reason of the earth's happening to be within the angle formed by the direction of the train with the radius-vector prolonged. 402. Vast Size of Comets. Comets are the most voluminous bodies in the solar system. The tail of the great comet of 1680 was found by Newton to have been, when longest, no less than 123,000,000 miles in length. The remarkable comet of DIMENSIONS OF COMETS. 235 1843, about three weeks after its perihelion passage, had a tail of over 108,000,000 miles in length. Other comets have had trains fromt fifty to one hundred million miles long. The heads of comets are generally tens, and often hundreds, of thousands of miles in diameter. That of the great comet of 1811 had a diameter of over 1,000,000 miles; that of Halley's comet, in 1836, a diameter of 350,000 miles, and that of Encke's comet, in 1828, a diameter of over 300,000 miles. The head of the great comet of 1843 was about 30,000 miles in diameter. 403. The Nuclei of comets, so far as they have been accurately determined, do not exceed a few hundred miles in diameter. For example, the great comet of 1811 had a nucleus of 428 miles, and that of 1798 one of 125 miles in diameter. Instances are cited of comets with nuclei of several thousand miles in diameter (e. g., the third comet of 1845, and the fourth comet of 1825); but there is little reason to doubt that in these cases, the apparent telescopic nucleus ordinarily observed was measured, instead of the true nucleus, which is only occasionally seen. When a comet is viewed with the naked eye, it usually offers the appearance of a star-like nucleus at the centre of the head. Telescopes resolve this, more or less, into a bright nebulous mass, which is the ordinary telescopic nucleus. But occasionally they show, in the case of a bright comet, within this a stellar point, distinguished by its brightness and appearance of solidity from the nebulosity about it. This is the true nucleus. The nucleus, so-called, of Donati's comet, is stated to have been 5,600 miles in diameter, but according to Bond, the true nucleus that was occasionally discernible in his telescope, was less than 500 miles in diameter. 404. Variation of Dimensions. The dimensions of comets are subject to continual variations. The tail increases in actual length as the comet approaches the sun, and attains its greatest size a certain time after the perihelion passage; after which it gradually decreases. The head, on the contrary, diminishes in size during the approach to the sun, and augments during the recess from him. These changes of dimension, both in the case of the head and of the tail of the comet, are often very great, and sometimes quite sudden and rapid. Encke's comet, at its return in 1828, in the course of two months, while its distance from the sun was diminished in the ratio of 1 to 3, underwent an apparent diminution of volume in the ratio of 1.6,000 to 1. The apparent nucleus of Donati's comet was 1,000 times less soon after the perihelion passage than when it was previously seen at a distance two or three times greater. The tail of the great 2omet of 1843 increased in length after the perihelion passage, at the rate of 5,000,000 miles per day; and that of Donati's comet increased in length for ten days after the perihelion passage, at the average.rate of 2,500,000 miles per day. 236 COMETS. PHYSICAL CONSTITUTION OF COMETS. 405. Small Mass and Density. The quantity of matter which enters into the constitution of a comet is exceedingly small. This is proved by the fact that comets have had no influence upon the motions of the planets or satellites, although they have, in many instances, passed near these bodies.. The comet of 1770, which was quite large and bright, passed in close pro(ximity to Jupiter's satellites, without deranging their motions in the least perceptible degree. Moreover, since this small quantity of matter is dispersed over a space of tens of thousands or millions of miles (if we include the tail), in linear extent, the nebulous matter of cornets must be incalculably less dense than the solid matter of the planets. In fact, the cometic matter, with the exception perhaps of that of the nucleus, is inconceivably more rare and subtile than the lightest known gas, or the most evanescent film of vapor that ever makes its appearance in our sky; for faint telescopic stars are distinctly visible through all parts of the comet, with, it may be, the exception of the nucleus, notwithstanding the great space occupied by the matter of the comet, which the light of the star traverses. The matter of the tail of a comet is even more attenuated than that of the general mass of the nebulosity of the head, but is apparently of the same nature, and derived fromn the head. 406. Nucleus and Nebulosity. The nucleus is supposed by some astronomers to be, in some instances, a solid, partially or wholly convertible into vapor under the influence of the sun; by others, to be in all cases the same species of matter as the nebulosity, only in a more condensed state; and by others still, to be a solid of permanent dimensions, with a thick stratum of condensed vapors resting upon its surface. Whichever of these views be adopted, it is a matter of observation that the nebulosity frequently receives fresh supplies of matter from the nucleus. It was the opinion of Sir William ITerschel, and it has been the more generally received notion since his time, that the nucleus of a comet is surrounded with a transparent atmosphere of vast extent, within which the nebulous envelope floats, as do clouds in the earth's atmosphere. But Olbers, and after him Bessel, conceives the nebulous matter of the head to be either in the act of flowing away into the tail under the influence of a repulsion from the nucleus and the sun, or in a state of equilibrium under the action of these forces and the attraction of the nucleus. 407. Luminosity of Comets. Observations with the polariscope have established that comets shine in a great degree by reflected light, This is especially true of the tail of the comet; the nucleus and nebulosity present feeble traces of polarization, and. FORMATION OF THE TAILS OF COMETS. 237 we must therefore conclude, emit a strong light of their own, or shine wholly by light radiantly reflected. If the head of a comet shone entirely by reflected light, and the amount of reflecting surface remained constantly the same, its apparent brightness would be inversely proportional to the product of the squares of the distances forn the sun and earth. By this rule, the head of Donati's comet should have been 188 times brighter on the 2d of October than on June 15th; whereas it was actually 6,300 times brighter. From which we may infer that the quantity of light emanating from it had increased in the proportion of 33 to 1. This increase of actual light was confined chiefly to thenebulosity of the head, and is probably attributable, in a great degree, to,n augmentation of the quantity of nebulous matter received from the nucleus. CONSTITUTION AND MODE OF FORMATION OF THE TAILS OF COMETS. 40S. Upon this topic we may lay down the following postulates: 1. The general situation of the tail of a comet with respect to the sun, shows that the sun is concerned, either directly or indirectly, in its formation. The changes which take place in the dimensions of a comet, both in approaching the sun and receding from it, conduct to the same inference. 2. Since the tail lies in the direction of the radius-vector prolonged beyond the head, the particles of matter of which it is made up must have been driven off by some force exerted in a direction from the sun. 3. This force cannot emanate from the nucleus, for such a force would expel the nebulous matter surrounding the nucleus in all directions, instead of one direction only. It is, however, conceivable that, as Olbers supposes, the nebulous matter is, in the first instance, expelled from the nucleus by its repulsive action, taking effect chiefly on the side towards the sun, and afterwards driven past the nucleus into the tail by a repulsion from the sun. 4. There seems, then, to be little room to doubt that the matter of the tail is driven from the head by some force foreign to the comet, and taking effect from the sun outwards. 5. This force, whatever may be its nature, extends far beyond the earth's orbit; for comets have been seen provided with tails of great length, though their perihelion distance exceeded the radius of the earth's orbit (e.g., the great comet of 1811). 6. It is natural to suppose that, like all central forces. the repulsive force exerted by the sun upon cometic matter varies inversely as the square of the distance. This law of variation has in fact been established by the investigations of Bessel and Professor Pierce, and confirmed by the author's determination of the form and dimensions of the tail of Donati's comet, upon the 238 COMETS. theory that it was made up of particles individually repelled by the sun with an intensity of force varying according to this law.* 409. Explanation of Situation and Curved Form of Tail. Let PCA (Fig. 100) be a portion of a comet's orbit, the FIG. 100. sun being at S; and suppose a particle to be expelled in the direction SAD, when the head is at A, and another particle to be driven off in the direction SBE, when the head is at B. Each particle will retain the orbital motion which obtained at the time of its departure, as it moves away from the sun; and thus, when the comet has reached the point C, instead of being at any points, D and E, on the lines SAD and SBE, will be respectively at certain points, a and b, farther forward. The line Cba, which, when the comet is at C, is the locus of all the particles that have been emitted during the interval of time in which the comet has been moving over the arc AC, is the tail. We here suppose the head to be a mere point. If we conceive the particles to be continually emitted from the marginal parts of the head, we shall have the hollow conical tail actually observed. It is easy to see that Cba, the line of the tail, must be a curved line concave towards the regions of space which the comet has left. Supposing the are AC to be so small, or its curvature so slight, that it may be considered as a straight line, and neglecting the change of velocity in the orbit, Ca will be parallel to AD, and Cb parallel to BE; whence RCa = CSA, and RCb = CSB. Thus the line joining any particle with the nucleus always makes an angle with the prolongation of the radius-vector, approximately equal * See Ameriaan Journal of Science, Vol. xxIx., pp. 79 and 383, and Vol. xxnL p. 54, etc. FORMATION OF THE TAILS OF COMETS. 239 to the motion in anomaly during the interval that has elapsed since the particle left the head. It follows from this, that if we suppose the velocity of the particles to be continually the same, and the motion in anomaly uniform, the deviations of the particles a and b from the line of the radius-vector SCR will be in the ratio of the distances Ca and Cb. But in point of fact the velocity increases with the distance, so that the curvature of the tail will be less than on the supposition just made. As to the amount of the deviation of the tail from the line of the radius-vector, it must depend upon the proportion between the velocity of the particles and the velocity of the head in its orbit; and it follows from the principle just established that unless the velocities of emission augment as rapidly as the velocity of revolution, the deviation in question will increase to the perihelion, and afterwards decrease, as it is in fact known to do. 410. Dispersion of the Comerntic Matter ili the Plane of thie Orbit. Observations made upon Donati's comet, have established that the nebulous matter was much more widely dispersed in the direction of the plane of the comet's orbit than in the direction perpendicular to the orbit; so that the transverse sections of the tail were approximately elliptical in form, and more elongated in proportion as their distance from the head was greater. The same fact was still more conspicuous in the case of the great comet of 1861, and is probably a general law. It is shown in the memoir above referred to, that this phenomenon had its origin in the case of Donati's comet, in an inequality in the force of repulsion exerted by the sun upon different portions of nebulous matter expelled from the nucleus. The limits between which the repulsive force varied were 0 and 1.21 (the intensity of the sun's ordinary force of attraction at the same distance being the unit). It is shown also that nearly one-hatlf of the tail, on the concave side, was made up of matter that was not actually repelled by the sun, but became widely separated from the head of the comet, after being expelled by a projectile force beyond the sphere of attraction of the nucleus, simply because it was subject to a diminished intensity of solar attraction. The concave edge of the tail consisted of matter subject to an attractive force equal to -0% of the full force of the sun's attraction. The greatest intensity of repulsive action (1.21) obtained at the convex, or preceding side of the tail. If we assume that the escaping particles did not receive any initial lateral velocity from a repulsive or projectile force ex erted by the nucleus, the limits of the effective solar repulsion and attraction, for the two edges of the tail, become 1.5 and 0.6 (instead of 1.21 and 0.45). In Fig. 101, the train of the comet as theoretically determined is compared with that actually observed. The full curve- runs through the positions of the particles that left the head at several 240 COM ETS. assumed dates, calculated for October, 5d. 7h. mean time at Greenwich, and is accordingly the outline of the train as theoretically determined for that instant. The dotted curved line is the 218 216: FIG 101. outline of the actual train as observed 1- hours later, when its form and dimensions were sensibly the same as at 7h. The broken line nearly in the middle of the theoretical train, runs through the calculated positions of several particles that left the head of the comet at different dates, and were neither attracted nor repelled by the sun, and therefore proceeded on in tangents to the orbit. The single straight streamers seen in connection with this and other comets (See Plate III.), must have been urged by a force of repulsion many times greater than the maximum limit of repulsion for the principal tail (1.21). 411. Coluniar Structur e of the Tail of Doanati's Comet. The tail of the comet of Donati. was seen on certain occasions to be traversed, for a part of its length, by bands of uneqtal brightness, diverging from the vicinity of the head (See Plate III.). This proves to have been a consequence of frequent alternations in the ejection of nebulous matter from the head of the comet; for it appears, as a result of the calculations above mentioned, that all the matter variously repelled which issued at any instant, must, at any subsequent date, have been arranged nearly in a straight line that produced would pass near the head. Fig. 102, shows, for the date of the calculations (October 5th), the lines made up of the particles that proceeded from the head of the comet, at the dates given, viz.: September 29th, September 26th, &c. The train may accordingly be considered as having been composed of a series of diverging bands, or columns of nebulous matter emanating from the head on successive days, or other equal intervals of time; which alternated in brightness when there were alternations in the quantity of matter discharged. CONDITION AND ORIGIN OF NEBULOUS ENVELOPES. 241 412. The Source of the Nebulous Stream, called the tail of the comet, has been generally supposed to be the envelope, or envelopes of the head; but at the present day the preponderating weight FIG. 102. of evidence is opposed to this view, and in favor of the theory that the envelope and tail are but different portions of one continuous stream of cometic matter emanating from the nucleus, or from the bright nebulosity contiguous to the nucleus proper. It appears to be an insuperable objection to the former hypothesis, that a small extent of the nebulous stream, in the immediate vicinity of the envelope from which it proceeds, contains as much luminous matter as the envelope itself, and yet the envelope usually continues in existence for many days. Some of the envelopes that were seen to rise apparently from the nucleus of Donati's Comet, did not become dissipated until two weeks after their first appearance. Besides it is certain that a considerable portion of the matter detached from the nucleus, does move in a continuous stream through the apparent envelope into the tail; for jets, or single streams, are frequently seen to proceed from the nucleus, on the side toward the sun, and after being bent back by the solar repulsion, to become merged in the general stream that seems to issue from the envelope. It is also possible to deduce the actual form and dimensions of both the envelope and tail, on the hypothesis of a single continuous stream proceeding from a certain portion of the nucleus exposed to the action of the sun.* CONDITION AND ORIGIN OF THE NEBULOUS ENVELOPES. 413. Successive Envelopes. As already intimated there are frequently two or more envelopes that appear to be indefinitely continued into the train (See Plate III.). These are detached in succession from the nucleus, and while receding continually * (See the American Journal of Fcic nce, Vol. XXVII., January, 1859.) 242 COMETS. from it and expanding, decline in lustre, and finally disappear; according to one of the above-mentioned hypotheses because they are dissipated by the repulsive action of the sun upon their particles, and according to the other because the supply of outstreaming matter at the nucleus falls off. The late Director of the Observatory of Harvard College, in his great work on the comet of Donati, states that no less than seven envelopes were detached in succession from the nucleus of the comet, at intervals of from four to seven days. Their rate of recess from the nucleus was about 1,000 miles per day. The great comet of 1861 presented a succession of eleven envelopes, rising at regular intervals on every second day. Their evolution and final dissipation were accomplished with much greater rapidity than the corresponding phenomena of the comet of 1858. 414. Expelling Force. Since the cometic particles which were distributed along the concave side of the tail of Donati's comet were not repelled by the sun (410), we must infer that they were not expelled from the nucleus by a force of repulsion, but were in all probability detached by some projectile force in operation at or near its surface. On the other hand, the cometic particles that were in a condition to be repelled by the sun, may have become detached from the nucleus under the operation of a force of repulsion exerted by its mass, or from its surface. We may conceive a repulsive force, exerted by both the nucleus and the sun, to be a consequence of the particles being more nearly in the condition of the ultimate molecules, in which there is reason to believe that they become subject to both a molecular and heat repulsion, operating at indefinitely great distances.* If we conceive the bright nebulous mass adjacent to the nucleus, which appears to be the fountain head of the nebulous stream that constitutes both the envelope and train of a comet, to be in a magnetic condition similar to that which has been attributed to the photosphere of the sun, it is to be observed that particles may become detached from the tops of magnetic columns simply in consequence of a diminution in the magnetic intensity of the nucleus and its photosphere; and such diminution of magnetic intensity should continually occur, from day to day, as the comet recedes from the sun, and consequently has a decreasing velocity in its orbit. For, according to the theory of cosmical magnetization, the intensity of the magnetic currents developed should be directly proportional both to the orbital velocity and the velocity of rotation.t A statical force of electric repulsion might also operate to detach particles, whether magnetic or not, in directions normal to the surface of the nucleus. The Projectile Force, whose existence we have here recognized, may have its origin in electric discharges along magnetic vaporous columns, like the similar force supposed to be in action upon the surface of the sun's photosphere (293). In support of this view it may be urged that, if we assume the hypothesis that the nebulous matter at the nucleus of a comet is made up of particles susceptible of magnetization, and capable of being expelled by discharges along lines of magnetic polarization, we are enabled to give an adequate explanation of diverse luminous phenomena presented by comets, that are wholly inexplicable upon all previous hypotheses. 415. Theoretical Process of Evolution of an Envelope. We would first remark that a rotatory motion of the nucleus, in conjunction with its orbita) motion, should, by the collision of the molecules with the ether of space, bring it into a magnetized state, with the poles in the vicinity of 90~ from the plane of the orbit. Now if we conceive the matter, disposed in magnetic columns, to be expelled in the lines of direction of the columns, and subsequently to be repelled by the sun, we have to observe that the lines of discharge will be nearly parallel to the surface of the nucleus near the magnetic equator, and that their angle of incli. * See American Journal of Science, Vol. xxxvm., p. 70. f See American Journal of Science, Vol. XLL, p. 62. ORIGIN OF THE NEBULOUS ENVELOPES. 243 nation to the surface will increase with the distance of the columns from the magnetic equator, or approximately from the plane of the orbit. The envelope should therefore consist of two portions proceeding from parts of the nucleus that lie on opposite sides of the magnetic equator. The nebulous streams issuing from the points on either side of the equator will pass to the other side, intersecting its plane at points more and more distant from the nucleus, until the initial directions of the streams become, at points at a certain distance from the equator, parallel to its plane, or to the plane of the orbit nearly. If we conceive the magnetic equator to lie in the plane of the orbit, and confine our attention to the streams proceeding from points on the meridian, whose plane contains the radius-vector, then at a point on this meridian about 35~ from the orbit the nebulous stream would issue in a direction parallel to the radius-vector, and at points that have a higher magnetic latitude its direction would diverge more and more from this line. All such streams would be bent back by the force of the sun's repulsion, and would form, collectively, an apparent envelope on the side towards the sun. This would have a parabolic form, if the discharge extend beyond 35~ of magnetic latitude, and the expelling force and the solar repulsion have each a constant intensity for all latitudes. But if the latter should increase, or the former decrease, with the latitude, the outline of the envelope would approach more nearly to the circular form, as was observed in the case of Donati's comet. 416. Phenomena confirmatory of the Theory. Various peculiarities of form, and diversities of brightness presented by Donati's comet, and several others, seem to indicate that each envelope does in fact consist of two portions, that do not in general originate simultaneously; and which in part pass from the one side to the other of the nucleus. The following are some of the peculiar features referred to: (1.) The spiral form, or awry position of each of the successive envelopes, when first seen distinctly separate from the nucleus. The explanation is that the discharge of cometic matter began from the one magnetic hemisphere sooner than from the other; from that which is most exposed to the sun's action, we may suppose. (2.) The depression, or deficiency of cometic matter about the vertex of the envelope, frequently noticed, especially in the later stages of the envelope. This has been in some instances represented as a notch in the envelopd. This deficiency of light at the vertex is obviously what should result if the discharge should relatively fall off at the magnetic latitudes (about 35~) from which the nebulous streams issue in directions parallel to the radius-vector; and we shall soon see that it is reasonable to expect that the discharge should begin to decline at these sooner than at higher latitudes. (3.) The remarkably dark band seen to extend nearly along the axis of the tail of Donati's comet, for a certain distance from the nucleus. This band was too dark to be explained by the supposition that the tail was hollow. Upon the present theory, the brightest portions of the tail, near the head, should have been in the plane perpendicular to the orbit, and through the radius-vector; that is, in the plane through the sun and the magnetic axis of the nucleus. A section of the tail and envelope in this plane would show the brightest parts of the two branches streaming away from the two magnetic hemispheres, on the side towards the sun, bending around past the nucleus, and separated there by a dark space. In the earlier and later stages of an envelope, the dark shade would be enhanced by the deficiency of the streams that would return along the axis (415). (4.) The great difference noticed in the brightness of the two branches of the train of Donati's comet, near the head. This may reasonably be ascribed to an inequality in the discharge of nebulous matter from the two magnetic hemispheres. This inequality of brightness was not changed by the earth's passage through the plane of the comet's orbit (on Sept. 8). G. P. Bond infers from this, that " the initial plane passing through the two branches would seem to have a strong inclination to that of the orbit." Theory, as we have seen, assigns it a position nearly perpendicular to the plane of the orbit. (5.) The remarkable shifting of the superior brightness and eccentric position from one branch of the tail of Donati's comet, near the head, to the other, about October 10th. At about that date, the plane through the sun, comet, and earth, was perpendicu. lar to the plane of the comet's orbit, and the earth should therefore have passed from one side to the other of the initial plane of the branches of the train. Cotem 244 COMETS. poraneously with these changes, the dark, axial stripe nearly disappeared, and reappeared at later dates. 417. Explanation of the Rise and Recess of Successive Envelopes. To understand how one envelope after another may rise and recede to a certain distance from the nucleus, we have to consider that masses of vaporous magnetic matter may rise, at certain intervals, from the nucleus to a certain height in its atmosphere, under the operation of the sun's rays; and that such matter should ascend most abundantly from the equatorial regions, where the sun is supposed to act most directly, and flow off towards the poles. - It will be seen, if we consider the diverse directions that would be assumed by the magnetic columns in different magnetic latitudes, that, as a necessary consequence, the nebulous streams proceeding from them would rise to a greater and greater height towards the sun, until the process of discharge reached the magnetic latitude of 35~. The combination of all the nebulous streams thus originating would present the appearance of a luminous envelope on the side of the nucleus towards the sun, the outer boundary of which would recede steadily from the nucleus. 418. Diversities in the Brightness of an Envelope. The great diversity often observed in the brightness of different parts of the same envelope, may be ascribed to intersections, on the line of sight, of the separate streams of cometic particles, and to varying velocities in different parts of the same stream. Besides the ordinary diversities which are thus satisfactorily explained, sudden interruptions of brightness are often observed at certain parts of an envelope; these may result from sudden variations in the intensity of the expelling force, or in the quantity of matter discharged. The dark spots sometimes seen are probably due to a deficiency of nebulous matter near the nucleus, on certain magnetic parallels. Such deficiency may result primarily from an intermission in the ascent of nebulous matter from the equatorial regions of the nucleus. As the ascended matter flows off towards the poles, any vacuity thus arising will gradually pass from one latitude to another, and the spot answering to it in the envelope will rise and expand with the envelope. THE FIXED STARS. 245 CHAPTER XIX. THE FIXED STARS. CONSTELLATIONS.-DIVISION INTO MAGNITUDES. 419. IN order to distinguish the fixed stars from each other, they are arranged into groups, called Constellations, which are imagined to form the outlines of figures of men, animals, or other objects, from which they are named. Thus, one group is conceived to form the figure of a Bear, another of a Lion, a third of a Dragon, and a fourth of a Lyre. The division of the stars into constellations is of very remote antiquity; and the names given by the ancients to individual constellations are still retained. The resemblance of the figure of a constellation to that of the animal or other object from which it is named, is in most instances altogether fanciful. Still, the prominent stars hold certain definite positions in the figure conceived to be drawn on the sphere of the'heavens. Thus, the brightest star in the constellation Leo is placed in the heart of the Lion, and hence it has sometimes been called Cor Leonis or the Lion's Heart: and the brightest star in the constellation Taurus is situated in the eye of the Bull, and therefore sometimes called the Bull's Eye; while that conspicuous cluster of seven stars in this constellation, known by the name of the Pleiades, is placed in the neck of the figure. Again, the line of three bright stars noticed by every observer of the heavens in the beautiful constellation of Orion, is in the belt of the imaginary figure of this bold hunter drawn in the skies. The three larger stars of this constellation are, respectively, in the right shoulder, in the left shoulder, and in the left foot. 420. Different Classes of Constellations. The constellations are divided into three classes: NTorthern Constellations, Southern Constellations, and Constellations of the Zodiac. Their whole number is 91: Northern 34, Southern 45, and Zodiacal 12. The number of the ancient constellations was but 48. The rest have been formed by modern astronomers from southern stars not visible to the ancient observers, and others variously situated, which escaped their notice, or were not attentively observed. The zodiacal constellations have the same names as the signs of the zodiac (Def. 25, p. 17): but it is important to observe that the individual signs and constellations do not occupy the same places 246 THE FIXED STARS. in the heavens. The signs of the zodiac coincided with the zodiacal constellations of the same name, as now defined, about the year 140 B. C. Since then the equinoctial and solstitial points have retrograded nearly one sign: so that now the vernal equinox, or first point of the sign Aries, is near the beginning of the constellation Pisces; the summer solstice, or first point of Cancer, near the beginning of the constellation Gemini; the autumnal equinox, or first point of Libra, at the beginning of Virgo; and the winter solstice, or first point of Capricornus, at the beginning of Sagittarius. It follows from this, that when the sun is in the sign Aries, he is in the constellation Pisces, and when in the sign Taurus, in the constellation Aries, &c. For the rest, it should be observed that the constellations and signs of the zodiac have not precisely the same extent. 421. ilodes of Designation of Individual Stars. The stars of a constellation are distinguished from each other by the letters of the Greek alphabet, and in addition to these, it' necessary, the Roman letters, and the numbers 1, 2, 3, &c.; the characters, according to their order, denoting the relative magnitude of the stars. Thus a Arietis designates the largest star in the constellation Aries; 3 Draconis, the second star of the Dragon, &c. Some of the fixed stars have particular names, as Sirius, Aldebaran, Arcturus, Regulus, &c. 422. Magnitudes. The stars are also divided into classes, or magnitudes, according to the degrees of their apparent brightness. The largest or brightest are said to be of the first magnituide; the next in order of brightness, of the second magnitude; and so on to stars of the sixth magnitude, which includes all those that are barely perceptible to the naked eye. All of a smaller kind are called telescopic stars, being invisible without the assistance of the telescope. The classification according to apparent magnitude is continued with the telescopic stars down to stars of the twentieth magnitude (according to Sir John Herschel), and the twelfth according to Struve. The following are all the stars of the first magnitude that occur in the heavens, viz.: Sirius or the Dog-star, Betelgeux, ERigel, Aldebaran, Capella, Procyon, Regulus, Denebola, Cor. Hydrce, Spica Virginis, Arcturus, Antares, Altair, Vega, Deneb or Alpha Cygni, Dubhe or Alpha Ursce Majoris, Alpherat or Alpha Andromedce, Fomalhaut, Achernar, Canopus, Alpha Crucis, and Alpha Centauri. It is the practice of Astronomers to mark more or less of these stars as intermediate between the first and the second magnitude; and in some catalogues some of them are assigned to the second magnitude. All of these stars, with the exception of the last four, come above the horizon in all parts of the United States. 423. Celestial Globe. There are two principal modes of CONSTELLATIONS.-DIVISION INTO MAGNITUDES. 247 representing the relative positions of the stars; the one by delineating them on a globe, where each star occupies the spot in which it would appear to an eye placed in the centre of the globe, and where the situations are reversed when we look down upon them; the other is by a chart or map, where the stars are generally so arranged as to be represented in positions similar to their natural ones, or as they would appear on the internal concave surface of the globe. The construction of a globe or chart, is effected by means of the right ascensions and declinations of the stars. Two points diametrically opposite to each other on the surface of an artificial globe are taken to represent the poles of the heavens, and a circle traced 90~ distant from these for the equator: another point 23~~ from one of the poles is then fixed upon for one of the poles of the ecliptic, and with this point as a geometrical pole a great circle described; the points of intersection of the two circles will represent the equinoctial points. The point which represents the place of a star is found by marking off the right ascension and declination of the star upon the globe. All the fixed stars visible to the naked eye, together with some of the telescopic stars, are represented on celestial globes of 12 or 18 inches in diameter. 424. Catalogue of Stars. The places of the fixed stars are generally expressed by their right ascensions and declinations, but sometimes also by their longitudes and latitudes. A table containing a list of fixed stars designated by their proper characters, and giving their mean right ascensions and declinations, or their mean longitudes and latitudes, is called a Catalogue of those stars. (See Table XC (a) ). NUMBER AND DISTRIBUTION OVER THE HEAVENS. 425. The number of stars visible to the naked eye, in the entire sphere of the heavens, is from 6,000 to 7,000; of which nearly 4,000 are in the northern hemisphere; but not more than 2,000 can be seen with the naked eye at any one hour of the night at a given place. The telescope brings into view many millions, and every material augmentation of its space-penetrating power greatly increases the number. As to the number of stars belonging to each different magnitude, astronomers assign from 20 to 24 to the first magnitude, from 50 to 60 to the second, about 200 to the third, and so on; the numbers increasing very rapidly as we descend in the scale of brightness; the whole number of stars already registered down to the seventh magnitude, inclusive, amounting to 12,000 or 15,000. The reason of this increase in the number of the stars, as we 248 THE FIXED STARS. descend from one magnitude to another, is undoubtedly that in general the stars are less bright in proportion as their distance is greater; while the average distance between contiguous stars is about the same for one magnitude as for another. It is easy to see that upon these suppositions the number of stars posited at any given distance, and having therefore the same apparent magnitude, will be greater in proportion as this distance is greater, and thus as the apparent magnitude is lower. 426. Comparative Brighltress. It is not to be understood that the classification of the stars into different magnitudes, is made according to any fixed definite proportion subsisting between the degrees of apparent brightness of the stars belonging to different classes. Stars of almost every gradation of brightness, between the highest and the lowest, are met with. Those which offer marked differences of lustre, form the basis of the classification; others, which do not differ very widely from these, are united to them. As a necessary consequence, there are some stars of intermediate lustre, which cannot be assigned with certainty to either magnitude. Thus, in the catalogue published by the Astronomical Society of London, 3 stars are marked as intermediate between the first and second magnitudes, and 29 between the second and third. Different astronomers also not unfrequently assign the same star to different magnitudes. As to the proportions of light emitted from the average stars of the different magnitudes, according to the experimental comparisons of Sir Winm. Herschel, they are, from the first to the sixth magnitude, approximately in the ratio of the numbers, 100, 25, 12, 6, 2, 1. 427. Distribution of the Stars. With the exception of the three or four brightest classes, the stars are not distributed indiscriminately over the sphere of the heavens, but are accumulated in far greater numbers on the borders of that belt of cloudy light in the heavens, which is called the milky way, and in the milky way itself, which the telescope shows to consist of an immense number of stars of small magnitude in close proximity. According to Struve, the total number of stars visible in the Herschelian telescope of 20 feet focus and 19 inches aperture, is a little over 20,000,000. 428. Stratum of the Milky Way. The great accumulation of stars in a zone of the heavens, encompassing the earth in the direction of a great circle, suggested to the mind of Herschel the idea that the stars of our firmament are not disseminated indifferently throughout the surrounding regions of space, but are for the most part arranged in a stratum, the thickness of which is very small in comparison with its breadth; the sun and solar system being near the middle of the thickness. If S (Fig. 103) represents the place of the sun, it will be seen that NUMBER A-ND DISTRIBUTION OVER THE HEAVENS. 249 upon this supposition the number of' stars il C the direction SC of the S~~rXZ~~k. Xl~~)c~thicknessof the stratum,,pX { e g g s_: w> ~p~ aS h~ iL3;r: ~~~17Pwill be less than in any I -A" other direction, and that the greatest number will FIG. 103. lie in the direction of the breadth, as SB. On one side of the point S, the stratum is supposed to be divided for a certain distance into two laminae, as shown in the figure, which represents a section of tile supposed stratum. This supposition is necessary to account for the two branches, with a dark space between them, into which the milky way is divided for about one-third of its course. Herschel undertook to gauge this stratum in various directions, on the principle that the distance through to its borders in any direction was greater in proportion as the number of stars seen in that direction was greater. He thus found that its actual form was very irregular; its section, instead of being truly that of a segment of a sphere divided for a certain distance into two lamina, as represented in Fig. 103, having the form represented in Fig. 104. He estimated the thickness of the stratum to be less than 160 times the interval between the stars, and the breadth to be nowhere greater than 1,000 times the same distance. These are his first results; we shall see in the sequel that they were materially modified by his subsequent investigations. Sir John Herschel conceives that the superior brilliancy and larger development of the milky way in the southern hemisphere, from the constellation Orion to that of Antinous, indicate that the sun and his system are at a distance from the centre of the stratum in the direction of the Southern Cross, and that the central parts are so vacant of stars that the whole approximates to the form Fr. 104. of an annulus. 250 THE FIXED STARS. ANNUAL PARALLAX AND DISTANCE OF THE STARS. 429. The Annual Parallax of a fixed star is the angle made by two lines conceived to be drawn, the one from the sun and the other from the earth, and meeting at the star, at the time the earth is in such part of its orbit that its radius-vector is perpendicular to the latter line; or, in other words, it is the greatest angle that can be subtended at the star by the radius of the earth's orbit. Thus, let S (Fig. 105) be the sun, s a fixed star, and E the earth, in such a position 8 that the radius-vector SE is perpendicular to Es the line of direction of the star, then ~E ~C__,_-CII — i<~. - j the angle SsE is the annual parallax of the star s. 430. Least Distance of the Stars. If the annual parallax of a star were known, we might easily find its dis\.. / tance fromrn the earth; for in the right-angled triangle SEs FIG. 105. we would know the angle SsE and the side SE, and we should only have to compute the side Es. Now, if any of the fixed stars have a sensible parallax, it could be detected by a comparison of the places of the star, as observed from two positions of the earth in its orbit, diametrically opposite to each other; and accordingly, the attention of astronomers furnished with the most perfect instruments, has long been directed to such observations upon the places of some of the fixed stars, in order to determine their annual parallax. But, after exhausting every refinement of observation, they have not been able to establish, until quite recently, that any of them have a measurable parallax. Now, such is the nicetv to which the observations have been carried, that, did the angle in question amount to as much as 1", it could not possibly have escaped detection by the methods of observation employed. We may then conclude that the annual parallax qf the nearest fixed star is less than 1". Taking the parallax at 1", the distance of the star comes out 206,265 times the distance of the sun from the earth, or about 20 millions of millions of' miles. The distance of the nearest fixed star must therefore be greater than this. A juster notion of the immense distance of the fixed stars, than can be conveyed by figures, may be gained from the consideration that light, which traverses the distance between the sun and earth in 8m. 18s., and would perform the circuit of our globe in ~ of a second, employs 31 years in coming from the nearest fixed star to the earth. ANNUAL PARALLAX AND DISTANCE OF THE STARS. 251 431. Determination of the Parallax of a Fixed Star. The long continued endeavor to detect an annual parallax of a fixed star, by the direct method of comparing the places of the star, determined at an interval of half a year, has at last been crowned with success. The parallax of a Centauri has been thus determined by Professor Henderson, from observations made in 1832 and 1833, with a large mural circle. Subsequent observations with a more efficient instrument by Maclear have furnished an angle of parallax differing but little from that obtained by Henderson. Its value is 0".913, which answers to a distance about -1 less than the least limit of distance of the stars, just determined. The parallax and distance of Sirius and of the polestar, have since been determined in a similar manner, but with less certainty. The result obtained for the parallax of the pole-star is 0".11, and for that of Sirius an angle a little greater. A parallax of 0".11 answers to a distance that light would require nearly 30 years to traverse. 432. Parallax of a Star found by the Differential Method. The honor of being the first to determine with certainty the parallax and distance of a fixed star belongs to Bessel. The star observed by him is that designated as 61 Cygni. It is a star of about the 6th magnitude, barely visible to the naked eye. When viewed through a telescope it is seen to consist of two stars of nearly equal brightness, at a distance from each other of about 16". These stars have a motion of revolution around each other, and the two move together at the same rate of 5".3 per year, as one star, along the sphere of the heavens. It is hence inferred that they are bound together into one system by the principle of gravitation, and are at pretty nearly the same distance from the earth. The great proper motion of this double star, as compared with other stars, led to the suspicion that it was nearer than any other; and thus to attempts to determine its parallax. The principle of Bessel's method is to find the difference between the parallaxes of the star 61 Cygni, and some other star of much smaller magnitude, and therefore supposed to be at a much greater distance, seen in as nearly the same direction as possible. This difference will differ from the absolute parallax of the double star by only a small fraction of its whole amount. It was found by measuring with a position micrometer (62) the annual changes in the distance of the two stars, and in the position of the line joining them. To make it evident that such changes will be an inevitable consequence of any difference of parallax in the two stars, conceive two cones having the earth's orbit for a common base, and their vertices respec- A tively at the two stars, and imagine their sur-.faces to be produced past the stars until they intersect the heavens. The intersections will be ellipses, but, by reason of the different distances of the two stars, of different sizes, as represented in Fig. 106; and they will be apparently described by the stars in the b course of one revolution of the earth in its D B orbit. The two stars will always be similarly situated in their parallactic ellipses: thus, P if one is at A the other will be at a; and after the earth has made one-quarter of a revolution, they will be at B and b; and after another quarter of a revolution at C and c, &c. Now it will be manifest, on inspecting the o figure, the ellipses being of unequal size, FIG. 106. that the lines of the stars will be of unequal lengths, and have different directions in the different situations of the stars, 252 THE FIXED STARS. A much smaller angle of parallax may be found, with the same degree of certainty, by this indirect method, than by the direct process explained in Art. 430; for since the two stars are seen in pretty nearly the same direction, they will be equally affected by refraction and aberration; and since it is only the relative situations of the two stars that are measured, no allowance has to be made for precession and nutation, or for errors in the construction or adjustment of the instrument. It is therefore independent of the errors that are inevitably committed in the determination of these several corrections, when it is attempted to find directly the absolute parallax, by observing the right ascension and declination at opposite seasons of the year. The measurements made with the micrometer in the hands of the most accurate observers, may be relied on as exact to within a small fraction of 1". For the sake of greater certainty Bessel made the measurements of parallactic changes of relative situation between the star 61 Cygni and two small stars instead of one,-the middle point between the two members of the double star being taken for the situation of this star. He found the difference of parallax to be for the one star 0".3584, and for the other star 0".3289: and assuming the absolute parallax of the two stars to be equal, found for the most probable value of the difference of parallax 0".3483. Whence he calculated the distance of the star 61 Cygni to be 592,200 times the mean distance of the earth from the sun; a distance which would be traversed by light in 9- years. The number of stars whose parallax and distance have been determined, more or less accurately, by both methods, now amounts to 12. The least parallax obtained is that of Capella, which is 0".05; but it must be regarded as quite uncertain. 433. Comparative Distances of Stars of Different Magnitudes. According to Peters, the mean parallax of stars of the second magnitude is 0".116, which answers to a distance that light would traverse in 28 years. From this result the mean parallax and distance of stars of each of the different magnitudes have been approximately deduced by means of the relative distances of stars of the different magnitudes, as determined by Struve on the assumption that the stars are uniformly distributed through space (at least in certain directions), and that the light from the stars of the different magnitudes varies according to a certain admitted law. The mean distance of stars of the first magnitude, as computed, is traversed by light in 15-5 years; and that of a star of the sixth magnitude in 120 years. Light requires 138 years to come from the most remote star visible to the naked eye. The same principle of computation of distances being extended to the telescopic stars, it appears that the stars just visible in the iHerschelian telescope of 20 ft. focus are separated from us by a distance that light takes 3,500 years to journey over. This is on the supposition that the rays of light do not experience any sensible degree of extinction in traversing the regions of space. NATURE AND MAGNITUDE OF THE STARS. 434. The vast distance at which the fixed stars are visible, and shine with a light not much inferior to the planets, leaves no room to doubt that they are all suns, or self-luminous bodies. VARIABLE STARS. 253 If it should be conjectured that some of the fainter stars might be bodies shining by reflected light, like the planets, the answer is, that if we were to suppose the existence of opake bodies, at the distance of the stars, so inconceivably vast in their dimnensions as to send a sensible light to the eye, if illuminated to the same degree as the planets, the stars of the smaller magnitudes are too remote from the brighter ones to receive sufficient light from them; for, the smallest measurable space in the field of the larger telescopes is, at the distance of the nearer stars, nearly as large as the earth's orbit. It is perhaps possible, that some of the faintest members of some of the double stars, as surmised by Sir John Herschel, may shine by reflected light. 435. Magnitude of the Stars. To be able to determine the magnitude of a star, we must know its distance, and also its apparent diameter. Now the distances of but few stars have as yet been found; and the discs of all the stars, even in the most powerful telescopes, are altogether spurious; so that in no instance have we the data, nor have we reason to expect that they will be hereafter obtained, for determining with certainty the magnitude of a fixed star. But we may infer from the quantity of their light as compared with that of the sun, and the mean distances of stars of the different magnitudes, as approximately determined (433), that the stars are in general much larger than the sun. According to the mean result of recent photometrical comparisons made by Messrs. G. P. Bond and Alvan Clark, between the bright star a Lyrae and the sun, if the sun were removed to 133,500 times its present distance it would send us the same quantity of light as this star. From this we may infer that if it were removed to the distance of the nearest star (430), it would appear as a star of the second magnitude; and that if it were removed to the mean distance of stars of the first magnitude, it would appear as a star of the sixth magnitude, and be just visible to the naked eye. It would seem then that the sun is much smaller than most, if not all, of the stars of the first magnitude; and is presumably also smaller than most of the stars of the other magnitudes. VARIABLE STARS. 436. There are many stars which exhibit periodical changes of brightness; these are termed Variable Stars. More than a hundred stars are now known to belong to this class. One of the most remarkable of the variable stars is o Ceti, or Mira. From being as bright as a star of the second magnitude, it gradually decreases until it entirely disappears; and after remaining for a time invisible, reappears, and gradually increasing in lustre, finally recovers its original appearance. The mean period of 254 THE FIXED STARS. these changes is 3311 days. The star remains at its greatest brightness about two weeks, employs about three months in waning to its disappearance, continues invisible for about five months, and during -the remaining three months of its period increases to its original lustre. Such has been the general course of its phases. But it does not always recover the same degree of brightness, nor increase and diminish by the same gradations. It is even related by Hevelius, that in one instance it remained invisible for a period of four years. A similar phenomenon has been noticed in the case of the star X Cygni. According to the testimony of Cassini, it was scarcely visible throughout the years 1699, 1700, and 1701, at those times when it should have been most conspicuous. On the other hand a variable star situated in the Northern Crown, sometimes fluctuates in its brightness very slightly for several years, and then suddenly resumes its regular variations, in the course of which it entirely disappears. The greater number of variable stars undergo a regular increase and diminution of lustre without ever becoming entirely invisible. Algol, or A Persei, is a remarkable variable star of this description. For a period of 2d. 14h., it appears as a star of the second magnitude, after which it suddenly begins to diminish in splendor, and in about 31 hours is reduced to a star of the fourth magnitude. It then begins again to increase, and in 31 hours more is restored to its usual brightness, going through all its changes in 2d. 20h. 49m. Besides the single variable stars, there are also a number of double stars, one or both the members of which are variable; as y Virginis, E Arietis, C Bootis, &c. 437. General Facts. Two general facts have been noticed with respect to the variable stars which are worthy of remark, viz.: that the color of their light is red, and that their period of increase of lustre is shorter than that of the decrease. The star Algol, offers an exception to both of these general facts. The ruddy color is especially observable in the case of the smaller variable stars. It is a curious and suggestive fact that a number of the variable stars present a hazy appearance at their minimum, as if some form of nebulous matter were interposed between them and the eye. 43S. Temporary Stars. There are also some instances on record of temporary stars having made their appearance in the heavens; breaking forth suddenly in great splendor, and without changing their positions among the other stars, after a time entirely disappearing. One of the most noted of these is the star which suddenly shone forth with great brilliancy on the 11th of November, 1572, between the constellations Cepheus and Cassiopeia, and was attentively observed by Tycho Brah6. It was then as bright as any of the permanent stars, and continued to VARIABLE STARS. 255 increase in splendor till it surpassed Jupiter when brightest, and was visible at mid-day. It began to diminish in December of the same year, and in March, 1574, entirely disappeared, after having remained visible for sixteen months, and has not since been seen. It was noticed that while visible the color of its light changed from white to yellow, and then to a very distinct red; after which it became pale, like Saturn. In the years 945 and 1264, brilliant stars appeared in the same region of the heavens. It is conjectured from the tolerably near agreement of the intervals of the appearance of these stars, and that of 1572, that the three may be one and the same star, with a period of about 300 years. The places of the stars of' 945 and 1264 are, however, too imperfectly known to establish this with any degree of certainty. Besides these three temporary stars, several others have made their appearance, viz.: one in the year 134 B. C., seen by Hipparchus; another in 389 A. D., in the constellation Aquila; a third in the 9th century, in Scorpio; a fourth in 1604, in Serpentarius, seen by Kepler; a fifth in 1670, in the Swan; and a sixth in 1848, in Ophiuchus. What is no less remarkable than the changes we have noticed, several stars, which are mentioned by the ancient astronomers, have now ceased to be visible, and some are now visible to the naked eye which are not in the ancient catalogues. 439. Explanation of Variable Stars. The most probable explanation of the phenomenon of variable stars is that they are self-luminous bodies rotating upon axes, and having, like the sun, spots developed periodically on their surface, under the varying action of revolving planets upon their photospheres. The range of the planetary action must be regarded as much greater than in the case of the sun. The fluctuations generally observable in the periods and in the maxima and minima of brightness of the variable stars, are analogous to the fluctuations that occur in the periods and maxima and minima of the sun's spots. Prof. Wolf has minutely investigated this correspondence of phenomena, in the case of certain stars, by constructing curves showing their variations of light in detail. The hazy appearance often presented by variable stars at their minimum, may result from the interposition of nebulous matter expelled from the star in the process of formation of the spots on its surface (293). The ruddy color frequently noticed may be ascribed to a lower temperature consequent upon a greater prevalence of spots, or to more intense electric discharges within the photosphere. In the case of the star Aigol the phenomena are precisely such as would result from the periodical interposition of an opake body. In those cases in which the period of the diminution of the light is a large iraction of the entire period of the star, as well as those in which there are occasional interruptions in the regular recurrence of the phenomena, the supposition of the interposition of an opake body will not answer. Temporary stars may be supposed to be suns which have entirely omitted the evolution of light for a long period of time, and then burst forth anew with a sudden and peculiar splendor, under the influence of a planetary action returning to its maximum at the end of a long period. Or they may possibly result from an encounter of two stars at the point of intersection of the vast orbits which they pursue in the regions of space. The remarkable fact, noticed by Sir John IIerschel, that all the temporary stars on record, of which the places are distinctly indicated, have occurred in or close upon the borders of the Milky Way, where, as we shall see, the stars are most condensed, lends some support to the latter hypothesis. 256 THE FIXED STARS. DOUBLE STARS. 440. Many of the stars which to the naked eye appear single, when examined with telescopes are found to consist of two (in some instances three or more) stars in close proximity to each other. These are called Double Stars, or JAiultt)le Stars. (See Fig. 107.) This class of bodies was first attentively observed by Sir William Herschel, who, in the years 1782 and 1785, published Castor. y Leonis. Rigel. Pole-star. 11 Monoc. d Cancri. FIG. 107. catalogues of a large number of them which he had observed. The list has since been greatly increased by Professor Struve, of Dorpat, Sir J. F. W. Herschel, and other observers, and now amounts to several thousand. 441. Degree of ProximiWnfty. Double stars are of various degrees of proximity. In a great number of instances, the angular distance of the individual stars is less than 1", and the two can only be separated by very powerful telescopes. In other instances, the distance is ~' and more, and the separation can be effected with telescopes of very moderate power. They are divided into different classes or orders, according to their distances; those in which the proximity is the closest forming the first class. 442. Comparative size. The two members of a double star are generally of quite unequal size (See Fig. 107). But in some instances, as that of the star Castor, they are of nearly the same apparent magnitude. Double stars occur of every variety of magnitude. Sirius is the largest of the double stars. It is attended by a minute companion star, at a distance of 10". This was first discovered by Clark, with his great telescope of 1S-in. aperture. In some instances one of the constituents of a double star is itself double. s Lyrve offers the remarkable combination of a double-double star. 443. Different Colors. It is a curious fact, that the two constituents of a double star in numerous instances shine with different colors; and it is still more curious that these colors are in general complementary to each other. Thus, the larger star is usually of a ruddy or orange hue, while the smaller one appears blue or green. This phenomenon has been supposed to be in some cases the effect of contrast; the larger star inducing the accidental color in the feebler light of the other. Sir John HIerschel cites as probable examples of this effect the two stars t Cancri, and y Andromede. But it is maintained by Nichol that this explanation cannot be admitted; for, if true, it ought to be universal, whereas there are many systems similar in relative magnitudes to the contrasted ones, in which DOUBLE STARS. 257 both stars are yellow, or otherwise belong to the red end of the spectrum. Again, if the blue or violet color were the effect of contrast, it ought to disappear when the yellow star is hid from the eye; which, however. it does not do. Thus, the star # Cygni consists of two stars, of which one is yellow, and the other shines with an intensely blue light; and when one of them is concealed from view by an interposed slip of darkened copper, the other preserves its color unchanged. The color, then, of neither of the stars can be accidental. It may be remarked in this connection, that the isolated stars also shine with various colors. For example, among stars of the first magnitude, Sirius, Vega, Altair, Spica are white, Aldebaran, Arcturus, Betelgeux red, Capella and Procyon yellow. In smaller stars the same difference is seen, and with equal distinctness when they are viewed through telescopes. According to Herschel, insulated stars of a deep red color, occur in many parts of the heavens, but no decidedly green or blue star has ever been noticed unassociated with a companion brighter than itself. 444. Discovery of Binary Stars. Sir William Herschel instituted a series of observations upon several of the double stars, with the view of ascertaining whether the apparent relative situation of the individual stars experienced any change in consequence of the annual variation of the parallax of the star. With a micrometer adapted to the purpose, (62), he measured from time to time the apparent distance of the two stars, and the angle formed by their line of junction with the meridian at the time of the meridian passage, called the Angle of Position. Instead, however, of finding that annual variation of these angles, which the parallax of the earth's annual motion would produce, he observed that, in many instances, they were subject to regular progressive changes which seemed to indicate a real motion of the stars with respect to each other. After continuing his observations for a period of twenty-five years, he satisfactorily ascertained that the changes in question were in reality produced by a motion of revolution of one star around the other, or of both around their common centre of gravity; and in two papers, published in the Philosophical Transactions for the years 1803 and 1804, he announced the important discovery that there exist sidereal systems composed of two stars revolving about each other in regular orbits. These stars have received the appellation of Binary Stars, to distinguish them from other double stars which are not thus physically connected, and whose apparent proximity may be occasioned by the circumstance of their being situated on nearly the same line of direction from the earth, though at very different distances from it. Sirnilar stars, consisting of more than two constituents, are called Ternary, Quaternary, -&c. Since the time of Sir W. Herschel, the observations upon the binary stars have been continued by Sir John Herschel, Sir James South, Struve, Bessel, M/dler, and other astronomers. According to Ma/dler the number of known binary and ternary stars is now about 600. Every year materially increases the list; and will probably (continue to do so for some time to come: for, while the changes of relative situation are in some instances 17 258 THE FIXED STARS, exceedingly slow, the actual number of such systems is probably a large fraction of the whole number of double stars; at least, if we confine our attention to double stars whose constituents are within ~' of each other. This may be interred from the fact, that the number of such double and. multiple stars actually ob~ serve(l, which anmounts to over 3000, is at least ten timhes greater than the number of instances of fertuitous juxtaposition that would obtain on the supposition of a unlifo>rm distribution of thle stars. Besides, there are a numnber of double stars not yet discovered to have a motion of revolution, whllch still give indications of a physical connectiion. Thus, their constituents are found to have constantly the same proper motion in the same direction; showing that they are in all probability moving, as one system through space. 4415. Periods aad Orbits of Binar t $trs, From the observations made upon some of the binary stars, astronomers have been enabled to deduce the form of their orbits, and approximately the lengths of their periods. The orbits are ellipses of considerable eccentricity. The periods are of various lengths, as will be seen from the followin( enumeration of soe of tl-lhose considered as best ascertained:?PI Bootis 650 years, v/ Virginis 171 years, p Ophillchi 92 years, ca Centauri 77 years, f Cancri 58 years, HEerculis 36 yearso Fig. 108 represents a portion of the Fig. 108o napparent orbit of the double star v Virginis, and shows the relative positions of the two members of the double star in various years. At the time of tleir nearest approach, in 18836, the inter~ val between them-) was a firaction of 1", and they could not be separated by the best telescopes, with a manifvying power of 1()00. Since then their distlulece has been co>ntinua/lly increasi lI. In 1844 it amounted to 2", and a power of from 200 to 300 was suffcient to separate thetm, The orbit represented in the PROPER MOTIONS OF THE STARS. 259 figure is the stereographic projection of the true orbit on a plane perpendicular to the line of sight. Thle actual distance between the members of a binary star has been found for 61 Cygni, and X Centauri. Bessel makes it for the first about two and a half times the distance of Uranus from the sun. It is important to observe, that the revolution of one star around another is a different phenomenon from the revolution of a planet around the sun. It is the revolution of one sun around another sun; of one solar system around another solar system; or rather of both around their common centre of gravitv. We learn from it the important fact, that the fixed stars are endued with the same property of attraction that belongs to the sun and planets. PROPER MOTIONS OF THE STARS. 446. It has already been stated that the fixed stars, so called, are not all of them rigorously stationary. By a careful comparison of their places, found at different times with the accurate instruments and refined processes of modern observation, it has been found that great numbers of them have a progressive motion along the sphere of the heavens, from year to year. The velocity and direction of this motion are uniformly the same for the same star, but different for different stars. One of the stars which has the greatest proper motion, is the double star 61 Cygni. During the last fifty years it has shifted its position in the heavens 4' 21"; the annual proper motion of each of the individual stars being 5".2. An isolated star, called e Indi, has a still greater proper motion. It changes its place 7".7 every year. The proper motions of some of the stars are either partially or entirely attributable to a motion of the sun an(l the whole solar system in space; but the motions of others cannot be reconciled with this hypothesis, and must be regarded as indicative of a real motion of these bodies in space. 447. M1otion of the Solar System through Space. The first successful attempt to explain the proper motions of the fixed stars on the hypothesis of a motion of the solar system through space, was made by Sir William Herschel. After a careful examination of these motions, he conceived that the majoritv of them could be explained on the supposition of a general recess of the stars from a point nealr that occupied by the star A Herculis towards a point diametrically opposite. Whence he inferred that the sun, with its attendant system of planets, was moving rapidly through space in a direction towards this constellation. Doubt has since been thrown upon these conclusions by Bessel and other astronomers; but they have recently been decisively reestablished by iM. Argelander, of Abo. The investigations of 260 THE FIXED STARS. Argelander, which were communicated to the Academy of St. Petersburgh in 1837, have since been confirmed by M. Otto Struve, of the Pulkowa Observatory, and other eminent observers. Taking, the mean of all the more recent determinations, we find the most probable situation of the point towards which the sun's motion is directed to be as follows: R. A. 260~ 14', N. Dec. 35~ 10'. This point is a little to the east and north of the star u in the constellation Hercules, and about 9~ distant from the point first supposed by Herschel. 448. Velocity of SunSs Motion through Space. O. Struve finds that for a star situated at right angles to the direction of the sun's motion, and placed at the mean distance of the stars of the first magnitude, the annual angular displacement due to the sun's motion is 0".339 (with a probable error of 0".( 25). So that, if we assume, according to the best determinations, 0".2o)9 for the hypothetical value of the parallax of a star of the first magnitude, it follows that at the distance of the star supposed the annual motion of the sun subtends an angle 1.623 times greater than the radius of the earth's orbit: which makes it nearly 150,000,000 of miles. This is at the rate of 4.7 miles per second. 449. Velocity of the proper motions of the stars. The above angle of 0".339 is about the greatest annual displacement which a star can experience in consequence of the sun's motion. Whence it appears that the whole of the proper motion of any star which is over and above this amount must certainly be due to a real motion in space. Thus, in the case of the star 61 Cygni, nearly 5" of its annual proper motion (5".23:) mnust result from an actual motion in space. This is 14.37 times greater than the parallax of this star (0".35). Accordingly, if we suppose the direction of its motion to be perpendicular to its lilne of direction from the sun or earth, its annual motion is 14.37 times greater than the radius of the earth's orbit, or at the rate of nearly 42 miles per second. As we have no means of ascertaining the actual direction of its motion, it is impossible to discover how much the velocity exceeds this determination. 450. Sun's motion comparatively slow. By comparing the particular motions presented by stars of different classes with the motion of the solar system, viewed perpendicularly at the distance of a star of the first magnitude, as above given, it is found that the former, at the mean, are 2.4 times greater than that of the sun; whence it follows that this luminary may be ranked among those stars which have a comparatively slow motion in space. NEBULA. 261 CLUSTERS OF STARS. 4,1. In many parts of the heavens stars are seen crowded to)gether into clusters, often in numbers too great to be counted. Some of these clusters are visible to the naked eye. One of the most conspicuous is that called the Pleiades. To the unaided sight it appears to consist of six or seven stars, but with a telescope of moderate power 50 or 60 conspicuous stars are seen grouped together within the same space, and more than 100 smaller ones are distinctly discernible. In the constellation Cancer is a luminous spot called Prcesepe, or the bee-hive, which a telescope of moderate power resolves entirely into stars. Within a space about.- square, more than 40 conspicuous stars are seen, besides many smaller ones. In the sword-handle of Perseus is another cloudy spot thickly crowded with stars, which become separately visible with a telescope of low power. One of the richest clusters in the northern hemisphere occurs in the constellation Hercules, between the stars 4 and s. It is visible to the naked eye, on clear nights, as a hazy mass of light; which is readily resolved into stars by a good telescope. Viewed through a telescope of high power it presents the magnificent aspect of an innumerable host of stars crowded together towards the centre into a perfect blaze of light. The richest and largest cluster in the whole heavens is seen in the constellation Centaurus, in the southern hemisphere. It is visible to the naked eye as a nebulous star, and is designated X Centauri. The telescope shows it to consist of an immense multitude of stars congregated together in the form of a magnificent globular cluster (see Fig. 1, Plate IV.). In the field of view of a large telescope, it has an apparent diameter nearly equal to that of the moon. NEBULAE. 452. With the aid of the telescope, a great number of faintly luminous spots, or patches, are seen scattered here and there over the sphere of the heavens. These are called NTebulke. Some of these nebulous objects are partially visible to the naked eye, but the great majority of them cannot be discerned without the assistance of a good telescope, and very many are beyond the reach of any but the most powerful instruments. 453. Number anid Distribution of Nebulae. The number of nebulse hitherto discovered, is over 5,000. They are very unequally distributed over the heavens, especially in the northern hemisphere. They are most abundant in the constellations Virgo, Leo, Comrra Berenices, Canes Venatici, and Ursa Major; 262 THE FIXED STARS. and occur in astonishing profusion in certain regions in this quarter of the heavens, as in the northern wing of Virgo. When the telescope is directed towards these regions it is observed that the nebnule follow each other in rapid succession, from the diurnal motion of the heavens; while, in some parts of the heavens, hours elapse after one of them has passed through the field before another enters. In the southern hemisphere there are two detached spaces of considerable extent, visible to the naked eye, called the iliagellanic Clouds, that shine with a nebulous light like the milky way, which are thickly sown with nebulae. 454. Diversity of Form antd Appearance. As seen through telescopes of' moderate power, the nebulte are, for the most part, round or ov;al in form; but, when carefully examined with the larger telesco)pes, they are found to present a great variety of aspects and forms. A large number are found to consist of a multitude of minute stars distinctly separate, and condensed about one or more points within the mass. Many others take on the appearance of incipient resolvability, characterized by the phrase star-dust, and are doubtless real clusters too distant, or too much condensed, to show their individual stars. Others still offer no appearance of stars, and remain the same cloud-like objects when the highest telescopic power is applied to them. These Irresolvable Nebuice were supposed by Sir William Herschel to be masses of actual nebulous matter (disseminated through space, but are now generally believed to be clusters, or beds of stars, like the rest; only too vastly remote to be revealed as such by any optical means yet employed. 445. Classification of Ne)nulae. The nebulae are classified according to their aspects and forms. as seen through the larger telescopes, as follows: (1) Globular Clusters. (2) Irregular Clusters, (3) Oval Nebulce, (4) Annular Nebulce, (5) Planetary Nebulce, (6) Stellar lVebulce, (7) Spiral NTebalce, (8) Irregular Nebulce, (9) Double Nebulce. 456. Globular Clusters take their name from their supposed actual form. Their component stars are so crowded together as to form an almost definite outline, and they run up to a blaze of light towards the centre, where their condensation is the greatest. The number of stars congregated in a single cluster is to be told only by thousands and tens of thousands; although their apparent size does not in any instance exceed the - part of the moon's disc. They are, in general, difficult of resolution, and appear in telescopes of moderate power as small, round, nebulous specks, resembling a comet without a tail. Fig. 3, Plate IV., represents a globular cluster to be seen in the constellation Pegasus. 457. Irregular Clusters. These are more or less irregular and indefinite in their outline. They are generally less rich in stars, and less condensed towards the centre than the globulax NEBUL2E. 263 clusters. Fig 2, Plate IV., represents an irregular cluster situate(l in the constellation Capricornus. The Pleiades, and Coma Berenices, are instances of' irregular clusters whose individual stars are seen in telescopes of low power. Irregular clusters occur of almost every degree of condensation, from a cluster which seems to be only a space of an irregular and ill-defined outline, somewhat more rich in stars than the surrounding regions, to the perfectly defined globular cluster highly condensed at the centre. 45S. Oval Nebulae. Nebulae having a distinct elliptic outline occur of various degrees of eccentricity, from moderately oval to an elongation almost linear (see Figs. 5, 6, and 7, Plate IV.). They are more condensed, though in very different degrees, in their central parts, and often present great arnd sudden variations of brightness from one portion of their niass to another.. This is very observable in Fig. 9, Plate V. Such nebulm, which retain their oval form in the field of the most powerful telescope, are doubtless spheroidal clusters, in their general form, though more or less complex in their internal structure. Many of them are either, wholly or partially resolvable into individual stars. Others afford to the eye only indistinct intimations of their stellar structure. In general the spheroidal clusters are far more difficult of resolution than globular clusters. 459. Dynainical Equilibri-ium of Sidereal Systems. It: cannot be doubted that the systematic organization of sidereal, systems has been determined under the operation of the princi — ple of universal gravitation; and it is plain that in the instance of globular and spheroidal clusters, a general state of equilibrium would be possible only upon the supposition that theindividual stars of each cluster revolve around some central' point. Such a general dynamical equilibrium of a cluster may however exist, and the internal structural condition be subjects at the same time to secular changes, from the varying combinations of individual orbital positions, and the disturbing actionsof some of the component stars on one another. 460. Annular Nebiula. A very small number of observed.: nebulae have the annular form (Fig. 12, Plate V.). A conspicuous example of this singular class of nebula may be seen with. a telescope of moderate power, midway between the stars i and, Lyr-e. The central vacuity is not perfectly dark, but filled. with a faint nebulous light. The telescope of Lord Rosse, hasresolved it into minute stars, and shown it to be frinlged on itsouter edge with filaments of stars (Fig. 11, Plate V.). Chacornac, of the Paris observatory, describes it as presenting, in Fbucault's great telescope of plated glass, the appearance of a hollow cylindrical bed of very small stars, with a thin stratum of minute stars stretching across the centre. 461. Planetary Nebulae have a round planet-like disc of 264 THE FIXED STARS. all equable lighlt throughout, or only slightly mottled, and often perfectly definite in outline. As many as 25 of these curious objects have been discovered. A large planetary nebula occurs near i Ursae Majoris. It is nearly 3' in diameter. There is a still larger planetary nebula in the constellation Bootes. If we suppose the former nebula to be at no greater distance than a Centauri, the nearest fixed star, its linear diameter must still be more than three times the diameter of the orbit of Neptune. Its actual distance must be vastly greater than here supposed, and its dlimensions correspondingly greater, unless its individual stars are very minute in comparison with the most distant isolated stars. If we suppose them to be of the same size as the more distant stars, its distance should be equally great, and its dimensions more than 1,000 times greater than the above determination. One of the planetary nebula has been resolved by Lord Rosse's telescope, and another shown to be an annular nebula This class of nebulae are generally supposed to be either cylindri cal beds of stars, or assemblages of stars in the form of hollow spherical shells. 462. Stellar Nebulae are those in which one or more stars are seen apparently connected with a nebulositv. This class of nebule comprises several varieties, the most important of which is that of the Vcbbutlo,s Stars. Nebulous stars are stars encircled by a faint nebulosity; in some cases terminating in a distinct outline, in others shading off gradually into the general light of the sky (Fig. 14, Plate V.). Fig. 16, Plate V., shows the appearance of a nebulous star in Gemini, as seen through Lord Rosse's telescope. The stars surrounded by these nebulous atmospheres have the same appearance as other stars; and their atmospheres offer no indication of resolvability into stars with the most powerful telescopes. Fig. 15, Plate V., is a remarkable stellar nebula in the constellation Cygnus. It consists of a star of' the Ith magnitude, surrounded by a very bright and perfectly round planetary nebula of uniform light, nearly 15' in diameter. Herschel regards it as constituting a connecting iink between planetary nebulae and nebulous stars. In the other varieties of stellar nebulke stars are seen occupying various positions, in apparent connectioll with nebulous masses which are generally of an oval form. Sometimes the nebulosity is spindle-shaped, with a star at each end. One variety has received the name of Cometic Nebuke, from their close resemblance to a comet with a spreading tail. Fig. 18, Plate V., represents a comnetic nebula in the tail of Scorpio. 463. Spiral Nebuila. The great telescope of Lord Rosse has revealed the remarkable tact that some of the nebule are made up of spiral convolutions proceeding from a common nucleus, or from two nuclei. r'The most conspicuous example of NEBULM. 265 this curious form is represented in Fig. 10, Plate V. It is situated near the star a, at the extremity of the tail of the Great Bear. The spiral nebulous coils diverge from two bright centres, about 5' apart. As seen in the field of his great reflecting telescope, they are described by Lord Rosse as "breaking up into stars." Another beautiful spiral nebula is situated in the northern wing of Virgo. In some of the instances cited by Lord Rosse, the spiral arrangement was only partially made out. 464. Irregular Nebulae. Under this head are classed all the remaining single nebulae that, as seen through the best telescopes, have no simple geometrical form. The majority of these are of great extent in comparison with other nebulae, and are devoid of all symmetry of form. They are also remarkable for the great irregularities observable in the distribution of their light, ineicating a singular complexity of internal structure. The Great Nebula iZn the sword handle of Orion is the most conspicuous example of this class of nebulae. It consists of irregular nebulous patches extending over a surface about 40' square, or about twice the size of the moon's disc. From its great magnitude and beauty, singularly grotesque form, and the wonderful variety of its light, it is the most remarkable of all the nebula. Onze portion, near the trapezium or sextuple star 8, is uncommonly bright, and is visible to the naked eye. Other portions are quite hazy and dim; and still other intervening parts are dark, and even absolutely black. Sir John Herschel describes the brightest portions as resembling the head and yawning jaws of some monstrous animal, with a sort of proboscis running out from the snout. The constitution of this singular nebu a remained enveloped in mystery from the time of its first discovery by HIuyghens, in 1656, until the telescope of Lord Rosse was directed upon it; when the brighter portion near the trapezium was distinctly perceived to consist of clustering stars. The elder Bond, with the great Cambridge refractor, subsequently succeeded in resolving the same part of the nebula. More recently G. P. Bond has detected indications of an arrangement of the separated stars in spiral lines. TzThe Great -Nebula in Andromeda is another remarkable irregular nebula. In the field of the Cambridge telescope it has the irregular outline and peculiar appearance represented in Fig. 8, Plate IV. Its extreme length is 2~~, and breadth over 1~. It is traversed, for a considerable portion of its length, by "two dark bands or canals." Certain parts offered, in the same telescope, decided indications of a stellar constitution. The brighter portion of this nebula is distinctly visible to the naked eye. As viewed with a telescope of moderate power, it has an elongated oval form, similar to Fig. 7, Plate IV. The Crab Nebula. Fig. 4, Plate IV., represents the appearance of this curious nebula as seen through Lord Rosse's tele 266 THE FIXED STARS. scope. It is described as studded with stars, mixed with a nebulosity probably consisting of stars too minute to be recognised, and exhibiting filaments extending out firom the southern portion of the nebula. In ordinary telescopes these outlying branches, which have suggested the name of crab nebula, are invisible, and the part seen has an oval form. The Dw.cmb-bell Nebula is so named from the fact that as seen through a telescope of moderate size, in which the brighter portion alone is visible, it has the apparent form of a dumb-bell. In Lord Rosse's telescope the nebula appears less regular in its form; and it is at the same time seen to consist of innumerable stars mixed with irresolvable nebulosity. When its fainter portions are included, its outer limit has an oval forrm (see Fig. 9, Plate V.), which shows the nebula as viewed through the smaller telescope of 3 feet aperture, constructed by Lord Rosse. 46a. Double Nebtulw. A considerable number of double nebulme occur in different parts of the heavens. M. D'Arrest, of Copenhagben, enumerates fifty whose constituents are not over 5' apart, and estimates that there may be as many as 200 such double nebulke. The two constituents are most commonly circular in their apparent form, and are probably real globular clusters. (Fig. 17, Plate V.) The individual members of most of these nebulam are probably physically connected. In one instance considerable changes have been recognised in the distance and relative position,of the nebulae in the interval from 1785 to 1862, which seem to indicate a motion of revolution of the one around the other. 466. Variability of Nebulve. Systematic observations have been made by Struve, D'Arrest, and other astronomers, with the view of ascertaining whether any of the nebuloe were subject to variations of brightness. The result is that in a small number of cases some degree of variability has been positively ascertained. One case is that of the nebula in Orion, in certain parts of which material chances of brightness have been observed. But the most marked case is that of a small and faint nebula, discovered by Hind, in 1852, in the constellation Taurus. It has since gradually faded from year to year, and in 1862 was barely discernible in the great Pulkowa refractor. It is now entirely invisible in the telescope with which it was first detected. It is an interesting fact that this diminution of brightne-ss has proceeded pari passu with that of a small star which presented itself almost in contact with the nebula. It has been observed also that there are many variable stars in a part of the nebula in Or-ion that is subject to change. Corresponding changes have been observed in the faint nebulous haze noticed around some of the variable stars; for instance, the new star that suddenly burst forth in May, 1866, in Corona Borealis, and then rapidly declined in brightness. DISTANCE AND MAGNITUDE OF NEBULAE. 267 DISTANCE AND MAGNITUDE OF NEBULIE. 467. Resolved Nebulae. Herschel undertook to determine the distance of resolved nebulae, by noting the space-penetrating power of the telescope which first succeeded in revealing their distinct stars. According to his determinations, therefore, the most remote of the resolved nebulae are at the same distance as the most remote of the isolated stars discerned in his large telescope. The theoretical space-penetrating power of his telescope was 2,080 times the mean distance of stars of the first magnitude. This should accordingly be the limiting distance of the resolved nebulae seen in Herschel's telescope. The corresponding limit for stars and nebulae, as seen in Lord Rosse's telescope, should be 3,120. But Struve, after determining the comparative distances of stars of the different photometric magnitudes, by comparing the actual number of stars of the different magnitudes, has been enabled to ascertain the actual space-penetratiig power of any telescope in which all the stars up to any particular magnitude could be seen. According to his determinations, the actual space-penetrating power of Herschel's telescope of 20 feet focus was 183; that of the 40 feet reflector was 368, instead of 2080 as deduced upon optical principles; and that of Lord Rosse's great telescope is 422, instead of 3,120, the theoretical determination. The unit of distance in these numerical values is the mean distance of stars of the first magnitude. According to Peters, this corresponds to a parallax of 0".21, and is traversed by light in 15.5 years. We mlay therefore conclude that light employs about 6,540 years in coming from the most remote- telescopic stars hitherto discerned to the earth. It traverses the distance from the nearest star (a Centauri) to the earth in 38 years. The resolvable nebulae require telescopes of various powers to reveal their individual stars, and must therefore be distributed at the same variety of distance as the isolated telescopic stars of similar magnitudes. 46S. Irresolvable Nebulae. Herschel also undertook to determine the probable distance of the more remote irresolvable nebulae. lie estimated that a certain cluster of stars (75 of Messier's catalogue), which at one-fourth of its distance would be visible to the naked eye, would be visible as a faint irresolvable nebula, in his great reflector, if it were removed to 48 times its actual distance, or to more than 35,000 times the distance of Sirius. Struve's investigation reduces this determination to 787 times the mean distance of stars of the first magnitude (467). The corresponding result for Lord Rosse's telescope would be only a small fraction greater. 469. Extinctioli of the Light of the Stars, in its passage 2N68 THE FIXED STARS. through space. The course of investigation followed up by Struve, at the same time that it affixed a much lower limit to the power of telescopes to pierce into the depths of space, conducted in explanation of this fact, to an important theoretical conclusion, viz., that the light of the stars is partially extinguished in its transit through space. lie estimated the amount of this extinction to be such that light, in its passage through a distance equal to that of a star of the first magnitude, loses Tl- of its intensity. Sir John Herschel controverts this theory of the distinguished Pulkowa astronomer, but makes no attempt to overthrow the principal argument upon which it rests. If we reject, with Herschel, the testimony of the stars relative to the power of telescopes to penetrate the depths of space in which they lie, we must then adopt the determinations obtained upon optical principles alone as the exponents of telescopic power; we must accordingly conclude that stars can be discerned with the most powerful telescopes when separated from us by a distance so vast that light requires 48,000 years to traverse it; and that nebulae might still be visible at a distance which light would require 500,000 years to pass over. At that distance, the united impression of the light of 10,000 stars upon the eye would only equal that from 1)00 single stars, so remote as to be just discernible in the most powerful telescope; and therefore clusters containing hundreds of thousands of stars should be visible at a much greater distance. 470. Magaitude of Nebulae. At the distance of 422 stellar intervals (the utmost actual reach of Lord Rosse's telescope) a linear extent of 10', in the heavens, answers to 1.23 times one of these intervals (467). Some of the planetary nebulae have an apparent diameter as great as 10', and as they are probably more remote than the most distant telescopic stars, their actual diameters are probably greater than 1.23 stellar units. The irregular nebulae have a much greater extent. For example, the more conspicuous portion of the nebula in Orion extends to 30', or 3.7 stellar intervals, in the east and west direction, and nearly as far in the north and south direction. The outlying branches run out much further. The extreme length of the nebula in Andromeda is no less than 18 times the same unit or the mean distance of stars of the first magnitude. Its extreme breadth is 71 units. We here suppose these two nebule to be at the distance of the most remote telescopic stars. As they are barely resolvable by the most powerful telescopes, their distance cannot be less than this, unless their component stars are smaller, or intrinsically less luminous than the more remote isolated stars. If the space-penetrating power of telescopes, as obtained upon optical principles, be adopted, the above numerical results must be increased seven-fold. COMPONENT STARS OF CLUSTERS. 269 NUMBER, MUTUAL DISTANCE, AND COMPARATIVE BRIGHTNESS OF THE COMPONENT STARS OF CLUSTERS. 471. Possible Number of Stars in a Nebula. Ve may obtain an approximate estimate of the number of stars that may be congregated together in a nebula that is completely resolvable by a powerful telescope, by considering that if the telescope just shows them distinctly separate, the apparent distance between two contiguous stars may be assumed to be less than 1". A space of one square minute should then contain more than 3,600 stars. The planetary nebula near the star A, in the constellation of the Great Bear (461), has an apparent extent of 7 square minutes. If it were just resolvable it should then contain more than 25,000 stars. As it is really irresolvable, the number of its individual stars must be still greater. Upon the same basis of calculation, the more conspicuous portion of the nebula in Orion, occupying, according to Sir John Herschel, nof a square degree, should contain more than 500,000 stars; and the similar portion of the nebula in Andromeda (90' long by 15' broad) not less than 4,000,000 stars. If we suppose this vast nebula to be one continuous bed of stars, of different sizes, for its entire extent, it must comprise the enormous number of 30,000,000 stars. It is true that these great nebulae where resolved, in their brighter portions, show distinct stars in numbers that can be counted; but the space intervening between them is full of a nebulosity that is probably composed of smaller stars too closely compacted to be separated by the telescope. 472. Limit of Distance between Stars in a Resolved Nebula. An angular space of 1", at a distance equal to 422 stellar intervals, corresponds to a linear distance 2,019 times the distance of the earth from the sun, or about 67 times the radius of Neptune's orbit. The distance between two contiguous stars of a nebula, that are just separated by a powerful telescope, cannot exceed this amount. If the light of the stars suffers no sensible extinction in its passage, and therefore telescopes really penetrate as far into space as the optical theory requires, this determination is only + of the actual value. Clusters whose individual stars are separated by the distance just determined, would, if posited at a less distance than the furthest reach of telescopes, be more readily resolved; while any that might be at a greater distance would be wholly irresolvable by any telescope yet constructed. 473. Explaniation of Inequalities of Brightness in a Nebula. Globular and irregular clusters (456-7,) are brighter and more difficult of resolution at the central than at the outer 270 THE FIXED STARS. portions of the cluster. This is what should result if they were composed of stars of equal size and equally spaced. But in some instances the increase of brightness towards the centre is too great to admit of this supposition; and we infer that the stars are there condensed into a smaller space. Oval and irregular nebulae are more difficult of resolution at the fainter than at the brighter parts. From this we may infer that the stars are larger or more luminous in the brightest portions of such nebulae; or that instances of close juxtaposition more frequently occur, in groups of two or three, which appear united as one, as suggested by Sir John Herschel. STRUCTURE OF THE SIDEREAL UNIVERSE. 4741. System of the Milky Way. We have already seen (428) that Sir William Herschel made the grand discovery that the sun is one of the individual stars of a vast bed, or organized system of stars, called the system of the milky way; that the sun is posited near its middle plane, and that its innumerable stars constitute the starry host which diversify our firmament. HIe at first conceived that his telescope penetrated to the outermost limits of the stratum, but later investigations, recently confirmed by the observations and researches of Bessel, Argelander, and Struve, have fully established that it extends in all directions beyond the reach of the most powerful telescopes; and that we can obtain no definite knowledge of its exterior form. Herschel's star-gauges afford positive information only with regard to the comparative densities of the fathomless starry stra, tumrn in different directions, within the range of telescopic vision. From these we learn that the individual stars are not uniformly distributed throughout the system, but are greatly condensed towards the medial plane. Struve, by an elaborate discussion, has established that the distance between neighboring stars decreases, according to a regular law, on both sides of this plane as the distance from it increases; the decrease being much more rapid at first, and the rate gradually declining with the increasing distance. Within this plane of greatest condensation there is also a line of greatest density, from both sides of which the density gradually decreases. A corresponding line of superior density exists in each plane of the starry stratum parallel to the principal plane. The axis of greatest condensation is nearly coincident with the line passing through the points of intersection of the galactic circle, or middle line of the milky way in the heavens, with the equator. These points lie in R. Asc. 6h. 4)m., and R. Asc. 18h. 40m., between the constellations Orion and Canis Minor, and between Serpentarius and Antinous. According to Struve the sun is on the north side of the plane of great. STRUCTURE OF THE SIDEREAL UNIVERSE. 271 ast condensation, and at an estimated distance from it equal to the distance of o Centauri from the sun and earth. It is also to one side of the axis of greatest density in the direction of the constellation Virgo, and at a distance nearly equal to the distance of the nearest stars of the second magnitude from the earth. The galactic circle, and therefore, also, the principal plane of the milky wav, passes through the points on the equator above-mentioned, and within about 30~ of the north and south poles of the heavens; through points in the constellations Cassiopeia and the Southern Cross. The north pole of the galactic circle, or of the whole system, lies in R. Asc. 12h. 38m., and Dec. 31~.5, between the constellations Corna Berenices and Canes Venatici. 475. The Galaxy, or Belt of the MIilky Way. The luminous belt in the heavens called the milky way, as seen by the naked eve, varies in breadth at different points between the limits 5~ and 16~, and has an average breadth of about 10~. It presents a succession of luminous patches, unequally condensed, intermingled with others of a fhinter shade. From the bright star a Cygni, in the northern hemisphere, it runs towards the southwest in two clustering streams, which reunite beyond the southern constellation Scorpio, at a distance of 120~ from the point of separation.. Near the place in which it crosses the equator, between Antinous and Serpentarius, the double stream attains its greatest width of 22~. The middle. point of crossing is the ascending node, on the equator, of the galactic circle. To give a more accurate idea of the system of the milky way, we must add that its principal plane, so called, is not strictly a single plane, but a broken plane, or two planes differing about 100 in their direction, and separating at the line of the nodes in the equator. The two condensed branches answering to the two separate streams in the heavens just noticed, lie on oppysite sides of this broken plane. The line of greatest density befbre referred to (474) also is not truly a right line, but has sensible inflexions; and there occur in its vicinity remarkable alternations of starry condensations and vacant spaces. Similar interruptions of continuity are observed in various directions through the mass. In some directions dark intervening spaces are seen, in which, according to Sir John Herschel, the telescope seems to penetrate to the very confines of' the starry stratum. In other directions, there appear to be vast starless regions lying between the more remote portions and outlying branches of the milky way, or other systems entirely detached t'roln it. 476. Relations of Clusters and Nebulae to the Systeum of tihe Milky Way. Globular and irregular clusters are far more abundant in the denser portions of the milky way than in other portions of equal extent. The irregular nebula, some of which have been resolved, are, for the most part, either portions or outlying branches of the system. Some of those which have 272 THE FIXED STARS. not been resolved, may possibly be independent systems exterior to that of the milky way. Oval nebulae, and the irresolvable nebulse generally, do not hold the same relations to our starry firmamtnent. They are mostly absent from that great belt in which the stars are so numerous and condensed, and the conspic(uous clusters abound, and are congregated towards its poles. The region richest in nebulae lies around its north pole. They are more uniformly disseminated and more widely dispersed over the zone which surrounds its south pole; and are at the same time less numerous. But on the other hand, as already intimated, there are two luminous tracts of the southern heavens, called the Jagyellanic Clouds, in which they occur in large numbers. In these they are found associated with groups and clusters of stars of ever'y form, and must be presumed to be no more remote than these resolved clusters. In the northern hemisphere they in general occur dissociated from resolved clusters, and may be much more remote. According to the estimate already obtained (468) their extreme limit of distance does not exceed twice that of the most distant isolated stars visible in telescopes. 477. Theoretical Inferences. The peculiarity that has just been noticed in the position of most of the oval and irresolvable nebula of the northern hemisphere, leads to the supposition that they may have originated in a different manner from the clusters and nehule that are chiefly accumulated in the denser portions of the system of the milky way, and undoubtedly are component parts of it; and that they may differ from these in some of the features of their physical constitution. The latter supposition acquires additional probability from a recent discovery that the character of the light received from some of the nebulae is in certain respects different from that of the light received from the sun and the stars. A spectral analysis of the light from some of these nebuhe, by two eminent physicists, has disclosed the remarkable fact, that it is not made up of rays of widely different refrangibilities, but is, the greater part of it, monochromatic; and that the spectrum is not crossed by dark lines, like that obtained from the light of the sun, or of a star. From this, the experimenters draw the conclusion that the nebule in question can no longer be regarded as clusters of suns, similar in constitution to the centre of our planetary system, but as objects having quite a different and peculiar composition; and that instead of being considered as made up of bodies having a solid nucleus, they must be regarded as enormous masses of luminous gas or vapor. The latter conclusion does not follow of necessity from the results of the experiments; they only show that the light from these nebulae comes from masses of pure gas or vapor, rendered luminous either by ignition or electric discharges, but afford no certain knowledge with regard to the existence of a solid nucleus. 47S. General IM1otion of Rlevolution of the Stars. MAldler, after an elaborate discussion of the proper motions of a large number of stars, has arrived at the conclusion tll.ht the collective body of stars visible to us has, together with the sun, a common movement of revolution around a centre situated in the group of the Pleiades. He estimates the period of revolution to be about 27 millions of years. A general circulation of the sun and the stars of our firmament around a common centre of attraction, must also be regwarded as highly probable upon physical grounds, but it cannlot be doubted DYNAMICAL CONDITION OF SIDEREAL SYSTEMS. 2 73 that the centre of attraction would lie in the principal plane of the milky way. The group of the Pleiades lies considerably to the south of this plane, and therefore in all probability the actual centre is situated to the north of the Pleiades, in the constellation Perseus, as suggested by Argelander. 479. Hypothleses respecting the Milky Way. Madler supposes that the stars of the milky way are arranged in several concentric rings of unequal thickness, and of varying dirensions in different directions, but lying nearly in the same plane. He conceives the sun to be eccentrically situated in the system, and at a short distance from the general plane of the rings; so that on one side the rings are seen distinctly separate. Professor Stephen Alexander, of Princeton College, has advanced the hypothesis that the milky way, and the stars within it, together constitute a spiral with several branches, and a central spheroidal cluster. The hypothesis of Sir William Herschel has already been considered (428 and 474). Another conception of the probable structure, and present dynamical condition of the system of the milky way, is briefly presented in a Note in the Appendix. GENERAL DYNAMICAL CONDITION OF SIDEREAL SYSTEMS. 4SO. Three different general conceptions may be formed of the possible nature of the motions of the individual members of a cluster or system of stars. (1.) They may all be in the act of falling in right lines towards their common centre of attraction. (2.) They may be in the act of receding from a centre about which they were originally collected, under the influence of some dispersing force. (3.) They may be revolving in separate orbits around their common centre of attraction, or possibly around different centres. First H/ypothesis.-This was proposed by Sir William Herschel. It accords with the different aspects presented by clusters condensed towards a centre, but cannot be applied to annular nebulae, some of which are known to consist of stars, nor to spiral formed clusters. It involves also the highly improbable supposition that there is in the condition of the system no provision for stability, but only for its inevitable destruction, in the final collision of all its constituent stars at its centre. Second Hypothesis.-The second supposition is advocated by Professor Alexander, who has propounded a systematic theory of the evolution of sidereal systems, under the operation of a certain supposed process of dispersion. Third Hypothesis. —The supposition that the individual stars 18 274 THE FIXED STARS. of a system are moving in separate orbits about a common centre of attraction, is that which is suggested by the analogy of our planetary system, as well as that of the revolution of binary and triple stars around their common centre of gravity. It is supported also by the results of Madler's investigations with respect to a general revolution of the system of the milky way about a centre (478). It implies the existence of the only causes of stability that can be conceived to be in operation; viz., a centre of attraction, and a motion of revolution around that centre. For the rotation of a cluster of separate stars around an axis, as one single body of matter, is mechanically impossible. In the history of such an organized system, from its beginning, there may be epochs of collision among its individual members, but when all such cases, inevitably resulting from correspondences of original position, have occurred, the motions which remain outstanding may ultimately tend to a permanent stability. NEBULAR HYPOTHESIS. 2T5 CHAPTER XX. THEORIES OF THE EVOLUTION OF SIDEREAL AND PIANETARY SYSTEMS. NEBULAR HYPOTHESIS. 481. Primnitive Nebulous Condition of all Systems. Although the telescope, by revealing the stellar constitution of many of the nebulee regarded by Sir William Herschel as giving no intimations of resolvability, has removed the supposed direct evidence of the existence of detached masses of nebulous matter disseminated through space, there still remains strong indirect evidence of a primitive nebulous condition of all worlds and systems of wonlds. Numerous correspondences of structural and dynamical features, and intimations of a progressive creation, lead to this conception as the only ground upon which they can reasonably be explained. Thus Laplace adduces five general phenomena as indications of a common origin of the system of planets circulating around the sun; and infers that they must all have originally formed portions of one vast nebulous body rotating about an axis. These are: 1. The planets all revolve in the same direction around the sun; viz.: from west to east. 2. Their orbits lie nearly in the plane of the sun's equator. 3. Their orbits are ellipses of small eccentricity. 4. The sun and all the planets, so far as the circumstances of their rotation are known, rotate about axes in the same direction that the planets revolve around the sun. 5. The satellites revolve around their primaries in the same direction that these revolve around the sun, and turn about their axes. They also revolve, as far as known, appro.imately in the plane of the equator of each primary; and describe ellipses of small eccentricity. The only known exception to the general direction of revolution occurs in the case of the satellites of Uranus, which have a common retrograde motion. Their orbits are also inclined to the plane of the ecliptic under a large angle (79~); but their common plane may still coincide with the plane of Uranus's equator, and the direction of their motion of revolution may be the same as that of the rotation of the primary. (See Note III.). The hypothesis proposed by Herschel in explanation of sidereal systems, and since extended by Laplace to the explanation of the solar system, is called the Nebular jypothesis. It is, comprehensively stated, that all worlds and systems of worlds have been slowly evolved from primordial nebulous masses, under the operation of the general forces and properties which the Creator has either permanently imparted to matter, or is incessantly renewing in it. DEVELOPMENT OF THE SOLAR SYSTEM. 482. Origin of the Planets and Satellites. The mechanical theory of dhe formation of the solar system propounded by Laplace, is briefly this: The rotating nebulous body from which the system has been evolved, in the progress of ages slowly contracted and condensed, by the gravitation of its parts towards the centre, and by the process of cooling at its surface. This contraction of necessity accelerated the rotation of the body, and augmented the centrifugal force: until 276 EVOLUTION OF SIDEREAL AND PLANETARY SYSTEMS. finally the increasing centrifugal force at the equator balanced the gravity. Whet this mechanical condition was reached at the surface, and for a certain depth where the influence of the cooling had especially prevailed, a vaporous zone became detached, and revolved independently of the interior mass. This zone, by concentration at special points, eventually separated into fragments; which, from the preponderating attraction of the larger fragment, or because of slight differences of initial velocity, became incorporated into one revolving body. This body would take up a motion of rotation in the same direction that it revolves; since the parts most remote from the sun would have the most rapid motion of revolution. By an indefinite continuation of the same process a succession of zones would become detached, and a system of vaporous bodies revolving around a central condensed mass would be formed. Each of these revolving bodies being also in the samo condition of rotation as the original nebulous mass, might pass through a similar succession of chances, and thus a system of satellites circulating around a primary, in the direction of the rotation, be developed. The solar system presents one instance, that of Saturn's ring, in which the detached vaporous zone condensed uniformly without separating into parts. The planetoids appear to afford an instance of the opposite extreme, in which the ring broke up into a great number of small fragments that continued to revolve separately. 483. Origin of Cometary Bodies. Laplace supposed the comets did not belong, originally, to the solar system, but wandered into its precincts from other systems, and so became permanently united with it by the bond of gravitation. But, with the evidence now afforded by accumulated facts, several considerations may be urged which tend to show that comets have been derived from the same nebulous body as the planets and satellites. The principal of these are the following: 1. The comets of short period form a class but little distinguished, in their orbital motions, from the planetoids. They revolve in the same direction, and in orbits having about the same average inclination to the ecliptic, as those of the planetoids. Their orbits are only somewhat more eccentric. 2. All the known comets that describe orbits whose aphelia lie within the limits of the solar system, or do not fall more than fifty millions of miles beyond the orbit of Neptune, revolve in the same direction as the planets. 3. If we compare all the comets whose elliptic orbits have been determined with more or less accuracy, among themselves, we find that the more eccentric orbits of the comets of long period are more inclined to the plane of the ecliptic than the less eccentric orbits of the comets of short period. If we consider the class of comets which recede to a distance of more than fifty millions of miles beyond the limits of the solar system, it appears that as many among them have a retrograde as a direct motion; while the majority move in orbits inclined under large angles to the ecliptic. These exceptional facts do not necessarily imply that this class of comets have an origin extraneous to the system; but rather that the mode of their evolution from the primary nebulous body was different from that of the planets and comets of short period. Now, besides the process of evolution supposed to have been in operation m the case of the planets, we may conceive, (1.) That certain portions of the body, near its surface, became, by mutual attraction of their parts and by cooling, condensed upon particular points into masses of sufficient density to revolve independently. Such masses, as they would have less initial velocities in proportion as they were more remote from the equator, would, in general, describe orbits more eccentric in proportion as they are more inclined to the ecliptic. Besides, the masses which became detached at the equator in the manner here supposed, must have separated from the general mass in the intervals between the epochs of the separation of the equatorial planetary rings, during which the velocity of rotation at the equator was less than that answering to a motion of revolution in a circle. The comets of the first two classes may have thus originated. If so, as they must have performed many revolutions within the attenuated mass of the nebulous body, they are now doubtless moving in orbits much more eccentric than those which they first described. (2.) That fragments may have been suddenly detached from the general nebu. lous mass, by the operation of some expelling force. If' we adopt the most prob. able hypothesis, that this force acted indifferently in all directions outward from DEVELOPMENT OF THE SOLAR SYSTEM. 277 the surface, and assume it to have been of sufficient intensity to impart, when exerted under certain obliquities to the surface, a velocity in the direction of the parallel of latitude considerably greater than the velocity of rotation at the place of discharge, then among the comets thus originating that come within our firmament, a retrograde may be as frequent as a direct motion. For, those which were detached with the higher velocities, either obliquely in the direction of the rotation or in the opposite direction, would move in too large orbits to become visible from the earth. If all the comets detached, however, could be seen, there should be a preponderance in the number of those having a direct motion. (See Note III. in Appendix.) PART II. PHYSICA L ASTRONOMY. CHAPTER XXI. PRINCIPLE OF UNIVERSAL GRAVITATION. 4S4. Force of Gravity. IT is demonstrated in treatises on Mechanics, that if a body move in a curve in such a manner that the areas traced by the radius-vector about a fixed point, increase proportionally to the times, it is solicited by an incessant force constantly directed towards this point. The following is a geometrical proof of this principle. Conceive the orbit to be a polygon of an infinite number of sides. Let ABCD (Fig. 109) be a portion of it; and S the fixed point about which the radius-vector describes areas proportional to the times, or equal n areas in equal times. Since the impulses are only c \ communicated at the angular points A, B, C, D, i \ D &c., of the polygon, the motion will be uniform A along each of the sides AB, BC, CD, &c.: and since D we may suppose thie times of describing these sides \ \ to be equal, we shall have the triangular area SAB equal to the triangular area SBC, arid SBC equal to SCD, &c. Produce AB and make Bc equal to AB, which may be taken to represent the velocity along AB; and join Cc. Cc will be parallel to the line of direction of the impulse that takes effect at B. Upon SB let fallthe perpendiculars Am, cn, Cr. Then, since AB - Be, Am = cn; and since the equivalent triangles SAB, SBC, have a common base SB, Am = Cr. It follows, therefore, that cn = Cr, and consequently, that Cc is parallel to BS. The impulse which the body receives at B is therefore directed from B towards S. In the same manner S it may be shown that the impulse which it receives F at C is directed from C towards S. The line of direction of the force passes, therefore, in every position of the body, through the point S. Now, by Kepler's first law, the areas described by the radiusvectors of the planets about the sun, are proportional to the times. It follows therefore from this law, that each planet is acted upon by a force which urges it continually towards the sun. PRINCIPLE OF U:NIVERSAL GRAVITATION. 279 This fact is technically expressed by saying that the planets gravitate towards the sun, and the force which urges each planet towards the sun is called its Gravity, or Force of Gravity, towards the sun. 4S5. Its Law of Variation, It is also proved by the principles of Mechanics, that if a body, continually urged by a force directed to some point, describe an ellipse of which that point is a focus, the force by which it is urged must vary inversely as the square of the distance. Thus, let ABG (Fig. 110) be the B supposed elliptic orbit of the body, CA and CB its semi-axes, and S the focus towards wh ch the force is con- D stantly directed. Also let P be one position of the body, PR a tangent to the orbit at P; and draw RQ parrallel to PS, Quv, HI, and CD, parallel to PR, Qx; perpendicular to SP, R' and join S and Q. CP and CD are semi-conjugate diameters. Denote them, respectively, by A' and B'; and denote the semi-axes, CA and / CB, by A and B Since HI is parallel to PR, and, by a well-known property of the ellipse, the angle RPS is equal to the angle HPT, PHI is equal to PI: and since HC = SC, and CE is parallel to HI, E is the middle of SL We have, therefore, PS + PI PS + PHIICA A PE - 2 + =C A = A. 2 2 Now the force at P is measured by 2Pu; and we may state the proportion Pu: Pv:: PE: PC:: A: A'; which gives Pv = PuA' By the equation of the ellipse referred to its centre and conjugate diameters, PG and DL, = (Pv x Gv)= B"2 (Pu A x Gv). A'2 A'2 A If we regard Q as indefinitely near to P, then Qu = Qv, and Gv = 2CP = 2A'; and therefore B'2 A' B'2 2B (Pu.2A')= 2Pu.. (a.) A'2 A A But Q: Qx:: PE: PF::CA: PF: and, by Analytical Geometry, CD x PF=CA x CB, or, CA PF:: CD: CB:: B': B. 2 -2 -2 2 B'2 Hence Qu: Qx::B': B, QU: QX::B'2 B2, and Q, = Qx B2 -2B'2 B'; -2 B2 Substituting in equation (a), Q - - 2Pu; whence Qz -2Pu. &Q -2 4k'a Now triangular area SQP=k=-SP x -; whence Qx = Substituting, there 2 SP'2 results 4k2 B2 A 1 =-.2Pu; or 2Pu = — —.4k2. (I). SP A B2 SP 280 PRINCIPLE OF UNIVERSAL GRAVITATION. To compare the intensities of the force at different points of the orbit, we must take the values of 2Pu, by which they are measured, for the same interval of time. On this supposition k is constant, and therefore the force is inversely proportional to the square of the distance SP. It therefore follows from Kepler's second law, viz.: that the planets describe ellipses having the centre of the sun at one of their foci; that the force of gravity of each planet towards the sun varies inversely as the square of the distance from the sun's centre. 4S6. It operates on all tile Planets alike. By taking into view Kepler's third law, it is proved that it is one and the same force, modified only by distance from the sun, which causes all the planets to gravitate towards him, and retains them in their orbits. This force is conceived to be an attraction of the matter of the sun for the matter of the planets, and is called the Solar Attractiom. To deduce this consequence from Kepler's third law, let i, t', denote the periodic times of any two planets; r, r', their distances from the sun at any assumed point of time; k', k', the areas described by their radius-vectors in any supposed unit of time; and A, B, and A', B', the semi-axes of their elliptic orbits. Then kt, k't, will be equal to the areas of the entire orbits; which are also measured by,rAB, 7rA'B'. Thus kt: k't':: AB: A'B', and k2t2 k l'2t'2: 2B2 A'2B 2. But, by Kepler's third law, t2: t'2 "A A'3. B 2 B'2 Dividing, and reducing, k2: k'2:::: A A/ that is, the squares of the areas described in equal times are as the parameters of the orbits. Now, let f, f, denote the forces soliciting the two planets. Then, by equation (I), Art. 485, A 2 1 8A'2 1 f= B2.4k2., and f =. 4k'2.; A 2 1 A' k 1 A B2 1 A' B'21 whence f:f':: f a,,7 r''2 B or f:f r From which it appears that the planets are solicited by a force of gravitation towards the sun, which varies from one planet to another according to the law of the inverse square of their distance. 4S7. Planets Endued with an Attractive Force. The motions of the satellites are in conformity with Kepler's laws; hence, the planets which have satellites are endued with an attractive force of the same nature with' that of the sun. The existence of a similar attractive power in each of the planets that are devoid. of satellites, is proved by the fact that the observed inequalities of their motions, and of those of the other planets, may be shown upon this supposition to be neces NEWTON'S THEORY OF UNIVERSAL GRAVITATION. 281 sary consequences of the attractions of the planets for each other. In like manner the inequalities in the motions of the satellites and their primaries, show that the satellites possess the same property of attraction as the sun. 4S~. "FThe Constitnuent Particles Attract each other. We learn from the motions produced by the action of the sun and planets upon each other, that the intensities of their attractive forces are, at the same distance, proportional to their masses, and that the whole attraction of the same body for different bodies, is, at the same distance, proportional to the masses of these bodies. From which we may infer that a mutual attraction exists between the particles of bodies, and that the whole force of attraction of one body for another, is the result of the attractions of its individual particles. Moreover, analysis shows, that in order that the law of attraction of the whole body may be that of the inverse ratio of the square of the distance, this must also be the law of attraction of the particles. The fact, as well as the law of the mutual attraction of particles, is also revealed by the tides and other phenomena referable to such attraction. 4S9. Theory of Universal Gravitatiorn. The celestial phenomena compared with the general laws of motion, conduct us therefore to this great principle of nature; namely, that all particles of matter mutually attract each other in the direct ratio of their masses, and in the inverise ratio of the squares of their distances. This is called the principle of Universal Gravitation. The theory of its existence was first promulgated by Sir Isaac Newton, and is hence often called lVewton's T/zeory of Universal Gravitation. The force which urges the particles of matter towards each other is called the Force of Gravitation, or the Attraction of Gravitation. In the following chapters our object will be to develop the most important effects of the principle of gravitation thus arrived at by induction. The perfect accordance that will be observed to obtain between the deductions from the theory of universal gravitation and the results of observation, will afford additional confirmation of the truth of the theory. 282 THEORY OF THE ELLIPTIC MOTION OF THE PLANETS. CHAPTER XXII. THEORY OF THE ELLIPTIC MOTION OF THE PLANETS. 490. Accelerating Force due to Sun's Attraction. Let the attraction of the unit of mass of the sun for the unit of mass of a planet, at the unit of distance, be designated by 1. The whole attraction exerted by the sun upon the unit of mass, at the same distance, will then be expressed by the mass of the sun (M); or, in other words, by the number of units which its mass contains. And the attraction F, at any distance r, will M. result from the proportion M: F:: r: 12, which gives F- r2 This, in the language of Dynamics, is the Accelerating Force of the planet, due to the attraction of the sun. M As y- expresses the attraction of the sun for a unit of mass of the planet, its attraction for the entire mass m of the planet will be expressed by m -r. This is the moving force of the planet, and since it is, at the same distance, proportional to the mass of the planet, the velocity due to its action is the same, whatever may be the mass. Attractive Force of Planet. The planet has also an attraction for the sun, as well as the sun for the planet, and the expression for its attractive force, or for the accelerating force animating the sun, will obviously be m The sun will then tend towards r the planet, as the planet towards the sun. But if the two bodies were to set out from a state of rest, the velocity of the planet would be as many times greater than the velocity of the sun, as the mass of the sun is greater than that of the planet. For the velocity of the planet would be to that of the sun as the attractive force of the sun is to the attractive force of the planet, M m that is, as oras: m. r r As the attractions of the particles of the sun and planet are mutual and equal, the attraction of the planet for the entire mass of the sun must be equal to the attraction of the sun for the entire mass of the planet. GENERAL PRINCIPLE OF REVOLUTION. 283 491. The Sun and any Planet revolve about their Common Centre of Gravity. To show this, we would remark, in the first place, that it is a principle of Mechanics that the mutual actions of the different members of a system of bodies cannot affect the state of the centre of gravity of the system. This is called the Prhncide of the Preservation of the Centre of Gravity. It follows from it that the common centre of gravity of the sun and any planet is at rest, unless it has a motion of translation in common with the two bodies, imparted by a force extraneous to the system. As we are concerned at present only with the relative motion of the sun and planet, such motion of translation, if it does exist, may be left out of account. Now, let S (Fig. 111) be the sun, and P any planet, supposed for P the moment to be at rest. If neither of the two bodies should receive a velocity in a di- / A rection inclined to PS, the line of their centres, they would move towards each other by virtue of their mutual attraction, and meet a at C their common centre of gravity.* But, if the body P have a projectile velocity given to it in any direction Pt, inclined to the line PS, it is susceptible of proof that its motion relative to the sun may be in an ellipse, as is observed to be the case with the planets. FIG. 111.:Now, while the planet moves in space, the line of the centres of the planet and sun must continually pass through the stationary position of the centre of gravity; and therefore, when the planet has advanced to any point p, the sun will have shifted its position to some point s on the line pC prolonged. Moreover, as the two bodies mutually gravitate towards each other, the path of each in space will be continually concave towards the other body, and therefore also towards the centre of gravity C, which is constantly in the same direction as the other body. Since the planet performs a revolution around the sun, the sun and planet must each continue to move about the point C until they have accomplished a revolution and returned to the line PCS. Also as the distance PS of the two bodies will be the same at the end as at the beginning of the revolution, as well as the ratio of their distances PC and SC from the centre of gravity, they will return to the positions, P, S, from which they set out, and will therefore move in continuous curves. Moreover, these curves are similar to the apparent orbit described by P around S. For, draw Sp' parallel and equal to st, and join Pp and Ss. Then, since sC: Cp:: SC: CP, Pp is parallel to Ss; and therefore Pp produced passes through p'. Whence, CP: C:: SP: Sp. Moreover, the angle PC2O PSp'. It follows, * The common centre of gravity of two bodies lies on the line joining their cen. tres, and divides this line into parts inversely proportional to the masses of the bodies. 284' THEORY OF THIE ELLIPTIC MOTION OF THE PLANETS. therefore, that the area PCp is similar to the area PSp'; and thus that the orbit of P around C is similar to the apparent orbit of P around S. The latter is kno wn from observation to be an ellipse. The former is therefore also an ellipse. As the distances of the sun and planet from their common centre of gravity are constantly reciprocally proportional to their masses, the orbit of the sun will be exceedingly small in comparison with the orbit of the planet. 49'g. Entire Accelerating lForce of Planet. If to both the sun and planet there should be applied a force equal to the accelerating force of the sun,, 9), but in an opposite dir rection, the sun would be solicited by two forces that would destroy each other, but the planet would now be urged towards the sun remaining stationary, with the accelerating force r, or a force the intensity of which was equal to the sum of the intensities of the attractive forces of the sun and planet,at the distance of the planet. Now, the application of a common force will not alter the relative motion of the two bodies. Hence, in investigating this motion, we are at liberty to conceive the sun to be stationary, if we suppose the planet to be solicitedl by the accelerating force?n. As the mass of the sun is r2 very much greater than that of any planet, but little error will be committed in neglecting the attraction of the planet, and taking into account only the sun's action M[. 493. General Theoretical Results. Analysis makes known the general laws of the motion of a body, when impelled by a projectile force, and afterwards continually attracted towards the sun's centre by a force varying inversely as the square of the distance. WVe learn by it that the body will necessarily describe some on.e of the conic sections around the sun situated at one of its foci. We learn, also that the nature of the orbit, as well as the length of the major axis, is wholly dependent, for any given distance of the planet, upon the intensity of the projectile force; but that the position of the axis, and the eccentricity of the orbit, depend also upon the angle of projection (that is, the angle included, at the commencement of the motion, between the line of direction of the projectile force and the radius-vector). As to the relative intensity of projectile force necessary to the production of each one of the conic sections, a certain intensity of force will produce a parabola; any less intensity, an ellipse or circle; and any greater, a hyperbola. 494. Theoretical Deteruianlation of Orbit of Planlet. If the velocity that would at a given distance be imparted by the sun's attraction in a second of time, which is the measure of its THEORETICAL DETERMINATION OF ORBIT. 28t' intensity at the given distance, be found, and also the distance of a planet at any time, as well as its velocity and the angle made by the direction of its motion with the radius-vector, the form, dimensions, and position of the planet's orbit can be computed. This is to determine the orbit a priori. The practice has been, however, to determine the various elements of a planet's orbit by observation (as already described, Chap. IX.). The elements being known, the equations of the elliptic motion, investigated on the principles of Mechanics, serve to make known the position and velocity of the planet at any time. The physical theory of the motion of a satellite around its primary is obviously the same as that of the motion of a planet around the sun. 495. Cenrtre of Gravity of the Solar Systemn. According to the principle of the preservation of the centre of gravity (491), the centre of gravity of the whole solar system must either be at rest, or have a motion of translation in space in common with the system, resulting from the action of a foreign force. We have already seen (447) that it has been ascertained from observa tion, that it is in fact in motion. The sun and planets; revolve around their common centre of gravity. The path of the sun's centre results from the joint ac. tion of all the planets, and is a complicated curve. As the quantity of matter in all the planets taken together is very small, compared with that in the sun (less than -700), the extent of the curve described by the centre of the sun cannot be very great. It is found by computation, that the distance between the sun's centre and the centre of gravity of the system can never be equal to the sun's diameter. 496. Centre of Gravity of a Planet and its Satellites. It is demonstrated in treatises on Mechanics, that if foreign forces act upon a system of bodies, the centre of gravity of the system will move just as the whole mass of the system concentrated at the centre of gravity would move, under the action of the same forces. It follows from this principle, that from the attraction of the sun for a primary planet and its satellites, their common centre of gravity will revolve around the sun, just as the whole quantity of matter in the planet and its satellites concentrated at this point would, under the influence of the same attraction. Moreover, the same considerations which show that the sun and planets revolve about their common centre of gravity, will also show that a primary planet and its satellites revolve about their common centre of gravity. It appears, therefore, that in the case of a planet which has satellites, it is not, strictly speaking, the centre of the planet that moves agreeably to the first and second laws of Kepler, but the common centre of gravity of the planet and its satellites; the planet and satellites revolving around the centre of gravity, as it describes its orbit about the sun. 286 THEORY OF THE ELLIPTIC MOTION OF THE PLANETS. The mass of the earth is to that of the moon as 82 to 1, while the distance of the moon is to the radius of the earth as 60C to 1: it follows, therefore, that the common centre of gravity of the earth and moon lies within the body of the earth. 497. Kepler's third Law not rigorously true. It appears from the physical investigation of the elliptic motion of the planets, that Kepler's third law is not strictly true. In consequence of the action of the planets upon the sun, the ratio of the periodic times of the different planets depends upon the masses of the planets, as well as their distances from the sun. If p and p' be the periodic times of any two of the planets, a and a' their mean distances from the sun's centre, and mr and m' their quantities of matter, that of the sun being denoted by 1, then, disregarding the actions of the other planets, theory gives /3 p2 p/2 a a 1-+m 1 +m' As m and m' are very small fractions, the error resulting from their omission will be very small. If we omit them, we shall have 2 p22 * /3 which is eper's third law. a' a; which is Kepler's third law. r1ERTURBATIONS OF ELLIPTIC MOTION OF THE MOON. 2S7 CHAPTER XXIII. THEORY OF THE PERTURBATIONS OF THE ELLIPTIC AVMOTION OF THE PLANETS AND THE MOON.' 498. WE have, in a previous chapter, given a general idea of the mode of determining, from theory and observation combined, the law and amount of the perturbations or inequalities of the lunar and planetary motions. We propose now to give some insight into the nature and manner of operation of the disturbing forces, and will commence with the perturbations of the moon produced by the action of the sun. 499. Components of Disturbing Force. We have already shown (209) how the intensity and direction of the disturbing force of the sun, in any given position of the moon in its orbit, may be determined. Let us now derive the disturbing forces that take effect in the three directions in which the motion of the moon can be changed; namely, in the direction of the radius-vector, of the tangent to the orbit, and of the perpendicular to its plane. Let E (Fig. 112) be the earth,,[ the' moon, and S the sun. Let the force exerted by the sun upon the moon be decomposed into two forces, one acting along the line MS' parallel to ES, and the other from M towards E. If the component along MS' were equal to the t~ force exerted by the sun upon the earth, the motion of the moon about the earth would not be changed by the action of these two forces. Hence, the difference between them will be the disturbing force in the direction MS'. The component along ME is another disturbing force. It is called the Addiititious Force, because it tends to increase the gravity of the moon towards the earth. The disturbing force along MS' will generally be inclined to the plane of the orbit, and may be decomposed into three forces, one in the direction of the N tangent, another in the direction of the radiusvector, and a third in the direction of the perpendicular to the plane. The first mentioned component is called the Tangential Force; the second is called the Ablatitious Force; and the FIG. 112. third we shall call the Perpendicular Force. The actual disturbing force in the direction of the radius-vector is equal to the difference between the addititious and ablatitious forces, and is called the Radial Force. This and the tangential and perpendicular forces constitute the disturbing forces, the direct operation of which is to be considered. 500. To obtain General Analytical Expressions for these Forces, let the distance of the sun from the earth (which for the present we shall suppose to be constant) be denoted by a, and the distances of the moon from the earth and sun, respectively, by y and z. Also let F = the force exerted by the earth upon the moon, P = the force exerted by the sun upon the earth, and Q = the force 238S PERTURBATIONS OF ELLIPTIC MOTION OF THE MIOOLN. exerted by the sun upon the moon. Then, if we denote the mass of the earth by 1, and take m to stand for the mass of the sun, we shall have (490), F m=, p y a2 z2 Let the force Q be represented by the line MIS (Fig. 112); and let its component parallel to ES, or MS' —R, and its component along the radius-vector, or ME=T. Q: T:: lIS: ME; or, z'T::z:y. whence, addititious force T = __... (82). In a similar manner we obtain R.a (83). The disturbing force in the direction of the sun 3 2 z a3 Now, let a, 8, y, denote the angles made by the line MIS' with the tangent, radius. vector, and perpendicular to the plane of the orbit. and we shall have for the ap. proximate components of the disturbing force R - P, along these lines: tangential force ma ( 1 ) cos a... (84); ablatitious force = ma ( 1 — )... 85); perpendicular force = ma ( - cos y....(86). Combining equ. (85) with equ. (82), we obtain for the radial force, radial force=my -- mama ( os/. / z- as The obliquity of the orbit of the moon to the plane of the ecliptic, affects but very slightly the value of the tangential and radial forces. If we leave it out of account, or suppose the moon's orbit to lie in the plane of the ecliptic, we shall have (Fig. 113) 3 = S'MIL = SEM, the elonlgation of the moon -= A, and a = complement of a, which gives tang. force =ma( sin (87); rad. force = my — lma - cos 0 (88). Eqcation (86) may be transformed into another which is better adapted to the purposes we have in view. Let MK[I (Fig. 112) represent the perpendicular to the plane of the moon's orbit, IMF the intersection of the plane SMKI with the plane FIG. 113. of the moon's orbit, and SI, IF, the intersections of a plane passing through S and perpendicular to EN, the line of nodes, with the plane of the ecliptic and the plane of the orbit. SF will be perpendicular to both IF and MF. Denote SIF, the inclination of the orbit to the ecliptic, by I, SEN the angular distance of the sun from the node by N, and SE and SM by a and z, as before. Now, in equ. (86) y stands for the angle S'MK, but S'MK = SMK (nearly), and cos SMK -= sin SMF = SF. SM INVESTIGATION OF THE DISTURBING FORCES. 289 SF = SI sin SIF, and SI = SE sin SEI; whence SF = SE sin SEI sin SIF = a sin N sin I: substituting, cos 7 = cos SMK = a sin N sin I - a sin N sin I SM z Thus we have perpen. force = ma as) n sin.... (89). The vlariable z may be eliminated from equations (87), (88), and (89), and ocher equations obtained, involving only the variables y and 5. Let ML (Fig. 112) be drawn through the place of the moon perpendicular to ES. Then, using the same notation as before, LS = z (nearly), EL = EM cos LEM = y cos 0. But LS = SE -- EL; whence z = a -y cos A, and z3 = a3 —- -a2y Cos 0: neglecting the terms containing the higher powers of y than the first, as they are very minute, y being only about -4 a. 1 1 1 3y cos. Thus,= z3 a3-3a2ycoS q5 a- 3 aa neglecting all the terms of the quotient that involve higher powers of y than the first. Substituting this value of 1 in equ. (87), we obtain, tangential force = 3my cos sin. or (App. For. 13), tangential force = 3my sin 20....(90). 2 a' Making the same substitution in equ. (88), and neglecting the term containing y', there results, radial force = my (1-3 cos%). a3 or (App. For. 9), radial force = - my (1 + 3 cos 2) (91). 2 a3 In equ. (89) we have to substitute, besides, the value of z, viz. a - y cos 0; then dividing and neglecting as before, we have perpen. force = 3my cos sin N sin I....(92). a 501. Variations of disturbingforces. If the disturbing forces retained constantly the same intensity and direction, the result would be a continual progressive departure from the elliptic place; but, in point of fact, these forces are subject to periodical changes of intensity and direction from several causes, from which results a compensation of effects, and an eventual return to the elliptic place. The causes of the variation of the disturbing forces are: (1.) The revolution of the moon around the earth. (2.) The elliptic form of the apparent orbit of the sun. (3.) The elliptic form of the orbit of the moon. (4.) The inclination of the two orbits. As the variations of the radial and tangential forces, resulting from the inclination of the orbits, are very minute, we shall leave them out of account, and in the consideration of the effects of these forces shall, for the sake of simplicity, regard the orbits as lying in the same plane. The first mentioned circumstance is the most prominent cause of variation, and gives rise to the more conspicuous perturbations. The other two serve to modify 19 0290 PERTURBATIONS OF ELLIPTIC 2MOTION OF THE MIOON. the variations of the forces resulting from the first, and occasion each a distinct set of periodical perturbations. 502. Tanlgelntial Force. Let us now investigate, in succession, the effects of each of the disturbing forces, commencing with the tangential force. The tangential force takes effect directly upon the velocity of the moon in its orbit; and as its line of direction does not pass through the earth, it disturbs the equable description of areas. It also affects the radius-vector indirectly, by changing the centrifu(al force. To understand the detail of its action we must inquire into the variations which it undergoes. If we regard y as constant in the expression for the tangential force, (equa. 90), which amounts to considering the moon's orbit as circular, the expression will become equal to zero when sin 2-,.0, and will have its maximum value when sin 2 p=1. It will also change its sign with sin 2,). It appears, therefore, that the tangential force is zero in the syzigies and quadratures, where it also changes its direction, and that it attains its maximum value in the octants. It will be seen, on inspecting Fig. 114, that it will be a retard-,c ing force in the first quadrant (AB). Accordingly, it will be an accelerating force in the second, a retarding force again in the third, and an accelerating force again in the fourth. This will also appear upon considering D _________ E \ the direction of tlhe disturbing force parallel to the line of the centres of the sun and \ / earth, in the various quadrants. In the nearer half of the orbit the sun tends to draw the moon away from the earth, and the force in question is directed towards the sun. In the more remote half the sun tends to draw the earth away from the moon, but we FIG. 114. may regard it, instead, as urging the moon from the earth by the same force; for the relative motion will be the same on this supposition. In the part of the orbit supposed, then, the disturbing force under consideration will be directed from the sun, as represented in Fig. 114. It appears, then, that the tangential force will alternately retard and accelerate the motion of the moon during its passage through the different quadrants, and that the maximum of velocity will occur in the syzigies, A, C, where the accelerating force becomes zero, and the minimum of velocity in the quadratures, B, D, where the retarding force becomes zero. On the supposition that the orbit is a circle, the arcs AB, BC, CD, and DA, would be equal, and the retardation of the velocity in one quadrant would be compensated for by an equal acceleration in the next, and at the close of a synodic revolution the velocity of the moon would be the same as at its commencement. As the velocity is greatest in the syzigies and least in the quadratures, and as the degree of retardation is the same as that of acceleration, the mean motion* must have place in the octants. Now, as the moon moves from the syzigy A with a motion greater than the mean motion, its true place will be in advance of its mean place, and will become more and more so till it reaches the octant, where the true motion is equal to the mean. The difference between the true and mean place will then be the greatest; for after that, the true motion becoming less than the mean, the mean place will approach nearer to the true, till at the quadrature they coincide. Beyond B, the true motion still continuing less than the mean, the mean place will be in advance of the true, and the separation will increase till at the octant the true motion has attained to an equality with the mean motion, after which, the mean motion being the slowest, the true place will approach the mean till at the syzigy C they again coincide. Corresponding effects will take place in the two remaining quadrants. We perceive, therefore, that the tangential force produces an inequality of longitude, which attains to its maximum positive and negative value in the octants, and is zero in the syzigies. * The expressions, mean motion, true motion, mean place, true place, are here to be understood only in relation to the perturbation under consideration. EFFECTS OF THE TANGENTIAL FORCE. 291 This is the inequality known in Spherical Astronomy by the name of ~aria. tion~ (2 i 7). 503. Modifications of the effects of the tangential force, that result from the elliptli foerlL of the sun's orbit. Suppose that at the moment when the moon sets out from conjunctiont the sun is in the apogee of its orbit: then it is plain that, during the whole revolution of the moon, the sun's disturbing force would be on the increase by rlteasonl of the diminution of the sun's distance, and that, in consequence, the retardation in the first quadrant would be less than the acceleration in the second, and the retardation in the third less than the acceleration in the fourth. So that, when the moon has again come round into conjunction, the acceleration will have overcompensated the retardation. This kind of action would go on so long as the sun appn'oaclhes the earth; but when it has passed the perifee of its orbit, and begun to recede from the earth, the reverse effect would take place, and a retardation of the moorn' orbital motion would happen each revolution. If the anomalistic revolution of the sun were an exact multiple of the synodic revolution of the moon, the acceleration in each revolution of the moon during the passage of the sun fiom the apogee to the perigee of its orbit, would be compen.sated for by an equivalent retardation in the revoiutiou of the moon, answering to the same distance of the sun in its passage from the perigee to the apogee; and the velocity of the moon would be the same at the close of an anomalistic revolution of the sun as at its commencement. But as this relation does not, in fact, subsist between the anomalistic revolution of the sun and the synodic revolution of the moon, a compensation between the accelerations and retardations, answering to the different revolutions of the moon, will not be effected until conjunctions shall have occurred at every variety of distance of' the sun in each halft of its orbit. Since the anomalistic and synodic revolutions are incommensurable, the sun will be, in reality, in every variety of position in its orbit at the time of conjunction, in process of time, so that eventually the original velocity in conjunction will be regained. It appears, therefore, that the variation of the moon's motion from one revolution to another, occasioned by the elliptic form of the sun's orbit, is periodic. Its period will be the interval of time in which the moon will perform a certain number of synodic revolutions, while the sun perforims a certain number of' anomalistic revolutions. Avoiding unnecessary precision, we find it to consist of but a moderate number of years. 504. Consequences of the elliptic foer1 of the moon's orbit. mre remark, in the first place, that the orbit being an ellipse, the areas AEB, BEC, CED, arid DEA (Firg. 1 4). will be unequal, and therefore, by the laws of elliptic motion, the arcs AB, BC, CD, and I)A, will be described in unequal tinies. It follows from this, that the retardation in the first qu;adrant will not be exactly compensated by the acceleration in the second, ald that the retardation in the third will not be exactly compensated by the acceleratioii in the fourth. Therefore, at the end of the synodic revolution the moon will have an excess or deficiency of velocity. Its mean motion will then vary fi'om one revolution to another, by reason of the ellipticity of its orbit. This variatiori will be periodic, like that just considered, and for similar reasons. The excess or deficiency of velocity at the close of any one revolution, will in time be compensated by an equal deficiency or excess occurring at the close of another revolution, when the sun has a certain different position with respect to the perigee of the iaooi's orbit. 505. Radial Force. Wre pass now to the consideration of the action of the radial force. The direct general effect of the radial force, is an alteration in the intensity of the moon's gravity towards the earth, and in its law of variation. Its specific effects are periodical variations in the magnitude, eccentricity, and position of the orbit. As it is directed towards the earth, it will not disturb the equable description of areas. To discover the variations of this force we have only to discuss the general analytical expression for it, already investigated. It is, radial force =my (1 —3 cos2 q). We shall have radial force = 0, when 1 - 3 cos2 ~ = 0, or when cos -- /~. This value of cos q answers to four points lying on either side of the quadratures, and about 35~ distant from them. When cos S is numerically greater than x/ the result will be negative, and when it is less than ~/~ the result will be positive. It follows, therefore, that the radial force increases the gravity of the moon in the 292 PERTURBATIONS OF ELLIPTIC MOTION OF THE MOON. quadratures, and for about 35~ on each side of them, and that during the remainder it'a synodic revolution it diminishes it. When the moon is in quadratures, cos S = 0, and radial force my.... (93). a3 In the syzigies, we have cos = -~ 1, which gives radial force =-I2y. (94). It appears, then, that the diminution of the moon's gravity in the syzigies is donble of its increase in the quadratures. We learn also from equations (93) and (94), that the radial force in the quadratures and syzigies varies directly as the distance; from which we conclude that the gravity of the mnoon varies at these points by a different law from that of the inverse squares. In the quadratures the gravity will be increased most at the greatest distance, where it is the least; and thus it will vary in a less rapid ratio than the square of the distance. In the syzigies it will be diminished most at the greatest distance, or where it is the least; and accordingly, at these points it will vary in a more rapid ratio than the square of the distance. 506. Moon's distance increased by radial force. With the aid of the Differential Calculus, we readily find that the mean diminution of the moon's gravity fiom the sun's action is2r; r rrepresenting in this case the mean distance of the 2a3 moon from the earth. The value of this expression is equal to about the 360th part of' the whole gravity of the moon to the earth. in consequence of this diminution, the moon must describe its orbit at a greater distance from the earth, with a less angular velocity, and in a longer time, than if it were acted on only by the attraction of the earth. 507. The radial force of the sun alters the eccentricity of the moon's orbit and differently in different revolutions of the moon, according to the position of the line of syzigies with respect to the line of apsides. When these lines are coincident the eccentricity is increased. For suppose PAMAN (Fig. 115) to be the elliptic orbit of the moon that would be described under the influence of a force varying inversely as the square of the distance. In going fiom the apogee to the perigee, the gravity will increase in a A- B lp greater ratio than that of the inverse square of the distance; the true orbit will therefore fall within the ellipse, and the perigean distance (EP') will be less than for the ellipse. Consequlently, the eccentricity will increase iJN so much the more as the major axis diminishes. On the other hand, in going from the FIG. 115. perigee to the apogee, the gravity will decrease in a greater ratio than the inverse square of the distance, and the moon will consequently recede further from the earth than if the orbit described was an ellipse. Therefore, in this half of the orbit the eccentricity will also be increased. When the apsides are in quadratures the eccentricity will be diminished; for the gravity will then vary fiorn the apogee to the perigee, and from the perigee to the apogee, in a less ratio than that of the inverse squares; and thereibre the results will be contrary to those just obtained. The eccentricity will have its maximum value when the apsides are in syzigies, and its minimum when they are in quadratures; for, in every other position of the line of apsides with respect to the line of syzigies, the radial force in the apogee and perigee will be less than in these positions (equa. 91), and therefore alter less the proportional gravity of the moon in the apogee and perigee. It is evident, fiom the gradual decrease of the radial force as we recede from the syzigies and quadratures, that the eccentricity will continually diminish in the progress of the apsides from the syzigies to the quadratures, and that it will continually increase from the quadratures to the syzigies. EFFECTS OF THE RADIAL FORCE. 293 Tle change in the eccentricity of the moon's orbit, thus produced, will be attended with a corresponding change in the equation of the centre, and thus of the longitude. And this change is the conspicuous inequality of' the moon, known by the name of Evection (217). 508. The, radial force also produces a notion of the line of apsides. If the moon were only acted upon by the attraction of' the earth its orbit would be an ellipse, and the motion from one apsis to another, or, in other words, from one point where the orbit cuts the radius-vector at right angles to the other, would be 180~. In point of fact, however, the gravity due to the earth's attraction is constantly either diminished or increased by the radial disturbing force of the sun, and therefore its true orbit must continualiy deviate from the ellipse that would be described under tile sole action of the earth's attraction. When from the action of this force there is a diminution of the force of aravity, the moon will continually recede friom the ellipse in question, its path will be less bent, and it must therefore move through a greater angular distance before the central force will have deflected its course into a direction at right angles to the radius-vector. Accordingly, it will move through a greater angular distance than 180~ in going from one apsis to another, and thus the apsides will advance. On the other hand, when the same force increases the force of gravity, the moon's path will fall within the ellipse, its curvature will be increased, and therefore it will be brought to intersect the radius-vector at right angles at a less angular distance. In this case, therefore, the apsides will move backward. Now, we have shown (505) that the radial disturbing force of the sun alternately diminishes and increases the moon's gravity to the earth. It follows, therefore. that the motion of the apsides will be alternately direct and retrograde; but since, as has been shown (505), the diminution subsists during a longer part of the moon's revolution, and is moreover greater than the increase, the direct motion will exceed the retrograde, and therefore in an entire revolution the upsides will advaince. The observed motion of the apsides of the moon's orbit is not, however, lvholly produced by the radial disturbing force. It is in part due to the action of the tanl gential force. This force alters the centrifugal force of the moon, and thus changes its gravity towards the earth, at the same time with the radial force. 509. Explacnation of the Anneal Equation. The elliptic form of the sun's orbit is tile occasion of a change in the radial force, from wrhich results a perturbation of longitude called the Annual Equation (217). The mean diminution of the moon's gravity, arising from the action of the sun, or the mlean radial tbrce, is equal to mr (506). Hence this diminution is inversely proportional to the cube of the sun's 2a distance from the earth. Therefore, as the sun approaches the perigee of its orbit, its distance from the earth diminishinig, the mean diminution of the moon's gravity to the earth will increase, and consequently the moon's distance frorn the earth wrill become greater, and its motion slower, than it otherwise would be. The contrary will take place while the suii is moving from the perigee to the apogee. 510. Perpendicular Force. The disturbing force perpendicular to tile plane of the moon's orbit, produces a tendency in the moon to quit that plane, from which there results a change in the position of' the line of the nodes, and a change in the inclination of the plane of the orbit to that of the ecliptic. If we examine the general expression for this force, viz.: perpen. force-3-my cos 0 sin N sin I, we see that for any given values of N and I, it will be zero in the quadratures, and have its greatest value in the syzigies; and that it will change its direction in the quadratures, lying, in the nearer half of the orbit, on the same side of its plane as the sun, and in the more remote half, on the opposite side. We perceive also that it will be zero for every value of A, or for every elongation of the imoon, when the angle N is zero, tbat is, when the sun is in the plane of the orbit; and will attain its maximum, for any given elongation, when the line of direction of the sun is perpendicular to the line of nodes. It will also be the less, other things being the same, the smaller is the inclination I. 511. Retrograde Motion of th/e Nodes. Let NM'R (Fig. 116) represent the orbit 29: PERTURBATIONS OF ELLIPTIC MOTION OF THE MOON. of the moon, and S the sun, supposed stationary, the line of the nodes being in quadra. tures; and let L, L' be the points of the orbit 90' distant from the nodes. The direction of the force. in the various points of the orbit, is indicated by the arrows drawn in the figure. When the moon is at any point M' between L and the descending node N', it will be drawn out of the plane in which it is moving by the N \ ltv\N disturbing force M'K', and compelled to move in such a line as M't'. The node N' will there-, fore retrograde to some point n'. When the moon is at any point M, moving from the ascending node N towards L, its course will be changed to the line Mt, lying, like the line M't', below the orbit, which being produced backward, meets the plane of the ecliptic in some point n behind N. The nodes, therefore, ietro-.S' grade in this position of the moon, as well as in the former. When the moon is in the half FIG. 116. Y'L'N of the orbit, lying below the ecliptic, the absolute direction of the disturbing force will be reversed, and thus its tendency will be the sarme as before, namely, to draw the moon towards the ecliptic. It follows, therefore, that throughout this half of the orbit, as in the other, the motion of the nodes will be retrograde. Accordilloly, when the nodes are in quadratures, or 90~ distant from the sun, they will retrograde during every part of the revolution of the moon. Suppose the sun now to be fixed on the line of nodes, or the nodes to be in syzigies. In this case the perpendicular force will be zero (510), and therefore there will be no disturbance of the plane of the moon's orbit. Next, let the situation of the sun be intermediate between the two just considered, as represented in Figs. 116 and 117. The effect of the disturbing force will be the same as in the first situation from the quadrature q (Fig. 116) to the node N', and from the quadrature q' to the node N. But throughout the arcs Nq, N'q', the direction of the force, and therefore the effects, will be reversed. The node will then retrograde, as before, while the moon moves over the arcs qN' and q'N, and advance while it is in the arcs Nq, N'q'. But as the force is greatest over the arcs qN', q'N, which contain the syzigies (510), and as these arcs are also longer than the ares Nq, N'q', the node will, on the whole, retrograde each revolution. The velocity of retrogradation will, how-ever, be less than when the nodes are in quadratures, and proportionably less as the distance of the sun from this position is greater. In the position represented in Fig. 117, a A~~- -59 direct motion wvill take place over the arcs q'N' and qN: but as Nq' and N'q, the arcs of retrograde motion, are of greater extent 4K./~N~x]/f//// \\ than q'N' and qN, and moreover contain the syzigies, tile retrograde motion in each revolution rmust exceed the direct, as before. N aittN~ If we suppose the sun to be situated on the other side of the Iine of nodes, the effect of the disturbing force will obviously be the same in any one position of the sun, as in the position diametrically opposite to it. It appears, then, that the line of the nodes has a retrograde motion in every possible position of the sun. 512. Effect of Sun's Motion. We have thus far supposed the sun to remainstationary in the various positions in which we have considered it, during the revolution of FIG. 117. the moon. It remains, then, to consider the effect of the sun's motion in this interval. And first, it is plain, that, as the sun advances from S towards N' (Fig. 116), the EFFECTS OF THE PERPENDICULAR FORCE. 295 arcs Nq, N'q' will increase, and the arcs qN' and q'N diminish; from which it appears, that during the advance of the sun from the point 90~ behind the descending node to this node, its motion in the course of each revolution of the moon will cause the retrograde motion of the node to be slower than it otherwise would be. While the sun moves from the ascending node to the point 90~ from it, the effect of its motion will obviously be just the reverse of this. During its passage from the descending to the ascending node, the effect will be the same in either quadrant as in that diametrically opposite. The variation in the intensity of the perpendicular force, conspires with the difference of situation of the sun and its motion during a revolution of the moon in diminishing or increasing, as the case may be, the velocity of retrogradation of the nodes. 513. Change of the inclination of the orbit. If we refer to Fig. 116 we shall see that when the nodes are in quadrature the inclination will diminish while the moon is moving from the ascending node N to the point L 90~ distant from it, and increase while it is moving from L to the other node N. In the other half of the orbit the tendency of the disturbing force is the same (511), and therefore while the moon is moving from N' to L' the inclination will diminish, and while it is moving from L' to N the inclination will increase. The diminutions and increments will compensate each other, and the original inclination will be regained at the close of the revolution. When the nodes are in syzigies there will be no change of inclination (510). In the situations of the sun represented in Figs. 116 and 117, the inclination will decrease from q to L and from q' to L', and increase from L to q' and from L' to q; the effects being the same as when the nodes are in quadratures over the arcs qL and LN in Fig. 116, and NL and Lq' in Fig. 117, and being reversed over LR -K L AI C): t AI I' / FIG. 116. FIG. 117. the arcs Nq and N'q' in Fig. 116, and qN and q'N' in Fig. 117. When the sun has the position represented in Fig. 116, the arcs of increase Lq' and L'q will be greater than the arcs of diminution qL and q'L'. The disturbing force will also be greater in the former arcs than in the latter. In the position supposed, therefore, there will be, on the whole, an increase of inclination every revolution. When the sun is in the position represented in Fig. 117, the arcs of diminution qL and q'L' will be the greater; and the force in them will also be the greater. In this case, therefore, there will be a diminution of the inclination each revolution of the moon. When the sun is on the other side of the line of nodes, the results will be the same as in the positions diametrically opposite. 514. Consequences of the sun's motion during the revolution of the moon. As the sun moves from S towards N' (Fig. 116) the arcs Lq', L'q, over which there is an increase of the inclination, will increase; and the arcs qL, q'L', over which there is a diminution, will diminish. The motion of the sun will, therefore, in ap 2') 6 PERTURBATIONS OF ELLIPTIC MOTION OF THE MOON. proaching the descending node, render the increase of the inclination each revolution of the moon greater than it otherwise would be. When the sun is receding from the ascending node, the corresponding arcs will experience corresponding changes, and therefore the diminution will now be less than if the sun were stationary. The results will be similar for the opposite quadrants on the other side of the line of nodes 515. Epochs of greatest and least Inclinations. Since the inclination diminishes as the sun recedes from either node, and increases as it approaches eithe'r node, it will be the least when the nodes are in quadratures, and the greatest when they are in syzigies. It is important to observe that the change of inclination which we have been considering is modified by the retrograde motion of the node; and thus, that, besides the variations of this element connected with the motions of the moon and sun, there is another extending through the period employed by the node in completing a revolution with respect to both the sun and moon. 516. Perturbations Per-iodic. The perturbations of the elliptic motion of the moon, comprising inequalities of orbit longitude, and variations in the form and position of the orbit, which have now been under consideration, depend upon the configurations of the sun and.moon, with respect to each other, the perigee of each orbit, and the node of the moon's orbit. Their effects will disappear when the configurations upon which they depend become the same. They are therefore periodical. 517. blihe Perturbations of the iiotions of a Planet, produced by the action of another planet, are precisely analogous to the perturbations of the motions of the moon, produced by the action of the sun. The disturbing forces a:re obviously of the same kind, and they are subject to variations from precisely similar causes. But, owing to the smallness of the masses of the planets and their great distances, their disturbing forces are much more minute than the disturbing force of the sun. From this cause, together with the slow relative motion of the disturbing and disturbed body, the motion of the apsides and nodes, and the accompanying variations of eccentricity and inclination, are very much more gradual in the case of the planets than in the case of the moon. Their periods comprise many thousands of years, and on this account they are called Secular Mlotions or Variations. In consequence of the greater feebleness of the disturbing forces, the periodical inequalities are also much less in amount. Moreover, as the motion of a planet is much slower than that of the moon, and as the variations of its orbit are more gradual than those of the lunar orbit, the compensations produced by a change of configurations are much more slowly effected, and thus the periods of the inequalities are much longer. 518. Acceleration of the Moon. The motions of the moon would be subject to no secular variations if the apparent orbit of the sun were unchangeable; but the secular variation of the eccentricity of the sun's orbit, which answers to an equal variation of the eccentricity of the earth's orbit, that is produced by the action of the planets, gives rise to a secular inequality in the motion of the moon, called the Acceleration of the Moon. This inequality was discovered from observation. Its physical cause was first made known by Laplace. MASSES AND DENSITIES OF THE PLANETS. 297 CHAPTER XXIV. RELATIVE MASSES AND DENSITIES OF THE SUN, MOON, AND PLANETS:- RELATIVE INTENSITY OF THE FORCE OF GRAVITY AT THEIR SURFACE. 519. Determination of the Masses of the Planets. The perturbations which a planet produces in the motions of the other planets, depend for their amount chiefly upon the ratio of the mass of the planet to the mass of the sun, and the ratio of the distance of the planet from the sun to the distance of the planet disturbed from the same body. Now, the ratio of the distances is known by the methods of Spherical Astronomy; consequently, the observed amount of the perturbations ought to make known the ratio of the masses, the only unknown element upon which it depends. This is one method of determining the masses of the planets. The masses of those planets which have satellites may be found by another and simpler method, viz.: by comparing the attractive force of the planet for either one of its satellites with the attractive force of the sun for the planet. These forces are to each other directly as the masses of the planet and sun, and inversely as the squares of the distances of the satellite from the primary and of the primary from the sun. Thus calling the forcesf F, the masses m, M, and the distances d, D, we have M f: F::clT: V M whence we obtain m: M::fd2: FD2. If we regard the orbits as circles, then d and D will be the mean distances, respectively, of the satellite from the primary, and of the primary from the sun, and are given in Tables II, III, and VI. The ratio off to F is equal to the ratio of the versed sines of the arcs actually described by the satellite and primary, in some short interval of time; since these are sensibly equal to the distances that the two bodies are deflected in this interval from the tangents to their orbits, towards the centres about which they are revolving: and since the rates of motion and dimensions of the orbits of the planet and satellite are known, these arcs and their versed sines are easily determined. Table IV exhibits the relative masses of the sun, moon, and 298 RELATIVE MASSES OF THE SUN, MOON, AND PLANETS. planets, according to the most received determinations, that of the sun being denoted by 1..5O0. Colputatioin of the Densities of the Platnets. The quantities of matter of the sun, moon, and planets, as well as their bulks, being known, their densities may be easily computed; for, the densities of bodies are proportional to their quantities of matter divided by their bulks. Table IV contains the densities of the sun, moon, and planets, that of the earth being denoted by 1. It will be seen on inspecting it, that the densities of the planets decrease from Mercury to Saturn; and that the four planets most distant from the sun are much less dense than the four which are nearest the sun. 521.'Tie Comparative Forces of Gravity at the surface of the sun, moon, and planets, may also readily be found, when the masses and bulks of these bodies are known. For supposing them to be spherical, and not to rotate on their axes, the force of gravity at their surface will be directly as their masses and inversely as the squares of their radii, or, in other words, proportional to their masses divided by the squares of their radii. The centrifugal force at the surface of a planet, generated by its rotation on its axis, diminishes the gravity due to the attraction of the matter of the planet. The diminution thus produced on any of the planets is not, however, very considerable. Tile method of determining the centrifugal force at the surface of a body in rotation, is given in treatises on Mechanics. (See Table IV.) FORM AND DENSITY OF THE EARTH. 299 CHAPTER XXV. FORM AND DENSITY OF THE EARTH:-CHANGES OF ITS PERIOD OF ROTATION.- PRECESSION OF THIE EQUINOXES, AND NUTATION. 522. NVE have already seen (105) that measurements made upon the earth's surface establish that the figure of the earth is that of an oblate spheroid, and that the oblateness at the poles is X9. 523. Density of the Earth. From the amount and law of variation of the force of gravity upon the earth's surface, ascertained by observations upon the length of the seconds' pendulum, it is proved that the matter of the earth is not homogeneous, but denser towards the centre, and that it is arranged in concentric strata of nearly an elliptical form and uniform density. The fact of the greater density of the earth towards its centre has also been established by observations upon the deviation of a plumb-line from the vertical, produced by the attraction of a mountain; the amount of the deviation being ascertained by observing the difference in the zenith distances of the same star, as measured with a zenith-sector on opposite sides of the mountain. To the north of the mountain the plummet was drawn towards the south, and the zenith distance of a star to the north of the zenith was diminished; while to the south of the mountain the plummet was drawn towards the north, and the zenith distance of the same star was increased by an equal amount: and thus the difference of the two measured zenith distances was equal to twice the deviaiion of the plumb-line from the true vertical in either of the positions of the instrument (allowance being made for the difference of latitude of the two stations, as determined from the distance between them and the known length of a degree). Such observations were made for the purpose of determining the mean density of the earth by Dr. Maskelyne, in 1774, on the sides of the mountain Schehallien in Scotland. The observed deviation of the plumb-line made known the ratio of the attraction of the mountain to that of the whole earth, and thus the relative quantities of matter in the mountain and earth. These being ascertained, and the figure and bulk of the mountain having been determined by a survey, the relative density of the earth and mountain became known by the principle men 300 FORM AND DENSITY OF THE EARTH. tioned in Art. 520, and thence the actual density of the earth; the density of the mountain having been found by experiment. The result was, that the mean density of the earth is 4.95. Later determinations make it 5.44. 524. Explaanatiou of Spheroidal Form of Earth. The spheroidal form of' the surface of the earth and of its internal strata is easily accounted for, if we suppose the earth to have been originally in a fluid state. The tendency of' the mutual attraction of its particles would be to give it a spherical form; but by virtue of its rotation, all its particles, except those lying immediately on the axis, would be aninmated by a centrifugal force increasing with their distance from the axis. If, therefore, we conceive of two columns of fluid extending to the earth's centre, one from near the equator, and the other from near either pole, the weight of the former would by reason of the centrifugal force be less than that of the latter. In order, then, that they may sustain each other in equilibrio, that near the equator must increase in length, and that near the pole diminish. As this would be true at the same time for every pair of columns situated as we have supposed, the surface of the whole body of fluid about the poles must fall, and that of the fluid about the equator rise. In this manner the earth would become flattened at the poles and protuberant at the equator. Upon a strict investigation it appears that a homogeneous fluid of the same mean density with the earth, and rotating on its axis at the same rate that the earth does, would be in equilibrium, if it had the figure of an oblate spheroid, of which the axis was to the equatorial diameter as 229 to 230, or of which the oblateness was -23. If the fluid mass supposed to rotate on its axis be not homogeneous, but be composed of strata that increase in density from the surface to the centre, the solid of equilibrium will still be an elliptic spheroid, but the oblateness will be less than when the fluid is homogeneous. 525. Possible Changes of Period of Rotation. The time of the earth's rotation, as well as the position of its axis, would change if any variation should take place in the distribution of the matter of the -earth, or in case of the impact of a ibreign body. If any portion of matter be, from any cause, made to approach the axis, its velocity will be diminished, and the velocity lost being imparted to the mass, will tend to accelerate the rotation. If any portion of matter be made to recede from the axis, the opposite effect will be produced, or the rotation will be retarded. In point of fact, the changes that take place in the position of the matter of the earth, whether from the washing of rains upon the sides of mountains, or evaporation, or any other known cause, are not sufficient ever to produce any sensible alteration in the circumstances of the earth's rotation on its axis. EARTH'S AXIS INVARIABLE. 301 526. Earth's Dimensions aand Axis Invariable. It is ascertained from direct observation, that there has in reality been no perceptible change in the period of the earth's rotation since the time of Hipparchus, 120 years before the beginning of the present era. We may therefore conclude, dc posteriori, that there has been no material change in the form and dimensions of the earth in this interval. lWere the axis of the earth to experience any change of position with respect to the matter of the earth, the latitudes of places would be altered. A motion of 100 feet might increase or diminish the latitude of a place to the amount of 1", an angle which can be measured by modern instruments. Now, in point of fact, the latitudes of places have not sensibly varied since their first determination with accurate instruments; therefore, in this interval the axis of the earth cannot have materially changed. Indeed, since the earth's surface and its internal strata are arranged symmetrically with respect to the present axis of rotation, it is to be inferred that this axis is the same as that which obtained at the epoch when the matter of the earth changed from a fluid to a solid state. 527. Physical Theory of Precession and Nutation. The motions of the earth's axis, along with the whole body of the earth, which give rise to the Precession of the Equinoxes and Nutation, are consequences of the spheroidal form of the earth, inasmuch as they are produced by the actions of the sun and moon upon that portion of the matter of the earth which lies on the outside of a sphere conceived to be described about the earth's axis. The physical theory of the phenomena in question is analogous to that of the retrogradation of the moon's nodes. The sun produces a retrograde movement of the points in which the circle described by each particle of the protuberant mass cuts the plane of the ecliptic, as it does of the moon's nodes; the effect produced is, however, exceedingly small, by reason of the inertia of the interior spherical mass connected with the external mass upon which the action takes place. The moon, in like manner, occasions a retrograde movement of the nodes of the same particles on the plane of its orbit. The actions of the sun and moon will not be the same each revolution of a particle. That of the sun will vary during the year with the angular distance of the sun from the node (510); and that of the moon will vary during each month with the distance of the moon from the node, and also during a revolution of the nodes of the moon's orbit by reason of the change in the inclination of the orbit to the equator. The mean effect of both bodies is the precession; the inequality resulting from the change in the sun's action during the year is the solar nutation; and the inequality consequent upon the retrogradation of the moon's nodes is the lunar nutation, or the chief part of it. 302 THE TIDES. CHAPTER XXVI. THE TIDES. 52S. THE alternate rise and fall of the surface of the ocean twice in the course of a lunar day, or about 25 hours, is the phenomenon known by the name of the Tides. The rise of the water is called the Flood Tide, and the fall the Ebb Tide. 529. Times of High and Low Water. The interval between one high water and the next is, at a mean, half a mean lunar day, or 12h. 25m. Low water has place nearly, but not exactly, at the middle of this interval; the tide, in general, employing nine or ten minutes more in ebbing than in flowing. As the interval between one period of high water and the second following one is a lunar day, or id. Oh. 50m., the retardation in the timne of high warter from one day to another is 50m., in its mean stiate. The timne of hioill water is mainly dependent upon the position of the moii,,lhii}g always, at any given place, about the same length of tilii afler the moon's passage over the superior or inferior meridian. As to the length of the interval between the two periods, at different places, in the open sea it is only from two to three hours; but on the shores of continents, and in rivers, where the water meets with obstructions, it is very different at different places, and in some instances is of such length tlhat the time of high water seems to precede the moon's passage. 5O0. Height of Tide. The height of the tide at high water is not always the same, but varies from day to day; and these variations have an evident relation to the pchases of the imoon. It is greatest soon after the syzigies; after which it diminishes and becomes the least soon after the quadratures. The tides which occur near the syzigies, are called the S2priig Tides; and those which occur near the quadratures are called the Necsp Tides. The highest of the spring tides is not that which has place nearest to new or full moon, but is in general the third following tide. In like manner the lowest of the neap tides is the third or fourth tide after the quadrature. The spring tides are, in general, from once and a half to twice the height of the neap tides. At Brest, in France, the former rise to the height of 19.3 feet, and the latter only to 9.2 feet. PHENOMENA OF TIIE TIDES. 303 On the Atlantic coast of the United States the spling tides exceed the neap tides in the proportion of 3 to 2. The tides are also ctffected by the declinations of the sun and moon: thus, the highest spring tides in the course of the year are those which occur near the equinoxes. The extraordinarily high tides which frequently occur at the equinoxes are, however, in part attributable to the equinoctial gales. Also, when the moon or the sun is out of the equator, the evening and morning tides differ somewhat in height. At Brest, in the syzigies of the summer solstice, the tides of the morning of the first and second day after the syzigy are smaller than those of the evening by 6.6 inches. They are greater by the same quantity in the syzigies of the winter solstice. The dcislance of thle moon from the earth has also a sensible influence tvpon the tides. In general, they increase and diminish as the distance diminishes and increases, but in a more rapid ratio. 531. Daily Retardation of High Water. The daily retardation of the time of high water varies with the phases of the moon. It is at its minimum towards the syzigies, when the tides are at their maximum; and at its maximum towards the quladratures, when the tides are at their minimum. It also varies with the distances of the sun and moon from the earth, and with their declinations. 532. Plhyical Theory of the Tides. The facts which have been detailed indicate that the tides are produced by the actions of the sun and moon upon the waters of the ocean; but in a greater degree by the action of the moon. To explain them, let us suppose at first that the whole surface of the earth is covered with water. We remark, in the first place, that it is not the whole attractive force of the moon or sun which is effective in raising the waters of the ocean, but the difference in the actions of each body upon the different parts of the earth; or, more precisely, that the phenomenon of the tides is a consequence of the inequality and non-parallelism of the attractive forces exerted by the moon, as well as by the sun, upon the different particles of the earth's mass. From this cause there results a diminution in the gravity of the particles of water at the surface, for a certain distance about the point immediately under the moon, and the point diametrically opposite to this, and an augmentation for a certain distance on the one side and the other of the circle 90~ distant from these points, or of which they are the geometrical poles: in consequence of which the water falls about this circle and rises about these points. That the actions of the moon upon the different parts of the earth's mass are really unequal, is evident from the fact that these parts are at different distances from the moon. To show that the inequality will give rise to the results just noted, let us suppose that the 304 THE TIDES. circle acbd (Fig 118) represents the earth, and M the place of the moon; then a will be the point of the earth's surface directly under the moon, b the point diametrically oppo97 7I \F site to this, and the right line de, perpendicular to MO,will represent the _ \m/ lo "circle traced on the earth's surface 900 distant from a and b. Now, the attraction of the moon for the general m mass of the earth is the same as if the whole mass were concentrated at the \a / centre 0. But the centre of the earth is more distant from the moon than the point a at the surface. It follows, therefore, that a particle of matter situated at a will be drawn towards the moon with a proportionally greater fbrce than the centre, or than the gen\ eral mass of the earth. Its gravity or \' I tendency towards the earth's centre will therefore be diminished by the amount of this excess. On the other hand, the centre is nearer to the moon than the _I point b. It is therefore attracted more FIG. 118. strongly than a particle at b. The excess will be a force tending to draw the centre away from the particle; and the effect will be the same as if the particle were drawn away from the centre by the same force acting in the opposite direction. The result then is, that this particle has its gravity towards the earth's centre diminished, as well as the particle at ca. If now we consider a particle at some point t near to a, the moon's action upon it (tr) may be considered as taking effect partially in the direction tk parallel to OM, and partially in the direction of the tangent or horizontal line ts. The component (rs) in the latter direction, will have no tendency to alter the gravity of the particle towards the earth's centre. The component (sr) in the direction tck, will obviously be less than the actual force of attraction tr; and the difference will be greater in proportion as the particle is more remote from a. This component will decrease gradually from a, where it is equal to the attractive force, while the attraction for the centre is less than for a by a certain finite difference: it is plain, therefore, that the component in question will be greater than the attraction for the centre, in the vicinity of the point at, and for a certain distance from it in all directions. The gravity of tile particles will therefore be diminished for a certain distance from this point. In a similar manner it may be shown that it will also be diminished for a certain distance from the point b. Let us now THEORY OF THE TIDES. 305 consider a particle at c, 90~ from the points a and b. The attraction of the moon for it will take effect in the two directions ce and cO. The force in the latter direction alone will alter the gravity of the particle; and this, it is plain, will increase it. The same effect will extend to a certain distance from c in both directions. A strict mathematical investigation would show that the gravity is diminished for a distance of 550 from a and b in all directions; and is augumented for a distance of 350 on each side of' the circle de, 900 distant firom the points a and b. These distances are represented in the Figure. This may be easily made out by means of the expression for the radial disturbing force of the sun in its action upon the moon (505), viz. m x y (1 - 3 cos2 I). If we consider m as denoting the mass of the moon, a the moon's distance from the earth's centre, y the distance of a particle of matter at some point t of the earth's surface from the earth's centre, and p the angular distance or elongation (MOt) of the sanme particle firom the moon, as seen from the centre of the earth, it will express the change in the gravity of a particle at tile earth's surface, produced by the moon's action. The points a and b will answer to conjunction and opposition, and the points c and d to the qandratures. Now we have already seen (505) that the gravity of thle moon is increased at the quadratures, and for 35~ on each side of them; and dirinished at the syzigies, and 55~ from them in both directions. It follows, therefore, that the same is true for particles of matter at the earth's surface. In consequence of the earth's diurnal rotation, the parts of the surface, at which the rise and fall of the water will take place, will be continually changing. Were the entire rise and fall produced instantaneously, the points of highest water would constantly be the precise points in which the line of the centres of the moon and earth intersects the surface, and it would al]ways be high water on the meridian passing through these points, both in the hemisphere where the moon is, and in the opposite one. On the west side of this meridian, the tide would be flowing; on the east side of it, it would be ebbing; and on the meridian at right angles to the same, it would be low water. But it is plain that the effects of the moon's action would not be instantaneously produced, and therefore that the points of highest water will fall behind the moon. 533. Comparative Effects of Sun and M[ooin. It is evident that the sun will produce precisely similar effects with the moon, and will raise a tide wave similar to the lunar tide wave, which will follow it in its diurnal motion. To show that the effects of the sun are less in degree than those of the moon, let us take the general expression for the change of the moon's gravity, arising firon the action of the sun, namely, M x Y (l — 3 cos )... (a). a3 From what we have seen in the previous article, this formula will serve to ex press the change in the gravity of a particle of matter upon the earth's surface, produced by the sun's action, if we take m = the mass of the sun, as befbre, at _ its distance expressed in terms of the radius of the earth as unity, y = the dis20 2j C0) THE TIDES. tance of the particle from the centre of the earth, and (b = its elongation from the sun, as seen firom the earth's centre. If we designate the corresponding quanrtities fbr the moon by m' a' y, a, we shall have for the change of the gravity of a particle, produced by tihe moon's action, 3x (I - 3 co(s2 )... (b). a13 For particles at equlal elongations from the sun and moon, we shall have p the same in expressions (a) and (b), and y may be regarded as the same without material error. For such particles, then, the alterations of the gravity, produced by the sun and moon, will bear the same ratio to each other as the quantities 1 and'a3 a Now, if we give to m, i?,'. a, a', their values, we shall find that the latter quantity is about 2: times greater than the former. Accordingly, the effect of tile moon's action, at corresponding elongations of' the particles, and therefore generally, is about 21- times gfreater than that of the sun. 534. Coinbineid Effects of Suit and lIoon. The actual tide will be produced by the joint action of the sun and moon, or it many be regarded as the result of the combination of the lunar and solar tide waves. At the time of the syzigies, the action of the sun and moon will be combined in producing the tides, both bodies tending to produce high as well as low water at the same places. But at the qlliadratures they will be in opposition to each other, the one tendingo to raise the surface of the water where the other tends to depress it, and vice versa. The tides should, therefore. be mluch higher at the syzigies than at the quadratures. Between the syzigies and the quadratures the two bodies will neither directly conspire with each other, nor directly oppose,each other, and tides of intermediate height will have place. The apoints of highest water will also, in the configuration supposed, neither be the vertices of the lunar nor of the solar tide wave, but certain points between them. This circumstance will occasion a variation in the length of the interval between the time of the moon's passage and the time of high water. Spring and lVeap Tides. The effect of the moon's action being to that of the sun's nearly as 21 to 1 (533), the spring tides will,be to the neap tides nearly as 21 to 1. For, let x- the effect of the moon, and y - the effect of the sun: then the ratio of x + y to x - y will be the ratio of the heights of the spring and neap -tides. Now, x= 2.34y, and thusx 2.3 y 2.5. x - y - 2.34y - y'We have already seen that the ratio obtained from observation is less than this. The height of the joint tide, as well as the interval between the time of high water and that of the moon's meridian passage, wvill vary not only with the elongation of the moon from the sun, but also with the distance and declination of the moon. For. COMPARISON OF THEORY WITH OBSERVATION. 307 expressions (a) and (b) above given, show that the intensities of the rnoon's and sun's actions vary inversely as the cube of their distance; and the changes of the declinations of the two bodies must be attended with a change both in the absolute and relative situation of the vertices of the lunar and solar tide waves. COMPARISON OF THE THEORY OF THE TIDES WITH THE RESULTS OF OBSERVATION. 535. The laws of the tides, which should obtain on the hypothesis of the earth being entirely covered with water, are found to correspond only paltially with those of the actual tides. The continents have a material influence upon the formation and propagation of the tide-wave. Thle actual phenomena of the tides have been carefully observed, for many years, at numerous points along the coast lines of continents and on the shores of islands: and the results of the observations have been subjected to a thorough discussion by several distinguished astrononmers and physicists. As one result of the discussion the determination has been effected of a system of Cotidal Lines; that is, a set of lines connecting those places at which high tide occurs at the same instant, from hour to hour. A chart has been constructed showing, at intervals of 1", 2h, 3h, &c., after the meridian transit of the moon at Greenwich, the cotidal lines of the Southern, Atlantic, and Pacific Oceans. These lines show the varvyin, form of the ridge of the tide-wave as it proceeds on its course and by the greater or less distance between them the rate of propagation of the wave in different oceans and in different parts of the same ocean. Along the coasts they are, for the most part, constructed from actual observations, but their exteiisions into the open sea are mostly inferential. 536. Tide-wave of the Atlantic Ocean. By examinirog the chart of cotidal lines we learn that the floodtide of the Atlantic Ocean is, for the most part, produced by a derivative tide-wave, sent off from the great wave which, in the Southern Ocean, follows the moon in its diurnal motion around the earth. At 6 hours after tile meridian transit of the moon at Greenwich, the derivative tide-wave stretches from the coast of Upper Guinea to the coast of Brazil, a little to the south of the narrowest part of the Atlantic. Three houlrs later it has advanced, by estimation, in mid-ocean, to about 24~ of north latitude; lnd( in 3 hours more, or 12 hours after the meridian transit of the moon at Greenwich, it has reached the Atlantic coast of the United States. It advances more rapidly in the open sea th1lan along the coasts, where the depth of the water is less. It is therefoie convex towards the north. Thus, at the hour just mentioned, it stretches nearly parallel to the general trend of our Atlantic 308 THE TIDES. coast, along its whole extent into the northern Atlantic, and there curves around to the south-east, so as to strike, at its eastern end, the N. W. coast of Africa (lat. 23~). The same wave does not reach the coast of Spain until more than two hours later. 537. Nature and Velocity of the Tide-wave. The tidalwave is of the nature of a wave of translcation. In this form of wave there is no oscillation proper; but the particles of the fluid, in a cross section perpendicular to the line of propagation, by the transit of the wave are raised, transferred forward, and brought to rest in the direction of the motion in a new place; with the same extent of transference of each particle throughout the whole depth of the wave. Whereas in ordinary oscillatory waves, such as those caused by the wind, the individual particles oscillate in vertical circles, or ellipses, and return to their original position. A wave of translation travels with a velocity equal to that acquired by a heavy body in falling freely by gravity through a height equal to half the mean depth of the fluid, reckoned fiom the top of the wave to the bottom of the channel. Its velocity is therefore directly proportional to the square root of the depth of the fluid. The rate of propagation of an oscillatory wave, on the other hand, is independent of the depth, and varies only with the breadth of the wave. The moon tends to draw the wave which it raises along with it in its diurnal course, at the rate of 1,000 miles per hour at the equator; but it appears that the tidal wave actually travels at a much less rapid rate. Setting out from the Eastern Pacific, where it lags about 2 hours behind the moon, it travels westward in about 12 hours to New Zealand. From thence to the Cape of Good Hope, passing south of Australia, it occupies 17 hours, and has an average velocity of about 470 miles per hour. From the Cape of Good Hope the portion of the wave that passes northward into the Atlantic traverses the distance to the coast of the United States in about 11 hours; which is at the average rate of 565 miles per hour. The tide-wave accordingly does not reach our Atlantic coast until about 40 hours after it originated in the South-eastern Pacific. The average velocity of 565 miles in the South and North Atlantic, answers to a depth of 21,500 feet, or about 4 miles. The average velocity in midocean is greater than this, and answers to a greater depth. The velocity of the tide-wave becomes rapidly reduced after the wave strikes the shallow waters of the coast, to 100 miles per hour; or even less than 50 miles per hour in bays and sounds. As a necessary consequence the breadth of the wave diminishes with its velocity. At a velocity of 565 miles per hour it has a breadth of 7,000 miles. When the velocity is reduced to 100 miles per hour the breadth is only 1,240 miles. TIDES OF THE ATLANTIC COAST. 309 TIDES OF THE ATLANTIC COAST OF THE UNITED STATES. 538. General Phenomena. The phenomena of the tides as they occur along the entire coast line of the United States, have been carefully deduced by the late Superintendent of the Coast Survey, from the systematic tidal observations carried on in connection with the Survey. The following are the more iunportant general results obtained from the discussion: 1. The cotidal lines, in the vicinity of the Atlantic coast, are nearly parallel to the general trend of the coast. The ridge of the tide-wave, as it approaches the coast, is therefore nearly parallel to the coast line. This wave, when it reaches the most prominent points of the coast, has a mean height of about 2 feet above the lowest point of the ebb-wave, or mean low-water level. 2. The coast is physically divided, by projecting headlands, into three great bays, each of which has its particular system of cotidal lines, running nearly parallel to the shore. These bays may be designated as the Southern, Middle, and Eastern Bays. Tihe Southern Bay lies between Cape Florida and Cape Hatteras; the AMiddle Bay between Cape Hatteras and Nantucket (eastern end); and the Eastern Bay between Nantucket and Cape Sable (Nova Scotia). The latter is supposed to be a portion of a greater bay, from Nantucket to Cape Race (Newfoundland). In the Southern B-ay, the mean rise and fall, or range of the tides along the shores, increases from about 2 feet at the capes to'7 feet at Port (Royal, at the head of the bay. In- the Middle Bay, the range increases from 2 feet to nearly 5 feet at Sanldy IIook and Cape May. In the Eastern Bay the tides are more complex, owing to greater irregularities in the shore line, and the influence of shoals. The heights increase rapidly from Nantucket to Cape Cod; the mean range being 2 feet at Nantucket and 9.2 feet at Provincetown. At Cape Ann (the northern cape of Massachusetts Bay) it is about the same. From Cape Ann northward to Portsmouth there is a decrease of about half a foot in the mean range of the tides. From thence, following the shore line towards the northeast, it increases at an augmenting rate until at the entrance of the Bay of Fundy the tide rises, on the average, 18 feet above low water. 539. Tides of Inner Bays. The tides of Delaware Bay, New York Bay, and Narragansett and Buzzard Bays, present, on a smaller scale, the same phenomena of increase in the height of the tides in ascending, as the three great bays, or undulations of the coast. On the contrary, in Chesapeake Bay, which widens and changes direction at right angles immediately from the entrance, the tides diminish in height, as a general rule, In going up the bay. 310 THE TIDES. The tide-wave, on entering Massachusetts Bay, increases somewhat, viz., from 9 feet above low water at the entrance to 10 feet at Plymouth and Boston. In the Bay of Fumndy, the tides rise to a much greater height than on any other part of the Atlantic Coast. At St. Johns, N. B., the mean rise and fall of the tide is 19.3 feet; and at Shadwood Point, at the head of the bay, no less than 36 feet. The ordinary spring tides attain, at the latter place, to the height of 50 feet. Special tides have been lnown to rise 20 feet higher. This remarkable accumulation of the tidal waters results from the great contraction in the width of the bay or channel into which the ascending wave is forced. 450. Tides of Chananels. In channels peculiar tides occur, in consequence of the meeting of the waves which enter the channels at their two extremities. Where the two flood waves meet in the same state, a tide equal to the sum of their two heiohts is produced by their superposition. At other points the tides are variously modified by the interference of the waves. Tides in Sounds present similar peculiarities. 541. Tidees of Long Island Sound. The great tidal wave from the Atlantic enters the Sound between Point Judith and Montauk Point; and another portion of this wave enters New York Bay, and passing through Hell Gate, meets the wave propagated through the Sound from the eastward. The point of meeting of the crests of the two waves is off' Sands' Point, at the head of the Sound. At Montauk Point the mean height of the tide-wave, above low water, is 2 feet, and at Sandy Hook 4.8 feet. At Sands' Point it is 7.1 feet; exceeding the sum of the heights at the two entrances by nearly 1 foot, owing to the narrowingo of the Sound. The mean racnge of the tide declines in both directions from Sands' Point. At Bridgeport it is 6.6 teet; at New Haven 5.8 feet; at New London and Stonington between 2- feet, and 21 feet; and at Point Judith 3 feet. The tide is propagated from Montauk Point to the head of the Sound in 3 hours. It travels from Fisher's Island to Sands' Point, 95 miles, in 2h. im.; or at the average rate of 41 miles per hour. This agrees approximately with the velocity as theoretically computed from the soundings taken by the Coast Survey, accordcling to the law of' propagation of a wave of translation (531). At Fisher's Island it is about 60 miles, and becomes reduced to 30 miles at the head of the Sound, where the depth of the water is less. In the East River the rate of propagation of the tide is only about 71 miles per hour. Owing to the retardation of the tide-wave in the shallow waters near the Connacticut shore, it is liearly parallel tc the shore, from the head of the Sound to a distance of some 20 miles east of New Haven harbor. Accordingly high water occurs at about the same hour alongo this extent of the shore. Farther to the east, the line of the tide-wave is inclined to the shore line, and the tides occur earlier.'S2t. Tidal Currents. The currents produced by the tides in the shallow waters of bays, sounds, and rivers, are not to be confounded with the transmission of the tide-wave. Their velocity is but a few miles per hour; and the turn of the current, or tide-stream, does not in general correspond to the turn of the tide, and may occur at quite a different hour. For example, at Montauk Point the ebbstream does not begin until half ebb-tide, and in New York Bay it begins at onesixth of' the ebb tide. Tidal currents owe their origin to the resistance opposed by shallow waters, and contracted channels, to the free propagatiou of the tide-wave, and to differences of hydrostatic level. They have the greatest velocity in narrow channels, as in the Race off Fisher's Soiund. and in Hell Gate. About the time of the turn of the tide, at the head of the Sol, nd, there is a certain interval of slac wcater there. Aftex the tide-wave begins to move in the opposite direction, the accumulative effect of TIDES OF THE PACIFIC COAST. 311 the resistances determines, in a certain interval of time, a sensible current, which shows itself first at the surface and in-shore, but soon becomes general. In midchannel, throughout the Sound, the outward motion of the water commences shortly after high water at the head of the Sound, and evidently depends upon it. A similar, but still more striking fact is observed in the Irish Clhannel. The turn of the stream, whether flood or ebb, is simnultaneous throughout the entire length of the channel. It is coincident with the time of high or low water at Morecambe Bay, north of Liverpool, where the tides coming round the extremities of Ireland finally meet. The times of slackwater throughout the channel, therefore, correspond with the times of high and low water at Morecambe Bay. In the Irish Channel there are two spots, in one of which the stream runs with considerable velocity without the tide either rising or falling, while in the other the water rises and falls from sixteen to twenty feet without having any visible horizontal motion at its surface. The average maximum drift qf the current in Long Island Sound, is 2.2 knots per hour. The average maximum current velocity opposite the west end of Fisher's Is. land is nearly 4- knots per hour; and at Hell Gate nearly 6 knots. In New York harbor it is 3.7 knots, and in the Bay 3 knots. The point of meeting of the two flood streams in the East River, is a little to the east of Throgs' Neck. To the east and west of that point, both the flood and ebb streams run in opposite directions. The 7mean duration of the flood streamz at different points of Long Island Sound varies between 4f- hours and 7I hours. The corresponding limits for the ebb stream are 5h. and 8h. The mean duration of slackwater varies between 0mn. and 45m1. It is at most places less than 10m. The duration of the ebb or flood stream, differs as much as 4 of an hour in successive tides; but commonly not more than 10m. The set of the currents is ordinarily nearly parallel to the shore. 543. Wides of Iliveers. The tide-wave that enters the mouth of a river is propagated according to the same laws as a wave that comes in at the entrance of a sound, or channel. The ve locity varies with the depth of water; and the height of the tide increases where the river contracts, and decreases where it widens. Thus, in a tidal river of considerable length, the tide may have various heights at different points. The ascending flood tide may also be encountered by the descending ebb tide. On the Hudson the tide rises at West Point, 55 miles from New York, 2.7 feet; at Tivoli, nearly 100 miles from New York, 4 feet; and at Albany, 2.3 feet. In the shallow parts of rivers, the tide-wave becomes converted into a tidal current, by which alone the tide is transmitted. In rivers the, duration of' the ebb tide is considerably longer than that of the flood. Thus, at Philadelphia and Richmond, the ebb continues 2~ hours longer than the flood tide. TIDES OF THE PACIFIC COAST. 544. Cotidal Lines. The cotidal lines of the Pacific eoast of the United States are approximately parallel to the coast. Thus, high tide occurs at about the same hour from San Francisco to Vancouver's Island. South of San Francisco the tidewave arrives at an earlier hour; at the southern extremity of California, about 29 hours earlier. 545. Diarn ail lnequttaity. The tides of the Pacific coast 312 THE TIDES. are remarkable for the great ineqzUality that prevails between the heizgts of tco successive tides, as measured from the high water mark of each tide to the next succeeding low water mark. The difference of level of the two successive, high tides is less conspicuous, but quite marked. The differences are greater for the ebb tlhan for the flood tides. These diurnal inequalities increase with the imoon's declination, north or south; and vanish entirely when the moon is in the equator. When the moon's declination is north, the highest of the two high tides of the twenty-four hours occurs at San Francisco about 1]~- hours after the moon's superior transit; and when the declination is south, the lowest of the two high tides occurs about this interval after the transit. When the moon has its greatest declination the mean range of the highest tide is nearly 7 feet, and of the lowest tide from 1~ ft. to 3 ft. The lowest tide sometimes amounts to only two or three inches. According to Professor Bache, the tides that occur on the western coast, near the maximum of the moon's declination and fcr several days on each side of it, result from the interference of a semi-diurnal and diurnal wave, which at the maximum of each are nearly equal in magnitude, the crest of the diurnal wave being at that period about eight hours in advance of that of the semi-diurnal wave. This diurnal wave exists only when the moon has a considerable declination. On the Atlantic coast the corresponding inequality at the time of the moon's greatest declination, is a small fraction of the height of the tide, and is generally not more than one foot. A similar remark may be made of the tides of the coast of Europe. TIDES OF THE GULF CF MEXICO. 546. On the northern coast of the Gulf of Mexico, from Florida westward, there is but one tide in the 24 hours, and the mean range of this tide is only from 1 foot to 11 feet. The seconld tide is doubtless obliterated by the interference of the semi-(diuralal flood-tide with a diurnal ebb-tide; as happens approxilmately on the Pacific coast (545). For some three to five (lays, about the time when the moon is crossing the equator, when the diurnal inequality should vanish, from the absence of the diurnal wave (545), there are generally two tides at the sarnme places on the coast, the rise and flal being quite small. The greatest rise and fall of the single day-tide occurs when the moon's declination is the greatest. The small height of the tides in the Gulf of Mexico is attributable chiefly to the fact that the width of the gulf is three or four times greater than that of the two channels through which the tide-wave enters it. TIDES OF THE COAST OF EUROPE. 313 TIDES OF THE MEDITERRANEAN. 54-7. The average height of the tide in the Mediterranean is said not to exceed 1~ feet, though at some ports, as Tunis and Venice, it sometimes amounts to 3 or 4 feet. The Mediterranean is of sufficient extent for the sun and moon to produce a sensible tide by their direct action. A derivative tide-wave, from the Atlantic Ocean, should also enter the Straits of Gibraltar, and spread out laterally as it advances; but the ebb and flow from this cause is said to be slight. TIDES OF INLAND SEAS AND LAKES. 548. Lakes and inland seas have no perceptible tides, or only very small tides, for the reason that their extent is not sufficient to admit of any sensible inequality of gravity, as the result of the action of the moon (532). A tide of nearly 2 inches has been detected at Chicago, on the southwestern shore of Lake MRichigan. TIDES OF THE COAST OF EUROPE. 549. The tide-wave advancing from the south, makes a considerable angle with the coast of Europe, and thus the tide occurs continually later in following the coast from the Straits of Gibraltar northward; and along its entire extent from two to twelve hours later than the correspondcling tides on the coast of the United States. Similar varieties of tidal phenomena occur on either coast. T/le hizhest tides prevail in the Bristol Channel, and the Bay of St. Mlalo, on the northwest coast of France. At the head of the Br'istol Channel, and of the Bay of St. Malo, the spring tides sometimes rise to the height of 50 feet. The mean range of spring tides is 26 feet at Liverpool, nearly 13 feet at Portsmouth, and about 20 feet at London Docks. On the coast of France, tlle height of the tides at different ports falls approximately between the same limits as on the coast of England. The lowest tides occur on the eastern coast of Ireland, to the north of the entrance to St. George's Channel. At Courtown, about 30 miles north of Tuskar, there is scarcely any rise or fall of the water. Fromrn that point the height of the tide increases about equally in every direction, friom 0 to 15 feet on the opposite coast. The remarkably low tides at that locality result from the filct that the tide stream is diverted by a promontory at the entrance of the channel to the opposite shore. ESTABLISHMENT OF THE PORT.-TIDE-TABLES. 550. The interval between the time of the moon's crossing the 314 THE TIDES. meridian and the time of high water at a given place is nearly constant. It varies only between moderate assignable limits. The mean interval on the days of new and full moon is called the establzshrment of the port. The average of the intervals during a month's tides, is called the mean, or correct establishrazent. The mean establishment of Boston is 11h. 27m.; of New Haven 11h. 16m.; of New York 8h. 13m.; of Charleston, S. C., 7h. and 2mrn.; of San Francisco 12h. 12m. 551. Calculatiotl of Tlime of High Water. When the mean establishment of a port is known, the time of high water on any day may be approximately determined. The hour of transit of the moon on the given day is to be taken from the Nautical Almanac and added to the mean establishment; the result will be the time of high water. If the time thus determined falls in the succeeding day, half a lunar day (12h. 25m.) is to be subtracted, as this is the mean interval between two successive tides. On the day of new or full moon, the time of the next high water after noon, will be approximately equal to the establishment of the port. In the annual Coast Survey Reports a table is published, giving the interval between the time of the moon's transit and the time of high water for different hours of transit, and for the principal ports onl the U. S. coast. If the time of the moon's transit onl any day be obtained from the Nautical Almanac, the interval correspondingl to this titme in the table, added to the time of transit, will give more accurately the time of high water. 552. A tide table for the coast of the United States, is published in the same Reports, givinog t)r numerous points of the coast the mean values of the interval between the time of the moon's transit and time of high water, the rise and fall of the tides, the rise and fall of the spring and neap tides, the duration of flood and of ebb tide, and the duration of the stand, or the period of time during whichl the surface of the water neither rises nor falls. A table is also given sthowing, for various ports, the rise and tall of tides corresponding to different hours of tile moon's transit; firom which, by taking the time of transit for any day from the Almaniac, the corresponding rise and fall of the tide may be obtained tbr any of the ports Ientioned in the table PART III. ASTRONOMICA:L PROBJLEMS. EXPLANATIONS OF THE TABLES. THE Tables which form a part of this work, and which are em[)loyed in the resolution of the following Problems, consist of Tales of the Sun, Tables of the Moon, Tables of the Mean Places of some of the Fixed Stars, Tables of Corrections for Refraction, Aberration, and Nutation, and Auxiliary Tables. The Tables of the Sun, which are from XVII to XXXIV, inclusive, are, for the most part, abridged from Delambre's Solar Tables. The mean longitudes of the sun and of his perigee for the beginning of each year, found in Table XVIII, have been computed from the formulae of Prof. Bessel, given in the Nautical Almanac of 1837. The Table of the Equation of Time was reduced from the table in the Connaissance des Tems of 1810, which is more accurate than Delambre's Table, this being in some instances liable to an error of 2 seconds. The Table of Nutation (Table XXVII) was extracted from Francceur's Practical Astronomy. The maximum of nutation of obliquity is taken at 9".25. The Tables of the Sun will give the sun's longitude within a fraction of a second of the result obtained immediately from Delambre's Tables, as corrected by Bessel. The Tables of the Moon, which are from X[XXIV to LXXXV, inclusive, are abridged and computed from Burckhardt's Tables of the Moon. To facilitate the determination of the hourly motions in longitude and latitude, the equations of the hourly motions have all been rendered positive, like those of the longitude. Sonme few new tables have been computed for the same purpose. The longitude and hourly motion in longitude will very rarely differ from the re sults of Burckhardt's Tables more than 0".5, and never as much as 1'. The error of the latitude and hourly motion in latitude will be still less. The other tables have been taken from some of the most approved modern Astronomical Works. (For the principles of the construction of the Tables, see Note 1., Appendix.) Before entering upon the explanation of each of the tables, it will be proper to define a few terms that will be made use of in the sequel. The given quantity with which a quantity is taken from a table3 is called the A rgument of this quantity. 316 ASTRONN OMICAL PROBLEMS. The angular arguments are expressed in some of the tables according to the sexagesimal division of the circle. In others, they are given in parts of the circle supposed to be divided into 100 1000, or 10000, &c., piars. Tables are of Single or Double Entry, according as they contain one or two arguments. The Epoch of a table is the instant of time for wvhich the quantities given by the table are comnputed. By the Epoch of a quantity, is meant the value of the quantity found for some chosen epoch, from which its value at other epochs is to be computed by means of its known rate of variation. Table I, contains the latitudes and longitudes from the meridian of Greenwich, of various conspicuous places in different parts of the earth. The longitudes serve to make known the time at any one of the places in the table, when that at any of the others is given. The latitude of a place is an important element in various astronomical calculations. Table II, is a table of the Elements of the Orbits of the Planets with their secular variations, which serve to make known the elements at any given epoch different from that of the table. From these the elliptic places of the planets at the given epoch may be computed. Table III., is a similar table for the Moon. Table TI. (a) gives the mean distances, &c., of the Planetoids. Tables IV, V, VI, VII, require no explanation. Table VIII, gives the mean Astronomical Refractions; that is, the refractions which have place when the barometer stands at 30 inches, and the thermometer of Fahrenheit at 50~. Table IX, contains the corrections of the Mean Refractions for + 1 inch in the barometer, and - 1~ in the thermometer, from which the corrections to be applied, at any observed height of the barometer and thermometer, are easily derived. Table X, gives the Parallax of the Sun for any given altitude on a given day of the year; for reducing a solar observation made at the surface of the earth to what it would have been, if made at the centre. Table XI, is designed to make known the Sun's Semi-diurnal Arc, answering to any given latitude and to any given declination of the sun; and thus the time of the sun's rising and setting, and the length of the day. Table XII, serves to make known the value of the Equation of'I;me, with its essential sign, which is to be applied to the apparent time to convert it into the mean. If the sign of the equation taken firom the table be changed, it will serve for the conversion of mean time into apparent. This table is constructed for the year 1840. Table XIII, is to be used in connection with Table XII, when the given date is in any other year than 1840. It furnishes the Secular Variation of the Equation of Time, from which the proportional part of its variation in the interval between the gi yven date and the epoch of Table XII is easily derived. EXPLANATION OF THE TABLES. 317 Table XIV, contains certain other Corrections to be applied t~ tbe equation of time taken from Table XII, when its exact value to within a small fraction of a second, is desired. Table XV, gives the Fraction of the Year corresponding to each date. This table is useful when quantities vary by known and uniform degrees, in deducing theii values at any assumed time from their values at any other time. Table XVI, is for converting Hours, Minutes, and Seconds into decimal parts of a Day. Table XVII, is for converting Minutes and Seconds of a degree into the decimal division of the same. It will also serve for the conversion of minutes and seconds of time into decimal parts of an hour. The last two tables will be found frequently useful in arithmetical operations Table XVIII, is a table of Epochs of the Sun's Mean Longitude, of the Longitude of the Perigee, and of the Arguments for findingy the small equations of the Sun's place. They are all calculated for the first of January of each year, at mean noon on the meridian of Greenwich. Argument I. is the mean longitude of the Moon minus that of the Sun; Argument II. is the heliocentric longitude of the Earth; Argument III. is the heliocentric longitude of Venus; Argument IV. is the heliocentric longitude of Mars; Argument V. is the heliocentric longitude of Jupiter, Argument VI. is the mean anomaly of the Moon; Argument VII. is the heliocentric longitude of Saturn; and Argument N is the supplement of the longitude of the Moon's Ascending Node. Argument I. is for the first part of the equation depending on the action of the Moon. Arguments I. and VI. are the arguments for the remaining part of the lunar equation. Arguments II. and III. are for the equation depending on the action of Venus; Arguments II. and IV. for the equation depending on the action of Mars; Arguments II. and V. for the equation depending on the action of Jupiter; and Arguments II. and VII. for the equation depending on thile action of Saturn. Argument N is the argument for the Nutation in longitude: it is also the argument for the Nutation in right ascension, and of the obliquity of the ecliptic. Table XIX, shows the Motions of the Sun and Perigee, and the variations of the arguments, in the interval between the beginning of the year and the first of each month. Table XX, shows the Motions of the Sun and Perigee, and the variations of the arguments from the beginning of any month to the beginning of any day of the month; also the same for Hours. Table XXI, gives the Sun's Motions for Minutes and Seconds. Tables XVIII to XXI, inclusive, make known the mean longitude of the Sun from the mean equinox, at any moment of time. Table XXII, Mean Obliquity of the Ecliptic for the beginning 318 ASTRONOMIC VL PROBLEMS. of each year contained in the table. It is found for any intermediate timne by simple proportion. Tables XXIII, and XXIV, furnish the Sun's Hourly Motion and Semi-diameter. Table XXV, is designed to make known the Equation of the Sun's Centre. When the equati n has the negative sign, its supplement to 12s. is given: this is to be added along with the other equations of longitude, and 12s. are to be subtracted from the sum. The numbers in the table are the values of the equation of the centre, or of its supplement, diminished by 46".1. This constant is subtracted from each value, to balance the different quantities added to the other equations of the longitude, in order to render them affirmative. The epoch of this table is the year 1840. Table XXVI, gives the Secular Variation of the Equation of the Sun's Centre, from which the proportional part of the variation in the interval between the given date and the year 1840, may be derived. Table XXVII, is for the Nutation in Longitude, Nutation in Right Ascension, and Nutation of the Obliquity of the Ecliptic. The nutation in longitude and nutation in right ascension, serve to transfer the origin of the longitude and right ascension from the mean to the true equinox. And the nutation of obliquity serves to change the mean into the true obliquity. Tables XXVIII to XXXIII, inclusive, give the Equations of the Sun's Longitude, due respectively to the attractions of the Moon, Venus, Jupiter, Mars, and Saturn. Table XXXIV, is for the variable part of the Sun's Aberration. The numbers have all been rendered positive by the addition of the constant 0".3. Table XXXV, contains the Epochs of the Moon's Mean Longitude, and of the Arguments of the equations used in determining the True Longitude and Latitude of the Moon. They are all calculated for the first of January of each year, at mean noon on the meridian of Greenwich. The Argument for the Evection is diminished by 30'; the Anomaly by 2~; the Argument for the VariaLtion by 90, and the mean longitude by 90 45'; and the Supplement of the Node is increased by 7'. This is done to balance the quantities which are added to the different equations in order to render them affirmative.'I'ables XXXVI to XL, inclusive, give the Motions of the Moon, and the variations of the arguments, for Months, Days, Hours, Minutes, and Seconds; and, together with Table XXXV, are for finding the Moon's Mean Longitude and the Arguments, at any assumed moment of time. Tables XLI to LIII, inclusive, give the various Equations of the Moon's Longitude. It is to be observed with respect to Table XLI, that the right hand figure of the argument is supposed to be dropped. But when the greatest attainable accuracy is desired, ii EXPLANATION OF THE TABLES. 319 can be retained, and a cipher conceived to be written after the numbers in tile columns of Arguments in the table. In Tables L, LI, LII, and LV, the degrees will be found by referring to the head or foot of the column. (See Problem II., note 2.) Table LIV is for the Nutation of the Moon's Loni-tude. Tables LV to LIX, inclusive, are for finding the Latitude of the Moon. Tables LX to LXIII, inclusive, are for the Equatorial Paral lax of the -Moon. Table LXIV furnishes the Reductions of Parallax and of the Latitude of a Place. The reduction of parallax is for obtaining the parallax at any given place from the equatorial parallax. The reduction of latitude is foi reducing the true latitude of a place, as determnined by observation, to the corresponding latitude on the supposition of the earth being a sphere. The ellipticity to which tile numbel s in the table correspond is 3 l o. Tables LXV and LXVI, Moon's Semi-diameter, and the Augmentation of the Semi-diameter depending on the altitude. Tables LXVII to LXXXV, inclusive, are for finding the HIonrly i Motions of the Moon in Longitude and Latitude. Table LXXXVI, Mean New Moons, and the Arguments for the Equations for New and Full Moon, in January. The time of mean new moon in January of each year has been diminished by 15 hours, the sum of the quantities which have been added to the equations in Table LXXXIX. Thus, 4h. 20m. has been added to equation I.; 10h. 10m. to equation II.; 10m. to equation III.; and 20n. to equation IV. Tables I~XXXVII and LXXXVIII, are used with the preceding in finding' the Approximate Time of 3!Mean New or Full Moon ill any given month of the year. T'able LXXXIX furnishes the Equations for finding the Approximate Time of New or Full Moon. Table XC contains the Mean Right Ascensions and Declinations of 50 principal Fixed Stars, for the beginning of the year 1840, with their Annual Variations. Table XCI is for finding the Aberration and Nutation of the Stars in the preceding catalogue. Table XCII contains the Mean Longitudes and Latitudes of some of the principal Fixed Stars, for the beginning of the year 1840, with their Annual Variations. Tables XCIII, XCIV, XCV, Second, Third, and Fourtlh Differences. These tables are given to facilitate the determination, firom the Nautical Almanac, of the moon's longitude or latitude for any time between noon and midnight. Table XCVI, Logistical Logarithms. This table is convenient in working propDortions, when the terms are minutes and seconds, or degrees and minutes, or hours and minutes,-especially when the first term is lb. or 60m 320 ASTRONOMIICAL PROBLEMS. To find the logistical logarithm of a number composed of min utes and seconds, or degrees and minutes, of an arc; or of minutes and seconds, or hours and minutes, of time. 1. If the number consists of minutes and seconds, at the top of the table seek for the minutes, and in the same column opposite the seconds in the left-hand column will be found the logistical logarithm. 2. If the number is composed of hours and minutes, the hours must be used as if they were minutes, and the minutes as if they were seconds. 3. If the number is composed of degrees and minutes, the degrees must be used as if they were minutes, and the minutes as if they were seconds. To find the logistical logarithm of a number less than 3600. Seek in the second line of the table from the top the number next less than the given number, and the remainder, or the complement to the given number, in the first column on the left: then in the column of the first number, and opposite the complement, will be found the logistical logarithm of the sum. Thus, to obtain the logarithm of 1531, we seek for the column of 1500, and opposite 31 we find 3713. PROBLEM I. To work, by logistical logarithms, a proportion the terms of which are degrees and minutes, or minutes and seconds, of an arc; or hours and minutes, or0 minutes and seconds, of time. With the degrees or minutes at the top, and minutes or seconds at the side, or if a term consists of hours and minutes, or minutes and seconds, with the hours or minutes at the top, and minutes or seconds at the side, take from Table XCVI. the logistical logarithms of the three given terms; add together the logistical logarithms of the second and third terms and the arithmetical complement of that of the first term, rejecting 10 from the index.* The result will be the logistical logarithm of the fourth term, with which take it from the table. Note 1. The logistical logarithm of 60' is 0. Note 2. If the second or third term contains tenths of seconds, (or tenths of minutes, when it consists of degrees and minutes,) and is less than 6', or 6~, multiply it by 10, and employ the loga rithm of the product in place of that of the term itself. The * instead of adding the arithmetical complement of the ogarithm of the first term, the logarithm itself may be subtracted from the sum of the logarithms of the other two terms. TO TAKE OUT A QUANTITY FROM A TABLE. 321 result oc,btained by the table, divided by 10, will be the fourth term of the proportion, and will be exact to tenths. Note 3. If none of the terms contain tenths of minutes or seconds, and it IS desired to obtain a result exact to tenths, diminish the index of the logistical logarithm of the fourth term by 1, and cut off the right-hand figure of the number found from the table, for tenths. Exam. 1. When the moon's hourly motion is 30' 12", what is its motion in 16m. 24s.? As 60m.. 0: 30' 12".. 2981:: 16m. 24s.. 5633: 8' 15". 8614 2. If the moon's declination change 10 31' in 12 hours, what will be the ehange in 7h. 42m.? As 12h... ar. co. 9.3010: 10 31'... 1.5973: 7h. 42m... 8917: 00 58'... 1.7900 3. When the moon's hourly motion in latitude is 2' 26'.8, what is its motion in 36m. 22s.? 2' 26".8 60 146".8 10 As 60m.. 0 1468.. 1468".. 3896: 36m. 22s.. 2174 890".. 6070 Ans. 1' 29" 0. 4. When the sun's hourly motion in longitude is 2' 28", what is its motion in 49m. 1ls.? Ans. 2' 1". 5. If the sun's declination change 16' 33" in 24 hours, what will be the change in 14h. 18m.? Ans. 9' 52". 6. If the moon's declination change 54".7 in one hour, what will be the change in 52m. 18s.? Ans. 47".7. PROBLEM II. To takefrom a table the quantity corresponding to a given value of the argument, or to given values of the arguments of the table 21 322 ASTRONOMICAL PROBLEMS. Case 1. WlThen quantities are given in the table for each sign and degree of the argument. With the signs of the given argument at the top or bottom, and the degrees at the side, (at the left side, if the signs are found at the top; at the right side, if they are found at the bottom,) take out the corresponding quantity. Also take the difference between this quantity and the next following one in the table, and say, 60': this difference:: odd minutes and seconds of given argument: a fourth term. This fourth term, added to the quantity taken out, when the quantities in the table are increasing, but subtracted when they are decreasing, will give the required quantity. Note 1. When the quantities change but little from degree to degree of the argument, the required quantity may often be estimated without the trouble of stating a proportion. Note 2. In some of the tables the degrees or signs of the quantity sought, are to be had by referring to the head or foot of the column in which the minutes and seconds are found. (See Table. L, LI, LII, and LV.) The degrees there found are to be taken, if no horizontal mark intervenes; otherwise, they are to be increased or diminished by 1~, or 2~, according as one or two marks intervene. They are to be increased, or diminished, according as their number is less or greater than the number of degrees at the other end of the column. Note 3. If, as is the case with some of the tables, the quantities in the table have an algebraic sign prefixed to them, neglect the consideration of the sign in determining the correction to be applied to the quantity first taken out, and proceed according to the rule above given. The result will have the sign of the quantity first taken out. It is to be observed, however, that if the two consecutive quantities chance to have opposite signs, their numerical sum is to be taken instead of their difference; also that the quantity sought will, in every such instance, be the numerical difference between the correction and the quantity first taken out, and, according as the correction is less or greater than this quantity, is to be affected with the same or the opposite sign. Exam. 1. Given the argument 7S' 6~ 24' 36". to find the corre sponding quantity in Table L. 7S' 6~ gives 00 43' 17".4. The difference between 0~ 43' 17".4 and the next following quantity in the table is 1' 7".3. 60': 1' 7".3:: 24' 36"': 27".6.* * The student can work the proportion, either by the common method, or by lc gistical logarithms, as he may prefer. In working this and all similar proportions by the arithmetical method, the seconds of the argument may be converted into the equivalent decimal part of a minute by means of Table XVII, (using the sec. onds as if they were minutes.) It will be sufficient to take the fraction to the nearest tenth 1-0 TAKE OUT A QUANTITY FRO.M A TABLE. 323 From 0~ 43' 17".4 Take 27.6 0 42 49.8 2. Given the argument 2" 180 41' 20", to find the corresponding quantity in Table XXV. 2s. 180 gives 1~ 52' 32".5. The difference between 1~ 52' 32".5 and the next following quantity in the table is 21".8. 60': 21".8 41' 20": 15".0. To 10 52' 32".5 Add 15.0 1 52 47.5 3. Given the argument 99' 20 13' 33", to find the correspond. ing quantity in Table XII. 9S' 20 gives 29.8s. The arithmetical sum of 29.8s. and the next following quantity in the table is 30.4s. 60': 30.4s.: 13~ 33': 6.9s. From 29.8s. Take 6.9 22.9s. Ans. - 22.9s 4. Given the argument 58" 8~ 14' 52", to find the corresponding quantity in Tahlbe LII. Ans. 12' 36".0. 5. Given the argument 11' 110 23' 10", to find the corresponding quantity in Table LVI. Ans. 11' 48'.0. 6. Given the argument 01' 260 20', to find the corresponding quantity in Table XII. Ans. - 41'.0. Case 2. /VWhen the argument changes in the table by more or less thain 10; or when it is given in lower denominations than siogns. Take out of the table the quantity answering to the number in the colvmn of arguments next less than the given argument. Take the difference between this quantity and the next following one, and also the difference of the consecutive values of the argument inserted in the table, and say, difference of arguments: difference of quantities:: excess of the given argument over the value next less in the table a fourth term. This fourth term applied to the quantity first taken out, according to the rule given in the preceding case, will give the quantity sought. Note. In some of the tables the columns entitled Diff. are made up of the differences answering to a difference of 10' in the argumnent. In obtaining quantities from these tables, it will be found m;ore convenient to take for the first and seernld terms of the pro 324 ASTRONOMICAL PROBLEMS. portion, respectively, 10', and the difference furnished by the table and work the proportion by the arithmetical method. (See note at bottom of page 268.) Exam. 1. Given the argument 0s. 240 42' 15", to find the corresponding quantity in Table LI. 0O' 24~ 30' gives 90 47' 14".3. The difference between 90 47' 14".3 and the next following quantity = 3 x 63".0 = 189".0. The argument changes by 30'. And the excess of 0s 24~ 42' 15" over 0~' 240 30', is 12 15". Thus, 30': 189".0:: 12' 15": 77".2. But the correction may be found more readily by the following proportion: 10': 63".0:: 12'.25: 77".2. To 90 47' 14".3 Add 77.2 9 48 31.5 2. Given the argument 10 12', to find the corresponding quan. tity in Table VIII. 10 10' gives 23' 13", and 5': 33":: 2': 13" the correction. From 23' 13" Take 13 23 0 3. Given the argument 61' 60 7' 23", to find the corresponding quantity in Table LV. Ans. 900 20' 53".5. 4. Given the argument 490 27', to find the corresponding quan tity in Table LXIV. Ans. 11' 19".8. Case 3. When the argument is given in the table in hundredth, thousandth, or ten thousandth parts of a circle. The required quantity can be found in this case by the same rule as in the preceding; but it can be had more expeditiously by observing the following rules. If the argument varies by 10, multiply the difference of the quantities between which the required quantity lies by the excess of the given argument over the next less value in the table, and remove the decimal point one figure to the left; the result will be the correction to be applied to the quantity taken out of the table. The same rule will apply in taking quantities from tables in which the differences answering to a change of 10 in the argument are given, although the argument should actually change by 50 or 100. If the argument changes by 100, mul tiply as above, and remove the decimal point two figures to the left. When the common difference of the arguments is 5, proceed as if it were 10, and double the result. In like manner, when the comr mon difference is 50, proceed as if it were 100, and double the result. TO TAKE OUT A QUANTITY FROM A TABLE. 325 Exam. 1. Given the argument 973, to find the corresponding quantity in Table XLV column headed 13. 970 gives 23".5. Tile difference is 1".2, and the excess 3. 1".2 From 23".5 3 Take.4 Corr..36 23.1 2. Given the argument 4834, to find the corresponding quantity in Table XLII, column headed 5. 4800 gives 2' 3".7. The difference is 6".8, and the excess 34. 6"'.8 34 From 2' 3".7 2.312.. Take 2.3 2 1.4 3. Given the argument 5444, to find the corresponding quatLtity in Table XLI. Ans. 15' 37".7. 4. Given the argument 4225, to find the corresponding quantity in Table XLIII, column headed 8. Ans. 0' 47".2. Case 4. When the table is one of double entry, or quantities a re taken from it by means of two arguments. Take out of the table the quantity answering to the values of the arguments of the table next less than the given values; and find the respective corrections to be applied to it, due to the excess of the given value of each argument over the next less value in the table, by the general rule in the preceding case. These corrections are to be added to the quantity taken out, or subtracted from it, according as the quantities increase or decrease with the arguments. Note I. If the tenths of seconds be omitted, the corrections above mentioned can be estimated without the trouble of stating a proportion, or performing multiplications. Note 2. The rule above given may, in some rare instances, give a result differing a few tenths of a second from the truth. The following rule will furnish more exact results. Find the quantities corresponding, respectively, to the value of the argument at the top next less than its given value and the other given argument, and to the value next greater and the other given argument.. Take the difference of the quantities found, and also the difference of the corresponding arguments at top, and say, difference of argu. mnents: difference of quantities:: excess of given value of the argument at the top over its next less value in the table: a fourth term. This fourth term added to the quantity first found, if it is less than the other, but subtracted from it, if it is greater, will give the required quantity. The error of the first rule may be dimin. 626 ASTRONOMICAL PROBLEMS. ished without any extra calculation, by attending to the difference of the quantities answering to the value of the argument at the side next greater than its given value and the values of the other argument between which its given value lies. Exam. 1. Given the argument 64 at the top and 77 at the side to find the corresponding quantity in Table LXXXI. 50 and 70 give 47".7. The difference between 47".7 and the next quantity below it is 1".4. The excess of 77 over 70 is 7, and the argumept at the side changes by 10. 1".4 7 From 47".7 Corr. due excess 7,.98, or 1".0. Take 1.0 Quantity corresponding to 50: id 77, 46.7 The difference between 47".7 and the adjacent quantity in the next column on the right is 3".3. The excess of 64 over 50 is 14, and the argument at the top changes by 50. 3".3 14.462 2 From 46".7 Corr. due excess 14,.924 Take 0.9 45.8 2. Given the argument 223 at the top and 448 at the side, to find the corresponding quantity in Table XXX. 220 and 440 give 16".0. The difference between 16".0 and the quantity next below it is 2".2. 2".2 8 2) 1.76 From 16".0 Corr. for excess 8,.88, or 0".9. Take 0.9 Quantity corresponding to 220 and 448, 15.1 The difference between 16".0 and the adjacent quantity m the next column on the right is 0".7. 0".7 3 To 15".1 Corr for excess 3,.21 Add.2 15.3 TO CONVERT DEGREES, MINUTES, ETC., INTO TIME. 327 3. Given the argument 472 at the top and 786 at the side, to find the corresponding quantity in Table XXXI. A11S. 9".7, 4. Given the argument 620 at the top and 367 at the side, to find the corresponding quantity in Table LXXXI. Ans. 55".2. 5 Given the argument 348 at the top and 932 at the side, to find (by the rule given in Note 2) the corresponding quantity in Table XXXII. Ans. 15".4. PROBLEM III. To convert Degrees, Minutes, and Seconds of the Equator into Hours, lIiinutes, -c., of Time. Multiply the quantity by 4, and call the product of the seconds, thirds; of the minutes, seconds; and of the degrees, minutes. Exam. 1. Convert 83~ 11' 52" into time. 830 11' 52" 4 5'- 32"m 47s- 28"' 2. Convert 340 57' 46" into time. Ans. 2h. 19m. 51sec. 4"'. PROBLEM IV. To convert Hours, Minutes, and Seconds of Time into Degrees, Minutes, and Seconds of the Equator. Reduce the hours and minutes to minutes: divide by 4, and -all the quotient of the minutes, degrees; of the seconds, minutes; and multiply the remainder by 15, for the seconds. Exam. 1. Convert 7h. 9m. 34sec. into degrees, &c. 7h. 9m. 34S. 60 4 )429 34 1070 23' 30" 4. Convert I lh. 24m. 45s. into degrees, &c. Ans. 171' Ill 15" 328 ASTRONOMICAL PROBLEMS. PROBLEM V.'The Longitudes of two Places, and the Time at one of them being given, tojind the corresponding Time at the other. When the given time is in the morning, change it to astronomical time, by adding 12 hours, and diminishing the number of the day by a unit. When the given time is in the evening, it is already in astronomical time. Find the difference of longitude of the two places, by taking the numerical difference of their longitudes, when these are of the same name, that is, both east or both west; and the sum, when they are of different names, that is, one west and the other east. When one of the places is Greenwich, the longitude of the other is the difference of longitude. Then, if the place at which the time is required is to the east of the place at which the time is given, add the difference of longitude, in time, to the given time; but, if it is to the west, subtract the difference of longitude from the given time. The sum or remainder will be the required time. Note. The longitudes used in the following examples, are given in Table I. Exam. 1. When it is October 25th, 3h. 13m. 22sec. A. M. at Greenwich, what is the time as reckoned at New York? Time at Greenwich, October, 24d' 15h. 13m' 22' Diff. of Long. 4 56 4 Time at New York..24 10 17 18 P. M. 2. When it is June 9th, 5h. 25m. 10sec. P. M. at Washington, what is the corresponding time at Greenwich? Time at Washington, June, 9d' 5h' 25m- 10sDiff. of Long. 5 8 6 Time at Greenwich 9 10 33 16 P. M. 3. When it is January 15th, 2h. 44m. 23sec. P. M. at Paris what is the time at Philadelphia? Longitude of Paris. Oh. 9m- 21'.6 E. Do. of Philadelphia,. 5 0 39.6 W. 5 10 1.2 Time at Paris, January, 15d. 2h" 44m. 23" Diff. of Long... 5' ]0 1 Time at Philadelphia,. 14 21 34 22 Or January 15th, 9h. 34m. 22sec. A. M. 4. When it is March 31st, Sh. 4m. 21sec. P. M. at New Haven, %Nhat is the corresponding time at Berlin? Ans. April 1st, lh. 49m. 43sec. A. M. TO CONVERT APPARENT INTO MEAN TIME. 329 5. When it is August 10th, 10h. 32m. 14sec. A. M. at Boston, what is the time at New Orleans? Ans. Aug. 10th, 9h. 16m. 4sec. A. M. 6. When it is noon of the 23d of December at Greenwich, what is the time at New York? Ans. Dec. 23d, 7h. 3m. 55sec. A. M. PROBLEM VI. The Apparent Time being given, tofind the corresponding Mean Time; or the Mean Time being given to find the Apparent. When the given time is not for the meridian of Greenwich, reduce it to that meridian by the last problem. Then find by the tables the sun's mean longitude corresponding to this time. Thus, from Table XVIII take out the longitude answering to the given year, and from Tables XIX, XX, and XXI, take out the motions in longitude for the given month, days, hours, and minutes, neglecting the seconds. The sum of the quantities taken from the tables, rejecting 12 signs, when it exceeds that quantity, will be the sun's mean longitude for the given time. With the sun's mean longitude thus found, take the Equation of Time from Table XII. Then, when Apparent Time is given to find the Mean, apply the equation with the sign it has in the table; but when Mean Time is given to find tile Apparent, apply it with the contrary sign; the result will be the Mean or Apparent Time required. This rule will be sufficiently exact for ordinary purposes, for several years before and after the year 1840. Whlen the given date is a number of years distant from this epoch, take also with the sun's mean longitude the Secular Variation of the Equation of Time from'Fable XIII, and find by simple proportion the variation in the interval between the given year and 1840. The result, applied to the equation of time taken from Table XII, according to its sign, if the given time is subsequent to the year 1840, but with the opposite sign if it is prior to 1840, will give the equation of time at the given date, which apply to the given time as above directed. Note 1. When the exact mean or apparent time to within a small fraction of a second is demanded, take the numbers in the columns entitled I, II, III, IV, V, N, in Tables, XVIII, XIX, XX, answering respectively to the year, month, days, and hours, of the given time. With the respective sums of the numbers taken from each column, as arguments, enter Table XIV, and take out the corresponding quantities. These quantities added to ile equation of time as given by Tables XII and XIII, and,ne 330 ASTRONOMICAL PROBLEMS. constant 3.0s. subtracted, will give the true Equation of Time, if the given time is Mean Time. When Apparent Time is given, it will be farther necessary to correct the equation of time as giver by the tables, by stating the proportion, 24 hours: change of equation for 10 of longitude:: equation of time: correction. Note 2. The Equation of Time is given in the Nautical Almanac for each day of the year, at apparent, and also at mean noon, on the meridian of Greenwich, and can easily be found for any intermediate time by a proportion. Directions for applying it to the given time are placed at the head of the column. The Equation is given on the first and second pages of each month. Exam. 1. On the 16th of July, 1840, when it is 9h. 35m. 22s. P M., mean time at New York, what is the apparent time at the same place? Time at New York, July, 1840, 16d' 9h. 35m- 229. Diff. of Long. 4 56 4 Time at Greenwich, July, 1840, 16 14 31 26 M. Long. 1840. 9-. 10~ 12' 49' July..... 5 29 23 16 16d.... 14 47 5 14h..... 34 30 31m.... 1 16 M. Long. 3 24 58 56 The equation of time in Table XII, corresponding to 38' 24' 58 56", is + 51m' 44s. Mean Time at New York, July, 1840, 16d. 9h. 35m- 228. Equation of time, sign changed, -5 44 Apparent Time,.... 16 9 29 38 P.M. 2. On the 9th of May, 1842, when it is 4h. 15m. 21sec. A. M. apparent time at New York, what is the mean time at the same place, and also at Greenwich? Time at New York, May, 1842, 8d' 16h' 15m' 21' Diff. of Long. 4 56 4 Time at Greenwich,.. 8 21 11 25 M. Long. 1842. 9s. 10~ 43' 18" May. 3 28 16 40 8d... 6 53 58 21h.. 51 45 lin.. 27 M. Long, 1 16 46 8. Equa. oftime,-3m.45s. TO CONVERT APPARENT INTO MEAN TIME. 331 Apparent Time at Greenwich, May, 1842, 8d' 21h. 1 1in 25' Equation of Time,.... -3 45 Mean Time at Greenwich,.. 8 21 7 40 Diff. of Long.... 4 56 4 Mean Time at New York,.. 8 16 11 36 Or, May 9th, 4h. lm. 36s. A. M. 3. On the 3d of February, 1855, when it is 2h. 43m 36s. appa. rent time at Greenwich, what is the exact mean time at the same place? Appar. Time at Greenwich, Feb., 1855, 3d. 2h. 43m. 36s M. Long. I. II. III. IV. V. N. 1855. 9" 10~ 34' 30" 433 279 806 889 866 863 Feb.. 1 0 33 18 47 85 138 45 7 5 3d. 1 58 17 68 5 9 3 0 o 2h. 4 56 3 10 13 12 47 1551 369 953 937 873 868 Appar. Time at Greenwich, Feb., 1855, 3d. 2hb 43m' 369' Equation of time by Table XII,. +14 8.6 100yrs.: 13s. (Sec. Var., Table XIII):: 15yrs.: 1.9s.... -1.9 Approx. Mean Time at Greenwich, 3 2 57 42.7 24h.: 6s. (change of equa. for 10 of long.):: 14m.: 0.ls... +0.1 II. III... 0.8 IN... 0.3 N....... 0.1 Constant...-3.0 Mean Time at Greenwich, 3 2 57 42.4 4. On the 18th of November, 1841, when it is 2h. 12m. 26sec. A. M. mean time at Greenwich, what is the apparent time at Philadelphia? Ans. Nov. 17th, 9h. 26m. 28s. P. M. 5. On the 2d of February, 1839, when it is 6h. 32m. 35sec. P. M., apparent time at New Haven, what is the mean time at the same place? Ans. 6h. 46m. 39s. P. M. 6. On the 23d of September, 1850, when it is 9h. 10m. 12sec. mean time at Boston, what is the exact apparent time at the same place? Ans. 9h. 18m. 1.0s. 332 ASTRONOMICAL PROBLEMS. PROBLEM VII. To correct the Observed Altitude of a Heavenly Body for Re fraction. With the given altitude take the corresponding refraction from Table VIII. Subtract the refraction from the given altitude, and the result will be the true altitude of the body at the given station. This rule will give exact results if the barometer stands at 30 inches, and Fahrenheit's thermometer at 50~, and results sufficiently exact for ordinary purposes in any state of the atmosphere. When there is occasion for greater precision, take from Table IX the corrections for + 1 inch in the height of the barometer, and -1~ in the height of Fahrenheit's thermometer, and compute the corrections for the difference between the observed height of the barometer and 30in. and for the difference between the observed height of the thermometer and 500. Add these to the mean refraction taken from Table VIII, if the barometer stands higher than 30in. and the thermometer lower than 50~; but in the opposite case subtract them, and the result will be the true refraction, which subtract from the observed altitude. Exam. 1. The observed altitude of the sun being 32~ 10' 25", what is its true altitude at the place of observation? Observed alt.. 32~ 10' 25" Refraction (Table VIII). -1 32 True alt. at the station,. 32~ 8 53 2. The observed altitude of Sirius being 20~ 42' 11", the barometer 29.5 inches, and the thermometer of Fahrenheit 70~, required the true altitude at the place of observation. The difference between 29.5 inches and 30 inches is 0.5 inches, and the difference between 70~ and 50~ is 20~. 0)bs. alt.. 20~ 42' 11".0 Refrac. (Table VIII), 2' 33".0; Bar.+lin.,5".12; ther.-l 10".310 Corr.for-0.5 in.,bar. -2.6.5 20 Corr. for + 20~, ther. -6.2 2.560 6.20 True refrac.. 2 24.2 True alt. 20 39 46.8 3. The observed altitude of the moon on the 1 1th of April, 1838, being 14~ 17' 20", required the true altitude at the place of observation. Ans. 14~ 13' 35'". 4. Let the observed altitude of Aldebaran be 48~ 35' 52'", the barometer at the same time standing at 30 7 inches, and the thermonmeter at 12~, required the true altitude. Ans. 48~ 34' 58".8. TO DEDUCE THE TRUE FROM THE APPARENT ALTITUDE. 333 PROBLEM VIII. The Apparent Altitude of a Heavenly Body being given, to fina its True Altitude. Correct the observed altitude for refraction by the foregoing problem. Then, 1. If the sun is the body whose altitude is taken, find its parallax in altitude by Table X, and add it to the observed altitude corrected for refraction. The result will be the true altitude sought. 2. If it is the altitude of the moon that is taken, and the horizontal parallax at the time of the observation is known, find the parallax in altitude by the following formula: log. sin (par. in alt.)=log. sin (hor. par.) +log. cos (app.alt.)-10; and add it, as before, to the apparent altitude corrected for refraction. 3. If one of the planets is the body observed, the following formula will serve for the determination of the parallax in altitude when the horizontal parallaxis known: log. (par. in alt.) = log. (hor. par.) + log. cos (appar. alt)- 10. Note 1. The equatorial horizontal parallax of the moon at any given time may be obtained from the tables appended to the work. (See Problem XIV.) But it can be had much more readily from the Nautical Almanac. The equatorial horizontal parallax being known, the horizontal parallax at any given latitude may be obtained by subtracting the Reduction of Parallax, to be found in Table LXIV. The horizontal parallax of any planet, the altitude of which is measured, may also be derived from the Nautical Almanac. Note 2. The fixed stars have no sensible parallax, and thus the observed altitude of a star, corrected for refraction, will be its true altitude at the centre of the earth as well as at the station of the observer. Note 3. If the true altitude of a heavenly body is given, and it is required to find the apparent, the rules for finding the parallax in altitude and the refraction are the same as when the apparent altitude is given; the true altitude being used in place of the apparent. But these corrections are to be applied with the opposite signs from those used in the determination of the true altitude from the apparent; that is, the parallax is to be subtracted, and the refraction added. It will also be more accurate to make use of equa. (a), p. 422, in the case of the moon. Exam. 1. The observed altitude of the sun on the 1st of May, 1837, being 260 40' 20", what is its true altitude? 334 ASTRONOMICAL PROBLEMS. Obs. alt.... 26` 40' 20" Refraction.. -1 56 True alt. at the station,. 26 38 24 Parallax in alt. (Table X),. + 8 True altitude... 26 38 32 2. Let the apparent altitude of the moon at New York on the 17th of March, 1837, Sh. P. M., be 66~ 10' 44"; the barometer 30.4in. and the thermometer 62~; required the true altitude. Appar. alt... 660 10' 44" Mean refrac. 0 25.7 Corr. for + 0.4in., bar. +- 0.3 Corr. for + 12~, ther. — 0.6 True refrac... 0 25.4 logarithms True alt. at N. York, 66 10 18.6. cos. 9.60637 Equa. par. by N. Almanac, 54' 13" Reduc. for lat. 40~, 4 Hor. par. at New York, 54 9... sin. 8.19731 Par. in alt... 21 52. sin. 7.80368 True altitude. 66 32 11 3. On the 18th of February, 1837, the true meridian altitude of the planet Jupiter at Greenwich was 560 54' 57", what was its apparent altitude at the time of the meridian passage, the horizontal parallax being taken at 1".9, as given by the Nautical Almanac? True alt... 56~ 54' 57'. cos. 9.7371 Hor. par. 1".9. log. 0.2787 Par. in alt. — 1.0. log. 0.0158 Refraction... + 37.9 Appar. alt... 56 55 34 4. What will be the true altitude of the sun on the 22d of September, 1840, at the time its apparent altitude is 39" 17' 50"? Ans. 390 16' 46". 5. Given 290 33' 30" the apparent altitude of the moon at Phil adelphia on the 15th of June, 1837, at 9h. 30m. P. M., and 58' 33' the equatorial parallax of the moon at the same time, to find t}he true altitude. Ans. 30~ 22' 41". 6. Given 15~ 24' 23" the true altitude of Venus, and 8" its horizontal parallax, to find the apparent altitude Ans. 15~ 27' 41'. TO FIND THE SUN' S LfONGITUDE, ETC, FROM TABLES. 335 PROBLEM IX. Tofind the Sun's Longitude, Hourly Motion, and Semi-diameter, for a given time, from the Tables. For the Longitude. When the given time is not for the meridian of Greenwich, reduce it to that meridian by Problem V; and when it is apparent time, convert it into mean time by the last problem. With the mean time at Greenwich, take from Tables XVIII, XIX, XX, and XXI, the quantities corresponding to the year, month, day, hour, minute, and second, (omitting those in the last two columns,) and place them in separate columns headed as in Table XVIII, and take their sums.* The sum in the column entitled Ml. Lon2g. will be the tabular mean longitude of the sun; the sum in the column entitled Long. Perigee will be the tabular longitude of the sun's perigee; and the sums in the columns I, II, III, IV, V, N, will be the arguments for the small equations of the sun's longitude, including the equation of the equinoxes in longitude. Subtract the longitude of the perigee from the sun's mean long. tude, adding 12 signs when necessary to render the subtractit n possible; the remainder will be the sun's mean anomaly. With the mean anomaly take the equation of the sun's centre from Table XXV, and correct it by estimation for the proportional part of the secular variation in the interval between the given year and 1840; also with the arguments I, II, III, IV, V, take the corresponding equations from Tables XXVIII, XXX, XXXI, and XXXII. The equation of the centre and the four other equations, together with the constant 3", added to the mean longitude, will give the sun's True Longitude, reckoned from the Mean Equinox. With the argument N take the equation of the equinoxes or Lunar Nutation in Longitude from Table XXVII. Also take the Solar Nutation in longitude, answering to the given date, from the same table. Apply these equations according to their signs to the true longitude from the mean equinox, already found; the result will be the True Longitude from the Apparent Equinox. For the Semi-diameter and Hourly Miotion. With the sun's mean anomaly, take the hourly motion and semldiameter from Tables XXIII and XXIV. * In adding quantities that are expressed in signs, degrees, &c., reject 12 or 24 signs whenever the sum exceeds either of these quantities. In adding arguments expressed in 100 or 1000, &c. parts of the circle, when they consist of two figures reject the hundreds from the sum; when of three figures, the thousands; and when of four figures, the ten thousands. 336 ASTRONOMICAL PROBLEMS. Notes. i. If the tenths of seconds be omitted in taking the equations from the tables of double entry, the error cannot exceed 2"; in case the precaution is taken to add a unit, whenever the tenths exceed.5. 2. The longitude of the sun, obtained by the foregoing rule, may differ about 3" from the same as derived from the most accurate solar tables now in use. When there is occasion for greater precision, take from Tables XVIII, XIX, and XX, the quantities in the columns entitled VI and VII, along with those in the other columns. With the sums in these columns, and those in the columns I, II, as arguments, take the corresponding equations from Tables XXIX and XXXIII. Also with the sun's mean anomaly take the equation for the variable part of the aberration from Table XXXIV. Add these three equations along with the others to the mean longitude, and omit the addition of the constant 3". The result will be exact to within a fraction of a second. Exam. 1. Required the sun's longitude, hourly motion, and se mi-diameter, on the 25th October, 1837, at llh. 27m. 38s. A. M mean time at New York. Mean time at N. York, Oct. 1837, 24d' 23h' 27m' 38"S Diff. of Long... 4 56 4 Mean time at Greenwich,.. 25 4 23 42 M. Long. Long. Perigee. I. III. I.IV. V. N. 1837... 9 10 55 47.2 9 10 8 5816 2801549321 32148895 October.. 8 29 4 54.1 46250 748 215397 63 40 25d.. 23 39 19.9 4810 66 107 35 5 4 4h... 9 51.4 6' 0 1 23m... 56.7 42s. 1.7 9 10 8 55882 94 872 753 416 939 7 3 50 51 7 3 50 51.0 Eq.Sun'sCent. 11 28 12 43.5 9 23 41 56 Mean Anomaly. I... 2.5 II. III... 9.0 Sun's Hourly Motion,.. 2' 29".7 II. IV... 7.7 II. V... 19.3 Sun's Semi-diameter. 16' 17".2 Const... 3.0 7 2 4 16.0 Lunar Nutation - 6.3 Solar Nutation - 1.2 Sun'strue long.I 7 2 4 8.5 2. Required the sun's longitude, hourly motion, and semi-diam eter, on the 15th of July, 1837, at 8h. 20m. 40s. P. M mean time at Greenwich. TO FIND THE APtARENT OBLIQUITY OF THE ECLIPTIC. 337 M. Long. Long. Peri. I. II.III IV. V. N. NVI. VII. 1837.. 9 10 55 47.219 10 8 5816"'80549.321348895787 600 July. 5 28 24 7.8 31 129496i806!263 41 271569 17 15d.. 13 47 56.6 2 473 38 62 20 3 21508 2 8h... 19 42.8 11 1 1 11 20m... 49.3 - 40s. 1.6!9 10 8 38 4291815 418 604392192418751 619 _- 13 23 28 25 3 23 28 25.3 Eq. Sun's Cent. 11 29 33 10.36 13 19 47 Mean Anomaly. I... 10.7 II. III... 6.6 Sun's Hourly Motion,. 2' 23' 1 III. IV... 5.0 1II. V... 7.7 Sun's Semi-diameter, 15' 45".4 I. VI.. 1.8 II. VII.. 0.2 Aber. 0.. 0.6 3 23 2 8.2 Lunar Nutation - 7.8 Solar Nutation + 0.8 Sun's true long. 3 23 2 1.2 3. Required the sun's longitude, hourly motion, and semi-diameter, on the 10th of June, 1838, at 9h. 45m. 26s. A. M. mean time at Philadelphia, (omitting the three smallest equations of longitude.) Ans. Sun's longitude, 2" 1901 1' 57"; hourly motion, 2' 23".3; semi-diameter, 15' 46".1. 4. Required the sun's longitude, hourly motion, and semi-diam eter, on the 1st of February, 1837, at 12h. 30m. 15s. mean astro nomical time at Greenwich. Ans. Sun's longitude, 10' 13~ 1' 44".6; hourly motion, 2' 32" 1 ~semi-diameter, 16' 14".7. PROBLEM X. Tofind the Apparent Obliquity of the Ecliptic, for a given time, from the Tables. Take the mean obliquity for the given year from Table XXII. Then with the argument N, found as in the foregoing problem, and the given date, take from Table XXVII the lunar and solar nutations of obliquity. Apply these according to their signs to the mean obliquity, and the result will be the apparent obliquity. Exam. 1. Required the apparent obliquity of the ecliptic on the 15th of March, 1839. 22 338 ASTRONOMICAL PROBLIAMS. N. 1839,. 3 March, 9 15d.. 2 M. Obliquity, 230 27' 36".9 14... +.1 Solar Nutation for March 15th, + 0.5 Apparent Obliquity,.. 23 27 46.5 2. Required the apparent obliquity of the ecliptic on the 12th if July, 1845. Ans. 23~ 27' 28".2. PROBLEM XI. Given the Sun's Longitude and the Obliquity of the Ecliptic, te find his Right Ascension and Declination.* Let w = obliquity of the ecliptic; L = sun's longitude; R = sun's right ascension; and D = sun's declination; then to find R and D, we have log. tang R = log. tang L + log. cos w - 10, log. sin D = log. sin L + log. sin c - 10. The right ascension must always be taken in the same quadrant as the longitude. The declination must be taken less than 90~; and it will be north or south according as its trigonometrical sine comes out positive or negative. Note. The sun's right ascension and declination are given in the Nautical Almanac for each day in the year at noon' on the meridian of Greenwich, and may be found at any intermediate time by a proportion. Exam. 1. Given the sun's longitude 205~ 23' 50", and the obiquity of the ecliptic 23~ 27' 36", to find his right ascension and declnation. L= 205~ 23' 50".... tan. 9.67649 w = 23 27 36.. cos. 9.96253 R= 203 32 5.. tan. 9.63902 L = 205 23 50... sin. 9.63235 — = 23 27 36... sill. 9.60000 D= 9 49 52 S... (. sin. 9.232352. The obliquity of the ecliptic being 230 27' 30", required * The obliquity of the ecliptic at any given time for which the sun's longitude i. known, is found by the foregoing Problem. TO FIND THE SUN S LONGITUDE AND DECLINAT[ON. 339 the sud's right ascension and declination when his longitude is 140 i'/ 25/". Alls. Right ascension 41~ t0' 30", and declination 160 8' 40" N. PROBLEM XII. Given the Sun's Right Ascension and the Obliquity of the E'clip tic, tofind his Longitude and Declination. Using the same notation as in the last problem, we have, to fina the longitude and declination, log. tang L = log. tang R + ar. co. log. cos w, log. tang D = log. sin R + log. tang w - 10. Exam. 1. What is the longitude and declination of the sun, when his right ascension is 142~ 11' 34", and the obliquity of the ecliptic 230 27' 40"? R = 1420 11' 34"... tan. 9.88979= 23 27 40.. ar. co. cos. 0.03747 L = 139 46 30.. tan. 9.92726 - R=142 11 34.. sin. 9.78746 w= 23 27 40.. tan. 9.63750 D= 14 53 55N.... tan. 9.42496 2. Given the sun's right ascension 310~ 25' 11", and the obliquity of the ecliptic 230 27' 35", to find the longitude and declination. Ans. Longitude 307~ 59' 57", and declination 180 17' 0"' S. PROBLEM XIII. The Sun's Longitude and the Obliquity of the Ecliptic bezng given, to find the Angle of Position. Let p = angle of position; w = obliquity of the ecliptic; and L = sun's longitude. Then, log. tang p = log. cos L + log. tang w - 10. The angle of position is always less than 900. The nort}lern part of the circle of latitude will lie on the west or east side of tile northern part of the circle of declination, according as the sign of the tangent of the angle of position is positive or negative. Exam. 1. Given the sun's longitude 24~ 15' 20", and the obliquity of the ecliptic 23~ 27' 32", required the angle of position. 940 ASTRONOMICAL PROBLEMS. I= 240 15' 20".. cos 9.95986 - 23 27 32.. tan. 9.63745 p =21 35 10.. tan. 9.59731 The northern part of the circle of latitude is to the west of the circle of declination. 2. When the sun's longitude is 120~ 18' 55", and the obliquity of the ecliptic 230 27' 30', what is the angle of position? Ans. 12~ 21' 17'; and the northern part of the circle of latitude lies to the east of the circle of declination. PROBLEM XIV. To findfrom the Tables, the Moon's Longitude, Latitude, Equatorial Parallax, Semi-diameter, and Hourly Motion in Longitude and Latitude, for a given time. When the given time is not for the meridian of Greenwich, reduce it to that meridian, and when it is apparent time convert it into mean time. Take from Table XXXV, and the following tables, the arguments numbered 1, 2, 3, &c., to 20, for the given year, and their variations for the given month, days, &c., and find the sums of the numbers for the different arguments respectively; rejecting the hundred thousands and also the units in the first, the ten thousands in the next eight, and the thousands in the others. The resulting quantities will be the arguments for the first twenty equations of longitude. With the same time, take from the same tables the remaining arguments with their variations, entitled Evection, Anomaly, Variation, Longitude, Supplement of the Node, II, V, VI, VII, VIII, IX, and X; and add the quantities in the column for the Supplement of the Node. For the Longitude. With the first twenty arguments of longitude, take from Tables XLI to XLVI, inclusive, the corresponding equations; and with the Supplement of the Node for another argument, take the corresponding equation from Table XLIX. Place these twenty-one equations in a single column, entitled Eqs. of Long.; and write beneath them the constant 55". Find the sum of the whole, and place it in the column of Evection. Then the sum of the quanti ties in this column will be the corrected argument of Evection. With the corrected argument of Evection, take the Evectior from Table L, and add it to the sum in the column of Eqs. of Long. Place this in the column of Anomaly. Then the sumn of the quantities in this column will be the corrected Anomaly. TO FIND TIIE I' CN' S LON- J',TUDE, ETC. 341 With the corrected Anomaly, take the Equation of the Centre from Table LI, and add it to the last sum in the column of Eqs. of Long. Place the resulting sum in the column of Variation. Then, the sum of the quantities in this column will be the corrected argument of Variation. With the corrected argument of Variation, take the variation from Table LII, and add it to the last sum in the column of Eqs. of Long.; the result will be the sum of the principal equations of the Orbit Longitude, amounting in all to twenty four, and the constants subtracted for the other equations. Place this sum in the column of Longitude. Then the sum of the quantities in this column will be the Orbit Longitude of the Moon, reckoned from the mean equinox. Add the orbit longitude to the supplement of the node, and the resulting sum will be the argument of Reduction. With the argument of Reduction, take the Reduction from Table LIII, and add it to the Orbit Longitude. The sum will be the Longitude as reckoned from the mean equinox. With the Supplement of the Node, take the Nutation in Longitude from Table LIV, and apply it, according to its sign, to the longitude from the mean equinox. The result will be the Moon's True Longitude from the Apparent Equinox. For the Latitude. The argument of the Reduction is also the 1st argument of Latitude. Place the sum of the first twenty-four equations of Longitude, taken to the nearest minute, in the column of Arg. II. Find the sum of the quantities in this column, and it will be the Arg. II of Latitude, corrected. The Moon's true Longitude is the 3d argument of Latitude. The 20th argument of Longitude is the 4th argument of Latitude. Take from Table LVIII the thousandth parts of the circle, answering to the degrees and minutes in the sum of the first twenty-four equations of longitude, and place it in the columns V, VI, VII, VIII, and IX; but not in the column X. Then the sums of the quantities in columns V, VI, VII, VIII, IX, and X, rejecting the thousands, will be the 5th, 6th, 7th, 8th, 9th, and 10th arguments of Latitude. With the Arg. I of Latitude, take the moon's distance from the North Pole of the Ecliptic, from Table LV; and with the remaining nine arguments of latitude, take the corresponding equations from Tables LVI, LVII, and LIX. The sum of these quantities. increased by 10", will be the moon's true distance from the North Pole of the Ecliptic. The difference between this distance and 90~ will be the Moon's true Latitude; which will be North o, South, according as the distance is less or greater than 90~. For t1e Equatorial Parallax. With the corrected argu-ments, Evection, Anomaly, and Varia 342 ASTRONOMICAL PROBLTEM.S tion, take out the corresponding quantities from Tables LXI, LXII, and IXIII. Their sum, increased by 7", will be the Equa. torial Parallax For the Semi-diameter. With the Equatorial Parallax as an argument, take out the moon's semi-diameter from Table LXV. For the Hourly Motion in Longitude. With the arguments 2, 3, 4, 5, and 6 of Longitude, rejecting the two right-hand figures in each, take the corresponding equations of the hourly motion in longitude from T'able LXVII. Find the sum of these equations and the constant 3", and with this sum at ihe top, and the corrected argument of the Evection at the side, take the corresponding equation from Table LXIX; also with the corrected argument of the Evection take the corresponding equa tion from Table LXVIII. Add these equations to the sunm just found, and with the resulting sum at the top, and the corrected anomaly at the side, take the corresponding equation from Table LXX; also with the corrected anomaly take the corresponding equation from Table LXXI. Add these two equations to the sum last found, and with the resulting sum at the top, and the corrected argument of the Variation at the side, take the corresponding equation from Table LXXII. With the corrected argument of the Variation, take the corresponding equation from Table LXXIII. Add these two equations to the sum last found, and with the resulting sum at the top, and the argument of the Reduction at the side, take the corresponding equation from Table LXXIV. Also, with the argument of the Reduction take the corresponding equation from'Table LXXV. These two equations, added to the last sum, will give the sum of tile principal equations of the hourly motion in longitude, and the constants subtracted for the others To this add the constant 27' 24".0, and the result will be the Moon's Hourly Motion in Longitude. For the Hourly Motion in Latitude. VWrith the argument I of Latitude, take the corresponding equation from Table LXXIX. With this equation at the side, and the sum of all the e"quations of the hourly motion in longitude, except the last two, at the top, take the corresponding equation from Table LXXXI. With the argument II of Latitude, take the corresponding equation from Table LXXXII. And with this equation at the side, and the sum of all the equations of the hourly motion in longitude, except the last two, at the top, take the equation from Table LXXX I 1. Find the sum of thv-se four equations and the TO FIND THE MOON S LONGITUDE, ETC. 343 constant 1". To the resulting sum apply the constant - 237".2. The difference will be the Moon's true Hourly Motion in Latitude. The moon will be tending North or South, according as the sign is positive or negative Note. The errors of the results obtained by the foregoing rules, occasioned by the neglect of the smaller equations, cannot exceed for the longitude 15", for the latitude 8"', for the parallax 7", for the hourly motion in longitude 5", and for the hourly motion in latitude 3"; and they will generally be very much less. When greater accuracy is required, take from Tables XXXV to XXXIX the arguments from 21 to 31, along with those from 1 to 20, and their variations. The sums of the numbers for these different arguments, respectively, will be the arguments of eleven small additional equations of longitude. Also, take from the same tables the arguments entitled XI and XII, along with those in the preceding columns. Retain the right-hand figure of the sum in column 1 of arguments, and conceive a cipher to be annexed to each number in the columns of arguments of Table XLI. The numbers in the columns entitled Diff.for 10, will then be the differences for a variation of 100 in the argument. For the Longitude. With the arguments 21 to 31, take the cor responding equations from Tables XLVII and XLVIII, and place them in the same column with the equations taken out with the arguments 1, 2, &c. to 20. Take also equation 32 from Table _XLIX, as before. Find the sum of the whole, (omitting the constant 55",) and then continue on as above. The longitude from the mean equinox being found, take the lunar nutation in longitude from Table LIV, and the solar nutation answering to the given date from Table XXVII. Apply them both, according to their sign, to the longitude from the mean equinox, and the result will be the more exact longitude from the apparent equinox, required. For the Latitude. With the arguments XI and XII, take the corresponding equations from Table LIX. Add these with the other equations, and omit the constant 10". Tile difference between the sum and 90~ will be the more exact latitude. For the Equatorial Parallax. With the arguments 1, 2, 4, 5, 6, 8, 9, 12, 13, take the corresponding equations from Table LX. Find the sum of these and the other equations, omitting the constant 7", and it will be the more exact value of the Parallax. For the Hourly Motion in Longitude. With the arguments 1, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, and 18, of longitude, along with the arguments 2, 3, 4. 5, and 6, heretofore used, lake the corresponding equations fre i, Table LXVII. Find the sum of the 344 ASTRONOMICAL PROBLEMS. whole, omitting the constant 3", and proceed as in the rule already given. To obtain the motion in longitude for the hour which precedes or follows the given time, with the arguments of Tables LXX, LXXII, and LXXIV, take the equations from Tables LXXV1 and LXXVII. Also, with the arguments of Evection, Anomaly, Variation, and Reduction, take the equations from Table LXXVIII. Find the sum of all these equations. Then, for the hour which follows the given time, add this sum to the hourly motion at the given time already found, and subtract 2".0; for the hour which precedes, subtract it from the same quantity, and add 2".0. It will expedite the calculation to take the equations of the second order from the tables at the same time with those of the first order which have the same arguments. For the Hourly Motion in Latitude. The moon's hourly mo tion in latitude may be had more exactly by taking with the arguments of Latitude V, VI, &c. to XII, the corresponding equations from Table LXXX, and finding the sum of these and the other equations of the hourly motion in latitude. To obtain the moon's motion in latitude for the hour which pre cedes or follows the given time, with the Argument I of Latitude, take the equation from Table LXXXIV, and with this equation and the sum of all the equations of the hourly motion in longitude except the last two, take the equation from Table LXXXV. Find the sum of these two equations. Then, for the hour which follows the given time, add this sum to the Hourly Motion in Latitude already found, taken with its sign, and subtract 1".3; and for the hour which precedes, subtract it from the same quantity, and add 1".3. It will also be more exact to enter Table LXXXI with the sum of all the equations of Tables LXXIX and LXXX, diminished by 1", instead of the equation of Table LXXIX, for the argument at the side. The numbers over the tops of the columns in Table LXXXI are the common differences of the consecutive numbers in the columns. The numbers in the last column are the common differences of the consecutive numbers in the same horizontal line. Exam. 1. Required the moon's longitude, latitude, equatorial parallax, semi-diameter, and hourly motions in longitude and latitude, on the 14th of October, 1838, at 6h. 54m. 34s. P. M. mean time at New York. Mean time at New York, October, 14d' 6h- 54m' 348. Diff. of Long. 4 56 4 Mean time at Greenwich, October, 14 11 50 38 1 2 3_4 5_6 7 8 91011121314 15116 171819 20 1838. 00153 3508 3868 3163 0329 7757 4579 0360 8583 211 175 354 319 670 576 178 492 315 715 870 October 74741 7419 3969 8343 1602 1569 5752 6550 6630 152 497 237 329 71 333 992 483 087 578 125 14d...03559 8449 3522 3731 4362 4837 0750 5074 0316 912 405 916 444 289 397 476 547 337 028 006 1lh... 125 298 477 131 154 171 26 179 11 32 14 32 16 45 14 17 19 12 1 0 50m. 38sec. 9 23 36 10 11 13 2 13 1 2 1 2 1 3 1 1 1 1 0 78587 9697 1872 5378 6458 4347 1109 2176 5541 309 092 541 109 078 321 664 5421 752 322 001 Evection. Anomaly. Variation. Longitude. Sup. of Node. II. V. I VI. VII. VIII. IX. X. 8 0 / I 0 I I/ 3 0 / If 8 0 I If a 0 I It 8 0 I 1838... 3 23 54 9 0 10 25 33 124 24 51 11 4 21 29.8 11 11 51 10 5 1 21 876 880 419 423 137 588 October 629 24.24 10 26 44 34 228 4 28 11 27 922.4 14 27 24 5 14 32 285 780 710 204 783 451 14d.. 4 27 6 53 5 19 50 42 5 8 28 47 521 17 35.4 41 18 4 24 59 442 513 367 438 466 069 lib..h.. 5 11 12 559 17 5 35 15 6 221.0 1 27 5 7 16 18 13 15 16 2 50m.... 2334 27 13 25 24 27 27.0 7 23 1 1 1 1 1 0 H 38sec.... 18 21 19 20.9 0 0 0 0 0 0 0 0 Sum of Equa.. 34 47 3 16 14 12 36 47 12 40 50.8 12 41 35 35 35 35 35 3 26 35 17 5 6 43 54 10 19 35511 5 11 59 27.3 11 27 1 26 3 29 3 655 227 545 116 438 110 H Reduction... 11 35.6 5 11 59 27 5 12 11 2.9 5 9 0 53 Arg. I of Latitude. Nutation in Longitude - 0.9 Moon's True Longitude... 5 12 11!?.O 4i Arg. Eqs. D's Long. Arg. Eqs. D'sLat. Arguments. )'s Eq. Par. I-ourly Motion in Longitude. Hourly Motion in Latitude.| O 0 - o t' " 0 Arguments. Equa. Arguments. Equa. 1 0 23 25.7 I. 87 57 33.1 Evection. 0 26.0 2 2 2.1 II. 1 20.0 Anomaly. 52 55.1 i 3 0 6.8 D's long. 10.5 Variation.. 33.3 2 of long... 5.0 I. 33.8 4 2 35.3 20 long. 14.9 Constant.. 7.0 3 do... 1.0 Pre. Eq. & Sm Eq. 46.9 5 0 23.3 V. 8.5 4 do... 0.0 II. 2.9 6 0 56.6 VI. 6.4 Moon's Eq. Par. 54 1.4 5 do... 0.3 Pre. Eq. & Sum Eq 1.4 7 0 22.0 VII. 31.3 6 do... 1.5 Constant. 1.0 8 0 10.8 VIII. 21.7 Moon's Semi diameter, 14'43" Constant. 3.0 9 1 48.5 IX. 19.8 86.0 10 10.7 X. 6.6 Sum 10.8 -237.2 11 18.0 Const. 10.0 Evec.& Sum Eqs. 0.2 12 24.5 Evection. 21.8 - 151.2 - 13 8.4 88 1 2.8 _ 14 14.2 90 0 0.0 Sum.... 32.8 -2 31".2 15 1.0 An. & Sum Eqs. 11.6 I 16 8.2 Moon'sLat. 1 58 57.2 Anomaly. 21.7 Moon's Hourly Motion in 17 11.7. Latitude, tending S., 2' 31'.2 18 3.3 Sum 66.1 0 19 0.8 Var. & Sum Eqs. 9.4 20 10.1 Variation.. 44.9 32 10.3 C Const. 55.0 Sum.... 120.4 Red. & Sum Eqs. 2.6 Sum 0 34 47.3 Reduction.. 4.1 Ev. 2 41 26.6 Sum 3 1t6 13.9..Sum.... 127.1 An. 9 20 33.5 2' 7".1 Sum 12 36 47-L4 Constant. 27 24.0 Var. 4 3.4 | Moon's Hourly l 29 31 1 Sum 12 40 50.8 Mot. in Long. 1 2 |3 4 5 6 7 8 9 10 11 12 13 14115 16 17 18 19 20 2122'23 24'2526!27'28!29i30 31 1838 ~..00153 3508 3868 31G3 0329 7757 4579 0360 8583 211 175 354319 670 576 178 492 315 715 870 72 46 38P6o75 32 07 04 0489 84 October. 74741 7419 3969 8343 1602 156915752 6550 6630.152 497 237:329 71 333 992 483 087 578 125 07 3265262308 07 4041 4768 14d... 035598449 3522 37314362 48370750 50740316 912 405 916 444 289 397 476 547 337 028 006 10!11 32 34153 48 91 54140 02 03 11h.. 125 298 477 131 154 171 26 179 11 32 14 32 16 45 14 17 19 0 051 2 3 21 0 50m. 38sec. 9 23 36 10 11 13 2 13 1 2 1 2 1 3 11 1 0 0 7858719697 1872 537816458 4347 1109 2176955411309iO 92 541 O9i078 321 6645421752322 001 8918940'21 53i90 080O863855 Evection. Anomaly. Variation. Longitude. Supp. of Node. II. V. IVI. VII.VIII. IX. X. XI. XII. I a l 0 I I Q,I 0 I I s 0 I II 8 0 I 0 1838.. 1 3 23 54 9 0 10 25 32.9 1 24 24 51 11 4 21 29.8 11 11 51 10.4 5 1 21 876 880 419 423 137 588 595 714 l October. 6 29 24 24 10 26 44 33.7 2 28 4 28 11 27 9 22.4 0 14 27 23.8 5 14 32 285 780 710 204 7831451 358 182 l 14d.. 4 27 6 53 5 19 50 41.6 5 8 28 47 5 21 17 35.4 41 18.3 4 24 59 442 513 367 438 466 069 541 437 1 11h.... I 5 11 12 5 59 17.2 5 35 15 6 2 21.0 1 27.4 5 7 16 18 13 15 16 2 19 15 Q 50m... 23 34 27 13.1 25 24 1 27 27.0 6.6 23 1 1 1 1 1 0 1 1 1 3 38sec. 18 20.7 19 20.9 0.0 0 0 o0 0 0 0o0 0 0 Sum of Equa. 34 47 3 16 14.0 12 36 47 12 40 50.9 12 41 35 35, 351 35 |35!35 35 3 26 35 17 5 6 43 53.2 10 19 35 51 5 1159 27.411 27 1 26.5 329 3 65512271545 116 4381 1101549 384 Reduction.. 11 35.6 5 11 59 27.4 5 12 11 3.0 5 9 0 53.9 Arg. I of Latitude. Lunar Nutation. 0.8 Solar Nutation... -0.8 Moon's True Longitude.. 5 12 11 14 C Ar. qs' la. s' lo~/ " I Ho otoI o ue I.I 1 It Wto i O " g Eqs.'s ILon. Hourly Motion in Longitude. Arguments. Equa. Eq. 2d ord. Hourly Motion in Latitude. Sum 0 34 47.4 CD 1 0 23 7 Ev. 2 41 26.6 Arguments. Equa."Eq.2d ord. I Args. Equa. Eq.2dord 1 0 23 25.7 Sum 120.2 1.55,, um.. 120.2 1.55 --., 2 2 2.1 Sum 3 16 14.0 Red. S Eqs. 2.6 0.04 3 0 6.8 uAn. 9 120 335 o....ofl0 Reduction 4.1 0.12 I. 33.8 0.59 4 2A35.3 n 9 20 33.5 Rdo...1 _.1 Pre. Eq. &c.146.9 0.44 4 do... 0. 23. 3 do. 1.0 Sum 2' 6.9 II. 2.9 Sum 323 4. 6 0 56.6 12 4 4 do. 0.0 Constant 27 24.0 F2.0 Pre.Eq.&c. 1.4 70 25.0 5 do.. 0.3. V. [0.36 0 10.8 6 do... 1.5 Moon's Hourly 29 V. 0.3 9 0 08 Sum 12 40 50.9 7 do. 0.02 Mot. in Long. 30.9 29' 30.9 VI. 0.29 9 148.5 ________- do. 0..0 10 10.7 8 do... 0.58 VIII. 0.03 11 18.0 9 do...0.12 Forthehourfollowing, 29 30.6 30. ~_ II. I0.0 1 i 12 24.5 10 do... 0.33 For the hour preceding, 29 31.2 X. 0.04 12 24.5 10 do. - - 0.33 ~~~~~~~~~~~~~~~~X. 0.0 __________________~~~~~~~~~~~~~~ 0.0 13 8.4 11 do.... 0.19 XI. 0.00 [I 0.0 4' 14 14.2 Arg. Eqs. D's Lat. 12 do... 0.65 XII. 0.04 0 15 1.0,, 13 do.... 0.04 16 8.2 I 8753.114 do. ~ 0.03 Arguments. )'s Eq. Par. Sum 85.79 1.03 8. 1. 87 57 33.1 17 11.7 I.20015 do.. 0.11 Const. -237.2 1.7 1 II. 1 20.0 18 3.3 16 do... I 0'06 I1 / I'd l)'s lcng. 10.5 19 3. D'slong. 10.5 17 do.... 0.14 1 oflong... 0 0.3 151.4 0.8 20 long. 14.9 20 10.8 18 do.. 0.16 2 do.... 1.6 V. ~~~8.5 t 21 3.2 V. 8.5 4 do... 0.0 2' 31".4 tending S. j VI. 6.4 22. VI. 6. Sum... 10.6 5 do.. 0.3 22 6.1 VII. 31.3 Evec. & Sum Eqs. 0.2 6 do... 0.4' 6 VIII. 21.7 24 6.1 X. 19.8 Evection.. 21.8 0.02 8 do... 0.5 Moon's Hourly -2 31.4 25 4.8 X. 66 9 do.. 3.2 Motion in Lat., 26 5.5 xi. Sum... 32.6 12 do.... 0.0 ~ 1.0 27 5.4 XI. An. & Sum Eqs. 11.6 0.06 13 do.... 1.7 F 1.3 27 /. 13.: 1.3 5.4 XII. 28 5.0 -- Anomaly...21.7 0.74 Evection,... 26.0.0 Sum.88 i 6.9' 29 4. um.88 1 6. Anomaly,... 52 55.1 90 0 0.0 iSm...[ veln.. 30 3.90 0.0 Sum.. 65.9 Variation,... 33.3 For the hour following, 231.7 31 4.6 iVar. & Sum Eqs. 9.4 0.06 32 10.3 Moon's Lat. 1 58 53.1 Variation...44.9 0.67 Moon's Eq. Par. 54 2.4 For the hourpreceding, 2 31.1 Sum 0 34 47.4 Sum....120.2 1.55 Moon's Semi-diameter, 14' 43".5 TO FIND THE MOON'S REDUCED PARALLAX, ETC. 349 Exam. 2. Required the moon's longitude, latitude, equatorial parallax, semi-diameter, and hourly motions in longitude and latitude, on the 9th of April, 1838, at 8h. 58m. 19s. P. M. mean time at Washington. Ans. Long. 6"' 190 45' 31".2; lat. 36' 21".9 S.; equat. par. 54' 36".3; semi-diameter 14' 52".7; hor. mot. in long. 30' 15" 2; and hor. mot. in lat. 2' 47".0, tending south.* PROBLEM XV. The Moon's Equatorial Parallax, and the Latitude of a Place, being given, to find the Reduced Parallax and Latitude. With the latitude of the place, take the reductions from Table LXIV, and subtract them from the Parallax and Latitude. Exam. 1. Given the equatorial parallax 55' 15", and the latitude of New York 400 42 40" N., to find the reduced parallax and latitude. Equatorial parallax,... 55' 15" Reduction,..... 5 Reduced parallax,... 55 10 Latitude of New York,. 40~ 42' 40" N. Reduction,.... 11 20 Reduced Lat. of New York, 40 31 20 2. Given the equatorial parallax 60' 36", and the latitude of Baltimore 390 17' 23" N., to find the reduced parallax and latitude. Ans. Reduced par. 60' 32", and reduced lat. 390 6' 9". 3. Given the equatorial parallax 57' 22", and the latitude of New Orleans 29~ 57' 45" N., to find the reduced parallax and latitude. Ans. Reduced par. 57' 19", and reduced lat. 290 47' 50". PROBLEM XVI. To find the Longitude and Altitude of the Nonageszmal Degree of the Ecliptic, for a given time and place. For the given time, reduced to mean time at Greenwich, find the sun's mnean longitude and the argument N from Tables XVIII, XIX, XX, and XXI. To the sun's mean longitude, apply according t..,;s sign the nutation in right ascension, taken from Table " The smaller equations were omitted in working this example. 350 ASTRONOMICAL PROBLEMS. XXVII with argument N; and the result will be the right ascension of the mean sun, (see Art. 127,) reckoned from the true equinox. Reduce the mean time of day at the given place, expressed astronomically, to degrees, &c., and add it to the right ascension of the mean sun from the true equinox. The sum, rejecting 3600 when it exceeds that quantity, will be the right ascension of the midheaven, or the sidereal time in degrees. Next, find the reduced latitude of the p]ace by Problem XV; and when it is north, subtract it from 900; but when it is south, add it to 90~. The sum or difference will be the reduced distance of the place from the north pole. Also, take the obliquity of the ecliptic for the given year from Table XXII.* These three quantities having been found, the longitude and altitude of the nonagesimal degree maybe computed from the following formulae: log. cos I (H - ) - log. cos I (H -+ x) = A... (1); log. tang I (IH-aw) + 10- log. tang I (H + ) = B.. (2); log. tang E =A + log. tang (S - 90)... (3); log. tang F = log. tang E + B... (4); N=E+F+900... (5); log. tang I~h = log. cos E + log. tang - (H + w) + ar. co. log cosF -20... (6). in which H = the reduced distance of the place from the north pole; = -the Obliquity of the Ecliptic; S = the Sidereal Time converted into degrees; N= the required Longitude of the Nonagesimal; h =the required Altitude of the Nonagesimal; E and F are auxiliary angles. We first find the logarithmic sums A and B. With these we determine the angles E and F by formulae (3) and (4), and with these again N and h by formulae (5) and (6). The angles E, F, are to be taken less than 1800; and less or greater than 900, according as the sign of their tangent proves to be positive or negative. Note 1. In case the given place lies within the arctic circle, we must take, in place of formula (5), the following: N= E — F + 900. I* f great precision is required, the apparent obliquity is to be used in place of the mean. (See Prob. X.) TO FIND THE LONG. AND ALT. OF NONAGESIMAL DEGREE. 351 Note 2. As the obliquity of the ecliptic varies but slowly from year to year, the values which have once been found for the logarithms A, B, and log. tang I (HI + w) (C), will answer for several years from the date of their determination, unless very great accuracy is required. Note 3. The angle h derived from formula (6), is the distance of the zenith of the given place from the north pole of the ecliptic. This is not always equal to the altitude of the nonagesimal Throughout the southern hemisphere, and frequently ill the northern near the equator, it is the supplement of the altitude. In employing this angle in the following Problem, it is, however, for the sake of simplicity, called the altitude of the nonagesimal in all cases. Exam. 1. Required the longitude and altitude of the nonagesimal degree of the ecliptic at New York, on the 18th of September, 1838, at 3h. 52m. 56s. P. M. mean time. The sun's mean longitude taken from the tables, for the given time, is 5S' 27~ 19' 17", and the argument N is 987. The nutation taken from Table XXVII with argument N is -1". Hence, the right ascension of the mean sun, reckoned from the true equinox, is 5' 27~ 19' 16". The given time of day, expressed astronomically, is 3h. 52m. 56sec.; which in degrees is 580 14' 0". The reduced latitude of New York, found by Problem XV, is 400 31' 20", and this taken from 900 leaves the polar distance 490 28' 40". The obliquity of the ecliptic, derived from Table XXII, is 230 27' 37". Given time in degrees,. 58~ 14' 0"' R. Asc. of mean sun,... 177 19 16 Sidereal time in degrees (S),.. 235 33 16 90 2)145 33 16 H.. 490 28' 40"...23 27 37 (S - 90) 72 46 38 Diff.. 26 1 3 Sum.. 72 56 17 2diff... 13 0 31 cos. 9.98870. tan.+ 10,19.36366 2 sum.. 36 28 8 cos. 9.90535. tan. C. 9.86871 A. 0.08335 B. 9.49495 - (S - 900) 72 46 38 tan. 0.50866 E 75 38 55. tan. 0.59201. cos. 9.39422 B. 9.49495 C. 9.86871 F 50 41 55 tan 0.08696. ar. co. cos. 0.19832 90 0 0 2 alt. non. 160 7' 54". tan. 9.46125 long. non. 216 20 50 alt. non. 32 15 48 352 AS rRONOMICAL PROBLEMS. 2. Required the longitude and altitude of the notiagejimal de. gree of the ecliptic at New York, on the 10th of May, 1838, at 1 lh. 33m. 56sec. P. M. mean time. Ans. Long. 200~ 12' 23", and alt. 37~ 0' 34". PROBLEM XVII. To find the Apparent Longitude and Latitude, as affected by Parallax, and the Augmented Semi-diameter of the Moon; the Moon's True Longitude, Latitude, Horizontal Semi-diameter, and Equatorial Parallax, and the Longitude and Altitude of the Nonagesimal Degree of the Ecliptic, being given. We have for the resolution of this Problem the following for mulae: log. x = log.P + log. cos h+ar. co. log. cos X- 10... (1); c = log. x + log. tang h - 10... (2); log. u=c + log. sin K-10... (3); log. u'= c + log. sin (K + u) - 10... (4); log. p = c + log. sin (K + u') - 10... (5); Appar. long. = true long. +p... (6); log. tang X' = log. p + ar. co. log. cos X + ar. co. log. u + log. sin (X - x) -10*... (7); log. v = log. P + log. cos h + log. cos' - 10... (8); log. z = log. v + log. tang h + log. tang X' + log. cos (-K + 2P)-30... (9); r =v- Z... (10); Appar. lat. = true lat. - r... (11); log. R' = log. p + ar. co. log. cos X + ar. co. log. u + log cos x' +log. R — 10... (12): in which P = the Reduced Parallax of the Moon; h = the Altitude of the Nonagesimal; X = the True Latitude of the Moon (minus when south); K = the Longitude of the Moon, minus the longitude of the No. nagesimal; p = the required Parallax in Longitude;' = the approximate Apparent Latitude of the Moon; r = the required Parallax in Latitude; R = the True Semi-diameter of the Moon; R' = the Augmented Semi-diameter of the Moon;, u, u', v', z, are auxiliary arcs. Formula (7) will be rendered more accurate by adding to it the ar. co. ces x - 10, and will generally give the apparent latitude with sufficient accuracy; thus rendering formula (8), (9), (10), and (11) unnecessary. TO FIND THE MOON'S APPARENT LONC AND LAT. 353 Formulae (1), (2), (3), (4), and (5), being resolved in succession, we derive the apparent longitude from formula (6); then the apparent latitude from equations (7), (8), (9), (10), (11); and lastly, the augmented semi-diameter from equation (12.) The latitude of the moon must be affected with the negative sign when south: and the apparent latitude will be south when it comes out negative. In performing the operations, it is to be remembered that the cosine of a negative arc has the same sign as the cosine of a positive arc of an equal number of degrees; but that the sine or tangent of a negative arc has the opposite sign from the sine or tangent of an equal positive arc. Attention must also be paid to the signs in the addition and subtraction of arcs. Thus, two arcs affected with essential signs, which are to be added to each other, are to be added arithmetically when they have like signs, but subtracted if they have unlike signs; and when one arc is to be taken from another, its sign is to be changed, and the two united according to their signs. An arithmetical sum, when taken, will have the same sign as each of the arcs: and an arithmetical difference the same sign as the greater arc. The use of negative arcs may be avoided, though the calculation would be somewhat longer, by using the true polar distance d, and the approximate apparent polar distance d', in place of X and X', substituting sin d for cos x, cos (d + x) for sin (X - x), sin d' for cos X', log. co-tang d' for log. tang X'; and observing that p is to be subtracted from the true longitude in case the longitude of the nonagesimal exceeds the longitude of the moon; that z, when it comes out negative, is to be added to v, which is always positive to the north of the tropic, otherwise subtracted; and that the parallax in latitude is to be applied according to its sign to the true polar distance. In seeking for the logarithms of the trigonometrical lines, it will be sufficient to take those answering to the nearest tens of seconds. Note 1. When great accuracy is not desired, u' may be taken for p, from which it can never differ more than a fraction of a second. Note 2. In solar eclipses the moon's latitude is very small, and formula (7) may be changed into the following: log.' log. p + ar. co. log. cos + ar. co. log. u +log. (X - x)- I 0 and cos X' omitted in formula (12) without material error. Formula (8), (9), (10), and (11), may also now be dispensed( with, unless very great precision is desired, and the value of X' given by the above formula taken for the apparent latitude. It is to be observed also, that in eclioses of the sun P is takern equal to the reduced parallax of the moon minus the sun's horizontal parallax. By this the parallax of the sun in longitude and latitude is referred to the moon, and the relative apparent places of the sun and moon are correctly obtained, without the necessity of 23 354 ASTRONOMICAL PROBLEMS. a separate computation of tile sun's parallax in longitude and latitude. Exam. 1. About the time of the middle of the occultation of the star Antares, on the 10th of May, 1838, the moon's longitude, by the Connaissance des Terns, was 247~ 37' 6".7; latitude 4~ 14 14'.7 S.; semi-diameter 15' 24".2; and equatorial parallax 56 31".7; and the longitude of the nonagesimal at New York was 2000 12' 23"'; the altitude 370 0' 34"; required the apparent lon gitude and latitude, and the augmented semi-diameter of the moon at New York, at the time in question. Equat. par. 56' 31".7 Moon's long. 247~ 37' 7" Reduction 4.6 Long. nonag. 200 12 23 P =56 27.1 K = 47 24 44 l - 37 0 34 X= —= 4 14 14.7 P.. 3387".1. log. 3.52983 h 37~ 0' 34".. cos. 9.90230 a. 3.43213 X. — 4 14 15 ar. co. cos. 0.00119 x. 45 12. 2712". log. 3.43332 h.,. 37 0 34.. tan. 9.87725 c. 3.31057 K... 47 24 44.. sin. 9.86701 U 25 5. 1505". log. 3.17758 c. 3.31057 K+u.. 47 49 49... sin. 9.86991 u'... 25 15. 1515".2. log. 3.18048 c. 3.31057 K~+u'.. 47 49 59,,. sin. 9.86993 13p. 25 15.3. 1515".3. log. 3.1]8050 True long.. 247 37 6.7 A ppar. long.. 248 2 22.0 p.... log. 3.18050 A —,:.. — 4 59 27. sin. 8.93957X.. *... ar. co. cos. 0.00119 Ut....ar. co. log. 6.82242..,. -5- 1 10, X, tan. 8.94368 TO FIND THE MOON S APPAR. LONG. AND LAT. 355 X'I,. 5~ 1' 10". ~. cos. 9.99833 a. 3.43213 v... 44 54.4. 2694"/.4. log. 3.43046 h........ tan. 9.87725 X'....... tan. 8.94368K +- p. 47 37 22. cos. 9.82867 z.-2 0.2. 120".2. log. 2.08006 - v-z... 46 54.6 v-z (sign changed) -46 54.6 T'rue lat. -4 14 14.7 Appar. lat... 5 1 9.3 S. p........ log. 3.18050 X....... ar. co. cos. 0.00119 u....... ar. co. log. 6.82242 XI......a.. cos. 9.99833 R... 15 24.2. 924".2. log. 2.96577 Augm. semi-diam. 15 29.4. 929".4. log. 2.96821 Exam. 2. About the middle of the eclipse of fhe sun on the 18th of September, 1838, the moon's longitude was 1750 29' 19".0, latitude 47' 47".5, equatorial parallax 53' 53".5, and semi-diameter 14' 41".1; and the longitude of the nonagesimal at New York was 216~ 20' 50", the altitude 32' 15' 48": required the apparent longitude and latitude, and the augmented semi-diameter of the moon. Equat. paral. 53' 53".5 Moon's long. 1750 29' 19:' Redliction, 4.4 Long. nonag. 216 20 50 53 49.1 K= —40 51 31 Sun's paral. 8.6 h = 32 15 48 P-53 40.5 X= 0 47 47.5 P.... 3220".5.. log. 3.50792 h.. 32~15' 48".. cos. 9.92716... 47 47.5.. ar. co. cos. 0.00004 x.. 45 23.5. 2723".5. log. 3.43512 h.. 32 15 48... tan. 9.8002:3 c. 3.25t:35 K.. -40 51 31 sin. 9.b1570t~.. — 18 45. 1i25". loug. 3.05LO05 — 356 ASTRONOMICAL PROBLEMS. c. 3.23535 K +u.. -4l 10' 16"... sin. 9.81844 — u' — 18 52.9. 1132".9. log. 3.05379c. 3.23535 K +u', — 41 10 24... in. 9.81844p — 18 52.9. 1132".9. log. 3.05379True long.. 175 29 19.0 Appar. long. 175 10 26.1 p.. log. 3.05379 X...... ar. co. cos. 0.00004 it...... ar. co. log. 6.94895 hX-x. 2'24".0. 144".0.. log. 2.15836 Appar. latitude 2' 24".9 N. 144".9. log. 2.16114........ log. 3.05379 X...... ar. co. cos. 0.00004 u...... ar. co. log. 6.94895 R.. 14' 41".1. 881".1.. log. 2.94502 Augm. semi-diam. 14 46.7. 886".7.. log. 2.94780 PROBLEM XVIII. To find the Mean Right Ascension and Declination, or Longitudme and Latitude of a Star, for a given time, from the Tables. Take the difference between the given year and 1840. Then seek in Table XV for the fraction of the year answering to the given month and days, and add it to this difference, if the given time is after the beginning of the year 1840; but if it is before, subtract it. Muitiply the sum or difference by the annual variation given in the catalogue, (Table XC, or XCII,) and the product will be the variation in the interval between the given time and the epoch of the catalogue. Apply this product to the quantity given in the catalogue, according to its sign, if the given time is after tile beginning of the year 1840, but with the opposite sign if it is before, andl the result will be the quantity sought. (See Poob. XXT. YVote ) l xam. 1. Required the mean right ascension and declination of the star Sirius on the 15th of August, 1842. Interval between given time and beginn. of 1840, (t,) 2.61 9yrs Anriuial variation of right ascension... 2.64 6is Variation of right ascension for interval t,.. 6.. TO FIND A STAR' S ABERR. IN RIGHT ASCENSION, ETC. 357 A similar operation gives for the variation of declination in the same interval, 1 1".65. Mean right ascen.,beginning of 1840, Table XC, 6h. 38m' 5.76" Variation for interval t,.... - + 6.93 Mean right ascension required,... 6 38 12.69 Mean declination,beginning of 1840,.. 16~ 30' 4".79 S Variation for interval t,..... + 11.65 Mean declination required,. 16 30 16.44 S. 2. Required the mean longitude and latitude of Aldebaran on the 20th of October, 1838. Interval between given time and begin. of 1840, (t) 1.200yrs. Annual variation of longitude,.... 50".210 Variation of longitude for interval t,... 60".2 A similar operation gives for the variation of latitude in the same interval 0".4. Mean longitude,beginning of 1840,. 2" 70 33' 5".9 Variation for interval t,... -1 0.2 Mean longitude required, 2 7 32 5.7 Mean latitude,beginning of 1840,. 5~ 28' 38".0 S. Variation for interval t,... + 0.4 Mean latitude required,. 5 28 38.4 S. 3. Required the mean right ascension and declination of Capella on the 9th of February, 1839? Ans. Mean right ascension 51' 4m- 48.748', and mean declination 450 49' 38".53 N. 4. Required the mean longitude and latitude of Aldebaran on the 16th of April, 1845? Ains. Mean longitude 2s' 7~ 37' 31".4, and mean latitude 50 28' 36".2. PROBLEM XIX. To find the Aberrations of a Star in Rzqht Ascension and Declination, for a qiven Day. (See Prob. XXI. iNote.) This problem may be resolved.for any of the stars in the catalogue of Tab'e XC by means of the following formulae 358 ASTRONOMICAL PROBLEMS. log. (aber. in right ascen.) = M + log. sin (O + q) - 10. log. (aber. in declin.) N- + log. sin (O + 0) - 10, in which M, N, are constant logarithms, O the longitude of the sun on the given day, and p, 8, auxiliary angles. M, N, and the anr g]es (P, 0, are given for each of the stars in the catalogue, in Table XCI. 0 may be derived from an ephemeris of the sun, or it may be computed from the solar tables by Problem IX. Exam. 1. What was the amount of aberration, in right ascension and declination, of rM Orionis on the 20th of December, 1837, the sun's longitude on that day being 8Ss 28~ 28'? Right Ascension. Table XCI, p. 6s 30 13' M. 0.1361 0. 8 28 28 0D +. 3 1 41.sin. 9.9998 Aberration = 1".37.... log. 0.1359 Declination. Table XCI, 0. 8' 28~ 23' N.. 0.7521 o. 8 28 28 0+-. 5 26 51. sin. 8.7399 Aberration = 0".31. log. 1.4920 2. Required the aberrations in right ascension and declination of a Andromedae on the 1st of May, 1838, the sun's longitude being 1Va 10~ 38'. Ans. Aberr. in right ascension - 1".07, and aberr. in declination - 11".69. PROBLEM XX. To find the Nutations of a Star in Right Ascension and Declina tion, for a given Day. This Problem may be solved by means of the formulae, log. (nuta. in right asc.) M' + log. sin (a +q-') - 10; log. (nuta. in declin.) N' + log. sin (a + 8') - 10; in which M', N', are constant logarithms, Q the mean longitude of the moon's ascending node, and p', 0', auxiliary angles. M', N', and the angles Ap', 8', are given for each of the stars in the cataloulle, in Table XCI.'TI'e mean longitude of the moon's ascend-,lng node is given for every tenth day of the year in the Nautical Almanaa, page'249, and may be easily found for any interr-nmediate TO FIND A STAR'S NUTATION IN RIGHT ASCEN., ETC. 359 day from the daily motion inserted at the foot of the column of longitudes. It may also be had by finding the supplement of the moon's node, for the given time, from the lunar tables, and subtracting it from 12'S 0~ 7'. (See Note to Prob. XXI.) Exam. 1. Wrhat was the amount of the nutation, in right ascen sion and declination, of a Orionis on the 20th of D)ecember, 1837, the mean longitude of the moon's node on that day being 18~ 54' Right Ascension. Table XCI,'. 6s' 00 15' M'.. 0.0481 a~. 0 18 54 Qa +'. 6 19 9.. sin. 9.5159 - Nutation = - 0".37. log. 1.5640Declination. Table XCI,'. 3S 20 37' N'.. 0.9657 a. 0 18 54; +8'. 3 21 31. sin. 9.9686 Nutation = 8".60. log. 0.9343 2. Required the nutations in right ascension and declination ot a Andromedae on the 1st of May, 1838. Ans. Nutation in right ascension - 0".54, and nutation in declination - 1".43. Note. When the apparent place of a star is desired with great accuracy, the solar nutations must also be estimated and allowed for. These may be determined by repeating the process for finding the lunar nutations, only using twice the sun's longitude in place of the longitude of the moon's node, and multiplying the results by the decimal.075. The calculation of the solar nutations in Example lst, is as folb tows: Right Ascension. Table XCI,'.. 6s' 00 15' M'.. 0.0481 20..5 26 56 2 0+q'. 11 27 11.. sin. 8.69140".05. log,.2.7395-.075 Solar Nutat. = - 0".00 360 ASTRON0)MICAL PROBLEMS. I)eclination. Table XCI,', 3s. 2~ 37' N'. 0.9657 20. 5 26 56 2 (- + -'. 8 29 33.. sin. 10.0000- 9".24.. 0.9657-.075 Solar Nutat. = - 0".69 Tn Example 2d, we find for the solar nutation in right ascension, - 0".08. and for the solar nutation in declination, - 0".51. PROBLEM XXI. To find the Apparent Right Ascenszon and Declination of a Star, on a given Day. Find the mean right ascension and declination for the given day by Problem XVIII; then compute the aberrations in right ascension and declination by Problem XIX, and the lunar and solar nutations in right ascension and declination by Problem XX. Apply the aberrations and nutations according to their signs, to the mean right ascension and declination on the given day, observing that the declination when south is to be marked negative, and the results will be the apparent right ascension and declination sought. Exam. 1. \Vhat was the apparent right ascension and declination of ca Orionis on the 20th of December, 1837? h. m. s. 0' it Table XC, M. right ascen. 5 46 30.71 M. dec. 7 22 17.14N Variations. - 6.59.. -2.42 5 46 24.12 7 22 14.72 Aberr... - 1.37.- +0.31 Lun. nutat.. -0.37.. +8.60 Sol. nutat.. 0.00.. - 0.69 App. right asc. 5 46 25.12 App.dec. 7 22 22.94N. 2. Required the aipmarent right ascension and declination of u Andromedae on.the 1st of May, 1838. Ans. Appar. right ascen. Oh. Orn. 0.90s., and appar. dec. W8~ 11' 39".92. NOTE.-T-. Prob. XVIII. use Table XC. (a) for calculations after 1860. Table XCI. will not give accurate results for dates after 1860. The method now adopted in solving Prob. XXI. is by means of tables published annually in theN. Almanac. TO FIND A STAR S ABERRATION IN LONGITUDE, ETC. 361 PROBLEM XXII. o Jfind the Aberrations of a Star in Longitude and Latitude, for a given Day. The formulae for the computation are, log. (aber. in long.) = 1.30880 + log. cos (6s. + 0 - L) + ar. co. log. cos X - 10; log. (aber. in lat.) - 1.30880 + log. sin (6s. + 0 - L) + log. sin X - 20; in whicl- = longitude of the sun on the given day; L = mean longitude of the star; and X = mean latitude of the star. Exam. 1. Required the aberrations in longitude and latitude of Antares on the 26th of February, 1838, the SUll's longitude on thal day being 118. 7~ 29'. By Prob. XVIII, L = 8S 7~ 30', and X- 40 32' S 6s. +. 17 7 29 Const. log. 1.3088 6s. +O-L 8 29 59. cos. 6.4637X.. 4 32. ar. co. cos. 0.0014 Aberr. in long. — 0".00. log. 3.7739 - Const. log. 1.3088 6s. + 0 - L 8S 29~ 59'.. sin. 10.0000 - X... 4 32.. sin. 8.8978 Aberr. in lat. - 1".61. log. 0.2066 - 2. Required the aberrations in longitude and latitude of Arcturus on the 5th of October, 1838, the sun's longitude being 66. 110 47'. Ans. Aberr. in long. - 23".34, and aberr. in lat. 1".85. Note. The nutation in longitude of a fixed star may be found after the same manner as the nutation in longitude of the sun. See Problem IX.) PROBLEM XXIII. To find the Apparent Longitude and Latitude of a Star, for a given Day. Find the mean longitude and latitude on the given day by Prob [em XVIII. Find also the aberrations in longitude and latitude by Problem XXII, and the nutation in longitude, as in Problem IX. Apply the aberration and nutation in longitude, according to their 362 ASTRONOMICAL PROBLEMS. signs, to the mean longitude, and the result will be the apparent longitude; and apply the aberration in latitude according to its sign, to the mean latitude, and the result will be the apparent latitude. Exam. 1. Required the apparent longitude and latitude of An tares on the 26th of February, 1838. Table XC, M. long. 8S' 70 31' 45".2 M. lat. 40 32' 51".6 S. Var... - 1 32.57.. 0.78 8 7 30 12.63..4 32 50.82 Aberr.. 0.00. -1.61 Nutat.. - 4.40 App. long. 8 7 30 8.23 App. lat. 4 32 49.21 S 2. Required the apparent longitude and latitude of Arcturus oil the 5th of October, 1838. Ans. Appar. long. 6-' 21~ 58' 37'".4, and appar. lat. 300 51' 19.1. PROBLEM XXIV. rTo compute the Longitude and Latitude of a Heavenly Body fiom its Right Ascension and Declination, the Obliquity of the Lclip. tic being given. This Problem may be solved by means of the following formulae: log. tang x -- log. tang D + ar. co. log. sin R; log. tang L=log. cos (x-w)+log.tangR + ar. co.log 90~). When it is not known whether this circumstance has place or not, the problem is susceptible of two solutions. The detail of the different cases is as follows: the data are A. b, and another arc or angle. Case 1. Given two sides and the included angle; or b, c, A. Equation 92 makes known m, 94 nm', which may be negative (what the calculation shows,) 96 a, 98 B, and equation 73, (page 399,) C, which is known in kind. Case 2. Given two angles and the adjacent side; or A, C, 1. Equation 93 makes known n, 95 n', which may be negative (what the calculation shows,) 97 B, 99 a; finally, equation 72 (page 399) gives c, which is known in kind. * Franeceur's Practical Astronomy. 26 402 APPENDIX. Case 3. Given two sides and an opposite angle; cr b, a, A. Equation 92 gives m, 96 itn', 94 c, 93 and 73 B and C; or clse, 93 gives n, 99 n', 95 C, 97 and 73 B and c. This problem admits in general of two solutions. In effect, the are m' or angle n' being given by its cos., may have either the sign + or -; there are then two values for c, and also for C. m' and n' enter into equations 97 and 98 by their sines, whence result therefore also two values of B. Case 4. Given two angles, and an opposite side; or A, B, b. JEquI(.ion 92 gives il, 98 Mn', 94 c, 96 a, and equation 73 mnakes knowl C; or else 93 gives n, 97 n', 95 C, 99 and 73 a and c.'['here are also two solutions in this case; for, m' or n' is given by a sin., and therefore two supplementary arcs satisfy the question. Th'lus c in 94, and a in 96, receive two values; same for C in 95, and a in 99, &c. Instead of solving the two right-angled triangles, into which the oblique-angled triangle is divided, by equations 92 to 99, we may employ Napier's rules, from which these equations have been obtained. Isosceles Triangles. When the triangle is isosceles, B = C, b = c, the perpendicular arc must be let fall from the vertex A, and the equations furnished by Napier's rules, become very simple. We find 101. sin ~ a = sin ~ A sin b 102. tan a = tan b cos B 103. cos b — cot B cot I A 104. cos e A-ncos a sin B The knowledge of two of the four elements A, B, a, b, which form the isosceles triangle, is sufficient for the determination of the two others. INVESTIGATION OF ASTRONOMICAL FORMULAE. Formulae for the Parallax in Right Ascension and Declination, and in Longitude and Latitude. (See Article 93, page 65.) Fig. 121 Let s (Fig. 121) be the true place ~.- a...~ of a star seen from the centre of tilhe ~in /....t~ aearth, s' the apparent place, seen from a point on the surface of which z is the zenith, the latitude being 1. Tile f*w displacement ss' = p is the parallax in altitude, which taklics effect in the vertical circle zs'; p is the PARALLAX IN RIGHT ASCENSION A.ND DECLINATION. 403 pole; thle hour angle zps =q is changed into zps', and sps' — o is the variation of the hour angle, or the parallax in r ight ascen ionZ; the polar distance ps d is changed into ps'; the differ ence S of these arcs is the parallax in declination or of polar dis. tance.* We have, (For. 73, p. 399,) sin s': sin ps (d) sin sps' (a): sin ss' (p), sin zps' (q + a): sin zs' (Z):: sin s': sin pz (900- 1). Multiplying, term by term, we obtain sin s' sin (q + a): sin d sin Z:: sin a sin s': sin p cos 1: sin p cos. whence, sin a sin d sin (q + ac). Or, substituting for p its value given by equa. (8,) p. 62, and replacing I by P, sin P cos I. sin a sin (q +- a)... (A). This equation makes known a when the apparent hour angle zps -= q + a, seen from the earth's surface, is given; but if we know the true hour angle zps = q, seen from the centre of the earth, developing sin (q + a), (For. 15, p. 395), and putting sin P cos 1 =m sin d sin a = m (sin q cos a + sin a cos q), or, dividing by sin a, 1 -= m (sin q cot a + cos q); whence, by transformation, tan 1t 1 -- m -- m sin q + m2 sin q cos q (very nearly.) I- m cos Restoring the value of m, sin P cos. (sin P cos l\ / tan a = sin d d sin q cos q. sin d sin )2 Putting the arc a in place of its tangent, and P in place of sin P, and expressing these arcs in seconds, (For. 47, p. 397,) there results, P cos 1 P COS 1 2 a- cl d q + cl sin cos q sin 1"... (B). The parallax in declination (3) is tile difference of the arcs ps (= d) and ps' (=d + 6.) Let zs = z, and zs' = Z. The trilgles zps and zps' give (For. 74 and 73), cosd - sinll I cos z cos (d + 3)-sin I cos %Z cos 1 siln z cos I sin Z * Franccur's Uranography, p. 418. APPENDIX. 02~. sin pzssin d sin q sin (d q+ 6) sin (q +t- a) sin z sin Z From the first equation we derive cos d sin Z - sin I cos z sin Z cos (d + 3) =-+ sin I cos Z sin z cos d sin Z - sin 1 (cos z sin Z - sin z cos Z' sin z cos d sin Z - sin I sill (Z - z) sin z or, (equ. 8, p. 62,) sin Z -- sill (cos d - sin P sin 1); sin z from the second, sinZ sin (d + 6) sin (q + a) sin z sinl d sin q substituting, sin (d + ~) sin (q + a) cos (d +- ) sin q a) (cos d - sin P sin sin' sin q cos (d + 6) sin (q + a) cos d sin P sin l\ sin (d + 6) sin q sin d sin d sin (q+ a) sin P sin (C) cot V 1 -) cot e o'+3 sin q sin d sin P sin I Put tan x - sin d then, cot (d + ) sin ( + (cot d - tan sin q sin d cos sinq sin d cos x sin (q + a) cos d cos x - sin d sin x sin q sin d cos x sin (q + a) cos (d + x) (D) sin q sin d cos x The apparent polar distance (d -+ 6) being computed by eithei of the formulae (C) and (D), we have 3 = (d + 3) - d. Formulae may be obtained that will give the parallax in declination without first finding the apparent declination, (except approximnately.) From equa. (C) we obtain sin P sin t sin q cot (d + 6) sin d sin (q + a) PARAILAX IN RIGHT ASCENSION AN]) DECLINATION. 405 and we also have cot d - cot (d + 6) =cos d cos (d + 6) sin 6 sin d sin (d + 6) Sill d sin (d + 6) the sum of these equations gives sin P sin I sin q\ sin 3 sin d cot(d s)1 sin (q + ) sin d sin(d +- )' Now, 1- sin q sin (q + a) -- sin q sin (q -+ a) sin (q - a) 2 sin - a cos (q + a) _ sin a cos (q + ) o - ____ -_ - (For. 22, 13) sin (q + c) sin (q + a) cos a cos (q + - a) sin P ccs I =sin d cos ~ a -' by equa. (A). Substituting, sin P sin I cos (q + - a) sin P cos 1 = cot (d + 6) 2 + sin d sill d cos + a sin 6 sin d sin (d + d)' or, sin 6 = sin P sin I sin (d + 6) - cos (d + 6) cos (q + 2a) sin P cos 1 COS (E) cos DC = sin P sin I [sin (d + 6) - tan y cos (d + d)1 cot I cos (q +- a)) making tan y cot I COS malilnC, tan = cos 2 a whence, sin = P sin in (d + 3 - y).. (F). Cos y To facilitate the calculation, the sines of 3 and P in eqs. (E) and (F), may be replaced by the arcs. To obtain an expression for the parallax in declination in terms of the true declination, develope win (d + 6 - y) in equation (F) which gives sin P sin 1 sin 6 -- [sin (d + 6) cos y- sin y cos (d + 6)]; Cos y developing sin (d + 5) and cos (d + 3), and reducing, we have sin P sin 1 sin 6 cs [sin (d- y) cos 6 +- cos (d -y) sin 6]; Cos y dividing by cos 6, sin P sin I tan =- [sin (d-y) + cos (d-y) tan 1], Cos y 106 APPENDIX sin P sin I c- sin (d-y) whence tan -- siln P sin 1 1 - cos (d-y) cosy illP sin P sin / sin (d-y) + sin cos y cos y sin (d- y) cos (d —y) (very nearly;) or, replacing tan a and sin P by B and P, expressing these arcs in seconds, (For. 47, p. 397), and reducing by For. 13, p. 395, P sin I. P sin + 2 sin y / P sin ((d-y) + P sin 2 (d-y)..(G.) c Coy - 2 " If the place of a body be referred to the ecliptic, similar formu le will give the parallax ilz latitude and lonzgitude, but as the ecliptic and its pole are continually in motion by virtue of the di urnal rotation of the heavens, it is necessary, in order to be able to determine the parallax in longitude at any given instant, to know the situation of the ecliptic at the same instant. This is ascertained by finding the situation of the point of the ecliptic 90~ distant from the points in which it cuts the horizon, and which are respectively just rising and setting, called the Nonagesimal Degree, or the Y Nonagesimnal. Fig. 122. Let K (Fig. 122) be the pole of the eclipticfb, p the -~. --- pole of the equatorfa;f is the vernal equinox, the origin of longitudes and of,~~(e(, / /right ascensions; hbs is the j C< ^/ eastern horizon, b the hor ~~~~~~~i | ~s oscope, or the point of the -iY~~5, ecliptic which is just rising; pz = 90O - 1 (the latitude of given place); Kp -= the obliquity of the ecliptic. The circle Kznv is at the same time perpendicular at n to the eclipticfb, and at v to the horizon hb; it is a circle of latitude and a vertical circle, since it passes through the pole K and the zenith z: b is 900 from all the points of the circle Knv; zn is the latitude of the zenith,fn its longitude; the point n is the ncnagesimal, since bn = 90~; nv is the altitude of this point, and the complement of -zn; nv measures the inclination of the ecliptic to the horizon at the given instant, or the angle b, so that b = nv = Kz; thusfn. = N the longitude of the nonagesimal, and nv = h the altitude of the nonagesimal, designate the situation of this point, and conseqlently ascertain the position of the ecliptic and its pole at the moment of observation.* * Francmur's Urarjography, p. 421 LONGITUDE AND ALTITUDE OF THE NONAGESIMAL. 407 The points m and d are those of the equator and ecliptic which are on the meridian; the arc fm, in time, is the sidereal time s. which is known; the arcfi = 90~, since the plane Kpi, passing through the poles K and p, is at the same time perpendicular to the ecliptic and to the equator; the arc z.i =fi —fm = 90~ -s; then the angle zpK = 180~-zpi = 180~ —ni = 90~+ s.* Now, in the spherical triangle pKz we know the sides Kp = w, zp- 90~- - H, and the included angle zpK = 90~ + s; and may therefore find Kz It thet altitude of the nonagesimal, and the angle pKz = nc =fc-fn = 90~- N = complement of the longitude N of the nonagesimal. Let S = sum of the angles Kzp and zKp, then, (For. 86, page 400,) cos 2 (H- W) =t2an j tan C =os (H1+ G,) cot 2 (90~ + s), is cos 2i (H - ) m1, tan 2S = I (H+ ) tan I (90~-s)~ but, tan'S =-tan (1 80o~-S), and tan - (90~-s) =-tan a (s-90~); substituting, and denoting (1800- S) by E, we have cos (H-) ( ). tanE = cos (H+.tans -90~) (H) Again, let D = zKp —Kzp, then, (For. 87,) tan {Dozsin 2 (H-w) tan-D = sin - (H + -- ) cot a (90" +s); whence, by transforming as above, and denoting (180~ -- D) by F we have sin (H —u) -9 tan F s tan (s - 90 )... (I). Now, iS + D =-pKz = 900-N; whence, CN 90~ —(IS + 11D) or, N=360~ +9-0"- (-S + KlD)- 1o — S+180~ — + — D + 90; consequently, N - E + F + 90~... (J), rejecting 3600 when the sum exceeds that number. Next, for the altitude of the nonagesiinal, we have, (For. 88,) tan 1hl = 2-, tan 2 (H + -), cos'D cos E cos tan 2 (1 + w)... (K). cos I N and h being known, to obtain the forimulcefor the parallax in longitude and latitutde, we have only to replace in the formulae * Franceur's Uranography, p. 421. 408 APPENDIX. for the parallax in right ascension and declination, the altitude l ol the pole of the equator by that 90~ —h of the pole K of the eclip tic, and the distance imn of the star s from the meridian by the distance nc to the vertical through the nonagesimal. Let us change then in formulae (A), (B), (C), (D), (E), (F), and (G),! into 90~-h, and q into fc -fnt L-N, L being the longitudefc of the star s. Besides, d will become the distance sK to the pole of the ecliptic, or complement of the latitude X =sc. Making these substitutions, and denoting the parallax in longitude by H, and the parallax in latitude by *r, we obtain in terms of' the apparent longitude and latitude, _ sin P sin h (Sin ]J-N - + )... (L) sin P sind sin d sin (L-N + I) ctd- sin P cos h (M)7 cot (d + " sin (L-N) csind. sin P cos h tan x (N)= sin.. (N), cot (d -+ )= sin (L- N ~+ ) cos (d + x) sin (L — N) sin d cos x sin r = sill P cos h sin (d + ir) - cos (d + *r) cos (L — N + - H) sin P sin h cos.2 tan h cos (L-N + 2HI) tan - * (Q)= cos 2}r sin P cos h sin r =*r- sin (d +,-y)... (R); cos y and in terms of the true longIitude and latitude, Psinh Psin h sin d sin d ) sin (L-N) cos (L-N) sin 1"... (S), Pcos h.i(d )+ Pcos /\ I Cos y Cos sin 2 (d-y) sin 1"... (T), tan h cos (L -N+ln) tan 2 - cos 2 r To facilitate the computation, sin H, sin %, and sin P, in formu. le (L), (P'), and (R), may be replaced by the arcs themselves. The distance d of the star from the pole of the ecliptic enters into these formulae in place of the latitude X. To find tile apparent distance d', we have d'cl - + ~; MOON'S AUGMFENTED SEMI-1)IAMETER. 409 fi)r the apparent latitude X', X'=X —; for the apparent longitude L', L'= L +. The logarithmic formule given on page 352, were derived from equations (L), (0), and (P), and the logarithmic formula on page 353 from equa. (0). To determine now the effect of parallax upon the apparent diameter of the moon. Let ACB (Fig. 71, p. 163) represent the moon, and E the station of an observer; also let R apparent semi-diarneter of the moon, and 1) = its distance. The triangle AES gives AS AS sin AES = or sin R = ES' D At any other distance D' we should have for the apparent semidiameter Rt', AS sin R'; sin R' D whence, sinll 1R -- Thus, if q/' = moon's apparent semi-diameter to an observer at the earth's surface, as at 0 (Fig. 34, p. 61), R = the same as it would be seen from the centre C, and S represents the situation of the moon, sin R' CS sin ZOS sin Z sin R OS sill ZCS sin z But we have, (see page 404,) sin Z _ (sin d -6) sin(q+a) sil z sinl' sill q or, In terms of the apparent longitude and latitude, (see page 40S,) sill Z sill (dL- +r) sin (L -N + ) sin z sin d sin (L - N) Hence, s It' sin t sill (d + c) sin (L - N + IT) sin dsin (L - N) Aberration in Longitudel and Latitude, and in Rilht Ascension and Declination.* (See Art. 100, page 70.) Aberration is caused by the motion of ligtht in conjunction with the motion of the earth. Light coInes to us from the sun in 8"'. 179'.8, during which time the earth describes ian arc a = 20'.41, *Francceur's UJranography, p. 442, &c. 410 rPPENDIX. of its orbit pbdin (Fig. 123,) supposed circular: p is the place of the earth. Let us take any plane whatsoever, which we will cap Fig. 123. relative, passing through the star and the sun, and let dd' be the intersection (i this plane and the ecliptic, with which it makes an angle k: let us seek the quantity ~ by which the aberration displaces the star in T the direction perpendicular to this plane. The question is to project on to a line perpendicular to the relative plane, the small constant arc a which the earth describes, this being the quantity that the star is displaced from its line of direction in a direction parallel to the line of the earth's motion, (see Art. 196 of the text:) this projection is p, variable according to the position of the relative plalze in relation to which it is estimated. The velocity along the tangent at p, makes with ph an angle 0 =pch = the arc pd'; a cos 6 is then the projection of this velocity on the line ph. The angle of our two planes being k, this projection will be reduced to a cos 0 sin k, when it is taken perpendicularly to the relative plane. Thuis, q = a sin k cos... (V). The aberration displaces the star from the relative plane by this quantity cp, k designating the inclination of this plane to the ecliptic, and 8 the are pd', reckoned from p the place of the earth to d' the point of intersection of these two planes. Let us give to the relative plane the positions which are met with in applications. Let us suppose at first that k = 90~, or sin k- 1; the relative plane will then be perpendicular to the ecliptic. Let h brz the vernal equinox; we have pd' = 7rp - lid'; tip is the luongl ud-e of the earth, or 180~ + that 0 of the sun; Ild' is the lonIo. acte I of the star; whence cp = - a cos (0 - i). Fig. 124. Now, let M (Fig. 124) b- ilhe true place K of the star, T' the stdo, as displaced by aberration, KM is the circle of true lati/i I /tude, KM' the circle of app'trent latitude, and MMnI': this arc ias its centre C -, ---— a —- on the axis which passes hbrough tlhe pole /K of the ecliptic; the longitude of tile star is then altered by the part 00' of the o ecliptic comprised between these two planes; and since 00' is to the arc )MM' as the radius 1 is to the radius CM = sin KM cos latitucl. x of the star, we have a aberr. in long.- cos (0 -- )... (W7). If the relative plane is hc, (Fig. 125,) perpendicular to dte circle ABERRATION IN RIGHT ASCENSION AND DECLINATION. 411 of latitude Kcd, the aberration p Fig. 125. I erpendicularly to it, will be the K aberration in latitude. Let kd be 7 -__ the ecliptic, and o the earth; the / angle k is measured by the arc cd \ ~ =X; the arc ok =: = - long. of k; and as kd = 90~, long. of & point =k1 - 90~: substituting in equation (V), we find aberr. in lat. - a sin X sin (0 - 1)... (X). These aberrations of the star produce a small apparent orbit, which is confounded with its projection on the tangent plane to the celestial sphere. Let us suppose the orbit to be referred to two co-ordinate axes passing through the true place of the star and lying in the tangent plane, of which one is parallel to the plane of the ecliptic, and the other perpendicular to this, or tangent to the circle of latitude at the star; and let aberr. in long., and cos X y aberr. in lat.; y will be the ordinate, and x (the aberr. in long., reduced to the parallel through the star) the abscissa: we have X a _ — =cos ( - 1), cos X cos X y = - a sin (O-i); or, -= - os ( - 1), a Y - sin(O - 1). a sin X Squaring the last two equations, and adding them together, 0 disappears, and we find y2 + x22sil2 =a2sin2X.. (Y), whatever may be the place of the earth. Such is the equation of the apparent orbit, which, as we perceive, is an ellipse of which the semi-axes are a and a sin X, and whose centre is the true place of the star. When the star is at the pole of the ecliptic, X 90~, and the ellipse becomes a circle of which the radius is a. When = O0, this ellipse is reduced to an arc 2a of the ecliptic. To find the aberration in right ascension, the relative plane must be perpendicular to the equator. Let kc be the equator, (Fig. 125,) p its pole, psd the relative plane, which is the circle of declination of the star s; kd the ecliptic, o the earth, k the vernal equinox, kc= R, sc -D. Aberration carries the star s out of the plane.,cd a distance a, which it is the question to determine. Equa. (V) is here q -- a sin d cos do = a sin d cos (kd - ko) - a sin d (cos kd cos ko + sin kd sin ko) -a sin d cos kd cos ko + a sin d sin kd sin ko 412 APPENDIX. but ko = long. of earth = 1SO~ + 0; we have also the angle k = the obliquity w of the ecliptic, and the right-angled spherical trian gle kcd gives, by Napier's rules, cot kd - cot R cos w, sin d sin kd = sin R. The 1st equa. multiplied by the 2d, gives sin d cos kd = cos R cos w, whence = - a (cos R cos w cos O + sii R sin 0). The displacement from M to M' (Fig. 124) conducts, as before, to the division of p by cos D, to have the con esponding arc of the equator: thus the aberration in right ascension is, u =- a sin R sec D sin 0 - a cos w cos R sec D cos 0 (Z). Taking the relative plane perpendicular to the circle of declination, we find for the aberration in declination, v — a sinD cos Rsin O-acos w(tan cosD - sin RsinD) cos 0... (a). These formulae may easily be adapted to logarithmic computa. tion: In formula (Z) let a sin R sec D = A, and a cos w cos R sec D - B; then, B u = - A (sin + -cos 0)... (Z'). B a w s U cos R see D Put tan qp = =- = cos W cot R.. (b) A a sin R sec D and we shall have sin qp u — A (sin 0+ o- cos 0) cos q sin O cos p + sin Pq cos 0 cos p A -- --- sin (0 + q). cos q) Restoring the value of A, and taking cos D for sec D, we obtain a sin R u..s sin (0 )... (c). cos D cos p9 The auxiliary arc qp is given by equation (b); it must be substituted in equation (c), with its sign, and we then obta n a. Tan p, and the co-efficient of sin (( + - ) are constant, for the same star, for a long period of time, since these quantities vary very slowly with w and the precession. Moreover, the co-efficient of sin (03-+ ) is the maximum value of u, since it answers to sin ( + 9) = 1. Thus we shall be able to cal-ulate in advance, for NUTATION IN RIGHT ASCENSION AND DECLINATION. 413 any designated star, the values of 0 and of the maximum of the aber ration in right ascension, or of the logarithm of this maximum. The results of these calculations for 50 principal stars are given in Table XCI, columns entitled M and p. If in equation (a) we make a sin D cos R = A', and a cos w (tan X cos D -sin R sin D) = B', we shall have the equati( n B' v =- A' (sin B + XA cos' ) in which A' and B' are constants. This equation is of the same form with equa.(Z'). We therefore have, in the same manner as for the right ascension, B' a cos w (tan X cos D - sin R sin D) tan- - At 0a sin D) cos R a sin w cos D - a cos w sin R sin D a sin D cos R sin w cot D cos R -cos w tan R... (d). A' a sin D cos R v= -cos sin(~+ O) - cos -x sin(O+).. (e). O is given by equation (d), and being substituted in equation (e), we shall have v. 0 and the co-efficient of sin (O + 0) are constant for the same star, and we can therefore calculate in advance the value of this arc, and of the co-efficient, which is the maximum of the aberration in declination. Columns entitled 0 and N, Table XCI, contain the quantities 0 and the logarithms of the maxima of the aberration in declination for 50 principal stars. For convenience in calculation, the angles q, 8, and the maxima, M, N, in Table XCI, have been rendered positive in all cases. This has been accomplished by adding 12-' to g and 0 whenever the calculation conducted to a negative value, and by adding 6'- to o + -, or 0 + 0, whenever the co-efficient had the sign -, (this sign being changed to +;) in this manner the sign of each of the two factors is changed, which does not alter the sign of the pro duct. Formulce for the Nutation in Right Ascension and Declination.* (See Article 124, p. 90.) In deriving these formulae, we must begin with borrowing certain results established by Physical Astronomy. It has been proved, in confirmation of Bradley's conjectures, that the phenomena of nutation are explicable on the hypothesis of the pole of the earth describing around its mean place (that place which, see page X Woodhouse's Astronomy, p. 357, &c. 414 APPENDIX.'7, it would hold in the small circle described around the pole of the ecliptic, were there no inequality of precession) an ellipse, in a period equal to the revolution of the moon's nodes. The major axis of this ellipse is situated in the solstitial colure and equal to 18".50; it bears that proportion to the minor axis (such are the results of theory) which the cosine of the obliquity bears to the cosine of twice the obliquity: consequently, the minor axis will be 131'.77. Let CdA (Fig. 126) represent such an ellipse, P being the mean place of the pole, K the pole of the ecliptic. CDOA is a circle Fig. 126. D,' C V' a X dleseribed with the centre P and radius CP. VL is the ecliptic.. Vw thle equator, KPL the solstitial colure. In order to determine the true place of the pole, take the angle APO equal to the retrogradation of ihe moon's ascending node from V: draw Oi perpenlicular t(, PAF, and the point in the ellipse, through which Oi passes, is the true place of the pole. This construction being admnitted, the nut,.ztions in right ascension and north polar distance mnay, Pp being very small, be thus easily computed. iAutation in North Polar Distance. Nutation in N. P. D. = P —. = Pr = Pp cos pPa, nearly, = Pp cos (APp + APo) - Pp cos (APp + R -o9 ) = Pp sin (APp + R), R denoting Ilhe rig rt ascension. NUTATION IN RIGH1T ASCENSION AND DECLINATION. 415 Nutation in Ri ht Ascension. The right ascension of the star' is, by the effect of nutation, changed firom Vw into V'ts. Now, V'ts = V' + V + ts, nearly, wlhence, Vw - V'ts- -V'v - ts - VV' cos VV'v - Pp sin Pp Si P sin Po' in which expression V'v (- VV' cos VV'v) is, as in the case of precession, common to all stars. In order to reduce farther the above expression, we have pPe = APp + APo = APp + R - 90~, sin APp and VV' - L =Pp si PK —; sin PK whence, - V'v - ts - Pp sill APp cot w - Pp sill (APp + R - 90~) cot N. P. D. = - P) sin APp cot ~ + Pp cos (APp + R) cot S, 3 representing the north polar distance, and w the obliquity of the ecliptic. But these forms are not convenient for computation. In order to rendcr them convenient, we inust, fiom the properties of the ellipse, deduce the values of Pp, anld of the tangent of APp, and then sul:)stitute such values in the above expressions: thus, PI) _ sec APp _ cos APO cos (1 2s -- ) - cos 2 PO0 sec AP6O cos APp cos APp) cos APp' sg designating the longitude of the moon's ascending node; P0 cos ~ whence Pp o APp cos APp tan APp _ pi _ Pd Pd Again,.... tan APO Oi PD PO; Pd Pd hence, tan APp p-P tan APO -= -0P tan (12' - 2 ) Pd:t tan ~. PO Now substitute, and there will result The Nutation in North Polar Distance PO cos ~2 =-_(- cos 3(sin APp cos R + cos APp sin R) cos APp( - PO (tan APp cos R cos a + cos s2 sin R) = — Pd cos R sin ~ + PO cos ~% sin R - - 6".887 cos R sin ~g + 9'.25() cos 2 sin R (J) 416 APPENDIX. which is the difference, as far as nutation is concerned, betweern the mean and apparent north polar distance. The apparent north polar distance, therefore, must be, had by adding the preceding quantity, with its sign changed, to the mean. Nutation in right ascension = Pd sin aS cot + PO cos a~ cos R cot 5 + Pd sin g2 sin R cot 6, which, as far as nutation is concerned, is the difference of the mean and apparent right ascensions: and, consequently, the above expiession must be subtracted firom the mean, in order to obtain the apparent right ascension; or, which is the same, must be added after a negative sign has been prefixed; in which case, we have, substituting for PO, Pd their numerical values, The Nutation in Rig,,ht A scension - - 6".887 sin ~ cot X — 9".250 cos s% cos R cot - 6".887 sin a sin R cot 8... (g). Formulae (f) and (g) are of the same form with (Z) and (a) for the aberrations in right ascension and declination, and therefore formulae may be derived from them similar to (c) and (e), adapted to logarithmic computation. The quantities corresponding to p, M, 0, N, have been calculated for the stars in the catalogue of Table XC, and inserted in Table XCI, in the columns entitled p', M', O', N'. The Solar Nittation arises from like causes as the Lunar, and admits of similar formula. As an ellipse, made the locus of the true place of the pole, served to exhibit the effects of the lunar nutation, so an ellipse, of different, and much smaller dimensions, may be made to represent the path which the true pole of the equator would, by reason of the sun's inequality of force in causing precession, describe about the mean place of the pole. Thus, in Figure 130, the ellipse AdC will serve to represent, tile locus of the pole, when AP = 0".545, Pd = 0".500, and APO, iilstead of being -A, is equal to 2 ~, or twice the sun's longitude, taken in the order of' the signs; the equations, therefore, for the solar nutation in north polar distance, and right ascension, analogous to eqs.f and g will be The Solar Nutation in North Polar Distance -- 0".500 cos R sin 2 0 + 0".545 sin R cos 2 O... (h). The Solar Nutation in Rig7ht Ascension =- 0".500 sin 2 0 cot w - 0".545 cos 2 ( cos R cot a - 0".500 sin 2 O sin R cot...(i). If the apparent place of a star should be required with great precision, it would be necessary to comnpute the solar mnutatioins from these formulae, and apply theni as corrections to the meal EFFECTS OF OBLATENESS OF THE EARTH' S SURFACE. 417 right ascension and declination. The calculation would be per formed after the same manner as for the lunar nutation; but it is much abridged by remarking that the form of the equations is the same as that of the equations for the lunar nutation, and that the co-efficients are very nearly the 0.075 of those of the latter equations. Thus we can make use of the same arcs p', 0', and log. maxima, M', N', repeat the calculation for the lunar nutation, taking 2 0 instead of;2, and multiply the nutations in right ascension and declination thus obtained by 0.075. The results will be the solar nutations required. (See Prob. XX.) F irmulce for computing the effects of the Oblateness of the Earth's Surface upon the Apparent Zenith Distance and Azimuth of a Star.* From the centre of the earth, an observer would see a star at I, Fig. 127. (Fig. 127,) and would have V for his v zenith: from the surface his zenith is / /Ne Z, and he sees this star at B; IB =p M? /XX \ is the parallax in altitude; the azi~~/ //. ~ P muth VZI is changed into VZB. If for a given time, we wish to calculate Pl; the apparent zenith distance BZ, and B the apparent azimuth VZB, we have first to resolve the spherical triangle IZP, in which we know the two sides ZP = co-latitude and IP = co-declination, and the included hour angle P; the azimuth VZI (= A), and the arc IZ (= n) will thus be known. But from the earth's surface, the star is seen at B: the azimuth VZB = VZI +- IZB =A + a; the zenith distance BZ = n +p, since, VZ (- i) being very small, we have sensibly IB + JZ = BZ. By reason of the want of sphericity of the earth, parallax then increases the true azimuth and zenith distance of a star by small quantities, a and p, which it is necessary to calculate. In the triangle VIZ we have cos IV = cos i cos n + sin i sin n cos A = cos n + k sin n; making cos i = 1, sin i = i, and i cos A=k. Now, k /_ i, and d fortiori cos k = 1, sin k = k; whence cos IV = cos n cos k + sin n sin k = cos (n - k). and IV =n - k =n - icos A. Thus we correct the calculated arc n by the quantity - i cos A, to have IV = z = n - i cos A... (j). If this value of z be introduced into equation (a), page 422, we * Francceur's Uranography, p. 426, &c. 27 418 APPENDIX. shall have p, and thence the apparent zenith distance Z = n +-p =BZ. Afterwards, to obtain IZB = a, or the parallax in azimuth, the triangles ZBV, ZBI give sin ZBV sin (A + a) sin ZBV sin a sin i sin (z +p)' sin n sinp' whence, by equating the values of sin ZBV, sin n sin a sin i sin (A + a) sin p sin (z + p) substituting for sin p its value sin H sill (Z +p) = sin H sin Z, (equa. 8, page 51,) and reducing, we have sin a sin (A + a) sin H sini sinn' and as i is very small, sin i sin (A + a-) does not differ sensibly from i sin A, and we thus have in seconds, (For. 47, page 397,) Hi sin A sin 1 " = n... (k). sin n Solution oJ Kepler's Problem, by which a Body's Place is found in an Elliptical Orbit.* (See Art. 199, p. 127.) Let APB (Fig. 128) be an ellipse, E the focus occupied by the sun, round which P the earth or any other planet is supposed to revolve. Let the time and planet's motion be dated from the apFig. 128. M A side or aphelion A. The conditiongiven is the time elapsed from the planet's qultting A; the result sought is the place P; to be'determined either by finding the value of the angle AEP, oi by * Woodhouse's Astronomy, p. 457, &c. SOLUTION OF KEPLER S PROBLEM. 419 cutting off, from the whole ellipse, an area AEP bearing the same proportion to the area of the ellipse which the given time bears to the periodic time. There are some technical terms used in this problem which we will now explain. Let a circle AMB be described on AB as its diameter, and suppose a point to describe this circle uniformly, and the whole of it in the same time as the planet describes the ellipse; let also t denote the time elapsed during P's motion from A to P; then if AM = t x 2 AMB, M will be the place of the point that moves period uniformly, while P is that of the planet; the angle ACM is called the Mean Anomaly, and the angle AEP is called the True Anomaly. Hence, since the time (t) being given, the angle ACM can always be immediately found, (see Art. 198, p. 127,) we may vary the enunciation of Kepler's problem, and state its object to be the dfindirg of the true anomaly in terms of the mean. Besides the mean and true anomalies, there is a third called the Eccentric Anomaly, which is expounded by the angle DCA, and which is always to be found (geometrically) by producing the ordinate NP of the ellipse to the circumference of the circle. This eccentric anomaly has been devised by mathematicians for the purposes of expediting calculation. It holds a mean place between the two other anomalies, and mathematically connects them. There is one equation by which the mean anomaly is expressed in terms of the eccentric; and another equation by which the true anomaly is expressed in terms of the eccentric. We will now deduce the two equations by which the eccentric is expressed, respectively, in terms of the true and mean anomalies. Let t = time of describing, AP, P = periodic time in the ellipse, a = CA, ae= EC, v Z- PEA, u - L DCA; (whence, ET, perpendicular to DT, = EC x sin u,) p =PE, qr = 3.14159, &c.; then, by Kepler's law of the equable description of areas, area PEA area DEA P t =P X x -P - (DEC +DCA) area of ellip. area circle qra( PEl. )C +A. D( 2 (EC. sin u + DC. A') -22 i 2 -ra P P 1 — (e sin u + u): hence, if we put - 27r 2( n, 120 APPENDIX. we have nt = e sin u + u.. (1), an equation connecting the mean anomaly nt, and the ec entric u. In order to find the other equation, that subsists between the true and eccentric anomaly, we must investigate, and equate, two values of the radius-vector p, or EP. First value of p, in terms of v the true anomaly, a(1 - e2) 1- e cosv Second, in terms of u the eccentric anomaly, p =a (I+ecosu)... (2). For, p2 = EN2 + PN2 = EN2 + DN2 x (1 - e2) = (ae + a cos U)2 +a sin2 u (1 -e2) = a2 e2+ 2e cos u + Cos2 U } + a2 (1 -e) sin2z u =a + 2e cosu +e2cosu2 u. Hence, extracting the square root, p =a (1 + e cos u). Equating the expressions (1), (2), we have (1 - e2) = 1 - e cos v) (1 + e cos u), whence, e +- cos u cos v i + e cos u' an expression for v in terms of u; but, in order to obtain a formula fitted to logarithmic computation, we must find an expression for tan 2: now, (see For. 12, p. 397,) tan / -cos v /= /(1- e) (1- cos u) 2 V 1 + cos v} V (1 +- e) (1 +- cos u) / ( - + tan 2.. (m). These two expressions (1) and (m), that is, nt = e sin u + u, v /1- e u tan- = / tan --- 2 V t1 + en 2 analytically resolve the problem, and, from such expressions, by certain formulae belonging to the higher branches of analysis, may v be expressed in the terms of a series involving nt. Instead, however, of this exact but operose and abstruse method of solution, we shall now give an approximate method of expressing the true anomaly in terms of the mean. MO is drawn parallel to DC. (1.) Find ihe half difference of SOLUTION OF KEPLER'S PROBLEM. 421 the angles at the base EM of the triangle ECM, from this expression, tan (CEM - CME)- tan 2 (CEM + CME) x e 2 ~L2 L + eL in which CEM + CME ACM. the mean anomaly. (2.) Find CEM by adding I (CEM + CME) and - (CEM - CMIE) and( use this angle as an approximate value to the eccentric anomaly DCA, from which, however, it really differs by Z EM O. (3.) Use this approximate value of / DCA = Z ECT in computing ET which equals the arc DM; for, since (see p. 419), t - P x DEA, and (the body being supposed to revolve area circle in the circle ADM) -- P x ACM, area AED = area ACM, area circle or, area DEC + area ACD - area DCM + area ACD; consequently the area DEC = the area DCM, and, expressing their values, ET x DC DM x DC -- ETx-xD, =and thus, ET- DM. 2 2 Having then computed ET = DM, find the sine of the resulting are DM, which sine = OT; the difference of the arc and sine (ET - OT) gives EO. (4.) Use EO in computing the angle EMO, the real difference between the eccentric anomaly DCA and the ZMEC; add the computed ZEMO to Z MEC, in order to obtain Z DCA. The result, however, is not the exact value of / DCA, since Z EMO has been computed only approximately; that is, by a process which commenced by assuming i MEC for the value of the Z DCA. For the purpose of finding the eccentric anomaly, this is the entire description of the process, which, if greater accuracy be required, must be repeated; that is, from the last found value of Z DCA = Z ECT, ET, EO, and / EMO must be again computed. Formula for calculating the Parallax in Altitude of a Heavenly Body from its True Zeliith Distance. (See Art. 88, p. 62.) In the actual state of astronomy, the true co-ordinates of the places of the heavenly bodies are generally known, or may be obtained by computation from the results of observations already made, and from these there is often occasion to deduce the apparent co-ordinates. For this purpose there is required an ex, pression for the parallax in altitude in terms of the true zenith distance. If we make Z=z+p in equation (8) p. 62, we shall have sin p sin p - sin H sin (z+p), or sin H -sin ( }-hence, sin p sin (z +p) + sin p 1 + s Si = 1 + sin ( +P) - sin (z ai) 422 APPENDIX. and sin p sin (z +p) -- sill p I —sin p = I - 1-sin -=- 1-sin (z +p-) sin (z +p) Dividing, 1 + sin H sin (z +p) + sin p. 1 —sin H sin (z +p) —sinp or, tang2 (45~+ ~ H) Ztang (App. For. 36, 29); tang ~ z whence, tang(' z +p)-tang I z tang2 (45~ +~ H).... (a). This equation makes known I z+p, from which we may obtain p by subtract ing Y z. Formulce for computing the Annual Variations in the Right Ascension and -De- P clination of a Heavenly Body. (See Art. g 119, p. 88.) Let VLA (Fig. 129) be the ecliptic,,, K its pole, PP'P" the circle described P.-::: x by the mean pole, P the mean pole, -: - and VQA the mean equator at any given time, P' the mean pole and V'Q'A' the mean equator a year afterwards, and E s a star. Draw P'r perpendicular to the Q declination circle Psa. We have an. var. in dec.=sa'-sa=Ps-P's=-Pr; but since PP'r may be considered as a?right-angled plane triangle, FIG. 129. Pr=PP' cos P'Pr=PP' sin QPa... (a). Regarding KPP' as a right-angled isosceles triangle, we obtain sin KPP' or I: sin KP':: sin PKP': sin PP'; whence, sin PP'=sin PKP' sin KP", or PP'=PKP' sin KP' (nearly)......(b): substituting in equation (a), there results, Pr=PKP' sin KP' sin QPa. PKP' = 50".24; KP' = obliquity of the ecliptic = w; QPa=VQ-Va=90~-R (R designating the right ascension of the star s). Thus, finally, an. var. in dec.= -50".24 sin w cos R..... (c). Next, we have an. var. in r. asc.=V'a' —Ya='a'a' —mb=V'm +ba'.......(d); but, V'm:=VV' cos VV'z —50".24 cos w; and since the right-angled triangles sP'r and sba' are similar, sin sr or sin sP' (nearly): sin P'r:: sin sa': sin ba'; whence, sin sa' sin sa' sin ba'=sin Pr s —n' or ba' = P'r —-s- (nearly). sin k's sin P 5 The triangle PP'r gives P'r=PP' sin P'Pr=PP' cos QPa —=PKP' sin KP' cos QPa (equa. b); and sin P's=cos sa'. Substituting, we obtain sin sa' ba =PKP' sin KP' cos QPao -- PKP' sin KP' cos QPa tang sa'. HELIOCENTRIC LONG. AND LAT. 42 2 Replacing PKP', KP', and QPa by their values, as above, and taking the declination sa for sa' and denoting it by D, there results, ba'=-50".24 sin Xo sin R tang D. Now, substituting in equation (d) the values of V'm, and ba', we have an. var. in r. asc.-50".24 cos w+50".24 sin (,), sin R tang D.... (e) The results of formulke (c, e, ) are to be used with their algebraic signs, if the reduction is from an earlier to a later epoch, otherwise with the contrary signs. The declination is always to be considered positive if North, and negative if South. V'm —50".24 cos — =50".24 cos 23~27'=46".0, is the annual retrograde motion of the equinoctial points along the equator. Formulce for computing the IHeliocentric Longitude and Latitude, and Radiuls-vector of a Planet, from its Geocentric Longitude and Latitude. (Referred to in Art. 177, p. 119.) The longitude of the node and the inclination of the orbit are supposed to be known. Let NP (Fig. 130) be part of the orbit of a planet, SNC the plane of the ev FIG. 130. ecliptic, N the ascending node, S the sun, E the earth, and P the planet; also, let P~r be a perpendicular let fall from P upon the plane of the ecliptic, and EV, SV, the direction of the vernal equinox. Let A = PET the geocentric latitude of the planet; I = PST its heliocentric latitude; G= VE E its geocentric longitude; L = VST its heliocentric longitude; S = YES the longitude of the sun; N - VSN the heliocentric longitude of the node; I = PNC the inclination of the orbit; r = SE the radius-vector of the earth; and v = SP the radius-vector of the planet. The point XT is called the reduced place of the planet, and ST its curtete distance. All the angles of the triangle SET have also received particular appellations; STE the angle subtended at the reduced place of the planet by the radius of the earth's orbit, is called the Annual Parallax, SET the Elongation, and ES, the Coavmutation. Let A = S E, E = SE,, and C = EST. Draw ST' parallel to ET: then A = 7rS-' = VSR VST' = VS, - VET = L-G; E = VET - VES = G- S; C = VSE - VSr = 180~ + VSE' - VSr = 180 + VES - VS = 180 + S - L T - L (putting T = 180~ + S). (1.) For the latitude.-The triangles EP7, SPT7, give Er tang X = Ptr = Sr tang 1, whence tang X _ Sr tang I Er; Su sin E out, Si:E:: sin E: sin C, or, - E sin 0' 424 APPENDIX. substituting, tang X sin E tang I sin O' whence, tang X sin C = tang I sin E.... (a). or, tang A sin (T- L) = tang I sin (G —S)....(b). Again, the triangle NPp gives, by Napier's first rule, sin Np = cot PNp tan Pp, or, sin (L - N) = cot I tan 1....(c). Either of the equations (b) and (c) will give the value of 1, when the longitude L is known. (2.) For the Longitude.-If we substitute in equation (b) the value of tang 1, given by equation (c), and replace (G- S) by E, we have tang X sin (T - L) -sin (L —N) tang I sin E; but T -L = (T - N)- (L - N) = D - (L - N), (denoting (T - N) by D); substituting, and designating L - N by x, tang A sin (D- x) = sin x tang I sin E; whence, tang X sin D cos x- tang A cos D sin x = tang I sin E sin x, or, tang X sin D - tang X cos D tang x = tang I sin E tang x, which gives tang = tang Atang A sin D. (d tang A cos D + tang I sin E Substituting the values of x, D, and E, we have, finally, tang A sin (T - N) tang (L - N) = n ~ II (e). tang A cos (T — N) + tang I sin (G- S) As N is known, the value of L will result from this equation. The co-ordinates employed to fix the position of a planet in the plane of its orbit, are its orbit longitude and its radius-vector, both of which result from the heliocentric longitude and latitude, the longitude of the node and the inclination of the orbit being known. In Fig. 130, V'NP represents the orbit longitude, and SP (= v) the radius-vec. tor, for the position P. Now, the triangle PSR gives Si Si SP — S or, v=.-S cos PSi' cos 8 and the triangle ESr gives sin A: sin EE r sill E sin A sin A whence, by substitution, r sin E r sin (G - S), = -....(f). sin A cos 1 sin (L-G) cos I The orbit longitude L' = NP + long. of node....(g): and to find NP, the triangle NPp gives cos PNp = cot NP tang Np, or tang NP tang Np; cos I and Np = long. of planet - long. of node. Formulce for computing the Geocentric Longitude and Latitude of a Planet front its Heliocentric Longitude and Latitude and Radius- Vector. Let S (Fig. 130) be the sun, E the earth, P the planet, 7r its reduced place, and V the vernal equinox. Denote the heliocentric longitude VSr by L, the heliocentric latitude PSir by l, and the radius-vector SP by v; and denote the geocentric longitude by G, and the geocentric latitude by A. Also let E = SET the elongation; C - ESiT the commutation; A = SnrE the annual parallax; and r = SE the radiusvector of the earth. N ow, VEn = SE7r + VES, or G = E + long. of sun. GEOCENTRIC LONG. AND LAT. 425 This equation will make known the geocentric longitude when the value of E is found. In the triangle PS- the side Sr -_ SP cos PS7r = v cos 1, and is therefore known, the side ES is given by the elliptical theory, and the angle C may be derived from the followilh-g equation: C = VSE - VSr = long. of earth —long. of planet; and to find E we have, by Trigonometry, ES + S7r: ES- Sr:: tan ~ (EirS + SEr): tan ~ (ErS- SE7r), or, r + v cos l:r — v cosl:: tang (A + E): tang~ (A - E); whence, 1 v cos r-v cos I r tang (A- E) = r + v tang (A + E) = v cos l tang - (A + E). 1v cos I Let tang 0= r: then, 1 - tang 0 tang - (AE — E) tang (A+ E); or, tang L (A - E) = tang (45~ -0) tang (A+ E).....(a) But, A + E = 180~- C, and E = (A + E)- (A-E) Next, to find the geocentric latitude. Sr tang I = P7 = Er tang X Sr tang X whence, r Eir tang I' Sr sin E but, S,: Er:: sin E: sin C, or E = sin C' sin E tang X and therefore = ta sin C - tang I or sin E tang I tang X sin C...(b). When a planet is in conjunction or opposition, the sines of the angles of elonga. tion and conlmutation are each nothing. In these cases, then, the geocentric latitude cannot be found by the preceding formula; it may, however, be easily determined in a different manner. Suppose the planet to be in conjunction at P, (Fig. 56, p. 120;) then, P7r P7r tang X = - ES + Sir But the triangle SPir gives Pr = v sin l and Sr = v cos 1, and ES r; v sin 1 hence, tang X = r _ v cos'..(c). To find thle distance of the planet from the earth, represent the distance by D; then, from the triangles PuTS and EPr, we have Pr = EP sin PEr = D sin X, and Pr = SP sin PSr = v sin l; v sin I (d.) whence, D= si -..'' The distance of a planet being known, its horizontal parallax may be computed from the equation sin H =.... (e.) (Art. 88). ,126 APPENDIX. CALCULATION OF AN ECLIPSE OF THE SUN.. (1). Of the circumstances of the general eclipse. It is a simple inference from what has been established in Art. 333, that an eclipse of the sun will begm and end upon the earth, at the times before and after conjunction, when the distance of the centres of the moon and sun is equal to P-p + z + d; that the total eclipse will begin and end when this distance is equal to P-p-i + —d; and the annular eclipse when the distance is equal to P-p-+d -d. The times of the various phases of the general eclipse of the sun may be obtained by a process precisely analogous to that by which the times of the phases of an eclipse of the moon are found. Let C (Fig. 131) be the centre of the sun, and C' the centre of the moon, at the time of' conjunction. We may suppose the sun to remain stationary at C, if we attribute to the moon a motion equal to its motion relative to the sun; for, on this supposition, the distance of the centres of the two bodies will, at any given period durin:g the eclipse, be the same as that which obtains in the actual state of the case. Let N'C'L' represent the orbit that would be described by the moon if it had such a motion, which is called the Relative Orbit Let CM be drawn perpendicular to it; and let Cf = Cf = P -p + + dl, and Cy = Cg' — P-p - + d, or P -p + - d, according as the eclipse is total or annular. Then, M will be the place of the moon's centre at the middle of the eclipse; f andf the places at the beginning and end of' the eclipse; and g and g' the places at the beginning and end of the total, or of the annular eclipse. We shall thus have, as in eclipses of the moon, FIG. 131. tn CMg= \ - - cosI. C'M= X sin I tang i -- I 3600s. X sin I cos I Interval from con. to mid. =.-(a). Interval from middle to beginning or end 3600s. cos I = M- o 1/(k'+ A cos I) (k'-X cos I)... (). Interval for total eclipse S300s. cos T..=- M m /(k" + I cos 1) ("-). cos I) Interval for annular eclipse 3600s. cos I — M =+0 A(k"+A oos 1) (k"' — X cos I). (d). 6 (k' - A cos I) Quantity = d.. (e). k' = P —p+~ +d, k" =P- p-p - +d, k"' = P —Pp+L- d... (f). The letters A, M, m, &c., represent quantities of the same name as in the formul] for a lunar eclipse; but they designate the values of these qulantities at the time of CALCULATION OF AN ECLIPSE OF THE SUN. 427 conjunction, instead of opposition. These values are in practice obtained from tables of the sun and moon, as in a lunar eclipse. The times of the different circumstances of a general eclipse of the sun may also be found within a minute or two of the truth, by construction, in a precisely similar manner with those of an eclipse of the moon (330). (2.) Of the phases of the eclipse at a particular place. The phase of the eclipse, which obtains at any instant at a given place, is indicated by the relation between the apparent distance of the centres of the sun and moon, and the sum, or difference, of their apparent semi-diameters; and the calculation of the time of any given phase of the eclipse, consists in the calculation of the time when the apparent distance of the centres has the value relative to the sum or difference of the semi-diameters, answering to the given phase. Thus, if we wish to find the time of the beginning of the eclipse, we have to seek the time when the apparent distance of' the centres of the sun and moon first becomes equal to the sum of their apparent semi-diameters. The calculation of the different phases of an eclipse of the sun, for a particular place, involves, then, the determination of the apparent distance of the centres of the sun and moon, and of the apparent semi-diameters of the two bodies at certain stated periods. The true semi-diameter of the sun, as given by the tables, may be taken for the apparent without material error. For the method of computing the apparent semidiameter of the moon, for any given time and place, see Problem XVII. According to the celebrated astronomer Dusejour, in order to make the observations agree with theory, it is necessary to diminish the sun's semi-diameter, as it is given by the tables, 3.5. This circumstance is explained by supposing that the apparent diameter of the sun is amplified by reason of the very lively impression which its light makes upon the eye. This amplification is called Irradiation. He also thinks that the semi-diameter of the moon ought to be diminished 2", to make allowance for an Inflexion of the light which passes near the border of this luminary, supposed to be produced by its atmosphere. It must be observed, however, that the astronomers of the present day do not agree either as to the necessity or the amount of the diminutions just spoken of. The determination of the apparent distance of the centres of the sun and moon may easily be accomplished, as will be shown in the sequel, when the apparent longitude and latitude of the two bodies have been found. Now, the true longitude of the sun, and the true longitude and latitude of the moon, may be found from the tables, (Probs. IX. and XIV.); and from these the apparent longitudes and latitudes may be deduced by correcting for tile parallax. But the customary mode of proceeding is a little different from this: the true longitude and latitude of the sun are employed instead of the apparent, and the parallax of the sun is referred to the mnoon; that is, the difference between the parallax of the moon and that of the sun is, by fiction, taken as the parallax of the moon. This supposititious parallax is called the moon's Relative Parallax. (See Prob. XVII.) We will first show how to find the approximate times of the different phases of the eclipse. Put T = the time of new moon, known to within 5 or 10 minutes. (Prob. XXVII.) For the time T calculate by the tables the sun's longitude, hourly motion, and semi-diameter, and the moon's longitude, latitude, horizontal parallax, semidiameter, and hourly motions in longitude, and latitude. Subtract the sun's horizontal parallax from the reduced horizontal parallax of the moon,*'and calculate the apparent longitude and latitude, and the apparent semi-diameter of the moon. From a comparison of the apparent longitude of the moon with the true Ion. gitude of the sun, we shall know whether apparent ecliptic conjunction occurs before or after the time T. Let T' denote the time an hour earlier or later than the time T, according as the apparent conjunction is earlier or later. With the sun and moon's longitudes, the moon's latitude, and the hourly motions in longitude and latitude, at the time T, calculate the longitudes and the moon's latitude for tile time T'; and lbr this time also calculate the moon's apparelt longitude and latitude. Take the difference between the apparent longitude of the moon and the true longitude of the sun at the time T, and it will be the apparent distance of the moon from the sun in longitude, at this time. Let it be denoted by n. Find, in like manner, the apparent distance of the moon from the sun in longitude at the time T', and denote * The reduced horizontal parallax of the moon is its horizontal parallax as reduced from the equator to the given place. (See Prob. XV.) 428 APPENDIX. it by n'. In the same manner as at the time T, we find whether apparent conjunotion occurs before or after the time T'. If it occurs between the times T and T', the sum of n and n', otherwise their difference, will be the apparent relative motion of the sun and moon in longitude in the interval T'-T, or T-T'; from which the relative hourly motion will become known. The difference of the apparent latitudes of the moon, at the times T and T', will make known the apparent relative hourly motion in latitude. With the relative hourly motion in longitude and the difiference of the apparent longitudes at the time T, find by simple proportion the interval between the time T and the time of apparent ecliptic conjunction; and then, with the apparent latitude of the moon at the time T and its hourly motion in latitude, find the apparent latitude at the time of apparent conjunction thus determined. Then, knowing the relative hourly motion of the sun and moon in longitude and latitude, together with the time of apparent conjunction, and the apparent latitude at that time, and regarding the apparent relative orbit of the moon as a r ght line (which it is nearly), it is plain that the time of beginning, greatest obscuration, and end, as well as the quantity of the eclipse, may he calculated after the same manner as in the general eclipse; the disc of the sun answering to the section of the luminous frustum mentioned in Art. 331, and the apparent elements iC l answering to the true. Let C (Fig. 132) represent the centre of the sun supposed stationary, CC' the apparent latitude of tile moon at apparent conjunction, N'C' the apparent relative orbit RE _of the moon, determined by its passing through the point C' and z/zt~'I /t`~f making a determinate anile with the ecliptic N'F, or by its passin"g through the situations of the moon at the times T and T'. Also, let RKFEI' be the border of the FIG. 132. sun's disc; f, f' the positions of the moon's centre at the beginning and end of the eclipse, determined by describing a circle around C as a centre, with a radius equal to the sum of the apparent serni-diameters of the suni and moon; and M (the foot of the perpendicular let fall from C upon NYC') its position at the time of greatest obscuration. If the eclipse should be total or annular, then g, g' will be the positions of the moon's centre at the beginning and end of the total or annular eclipse; these points being determined by describing a circle around C as a centre, and with a radius equal to the difference of the apparent semi-diameter of the sun and moon. The results will be a closer approximation to the truth, if the same calculations that are made for the time T' be made also for another time T". The various circumstances of the eclipse rmay also be had by construction, after the same manner as in a lunar eclipse (330). In order to be able to observe the begiiining or end of a solar eclipse, it is necessary to know the position of the point on the sui's limb where the first or last contact takes place. The situation of these points is designated by the distance on the limb, intercepted between them and the highest point of' the limb, called the Vertex. The contacts will take place at the points t, t', (Fig. 132,) on the lines Cf CQf'. To find the position of the vertex, with the sun's longitude found for the beginning of the eclipse, calculate the angle of position of the sun at that time, (see Prob. XIII.), and lay it off to the right of the circle of latitude CK, when the sun's longitude is between 90" and 270", and to the left when the longitude is less than 90" or more than 270". Suppose CP to be the circle of declination thus determined. Next, let Z (Fig. 6, p. 13) be the zenith, P the elevated pole, and S the sun: then in the triangle ZPS we shall know ZP the co-latitude, ZPS tire hour angle of the sun, and we may deduce PS, the co-declination of the sun, fromi the longitude of the sun as derived from the tables (equ. 24). These three quantities beingo known, ZSlP, the angle made by the vertical th'rough the sun with its circle of declination, may be computed; and being laid off in the figure to the right or left of CP (Fig. 132), according as the time of beginning is before or after noon, the point Z or Z', as the case may be, in which the vertical intersects the limb RKK', will be the vertex, and the CALCULATION OF AN ECLIPSE OF THE SUN. 429 are Zt, Z't, on the limb, will ascertain the situation of t, the first point of contact, with respect to it. The situation of the last point of contact may be found by the same mode of proceedinrg. Let us now show how to find the exact times of the beginning, greatest obscuration, and end of the eclipse, the approximate times being known. Let B designate the approximate time of beginning, taken to the nearest minute. Calculate for the time B by means of the tables, the sun's longitude, hourly motion, and semidiameter; also the moon's longitude, latitude, horizontal parallax, semi-diameter, and hourly motions in longitude and latitude. Then, making use of the relative parallax, calculate the apparent longitude, latitude, and semi-diameter of the moon. Subtract the apparent longitude of' the moon from the true longitude of the sun; the difference will be the apparent distance of the mnoon from the sun in longitude: let it be denoted by a. Denote the apparent latitude of the moon by X. Now, let EC (Fig 133) represent an arc of the ecliptic, and K its pole; and let S be the situation of the sun, K and M the apparent situation of the moon at the time B. Then MS is the apparent distance of the centres of the two bodies at this time. Denote it by A. Sm=a, and Mlm - X. The right-angled triangle MSm being very small, may be considered as a plane triangle, and we therefore have, to determine A, the equation A 2=a2 + X2.....(g)*M Having computed the value of A, we find, by comparing it with the sum of the apparent semi-diameters of the sun and moon, whether the beginning of the eclipse occurs before or after the approximate time B. EB S Fix upon a time some 4 or 5 minutes before or after B, according as the beginning is before or after, and call it FIG. 133. B'. With the sun and moon's longitudes, the moon's latitude, and the hourly motions in longitude and latitude, at the time B, find the longitudes and the moon's latitude at the time B', and compute for this time the apparent longitude, latitude, and semi-diameter of the moon. Subtract the apparent longitude of the moon from the true longitude of the sun, and we shall have the apparent distance of the moon from the sun at the time B'. Take the difference between this and the same distance a at the time B, and we shall have the apparent relative motion of the sun and moon in longitude during the interval of time between B and B'. Then find, by simple proportion, the apparent relative hourly motion in longitude, and denote it by k. Take the difference between the apparent latitudes of the moon at the times B and B', and it will be the apparent relative motion of the sun and moon in latitude, in the interval; from which deduce the apparent relative hourly motion in latitude, and call it n. Now, put t = the interval between the approximate and true times of the beginning of the eclipse, and suppose S and M (Fig. 133) to be the situations of the sun and moon at the true time of beginning. In the time t, the apparent relative motions in longitude and latitude will be, respectively, equal to kt and nt, and accordingly we shall have Sm= a —kt, Mm =A+ nt. The small right-angled triangle S-M~m may be considered as a plane triangle; the hypothenuse SM —= —=the sum of the apparent semi-diameters of the sun and moon, minus 5".5 (p. 427). We have then, to find t, the equation (a - kt)2 + (X + nt)2 =q2, or, developing and transposing, (n2 + k2) t2 _-2 (ak-Xn) t = 2 _- (,a24 +2) = t2_ A 2; * In place of equation (g) the following equations may be employed in logarith, mic computation: X a tang =-a-' a = cos where 0 is an auxiliary arc. 4-I30 APPENDIX. making A= 2_ A 2 and B = ak - Xn, we have (n2 + k2) t2 - 2Bt -- A, and t B +A (n2 + k2) (h) and t* The negative sign must be prefixed to the radical, for, if we suppose A to be equal to zero, t must be equal to zero. Multiplying the numerator and denomi. nator by B + V B2 + A (n2 + k2), and restoring the value of A we obtain 3600s. (A 2_-L2) (in seconds), t... (i). -B + i B + (L,2_ - 2 ) (n2 + k2 ) Although this equation has been investigated for the beginning of the eclipse, it is plain that it will answer equally well for the determination of the other phases, if we give the proper values and signs to zQ, a, 1, n, and k. k is positive before conjunction and negative, after it, and the radical quantity is negative after conjunction; n is negative, when the moon appears to recede from the north pole of the ecliptic; A has the sign -, when it is south; a is always positive.* The value of t taken with its sign is to be added to the time B. The values of the quantities a, A, n, and k, are found for the other phases after the same manner as for the beginning. To obtain the value of d, at the time of greatest obscuration, find the relative motions in longitude and latitude (k and n), during some short interval near the middle of the eclipse, which is the approximate time of greatest obscuration; then compute the inclination of the relative orbit by the equation n tangI= k _- (j). after which p will result from the equation m = A cosI... (k). X is the moon's latitude at the time of apparent conjunction, which is easily calculated, by means of the values of k and n, and the apparent longitude and latitude of the moon, found for some instant near the time of apparent conjunction. For the beginning and end of the total eclipse, we have, =- appar. semi-diam. of moon - appar. semi-diam. of sun + 1".5; and for the beginning and end of the annular eclipse,.- appar. semi-diam. of sun - appar. semi-diam. of moon1".5. If the value of 4; given by equation (k), be substituted in equation (i), this equation will make known the time of greatest obscuration; but this may be found more conveniently by a different process. Let NCF (Fig. 134) represent a portion of the ecliptic, EML a portion of the relative orbit passed over about the' time of greatest obscuration, C the stationary position of the sun's centre, and M the 1. place of the moon's centre at the instant of 1 - its nearest approach to C. Also, let a = CR the apparent distance of the moon from / the sun in longitude at the time of the nearest approach of the centres, X' = RIMl the - C,. N moon's apparent latitude at the same time, kIc - k the apparent relative motion in FIG. 134. longitude in some short interval about this time, and n =- I1 the moon's apparent motion in latitude during the same interval. The right-angled triangles MIzk and CMR are similar, for their sides are respectively perpendicular to each other; whence Mk: MR:: ckn: CR; kn n and CR = MR ~-r' or, a -' k -... (1). * Developing the radical in equation (h), and neglecting all the terms after the second, as being very small, we obtain for the beginning and end of the eclipse the more convenient formula 1 800s. ( 2 _ 2) =. CALCULATION OF AN ECLTPSE OF THE SUN. 431 If the moon's apparent latitude be found for the approximate time of greatest obscuration, and substituted for X' in equation (1), this equation will give very nearly the apparent distance (X) of the two bodies in longitude at the true time of' greatest obscuration. With this, and the apparent distance at the approximate time of greatest obscuration, together with the relative apparent motion in longitude, the true time of greatest obscuration may be found nearly by simple proportion. A more accurate result may then be had by finding the moon's apparent latitude for the time obtained, substituting it for A' in equation (1) and then repeating the calculations. A simpler, though less accurate method than that already given, of finding the times of beginning and end of the total or annular eclipse, is to compute the half duration of the total or annular eclipse, and add it to, and subtract it from, the time of greatest obscuration. This interval may easily be determined, if we can find the rate of motion on the relative orbit, and the distance passed over by the moon's centre during the interval. Let g, g' (Fig. 134) be the places of the moon's centre at the instants of the two interior contacts, and AIn, the distance passed over in some short interval (L). Let 0 = < Mnk the complement of the inclination of the relative orbit, k - Mks, I' = Mn, and t = half duration of total or annular eclipse. The triangles Mnkh, CRM, give Mn =, or k'r.- (m): sin MaI/k sin 0 and tang RCM- = tang Mhnk =, or, tang 0. ()..CR a Finding the value of 0 by tile last equation, and substituting it in equation (m), we obtain the value of k'; and tlhen, to find t, we have L xMlq k':L:: Mg:t, ort L x k' Mg-= V-i/-2. whence, t: _ L~/,- A L v/'(P+ A)(i- A) m? (o) k' k' The apparent distance of the centres of the two bodies at the time of greatest obscuration being known, the quantity of the eclipse may be readily found. We have but to subtract the apparent distance finrom the sum of the apparent semi-diameters, and state the pr oportion. as tile suli's apparent diameter: the remainder:: 12 digits: the digits eclipsed. (For a mrore particular description of the method of calculating a solar eclipse, see Prob. XXX.) CALCULATION OF AN OCCULTATION. The calculation of an occultation is very nearly the same as. that of a solar eclipse. The only difference is in the data. The star has no diameter, parallax, or motion in longitude; and as it is situated without tile ecliptic, we have, in place of the latitude of the moon, employed in solar eclipses, the difference between the latitude of the moon and that of' the star, and in place of the difference between the longitudes of the two bodies and their relative hourly motion in longiK tude, tllese quantities reterred to all arc passing' through the star and parallel to the ecliptic. Thllus, if EC (Fig. 133) represent the ecliptic, K its pole, s the situation of the star, M that of the mloon, and sai' an are passing through s and parallel to the arc EC, we have in place of mM, m'M = B f C mM —nmm', and in place of Sin, smin'. The hourly variation of Smz must also be reduced to the are san'.'like reduction of the difference of longitude of the moon s' - l ---' C and star, to the parallel to the ecliptic, passing through the star, is effected by multiplying the difference by the coA sine of the latitude of the star. For, let AB (Fig. 135) be FIG. 135. an arc of the ecliptic, and A'B' the corresponding arc of a 432 APPENDIX. circle parallel to it, then, since similar arcs of circles are proportional to their radii we have BC: B'C':: AB: ='B AB.B'C' BC But, B'C' = Ca = B'C cos BCB' = BC cos BB': hence, A'B' = AB.BC cos BB' _ AB cos BB'. BC The reduction of the relative hourly motion in longitude to the parallel in question, is obviously effected in the same manner. NOTE I. CONSTRUCTION OF TABLES. The determination of the place of the sun or moon, or of a planet, may be greatly facilitated by the use of tables. The principle and mode of construction of tables adapted to this purpose are nearly the same for each body. We will first explain the mode of constructing tables for facilitating the computation of the sun's longitude. We have the equation True long. = mean long. + equa. of centre + inequalities + nutation. If, then, tables can be constructed which will furnish by inspection the mean longitude, the equation of the centre, the amounts of' the various inequalities in longitude, and the nutation in longitude, at any assumed time, we may easily find the true longitude at the same time. (1.) For the mean longitude.-The sun's mean motion in longitude in a mean tropical year, is 360~. From this we may find by proportion the mean motions in a common year of 365 days, and a bissextile year of 366 days. With these results, and the mean longitude for the epoch of Jan. 1, 1850 (see Table II.), we may easily derive the mean longitude at the beginning of each of the years prior and subsequent to the year 1850. The second column of Table XVIII. contains the mean longitude of the sun at the beginning of each of the years inserted in the first column. The third columnl of this table contains the mean longlitude of the perigee at the same epochs: it was constructed by means of the mean longitude of the perigee found for the beginning of the year 1800, and its mean yearly motion in longitude.' Having the sun's mean daily motion in longitude, we obtain by proportion the motion in any proposed number of months, days, hours, minutes, or seconds. Table XIX contains the respective amounts of the sun's motion friom the commencement of the year to the beginning of' each month; Table XX, the sun's mean motion from the beginning of any month to the beginning of any day of the month, and his motion fobr hours from 1 to 24; and Table XXI, the same for minutes and seconds from 1 to 60. With these tables the sun's mean motion in longitude in the interval between any given time in any year and the beginning of the year may be had: and if this be added to the epoch for the given year,'taken out from Table XVIII, the result will be the mean longitude at the given time. (See Problem IX.) Tables XIX and XX also contain the motions of the sun's perigee, from which and the epoch given by Table XVIII results the longitude of the perigee at any proposed time. The longitude of the perigee is given in the Solar Tables for the purpose of making known the mean anomaly, the mean anomaly being equal to the mean longitude minus the longitude of the perigee. (2.) For the equation of the centre.-To find the equation of the centre of an orbit we have the following equation: Equa. of centre = A sin 0 + B sin 29 + C sin 30 + &c.; * The quantities in Table XVIII are called Epochs. The Epoch of a quantity is its value at some chosen epoch. CONSTRUCTION OF TABLES. 433 in which A, B, C, &c., are constants that rapidly decrease in value, and which may be determined for any particular orbit, and!J the Iman anorlaly. Now, by giving to the mean anomaly 0 in this equatiorn a series of values increasing by small equal differences (of 1~, for instance,) from zero to 36()~, and computing the corresponding values of the equation of the centre, then registering in a column the different values assigned to 0, and in another column to the rigrht of this the computed values of the equation of the centre, we shall obtain a table which will give on inspection the equation of the centre corresponding to any particular mean anomaly. In this manner was constructed Table XXV. In this table, however. for the sake of compactness, the values of the equation, instead of being registered in one column, are put in as liany different columns as there may be different nunrlbers of signs in the value of the mean anomaly; each column answering to the particular number of signs placed at the head of it. If the equation of the centre at an assumed time be required, find the mean anomaly by the tables, and with the value found for it take out the equation of the centre from Table XXV. The given quantity with which a quantity is taken from a table is called the Ai gument of that quantity. Accordingly the mean anomaly is the argumnent of the equation of the centre in Table XXV. (3.) For the ifequalfities.-The equations of the inequalities, as we have already stated, are of the form C sin A, the argument A being the difference between the ]ongitude of the disturbing planet and that of the earth, or some multiple of this difference. With the equations of the inequalities a table of each inequality may be constructed, upon the same principles as Table XXV. But, as the expression for the whole perturbation in longitude (Art. 212), produced by any one planet, involves only two variables, the longitude of the earth and the longitude of the planet, it is thought to be more convenient to have a table of double ei'rly, which will give the amount of the perturbation by means of the two variables as arguments. Such a table may be constructed, by assigning to the longitude of the earth and the iongitude of the disturbing planet a series of values increasing by a common difference, and computing with each set of the values of these quantities, the corresponding amount of the perturbation. In connection with the tables of the perturbations, we must have tables that make known the values of the arguments at any given time. Now, the mean!olngitude of the sun may be found by the solar tables (XVIII to XXSI, and thence the mean heliocentric longitude of the earth by subtracting 1850; and the mean longitude of the disturbing planet may be had from similar tables. The columns of Table XVIII, marked I, II, III, IV, V, VI, VII, contain the arg'uments of all the perturbations for the beginning of each of the years registered in the first column, expressed in thousandth parts of a circle. Tables XIX and XX contain the variations of the arguments for months, days, and hours. Their variations for minutes and seconds are too small to be taken into account. W ith these tables, and Table XV1II, the values of the arguments at any given time may be found, and by means of the arguments the perturbations may be taken from Tables, XXVIII, XXIX, XXXII, XXXI, XXX, and XXXIII. (4.) For the nutation. —The formula for the lunar nutation in longitude, is 17".3 sin N —0".2 sin 2 N, in which N denotes the supplement (to 3G60) of the longitude of the moon's ascending node. With this formula the second column of the Table XXVII was constructed. The value of N, in thousandth parts of a circle, resultj from Tables XVIII, XIX, and XX. The solar nutation is also given by Table XXVII. Tables may also be constructed that will facilitate the computation of the radiusvector. We have True rad.-vector = elliptic rad.-vector + perturbations. A table of the elliptic radius vector may be formed by means of the polar equation of the orbit, and tables of the perturbations from their analytical expressions (Art. 214). The tables of' the perturbations will have the same arguments as the tables of the perturbations of longitude. Lunar and planetary tables are constructed upon the same principles as the solar tables we have been describing, which serve to make known the orbit longitude and radius-vector. But other tables are necessary in the case of these bodies, for the computation of the ecliptic longitude and the latitude. 28 434 APPENDIX. The difference between the orbit longitude and the ecliptic longitude is called thk Reduction to the ecliptic. A formula for the reduction has been investigated, in which the variable is the difference between the orbit longitude and the longitude of the node (or, what amounts to the same, the orbit longitude plus the supplement of the longitude of the node to 360~). If this formula be reduced to a table, by taking the reduction from the table and adding it to the orbit longitude, we shall have the ecliptic longitude. Table LIII is a table of reduction for tie moon. For the latitude, we have the equation True lat. - lat. in orbit + perturbations. We have already seen (Art. 201) that sin (lat. in orbit)= sin (orbit long. -long. of node) sin (inclina.) A table constructed from this formula will have for its argument the orbit lon gitude minus the longitude of the node, which is also the argument of the reduction (See Table LV.) The tab)les of the perturbations in latitude are constructed upon the same principles as the tables of the perturbations in longitude and radius-vector. NOTE II. (Refcerred to on p. 115.) The fact stated in the text (Art. 273) that the penumbra of the solar spot does not begin to be formed until the spot, which at first has an umbral blackness, has attained a measurable size, is cited by astronomical writers as a circumstance strongly favoring the hypothesis of the origination of the spot in a disturbance from below. But this fact may be reconciled with the opposite hypothesis advocated iu the text, if we reflect that the penumbral lies at a considerable depth below the luminous envelope, and that the process of discharge and ascent of a column of photospheric matter (Art. 293), which results in disclosing to view a portion of the penumbral envelope, should occupy a certain interval of time in passing down to it. During this interval this envelope may have an umbral blackness, and it may owe its subsequent visibility, as distinct from the umbra, entirely to the fact that it acquires a luminosity in consequence of the electric discharges that attend the process of spot evolution, which has penetrated to its depth in the atmosphere of the sun. This view is supported by the fact that it furnishes a simple explanation of the decrease in the brightness of the penumbra from the umbra to its outer margin. We have only to observe that the process of expulsion and ascent of vaporous matter, which begins at a certain point of the photosphere, at the same time that it proceeds downward, spreads laterally. and that when it has penetrated to the depth of the penumbral envelope at the point below that where it originated, an opening of a certain size will have been formed in the luminous envelope, and except below the central portions of this opening, the lower envelope will still be in a comparatively quiescent condition, and retain its natural depth of shade. Subsequently the same process of evolution is repeated at this envelope; an opening is made in it that has the umbral depth of shade, and this is surrounded by a region of luminous activity, which is the penumbra of the spot, and is fringed by a dark border consisting of the part which the descending action has not yet reached. In the case of the larger spots and of long continuance, the same process penetrates to the third envelope, and the former umbra shows, in its turn, a black central nucleus, surrounded by a fringe of a shade perceptibly less dark. Upon the present hypothesis with regard to the mode of development of the spots, the principal varieties, consisting of a spot without a penumbra, a spot without an umbra, a spot without the central black nucleus at the centre of the umbra, and a spot with this nucleus, are but the varieties that present themselves according as the process of discharge, beginning at the surface of the photosphere, penetrates only through the first envelope, or through the first and as far as the second, or through DEVELOPMENT OF SUN'S SPOTS. 435 the first and second, or through the first, second, and third. Upon the old hypothesis, it is necessary, in order to explain these diverse phenomena, to assume that there are three possible centres of explosive action, posited below the successive envelopes. So long as the active evolution continues at the lower envelopes, the ascending vaporous column, expanding as it rises, should check any eventual tendency of the opening in the luminous envelope to close. When the activity subsides at the penumbral stratum, and the opening in it begins to close, this should be followed by a similar collapse in the regions above it; and so the contraction of a spot should generally begin in accordance with observation at the umbra, and be fcllowed by the encroachment of the luminous margin upon the penumbra. But it is conceivable, also, that the closing up of a spot lmay result from a condensation into luminous clouds of portions of the expanded matter ascending within the region of the spot; and that the luminous veil that is often seen to form over a spot, and the bridges of light which suddenly span the umbra, are the first indications of the beginning of such condensations. To give a more complete exposition of the author's theory of the development of solar spots, the following general conclusions are added to those given in the text. 1. The spots are, for the most part, due to diminutions occurring in the magnetic intensity that obtains in the photosphere of the sun. 2. Each planet operates on the photosphere by repulsive impulses, and develops electro-magnetic currents circulating in a direction opposite to that of the rotation. The effective currents thus originating are differential, and result from the fact, that on the left or east side of the line from the planet to the sun's centre the motion of the surface is in a direction opposite to that in which the impulses are propagated, and on the other side in the same direction. 3. The general tendency of such new currents should be to increase the magnetic intensity at the surface of the photosphere; but it is possible that in peculiar conditions of the photospheric matter, as to density and other qualities, the superficial currents developed by planetary action may prevail over those set in motion below the surface, and the opposite magnetic effect be produced. 4. The motion of the sun through space also develops similar magnetic currents, which should have a similar effect. These currents may be considered as originating on that side of the sun where the absolute motion of a point of its surface is the most rapid. 5. If the sun were stationary the motion of revolution of a single planet would have but little direct effeet to change its magnetic action on the sun as a lwhole; except so far as this nay vary by reason of the varying distance of' tihe planet. But in point of fact the effective action of a planet will change with its distance in longitude from the point towards which the sun is moviing in space. It wvill be at its maximum when the planet is in heliocentric conjunction with this point, and at a minimum when it is ill opposition to it. In the former case its heliocentric longitude will be 253~, and in the latter 73~. 6. The joint magnetic action of two planets varies with their relative position; it has its maximum value when the planets are in conjunction, and its minimum when they are in opposition. 7. The epochs of the conjunction of one planet with another, or of a planet with the point towards which the sun is moving, are, in general. the epochs of minimum of spots, since the magnetic intensity at the surface of the photosphlere is onil the increase before every such epoch. The approximate coincidence of planetary conjunctions with special epoclis of minimum of' spots is a recognized iact in the case of Jupiter and the earth, and Venus and the earth. On the other hand, the opposition of t wo planets tends to determine a maximum of spots. S. Jupiter and Venus are the most influential planets. The period of the spots is determined mainly by the motion of revolution of Jupiter, but appears to be modified bS the varying planetary configurations, and also by changes occurring in the physical condition of the sun's photosphere. The varying action of Jupiter, in the course of a revolution, has been attributed to its variations of distance from the sun, but it seerns improbable that effects so marked should result from so slihght a cause. The mean period of' the spots is, according to Wolf, nearly one year, and according to Schlwabe, nearly two years less than Jupiter's period of revolution. This difference canlnot be explained by any mere recurrence of planetary configurations. In fact, 436 APPENDIX. epochs of maximum and minimum of spots have occurred under every variety o! configurations; and also when Jupiter has had every variety of position in its orbit The following Table shows the mean positions of Jupiter and Saturn at various such epochs, from 1750 to 1860; together with the relative numbers showing, according to Wolf, the spot-condition of the sun. Mean Heliocentric Mean Heliocentric Longitude. Longitude. Epochs of Relative Juiter. aturn Epochs of Obs'd Min. Jupiter. Saturn. ~ ~/Maxima. Numbers. JutrMinima. -Mean Min. Jiter.. 1750.0 68.2 4~ 23 1 1744.5 + 0.558 1970 1640 1761.5 75.0 353 1 2 1755.7 + 0.639 177 301 170.0o 79.4 251 11 6 1766.5 + 0.8'20 145 73 1779.5 99.2 179 232 1775.8 -1.499 67 187 1788.5 90.6 93 342 1784.8 -3.618 340 297 1804.0 70.0 203 172 1798.5 -1.037 36 105 1816.8 45.5 232 329 1810.5 -0.156 41 251 1829.5 53.5 258 124 1823.2 + 1.425 66 47 18s37.2 111.0 131 218 1833.8 + 0.906 28 177 1848.6 100.4 117 358 1844.0 -0.013 338 301 1860.5 98.6 119 143 1856.2 + 1.068 348 91 It xvill be seen that since 1761 the maxima have occurred when Jupiter was in the second or third quadrant of longitude; and that since 1755, the minima have occurred when he was in the other two quadrants. But previous to those dates the condition of things was reversed, and the transition from the one condition to the other was gradual. This fact seems to necessitate the supposition that the physical condition of the sun's photosphere is liable to changes, by reason of which the ordinary effect of the planets is, at intervals, wholly reversed. 9. The highest maxima of spots have occurred when Jupiter was in the second quadrant of longitude; that is, after he ilas passed a certain distance beyond the point (long. 73~) where his magnetic action on the sun is most directly opposed to the effect of the sun's motion through space, and advanced towards the aphelion of the orbit, where the direct magnetizing tendency of the planet is the least. It should here be remarked that the author has undertaken, in other publications, to show that the earth derives its magnetic condition from the collision of the molecules of its revolving and rotating mass with the ether of space, and that the sun and the planets should each be in a magnetized condition from a similar cause; also that in the new terrestrial magnetic currents being continually developed by the earth's motions of revolution and rotation combined, in connection with those generated in the photosphere (upper atmosphere) of the earth by ethereal impulses, and streams of nebulous matter, proceeding from the sun, we have the principal operative causes of the disturbances experienced by the magnetic needle on the earth's surface. NOTE III. (Referred to on page 275.) A remarkable analogy in the periods of rotation of the primary planets was discovered a few years since (1848) by Daniel Kirkwood, of Pottsville, Pennsylvania. Ti)is analogy is now generally known by the name of irkiwood's Law, and is as follows: " Let P be the point of equal attraction between any planet and the one next interior, the two being in conjunction: P' that between the same and the one next exterior. Let also D = the sum of the distances of the points P, P' from the orbit of the planet; which I shall call the diameter of the sphere of the planet's attraction; D' -- the diameter of any other planet's sphere of attraction found in like man. ner; KIRKWOOD'S ANALOGY. 437 -- = tnd number of sidereal rotations performed by the former during one sidereal revolution around the sun; n = the number performed by the latter; then it will be found that 2.:,:: D3; D'S; or nA =' (D ). That is, the square of the number of rotations m.ade by a planet during one revotution round the sun, is yroportioned to the cube of the diameter of its sphere of attraction; or,-is a constant quantity for all the planets of the solar system. The analogy thus announced has been subjected to a rigid mathematical examination by Sears C. Walker, with the following result: "We may therefore conclude," says he, "that whether Kirkwood's Analogy is or is not the expression of a physical law, it is at least that of a physical fact in the mechanism of the universe." (See the American Journal of Science, New Series, vol. x. pp. 19-26.) There are but three planets, viz., Venus, the Earth, and Saturn, for which all the elements embraced in this lawv are known. The diameters of the spheres of attraction of Mercury and Neptune are, from the nature of the case, incapable of determination. The mass of the one planet into which the planetods are supposed once to have been united is not known with certainty, as there may be planetoids yet undiscovered, and its period of rotation is hypothetical only. The diameters of the sphleres of attraction of Mars and Jupiter can only be approximately determined; and the period of rotation of Uranus is unknown. The interest naturally awakened by the announcement of so important a discovery was heightened by the faiet, that it was at once perceived that it furnished a new and powerful argument in support of the nebular hypothesis (or cosmogony) devised by Laplace. (See a paper on this subject by Dr. B. A. Gould, Jr., in the Journal of Science, New Series, vol. x. p. 26, etc.) NOTE IV. (Referred to on page 277.) It remains to deduce, if possible, the known law of the distribution of the inclinations of the cometary orbits. This law is, that the number of orbital inclinations of different values increases with the angle from V0 to 50~ or 60~, and then decreases; as appears from the following tabular statement, from which the comets of short period have been deducted. 0 to 10~ 100 to 20~ 200 to 8o 800 ~to 40~1400 to 50a500 to 60' 600~ to 700 70 to SO~ " 0 to 80~ 90 11 15 16 28 34 32 23 22 14 Let us first consider the case of a discharge from the equator, and conceive, for the present, the direction of discharge to be tangent to the surface. Since the orbits of the class of comets under consideration are very eccentric, the initial velocity must be very much less than the velocity of rotation at the equator. If we fix upon a maximum limit (V) to this velocity, at any assumed epoch, and from this and the velocity of rotation at the equator (v) deduce the direction of the expelling force, and the velocity (v~,) due to its action, it will be seen: (1.) That this force will take effect in directions opposed to that of the rotation, and inclined to it under angles differing but little from 180", whatever direction may be assumed tbr V, the resultant or effective velocity. (2.) That the velocity, v', due to the expelling force, will be either equal to the velocity of rotation, v, or a little greater, or a little less. 438 APPENDIX. Under these circumstances, nebulous masses may be projected in every variety of direction in the tangential plane, which would move in planes having any angle of inclination, and describe orbits of every variety of eccentricity greater than that answeringo to the assumed maximum initial velocity, V. Now, if we conceive the point at which a discharge occurs to lie in any latitude (1), the velocity of rotation will be less in the proportion of cos g to I; and making use of the parallelogram of velocities as before, and retaining the same effective velocity V, we find that for any assumed direction of V, the line of direction of the expelling force will deviate more from that of direct opposition to the motion of rotation than at the equator; and that the deviation will increase with the latitude. For any latitude (1) the orbits described by the masses detached may have every variety of inclination from 3 to 90~; but the larger inclinations will result from an action of the expelling force exerted in a direction inclined under a smaller angle to the meridian in proportion as I is greater. If we conceive this force to act in some direction oblique to the surface, instead of tangentially, the velocity due to its action will be replaced by its tangential component, which is now to be taken equal to V. The aphelion of the orbit will also now be removed to a certain distance from the nebulous body, instead of being within its surface at the point of discharge. In view of what has now been stated, it may be seen that the actual Iaw of distribution of the inclinations aigh/t resvlt if the friequeacy of discharge uwere to decrease with the cangle included between the line of direction of the operatng forice acd the meridian. This theoretical result suggests, as the possible origin of the separation of frag. ments from the surface of the nebulous body, the flow of electric corrqents in all directions from points near the equators; similar to the currents we have conceived to be developed by planetary action on the sun's photospheric surface, and to give rise to the solar spots (293). Such currents, in proportion to the resistance they experience, would develop statical electricity, the repulsive action of which might occasion discharges in the direction of the radial currents and obliquely upwards. Upon this conception, if we consider the disengagements that may occur from any point of any one meridian. and bear in mind that the electric currents supposed may proceed, indifferently, from all points near the equator, it will be observed that the frequency of the detachment of fragments from the point in question will decrease in proportion as the radial current, or direction of the expelling force, makes a less angle with the meridian. For an arc of the equator (say 10~) will subtend, at the point considered, a greater angle in proportion as it is nearer to the meridian. At each point of the meridian, therefore, the liability to expulsive action should be greatest in directions nearly perpendicular to the meridian; for which directions the resulting orbital inclinations would be nearly equal to the latitude of the point. The effective directions of expulsion Mwould fall, for each latitude, I, on the meridian, between P1 and 90~. The complete result, for all points on the meridian, should then be that the number of inclinations would augment with the angle of inclination up to a certain large value of this angle, and then decrease; in accordance with the observed law. The agreement would apparently be more exact if, as would naturally be supposed, the frequency of discharge became less at the higher latitudes. We must suppose that the masses discharged which have since become perma nent members of the solar system, must, after being projected into space, have become condensed sufficiently before returning within the attenuated mass of' the nebulous body, to have pursued their course unaffected by the resistance of the medium traversed, except within certain limits. It appears that whether we consider the movements of the magnetic needle upon the earth, resulting from solar action, or the development of spots on the sun's surface by planetary influence, or the rise of nebulous envelopes from the nucleus of a large comet, under the operation of the sun, or the origination of cometary bodies, we are conducted, in each instance, to the primary conception of electric currents radiating from points near the equator of the body subject to external influence, as playing an important part in the production of the phenomena observed. Analogy would then lead us to infer that the exciing cause of the electric currents supposed to have furnished the operating cause of the detachment of cometary masses from the same nebulous body from which the planets have been derived, has been an action of the planets upon the surface of this body, similar to that which has ORIGIN OF SIDEREAL SYSTEMS. 439 been operative in the other cases. The same process may have continued in operation down to the present epoch, originating, in the later ages, meteorites rather than true cometary bodies. In fact we have seen that a continual process of dis. charge of nebulous magnetic matter from the sun is in operation in the present age. (Art. 293.) NOTE V. ORIGIN OF SIDEREAL SYSTEMS. We propose, in the present note, to develop very briefly a theory of the possi. ble evolution of all sidereal systems from primordial, rotating, nebulous masses. Fulndamental hypothesis, and circuList'ances of evolution. Let us assume that the component stars of every cluster were originally integrant portions of a vast nebulous body, and that this body had a motion of rotation about an axis. It is obvions that every portion of the rotating mass that might become detached would thenceforward tend to revolve independently about its centre in a certain orbit. Every such orbit would cross the plane of the equator in two points, or nodes, unless the detachment occurred in that plane, in which event the orbit would lie vwithin it. Again, if we conceive the separation to have taken place without bringing disturbing forces into play, the detached portion of the mass should have an initial motion in a direction parallel to the equatorial plane. and an initial velocity proportional to the cosine of the angular distance from that plane. If we suppose impulsive or repulsive forces to have been in energetic operation, we may approximately determine the nature and amount of their influence upon these initial circumstances, and thence upon the orbit subsequently described. Case I. A simultaneous disruption of the whole amass of the nebntlons bodyJ. We will regard the original nebulous body as sensibly spherical in form, and first conceive the disruption to have occurred at the same epoch in all its parts; or at epochs separated by small intervals in comparison with the vast duration of one rotation period. General results. The first result will be the formation of a globular cluster of stars, separated either by equal or unequal intervals of space. We will confine our attention to the case of equal intervals. All the stars of each spherical layer will then set out on their various courses at the same epoch. If we consider those which lie on any one meridian of the outer layer, their initial velocities will de. crease proportionally to their angular distance from the equator, and they will therefore set out in elliptic courses that will be more eccentric in proportion as this distance is greater. In case the disruption occurs at the period when the centrifugal force of rotation is equal to the force of gravity, at the equator of the nebu. lous body, the equatorial stars will move in circles, and the others in orbits of every degree of eccentricity, from a circle at the equator to a right line at the poles. The stars in question will all pass, at the end of one quadrant of a revolution, through the plane of the equator at various points of the line perpendicular to the plane of the meridian from which they set out. Another merician of the outer surface would give a new set of stars, with a new common line of nodes for all their orbits. As a general result then of the orbital motions communicated to the stars of the outer spherical layer, this layer will assusne the form of ac, oblate s7heroid, owith itAs shorter axis coincident with the line of the original axis of rolation. If we consider the next spherical layer of stars, these will all have taken up their independent movements sinultaneonsly with those at the surface, and as a general result the whole layer will assume the oblate spheroidal form, like the first. The same will be true of each successive layer, and the contractions will proceed simultaneously, while the order of the layers will remain unchanged. Periods of reols6tion. Since the initial velocities, from the surface inwarcd are proportional to the distance from the axis of rotation, and the attractive forces by which the stars are solicited at the outset, are proportional to the distance, from the centre, it follows that all stars proceeding from points at the same angular distance from the equator will accomplish their revolutions in the same period of time. ,40 A PPENDIX. This will be true whether the accelerating force soliciting each star, in its orbital motion, varies inversely as the square of the distance from the centre, or directly as the distance; and must be approximately true if the actual law of variation lies between these two. Now if in the contraction of the starry layers they all retained their spherical form, each star would be subject to the attraction of the same spherical mass, during an entire revolution, and therefore the law of variation of its accelerating force would be that of the inverse squares; but inl point of fact the layers in question contract into spheroids continually increasing in oblateness, and hence the acceleratino force must vary according to a less rapid law. Upon. the supposition that the law of the inverse squares obtains, the period of revolution of an equatorial star in a circle, would be to that of a star from very near the pole, in a very eccentric ellipse, as 2.8 to 1. On the other hand, if the force varied directly as the distance, the two periods w-ould be equal. The actual ratio should then lie between 2 8 and 1; and may be assumed to be not far fiom 2. Thlle equatorial star would complete a half revolution in an interval of time equal to the duration of one oscillation of an ideal celestial pendulum, having a length equal to the radius of the globular cluster at the epoch of its formation, and solicited by the force of gravitation in operation at the surface of the cluster. Either pole of the cluster would contract to the equatorial plane, anld attain to its limit of expansion on the opposite side, while such an ideal pendulum is completing a half oscillation, or thereabouts. Every such dynamical cluster has its cast cycle, at the close of which all its stars return approximately to their original positions. Such a state of things would be realized in about the interval of two periods of revolution of the equatorial stars, supposed to move in circles, or in four oscillations of tile representative pendulum. llic''eac'e of dlei',ity. As the contraction of the original globular cluster proceeds, the density continually increases, and attains, at any part of the equatorial plane: its greatest value at the epoch when thle stars are crossing the plane in that reogon. The conlensations not only augment, as the contraction goes on, in directions towards the equatorial plane, but also towards tile centre; since all the stars, except those moving in the plane of the equator, tend towards the centre, in their orbital miotions. ThIe greatest density wvill be attained at the centre; and at the epoch whein the polar stars have reached its vicinity. P.esibl' colli'ions. In all the movements of the stars of the supposed cluster, the only collisions that could directly ensue would be in those cases in which two stars set out from two points of a meridian, at the same distance fromn the equator, at the samine epoch. Each of the two stars should, at the end of a quarter revolution, reach the plane of the equator at the same point. Such exact coincidences of position and date of origin should, however, rarely occur. Butl frequent close approximllations of two or nmore stars may well occur, and eventuate in tl-e fobnatltion of dozcble or tlriple stars, revolving around each other.,S)l/,roii:l cblters. We are led, by the theoretical views that have now been presented, to reogard the oval nebulie, or spheroidal clusters, seen in the heavens, as oritiztal y/l/obez r clts'ters' inz somze of their dlffre'tlt sctces of p;2iIieroiddl co,;deesatioz. Upon this hypothesis they should in general be more condensed and more difficult of' resolution than existing globular clusters (as is observed to be the case). The amllount of oblateness, with the attendant condensation, would be an index of the age of the cluster. Present globular clusters wsould be just at the beginning of their first, or of a new grand cycle; and destined in future ages to pass through the continued series of spheroidal forms that we have signalized. LveeLcliies of briy/hlness in different p(.rts of slphereoidl cinLers. Instances of such inequalities, not resulting from a mere central condensation, are observable in Fig. 6,'late IV, and in Fig. 9, Plate V. They may be attributed to inequalities of density existing at certain stages of the contraction of the original globular cluster. For example, at the end of a half revolution of the polar stars they would be condensed about tile contracted poles, which would have passed to the opposite sides of the equatorial plane, and the other stars of each original spherical layer would be condensed in the resulting spheroidal layer, but in a decreasing degree towards the equator. This theoretical condition answers to the dumb-bell nebula (Fig. 9, Plate V). The differences observable in the distribution of the light on opposite sides of the equatorial plane (or of the larger axis of the faint elliptic outline) are such as might result from a want of entire correspondence in the epochs of thle initial orbital motions of the stars on opposite sides of that plane. Fig. 6, ORIGIN OF SIDEREAL SYSTEMIS. 441 Plate IV, answers to a similar period in the structural history of a cluster; but the inner layers have experienced a more marked condensation towards the centre. This would he the result if the initial velocities of the stars of these layers were, from some special cause, materially less than the normal values. Under the same circumstances these layers would be more nearly spherical in their form than in the normal type above considered. Case IAI. A simtlltaneous disruption of the nebtlous body along a limnited number of imeridians. We will now suppose the disruptive evolution of the starry masses to be confined to certain meridians, and the regions contiguous to them on either side, and enquire into the subsequent form and internal condition of the cluster. It will readily be seen that, since the initial velocities and the periods of revolution decrease from the equatorial to the polar stars, the stars proceeding from any one of these meridians, will, as they follow their natural orbits, take on collectively a spiral arrangement. This will be best seen when viewed perpendicularly to the equatorial plane. At the close of a half revolution of the polar stars, the spiral, thus viewed, would occupy the second quadrant of revolution of the entire set of stars considered. As the revolution proceeded, the angular extent of the spiral would continually increase. The stars proceeding from the meridian contiguous to that first supposed, would form a similar spiral contiguoas to that just considered; and the entire collection of stars setting out from the one meridional region of disruption would form a spiral band, increasing in width from its inner to its outer end. The similarly situated stars of the successive layers, proceeding inwards, would form shorter spirals, that would be combined with the others and add to the width of the spiral band. A similar spiral band would result from each of the other collections of detached stars. Spiral nebzlce. According to what has just been shown, the spiral formation is a natural consequence of a cotemporaneous evolution of stars from various points of the same meridian. The spiral arrangement of stars should therefore exist in every cluster, and be more or less discernible, unless the disruption was general and nearly simultaneous throughout the original nebulous mass. Accordingly spiral lines and fringes of stars are in many instances observable on the borders of spheroidal and irregular clusters. The theory of the origin of the true spiral nebulhe has been sufficiently indicated. The length of the spiral coils in Fig. 10, Plate V, indicates that the equatorial stars of the cluster have completed three-quarters of a revolution, and the polar stars one revolution and a half. The secondary condensation, on the right of the figure, may be ascribed to the circumstance of the stars that set out from points on the meridian near the equator, being at the present epoch in the vicinity of one of the nodes; three-quarters of a revolution having been completed. The nebula seems to consist of two spiral bands or coils, made up of an indefinite number of spiral filaments, or smaller bands. The line of sight is probably nearly perpendicular to the line of the two centres of condensation, and oblique to the equatorial plane. The great comparative dispersion of the filaments of the lower coil may be in part attributable to the detachment of stars beginning on one meridian, and extending gradually around to successive meridians. Case lIli. An irregular disOruption. If irregular deviations from the normal type of evolution that has been considered occur, *the result should be the formation of irregular clusters dillering more or less from true globular or spheroidal clusters. By reason of the want of correspondence in the epochs of detachment on different meridians of the nebulous body, such clusters should be less condensed, and with less regularity than the regular clusters. Case ]tV. A1 disru)ption beginning at the equator, and extending gradually towards the poles. W hen this deviation from the normal type occurs, the obvious result will be that the arrival of the stars at the equatorial plane will be delayed, in proportion to their angular distance from it, at the outset; and therefore that the contraction of each of the original spherical layers will take place more slowily The condensation towards the equatorial plane will also go on more slowly. The law of retardation of the dates of initial movement, for stars at increasing distances from the equator, may theoretically be such as to determine any known law of decrease of density along each spheroidal layer, from the equator to the axis. Systel;z o'f the JiiikJy Way. It is accordingly conceivable that the observed law of decrease of the density of the system of the milky way, as the distance from its principal plane increases, may have resulted from the operative cause just con 442 APPENDIX. sidereald. If the same law of evolution prevailed cotemporaneously, or approxi. mately so, throughout the mass of' the nebulous body from which this system is supposed to have been derived, the result would be the formation of spheroidal layers of stars in which the density would vary according' to a common law. The density would therefore decrease outwardly from the principal plane of the milky way at the same rate for stars at all distances. Struve found this to be nearly true for all stars beyond the 6th or 7th magnitude. l1iotion of revolution of the sun. Since the sun is now near the equatorial plane, and moving away from it (Art. 474), we must suppose that it has at least completed either one-quarter or three-quarters of a revolution in its vast orbit. The position of the centre of the system is too imperfectly known (Art. 478) to make it possible to determine with certainty which of the two nodes it is now leaving. Taking the average velocity of the sun at 42 miles per second, as given in the text, and supposing the distance of the centre of the system to be equal to the exterior limit of stars of the 6th magnitude, the period of revolution of the sun should be 35 millions of years. WIMidler's estimate of the distance of the centre of the system of the milky way places it a little beyond the exterior limit of stars of the 5th magnitude. The above larger estimate answers nearly to Struve's determination of the sun's orbital velocity (Art. 448). PcGSti:c: ear features of the MilIky l1azy. The variable breadth of the belt of the milky way, its bifurcation, and alternations of bright and dark patches, may have proceeded from a want of correspondence in the dates of evolution of stars on different meridians of the original nebulous mass, as well as abnormal interruptions of the process of separation at certain parts of particular meridians. Thus, if on a certain meridian, the separation into stars should not have occurred for a certain distance from the equator, on either side, it would follow that just before or after - of a revolution of the stars that set out from the points nearest the equator, the stars from greater latitudes would be concentrated upon two points at a short distance from the equator, on either side. If the samne initial circumstances prevailed over a series of meridians, there would be formed two bands of greatest condensation at a certain distance from the equator, on opposite sides. The same bands would manifest themselves before and after I of a revolution. Primeary coedition, and subsequent 1piocesses of chanlye, of fraygments disueziied from the primitive nebatloss mass. Since the parts of any such fragment unequally distant from the axis must have had, at the time of separation, unequal velocities of rotation, it must have taken up a rotation about an axis of its own, and tended to assume the form of a sphere, or spheroid somewhat flattened at the poles. It should therefore, at some subsequent date, have broken up into a cluster of stars, or into a planetary system revolvingo around a central sun. If the mass detached should have been of comparatively great extent, it may have separated into a combination of stars and clusters of stars, as in the Magellanic Clouds; or into irregular beds of stars, as in the irregular nebulae. The concentration attending the formation and subsequent contraction of such systems should have occasioned a vacuity of stars in the spaces contiguous to them. Isresolvable nelulce. If, as we have already been led to suppose, the process of evolution of the system of the milky way from a, primitive nebulous mass, extended gradually from the equator towards the poles, the vast nebulous mass left detached at the outer polar regions, and subject to peculiar conditions, may have become disunited into large masses, from which clusters have been derived that are now at an earlier stage of development, and at a greater distance than the telescopic stars and clusters. Annular nebulce, which are the rarest objects in the heavens, may have resulted from the matter of the polar regions of a nebulous body being mostly drawn to surrounding points of condensation, or not having' yet condensed into true stars, or into stars comparatively minute. Planetar/y nebulce probably belong to the same type of development as annular nebulae; since some of them have been resolved into annular nebulae. Their equable light may result from the process of star-formation having penetrated only a certain depth into the original nebulous mass. Genzeral considerations. We may conclude from the previous theoretical discussion, that if we assume all systems of stars to have been derived, by separation, from rotating nebulous bodies of vast extent, according to one or the other of a ORIGIN OF SIDEREAL SYSTEMS. 443 certain small number of types of evolution, the forms and internal conditions that would be inevitably passed through, in the progress of ages, would be the counterparts of the various forms and apparent structural conditions of the clusters and nebula-, actually observed. We may suppose, it is true, in explanation of the single case of the spheroidal clusters, that the rotating bodies took on a decidedly spheroidal form before disintegration, and that the derived clusters have now, sensibly; their original form. Upon this view we must still admit that these clusters are destined, in the lapse of future ages, to pass through changes similar to those we have deduced for globular clusters. But spiral, and some other forms of nebula, cannot but have passed through vast ages of development, and in the light of this indication of the age of the heavens, it seems improbable that spheroidal clusters should be of comparatively recent origin. As to the real nature of the process of separation of starry fragments from rotating nebulous masses, we are led, on physical grounds, to conceive that it must have consisted in a concentration upon special points of the mass at certain favorable epochs, rather than in a violent separation. It would seem that such a process beginning at any part of the nebulous body, should tend to extend indefinitely through it. But it is to be observed that the propagation of a force through such a mass would of itself require a vast period of time. It is possible that the want of correspondence in the epochs of separation, in different parts of the body, we have inferred existed in the development of the system of the milky way, and to a less extent also in spiral nebulae, may have resulted from the far greater comparative size, in these instances, of the original nebulous mass. TABLE I. Latitudes and Longitudes of Places. Names of Places. Latitude. Longitude from ILong. from Longitude from Greenwich in P src. Greenwich in Washington Time. in Time. h. m. s. h. m. s. Abo, Ohs............. 60 26' 56'.8 N. 22~ 17' 11.4 E. -1 29 8.8 - 6 37 20.0 Albany, Obs...........42 39 50.0 N. 73 44 39.0W. +4 54 56.6/- 0 13 12.6 Altona, Obs...........53 32 45.3 N. 9 56 32.3 E. —0 39 46.1- 5 47 57.4 Ann Arbor, Obs.......42 16 48.0 N. 83 43 3 0W.W +5 34 52.2 + 0 26 41.0 Astor Point, Oregon... 46 11 27.6 N. 123 49 31.7 W. +8 15 18.1 + 3 7 6.9 Athens, Obs..........37 58 20.0 N. 23 43 47.8 E. -1 34 55.2 - 6 43 6.4 Baltimore, Waish. Mt... 39 17 47.8 N. 76 36 38.6W. +5 6 26.6- 0 1 45.4 Berlin, Obs............52 30 16.7 N. 13 23 52.8 E. -0 53 35.5 — 6 1 46.7 Boston, State House....42 21 27.6 N. 71 3 30.0W. +4 44 14.0 - 0 23 57.2 Cambridge, Obs........42 22 48.6 N. 71 7 24.9W. +4 44 29.7- 0 23 41.5 Cape of Good Hope, Obs 33 56 3.0 S. 18 28 45.0 E. -1 13 55.0 - 6 22 6.2 Cape Horn............55 58 41 S. 67 10 53.0W. +4 28 43.5- 0 39 27.7 Charleston, St. Mich.'sCh 32 46 33.1 N. 79 55 37.6W. +5 19 42.5 + 0 11 31.3 Copenhagen, Obs.......55 40 53.0 N. 12 34 57.0 E. -0 50 19.8 - 5 58 31.0 Dorpat, Obs...........58 22 47.1 N. 26 43 23.4 E. -1 46 53.6- 6 55 4.8 Dublin, Obs...........53 23 13.0 N. 6 20 30.0W. t0O 25 22.0 - 4 42 49.2 Edinburgh, Obs........55 57 23.2 N. 3 10 45.0W. +0 12 43.0 - 4 55 28.2 Galveston, Cath.......29 18 17.3 N. 101 33 33.0W. +6 46 14.2 + 1 38 3.0 Gotha, Obs............50 56 5.2 N. 10 43 54.9 E. -0 42 55.7 -- 5 51 6.9 Gottingen, Obs.........51 31 47.9 N. 9 56 31.5 E. -0 39 46.1 - 5 47 57.3 Greenwich, Ohs........51 28 38.0 N. 0 0 0.0 0 0 0.0 -- 5 8 11.2 Hamburg, Obs.........53 33 5.0 N. 9 58 23.4 E. —0 39 53.6 - 5 48 4.8 Hamilton Coll., Obs.... 43 3 16.5 N. 75 24 16.8W. +5 1 37.1- 0 6 34.1 Kazan, Obs...........55 47 23.1 N. 49 6 36.6 E.-3 16 26.3 — 8 24 37.5 Kdnigsberg, Obs.......54 42 50.7 N. 20 30 5.4 E.-3 22 0.4- 630 11.2 Liverpool, Lassell Obs... 53 25 28.0 N. 2 54 40.5W. +0 11 38.7- 4 56 32.5 London, Obs.......... 51 31 29.8 N. 0 9 16.5W. +0 0 37.1-5 7 34.1L Madras, Obs...........13 4 8.1 N. 80 14 15.0 E. —5 20 57.0 — 10 29 8.2 Madrid, Obs...........40 24 27.7 N. 3 41 21.0W. +0 14 45.4 — 4 53 25.8 Marseilles, Obs........43 17 49.0 N. 5 22 14.8 E. —0 21 29.0 - 5 29 40.2 Milan, Obs............45 28 0.7 N. 9 11 39.6 E. -0 36 46.6- 544 57.8 Moscow, Ohs..........55 45 19.8 N. 37:'4 14.4 E. — 2 30 17.0 - 7 38 28.2 Mount Desert, Maine... 44 21 3.9 N. 68 13 15.5W. +4 32 53.3- 0 35 17.9 Naples, Obs...........40 51 46.6 N. 14 14 42.9 E. -0 56 58.9- 6 5 10.1 New Haven, Sheff. Obs.. 41 18 36.5 N. 72 55 30.0W. +4 51 42.0- 0 16 29.2 New Orleans..........29 57 45.0 N. 90 6 49.0W. +6 0 27.0 + 0 52 15.8 New York, City Hall... 40 42 43.2 N. 74 0 3.1W. +4 56 0.2 - 0 12 11.0 Palermo, Obs..........38 6 44.0 N. 13 21 2.6 E. — 0 53 24.2- 6 135.4 Paramatta, Obs....... 33 48 49.8 S. 151 1 33.7 E. -3 39 31.4 - 8 47 42.6 Paris, Obs............ 48 50 13.2 N. 2 20 9.4 E. —0 9 20.6- 5 17 31.8 St. Petersburg, Obs.... 59 56 29.7 N. 30 18 22.2 E. — 2 1 13.5 - 7 9 24.7 Philadelphia, Obs......39 57 7.5 N. 75 9 23.4W. —5 15 44.8 — 0 733.6 Point Venus, Otaheite.. 17 29 21.0 S. 149 28 55.0W. +9 57 56.0 + 449 45.0 Princeton, Seminary... 40 20 40.0 N. 74 39 34.3W. +4 58 38.3 - 0 9 32.9 Pulkowa, Obs.........59 46 18.7 N. 30 19 39.9 E. — 2 1 18.7 - 7 9 29.9 Quebec, Obs...........46 48 30.0 N. 71 12 15.0W. +4 44 49.0 — 0 23 22.2 Rio de Janeiro, Obs....22 53 51.0 S. 43 3 39.0W. +2 52 14.6 — 2 15 56.6 Rome, Obs............41 53 52.2 N. 12 28 40.5 E.-0 49 54.7 — 5 58 5.9 San Francisco, Tel. Hill.. 37 47 59.2 N. 122 23 19.4W. +8 9 32.5 + 3 1 21.3 Santiago de Chile, Ohs.. 33 26 25.4 S. 70 38 14.5W. +4 42 33.0- 0 25 38.2 Savannah, Exch........32 4 53.4 N. 81 5 14.3W. +5 24 20.9 + 0 16 9.7 Stockholm, Obs........59 20 31.0 N. 18 3 42.0 E.I-1 12 14.8 — 6 20 26.0 Vienna, Obs...........48 12 35.5 N. 16 23 7.9 E. -1 532.5- 613 43.7 Washington, Obs......38 53 39.3 N. 77 2 48.0W. + 5 8 11.2 — 0 0 0.0 2 TtABLE II. Elements of' tlhe Planetary Orbits. Epoch, January 1, 1850, mean noon at Paris. Planet's Name. the Inclination to, r Longitude of Sec.Var. Longitude of Sec Var' Plthe Ecliptic:n to Se a AscendingNode. Perihelion. Mercury.....7~ 0' 8" + 6'.3 46~ 33' 9" 71' 4" 75: 7' 14" 93' 11' Venus... 3 23 31 + 4.575 19 52 54 4 129 27 14 8226 Earth 100 21 21 102 50 lars........ 1 51 5 - 2.4 48 23 43 46 39 333 17 54 110 24 Jupiter...... 1 18 40 -23 98 54 20 57 14 11 54 53 94 49 Saturn.......2 29 28 -15 112 21 44 51 10 90 6 12 115 55 Uranus.... 0 46 30 + 3 73 14 14 23 39 168 16 45 87 32 Neptune 1 46 59 130 6 52 47 14 37 M. Distance Mean Distance from Sun, or from Sun in Eccentricity Sec. Variation. Semi-Axis. Miles. Mercury......... 0.3870987 35,353,000 0.2056048 + 0.000020294 Venus................ 0.7233322 66,060,000 0.0068433 -0.000053843 Earth................. 1.0000000 91,328,000 0.0167711 -0.000042582 Mars................. 1.523691 139,156,000 0.0932611 + 0.000095284 Jupiter............... 5.202798 475,161,000 0.0482388 + 0.000159350 Saturn................ 9.538852 871,164,000 0.0559956 -0.000312402 Uranus................ 19.182639 1,751,912,000 0.0465775 -0.000025072 Neptune.............. 30.03697 2,743,216,000 0.0087195 Mean Longitude M. Sidereal Period Motion in Mean M ean Daily MoR~eat the Epoch. in Mean Solar Long. in 1 yr. of' tion in Days. 365 days. Longitude. d. Mercury.......... 327~ 15' 20" 87.9692580 530 43' 3",40 5' 32".6 Venus......... 245 33 15 224.7007869 224 47 30 1 36 7.8 Earth.... 280 46 43.5 365.2563744 359 45 41 0 59 8.33 Mars............ 83 40 31 686.9796458 191 17 9 0 31 26.7 Jupiter.......... 160 1 20 4332.5848212 30 20 32 10 4 59.3 Saturn........... 14 50 41 10759.2198174 12 13 36 10 2 0.6 Uranus........ 28 26 41 30686.8208296 4 17 45'0 0 42.4 Neptune....3..... 35 8 58 60126.72 2 11 58 10 0 21.7 TABLE III. —Elements of.lfoon's Orbit. Epoch, January 1, 1801, mean noon at Paris. Mean inclination of orbit................................. 50 8' 40" Miean longitude of node at epoch.......................... 13 53 17. 7 Mean longitude of perigee at epoch........................ 266 10 7.5 Mean longitude of moon at epoch.......................... 118 12 8.3 Mean distance from earth................................ 60.2665590 Eccentricity.......................................... 0.05490807 d. h. m.s. d. Mean sidereal revolution.....................27 7 43 11.5 = 27.32166142 Mean tropical:......................27 7 43 4.7 = 27.32158242 5Mean synodical ".....................29 12 44 2.9 = 29.53058872 Mean anomalistic...................... 27 13 18 37.4 27.55459950 Mean nodical..................... 27 5 5 36.0 = 27.21222222 d. d. Mean revolution of nodes: sidereal.... = 6793.432; tropical.... = 6798.33557 Mean revolution of perigee: sidereal... = 3232.57534; tropical.. = 3231.4751 TABLE IV. Dia(le'ers, TFolzumes, Mlfasses, etc., of Sun, lfoon, and Planets. Apparent I)iameter. Equatorial Equatorial Diameter in Volum e. Least. Dt ance. Greatest. Dia t iles. Mercury.. 4".5 6".7 12".9 0.3732 2,958 0.0518 Venus... 9.7 17.0 66.3 0.9525 7,549 0.8686 Earth.... 1.0000 7,925.6 1.0000 Mars..... 4.1 11.1 30.1 0.6201 4,915 0.2345 Jupiter... 30.8 37.2 50.6 11.1401 88,294 1303.91 Saturn... 14.6 16.1 20.3 9.0621 71,823 667.54 Uranus... 3.5 3.9 4.3 4.1864 33,124 73.369 Neptune... 2.6 2.7 2.9 4.5383 35,910 93.470 Sun.... 31' 32 32' 3.4* 32' 36 107.263 850,123 1,240,285.0 aoon.... 28 48 31 7.0 33 32 0.2725 2,160t 0.0203 Mass. Density. Gravity at Sidereal Lightand Equator. Rotation. i eat. h. m. s. Mercury.............. 2.020 0.751 24 5 28 6.680 Venus................ 4. 4)24u6 0.903 0.865 23 21 24 1.911 Earthl................. 340Oo 1.000 1.000 23 56 4 1.000 Mars.................. 2 0.447 0.273 24 37 20.431 Jupiter................ Tl-a 0.229 2.410 9 55 26.037 Saturn............. -- 0.134 1.089 10 29 17.011 Uranus................ 24 0o 0.178 0.746.603 Neptune............... 0.179 0.812 d. h. m.001 lun................... 1 0.253 27.292 25 4 29~ SMoon.................. 2~tbs7766 0.602 0.164 27 7 43 TABLE V. Elemernts of the Retrograde Illotion of the Planets. Are of Retro- Duration of Retro- Elongation at the Synodic gradation. gradation. St4tions. Revolution. d. h. d. h. d. Mercury. 9~ 22' to 15~ 44' 23 12 to 21 12 140 49' to 20~ 51' 115.8775 Venus....14 35 to 17 12 40 21 to 43 12 27 40 to 29 41 583.9214 Mars.... 10 6 to 19 35 60 18 to 80 15 128 44 to 146 37 779.9364 Jupiter... 9 51 to 9 59 116 18 to 122 12 113 35 to 116 42 398.8841 Saturn... 6 41 to 6 55 138 18 to 135 9 107 25 to 110 46 378.0919 Uranus.. 3 36 151 103 30 369.6563 Neptune. 367.4888 * The value 32' 0" used in the text (269) is the above value corrected for irradiation, and agrees with that obtained from observations of the transits of Mercury over the sun's disc. 1 The value of the moon's diameter (2161.6 m.) obtained in the text, is probably about 2 miles too large, as, according to Airy, a correction of 2" should be applied to the moon's measured diameter for the effect of irradiation, and 1" answers to about 1 mile. t This is the mass of Mercury adopted by Le Verrier. Many astronomers still retain Encke's determination, which is T4so —-T. This gives for the density of Mercury 1.246. ~ This is Faye's recent determination. According to Carrington, the most pro bable value is 24d. 23h. 30m. Spdrer's value is 25d. 5h. t7m. 4 TTABLE VI. Elements of the Orbits of the satelites. The distances are expressed in equatorial radii of the primaries. I. Satellites of JpPifer. Inclinati(,n to Epoch Mass; that of Satellites. Men Sidereal evolution. ) it of Foch ass; that of Distance. Jeouf El'ts. Jupiter-. d. h. m. s. 1.............. 6.04853 1 18 27 38.506 3~ 5' 30" Jan. 1, 0.00)017328 2.............. 9.62347 3 13 14 36.393 Variable. 1801. 0.000023235 3.............. 15.35024 7 3 42 33.362 Variable. G. T. 0.000088497 4.. 26.99835 16 16 11 49.702 2 58 48 0.000042659 IT. Scatei'tes of atturn. Mean Sidereal M1. Long. Eccentricity and Epoch Name and Order. Distance. Revolution. at the Epoch. Perisaturnium. of El'ts. d. h. m. s. 1. Mimas.... 3.3607 0 22 37 22.9 256~ 58' 48" 1790.0 2. Enceladus.. 4.3125 1 8 53 6.7 67 41 36 1836 0 3. Tethys.... 5.3396 1 21 18 25.7 313 43 48 0.04(?); 54~(?) Ditto. 4. Dione..... 6.8398 2 17 41 8.9 327 40 48 0.02 (?); 42 (?) Ditto. 5. Rhea...... 9.5528 4 12 25 10.8 353 44 0 0.02(?); 95 (?) Ditto. 6. Titan...... 22.1450 15 22 41 25.2 137 21 24 0.0293; 256 38' 1830.0 7. HIyperion 28 21 7 7 Apsides of T'itan have a 7. Hyperion 28 21 7 ~motion in long. of 30' 8. Iapetus.... 64.3590 79 7 53 40.4 269 37 48 28"per an. 1790.0 The longitudes are reckoned in the plane of the ring from its descending' nodo with the ecliptic. The first seven satellites move in or very nearly in its plane; the orbit of the 8th lies about half way between the plane of the ring and that of the planet's orbit. III. Saltelites of U[rcnus. Satellites. Satellites. M. Die. Sidereal Passage through tance. Revolution. Asc. Node, G. T. Inclination to Ecliptic. d. h. m. The orbitR are in1. Ariel........ 7.44 2 12 28 dined at an angle of 2. Umbriel...... 10.37 4 3 27 about 79~ to the eclip3............ 13.12 5 21 25 d. h. m.`tic in a plane whose as4. Titania....... 17.01 8 16 56 1787, Feb. 16 0 10lconding node isin long. 5........... 19.85 10 23 3 165~ 30' (Equinox of 6. Oberon...... 22.75 13 11 7 1787, Jan. 7 0 28 1798). Their motion is 7.............. 45.51 38 1 48 retrograde. Their orbits 8.............. 91.01 107 16 39 are nearly circular. IV. Satellite o Neptune. Period, 5 8;7 d.; AM. Distance, 12 radii of Planet. TABLE VII. Saturn's Ring. Outer diameter of outer ring..................40".095....... 169,341 miles. Inner diameter of outer ring.............................. 149,060 " Breadth of outer ring................................... 10,149 Breadth of inner ring.................................... 16,484 " Interval between rings.................................... 17723 Breadth of double ring.................................... 28,356 Distance of ring from planet.............................. 18,327 " Thickness of the rings not exceeding........................ 210 " TABLE II (a). 5 The Planetoids. N'ames, particulars of discovery, mean distances, etc. Mean Sidereal No. Name. Date of Discovery. Discoverer. Distance. Period. Eccentricity. Yrs. -. Ceres...... 1801, Jan. 1. Piazzi. 2.7660 4.600 0.08024 2. Pallas..... 1802, March 28. Olbers. 2.7700 4.6J0 0.23969 3. Juno....... 1804, Sept. 1. Harding. 2.6687 4.362 0.25590 4. Vesta..... 1807, March 29. 01lbers. 2.3607.3.627 0.09012 5. Astroea.... 1845, Dec. 8. IHencko. 2.5775 4.136 0.18999 6. Hebe...... 1847, July 1. FHencke. 2.4254 3.777 0.20115 7. Iris........ " Aug. 13. Hind. 2.3862 3.686 0.23125 8, Flora......' Oct. 18. Hind. 2.2014 3.266 0.15670 9. Metis...... 1848, April 25. Graham. 2.3862 3.686 0.19320 10. Hegeia..... 1849, April 12. De Gasparis. 3.1494 5.589 0.10056 11. Parthenope. 1850, May 11. De Gasparis. 2.4526 3.841 0.09888 12. Victoria.... "Sept. 13. Hind. 2.3344 5.567 0.21890 13. Egeria..... " Nov. 2. De Gasparis. 2.5756 4.133 0.08775 14. Irene...... 1851, May 19. Hind. 2.5895 4.167 0.16525 15. Eunomia... July 29. De Gasparis. 2.6429 4.297 0.18801 16. Psyche..... 1852, March 17. De Gasparis. 2.9263 5.006 0.13575 17. Thetis..... " April 17. Luther. 2.4737 3.890 0.12686 18. Melpomene. " June 24. Hind. 2.2958 3.479 0.21723 19. Fortuna.... " Aug. 22. Hind. 2.4414 3.815 0.15792 20. Massilia.... " Sept. 19. De Gasparis. 2.4093 3.740 0. 14383 21. Lutetia... " Nov. 15. Goldschmidt. 2.4354 3.081 0.16204 22. Calliope.... " Nov. 16. Hind. 2.9091 4.962 0.10361 23 Thalia..... " Dec. 15. Hind. 2.6250 4 263 0.23180 24. Themis... 1853, April 5. De Gasparis. 3.1420 5.570 0.11701 25. Phocea..... "April 6. Chacornac. 2.4023 3.723 0.25335 26. Proserpine.. " May 5. Luther. 2.6556 4.329 0.08752 27. Euterpe.... " Nov. 8. Hind. 2.3473 3.596 0.17290 28. Bellona..... 1854, March 1. Luther. 2.7784 4.631 0.15039 29. Amplhitrite.. " larch 1. Marth. 2.5548 4.084 0.07238 30. Urania.... " July 22. Hind. 2.342 3.635 0.12718 31. Euphrosyne " Sept. 1. Ferguson. 3.1561 5.607 0.21601 32. Pomona.... " Oct. 26, Goldschmidt. 2.5831 4.160 0.08240 33. Polyhymnia. " Oct. 28. Chacornac. 2.8646 4.848 0.33769 34. Circe...... 1855, April 6. Chacornac. 2.6839 4.397 0.10961 35. Leucothea..' April 19. Luther. 3.0060 5.215 0.21363 36. Atlanta... " Oct. 5. Goldschmidt. 2.7487 4.557 0.29788 37. Fides...... " Oct. 5. Luther. 2.6422 4.295 0.17489 38. Leda...... 1856, Jan. 12. Chacornac. 2.7399 4.535 0.15552 39. Laetitia..... " Feb. 8. Chacornac. 2.7710 4.613 0.11081 40. Harmonia.. " March 31. Luther. 2.2679 3.415 0.04608 41. Daphne.... May 22. Goldschmidt. 2.7674 4.605 0.27034 42. Isis........ " May 23. Pogson. 2.4401 3.812 0.22566 43. Ariadne.... 1857, April 15. Pogson. 2.20.38 3.272 0.16756 44. Nysa..M... " May 27. Goldschmidt. 2.4242 3.774 0.14933 45. Eugenia... " June 28. Goldschmidt. 2.7159 4.476 0.08200 29 6 TABLE II (a)-CoNTINUED. Men Sidereal No. Name. Date of Discovery. Discoverer. DiManc Period. Eccentricity. Yrs. 46. Hestia..... 185, Aug. 16. Pogson. 2.5178 3.995 0.16184 47. Melete...... Sept. 9. Goldschmidt. 25976 4.189 0.23686 48. Aglaia...... " Sept. 15. Luther. 2.8831 4.896 0.12788 49. Doris...... " Sept. 19. Goldschmidt. 3.1044 5.470 0.07580 50. Pales... " Sept. 19. Goldschmidt. 3.0861 5.421 0.23783 51. Virginia.... " Oct. 4. Ferguson. 2.6486 4 310 0.28695 52. Nemausa... 1858, Jan. 22. Laurent. 2.3779 3.667 0.06285 53. Europa.... Feb. 6. Goldsehlmidt. 3.0999 5.458 0.00450 54.Cypso.... April 4. Luther. 2.6102 4.217 0.21 2639 55. Alexandra.. " Sept. 10. Goldschmidt. 2.7076 4.553 0.19941 56. Pandora... " Sept. 10. Searle. 2.7692 4.608 0.13895 57. Mnemosyne 1859, Sept. 22. Luther. 3.1597 5.616 0.10752 58. Concordia.. 1860, March 24. Luther. 2.6979 4.43 1 0.04103 59 Dlanae..... " Sept. 9. Goldschmidt. 2.9746 5.131 0.16308 60. Olympia... " Sept. 12. Chacornac. 2.7147 4.472 0.11883 61. Erato...... " Sept. 14. Forster. 3.1296 5.537 0.16964 62. Echo...... " Sept. 14. Feroguson. 2.3939 3.729 0.1 8543 63. Ausonia 1861, Feb. 10. De Gasparis 2.3972 3.712 0.12732 64. Angelina.... " March 4. Tempel. 2.6783 4.385 0.12482 65. Cybele..... March 8. Tempel. 3.4205 6.658 0.12030 66. MAnia..... " April 9. I. P. Tuttle. 2.6539 4.322 0.15422 67. Asia....... " April 17. Pogson. 2.4209 3.769 0.18443 68. Ilesperia.. " April 29. Schiaparelli. 2.9949 5.186 0.17452 69. Leto.......' April 29. Luther. 2.7748 4.622 0.18566 70. Panopea... " May 5. Goldschmidt. 2 6132 4.2,) 4 0.18309 71. Feronia....'" May 29. C. HI. F. Peters 2.2660 3.411 0.11977 72. Niobe..... " Aug. 13. Luther. 2.7555 4.574 0.17374 73. Clytie.....1862, April 7. Tuttle. 2.6648 4.350 0.04424 74. Galatea.... " Aug. 30. Temple. 2.77 77 4.629 0.23820 75. Eurydice... " Sept. 22... C.H.F. Peters 2.6698 4.262 0.30690 76 Freia...... " Oct. 21. D'Arrest.. 3.3877 6 235 0.187772 77. Frigga..... Nov. 15. C. H.F. Peters 2.67 19 4.368 0.13582 7 8. Diana..... 1863, Mlarch 15. Luther. 2.6228 4.248 0 20549 79. Eurvnome.. " Sept. 14. WVatson. 2.44:33 3 819 0.19509 80. Sappho..... 1864, May 2. Pogson. 2.2963 3.480 0.20022 81. Terpsichore. " Sept 30. Tempel. 2.8563 4.827 0.21175 82. Alemene... N" ov. 27. Luther. 2.7603 4.586 0.22599 83. JBeatrix.... 1865. April 2). De Gasparis. 2.4287 3.785 0.08410 84. Clio....... " Aug. 25. Luther 2.3ti74 3.6i43 0.23754 85. To........ " Sept. 19. C. H. F. Peters 2.6594 4.337 0.19395 86. Semele.... 1866, Jan. 4. Tictjen. 3 0908 5.434 0.20493 I 87. Sylvia. " May 17. Pogson.................. 88. Thisbe..... June 15. C. H. F. Peters 2.75(03 4.561 0.1(667 89. Aug. 6. 2.5341 4.032 0.20499 TABLE VIII. 7 Mean Astronomical Refractions. Barometer 30 in. Thermometer, Fah. 50". Ap.Alt. Refr. V p. Alt. Refr. Ap. Alt. Refr. Alt. Refr. 00 0' 33'51 4~ 0' 11 52" 12 0' 4' 28.1" 420 1'.4.6 5 32 53 10 11 30 10 4 24.4 43 1 2.4 10 31 58 20 11 10 20 4 20.8 44 1 0.3 15 31 5 30 10 50 30 4 17.3 1:5 0.58.1 20 30 13 40 10 32 40 4 13.9 46 56.1 25 29 24 1 50 1015 50 4 10.7 47 54.2 30 28 37 5 0 9 58 13 0 4 7.5 48 52.3 35 2751 10 942 10 4.4 49 50.5 40 27 6 20 9 27 20 4 1.4 50 48.8 45 26 24 30 9 11 30 3 58.4 r1 47.1 50 25 43 40 8 58 40 3 55.5 52. 45.4 55 25 3 50 845 50 3 52.6 53 43.8 0 24 25 1 6 0 8 32 14 0 3 49.9 54 42.2 5 23 48 1 10 8 20 10 3 47.1 55 40.8 10 2313 20 8 9 20 3 44.4 56 39.3 15 22 40'30 7 58 30 3 41.8 57 37.8 20 22 8 40 7 47 40 3 39.2 58 36.4 25 21 37 50 7 37 50 3 36.7 59 35.0 30 21 7 7 0 7 27 15 0 3 34.3 60 33.6 35 20 38 110 7 17 15 30 3 27.3 61 32.3 40 20 10 20 7 8 16 0 3 20.6 62 31.0 45 19 43 30 6 59 16 30 3 14.4 63 29.7 50 19 1 7 40 6 51 17 0 3 8.5 64 28.4 55 18 52 50 6 43 17 30 3 2.9 65 27.2 2 0 18 29 8 0 6 35 1 0 2 57.6 66 25.9 5 18 5 10 6 28 19 2 47.7 67 24.7 10 17 43 20 21 20 2 38.7 68 23.5 15 17 21 30 6 14 21 2 30.5 69 22.4 20 17 0 40 6 7 22 2 23.2 170 21.2 25 16 40 50 6 0 23 2 16.5 71 19.9 30 16 21 ii 9 0 5 54 24 2 10.1 72 18.8 3.5 16 2 11 10 5 47 1 25 2 4.2 73 17.7 40 15 43 20 5 41 ii 26 1 58.8 74 16.6 45 15 25 30 5 36 27 1 53.8 75 15.5 50 15 8 4( 5 30 28 1 49.1 76 14.4 55 14 51 50 i 5 25 /1 29 1 44.7 77 13.4 3 0 14 35 1 0 0 5 40 30 1 40.5 78 12.3 5 14 19 1 0 5 1 5 31 1 36.6 79 11.2 10 14 4 20 5 1n i' 32 1 33.0 80 10.2 15 13 50 30 5 5 l 33 1 29.5 81 9.2 20 13 35 1 40 5 ( 34 1 26.1 82 8.2 25 13 21 1 50 4 56 35 1 23.0 83 7.1 30 13 7 11 0 4 51 36 1 20.0 84 6.1 35 12 53 lj 10 4 47 3 37 1 17.1 85 5.1 40 12 41 20 4 43 3 3I 1 14.4 86 4.1 45 12 28! 30 4 39 39 1 1 1 87 3.1 50 12 16 i 40 4 35 40 1 9.3 88 2.0 f55 12 3 1 50 4 31 L1 1 6.9 89 1.0 8 TABLE IX Corrections of Mean Refractions. Ap. Alt diffor dif. fr I ApAlt.i Dif. for Dif. for Ap. At. for D if. for A D if. for Dif. for 4 —i 11. 1OF. +-4IB. -10 F. -+FLu -]OF. — lB. 10 F. 0 o o. 00 74 8.1 4 0 24.1 1.70 12 0 9.00 0.556 42 2.16 0.130 5 71 7.6 10 23.4 1.64 10 8.86.548 43 2.09.125 10 69 7.3 20 22.7 1.58 20 8.74.541 44 2.02.120 15 67 7.0 30 22.0 1.53 30 8.63.533 45 1.95.116 20 ]o 6.7 40 21.3 1.48 40 8.51.524 46 1.88.112 25 63 6.4 50 20.7 1.43 50 8.41.517 1 47 1.81.108 30 61 6.1 5 0 20.1 1.38 13 0 8.30.509 48 1.75.104 35 59 5.9 10 19.6 1.34 10 8.20.503 49 1.69.101 40 58 5.6 20 19.1 1.30 20 8.10.496 50 1.63.097 45 56 5.4 | 30 18.6 1.26 30 8.00.490 51 1.58.094 50 55 5.1 40 18.1 1.22 40 7.89.482 52 1.52.090 55 53 4.9 50 17.6 1.19 50 7.79.476 53 1.47.088 1 0 52 4.7 6 0 17.2 1.15 14 0 7.70.469 54 1.41.085 5 50 4.6 10 16.8 1.11 10 7.61.464 55 1.36.082 10 49 4.5 20 16.4 1.09 20 7.52.458 56 1.31.079 15 48 4.4 30 16.0 1.06 30 7.43.453 57 1.26.076 20 46 4.2 40 15.7 1.03 40 7.34.448 58 1.22.073 25 45 4.0 50 15.3 1.00 50 7.26.444 59 1.17.070 30 44 3.9 7 0 150 0.98 /15 0 7.18.439 60 1.12.067 35 43 3.8 10 14.6.95 115 30 6.95.424 61 1.08.065 40 42 3.6 20 14.3.93 16 0 6.73.411 62 1.04.062 45 4-0 3.5 30 14.1.91 16 30 6.51.399 63.99.060 50 39 3.4 40 13.8.89 1 17 0 6.31.386 64.95.057 55 39 3.3 50 13.5.87 17 30 6.12.374 65.91.055 2 0 38 3.2 8 0 13.3.85 18 0 5.94.362 66.87.0,52 5 37 3.1 10 13.1.83 19 5.61.340 67.83.050 10 36 3.0 11 20 12.8.82 1 20 5.31.322 68.79.047 15 36f 2.9 30 12.6.80 21 5.04.305 69.75.045 20 35 2.8 40 12.3.79 22 4.79.290 70.71.043 25 1 34 2.8 50 12.1.77 23 4.57.276 71.67.040 30 33 2.7 9 0 11.9.76 1124 4.35.264 72.63.038 35 33 2.7 10 11.7.74 25 4.16.252 73.59.036 40 32 2.6 20 11.5.73 26 3.97.241 74.56 033 45 32 2.5 30 11.3.72 127 3.81.230 75.52.031 50 31 2.4 40 11.1.71 28 3.65.219 76.48.029 55 30 2.3 50 11.0.70 29 3.50.209 77 45.027 3 0 30 2.3 10 0 10.8.69 30 3.36.201 78.41.025 5 29 2.2 10 10.6.67 31 3.23.193 79.38.023 10 29 2.2 20 10.4.65 32 3.11.186 80.34.021 15 28 2.1 30 10.2.64 33 2.99.179 81.31.018 20 28 2.1 40 10.1.63 34 2.88.173 1 82.27.016 25 27 2.0 50 9.9.62 35 2.78.167 83.24.014 30 27 2.0 11 0 9.8.60 36 2.68.161 84.20.012 35 26 2.0 10 9.6.59 37 2.58.155 85.17.010 40 26 1.9 20 9.5.58 38 2.49.149 86.14.008 45 25 1.9 30 9.4.57 39 2.40 87.10.006 50 25 1.9 j 40 9.2.56 40 2.32.139 88.07.004 55 25 1.8 L 50 9.1.55 41 2.24.134 89.03.002 TABLE X. Parallax of the Sun, on the first day of each Month: the mean horizontal Parallax being assumed = 8".60. Alti- Jn. Feb. March. April. May. June. July tude. Jan' Dec. Nov. Oct. Sept. Aug. 0 8.75 8.73 8.67 8.60 8.53 8.48 8.46 5 8.73 8.69 8.64 8.56 8.50 8.44 8.42 10 8.62 8.59 8.54 8.47 8.40 8.35 8.33 15 8.45 8.43 8.38 8.30 8.24 8.19 8.17 20 8.22 8.20 8.15 8.08 8.01 7.97 7.95 25 7.93 7.91 7.86 7.79 7.73 7.68 7.67 30 7.58 7.56 7.51 7.45 7.39 7.34 7.33 35 7.17 7.15 7.11 7.04 6.99 6.94 6.93 40 6.70 6.68 6.64 6.59 6.53 6.49 6.48 45 6.19 6.17 6.13 6.08 6.03 5.99 5.98 50 5.62 5.61 5.58 5.53 5.48 5.45 5.44 55 5.02 5.01 4.98 4.93 4.89 4.86 4.85 60 4.37 4.36 4.34 4.30 4.26 4.24 4.23 65 3.70 3.69 3.67 3.63 3.60 3.58 3.57 70 2.99 2.98 2.97 2.94 2.92 2.90 2.89 75 2.26 2.26 2.25 2.23 2.21 2.19 2.19 80 1.52 1.52 1.51 1.49 1.48 1.47 1.47 85 0.76 0.76 0.76 0.75 0.74 0.74 0.74 90 0.00 0.00 0.00 0.00 0000 0.00I TABLE XI. Semi-diurnal Arcs. Declination. Lat. 10 50 100 15~ 200 25o 30o 0 h m h m h m h m h m h m h m 5 60 6 2 6 4 6 5 6 7 6 9 6 12 10 6 1 6 4 6 7 6 11 6 15 6 19 6 24 15 6 1 6 5 6 11 6 16 6 22 6 29 6 36 20 6 1 6 7 615 6 22 6 30 6 39 6 49 25 6 2 6 9 6 19 6 29 6 39 6 50 7 2 30 6 2 6 12 6 23 6 36 6 49 7 2 7 18 35 6 3 6 14 6 28 6 43 6 59 7 16 7 35 40 6 3 6 17 6 34 6 52 7 11 7 32 7 56 45 6 4 6 20 6 41 7 2 25 7 51 8 21 50 6 5 6 24 6 49 7 14 7 43 8 15 8 5 4 55 6 6 6 29 6 58 7 30 8 5 8 47 9 42 60 6 7 6 35 7 11 7 51 8 36 9 35 12 0 65 6 9 6 43 7 29 8 20 9 25 12 0 10 TABLE XII. Equation of Time, to convert Apparent Time into MIean 7ime Argument, MlIean Longitude of the Sun. O- s Is IIs IIIs IVs Vs 0 m in. sec. min. sec. min. sec. min. sec. min. sec. min. sec. o + 6 58.4 -1 29.7 -3 38.7 + 1 27.0 + 6 4.1 + 2 49.7 1 6 39.7 1 42.0 3 34.2 1 40.1 6 6.3 2 34.5 2 6 20.9 1 53.8 3 29.1 1 53.1 6 8.0 2 18.9 3 6 2.1 2 5.2 3 23.5 2 6.0 6 9.1 2 2.8 4 5 43.3 2 15.9 3 17.3 2 18.9 6 9.5 1 46.4 5 5 24.5 2 26.1 3 10.7 2 31.7 6 9.3 1 29.5 6 5 5.7 235.9 3 3.5 2 44.3 6 8.5 1 12.3 7 4 46.9 2 45.0 2 56.0 2 56.7 6 7.2 0 54.6 8 4 28.2 2 53.6 2 47.9 3 8.9 6 5.2 0 36.6 9 4 9.6 3 1.8 2 39.5 3 20.8 6 2.5 + 0 18.2 10 3 51.1 3 9.3 2 30.5 3 32.5 5 59.3 -0 0.4 11 3 32.6 3 16.3 2 21.2 3 43.9 555.4 0 19.5 12 3 14.3 3 22.8 2 11.5 3 55.0 5 51.0 0 38.8 1 3 2 56.2 3 28.6 2 1.4 4 5.8 5 45.8 0 58.4 14 2 38.3 3 33.9 1 51.0 4 16.3 5 40.1 1 18.2 15 2 20.5 3 38.6 1 40.1 4 26.5 5 33.7 1 38.3 16 2 3.0 3 42.7 1 29.0 4 36.3 5 26.7 1 58.5 17 1 45.7 3 46.3 1 17.6 445.7 5 19.2 2 19.1 18 1 28.6 3 49.2 1 5.9 4 54.7 5 11.1 2 39.8 19 1 11.7 3 51.5 0 54.1 5 3.3 5 2.3 3 0.7 20 0 55.2 3 53.3 0 42.0 5 11.3 4 53.0 3 21.6 21 0 39.1 3 54.4 0 29.6 5 18.9 4 43.1 3 42.8 22 0 23.3 3 55.0 0 17.1 5 26.0 4 32.7 4 4.0 23 + 0 7.8 3 55.0 -0 4.4 5 32.6 4 21.6 4 25.3 24 -0 7.3 354.5 + 0 8.4 538.6 4 10.1 446.6 25 0 22.0 3 53.3 0 21.5 5 44.2 3 57.9 5 8.1 26 0 36.3 3 51.5 0 34.5 5 49.3 3 45.3 5 29.5 27 0 50.3 349.2 0 47.6 5 53.9 3 32.1 5 51.0 28 1 3.8 3 46.2 1 0.7 5 57.8 3 18.5 6 12.3 29 1 16.9 3 42.8 1 13.8 6 1.2 3 4.3 6 33.7 30 — 1 29.7 3 38.7 - + 1 27.0 + 6 4.1 + 2 49.7 - 54.9 TABLE XIII. Secular Variation of Equation of Time. Argument, Sun's Mean Longitude. Os Is IIs IIIs IVs VS sec. sec. sec. sec. seec. sec. sec. 0 -3 + 4 4+11 +14 -t13 +9 3 2 5 11 14 13 8 6 1 6 12 14 12 B 9 1 6 12 15 12 7 12 0 7 12 14 12 15 + 1 8 13 14 11 6 18 2 8 13 14 11 6 21 2 9 14 14 10 5 24 3 9 14 14 10 5 27 1 10 14 14 9 4 30 + 4 + 11 + 14 t 1-3 + 9 1+4 TABLE XII 11 Equation of Time, to convert Apparent Time into Mean Time. Argument, Mean Longitude of the Sun. VIs VIis VIIIs IXs Xs XIs o min. sec. mi. sec. min. sec. min. sec. mnz. sec. min. sec. 0 - 6 54.9 -15 18.9 -13 58.7 - 1 30.6 + 11 30.0 + 14 3.1 1 7 16.1 15 27.9 13 43.0 1 0.2 11 47.0 13 56.0 2 7 37.2 15 36.1 13 26.3 - 0 29.8 12 3.3 13 4S.4 3 7 58.3 15 43.7 13 8.9 + 0 0.6 12 18.7 13 40.1 4 8 19.1 15 50.5 12 50.5 0 31.0 12 33.4 13 31.1 5 8 39.8 15 56.5 12 31.4 1 1.3 12 47.2 13 21.6 6 9 0.2 16 1.8 12 11.6 131.4 13 0.1 13 11.4 7 9 20.5 16 6.3 11 51.1 2 1.3 13 12.2 13 0.7 8 9 40.6 16 9.9 11 29.9 2 31.0 13 23.5 12 49.4 9 10 0.3 16 12.9 11 7.9 3 0.5 13 33.9 12 37.4 10 10 19.8 16 15.1 10 45.4 3 29.7 13 43.6 12 25.0 11 10 38.9 16 16.5 10 22.0 3 58.6 13 52.3 12 12.2 12 10 57.8 16 17.0 9 5S.1 4 27.1 14 0.2 11 58.9 13 11 16.2 16 16.6 9 33.5 4 55.2 14 7.3 11 4.5.1 14 11 34.4 16 15.4 9 8.4 5 22.9 14 13.5 11 30.9 15 11 52.1 16 13.4 8 42.6 5 50.2 14 18.9 1116.3 16 12 9.5 16 10.4 8 16.4 6 17.1 14 23.4 11 1.1 17 12 26.5 16 6.7 7 49.6 6 43.5 14 27.2 10 45.6 18 12 42.9 16 2.1 7 22.5 7 9.3 14 30.0 10 29.7 19 12 58.9 15 56.6 6 54.9 7 34.6 14 32. I 10 13.5 20 13 14.4 15 50.1 6 27.0 7 59.3 14 33.3 9 56.9 21 13 29.5 15 42.9 5 58.5 8 23.4 14 33.7 9 40.1 22 13 44.1 15 34.8 5 29.7 8 46.9 14 33.3 9 23.0 23 13 58.0 15 25.8 5 0.5 9 9.8 14 32.2 9 5.7 24 14 11.4 15 16.0 4 31.0 9 32.0 14 30.2 8 48.0 25 1424.1 15 5.2 4 1.4 9 53.5 14 27.5 8 30.2 26 14 36.3 14 53.6 3 31.6 10 14.3 14 24.0 8 12.2 27 14 47.9 14 41.1 3 1.5 10 344 14 19.9 7 54.0 28 14 58.8 14 27.7 2 31.3 10 53.8 14 15.0 7 35.5 29 15 9.22 1413.6 2 1.0 1112.3 14 9.4 717.0 30 -1518.9 1-13 58.7 - 130.6 +-t1 30.0 - 14 3.1 658.4 TABLE XIII. Secular Variation of.:quation of Time. Argument, Sun's Mean Longitude. VIs VIIs VIIs IXS Xs XIS sec. sec. sec. sec. gec. sec. 0 +4 -2 -10 -15 -15 -10 3 3 3 10 15 14 10 6 3 4 11 15 14 9 9 2 4 12 15 14 8 12 1 5 12 15 13 8 15 +-1 13 15 13 7 18 0 7 13 15 12 6 21 0 7 14 15 12 5 24 - 8 14 15 11 5 27 2 9 15 15 11 4 30 -2 10 -15 -15 i-10 - .12 TABLE XIV. Perturbations of Equation of Time. III. II. 0 o 100 200 300 400 SOC 600 700 800 900 1000 sec. sec. sec. sec. sec. sec. sec. sec. sec. sec. sec. 0 1.4 0.8 1.0 1.7 1.7 1.2 0.7 0.4 0.6 1.4 1.4 100 1.2 1.4 1.1 1.0 1.6 1.8 1.1 0.7 0.6 0.7 1.2 200 0.9 1.0 1.2 I 1.2 1.2 1.5 1.7 1.1 0.5 0.7 0.9 300 0.7 1.1 1.1 0.9 1.2 1.4 1.5 1.6 1.2 0.5 0.7 4100 0.5 0.6 1.2 1.2 0.S 1.0 1.6 1.7 1.5 1.2 0.5 500 1.0 0.5 0.6 1.2 1.4 0.8 0.8 1.5 1.9 1.5 1.0 600 1.7 1.0 0.4 0.5 1.2 1.4 0.9 0.6 1.3 2.0 1.7 700 1.9 1.8 11 0.4 04 1.1 1.6 1.1 0.7 1.2 1.9 800 1.2 1.8 1.8 1.2 0.4 0.3 1.0 1.6 1.2 0.7 1.2 900 0.7 1.1 1.7 1.8 1.2 0.6 0.2 0.8 1.6 1.3 0.7 1000 1.4 0.8 1.0 1.7 1 1 0.7 0.47 0.6 1.4 1.4 II. - IV. sec. sec. sec. sec. sec. sec. sec. sec. sec. sec. sec. 0 0.6 0.7 0.5 0.3 0.2 0.6 0.7 0.5 0.2 0.1 0.6 100 0.2 0.7 0.6 0.5 0.2 0.5 0.6 0.9 0.5 0.2 0.2 200 0.2 0.4 0.6 0.5 0.4 0.3 0.4 0.6 0.5 0.5 0.2 300 0.4 0.2 0.5 0.5 0.5 0.4 0.4 0.4 0.5 0.5 0.4 400 0.5 0.4 0.4 0.4 0.4 0.4 0.5 0.5 0.4 0.4 0.5 500 0.4 0.5 0.5 0.5 0. 0.4 0.3 0.4 0.5 0.3 0.4 600 0.3 0.3 0.5 0.6 0.4 0.4 0.3 0.5 0.7 0.4 0.3 700 0.4 0.2 0.3 0.6 0.6 0.4 0.2 0.2 0.7 0.7 0.4 800 0.6 0.3 0.2 0.3 0.7 0.6 0.3 0.2 0.3 0.8 0.6 900 0.8 0.5 0.3 0.1 0.4.7 0.5 0.3 0.1 0.5 0.8 1000 1 0.6 0.7 0.5 0.3 0.2 0.6 0.7 0.5 0.2 0.1 0.6 II. V. sec. sec. sec. sec. Isc. sec. sec. sec. se. sec. sec. 0 1.0 1.0 1.1 1.2 1.1 1.2 0.7 0.4 0.6 0.9 1.0 100 0.9 0.9 0.8 1.0 1.3 1.3 1.0 0.7 0.4 0.5 0.9 200 0.5 0.7 0.7 0.8 1.0 1.0 1.1 1.2 0.9 0.3 0.5 300 0.2 0.5 0.7 0.7 0. 1.2 1.5 1.5 1.1 0.5 0.2 400 0.3 0.2 0.5 0.7 0.7 0.9 1.3 1.4 1.4 1.0 0.3 500 0.8 0.3 0.2 0.5 0.7 0.7 1.0 1.4 1.4 1.4 0.8 600 1.3 0.7 0.3 0.3 0.5 0.7 0.9 1.1 1.4 1.6 1.3 700 1.5 1.1 0.7 0.3 0.4 0.5 0.8 1.0 1.2 1.4 1.5 SO0 1.3 1.3 1.0 0.7 0.4 0.4 0.6 0.8 1.0 1.2 1.3 900 1.1 1.2 1.2 1.0 0.8 0.6 0.5 0.6 0.9 1.1 1.1 1000 1.0 1.0 1.1 1.2 1 1.1 1.0 0.7 0.4 10.6 0.9 1.0 Moon and Nutation. sec. sec. sec secI sec I sec. sec. sec. sec. sec sec. 0.5 0.8 1.0' 1.0 08 0.5 0.2 0.0 0.0 0.2 0.5 N. 0.1 0.1 0.2 0.20.2 0 0.2 0.2 0.2 0.2. 0.1 0.1 Ccastant 3s.0 TABLE XV. 13 For converting any given day into the decimal part of a yeas of 365 days. Day Jan. Feb. March April May June 1.000.085.162.247.329.414 2.003.088.164.249.331.416 3.006.090.167.252.334.419 4.008.093.170.255.337.422 5.011.096.173.258.340.425 6.014.099.175.260.342.427 7.016.101.178.263.345.430 8.019.104.181.266.348.433 9.022.107.184.268.351.436 10.025.110.186.271.353.438 11.027.112.189 274.356.441 12.030.115.192.277.359.444 13.033.118.195.279.362.446 ]4.036.121.197.282.364.449 15.038.123.200.285.367.452 16.041.126.203.288.370.455 17.044.129.205.290.373.458 18'046.132.208.293.375.460 19.049 34.211.296.378.463 20.052.137.214.299.381.466 21.055.140.216.301.384.468 22.058.142.219.304.386.471 23.060.145.222.307.389.474 24.063.148.22.310.392.477 25.066.151.227.312.395.479 26.068.153.230.315.397.4S2 27.071.156.233.318.400.485 28.0'74.159.236.321.403.488 29.077.238.323.405.490 30.079.241.326.408.493 31.082.244.411 14 TABLE XV., Continued. Flor converting any given day into the decimal part of a veam of 365 days. Day July August Sept. Oct. Nov. Dec. 1.496.581.666.748.833.915 2.499.584.668.751.836.918 3.501.586 671.753.838.921 4.504.589.674.756.841.923 5.507.592.677.759.844.926 6.510.595.679.762 846.929 7.512.597.682.764.849.931 8.515.600.685,767.852.934 9.518.603.688.770 855.937 10.521.605.690.773.858.940 11.523.608.693.775.860.942 12 526 611.696.778.863.945 13.529 614.699.781.866.918 14.532.616.701.784.868.951 15.534.619.704.786.871.953 16.537.622.707.789.874.956 17.540.625.710.792 877.959 18.542.627.712.795 879.962 19.545.630.715.797 882.964 20.548.633.718.800 885.967 21.551.636.721.803 888.970 22.553.638.723 805 890.973 23.556.641.726.8(8 893.975 24.559.644.729.811 896.978 25.562.647.731.814.899.981 26 564.649.734.816.901.984 27.567.652.737.819.904.986 28.570.655.740.822.907.989 29.573.658.742.825.910.992 30.575.660.745.827.912.995 31.57z8.663.830.997 TABLE XVI. 15 FFor converting time into decimal parts of a day. HIours Minutes Seconds h. m. m. s. s. 1.04167 1.00069 31.02153 1.00001 31.00036 2.08333 2.00139 32.02222 2.00002 32.00037 3.12500 i 3.00208 33.02292 3.00003 33.00038 4.16667 4.00278 34.02361 4.00005 34.00039 5.20833 5.00347 35.02430 5.00006 35.00040 6.25000 6.00417 36.02500 6.00007 36.00042 7.29167 7.00486 37.02569 7.00008 37.00043 8.33333 8.00556 38.02639 8.00009 38.00044 9.37500 9.00625 39.02708 9.00010 39.00045 10.41667 10.00694 40.02778 10.00012 40.00046 11.45833 11.00764 41.02847 11.00013 41.00047 12.50000 12!.00833 42.02917 12.00014 42.00049 13.54167 13 1.00903 43.02986 13.00015 43.00050 14.58333 14 1.00972 44.03056 14.00016 44.00051 15.62500 15 01042 45.03125 1.5.00017 45.00052 16.66667 16.01111 46.03194 16.00018 46.00053 17.70833 17.01180 47.03264 17.00020 47.00054 18.75000 18.01250 48.03333 18.00021 48.00056 19.79167 19.01319 49.03403 19.00022 49.00057 20.83333 20.01389 50.03472 20.00023 50.00058 21.87500 21 01458 51.03542 21.00024 51.00059 22.91667 22.01528 52.03611 22.00025 52.00060 23.95833 23 01597 53.03680 23.00027 53.00061 24 1.00000 24.01667 54.03750 24.00028 54.00062 25.01736 55.03819 25.00029 55.00064 26.01805 56.03889 26.00030 56.00065 27.01875 57.03958 27.00031 57.00066 28.01944 58.04028 28.00032 58.00067 29.02014 59.04097 29.00034 59.00068 30.02083 60.04167l 30.00035 60.00069 16 TABLE XVII. For converting Minutes and Seconds of a degree, into the decimal division of the same. Fr ~,,,Minutes Seconds 1.01667 31.51667 1.00028 31.00861 2.03333 32.53333 2.00056 32.00889 3.05000 33.55000 3.00083 33.00917 4.06667 34.56667 4.00111 34.00944 5.08333 35.58333 5.00139 35.00972 6.10000 36.60000 6.00167 36.01000 7.11667 37.61667 7.00194 37.01028 8.13333 38.63333 8.00222 38.01056 9.15000 39.65000 9.00250 39.01083 10.16667 40.66667 10.00278 40.0111111.18333 41.68333 11.00306 41.01139 12.20000 42.70000 12.00333 42.01167 13.21667 43.71667 13.00361 43.01194 14.23333 44.73333 14.00389 44.01222 15.25000 45.75000 15.00417 45.01250 16.26667 46.76667 16.00444- 46.01278 17.28333 47.78333 17.00472 47.01306 18.30000 48.80000 18.00500 48.01333 19.31667 49.81667 19.00528 49.01361 20.33333 50.83333 20.00556 50.01389 21.35000 51.85000 21.00583 51.01417 22.36667 52.86667 22.00611 52.01444 23.38333 53.88333 23.00639 53.01472 24.40000 54.90000 24.00667 54.01500 25.41667 55.91667 25.00694 55 01528 26.43333 56.93333 26.00722 56.01556 27.45000 57.95000 27.00750 57.01583 28.46667 58.96667 28.00778 58.01611 29.48333 59.98333 29.00806 59.01639 30.50000 60 1.00000 30.00833 60.01667 TABLE XVIII. 17 Stun's Epochs. Years. M. Long. Long. Peri. I. II. III. IV. V. N. VI. VII. 1830 9 10 37 46.9 9 10 0 54 228 279 169 598 758 519 989 362 1831 9 10 23 27.4 9 10 1 55 588 278 793 130 842 573 235 396 1832 B. 9 10 9 7.99 10 2 57 948 278 418 661 926 627 482 430 1833 9 10 53 56.8 9 10 3 59 342 280 47 194 11 681 764 464 1834 9 10 39 37.3 9 10 5 0 702 279 671 725 95 734 11 498 1835 9 10 25 17.8 9 10 6 2 62 279 296 256 179 788 257 532 1836 B. 9 10 10 58.4 9 10 t 3 422 278 920 788 264 842 504 566 1837 9 10 55 47.2 9 10 8 5 816 280 549 321 348 895 787 600 1838 9 10 41 27.8 9 10 9 6 176 279 173 852 432 949 33 634 1839 9 10 27 8.3 9 10 10 8 536 279 798 383 517 3 279 668 1840 B. 9 10 12 48.8 9 10 11 9 896 278 422 915 601 56 526 702 1841 9 10 57 37.7 9 10 12 11 1290 280 51 447 685 110 809 736 1842 9 10 43 18.2 9 10 13 12 650 279 676 979 770 164 55 770 1843 9 10 28 58.8 9 10 14 14 10 279 300 510 854 218 301 804 1844 B. 9 10 14 39.3 9 10 15 15 370 278 924 41 938 272 548 838 1845 9 10 59 28.2 9 10 16 17 764 280 553 574 23 325 831 872 1846 9 10 45 8.7 9 10 17 19 124 280 177 106 107 379 77 906 1847 9 10 30 49.2 9 10 18 20 484 279 802 637 191 433 324 940 1848 B. 9 10 16 29.8 9 10 19 22 844 278 427 168 276 487 570 974 1849 9 11 1 18.69 10 20 23 238 280 55 700 360 540 853 8 1850 9 10 46 59.2 9 10 21 25 598 280 680 231 444 594 99 41 1851 9 10 32 39.7 9 10 22 26 958 279 304 762 529 648 346 75 1852 B. 9 10 18 20.2 9 10 23 28 319 278 929 294 613 701 592 109 1853 9 11 3 9.1 9 10 24 29 713 280 557 827 697 755 875 143 1854 9 10 48 49.6 9 10 25 31 73 280 182 358 782 809 121 177 1855 9 10 34 30.2 9 10 26 32 433 279 806 889 866 863 368 211 1856 B. 9 10 20 10.7 9 10 27 34 793 279 430 421 950 916 614 245 1857 9 11 4 59.6 9 10 28 35 187'281 60 953 35 970 897 279 1858 9 10 50 40.1 9 10 29 37 547 280 684 485 119 24 144 313 1859 9 10 36 20.79 10 30 39 907 279 308 16 203 78 390 347 1860 B. 9 10 22 1.2 9 10 31 40 267 279 933 547 288 131 636 381 1861 9 11 6 50.1 9 10 32 42 661 281 562 80 372 185 919 415 1862 9 10 52 30.6 9 10 33 43 21 280 186 612 456 239 166 449 1863 9 10 38 11.1 9 10 34 45 381 280 810 143 541 292 412 483 1864 B. 9 10 23 51.7 9 10 35 46 741 279 435 674 625 346 659 517 1865 9 11 8 40.5 9 10 36 48 135 281 64 207 709 400 941 551 1866 9 10 54 21.1 9 10 37 49 495 280 688 738 194 453 188 585 1867 9 10 40 1.6 9 10 38 51 855 280 313 270 878 507 434 619 1868 B. 9 10 25 42.2 9 10 39 52 215 279 937 801 962 561 681 653 1869 9 11 10 31.0 9 10 40 54 609 281 566 334 47 615 963 687 1870 9 10 56 11.6 9 10 41 56 969 280 190 865 131 668 210 721 1871 9 10 41 52.1 9 10 42 57 329 279 814 396 216 742 457 755 1872 B. 9 10 27 32.6 9 10 43 59 690 278 439 928 300 775 703 789 1873 9 11 12 21.5 9 10 45 0 84 280 67 461 384 829 986 823 1874 9 10 58 2.0 9 10 46 2 444 280 692 992 469 883 232 857 1875 9 10 43 42.6 9 10 47 3 804 279 316 523 553 937 479 891 1876 B. 9 10 29 23.1 9 10 48 5 164 279 940 55 637 990 725 925 1877 9 11 14 12.09 10 49 6 558 281 570 587 722 44 8 959 1878 9 10 59 52.5 9 10 50 8 918 280 194 119 806 98 255 993 1879 9 10 45 33.1 9 10 51 10 278 279 818 650 890 152 501 27 1880 B. 9 10 31 13.6 9 10 52 11 638 279 443 181 975 205 747 61 1881 9 11 16 2.5 9 10 53 13 32 281 72 714 59 259 30 95 1882 9 11 1 43.0 9 10 54 14 392 280 696 246 143 313 277 129 1883 9 10 47 23.5 9 10 55 16 752 280 320 777 228 366 523 163 1884 B. 9 10 33 4.1 9 10 56 17 112 279 945 308 312 420 770 197 18 TABLE XIX. Suit's MIIotions for Months. MNonths M. Long. Per. I II III IV V1 N VI V11i 8 0 January 0 00 0.0 0 0 0 0 0 00 0 0 February 1 0 3318.2 5 47 85 138 45 7.5 125 3 Corn. 1 28 9 11.4 10 993 162 263 86 14 9 141 6 March Bis. 1 2~ 8 19.8 10 27 164 267 87 14 9 178 61 Arl Corn. 2 28 42 29.7 15 42 246 401 1.3) 21 13 266 8 Apil Ils. 2 29 41 38.0 15 7,6 249 405 132 21 13 302 8 Ma Corn. 3 28 1639.6 20 59 329 534 175 28 18 35511 1 May 3 29 15 47.9 20 92 331 538 176 28 18 391 11 June Corn. 4 28 49 57.9 26 110 414 672 220 35 22 480 14 Bis. 4 2949 6226 144 416 676 221 35 23 516 14 JCorn. 5 28 24 7.8 31 129 496 806 263 41 27 569 17 liy Bis. 5 29 23 16.1 31 163 499 810 265 42 27 605 17 Aug. Comn. 6 2857 26.1 36 182 580 943 309 49 31 694 20 Au. Bis. 6 29 56 34.4 36 216 583 948 310 49 31 730 20 Sep. Corn. 7 29 3044.2 41 233 665 81 354 56 36 819 23 Sep. 8 0 29 52.6 41 268 668 86 355 56 36 1855 23 Oc. Corn. 8 29 4 54.1 46 250 748 215 397 63 40'908 25 Oc. Bis. 9 0 4 2.5 46 284 750 219 399 63 40 944 2"5 Nov. Corn. 9 2938 12.5 51 300 832 353 443 70 45 33 28~ Bis. 10 0 37 20.7 51 333 835 357 444 70 45 69 28 Dec. {Corn. 10 29 12 22.3) 56 313 915 486 486 77 49 121 31 Bis. 11 0 11 30.6) 56 347 917 I491 488 77 49, 158 131 TABLE XX. Sun's Motions for Daqs and Hours. -Days~ M. Long. Per. I II 111 IV V N VI VII Hrs. Lono I 2 00 0000 00 1 ~~~~~~~~~~~ VI I 0 00.0 00 0 0 0 1 2 27.8 1 0 0 20 59 8.3 0 234 3 4 1 00 36 0 2 455.7 3 0 0 3 1 58 16.7 0 68 5 9 3 00 73 0 3 723.5 4 01 4 2 57 25.0 0 101 8 13 4 1 0 109 0 4 9 51.4 6 0 1 5 3 5633.3 1135111816 1 1145 0 5 1219.27 1 1 6 4 55 41.6 1 169 14 22 7 1 1 181 0 6 14 47.1 8 1 II1 7 5 54 50.0 1 203 16 27 9 1 1 218 1 7 17 14.9 10 1 I) 8 6 53 58.3 1 236 19 31 10 2 1 254 1 8 19 42.8 11 1 1 9 7 536.6 127022 3612 2 1290 1 9 22 10.6 13 12 110 8 52 15.0 1 304 25 40 13 2 1 327 1 10 24 38.5 14 1 2 ii 9 5123.3 2338 27 4415 2 1363 111 276.3 16 12 12 10 50 31.6 2.371 3 0 49 16 2 2 399 1 12 29 34.2 17 1191 13 11 49 40.0 2405 33 5317 32 435 113 322.01 IS1 114 12 48 48.3 2 439 36 58 19 3 2 472 1 14 34 29.9 20 2 3' 15 13 47 56.6 2 473 38 62 20 3 2 508 2 15 36 57.7 21 2 3 16 1447 4.9 2506 41 67 223 2Q5442 16 3925.6 23 23' 17 15 4613.3 3 540 44 71 23 4 2 581 2 17 41 53.4- 24 211 18 1645 21.6 3574 477625 4 2617 2 18 44 212 25 2'3 19 17 44 29.9 3 608 49 80 26 4 3 653 2 19 46 49.1 27 2 41 20 1843 38.3 3641 5285 28 436902 120 4916.9 282 4 21 19 42 46.6 3 675' 55 89 29 5 3 726 2 21 51 44.8 30 12 4~ 22 2041 54.9 4709 58 93 31 5 3 762 2 22 54 12.6 31 24 23 21 413.3 4743 60 98 32 5 3 798 22356 40.5 323 4 24 22 40 11.6 4 777 63 102 33 5 39 835 2'24 59 4. 34 24 P5 23 39 19.9 4 810 66 107 35 5 4 871 2 )iio 24 38 28.2 4)844 68 111 36 6 4 907 2 27 25 37 36.6 4 878 71 116 38 6 4 943 2 128 26 36 44.9 5 912~ 74 120 39~ 6 4 980 2 129 27 35 53.2 5 1945 77)125 41 6 4 16 3 30 2835 1.6 5 979 79'129 42 7 14 1521 3 1 I.31129 34 9.9 5 13 82/11394 44 7 14 891 3f ___ TABLE XXI. TABLE XXII. 19 Sun's Motions for Minutes and Seconds. Mean Oblicpitic. the Ecliptic. in. Long. Min. Long. Sec. Lon. l Sec.'Lon. Years 23 27 I_.._ f___ _. If 1 0 2.5 31 1 16.4 1 0.0 31 1.3 1835 38 80 2 4.9 32 1 18.8 2 0.1 32 1.3 1836 38.35 3 7.4 33 1 21.3 3 0.1 33 1.4 1837 37.89 4 9.9 34 1 23.8 4 0.2 34 1.4 1838 37.43 5 12.3 35 1 26.2 5 0.2 35 1.4 1839 36.98 61 14.8 36 1 28.7 6 0.2 36 1.5 1840 36.52 7 17.2 37 1 31.2 7 0.3 37 1.5 1841 36.06 8 19.7 38 1 33.6 8 0.3 38 1.6 1842 35.61 9 22.2 39 1 36.1 9 0.4 39 1.6 1843 35.15 10 24.f6 40 1 38.6 10 0.4 40 1.6 1844 34.69 11 27.1 41 1 41.0 11 0.5 41 1.7 1845 34.23 12 29.6 42 1 4-3.5 12 0.5 42 1.7 1846 33.78 13 32.0 43 1 46.0 13 0.5 43 1.8 1847 33.32 14 34.5 44 1 48.4 14 0.6 44 1.8 1848 32.86 15 37. 0 45 150.9 15 0.6 45 1.8 1849 32.41 16 39.4 46 1 53.3 16 0.7 46 1.9 1850 31.95 17 41.9 47 1 55.8 17 0.7 47 1.9 1851 31.49 18 44.4 48 1 58.3 18 0.7 148 2.0 1852 31.04 19 46.8 49 2 0.7 19 0.8 ] 49 2.0 1853 30.58 20 49.3 50 2 3.2 20 0.8 50 2.0 1854 30.12 21 51.7 51 2 5.7 21 0.9 51 2.1 1855 29.66 22 54.2 52 2 8.1 22 0.9 52 2.1 1856 29.21 23 56.7 53 2 10.6 23 0.9 53 2.2 1857 28 75 24 59.1 54 2 13.1 24 1.0 154 2.2 1858 28 29 25 1 1.6 55 2 15.5 25 1.0 55 2.3 1859 2784 26 1 4.1 56 218.0 26 1.1 56 2.3 1860 27.38 27 1 6.5 57 2 20.5 27 1.1 57 2.3 1861 26 92 28 1 9.0 58 o2 22.9 28 1.1 58 2.4 1862 26.47 29 1 11.5 59 2 25.4 29 1.2 59 2.4 1863 26.01 30 1 13.9 60 2 27.8 30 11.21 60 2.5 2864 25.55 TABLE XXIII. Sun's Hourly Motion. Argument. Sun's Mean Anomaly. 8Os Is IIs IIIs IVs Vs 0 2 32.92 2 32.20 I 30.29 2 27.74 2 25.32 2 23.60 30 10 2 32.84 2 31.67 2 29.46 2 26.89 2 24.64 2 23.2 6 20 20 2 32.59 2 31.02 2 28.611 2 26.07 224.06 2 23.05 10 30 2 32.20 2 30.238 2 27.74 2 25.32 1 2 23.60 2 22.99 0 L XIs Xs IXs I VIlIIs VIIs VIs TABLE XXIV. Sun's Semi-diameter. Argument. Sun's Mean Anomaly. Os|Is IDIs | IIs IIIs s o II, I sI 0 16 17.3 16 15.0 16 8.8 16 0.6 15 52.7 15 47.0 30 10 16 17.0 1 16 13.3 16 6.2 15 57.8 15 50.5 15 45.9 20 20!6 16.2 16 11.2 16 3.4 15 55.1 15 48.6 15 45.2 10 30 16 15.0 16 8.8 16 0.6 15 52.7 15 47.0 15 45.0 0 I I xi, I [ Xs vIIe I V:I'" vis 20 TABLE XXV. Equation of the Sun's Centre. Argument. Sun's Mean Anomaly. _ - Is Is HIs | IVs Ve o 0 t o, o 0 " 0 0' " 0o " 0 11 2959 13.9 0 5758.5 140 10.7 1 54 34.1 1 38 4.8 05552.6 10 0117.3 059 43.9 141 8.9 15430.5 137 2.4 054 8.7 2 0 320.6 1 128.0 142 5.1 1 54 24.8 1 3558.1 05224.0? 0 523.9 13 10.9 1 4259.3 1 54 17.0 1 3452.2 05038.2 4 0 727.0 1 452.6 14351.8 1 54 7.1 13344.6 04851.6 5 0 930.0 1 633.0 14442.1 1 5355.2 1 32 35.4 047 4.2 6 011 32.8 18 12.3 1 4530.4 1 5341.0 1 3124.4 04516.0 7 013 35.4 1 950.1 1 46 16.8!1 53 24.9 1 3011.9 0 43 26.9 8 01537.7 1 1126.5 147 1.2 1 1 53 6.7 1 2857.7 04137.0 9 017 39.6 1 13 1.7 1 4743.5 1 5246.5 1 2742.0 039 46.5 10 019 41.2 1 1435.3 148 23.9 1 52 24.2 1 2624.8 03755.3 11 0 21 42.4 1 16 75 149 2.2 1 51 59.8 1 25 5.9 036 3.3 12 0 23 43.1 1 17 38.2 149 38.4 1 51 33.4 1 23 45.7 0 3410.8 13 02543.4 1 19 7.5 15012.6 1 51 5.0 1 2223.8 03217.7 14 0 2743.2 1 2035.2 1 0 44.7 1 50 34.5 1 21 0.6 03023.8 15 0 29 42.31 22 1.5 1 51 14.9 1 50 2.2 1 1936.0 02829.6 16 0 31 40.9 1 23 26.0 1 51 42.9 1 49 27.7 1 18 9.9 02634.8 17 0 33 38.9 1 2448.9 1 52 8.7 1 48 51.3 1 1642.4 024 39.6 18'O 35 36.2 1 26 10.3 152 32.5 1 48 13.0 1 15 13.7 02243.9 19 0 37 32.9 1 2730.0 1 52 54.3 1 47 32.7 1 1343.5 02047.9 20 0 39 28.8 1 28 48.0 1 5313.9 14650.4 1 1212.1 01851.4 21 0 41 23.9 1 30 4.2 153 31.4 1 46 6.3 1 1039.3 01654.6 22 0 43 18.1 1 3118.8 153 46.8 1 45 20.3 1 9 5.4 01457.5 23 045 11.5 1 3231.7 154 0.1 1 44 32.2 1 730.3 013 0.1 24 047 4.0 1 3342.7 1 5411.2 14342.4 1 554.0 011 2.6 25 048 55.6 1 3452.0 15420.4 1 4250.7 14 16.5 0 9 4.8 26 050 46.3 1 3559.4 1 5427.2 14157.1 1 237.8 0 7 6.9 27 0 52 36.0 137 5.1 15432.1 141 1.7 1 058.0 05 87 28 05424.6 1 38 8.8 15434.9 1 40 4.5 0 5917.3 0 310.5 29 056 12.1 1 3910.8 1 54 35.4 11 39 5.6 0 57 35.4 0 1 12.2 30 0 57 58.5 1 40 10.7 154 34.1 11 38 4.8 0 5552.6 TABLE XXVI. Secular Variation of Equation of Sun's Cenwre. Argument. Sun's Mean Anomaly.,Os S IS 1 IVs IIsV 0 "If" 0 -0 9 1-15 -17 -15 -8 2 11 9 15 17 14 8 4 1 10 16 17 14 7 6 10 16 17 14 7 8 1 2 11 1 16 17 13 6 19 3 11 16 17 13 6 12 4 12 17 17 12 5 14 4 12 17 16 12 5 16 5 13 17 16 12 4 18 5 13 17 16 1 3 20 6 13 17 16 1 3 22 7 14 17 16 10 2 24 7 14 17 15 10 2 26 8 15 17 15 9 1 28 8 15 17 15 9 1 30 -9 1 -17 15 -- 8 — 0 TABLE XXV. 21 Equation of the Sun's (entre Argument. Sun's Mean Anomaly. VI-' VIIs VIIIs IXs X XIs 11ls 1 118 118 11s 118s o o 0 o 0 t o' If 0 29 59 13.9 29 2 35.2 28 20 23.0 28 3 53.7 28 18 17.1 29 0 29.3 1 29 57 15.6 29 0 52.4 28 19 22.2 28 3 52.3 28 19 17.0 29 2 15.7 2 29 55 17.3 28 59 10.5 28 18 23.3 28 3 52.8 28 20 19.0 29 4 3.2 3 29 53 19.1 28 57 29.8 28 17 26.1 28 3 55.6 28 21 22.7 29 5 51.8 4 29 51 20.9 28 55 50.0 28 16 30.7 28 4 0.5 28 22 28.4 29 741.5 5 29 49 23.0 28 54 11.4 28 15 37.1 28 4 7.4 28 23 35.8 29 9 32.2 6 294725.2 2852 33.8 281445.4 28 416.6 282445.1 2911 23.8 7 29 45 27.7 28 50 57.5 28 13 55.6 28 427.7 28 25 56.1 29 13 16.3 8 29 43 30.3 28 49 22.4 28 13 7.5 28 441.0 28 27 9.0 29 15 9.7 9 29 41 33.2 28 47 48.5 28 12 21.5 28 4 56.4 28 28 23.6 29 17 3.9 10 29 39 36.4 28 46 15.7 28 11 37.4 28 5 13.9 28 29 39.8 29 18 59.0 11 29 37 39.9 28 44 44.3 28 10 55.1 28 5 33.5 28 30 57.8 29 20 54.9 12 29 35 43 9 28 43 14.1 28 10 14.8 28 5 55.3 28 32 17.5 29 22 51.6 13 293348.2 2841 45.4 28 936.5 28 6 19.1 283338.9 292448.9 14 29 31 53.0 28 40 17.9 28 9 0.0 28 6 44.9 28 35 1.8 29 26 46.9 15 29 29 58.2 28 38 51.8 28 8 25.6 28 7 12.9 28 36 26.3 29 28 45.5 16 2928 4.0 283727.2 28 753.2 28 743.1 283752.6 293044.6 17 29 26 10.1 28 36 4.0 28 7 22.8 28 8 15.2 28 39 20.3 29 32 44-4 18 292417.0 283442.1 28 654.4 28 849.4 284049.6 293444.7 19 29 22 24.5 28 33 21.9 28 6 28.0 28 9 25.6 28 42 20.3 29 36 45.4 20 29 20 32.5 28 32 3.0 28 6 3.6 28 10 3.9 28 43 52.5 29 38 46.6 21 29 18 41.3 28 30 45.8 28 5 41.4 28 10 44.3 28 45 26.1 29 40 48.2 22 29 1650.8 282930.1 28 521.1 28 11 26.6 2847 1.3 294250.1 23 29 15 0.9 28 28 15.9 28 5 2.9 28 12 11.0 284837.7 294452.5 24 29 13 11.8 28 27 3.4 28 4 46.8 28 12 57.4 28 50 15.5 29 46 55.0 25 29l1 23.6 28 25 52.4 28 4 32.6 28 13 45.7 28 51 54.8 29 48 57.8 26 29 9 36.2 28 24 43.2 28 4 20.7 28 14 36.0 28 53 35.2 29 51 0.8 27 29 7 49.5 28 23 35.6 28 4 10.8 28 15 28.5 28.5 16.9 29 53 3.9 28 29 6 3.8 28 22 29.7 28 4 3.0 28 16 22.7 28 56 59.8 29 55 7.2 29 29 4 19.1 28 21 25.4 28 3 57.3 28 17 18.9 28 58 43.9 29 57 10.5 30 29 235.2 282023.0 28 353.7 28 18 17.1 29 02 9.3 2959 13.9 TABLE XXVI. Secular Variation of Equation of Sun's Centre. Argument. Sun's Mean Anomaly. VIs VIis VIIIs IXs Xs XIs 0 +0 + 8 + 15 +17 +15 +9 2 1 9 15 17 15 8 4 1 9 15 17 15 8 6 2 10 15 17 14 7 8 2 10 16 17 14 7 10 3 11 16 17 14 6 12 3 11 16 17 13 6 14 4 12 16 17 13 5 16 5 12 16 17 12 4 18 5 12 17 17 12 4 20 6 13 17 16 11 22 6 13 17 16 1 1 2 24 7 14 17 16 10 2 26 7 14 17 16 10 1 28 8 14 17 15 9 1 30 + 8 + 15 + 17 + 15 + 9 + 0 30 22 TABLE XXVIL. Nutations. Argument. Supplemnent of the Node, or N. Solar Nutatzon. N.I Lonncr R As. Obliq. N. LonR. Asc. Obliq. Long. OCbliq. +o + oo. +0. + 9.2 500 - 0. - 0.0 -9.3 Jan... 10i 1.0 1.0 9.1 510 1.1i 1.1 9.3 1 + 0.5 - 0.5 20' 2.1 2.1 9.1 520 2.2 2.0 9.3 11 0.8: 0.4 30 3.2 3.01 9.0 530 3.3 2.9 9.2 21 1.1 0.2 40 4.2 4.0 8.9; 540W 4.4 3.9 9.0 31 1.2 — 0.1 50 + 5.2 + 4.9 + 8.7: 550 - 5.5 - 4.8 -,8.9 Feb. 60 6 2 6.0 8 5' 560 6.5 5.7 8.7 10 1.2 + 0.1 701 7.2 6.9 8.3 570 7.5 6.6 8.4 20 1.0 0.3 80 8.2 7.8 8.1 580 8.5 7.5 8.1 9(0 9.11 8.7 7.8 590 9.5 8.4 7.8 arc 100 + 10.0 + 9.4 + 7.5 600 -10.4- 9.1 7.5 2 + 03 0 110 10.8 10.3 7.1 610 11.2 9.9 7.1 22 -0.1 0.5 120 11.6 11.1 6.7 620 12.0 10.6 6.7 A 1301 12.4 11.7 6.31 630 12.8 11.4 6.3 Aprl. 0.5 0.5 140 13.1 12.4 5.9 640 13.5 12.0 5.9 11 0.8 0.21 150 + 13.8 + 13.0 + 5.5 650 14.2 -12.6 -5.4 21 1.1 0.2 60 14.4 13.6 5.0 660 14.8 13.2 4.9 M 1170 15.0 14.1 4.5 670 15.3 13.8 4 ay. 12 + 01 180 15.5 14.5 4.0 680 15.8 14.2 3.9 1I 1.2 0.1 190 15.9 14.8 3.5 690 16.2 14.7 33 21 1.1 0. 31 0.8 0.41 20 16.615.4- 2.4 710 16.9 15.3 2.2 220 1;6.9 15.6| 1.8l 720 17.1 15.4 1.6 J.nn1 30 + 0.4 0.5 230l 17.1l 15.71 7:91 630 17.21 157 1.1 10.1 4 0.5, 50 + 17.3 + 15.9 + 0.1 7501-17.3 -15.9 + 0.1. 260 17.3 15.91 - 0.5 760 17.2 15.9 0.7 Jul 2'70 3 17.2 1 t.1 770 17.1 15.7 1.21 10 0.7 0.4 280 17. 15.6 1.6 780 16.9 15.4 1.8 20 1.0 0.3 171lj630 1. -0.1 290 16.9 15.4 2.2 790 16.61 15.3 24. 3001 + 16.6 + 15.1 - 800.316 15.0 + 2.91 1 + 3101 16.2 14.8 3.3 810 15.9 1.7 3 9 1 0 32()0 15.8 14.5 3.9 820 15.5 14.2 4)1 29 1. 0.4 330 15.3 14.11 4.4 830 15.0 13.8 29 0.9 0.4 340 14.8 13.6 4.9 840 14.4 13.2 5.0 Sept. 3501 14.21 +. 13.0 - 5.411 8501 -13.8 -12.6 +5.5 8 0.6 0.5 360 13.65 12.4 5.9 860 13.1 12.0 5.91 8 1+ 0.2 05 28 - 0.2- 0.5 370 12.8 11.7 6.31. 870 12.4 11.4 6.311 38so0 12.0 11.1 6.7 880ss 11.6 10.6 6.7 Oct. 390 11.2 10.3s 7.1 890 10.8 9.9 7.1 8 0.6 0.5 400 + 104 + 9.4 - 7.5 900 -10.0 9.1 + 7.5 18 1].0 0.3 410 9.5 8.7 7.8 910 9.1 8.4 7.8 28 1.2 0.2 420 8.5 7.8 8.1 920 8.2 7.5 8.1 Nov. 1430 7.5 6.9 8.4 930 7.2 6.61 8.3 7 1.2 4- 0.0 440 6.5 6.0 8.7 940 6.2 5.7 8.51 17 1.2 0.2 i450 + 551 + 4.9- 8.9 9501_ 5.21 - 4.8 +8.7 27 1.01 0.4 1460 4.4 4.01 9.0 960 4.2 3.9 8.9 Dec. 470 3.3 3.0 9.2 970 3.2 2.9 9.0 7 0.6l 0.5 480 2.2 2.11 9.3 980 2.1 2.0 9.1 17 -0.2 0.5 1490 1. 1 0'. 9.3 990 1.0 1.0 9.1 27 -t 0.31 0.5 l0oo + 0.0 + 00 9.331000 - 0.0 0.0 + 2.I2 37 + 0o - 0.5 TABLE XXVIII. TABLE XSXIX'3 Lunar Equation, I st pat: t. Lunar Equation, 2d part. Argument I. Arguments I. arid VI. I JEqua I Equ VI 0 50'100 150 200250 300350 400 450500 0 7.5 500 7.5 0 1.3 1.2 1.2 1.1 1.0 1.0 1.0 1.1 1.2 1.2 11.3 10 8.0, 510 7.0 5011.5 1.5 1.5 1.3 1.1 1.0 0.9 1.0 1.1 1.111.1 20 8.4 520 6.6 100 I 1.7 1.8 1.7 1.4 1.2 1.1 1.0 0.90.9 0.9 0.9 30 8.9 530 16.1 150 1.9 1.9 1.8 1.6 1.4 1.3 1.0 0.8 0.8 0.8 0.7 40 9.4 540 5.6 20011.9 2.0 2.0 11.7 1.5 1.4 1.010.8 0.8 0.8 0.7 50 9.8 550 5.2 250 22.0 2.0 1.8 1.6 i.5 1.1i0.9 0.7 0.7 0.6 60 10.3 560 4.7 300 1.9 1.9 1.9 9 1.7 1.6 1.2 1.0 0.8 0.7! 0.7 70 10.7 570 4.3 350 1.8 1.9 1.9 1.9 1.7 1.61.4 1.0 1.0 0.9 0.8 80 11.1 580 3.9 400 1.G 1.7 1.S 1.9..7 1.6 1.4 1.2 1.1l1.0l1.0 90 11.5 590 13.5 450 1.5 1.5 1.6 1.7 1.7 1.7 1.6 1.4 1.2 1.2 1.1 100 11.9 60013.1 500 1.3 1.4 1.4 1.5 1.7 1.7 1.7l1.5 1.4 1.4 1.3 110 12.3 610 2.7 550 1.1 1.2 1.2 1.4'1.6 1.7 1.' j7 1.6 1.5 1.5 120 12.6 620 2 600 1.01 1 11.2 1.4 1.6 1.8 1.8 1.8 1. 1 1.6 130 13.0 630 2.0 65010.8 0.9 1.0 1.1,1.3 1.5 1.7 1.8 1.9 1.9 1.8 140 13.3 640 17 700 0.7 0.7 0.8 1.1 1.2 1.4 1.7 1.9 1.9 1.9 1.9 150 |13.6 650 1.4 750 0.6 0.6 0.7 1.0 1.1 1.3 1.6 1.9 1.9 2.0 2.0 160 13.8 660 1.2 800 0.7 0.7 0.7 0 9 1.1 1.2 1.5 1.8 12.0 1.9 1.9 170 14.1 670 0.9 85010.7 0.8 0.8 0.9 0.9 1.1 1.4 1.7 1.8 1.8 1.9 1801 14.3 680 0.7 900 0.9 0.90.910.9 1.0 1.1 1.211.5 1.7 1.7 1.7 190 14.5 690 0.5 950 1.1 1.0.1.1 1.01.0 1.0 1.1 1.3 1.4.0 1.6 1.5 200 14.6 700 0.4 0 1.3 1.2 1.2. 1.0 1.0 1.1 1.2i1.2 1.3 210 14.8 710 0.2 I. 220 14.9 720 0.1 240 15.0 740O 0.0 250 15.0 750 0.0 0 1.3 1.4 1.4 1.5 1.6 1.6 1.6 1.5 1.4 1.4 1.3 1 260 15.0 760 0.0 50 1.1 1.1 1.2 1.3 1.5 1.5 1.7 1.61.5 1.5 1.5 270 14.9 770 0.1 10010.9 0.9 0.9 1.1 1.3 1.5 1.6 1.7 1.7 1.7 1.7 280 14.9 780 0.S 150 0.7 0.8 0.80.9 1.2 1.4 1.6 1.9 18 8 1.9 1290 14.8 790 0.21 20010.7:0.7 0.6 0.8 1.1 1.2 1.6 1.8 l.8 1.8 1.9 1300 14.6 800 0.4 250 0.6 0.6 0.7 0.7 1.0 1.1 1.5 1.7 1.9 11.9 2.0 310 14.5 810 0.5 300 0.7 0.7 0.7 0.7 0.9 1.0 1.4 1.6 1.8 1.9 1.9 320 14.2 820 0.7 350 0.810.7 0.7 0.8 0.9 1.0 1.4 1.6 1.6 1.7 1.8 320 14.1 830 0.9 400 1.0 0.9 0.8 0.8 0.9 1.0 1.1.4.6 1. 1340 13.8 84011.2 450 1.1 1.1 1.0 0.0.9 0.9 1.0 1.2 1.4 1.4 1.5 3510 13.6 85011.4 50011.311.2 1.2 1.1 0.9 0.9 0.9 1.1 1.2 1.2 1.3 1160 13.3 sf860 11.7 550 1.51 1.4 |1.21.0 0.9 0.9 0.9 1.0 1.1 1.1'370 13.0 87012.0 600 1.61116 1.5 1.4 1.2 1.0108 0.8 0.8 0.9 1.0 3:80 12.6 880 2.4 65C 1.8 1 1.6 1.6 13 1.1 0.9 0.8 0.7 0.7 0.8:390 12.3 890 2.7 700 1.911.8 1.811.61.4 1.2 0.9 0.7 0.7 0.7 0.7 100 11.9 900 3.1 750 2.0 1.9 1.9 1.761.5 1.3 1.0 0.7 0.710.6 0.6 410 11.5 910 3.5 800 1.9 1.8 1.SI 1.8 1.6 1.4 1.1 0.80.610.7 0.7 420 11.1 920 3.9 850 1.91.8 1.8! 1.8 1.6 1.5 1.210.9 0.8 0. 0.7'430 10.7 930 4.3 1900 1.7 71.711.711.6i 1.5 1.3 1.1 0.9:0.9 0.9 440 10.3 940 4.7 950: 1.5 1.5:1.511.6 1.7 1611.5 1.3 121 1. 1.1 450 9.8 950 5.2 0 1.3 1.4 1.401i.5l.6 1.6 1.611.5 1.41 1.4 1.3 4i |0 9.-4 960 5.6 Constant 1 ".3 470 89. 970 6.1 480 814 98016.6 4940 810 990 7.0 (0 1.50 100017.5!0.952 1. 4 5,. 1 i 24 TABLE XXX. Perturbations produced by Venus. Arguments II and III. III.. or10 20 30 1 40'50 160 70 80 90 100 110 120 0 21.6 20.8 19.8 19.0 17.9 16.8 15.9 14.7 14.0 13.2 12 8 i12 5 122. 209 23.1 22.7 21.6 21.0 20.1 19.3 18.4 17.4 1 6.4 15.5 14.5 13.8 13.4 40 23.5 23.2 22.9 22.7122.0 21.1 20.4 19.. 18.7 17.9 16.9, 16.1 15.3i 60 22.2 22.5 23.1 22.7 22.8 22.5 21.9 21.3 20.5 19.9 19.1 18. 17.4 i 80 20.0 29.7 21.4 21.7 22.1 22.3 22.2 22.2 21.7 21.3 20.7 1 19.9 19.3 100 17.6 13.6 19.2 19.9 20.5 21.0 21.6 21.7 21.6 21.6 21.5 21.1 29.5 120 15.3 16.0 16.9 17.7 18.4 19.2 19.8 20.2 20.7 20.8 21.1 21.1 20.8 140 113.6 14.2 14.8 15.5 16.2 17.0 17.6 18.3 19.0 19.4 20.0 20.0 20.4j 160 112.7 13.2 13.6 14.1 14.6 15.0 15.7 16.4 17.0 17.3 18.1 18.7 19.2 180 112.7 12.9 13.1 13.5 13.9 14.0 14.5 14.8 15.0 15.8 16.4 16.8 17.2 200 113.2 13.2 13.2 13.4 13.7 13.8 14.1 14.2 14.5 14.5 14.8 15.2 16.0 220 13.5 13.6 13.9 14.1 14.1 14.1 14.2 1143 14.5 14.6 14.6 14.7 14.8 240 13.6 13.8 14.1 14.4 14.6 14.8 14.8 14.9 15.1 15.1 15.1 14.9 14.8 260 12.8 13.3 13.8 14.2 14.6 15.0 15.3 15.6 15.5 15.5 15.6 15.6 15.6 280 11.5 12.3 13.0 13.4 14.0 114.6 15.1 15.4 16.0 16.2 16.2 16.3 16.2 300 10.1 10.9 11.3112.1 12.9 13.7 14.2 14.9 15.4 16.0 16.4 16.5 16.7 320 8.2 8.8 9.6 10.6 11.3 12.0 12.9 13.7 14.3 15.0 15.8 16.3 16.8 340 6.9 7.5 8.1 8.4 9.4 10.1 11.1 11.9 12.7 13.6 14.4 15.2 16.0 360 6.5 6.5 6.8 7.4 8.0 8.4 9.1 9.9 10.8 11.5 12.6 13.4 14.4380 6.8 6.5 6.3 6.4 6.7 7.0 7.6 8.2 8.9 9.6 10.6 11.4 12.4) 400 7.5 7.1 6.7 6.4 6.2 6.4 6.5 6.9 7.5 7.9 8.7 9.4 10.3 420 9.1 8.4 7.6 7.1 6.7 6.5 6.3 6.2 6.7 6.8 7.2 7.8 8.4 440 10.6 9.8 9.0 8.6 7.9 7.2 6.7 6.4 6.4; 6.4 6.6 6.8 7.1 460 1 12.1 11.5 10.5 9.6 9.0 8.5 8.0 7.3 6.8 6.6 6.5 6.4 6.51 480 13.6 12.8 11.9 11.0 10.4 9.6 8.8 8.2 7.7/ 7.2 6.8 6.4 6.51 500 15.1 14.4 13.4 12.4 11.6 110.S 10.11 9.3 8.6 8.1 7. 5 7.1 6.8 520 116.5 15.6 14.8 13.9 13.1 12.3 11.3 10.5 9.7/ 9.1 8.6 7.9 7.4 540 1 8. I 17.5 16.4 15.5 14.5 13.7 12.8 1 1.8 11.1 10.4 9.7 8.9 8.2 560 2)0.4 19.3 1 18.2 17.6 16.5 15.4 14.4 13.4 12.7 11.6 10.8 10.2 9.2 580 22.8 21.7 20.7 19.7 18.4 17.6 16.6 15.5 14.3 13.4 12.5 11.6 10.6 600 25.2 24.1 23.1 122.2 21.2 19.9 18.6 17.8 16.6 15.6 14.5 13.4 12.6 620 27.3 26.5 25.6 24.7 23.5 22.5 21.6 20.4 19.0 18.1 16.8 15.7 14.7, 640 29.0 28.5 27.7 26.9 26.2 25.1 24.1 122.9 21.8 20.8 19.6 18.4 17.2 660 29.8 29.6 29.2 28.5 28.1 27.4 26.5 25.6 24.5 23.4 22.5 21.2 19.8 680 29.7 29.6 29.5 29.5 29.1 28.8 28.2 27.6 27.0 26.0 25.0 23.8 22.8 700 28.8 29.2 29.3 29.5 29.5 29.5'9.2 28.2 28.4 27.27.8 127.2 26.4 25.2 720 26.9 27.6 28.3 29.0 29.2 29.4 29.4 29.3 129.1 28.9 28.4 27.9 27.3 740 24.7 25.7 26.6 27.3 27.9 28.5129.1 29.0 29.2 29.3 29.1 28.8 28.4 760 22.2 23.5 24.3 25.3 26.2 27.0 27.6 28.3 28.6 28.7 28.9 I 29.1 29.0 780 19.6 21.0 22.0 23.2 24.2 25.1 25.9 26.7127.3 27.8128.4 28.5128.7 800 17.2 18.5 19.3 20.9 21.8 22.9 23.9 25.0 25.8 264 26.9 27.6 28.1 820 15.2 15.9 17.0 18.4 18.9 20.7 21.7 22.8 23.8 24.8 25.6 26.2 26.6 840 13.2 14.0 15 0 16.0 17.0 18.2 18.8 20.3 21.7 22.7 23.6 24.5 125.3 860 11.5 12.2 13.0 13,9 14.9 15.9 17.1 18.0 18.9 20.3 21.4 22.6 23.5 88O 11.0 11.2 11.5 12.2 13.0 13.7 14.8 15.7 16.8 18.1 19.1 20.2 21.1 900 11.2 10.2 10.9 11.5 12.5 12.1 12.8 13.7 14.5 15.5 16.6 17.9 18.5 920 12.1 11.6 11.5 11.1 11.2 11.3 1 11.7 12.1 12.7 13.4 144 15.2 16.4 940 14.0 13.3 12.6 12.3 11.6 115 11.3 11. 11.6 12.0 12.8 13.3 14.2 960 16.7 15.6 14.6 13.7 13.1 12.5 11.9 11.7 11.6 11.4 11.7 12.1 12.6 980 19.5 18.3 17.3 16.4 15.2 14.2 13.4 12.7 1 1.2 12.0 11.9 11.8 11.8 1000 21,6 20.8 19.8 19.0 17.9 16.8 15.9 14.7 14.0 13.2 12.8 12.5 12.2 0I 10 20 30 40 5) 60 70 80 90o 00110120 TABLE XXX. 25 Perturbations produced by Venus. Arguments II and III. III. H. 1120 1301 1401 150 160 170 180i 190 200 210 220 230 230 240 0 12.2 12.2 12.3 12.4 12.8 13.3 13.9 14.7 15.6 16.5 17.7 18.8 20.1 20 13.4 12.9 12.6 12.3 12.2 112.4 12.9 13.3 14.0 14.6 15.5 16.4 17.3 40 15.3 14.4 14.0 13.5 13.0 12.9 12.6 12.6 13.1 13.5 14.0 14.4 15.4 60 117.4 16.7 16.0 15.2 14.5 14.0 13.6 13.3 13.2 13.2 13.4 113.5 14.1 801 19.3 18.7 17.7 17.1 16.4 15.9 15.4 14.6 14.3 13.9 13.8 13.7 13.6 100 20.5 20.2 19.5 18.9 18.2 17.5 17.1 16.3 15.9 15.4 14.8 114.6 14.3 120 1 20.8 20.7 20.4 20.0 19.7 19 2 18.5 118.0 17.3 16.9 16.5 16.2 15.6 140 20.4 2 20.4 20.2 20.0 2 0.1 19.7 19.5 19.3 18.S 1S.2 17.7 17.4 17.0 160 19.2! 19.1 19.4 19.7 19.5 19.6 19.3 19.6 19.2 19.0 18.7 18.4 18.1 180 17.2 117.7 18.5 18.5'i18.5 18.8 18.4 18.8 119.0 19.0 18.9 18.6 18.5 200 ]16.0 16.2 16.6 16.8 17.5 17.6 17.7 1.9 1 8.1 18.2 18.3 18.3 18.3 220 14.8 15.0 15.3 15.7 16.1 16.2 16.6 116.8 17.1 17.5 17.1 17.4 17.5 240 14.81 14.7 14.8 15.0 15.1 15.4 15.7 15.8 16.0 16.1 16.1 16 9 16.t 260 15.6 115.7 15.3 14.8 15.0 15.0 15.1 15.0 15.1 15.2 15.2 15.1 15.3 280 16.2 16.2 16.2 15.9 15.8 15.8 15.5 15.4 15.1 14.9 14.8 14.7 15.0 300 16.7 17.0 17.1 16.9 16.9 16.6 16.5 16.3 15.9 15.7 15.2 14.9 14.8 320 16.8 17.3 17.5 17.6 17.7 17.6 17.5 17.2, 17.0 16.8 16.5 16.1 15.6 340 16.0 16.4 17.2 11'.8 17.9 18.1 18.3 18.2 18.2 17.9 17.5 17.3 116.8 360 14.4 15.2 16.01 1.6.7 17.4 18.1 18.4 18.6 18.8 18.8 18.8 1S.7 18.4 380 12.4 13.4 14.3 15.3 16.1 16.9 17.5 18.1 18.6 19.1 1]9.3 19.5 19.5 400 10.3 11.2 12.3 13.2 14.2 15.1 16.0 16.8 17.8 18.4 18.8 19.3 19.8 420 8.4 9.2 10.0 11.0 12.2 13.0 14.1 15.0 15.9 16.9 17.7 18.5 19.0 440 7.1 7.6 8.4 9.0 9.9 10.9 11.8 112.9 13.8 11.9 16.0 1G.7 17.8 460 6.5 6.8 7.2 7.4 8.1 9.0 9.7 10.6 11.7 12.6 13.8 14.6 15.9 480 6.5 6.5 6.4 6.6 7.0 7.5 8.2 8.8 9.6 10.4 11.5 12.5 13.5 500 6.8 6.7 6.5 6.3 6.5 6.6 7.0 7.4 S.2 8.6 9.4 10.4 11.3 520 7.4 7.0 6.8 6.b/ 6.3 6.1 6.3 6.6 7.0 7.5 8.0 8.8 9.3 54:0 8.2 7.6 7.2 6.8 6.5 6.3 6.2 6.0 6.2 6.5 6.9 7.4 7.9 560 9.2 8.6 7.9 7.5 6.8 6.6 6.3 6.1 6.0 6.1 6.2 6.5 6.9 580 10.6 9.8 9.1 8.4 7.7 7.3 6.6 6.3 6.1 5.9 5.7 5.9 6.0 600 12.6 11.4 10.5 9.5 8.7 8.11 7.4.0 6.4 6.1 5.8 5.5 5.6 620 14.7 13.5 12.4 11.4 110.4 9.5 8.7 7.9 7.3 6.7 6.2 5.6 5.2 640 17.2 16.2 14.9 13.7 12.5 11.4 10.4 9.5 8.7 7.8 7.0 6.5 5.9 660 19.8 19.0 17.6 16.5 15.1 13.9 12.8 11.5 10.5 9.G 8.6 7.7 6.9 680 22.8 21.7 20.4 19.3 18.1 16.8 15.7 14.2 13.0 11.9 10.7 9.6 8.6 700 2;5.2 24.3 23.3 22.1 20.7 19.7 18.5 17.3 16.0 14.3 13.4 12.1 11.0 720 27.3 26.4 25.7 24.5 23.7 22.5 21.1 20.2 18.8 17.7!6.4 15.3 13.99 740 128.4 27.7 27.4 26.6 25.9 24.9 24.0 22.8 21.5 20.6 19.2 18.1 16.8 760 29.0 28.7 28.3 27.8 27.3 26.8 25.9 25.2 24.3 2 3.0'21.7 20.7 19.7 780 128.7 28.7 8 28.8 28.7 28.3 28.0 27.2 26.1 26.1 25.2 124.3 23.3 22.2 800 28.1 28.3 28.4 28.5 28.5 28.4 28.2 27.3 27.3 26.7 25.9 25.1 24.4 820 26.6 27.3'7.8 28.1 28.3 28.1 28.1 S8.0 27.9 27.7 27.2 26.5 25.9 840 25 3 26.2 26.7 27.2 27.5 27.9 28.1 28.1 27.9 27.9 27.6 27.3 27.2 860 23.5 24.5 25.1 25.9 26.6 27.1 27.4 27.7 27.93 28.0 27.9 27 7 27.5 880 21.1 22.4 23.3 24.2 25.1 25.8 26.5 27.0 27.3 27.5 27.8 28 0 27.7 900 18S.5 20.1 21.3 22.11 23.1 24.7 25.0 25.7 26.3 26.9 27.3 27.5 2 7.6 920 16.4 17.7 18.4120.0 121.0 22.O 23.0 23.9 24.9 25.7 26.2 26.9 27.3 940 14.2 14.9 16.1 17.5'18.2 19.6 20.S 21.9 23.0 23.9 24.7 25.7 26.1 960 12.6 13.3 14.1 1414 15.9 17.0 17.9 19.5 20.5 21.7 22.7 23.9 24.7 980 11.8 12.1 12.7 13.3 114.1 14.8 15.6 16.8 17.6 19.3 20.2 21.4 22.6 1000 12.2 12.2 12.3 12.4 112.8 13.3 13.9 14.7 15.6 16.5 17.6 18.8 20.1 20 130 140 1501 160 170 180 190 200 210 220 230( 24 [ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _",9o_ 26 T ALE XXX. Perturbations pr-oduced by Venus. Arguments I1. and III. II'. 2II. 240 2.50 260 270 280 290 300 310 320 330 340 350 360 0 20.1 21.1 22.2 23.4 24.3 25.2 25.8 26.6 27.2 27.6 27.7 27.6 27.6 20 17.3 18.6 19.7 20.9 21.9 23.0 24.2 24.9 25.8 26.6 27.0 27.4 27.7 40 15.4 16.5 17.3 18.3 19.4 20.5 21.6 22.7 23.7 24.9 25.5 26.3 26.9 60 14.1 1415.2 16.3 17.2 18.1 18.9 20.3 21.2 22.3 23.4 24.5 25.2 0 1 3.6 14.0( 14.5* 14.9 15.5 16.3 17.3 18.2 19.0 20.0 21.1 22.0 23.1!09i 14.3 14.3 14.3 14.4 14.6 15.0 15.5 16.2 16.9 17.7 18.9 19.8 20.8'120 15.6 15.2 14.8 14.8 1.5.0 14.9 15.0 15.2 1.5.9 16.3 17.0 17.7 18.5 i40 17.0 16.6 16.4 15.8 15.5 15.4 15.6 15.6 15.5 15.6 16.1 16.7 17.1 I 6:18.1 17.7 17.5 i7.3 16.9 16.6 16.3 15.9 16.1116.3 16.3 16.2 16.5 ld0!18.5 18.5 3 18.1 17.9 117.6 17.5 17.3 17.0 16.9 16.7 16.8 16.9 200 1 18.3 18.41 18.2 18.2 18.2 j 1S.2 1 18.1.1 11 7.8 7 17.6 17.5 17.7 g20 1 17.5 17.6 17.8 17.8 18.0 18.0 18.0 1 183 18.4 18.3 8.3 240' 164 16.5 16.7 16.9 17.1 17.3 17.3 17.7 17.5 18.0 18.3 18.4 18.6 260 15.3 1i5.5 15.5 15.6 15.8 1.6.1 16.4 16.6 16.8 16.9 17.4 17.7 18.2 280 15.0 14914.9 14.9 14.9 14.7 15.0 15.3 15.5 15.9 16.1 16.4_ 16.8 300 14.8 14.6 14.6 14.2. 14.0 14.0 13.9 13.9 14.2 14.5 14.8 15.0 15.5 320 15.6 15.3 14.7 14.5 14.4 13.1 13.6,13.4 13.3 13.1 13.4 13.6 13.8 340 16.8 16.6 16.0 15.5 15.2 14.5 14.3 13.1 13.0 12.7 12.6 12.6 360 18.4 179 17.5 17.0 16.5 15.) 15.4 14.9 1 4.3 13.7 13.0 12.6 12.3 380 19.5 19.2 18.9 18.5 117.9 17.7 16.9 16.4 15.8 15.0 14.5 13.6 13.1 400 19.8 19.8 20.1 19.7 19.4 19.1 18.6 18.1 117.5 17.0 16.1 15.2 14.8 420 19.0 19.5 20.0 20.3 20.3 20.3 20.1 19.4 19.0 18.9 18.1 17.3 116.5 440 17.8 18.7 19.2 19.7 20.1 20.4 20.7 20.7 20.5 20.2 119.8 19.5 18.C 460 15.9 16.8 17.6 18.6 19.2 19.9 20.3 20.6 21.0 120.9 20.9 20.8 120.3 480 13.5 14.6 15.5 16.6 17.7 18.5 1 19.3 19.9 20.5 20.8 21.1 21.2 21.2 500 111.312.4 113.4 14.4 15.5 15.5 17.7 1S.6 19.1 19.9 20.7 21.0 21.4 520 9.3 |10.2 11.2 12.2 13.3 14.2 15.4 16.4 17.6 18.4 19.2 19.8 20.6.540 7.9 8.6 9.4 10.1 11.1 112.1 13.1 14.2 15.3 16.3 17.4 18.3 19.2 560i 6.9 7.2 7.8 8.4 9.2 110.1 11.0 1.1.9 13.1 14.1 15.2 16.2 17.2 580 6.0 6.3 6.6 7.0 7.61 8.11 9.1 9.9 10.9 11.9 12.9 14.1 15.0 600 5.6 5.6 1 5.8 6.1 6 5 6.8 7.4 8.1 8.8 9.9 10.7 11.8 1 2. 6201 5.21 5.4 5.3 5.3 5.5 5.9 6.3 6.6 7.2 8.0 8.7 9.5 10.6 640 5.91 5.6 5.2 4.9 5.0 5.0 5.21 5.5 5.8 6.4 7.0 7.61 8.5 6601 6.9 6.3| 5.7 5.4 5.0 4.8 4.51 4.7| 4.9 5.1 5.5 6.0 6.8 680 8.6 7.6 6.9 6.2 5.6. 5.1 4.8 4.6 4.2 4.2 4.5 4.6 5.1 700 11.0 10.0 S.7 7.8 6.8 6.3 5.6 5.0 4.6 4.2 4.2 4.0 4.2 720 13.9 12.5 11.2 10.3 9.1 7.9 7.1 6.2 5.6 4.8 4.5 4.2 3.8 740 16.8 15.5 14.4 13.0 11.7 10.5 9.4 8.4 7.2 6.5 5.6 5.0 43 760 19.7 18 5 17.2 15.9 14.7 13.5 12.2 1.'8 9.8 8.9 7.6 6.7 5.9 9 780 122.2 21.2 20.1 19.0 17.6 16.3 15.1 14.0 12.6 11.6 10.2 9.21 8.1 8001 24.4 23.4 22.2 21.3 20.3 19.2 18.0 16.7 15.4- 14.3 13.2 11.9 10.8 820, 25.9 1 25.1 124.4 23.3 122.3 / 21.6 20.4 19.4 18.2 17.2 15.9 14.6 13.6 40 27.2 26.6 5.8 25.0 24.3 23.. 22.4 21.6 20.5 1 1.4 118.4 17.3 16.4 860 27.5 27.1 26.8 26.4 25.5 24.8 24.31 23.3 22.2 21.5 20.5 19.6 18.4 880 27.7 27.5 27.2 27.0 26.5 26.0 25.5 124.7 24.1 23.2 22.0 21.41 20.4 900 27.6 27.8 127.9 27.6 1 27.1 26.7 126.5 25.7; 25.3 24.6 23.9 23.0 22.0 920 1.27.3 1 27.5 27.5 27.6 27.7 275 2 7. 26.7 26.3 25.7 25.1 24.3 23.6 940 26. 1 26 7 2'7.2 27.4- 21.77 27.7 27.6'7.5 27.1 26.6 26.2 25.6 1 2.5 96O 24.7 25.4 26.2 26.6 127.2275 277 27.7 27.6 27.4 27.1 27.0 26.2 980 1 2.2 23.7,24.6,,.3 5.5.9 2 6. 2"7. 2 27.5 27.7 27.8 27.6 27.5 27.1 10001 201 21.1 22" 23.41 24.31 a 2a58 26.6, 27.2 27.6 27.7127.6 27.6 L 4i 0 21i3 i.20,3 270 280 90 i 330 310 320 330 340 j 350 360 L.1 TABLE XXX. 27 Pertturbations produced by Venus. Arguments II. and III. III. II. 360 370 380 390 400 410 420 430 440 450 4601 0 470480 0 27.6 27.7 127.3 26.7 26.2 25.5124.7 23.8 23.1 22.3 21.3 20.2 19.3 20 27.7 27.s 827.8 7.6 27.4 26.8 26.2 25.6 24.8 24.0 23.1 22.0 20.9 40 26.9'27.3 27.6! 27.9 27.9 27.7 27.5 27.1 26.3 25.6 24.9 24.01 23.2 60 25.3 26.0 i 26.8 27.1 27.5 27.9 27.8 27.7 27.3 27.1 26.7 25.9 25 0 80 23.1 24.0 25.1 25.9 26.5 27.3 127.5 27.9 28.2 28.0 27.6 27.5 27.2 100 20.8 m 21.8 22.6 23.6 24.6 25.5 26.2 26.7 27.2 27.5 27.6 27.8 27.4 120 18.5 19.6 20.6 21.5 j 22.4 23.2 24.1 25.1 25.8 26.4 26.9 27.3 1 27.5 140 17.1 17.9 18.6 19.3 20.3 21.3123.0 22.9 23.7 24.7 25.5 26.0126.7 160 16.5 17.1 i7.4 18.1 18.8 19.3 20.1 21.0 21.9 22.6 23.5 24.2 25.1 180 16.9 17.0 17.1 17. 18.0 118.4 118.9 19.4 20.1 20.7 21.2 22.2 23.0 200 17.7 17.5 17.7 17.7 17.6118.1 18.3 18.7 19.2 19.7 20.1 20.8 21.5 220 18.3 18.2 18.3 18. 3 18.3 118.3 1S.6 18.7 18.9 19.3 19.5 20.0 20.4 240 18.6 18.8 18.9 18.9 18.9i 19.0 19.2 19.1 19.2 19.5 19.6 19.7 19.9 260 18.2 18.5 18.7 18.819.0 19.3 119.5 19.6 19.9 19.9 20.0 20.1 20.2 280 16.8 17.4 17.9 18.73 187 19.1 19.3 19.8 20.0 20.2 20.4 20.6 20.8 300 15515.8 16.2 16.6 17.6 18.1 18.5 19.2 19.4 19.9 20.6 20.8 20.9 320 13.8 14.2 14.6 15.1 15.6 16.2 16.8 17.7 18.3 18.9 19.5 20.1 20.8 340 12.G 612.9 13.0 13.3 13.7 14.4 14.9 15.5 16.2 17.1 18.0 18.6 19.4 360 12.3 12.1 11.9 12.0 12.3 12.5 13.0 13.4 14.2 14.9 15.7 16.5 17.3 380 13.1 112.5 11.9 11.6 11.5111.4 11.6 11.7 12.3 12.7 13.3 14.0 15.0 400 14.8 13.9 13.1 12.5 11.7 11.2 11.1 10.9 11.0 11.1 11.4 12.0 12.6 420 16.5 15.7 15.1 14.3 13.4 12.5 11.7 11.1 10.8 10.S 10.5 10.6 10.7 440 18.6 17.9 17.1 16.1 15.6 14.4 13.5 12.8 11.9 11.1 10.6 10.3 10.2 460 20.3 19.8 19.3 18.5 17.6 16.8 15.9 14.7 13.7 12.9 12.0 11.1 10.9 480 21.2 21.1 120.8 o20.3 19.7 19.1 18.3 17.4 16.4 15.0 14.1 13.2 12.2 500 21.4 21.4 21.4 21.3 21.1 20.8 20.0 19.5 18.8 17.S 17.0 15.7 14.4 520 20.6 21.2 21.7121.7 21.5 21.5 21.4 21.1 20.5 19.8 19.1 18.2 17.6 540 19.2 20.0 20.7 21.1121.S 22.0 21.8 21.7 21.5 21.2 20.9 20.3 19.6 560 17.2 18.4 19.0 20.0i 20.821.1 22.7 21.9 22.2 22.1 21.9 21.7 21.1 580 15.0 16.0 17.3 18.2 19.1 19.9 20.8 21.1 21.7 22.0 22.2 22.3 22.1 600 1.8 13.9 15.1 1.5.9 17.2 18.0 19.0 19.9 20.6 21.3 21.8 22.0 22.4 620 10.6 11.5 12.7 13.7 14.9 16.0 17.1 18.3 19.1 19.9 20.8 21.3 22.0 6401.5 9.5 10.4 11.3 12.3 13.7 14.9 16.0 17.1 18.1 19.0 19.9 20.7 660 6.8 7.4 8.2 9.1 10.1 11.1 12.2 13.6 14.6 15.S 17.1 18.1 19.0 680 5.1 5.7 6.4 7.1 7.9 8.7 9.7 11.0 121131 14.1 15.7 16.8 700 4.1 4.4 4.7 5.1 5.8 6.7 7.4 8.4 9.4 10.6 11.5 13.0 14.1 720 3.8 3.8 3.8 4.0) 4.4' 4.8 5.4 5.9 6.9 8.0 9.1 10.1 11.5 740 4.3 3.9 3.8 3.7 3.6 3.81 3.9 4.4 4.9 5.7 6.4 7.4 8.9 760 5.9 5.1. 4.0 3.61i 3.4 3.4 3.5 3.91 4.3 4.7 5.2 5.9 780 8.1 7.1 6.1 5.3 4.61 4.1 3.7 3.3) 3.3 3.1 3.4 3.6 41 800 10.8 9.7 8.5 7.5 6.5 5.6 4.9 4.2 3.8 3.4 3.2 3.1 3.1 820 13 6 12.5 11.2 10.1 9.0 8.0 6.9 6.1 5.3 4.7 3.9 3.7 3.1 840 16.415.1 13.7 12.9 11.71 10.6 9.5 8.6 7.5 6.6 5.7 4.9 4.4 860 18.4 1 7.5 16.6 15.4 14.3 13.1 12.1 11.1 10.0 9.1 7.9 7.01 6.3 880 20.4119.6 18.7 17.5 1.6 15.6 1 14.5 13.6 12.5 11.5 10.4 9.5 8.6 9001 22.0 21.1 20.2 19.4 18.7 17.7 16.5 115.7 14.7 113.8 12.5 11.9 10 9 920 23.6 22.7 21.7 21.1 20.1 19.4 18.4 17.5 16.7 15.6 14.8 13.9 13.1 940 25.5 24.1 j23 4 22.4 21.4 i 20.6 j 19.9 19.0 8.2 17.3 16.6 15.7 14.81 960616.2 25.61 24.7 121-.1 23.3 22.3 21.3 20.6 19.0 18.9 17.9 17.1 16.31 980 27.1 26..3 5.5 2 4. 9 123.8 23.4 22.2 21.0 20.4 19.4 I8.47 4... ~il?~ql TABLE XXX. Perturbations produced by Vcnus. Arguments II and III. II. I. 1480 490 500 510 520 530 540 550 560 570 5)O G590 600 0 9 19.3 118.3 17.4 16.6 15.7 15.0 14.2 13.6 13.1 12.3 11.7 1 11.3 l0 8 20 20.9 20.2 19.1 18.2 17.1 16.2 15.5 14.7 14.1 13.3 12.7 12.2 11.5 40 23.2 22.0 20.8 20.1 18.9 17.9 17.1 15.9 115.1 14.4 13.7 13.0 12.3 6i0 5. 0 210 23.2 {22.0 20.7 19.9 18.9 17.7 16.8 15.8 14.9 14.0 13.3 80o?:7.2 26.4 25.6 24.1 23.2 22.1 20.8 20.0 18.7 17.9 16.6 15.6 1 4.0 100 "7.4 27 2 26.8 26.3 25.4 24.5 23.5 22.2 20.9 20.0 18.6 17.6 1G.G.2 7'7.5 27.5 027.6 2 7.1 2G.8 26.3 25.4 21.6 23.7 22.4 21.0 0.1 1 8 6. 25. 4 24.6 23.9;26.7 27. 0't 2 27.''74 27.3 27.r1 26.9 -.. 65.4 2.6 23 9 22.6 21 I [60 21 5. 25.6 26.1 26.7 26.9 27.3 27.1 27.0 26.9 26.4 25.5 24.7 23.9 110 23.0 3 8 24.5 25.0 25.7 26'.3 26.7 26.8 27.0 26.8 26.6'26.2 25. 200 21.5 22.2 22.8 23.5 24.1 2 4.7 i 25.5 25.S 26.3 2G.6 2.6 26.6 26.6 6 220 20.4 21.0 21.5 22.02202.6 23.2 23.S 24.5 25.0 25.4 25. 26.26.0 26. 240 19.9 20.4 20.8 21.2121.6 2. 22.2 2.6 23 1 23.3 23.9 24.2 24.6 260 20.2 20.3 20.6 21.2 21.4 21.7 21. 9 22.2 223 22.7 23.1 23.3 23.6 280 20.S 20.8 21.0 21.1 21.3 21.4 21.5 21.8 22.0 22.2 22.7 23.0 23.3 300 20.9 21.0 21.5121.72 21.7 22.0122.0 22.1 221 22.2 224 22.6 22.8 320 20.8 21.2 21.5 1.6 22.0 22.3 22.5 22.5 22 6 22.7 22.8 22.8 22.9 340 19.4 20.2 20.8 21.5 21.9 22.1 22.6,23.0 23.2 23.4 23.3 23.4 23.5 360 17.3 1S.4 19.5 20.0 20.6 21.5 22.2 22.7 23.0 23.7 23.7 24.0 24.2 380115.0 15.9 16.9 17.8 18.6 19.6 20.6 21.5 22.3 22.9 23 5 23.9 24.5 400 12.6 13.2 14.2 15.4 16.2 17.3 18.] 19.2 20.3 21.4 224 23.0 23.7 420 10.7 11.2 12.0 12.5 13.5 [4.5 15.6 16.7 17.7 18.7 201121.0 22.0 440 10.3 10.2 10.3 10.5 11.3 12.0 12.9 13.6 14 7 16.0 17.0 18.3 19.5 460 10.9 10.1 9.9 9.9 9.9 L0.1 10.7 11.3 12.2 L3.0 140 15.1 16.5 480 12.2 11.4 10.7 10.1 9.7 9.5 9.7 9.9 10.2 10.7 11.7 12.5 13.4 500114.4 13.6 12.5 11.6 10.9 10.2 9.8 9.4 9.3 9.6 9.8 10.2 11.1 520 17.6 16.2 15.1 13.9 12.9 [1.9 10.9 10.3 9.8 9.5 9.2 9.2 9.6 540 19.6 18.6 18.0 16.7 15.4 14.5 12.2 12.3 L1.3 10.5 10.1 9.5 9.3 560 21.1 20.4 19.8 19.0 18.2 17. 1 16.0 14.8 13.7 12.7 11.7 10.9 10.2 58 2.1 21.8 21. 21.5 20.9 20.3 L9.3 18.6 17.3 16.5 [15.4 14.0 129 12.2 600 22.4 22.4 22.2 22.2 21.5 21.2 20 6 19.5 19.1 7.7 16.8 15.8 14.4 620 22.0 22.3 22.4 22.4 22.3 22.3 219 21.5 20.9 20.0 19.3 18.0 16.9 640 1 20.7 21.7 22.0 22.3 22.6 22.5 22 6 22.4 22.0 121.6 21.1 20 3 19.6 66, 0 19.0 20.0 120. 21.3 22.1 22.3 22 6 22.s 22.7 22.6 22.2 21.8 21.3 680 16.8 18.0 [9.0 19.9 20.8 121.5 522 1 22.6 22.7 23.0 23.0 22.8 22. t 700 14.1 15.2 16.8 17.9 18.8 20.0 22 1 21.5 2.2 22.6 22.9 23 0 23.2 720 11.5 12.7 13.9 15.0 16.4 1'7.9 18.6 19.7 20.8 21.6 22.3 122 7 23.0 I 40 8.9 9.8 10.9 12.2 13.6 0 14.8 16.2 17.5 18.7 19.5 20.6 21.6 22.3 I,760 5.9 6.8 8.0 9.3 10.3 11.8 113.2 14.5 15.9 17.4 18.2 19.5 20.5 7801 4.1 4.9 5.6 6.4 7.5 8.6' 9.9 11.1 12.6 14.0 15.6 16.8 18.1 8001 3.1 3.3 4.4 4.8 5.5 6.1 6.9 7.9 9.4 10.7 12.1 13.4 14.9 8201 3.1 3.1 3.2 3.1 3.6 3.9 4.8 5.7 6 5 75s 8.7 10.0 11.5 840 4.4 3.7 3.5 3.2 3.2 3.11 3.4 3.7 4.1 5.0 6.2 7.0 8.2' 86 0; 6.3 5.5 416 4.6 4 3.3 3.2 3.4 3.4 4.0 4.5 5.6 S808 8.6 7.6 6.7 5.9 5.2 4.5 4.1 3.8 3.5 3 4 3.4 3.6 3.9 g9oo0 10.9 10.0 9.1 8.3 7.2 6.5 5.8 5.1 4.4 421 3.8 3.6 3.6 920 13.1 12.1 11.2 10.3 9.6, 8.7 7.7 6.9 6.3 5.8 5.1 4.6 4.2 940 i 14.8 14.1 13.1 12.4 11.5 10.8 9.81 9.1 8.3 7.6 6.8 6.5 5.9 C960 16.31I5.4 14.6 14.0 13.2 12.6 11.7 tl1.0 10.1 9.6' 8.8 8.1 7.5 9)80 17.7 16.8 16.2 115.2 14.5 13.9 13.1 1t2.5 11.8 11.2 10.5 9.7 9.3 1(000 19.3 18.3 17.4 16.6 15.7 15.0 14 2 13. 13.1 12.3 11.7 11.3 10.S 40 1490 500 510 520 530 540 2550 5 I 570 1 580 590 3.3 480 490 500 510 520 530 540 150 50 60 1570 580 590 6 1)( TABLE XXX. 29 Perlurbations produced by Venus. Arguments II. and III. 11I 60 600 610 620 630 1 640 650 660 670 680 690 700 710 720 0 10.S 10.2 951 91 8.4 7.91 7.4 7.0 6.6 6.3 5.9 5.5 5.4 20 11.511.3 10. 10. 4 9.8 9.41 8.9 8.5 7.9 7.7 7.3 6.7 6.6 40 19.3.0 11.5 11.0 10.7 10.3 110.0 9.6 9.3 8.9 8.5 8.1 7.8 60 13.3'11.7 112.1 11.2 10.91 10.5 10.2 10.0 9.8 9.5 9.2 8.9 80 14.S 13.6 12.9 12.4 11.8 11.3 10.9 10.7 10.3 9.9 9.8 9.8 9.6 0 1.G [G 15.4 14.4 13.4 12.6 1 12.1 11.5 11.0 10.6 10.2 10.0 9.9 9.6 1209 1 S. 17.7 16.4 15.3 14.3 13.2 12.4 11.6 11.2 10.6 10.1 10.1 9.6 140 21.11 20.1 18.9 17.7 16.5 15.2 14.2 13.0 12.3 11.6 11.1 10.3 9.9 160 23.9 22.9 21. 5 20.4 19.2 17.9 16.6 15.3 14.1 13. 1 12.0 11.2 10.5 180 2 I5.G 21-.8 23.9 22.9 1 21.6 20.6 19.1 18.0 16.7 15.5 14.3 12.9 12.0 200 26.4 2G.0 25.6 24.9 24.0 22.9 21.7 20.8 19.3 18.1 16. 15.5 14.4 220 26.2 "26.3 26.1 25.8 25.3 24.9 24.1 23.1 21.2 20.9 19.7 18.3 17.1 24/-0 21 25.1 25.1 25.3 i 25.2 25.1 24.7 24.3 24.0 23.0 21.9 21.3 20.2 260 | 2.3.6 2 3. 9 24.2 24.5 14.7 24.8 24.9 24.6 24.3 23.8 23.4 4 -.9 21.6 280 2 2 23. 636 23.9 2z4.2 24.7 24.8 25.0 124.9 24.9 24.8 -.4 24.0 23.5 300 1 2.S8 3.0 23.3 23.4 23.8 24.0 24.1 24.5 24.5 24.6 24.5 24.4 24.0 320 2.9 3.0 23.1 23.2 23.4 23.3 23.6 23.8 24.0 23.9 24.0 24.2 24.2 3101 2:3.5 | 23.5 /2 3.5 23.4 23.5 1 23.6 23 6 23.5 23.5 23.6 23.9 23.8 23.8'360! 2'4' 2.4.2 24.3 24.2 24.2 24.0 23.7 23.9 24.0 237.23.7 23. 6 23.6 380 24.J 5 24.6 24.8 25.1 24.8 24.9 25.0 24.9 24.6 24.5 24.5 24.5 3 | 24.0 400 3. 7 |4. 3 24.7 25.0 25.4 25.7 25.7 25.5 25.5 25.4 25. 25 2 1.8 24.6 420 22.0 23.0 23.7 24.6 25.0 25.7 26.1 26.2 26.3 26.5 26.2 26.0 25.9 440 19.5 20.S 21.7 22.7 23.7 24.6 25.4 26.0 126.5 26.7 26.9 27.0 26.9 460 16.5 17.8 19.0 20.1 21.4 2.3 1 23.5 24.8 125.4- 26.1 26.7 27.1 27.3 480 13.4 14.5 15.6 17.0 18.5 19.7 20.9 22.1 23.2 24.4 25.4 26.2 26.8 500 111.1 12.0 13.0 13.8 114.9 16.3 17.9 19.1 20.5 21.6 22.9 24.2 25.1 520 9.6 9.8 10.5 11.5112.4 13.4 14.4 15.5 17.1 18.4 19.9 21.2 22.3 540 9.3 9. 1 9.2 9.6 110.3 11.0 11.9 12.8 13.9 15.1 16.5 17.9 19.4 560 110. 9.7 9.3 9.1 9.1 9.4 10.0 10.611.5 12.4 13.3 14.5 16.0 580 112.2 11.3 110.4 9.9 9.4 9.0 9. 9.3 9.7 10.4 11.0 1.2.0 12.7 600 14.4 13.3 12.5 11.6 10.8 110.1 9.6 9.4 9.1 9.43 9.9 10.0 10.8 620 16fi.9 16.1 14.9 13.7 12.7 12.0 11.1 10.4 9.8 9.5 9.5 9.3 9.7 640 19.6 18.4 17.4 1G.3 15.2 14.2 13.1 12.1 11.3 10.6 10.1 9.6 9.5 660'21.3 20.6 19.9 18.7 17.8 16.7 15.6 14.4 13.4 12.4 11.7 11.0 10.2 1680 122.4 22.0 21.5 20.S 20.2 19.0 18.1 17.0 15.8114.7 13.7 12.8 12.0 700 23.2 23.2 22.6 2 2.221.7 21.0 20.5 19.3 18.3 17.3 16.0 15.0 14.1 720 23. 0 23.3 i 23.2 29 4 23.1 102.4 21.9 21.3 20.8 119.5 18.5 17.6 16.4 740 22.3 22.8 23.2 23.4 23.6 23.6 23.3 22.8 22.2 21.6 21.1 19.9 18.8 760 20.5 21.4 22.51 22.8 23.3 23.7 2316 23.8 23.5 23.3 22.7 21.8 21.3 780 18.1 19.2 20.4 21.3 22.3 23.0 23.3 23.7 23.8 24.0 23.8 23.5 23.0 800 114.9 16.4 17.7 19.1 20.1 21.2 21.1 22.9 23.4 23.8 24.1 24.2 23.9 82o0 11.5 12.9 114.3 15. 17.8:18.7 20.0 20.9 1 2.0 22.7 23.5 23.9 24.0 840 8.2o 9.5 10.8 12.2i13.8 15.2 16.6 18.1 19.5 20.6 21.7 22.6 23.3 860 5.6 6.8 7.7 8.8 i 10.2 11.5 13.2 14.7 16.0 17.4 19.0 20.2 21.93 880 3.9 4.4 5.2 6.1 7.2 8.2 9.7 10.91 12.5 14.1 15.4 16.8 18.2 900 3.6 3.6; 3.9 4.2 5.0 5.'7 6.6 7.8 9.1 10.3 11.8 13.4 14.8 920 4.2 3.S8 3.91 3.9 4.0 4.3 4.7 5.4 6.4 7.3 8.6 9.8 1.2 940 5.9 5.1 4.6 4.41 4.2 4.3 4.3 4.3 4.9 5.3 6.3 7.01 8.0 960 7. 5 6.9 6.3 5.8 5.3 4.7 4.7 4.6 4.6 4.6 4.9 5.4 6.0 980 9.3 8.7 7.9 7.4 6i8 6.4 6.0 5.6 5.2 5.0 4.9 5.1 5.1 1000 10.8 10.2 9.5 9.1 8.4 7.9 7.4 7.0 6.6 6.3 5.9 5.5 5.4,. 60 60 1 620 630 6 680 90 700 0 720 640 650 670 I 680 70 30 TABLE XXX. Perturbations produced by Venus. Arguments II. arid III. IIJ. IL 720 730 740 750 1760 1770 780 790 800 810 820 830 841 0 5.4 5.5 5.8 6.0 6.3 6.8 7.6 8.4 9.3 10.4 11.7 112.9 14.3 20 6.6 6.3 6.0 6.1 6.1 6.2 6.5 6.9 7.7 8.3 9.4 10.2 11.2' 40 7.8 7.41 7.1 7.0 6.7 6.6 6.8 6.8 6.9 7.2 7.7 8.5 9.3 60 8.9 8.8 8.3 8.11 7.8 7.6 7.4 7.4 7.3 7.4 7.4 7.7 8.3 80 9.6 9.51 9.1 9.1.0 8.8 8.4 8.2 8.1 8.1 8.0 8.1 8.2 100 9.6 9.5 9.61 9.5 9.51 913 9.3 9.2 9.2 9.0 8.7 8.7 8.7 120o 9.6 9.6 9.5 9.3 9.4 9.6 9.6 9.5 9.5 9.6 9.6 9.6 9.6 140 9.9 9.51 9.6 9.4 93. 9.3 9.0 9.3 9.5 9.8 9.7 9.8 10.0 160 10.5 9.9 1 9.5 9.1 8.9 9.0 8.9 9.0 9.0 9.0 9.5 9.6 9.9 180 12.0 11.0 10.1 9.7 9.1 8.8 8.7 8.3 8.5 8.7 8.8 9.0 9.1 200 14.4 13.3 12.0 11.0 10.1 9.4 8.9 8.5 8.2 8.0 8.0 8.3 8.5 220 117.1 15.7 14.6 13.2 12.0 10.9110.2 9.2 8.7 8.3 7.9 7.7 7.7 240 20.2 19.1 17.8 16.5 14.5 13.4 12.2 11.1 10.0 9.4 8.4 8.0 7.7 260 21.6 21.1 20.1 19.2 17.3 15.9 14.6 13.4 12.4 11.3 10.1 9.1 8.6 280 23.5 22.7 21.6 21.0 19.S 18.8 17.3 ].6.1 15.0 13.5 12.5 11.5 10.2 300 24.0 23.4 23.2 22.4 21.4 20.5 19.8 18.7 17.5 16.1 15.0 13.7 12.4 320 24.2 23.9 23.5 23.1 22.7 22.2 21.2 20.6 19.6 18.6 17.5 16.3 15.1 340 23.8 23.9 23.7 23.5 23.2 22.8 22.3 21.4 020.9 20.5 19.2 18.6 17.4 360 23.6. 23.6 23.6 23.3 23.3 23.1 22.9 22.4 22.0 21.4 20.4 19.9 18.9 380 24.0 24.0 23.7 23.5 23.3 23.1 23.1 22.7 22.4 292.2 21.6 20.8 20.0 400 24.6 24.4 24.4 24.0 23.8 23.41 23.2 123.0 21 2. 22.4 22.1 21.6 21.3 420 25.9 I 25.6 25.2 24.8 24.7 24.3 23.9 2 3.6 223.3 22.9 22.7 22.3 21.7 440 26.9 26.6 26.4 21.2 25.9 25.5 25.2 24.9 24.5 23.8 23.4 123.0 22.8 460 27.3 27.6 27.6 27.4 27.0 26.9 26.5 26.1 25.6 25.0 |24.6 241.2 23.7 480 26.81 27.4 27.6 28.0 28.1 28.2 27.7 1 27.4 ) 7.3 16.6 26.2 25.7 25.1 500Q25.1 26.1 26.8127.5 28.1 |28.2 28.6 128.5 28.4 28.3 27.G61 27.2 26.7 520 22.3 23.9 24.8 25.9 26.8 27.5 28.1 128.5 28.7 29.0 28.8 28.6 28.4 540 1 19.4 20.7 22.1 2 3.4) 24.G 2`5.6 26.5 27. 28.0 1 28.7 28.9 29.1 29.2 560 16.0 17.3 18.6 19.9 21.4 22.9 24.1 25.51 2614 27.3 i 8.2!2 6S. 29.2 580 12.7) 14.1 15.5 16.S 8 18.0 1 19.3 20.9 22.2 1 3.5 24.9 26.1 27.0 127.8 600 10.81 11.6 12.7 13.61 14.9 16.2 17.5 18.7 20.2 21.8 23.0 24.4 25.5 620 9.7110.0 10.5110.7 12.2 13.2 1.41 115.6 17.0 18.3 19.6 21.2 22.6 640 9.51 9.4 9.6 10.1 10.4 11.1 |12.0 13.0 14.0 15.2 16.5 17.9 19.2 660 10.2 10.0 9.7 9.5 9.5 9.9 10.4 11.0 11.7 12.7 13.8 14.9 16.2 680 12.0 1 1.2 10.5 10.0 9.7 9.5 9.6 10.0 10.4 11.0 11.6 12.5 13.8 700 14.1 13.1 12.3 11.3 10.7 10.11 9.7 9.7 9.9 9.9 10.4 10.9 11.5 720 116.4 15.3 14.4 13.3 12.2 11.6 10.9 10.21 10.1 9.9 10.0 10.1 10.4 740 18.8117.7 1 16.7 15.6 14.4 13.5 12.4 111.5 11.1 10.7 10.1 10.0 10.3 760 21.3 1 20.11 19.2 18.1 16.6 15.6 14.7 13.6 12.8 11.9 11.3 10.7 10.3 780 23.0 22.3 21.5 20.5 19.4 18.4 17.21 115.8 14.9 14.0 13 0 12.2 1 1.3 800 23.9 23.9 23.4 22.6 21.9 20.7 19.8 1S.8 17.5 16.2 15.1 14.2113.4 820 24.0 24.5 24.2| 23.9 23.3 22.6 22.3 21.3 |10.3 19.4 18.3 17.3 16.2 840 23.3 24.0 24.3 24.5 24.4 24.3 23.8 23.4 22.7 21.7 20.8 19.6 18.3 860 121.3 22.3 23.3 23.9 24.2 24.7 24.5, "11.5 24.3 23.6 23.1 21.9 21.0 880 18.2 19.7 20.9 22.0 22.8 23.8 24.1 24.6 24.8 24.7 24.5 24.0123.5 900 14.8 16.1 17.6 19.0 20.6 21.5 22.5 23.2 24.1 24.5 124.2 24.8 24.1 1 920 11.2)12.6 14.0 15.5 l17.0 18.4 19.9 21.0 22.0 22.9 23.5 24.5 24.5 940 8.0 9.3 10.7 12.0 13.3 1 14 8 16.41 17.6 19.1 20.4 21.4 22.4 1 23.2 960 6.0 6.91 7.8 8.6 110.2 111.5 12.7 14.1 15.6 16.9 18.1 19.5 20.7 980 5.1 5.5 6.0 6.71 7.71 8.5) t7j 10.9 112.2 13.6 14.8 16.1 17.61 100i0 5.4 5.5 5.8 5.8 6.3 6.8.6 8.4 9.3 10.5 11.7 12.9 14.3 1000 5 7' i 63j - j 1 720 730 740 ~750 ) 769 770 780 790 800 810 820 830 840 TABLE XXX. 31 Perturbations produced by Venus. Arguments II. and III. III.. 840 850 860 1 870 880 890 900 1910 920 930 940 950 960 0 14.3 15.5 16.9 18.2 19.2 20.2 21.4 22.5 23.0 23.51 24.0 24.2 24.2 20 1 1.2 12.4 13.6 14.9 16.2 17.3 18.6 19.6 20.5 21.5 22.4 23.1 123.6 40 9.3 10.2 10.9 11.8 13.3 14.2 15.5 116.6 17.8 18.8 19.7 20.7 21.6 60 8.3 8.7 9.5 10.1 10.8 111.6 12.7 13.8 14.9 15.9 17.0 18.1 19.1 80 8.2 8.3 8.6 8.9 9.6 10.3 10.7 11.6 12.5 13.3 14.5 15.2 16.2 100 8.7 8.7 8.9 9.0 9.1 9.4 9.9 10.4 11.0 11.7 12.4 12.9 14.0 120 9.6 9.5 9.3 9.6 9.6 9.7 9.9 9.8 10.4 10.9 11.3 11.8 12.3 140 10.0 10.2 10.1 10.2 10.1 10.3 10.4 10.5 10.5:0.6 10.9 11.4 11.5 160 9.9 10.0 10.2 10.4 10.6 11.0 11.0 10.9 11.0 11.3 11.3 11.3 11.6 180 9.1 9.6 9.9 10.1 10.4 10.7 11.0 11.3 11.5 11.7 11.7 11.9 12.2 200 8.5 8.8 9.1 9.5 9.7 10.0 10.5 11.0 11.2 11.6 12.0 12.2 12.4 220 7.7 7.7 8.1 8.4 8.8 9.2 9.7 10.1 10.6 11.0 11.4 11.8 12.3 240 7.7 7.3 7.4 7.4 7.7 8.0 8.4 9.0 9.6 10.0 10.5 11.0 11.5 260 8.6 7.9 7.4 7.2 7.1 7.1 7.3 7.6 8.1 8.5 9.3 1.0.0 10.4 280 10.2 9.2 8.3 7.9 7.4 7.1 7.0 6.9 7.0 7.3 7.7 8.5 8.8 300 12.4 11.4 10.4 9.3 8.5 7.8 7.4 6.9 6.7 6.8 6.8 7.0 7.5 320 15.1 13.9 12..5 11.4 10.5 9.7 8.6 7.8 7.4 7.0 6.6 6.5 6.7 340 17.4 16.4 15.2 13.9 12.7 11.6 10.6 9.7 8.7 8.0 7.3 6.8 6.6 360 18.9. 18.1 17 4 16.3 15.1 13.8 12 8 11.7 10.6 9.8 8.8 8.0 7.4 380 20.0 19.6 18.8 17.7 16.9 13.0 15.1 13.9 12.7 11.8 10.8 9.8 8.9 400 21.3 20.6 19.6 19.4 18.4 17.6 16.5 15.7 14.8 13.7 12.8 11.8 10.9 420 21.7 21.1 20.8 20.3 19.3 18.9 18.2 17.2 16.3 15.3 14.5 13.7 12.6 440 22.8 22.1 21.6 20.8 20.6 19.7 I9.0 18.6 17.7 16.6 15.9 15.1 14.2 460 23.7 | 23.3 22. 7 22.0 21.6 20.9 20.2 19.5 18.5 18.1 1 7. 13 16.7 15.7 180 25.1 24.4 23.9 2:3.3 22.8 22.0 21.4 20.9 20.2 19.3 18.3 17.7 16.9 500 26.7 26.3 25.7 24.9 24.3 i 23.6 23.0 22.3 21.4 20.7 1 20.3 19.1 18.1 520 28.4!27.8 27.3 26.8 26.3 25.6 24.7 23.9 23.3 22.6 21.8 20.8 20.1 540 29.2 29.2 28.9 28.5 27.8 27.4 26.8 26.1 25.3 24.4 23.7 23.0 22.0 560 29.2 29.3. 29.5 29.6 29.3 29.1 28.8 28.0 27.4 26.9 26.1 25.1 24.3 580 27.8 28.6 29.0 29.4 29.6 29.8 29.8 29.3 28.0 28.7 27.9 27.3 26.6 600 25. 5 26.7 27.6 28.4 28.9 29.2 29.6 29.9 29.9 29.8 29.3 29.0 28.5 620 22.6 23.8 25.0 2 6.2 27.1 27.9 28.8 29.3 29.6 1 29.8 30.1 29.8 19.6 640 19.2 20.6 21.6 23.3 24.6 25.2 26.6 27.8 28.3 28.9 i29.4 29.7 129.9 660 16.2;17.5 18.8 20.2 21.1 22.9 24.0 25.1 26.2 27.1 28.2 28.8 29.2 680 13.8 14.7 15.8 16.9 18.4 19.9 20.6 22.3 23.6 24.9 25.8 26.7 27.5 700 11.5 12.3 13.4 14.6 15.6 16.7 18.0 119.5 20.7 22.0 23.1 24.2 25.1 720 10.4 11.0 11.4 12.3 13.3 14.3 15.6 1 16.4 17.7 19.3 19.9 21.U6 2.U 7-'0 10.3 10.4 10.5 11.0 11.4 12.2 13.3 14.2 15.3 16.5 17.4 18.8 19.5 760 10.3 10.0 10.2 10.3 10.7 11.0 11.5 12.2 13.1 14.2 15.1 16.0 17.3 780 11.3 10.8 10.6 10.2 10.2 10.5 10.71 11.1 11.5 12.3 13.2 14.0 15.0 800 13.4 12.5i11.7 11.0 10.6 10.3 10.3 110.4 10.7 11.0 11.6 11.3 12.2 820 16.2 15.2 14.4 13.5 13.5 11.9 11.4 11.0 10.9 10.8 10.8 11.2 11.4 840 18.3 17.1 16.2 14.9 14.1 13.0 12.4 11.7 11.2 10.7 10.6 11.1 11.2 860 21.0 20.2 18.7 17.7 16.6 15.4 14.3 13.3 12.5 11.9111.4 11.0 10.9 880 23.5 122.4 21'.3 20.4 19.3 18.0 17.0 15.9 14.8 13.7 12.8 12.0 12.6 900 24.5 24.2 23.8 22.7 21.9 1".9 19.7 18.6 17.2 16.4 15.3 14.1 13.3 920 24.5!124.8 24.71243 24.1 23.22 22.3 21.3 20.0 19.3 18.0 16.7 15.7 94-0 23.2 24.0 2 4.5 24.6 24.5 24.5 24.2 23.5 22.7 21.8 20.6 19.5 118.4! 960 20.7121.9 22.8 23.6 24.0 24.5 24.5 24.2 1 24.3 23.7 22.9 22.1 21.0 980 17.6 18.71 20.1 21.2 22.2 23.1 23.6 24.0 24.3 24.3 24.3 23.7 23.0 110001) 4.3 15.5169 182 19.2 20.2 21.4 22.5 23.0 23.5 4.0 24. 24.2 840 S0 869 8701 880 890 9 00 910'320 930 940 1950 960 32 TABLE XXX. XXXI. Perturbations by Venus. Perturbations by Mars. Arguments II and III. Arguments II and IV. III. IV. I. 960 970 980 990,1000- 0! 10 20 30 40 1 50 60 70.i i-., I —__ _. -— I 1o 2 3. 71 197 11 o.8.~' 0 24.2 23.7 23.1 22.5 21..6 9.5 10.2 10.8 11.2 11.5 11.7 11.s 11..5 20 23 23 23.7 241 23.4 23.1 8.3 9.1 9.8 10.5 10.9 11.2 11.5 11.6 40 21.G6 224.- 22.9, 23.5 1 23.5 7.1'7.9 8.8 1 9.4 10.0 10.6 1 10.8 11.2 60 19. 1 20.1 20.7 21.5 22.2 5.8 6.7 7.6 8.4 9.1 9.8 10.3 10.5 80 16.2 17.3 18.4 19.7 20.0 4.3 5.3 6.4 7.2 8.0 8.9 9.3 9.9 100 14.0 14.8 15.61 16.5 17.6 3 31 4.2 5.0 5.9 6.S 7.6 S.4 9.1 120 12.3 12.9 13.7 14.3 15.3 2.4 3.1 3.9 4.8 5.6 6.4 7.3 S.0 140 11.5 12.0 12.6 12.8 13.6 2.1 2.4 2.9 3.8 4.6 5.5 6.3 7.0 160 11. 11.8S 2.1 12.3 12.7 2.0 2.2 2.4: 2.7 3.5 4.4 5.1 5.9 180 1S2.o 12.2 12.3 12.5 12.7 1.9 2.0 2.31 2.6 2.9 3.4 3.9 4.9 200 12.4 12.7 12.8 13.1113.2 2.3 2.2 2.2' 2.4 2.7 3.0 3.4 3.8 220 ].2.3 12.7 13.0 13.3 13.5 3.0 2.6 2.51 2.4 2.5 2.7 3.1 3.5 240 11.5 12.1 12.4 13.1 13.6 3.7 3.3 3.0' 2.9 2.7 2.8 2.9 3.2 260 10.4 11.0 11.5112.2 12.8 4.8 4.1 3.7 3.5 3.1 3.1 3.0 3.1 280 8.8 9.6 10.4: 10.7 11.5' 5.5 5.1 4.6 4.1 3.8 3.5 3.5 3.4 300 7.5 7.9 8.6 9.0 10.1l 6.2 5.8 5.61 5.0 4.8 4.2 3.9 3.8 320 6.7 6.8 7.3 7.8 8.3 i.9 6.6 6.1 5.9 5.4 5.1 4.7 4.3 340 6.6 6.4 6.6 6.7 6.2 7.2 7.1 6.9 6.5 6.2 5.8 5.5 5.1 360 7.4 6.9 6.5 6.5 6.5 7.5 7.4 7.1 7.0 6.8 6.4 6.2 5.8 380 8.9 8.2 7.5 6.9 6.8 3 7.5 7 6 7.3 7.3 7.2 7.1 6.7 6.5 4'0 10.9 10.0 9.0 8.3 7.5 7.3 7.3 7.5 7.4 7.4 7.4 7.1 7.0 4-20 12.6 11.6 10.7 9.9 9.1 6.9 7.0 7.3 7.4 7.4 7.4 7.3 7.5 440 14.2 13.3 12.5 11.6 10.6 6.5 6.8 6.8 7.1 7.2 7.3 7.3 7.4 460 15.7 14.8 13.9 13.0 12.1 6.2 6.2 6.5 6.7 6.8 7.1 7.1 7.3 4-30 16.9 16.3 15.5 14.5 113.6 5.8 5.9 6.0 6.2 6.4 6.5 7.0 6.9 500 13.1 17.6 16.6,15.8 15.1 1 5.3 5.4 5.7 5.8 6.0 6.0 6.3 6.6 520 20.1 19.2 18.1 17.4 16.51 5.1 5.1 5.1 5.3 5.4 5.6 5.8 6.0 540 22.0 21.0 1 20.2 19.2 18. 1 4.7 4.8 4.81 -4.8 5.0i 5.1 5.4 5.5 560 24.3 23.5 22.6 21.5 20.6 4.4 4.5 4.6 4.6 4.7 4.8 4.8 5.0 530 26.6 25.7 24.9 23.8123.0l 4.2 4.3 4.4 4.3 4.5 4.4 4.4 4.5 600 28.5 27.8 27.0 26.3 25.41 4.0 4.2 4.3 4.2 4.2 4.2 41.2 4.3 620 2 9.6 29.2 28.8 28.2 27.41 4.2 4.0 4.1 4.0 4.0 4.0 40 3.9 640 29.9 30.0 29.9129.5 29.5 4.3 4.2 4.1 4.0 4.1 4.0 3.9 3.9 660 29.2 29.5 29.7 29.8 29.9 4.6 4.4 4.3 4.1 4.1 4.1 4.0 3.8 680 27.5 28.6 28.9 29.2 29.7 4.8 4.6 4.5 4.3 4.2 4.1 4.0 3.9 700 25.1 26.4 27.3 27.8 28.71 5.3 5.0 4.81 4.5 4.6 4.0 4.1 4.1 720 22.6 23.9 25.0 261 26.Sl 5.8 5.5 5.1 5.0 4.7 4.5 4.1 4.1 740 19.5 21.3122.5 23.6 24.6' 6.5 6.1 5.7 5.4 5.2 4.9 4.6 4.3 760 17.3 18.6 19.4 21.0 l 22.1 7.4 6.7 6.4 6.0 5.6 15.3 5.1 5.0 780 15.0 15.8 17.1 18.5 19.3 8.2 7.6 6.91 6.5 6.4 5.8 5.6 5.3 800 12.2 14.1114.8 15.9 17.0 9.2 8.5 8.0 7.3 6.8 6.5 6.1 5.8 820 11.4 12.0 12.5 13.4 15.4 10.1 9.6 8.8 8.2 7.6 7.1 6.7' 6.5 840111.2 11.3 11.7 12.2 13.2 10.9 10.41 9.8 9.1 8.4 7.9 7.5 6.9 860 10.9 10.8 10.9 11.2 11.5 11.7 11.0 110.4 10.0 9.44 8.7 8.2 7.7 880 12.6 11.3 11.1 10.8 11.0 12.3 11.9' 11.3 10.6 10.2 19.7 8.91 8.4 3900( 13.3 12.3 1 12.9 1 11.3 111.2 12.4 12.2 11.8 11.6 110.8 10.3 9.7 920 115.7 14.6 1 13.71 12.8 1 12.1' 12.3 12.3 12.2 11.9 11.6 11.0 10.5 9.9 940 118.4 17.3 16.2 14.5 14.0 12.1 12.1 12.2 12.2 11.8 11.4 11.0 10.6 960 121.0 20.0 18 9 17.9 16.7 11.4 11.9 11.9 12.0 12.0 11.7 14- 1.11.0 O980 23.0 22.4 21.41 20.3 19.5 10.6 11.1 11.6 11.8 11.9 11.9 11.7 11.4 i10i)i 1 24.2 23.7 23.1 1 22.5 21.6 9.5 10.2 10.8 11.2 11.5 11.7 11.8.5 K 096 1 9701 980 1 990 1o0o- 0 10 20: 30 140, 50 60 70 _. TATLE XXXTI. Perturbations produced by Maars Arguments 11 and IV. IV. IT. 70 8) 90 100 110 120 130 140 1 10 160 170 180 1190 1 200 0 11.5 11.2 11.0 10.6 10.1 9.9j 9.5 9.0 8.6 8.2 8.1 7.8 7.6 7.4 20 11.6 11.4 11.0 10.91 10.6 10.2 9.7 9.1 9.1 8.8 8.4 8.1 7.9 7.8 40 11.2 11.3 11.2 11.0! 10.8 10.5 10.3 9.8 9.4i 9.3 9.1 8.7 8.4 8.2 60 10.5 10.9 11.1 10.91 11.0 10.9 10.4 11o0.0 9.7 9.5 9.2 8.8 8.7 8.4 80 9.9 10.0 10.5 10.9 10.8 10).7 10.4 10.3 10,.0 9.7 9.3 9.0 8.8 8.6 100 9.1 9.5 9.8 10.1!10.6 10.5 10.4 10.3 10. 1 9.9 9.6 9.3 9.0 8.8 120 8.0 S.S 9.3 9.5 9.9 10.2110.2 10.1 10.0 9.8 9.6 9.4 9.1 8.9 140 7.0 7.9 8.4 9.0 9.3 9.6 9.9 9.9 9.9 9.7 9.7 9.4 9.3 8.9 160 5.9 6.5 7.2 8.0 8.5 8.9 9.2 9.6 9.5 9.6 9.5 9.5 9.3 9.1 180 4.91 5.6 6.4 6.9 7.7 8.3 8.6 8.9' 9.4 9.3 9.3 9.3 9.2 9.1 200 3..S 4.6 5.3 6.0 6.7 7.4 7.9 8.3 8.0 8.9 9.1 9.0 9.0 8.9 220 3.5 3.9 4.4 65.1 5.8 6.4 7.1 7.6 7.9 8. 8.6 8.81 8.8 8.7 240 3.2 3.6 4.0 4.4 5.O 5.5 6.2 6.8 7.4 7.6 8.1 8.4 8.4 8.5 260 3.1 3.2 3.8 4.11 4.5 4.9 5.4 5.9 6.6 7.1 7.5 7.7 8.0 8.2 280 3.4 3.4 3.5 3.8 4.2 4.5 4.9 5.5 5.6/ 6.2 6.8 7.1 7.5 7.8 300 3.8 3.7 3.7 3.7 3.9 4.4 4.7 4.9 5.4 5.7 6.0 6.6 6 9 7.3 320.3 4.2 4. 44.0 4.1 4.2 4.4 4.7 5.0 5.4 5.8 6.0 6.4 6. 340 5.1 4.9 4.6 4.4 4.4 4.3 4.5 4.5 5.0 5.2 5.5 5.8 6.0 63 360 5.8 5.6 5.3 5.01 4.8 4.8 4.7 4.8 4.9 5.1 5.4 5.5 5.9 6.1 380 6.5 6.4 5.9I 5.7 5.5 5.4 5.1 5.1 5,1 5.1 5.4 5.5 5.7 5.8 400 7.0 6.7 6.7 I 6.3 6.1 5.9 5.7 5.6 5.5 5.5 5.5 5.6 5.7 5.9 420 7.4 7.2 6.9 7.1 6.7 6.4 6.31 6.1 6.0 5.9 5.9 5.8 5.8 6.1 440 7.5 7.4 7.4 7.0 7.1 7.4 6.8 6.7 6.5 6.3 6.3 6.4 6.2 6.3 4607.3 741 7.4 7.5 7.4 7.3 7.3 7.2 7.1 7.1 6.7 6.7 6.7 6.7 480 6.9 71 7.3 7.4 7.5 7.3 7.6 7.5 7.4 7.5 7.4 7.2 7.1 7.1 500 |6.6i 6.8 |6.9 |7.2 |7.3 7|.5 |7.5 |7.6 |7.8 7.7 7.8 7.7 7.6 7.4 520 6.0 6.3 6.5 6.7 7.1 7.2 7.5 7.5 7.7 7.8 7.9 7.6 7.9 7.9 540 5.5 5.7 6.0 6.3 6.6 6.9 7.1 7.3 7.4 7.7 7.9 8.0 8.2 8.3 5601.0 5.2 5.4 5.8 5.9 6.2 6.6 6.9 7.1 7.4 7.7 7.8 8.1 8.2 580 4.5 4.7 4.9 5.0 5.3 5.71 6.0 6.6 6.8 7.1 7.2 7.5 7.9 8.2 600 4.3 4.3 4.4 4.6 4.6 5.0 5.3 5.6 5.9 6.5 6.9 7.0 7.4 7.7 620 3.9 4.0 4.0 4.1 4.3 4.4 46 4.9 5.3 5.4 6.1 6.6 6.9 7.4 640 3.9 3.8 3.8 3.81 3.9 3.9 4.1 4.3 41.5 5.0 5.2 5.8 6.3 6.7 660 3.8 3.7 3.7 3.6 3.6 3.7 3.8 3.9 4.1 4.2 4.5 5.0 5.3 6.0 680 3.9 3.8 3.6 3.4 3.5 3.4 3.5 3.5 3.6 3.7 3.8 4.2 4.6 4.9 700 4.1 3.9 3.8 3.6 3.5 3.3 3.3 3.2 3.2 3.2 3.5 3.6 3.8 4.2 7201 4.1 4.1 4.0 3.8 3.6 3.5 3.31 3.2 3.3 3.2 3.0 3.2 3.4 3.6 7401 4.3 4.3 4.2 4.0 3.8 3.7 3.5 3.2 3.0 3.0 2.9 2.8 2.9 3.1 760 5.0 4.7 4.4 4.3 4.1 3.8 3.7 3.4 3.1 3.0 2.9 2.7 2.7 2.8 780 5.3 5.1 4.7 4.6 4.4 4.4 4.0 3.8 3.4 3.2 2.9 2.8 2.7 2.5 800 5.8 5.5 5.4 4.8 4.7 4.7 4.5 4.2 3.9 3.5 3.3 2.9 2.8 27 820 6.5 6.1 5.8 5.6 5.0 5.0 4.9 4.6 4.3 4.1 3.6 3.3 3.0 29 840 6.9 6.7 6.3 6.1 5.8 5.3 5.2 4.9 4.9 4.5 4.2 3.9 3.5 3.1 860 7.7 7.4 6.9 1 6.6 6.2 6.2 5.5 5.4 5.2 5.0 4.8 4.4 4.1 3.6 880 88 1 7.9 7.6 7.1 6.9[ 6.4[ 6.41 5.8] 5.7[ 5.4 5.2 5.0 4.6 4.3 880 901) 9.31 8.7 8.3 7.7 7.4 7.1 6.7 6.6 6.1 6.0 5.6 5.4 5.21 4.9 99 9.9 9.3 8.8 8.4 7.9 7.7 7.3 6.9 6.6 6.3 6.2 6.1 5.6 5 4 940 10.6 10.1 9.5 8.9 8.7 8.2 7.8 7.6 7.2 7.1 6.5 6.5 6.3 5.9 9601 11.0 |10.7 |10.3 |9.7 |9.1 |8.7 |8.4 |8.0 |7.8 |7.4 7.2 6.9 6.7 6.6 980 11.4 11.0 10.6 10.2 9.8 9.2 8.9 8.4 8.11 8.0 7.6 7.3 7.2 6.9 10(10 11.5 11 2 11.0 10.6 10.0 9.9 9.5 9.0 8.6 8.2 8.1 7.4 7.6 7.4 70 80 1 90 100 110 l11201 1301 149 150 160 170 180 190 200 34 TABLE XXXI. Perturbations produced by 1lMars. Arguments II. and IV. IV. II. 200 i 210 220 1 230 240 250 260 270 280 291 300 310 320 I- - - -.-. —0 7.4 7.2 7.0 6.6 6.4 6.2 5.7 5.3 4.9 4.7 4.1 3.8 3.4 20 7.8 7.2 7.3 7.2 7.0 6.6 6.3 6.0 5.7 5.3 5.0 4.4 3.9 40 8.2 8.1 7.6 7.5 7.3 7.2 6.8 6.6 6.2 5.9 5.6 5.2 4.7 60 8.41 8.0 7.9 7.8 7.6 7.5 7.3 7.1 6.8 6.4 6.1 5.8 5.4 8O 8.6 8.51 8.2 8.0 7.6 7.7 7.6 7.4 7.1 7.0 6.7 6i.3 6.0 o00 8.8 8.5 8.6 8.4 8.2 7.6 7.7 7.8 7.6 7.3 7.2 6.9 6.6 120 8.9 8.7 8.4 8.4 8.3 8.3 8.o0 7.9 7.7 7.6 7.5 7.3 7.0 140 8.9 8.7 8.4 8.3 8.2 8.1 8.3 8.0 7.9 7.8 7.7 7.5 7.4 160 9.1 8.9 8.7 8.4 8.3 8.3 8.2 8. 1 8.0 7.9 7.9 7.7 7.6 180 9.1 8.8 8.7 8.5 8.4 8.2 8.0 8.0 8.1 7.9 7.8 8.0 7.8 200 8.9 8.8 8.6 8.4 8.4 8.3 8.1 8.0 7.9 7.8 7.8 7.9 7T.9 220 8.7 8.7 8.6 8.4 8.2 8.1 8.0 7.9 7.8 7.7 7.7 7.6 7.7 240 8.5 8.4 8.5 8.3 8.1 8.0 7.8 7.8 7.8 7.8 7.8 7.8 7.6 260 8.2 8.2 8.1 8.1 8.1 7.8 7.8 7.7 7.6 7.6 7.6 7.5 7.4 280 7.8 7.8 8.0 7.8 7.9 7.9 7.7 7.5 7.5 7.3 7.3 7.4 7.3 300 7.3 7.6 7.5 7.6 7.7 7.6 7.6 7.6 7.4 7.3 7.1 7.0 7.1 320 6.6 7.1 7.3 7.4 7.4 7.3 7.4 7.4 7.3 7.1 7.0 7.0 6.8 340 6.3 6.4 6.7 7.2 7.1 7.2 7.2 7. 1 7.1 7.0 6.9 6.8 6.8 360 6.1 6.2 6.41 6.5 6.9 6.9 7.0 7.0 6.9 6.8 6.7 6.6 6.5 380 5.8 6.1 6.3 6.41 6.6 6.7 6.6 6.6 6.7 6.8 6.7 6.6 6.5 400 5.9 6.0 6.2 6.3 6.4 6.5 6.6 6.6 6.5 6.6 6.6 6.5 6.4 420 6.1 6.3 6.2 6.4 6.3 6.4 6.5 6.6 6.5 6.5 6.6 6.5 6.4 440 6.3 6.4 6.4 6.6 6.5 6.6 6.5 6.5 6.5 6.56.3 fi6.3 6.2 460 6.7 6.5 6.5 6.6 6.7 6.9 6.7 6.6 6.6 6.6 6.5 6.3 6.2 480 7.1 7.1 7.0 6.9 6 9 6.9 7.0! 7.() 6.8S 6.7 6.6 6.5 6.3 500 7.4 7.5 7.4 7.41 7.3 7.2 7.3 7.2 7.1 6.9 6.8 6.81 6.6 520 7.9 7.8 7.8 7.8 7.81 7.6 1 7.6 7.5 7.5 7.4 7.1 7.0 6.9 540 8.3 8.3 8.3 8.2 8.2 8.1 8.0 7.9 7.9 7.8 7.6 7.5 7.2 560 8.2 8.6 8.4 8.6 8.7 8.5 8.5 8.4 8.2 8.3 8.2 8.0 7.6 580 8.2 8.3 8.6 8.8 8.8 9.0 8.9 8.9 8.7 8.7 8.6 8.4 8.4 600 7.7 8.1 8.5 8.6 8.9 9.1 9.1 9.2 9.2 9.1 9.0 8.8 8.7 620 7.4 7.6 8.0 8.51 8.7 9.0 9.2 9.5 9.5 9.5 9.4 99.3 9.2 640 6.7 7.2 7.5 7.91 8.31 8.7 9.0 9.3 9.5 9.81 9.7 9.7 660 6.0 6.3 7.0 7.31 7.7 8.2 8.7 9.0 9.4 9.7 9.8 10.1 10.0 680 4.9 5.6 6.0 6.6 7.1 7.7 8.11 8.S1 -9.0 9.31 9.81 10.0 10.2 700 4.2 4.5 5.2 5.86. 6.8 7.4i 8.0 8.5 8.9 9.2 9.8110.1 720 3.6 3.9 4.3 1 4.7 5.3 5.9 6.6 7.0 7.8 8.3 8.8 9.1 9.71 740 3.1 3.3 3.6 3.9 4.4 4.8 5.6 6.2 6.9 7.5 8.0 8.7 9.2! 760 2.8 2.81 3.0 3.3 3.6 4.0 4.4 5.1 5.8 6.5 7.2 "8 8.4 780f 2.5 2.61 2.5 2.7 3.1 3.3:3.7 4.1 4.8 5.4 6.1 6.3 7.6 800 2.7 2.5 2.5 2.5 2.5 2.7 3.0 3.4 3.8 4.4 5.0 5.6 6 6 820 2.9 2.6 2.41 2.3 2.2 2.3 2.6 2.81 3.1 3.4 4.1 4.7 5.4 840 3.1 2.8'2.6 2.4 2.3 2.2 12.3 2.4 2.6 2.8 3.2 3.8 4.3 860 3.6 3.3 3.0 2.7 2.4 2.3 2.1 2.2 2.3 2.5 2.7 3.0 3.4 880 4.3 3.8 3.6 3.2 2.8 2.5 2.3 2.1 2.0 2.2 2.3 2.5 2.6 900 4.9 4.6 4.2 3.6 3.4 2.9 2.6 2.3 2.2 2.2 2.1 2.2 24 920 5.4 5.1 4.6 4.5 3.9 3.5 3.2 2.9 2.6 2.2 2.0 2.1 2.2 940 5.9 5.7 5.3 4.9 4.7 4.3 3.8 3.4 3.0 2.7 2.4 2.1 2.0 960 6.5 6.2 5.9 5.5 5.1 4.9 4.5 4.0 3.4 3.1 2.8 2.4 2.3 980 6.9 6.8 6. 11 6.1 5.8 5.4 514.8 4.3 39 3.5 3.0271 1000 37.4 7.0 7.0 6.6 6.4 6.2 5.1 5.3 1.9 4.7 4.1 3.8 3 4 200 210 220 230 240 250 2601 270. 280 290 300 310 1.32j 82 0 I I ~ 1 I I I~~~~~~~~~~~~ TABLE XXXI. 35 Pertztrbations produced by Ifaxrs. Arguments II. and IV. IV. 11. 320 330 340 350 360 370 380 390 4:00 410 420 430 410) 1 0 3.4 2.8: 2.6 2.4 2.2 2.3 2.3 2.5 2.7 2.9 3.4 4.0 45 20 3.9 3.5 3.1 2.7 2.6 2.4 2.4 2.3 2.5 2.7 3.0 3.3 S.S 40 4.7 4.2 3.9 3.5 3.0 2.8 2.7 2.6 2.5 2.6 2.8 2.9 3.2 6C 5.4 5.0. 4.6 4.2 3.8 3.4 3.1 2.8 2.8 2.7 2.7 2.7 3.0 8 6.0 5.7 5.4 4.8 4.4 4.0 3.6 3.4 3.1 2.9 2.9 2.9 2.9 100 6.6 6.3 5.9 5.6 5.2 4.8 4.3 4.0 3.7 3.5 3.2 3.0 3.0 120 7.0 6.9 6.4 6.1 5.8 5.3 5.2 4.6 4.3 4.0 3.8 3.6 3.4 1410 7.4 7.2. 6.9 6.6 6.5 6.1 5.6 5.4 5.0 4.6 4.3 4.0 3.9 160 7.6 7.5 7.3 7.0 6.8 6.6 6.2 5.9 5.5 5.3 4.9 4.6 4.4 180| 7.8 7.7' 7.5 7.4 7.3 6.9 6.7 6.5 6.2 5.8 5.6 5.3 5.0 200 7.9 7.8 7.7 7.6 7.5 7.3 7.1 6.9 6.6 6.4 6.1 5.6 5.5 o220 7.7 7.7 7.7 7.8 7.7 7.5 7.3 7.2 7.0 6.7 6.5 6.2 5.9 240 7.6 7.6 7.6 71 7.7 7.6 7.5 7.3 7.2 7.1 6.9 6.6 6.4 260 7.4 7.3 7.5 7.5' 7.5 7.6 7.6 7.5 7.5 7.3 7.1 7.0 6.7 280 7.3 7.4 7.3 7.3 7.4 7.4 7.3 7.4 7.3 7.5 7.2 7.1 6.9 300 1 7.1 7.1 1 7.1 7.0 7.2 7.3 7.3 7.3 7.2 7.2 7.31 7.2. 7.1 320 6.8.8 6.9 6.9 6.8 7.0 7.1 7.1 7.1 7.1 7.12 340 6.8 |6.71 6.6 6.6 6.6 6.8 6.9 6.9 7.0 7.0 6.9 6.9 6.9 360 65 6.5 6.4 6.3 6.4 6.5 6.6 6.7 6.8 6.8 6.8 6.8 6.9 380 6.5 6.3 6.3 6.2 6.2 6.2 6.3 6.3 6.4 6.5 6.6 6.7 6.7 4001t 6.4 6.2, 6.2 6.0 6.1 6.0 6.0 6.0I 6.0 6.1 6.2 6.3' 6.4 420 6|.4 6| 2!6.1 6.01 5.9 5.8 5.9 5.9 5.9 5.9 5.9 6.0 6.0 440( 6.2 6.1I 66.0 5.8 5.8 5.7 5.6 5.6J 5.6 5.7 5.7 5.8 5.9 460( 6.2 1 6.0 5.9 5.8 5.'7. 5.5 5..5 5.4 5.5 5.4 5.5 5.3 5.4 48i0 6.3 1 6.2 6.0 5.7 5.6 5.5 5.4 5.3 5.2 5.2 5.2 5.3 5.3 500( 6.61 6f.4 6.2 6.0 5.7 5.4 5.3 5.2 5.1 5.1 |5.1 5.0 5.0 520 6.9 6.7 6.4 6.1 6.1 5.7 5.5 5.1 5.1 5.0 4.9 5.0 4.9 540 7.' 7.1 6.7 6.5 6.2 6.1 5.8 5.5 5.2 5.0 4.9 4.8 4.8 560 7.6 7.4 7.3' 7.0 6.6 6.3 6.0 5.8 5.4 5.3 5.0 4.7 4.7 580 84 8.0 7.8 7.5 7.0 6.8 6.3 6.2 5.9 5.5 5.3 5.0 4.9 600 8.7 8.6 8.3 8.0 7.8 7.4 7.0 6.6 6.3 6.0O 5.6 5.3 5. 620 9.2 9.1 8.9 8.6 8.4 8.1 7.6 7.2 6.8 6.5 6.1 5.7 5.3 640 9.7 9.6 9.4 9.3 9.0 8.7 8.21 7.8 7.4 7.0 6.6 6.3 5.8 660 10.0 10.0 9.9 9.8 9.6 9.3 8.9 8.5 8.2 7.7 7.2 6.8 6.4 680 10.2 10.4 10.3 10.2 10.1 9.9 9.6 9.3 9.0 8.5 8.1 7.5 7.1 700 10.1 10.3 10.5 10.6 10.4 10.3 10.1 9.8 9.6 9.3 8.9 8.3 7.8 720 9.7 10 O.1 I 0.3 10.6 10.7 10.6 10.5 10.5 10.2 10.0 9.6 9.2 8.6 740 9.2 9.6 1o0.0 10.3 10.6 10.7 10.8 10.9 10.6 0.5 10.2 9.9 9.4 760 8.4 9.0 9.5 9.8 10.2 10.6 10.9 11.0 11.0 11.0 10.7 10.5 10.3! 780 7.6 8.2 8.9 9.4 9.9 10.3 10.6 10.9 11.1 11.2 11.0 1-0.8 10.7 800 6.6 7.3 7.9 8.5 9.2 9.8 10.1 10.6 10.8 11.11 11.3 11.1 11.0 820 5.4 6.0 7.0 7.6 8.2 8.9 9.6 10.0 10.5 10.8 11.0 11.3 11.3 840 4.3 5.0 5.6 6.5 7.2 7.9 9.8 9.2 9.9 10.3 10 7 1.9 11.2 860 3.4 4.0 4.6 5.3 6.1 6.9 7.5 8.4 9.1 9.6 10.1 10.7 10.9 880 2.6 3.1 3.7 4.3 5.0 5.7 6.6 7.1 8.1 8.7 9.4 9.8 10 4 900 2.4 2.7 3.0 3.4 4.0 4.6 5.4 6.1 6.9 7.6 8.4 9.1 9. 920 2.2 2.3 2.3 2.8 3.3 3.7 4.3 5.0 5.8 6.5 7.2 8.0o 8.7 940 2.0 2.1 2.3 2.31 2.7 2.9 3.4 O4.1 4.7 5.5 6.1 7.0 7.7 960 2.3 2.2 2.2 2.3' 2.3 2.5 2.8 3.2 3.9 4.5 5.1 5.7 6 5 980 |.7 2.4 2.2 2.3 2.3 2.4 2.5 2.8 3.0 3.6 6 4.1 4. 5 5! 1000 3.4 2.8 2.61 2.4! 2.0. 2.3 2.33 2.5 2 7 2.9 3.4 4. 4 5 I - 1 —-.49.90.310.6-0.911. 1 —. 11.1 10.$ i. 20.4330 3403.00 360 370 380 390 400 411100 14 430 4140 3. 2 330 340. 6 5 36 TAIBLE XXXI. Perturbations produced by Mars. Arguments II and IV. IV.. 440) 450 460 470 480 490 500 510 1 520 530 540 550 560 0 4.5 5.2 5.9 6.6 7.3i 8.0 8.5 9.0 9.5 10.0 10.4 10.7 10.9 20' 3.8 4.3 4.9 5.6 6.2 6.9 7.6 8.2 8.8 9.3 9.7 10.0 11.4 401 3.2 3.7 4.2 4.8 54 5.9 6.6 7.3 7.9 8.4 8.9 9.4 9.8 601 3.0 3.2 3.6t 4.0 4.5 5.1 5.7 6.3 6.9 7.5 8.0 8.6 9.1 80 2.9 3.1 3.3 3.5 3.3 4.4 4.9 5.4 5.9 6.5 7.1 7.7' 8.2 100 3.0 3.1 3.2 3.5 3.61 3.81 4.2 4.8 5.3 5.9 6.4 6.9 7.4 120 3.41 3.3 3.3 3.4 3.5 3.61 3.9 4.2 4.7 5.1 5.6 6.0 6.6 140 3.9 3.8 3.6 3.6 3.6 3.7 4.0 4.0 4.2 4.6 5.0 5.4 5.9 160 4.4i 4.2 3.9. 1 3.8 3.7 4.0 4.1 4.2 4.5 4.6 4.9 5.3 1SO 5.01 4.8 4.4 4.2 4.2 4.2 4.0 4.11 4.3 4. 4.4 4.7 5.0 200 5.5 5.2 5.1 4.8 4.6 4.5 4.5 4.41 4.5 4.5 4.7 4.6 4.8 220 5.9: 5.7 5.5 5.3 5..1 4.9 4.9 4.8 4.7 4.8 4.8 4.9 5.0 240 6.4, 6.2 5.9 i 5.8 5.6 5.41 5.3 5.2 5.1 5.1 5.1 5.2 5.2 260 6.71 6.6 6.4 6.1 6.0 5.9 5.8 5. 5.6 5.5 5.4 5.4 5.4 280 6.9 6.8 6.7 6.5 6.3 6.2 6.1 6.0 5.9 5.91 5.9, 5.8 5.8 300 7.1 7.0 6 6.8 6.6 6.5 6.41 6.3 6.2 6.2 6.' 6.2 6.2 320 7.2 7.1 9 6.8 6.8 6.7 6.6 6.5 6.5 6.5 65 6.6 6.6 340 6.91 6.9 7.0 6.9, 6.9 6.8 6.7 6.8 6.7 6.6 6.61 6.8 6.9 360 6.9 6.8 6.8 6.8 6.8 6.7 6.7 6.6 6.6 6.8 6.8 6.8 6.9 380 6.7 6. 6 6.5 6.6 6. 6 6.6 6.61 6.7 6.7 6.7 6.8 6.9! 6.9 400. 6.4 6.4 6.3 6.3 6.4 6.5 6.5 6.5 6.6 6.7 6.7 6.8 6.5 6. 420 6.0 6.2 6.3 6.3 6.2 6.2 6.3 6.3 6.31 6.3 6.5 6.6' 6 7 440i 5.9 5.9 6.0, 6.0 6.0 6.0 6.01 6.1 6.01 6.1 6.2 6.41 460 5.4 5.5 5.7 5.8 58' 5.8 58 5.8 5.8 5.8 5.9 6.01 (G. 480 5.3 5.3 5.5 5.5 5.51 56 6.5 5.6 5.4 35.6 5.7 5.S 500 5.0 5.0 5.1 5.2 5. 5.3 5.3 5.2 5.2j 5.21 5.3 5.4 5.4 520 4.9 4.9 4.9 4.8 5.0 5.1 5.1 5.1 5.1 5.1 5.0 5.0 5.1 540.8 4. 4.8 4.7 4.8 4.8 4.9 4.9: 5.01 4.91 4.81 4.8 4.9 4.8 560 4.7 4.6 4.6 4.7 4 4.7 4.7 4.7 4.7 4.6 4. 6 5801 4.9 4.6 4.5 4.5 4.6 4.5 4.4 441 4.5 4.5 4.& 1 4.4 44 600 5.1 4.9 4.6 4.5 4.4 4.4 4.41 4.3 43 4.3 4.31 4.3 4.3 620 5.3 5.1 4.9 4.7 4.6 4.4 4.3i 4.11 4.2 4.2 4.21 4.21 4.1 640 5.8 5.4 5.2 5.0 4.7 4.6 4.41 4.1! 4.1 4.1 4.2 4.2 4.0 660 6.4 6.0 5.7 5.4 5.0 4.8 4.7 4. 1 4.31 4.2 4.2 4. 1 4.0 680 7.1 6.6 6.2 5.7 5.4 5.1 4.91 4.7 4.51 4.4 4.3 4.0 3.9 700 7.8 7.2 6.8 6.4 6.0 5.6 5.3 5.0 4.7 4.6 4.6 4.3 4.1 720 8 6 8.0 7.6 7.1 6.6 6.2 5.7 5.5 5.2' 4.9 4.6 4.6 4.31 740 9.4 9.0 34 8.0 7.4 6.9 6.31 6.0 5.6 5.3' 5.0 4.7 4.5 760 10.3 9.7 9.3 8.6 8.1 7.6 7.21 6.5 6.2 5.8 6.5 5.2 4.9 780E 10.7 10.5 i 9.9 9.6 9.0 8.5 7.8 7.4 7.0 6.4 6.1 5.57 5.5 800 11.0 11.0 10.6 10.2 9.9 9.3 8.8 8.1 7.7 7.3 6.7 6.3 5.8 820 11.3 11.1 10.9,10.6 10.3 10.0 9.6 9.1 8.5 7.9 7.4 7.0 6.6 840 11.2 11.3 11.2 11.1 11.0 10.7 1 1.2 9.9 9.1 8.8 8.21 7.7 7.3, 860 10.9 11.1 11.4 11.3 11.3 11.2 10.7 10.4 9.9 9.6 9.2 8.5 7.9 880 10.4 10.8 11.0 11.3 11.2 11.2 11.2 10.9 10.5 10.3 9.8 9.3 8 7 900 9.7 10.1 10.6 11.0 11.2 11.2 11.2111.0 10.9 10.7 10.2 10.0 9.4 920 87 9.3 9.9 10.3 10.8 11.0 11.1111.2 1 1.2 11.0 10.7 10.4 10.1 940 77 8.2 8.8 9.5110.1 10.4 1).9 11.0 11.2 11.2 11.0 10.7 10.5 960 q.5 7.3 8.1 8.6 9.3 9.8 10.21 10.6 10.8 11.1 11.2 10.9 10.8 980 5.5 6.2 7.0 7.7 8.3, 8.91 9.5 10.0 10.4 10.6 10.8 1 1. 10.91 10001 4.5 5.2 5.9 6.6 7.3 8.0 8.5 10.0 1 0.4 10.7 10.9 - 450 460 470 40 490 500 i 560 1 440 1 450t 460 47! 40 500 510 5 21 530 1 540 51 5-0 560 TABLE XXXI. 37 Perturbatiorls produced by Mars. Arguments II and IV. IV. II. 560 570 580 590 600 610 620 630 640 650 660 670 680 0 10 9 10.8 10.6 10.4 10.3 10.0 9.7 9.2 8.9 8.5 8.1 7.9 7.7 20 11.4 10.6 10.7 10.6 10.4 10.2 9.9 9.7 9.3 9.0 8.8 8.5 8.1 40 9.8 10.1 1]0.4 10.4, 10.5 10.3 10.2 9.9 9.6 9.4 9.1 8.9 8.5 60 9.1 9.4 9.8 10.2110.2 10.3 10.2 10.1 9.9 9.6 93 9.0 8.8 80 8.2 8.7 9.0 9.31 9.6 9.8 10.0 9.9 9.8 9.7 9.5 9.3 91 100 7.4 7.9 8.4 8.7 9.0 9.4 9.6 9.7 9.8 9.7 9.7 9.5 9.2 120 6.61 6.9 7.6 8.1 8.3 8.6 9.0 9.2 9.4 9.5 9.5 9.4 9.3 140 5.9 6.3 6.8 7.21 7.7 8.0 8.3 8.7 8.9 9.1 9.2 9.3 9.3 160 5.3 I5.8 6.0 6.5 1 6.9 7.4 7.7 8.0 8.4 8.5 8.8i 8.9 9.0 180 5.0 5.2 5.6 6.0! 6.3 6.71 7.1 7.2 7.7 8.1 8.1 8.4 8.6 200 4.8 5.0 6.3 5.4 5.8 6.1 6.5 6.7 7.1 7.3 7.7 7.8 8.0 220 5.0 5.0 5.1 5.3' 5.5 5.7' 6.0 6.3 6.6 6.8 7.0 7.3 7.5 240 5.2 5.2 5.3 5.3 5.41 5.5 5.7 5.9 6.11 6.4 6.6 6.s 7.1 260 5.4.5.5 5.5 5.5s 5.5 5.5 5.5 5.7 5.8 6.0 6.3i 6.4 6.5 280 5.8 5.8 5.8 5.9 5.8 5.8 5.8s 5.9 5.9 5.91 6.0 6.1 6.2 300 6.2 6.1 6.2 6.1 6.1 6.1 6.2 6.1 6.0 5.91 5.9 6.0 6.1 320 6.6 6.5 6.6 6.6 6.5 6.5 6.6 6.5 6.5 6.3 6.1 6.0 6.0 340 6.9 6.9 6.9 7.0 7.0 6.9 6.8 6.9 6.9 6.8 6.6 6.5 6.3 360 6.9 7.0 7.2 7.3 7.3 7.3 7.4 7.3 7.3 7.1 7.1 7.0 6.7 380 6.9 7.0 7.2 7.4 7.5 7.6 7.7 7.71 7.7 7.6 7.5 7.4 7.2 400 6.8 7.0 7.11 7.3 7.6 7.9 8.0 8.0 8.1 8.1 8.1 7.9 7.8 420 6.7 6.9 7.0 7.2 7.6 7.8 8.0 8.2 8.3 8.4 8.4 8.5 8.4 440 6.4 6.6 6.9' 7.0 7.3 7.5 7.9 8.2 8.4 8.6 8.8 8.8 8.9 460 6.1 6.2 6.5 6.9 7.1 7.2 7.6 8.0! 8.41 8.7 9.0 9.1 9.2 480 5.8 5.9 6.0 6.2 6.7 7.1 7.2 7.6 7.9i 8.5 8.9 9.2 9.3 500 5.4 5.5 5.6 5.9 6.1 6.4 6.9 7.2, 7.7 7.9 8.4 9.0 9.4 520 5.1 5.2 5.2 5.3 5.6 5.9 6.3 6.7 7.0 7.6 8.0 8.4 9.0 540 4.8 4.8 4.8 5.0 5.1 5.4 5.6 6.0 6.41 6.7 7.5 8.1 8.5 560 4.6 4.5 4.5 4.5 4.7 4.8 5.0 5.3 5.8 6.2 6.6 7.1 7.8 580 4.4 4.3 4.3 4.3 4.3 4.3 4.5 4.7 5.2 5.5 5.9 6.4 6.9 600 4.3 4.3 4.2 4.1 4.0 4.0 4.1 4.2 4.5 4.8 5.1 5.7 6.2 620 4.1 4.0 4.0 3.9 3.9 3.8 3.8 3.8 3.8 4.0 4.4 4.9 5.4 640 4.0 3.9 4.0 3.8 3.8 3.8 3.7 3.5 3.5 3.6 3.8 4.0 4.5 660 4.0 4.0 3.9 3.8 3.7 3.5 3.5 3.4 3.3 3.3 3.4 3.5 3.7 680 3.9 4.0 3.9 3.8 3.6 3.5 3.4 3.3 3.2 3.1 3.1 3.1 3.1 700 4.1 3.9 3.9 3.9 3.7 3.5 3.4 3.31 3.2 3.0 3.0 3.0 2.9 720 4.3 4.1 4.0 3.9 3.8 3.8 3.5 3.4 3.1 2.9 2.9 2.7 2.7 740 4.5 4.2 4.2 4.2 4.0 3.7 3.6 3.4 3.3 3.0 2.8 2.6 2.5 760 4.9 4.7 4.5 4.3 4.2 4.11 3.8 3.6 3.3 3.1 2.91 2.8 2.5 780 5.5 5.1 4.9 4.5 4.4 4.3 4.1 3.9 3.8 3.4 3.2 3.0 2.7 800 5.8 5.6 5.2 5.0 4.6 4.5 4.4 4.3 4.1 3.8 3.5 3.1 2.8 820 6.6 6.1 5.8 5.5 5.3 5.0 4.8 4.61 4.4 4.2 4.0, 3.6 3.3 840 7.3 6.8 6.5 6.1 5.7 5.5 5.2 5.0 4.7 4.6 4.3 4.1 3.8 860 7.9 7.5 7.0 6.7 6.4 5.9 5.8 5.4 5.1 5.0 4.8 4.6 4.4 880 8.7 8.2 7.8 7.3 6.9 6.6 6.3 6.0 5.7 5.4. 5.2 5.0 4.7 900 9.4 9.0 8.5 8.0 7.6 7.2 6.8 6.6 6.3 5.9 5.6 5.4 5.2 920 10.1 9.8 9.2 8.7 8.3 7.8 7.4 7.0 6.7 6.4 6.0 5.8 517 940 10.5 10.2 9.8 9.4 8.8 8.5 8.0 7.6 7.3 6.9 6.6 6.21 61 960 10.8 10.5 10.2 10.0 9.5 9.1 8.6 8.2 7.8 7.51 7.1 6.8 66 980 10.9 10.7 10.3 10.2 9.9 9.6 9.2 9.0 8.5 8.0 7.7 7.41 72 11000 10.9 10.8 10.6 10.4 10.3 10.0 9.7 1.98.5 8.1 7.9 7.7 560 570 580 590 6 00 1 filO 6 fi20 3120 64QI fi50 6 CiO 670 1 680 31 3 8 TABLE XXXI. Perturbations produced by Alars. Arguments II. and IV. IV. 6. 80 69i0 700 710 720 730 740 750 760 1 770 780 790 800 7.7 7.4 6.9 6.8 6.7 6.4 6.1 5.8 5.5 5.2 4.8 I 4.4 3.7 20 8.1 7.8 7.4 7.0 7.1 6.9 6.7 6.4 6.1 5.8 5.5 5.1 4.7 4-0 8.51 8.3 7.8 7.5 7.2 7.1 7.0 6.9 6.6 6.4 6.1 5.8 5.3 6 88. 8.6 8.3 8.1 7.8 7.6 7.5 7.4 7.1 6.91 6.7 1 6.3! 6.0 80, 9.1i S.9 8.7 4 8.1 8.0 7.8 7.6 7.4 7.3 7.1 69 6.5 iO) 9.2; 8.9 8.8 8.7 8.6G 8.3! 8.0 7.7 7.6 76 7.61 7.3 7.0 9.3 9.2 90 87.6 8.4 8. 8.1 7.9 7. 7.7 7.6 7.5 O 93 9.21 9.0 9.0 8.7 S.5 8.4 8.3 8.0 7.8 7.7 7.7 7.7 1i10 9.0 9.0 8.9 8.8 8.7! 8.6 8.5 8.4 8.2 8.0 7.9 7.8 7 8 S01 8.6 8.6 8.7 8.7 8.7 8.6 8.5 8.3 8.3 S.0 8.2 7.8 7.9 200 S.O 8. S.31 8.3 8.5 S.4 8.4 8.4 8.2 8.1 8.1 8.1 7.9 720 7.5 7.7 7.9 8.1 8.2 8.2 8.1 8.2 8.2 8.0 8.1 8.0 8.0 24() 7.1 7.2 7.4 7.5 7.6 7.7 7.8 7.8 7.9 8.0 8.0 7.8 7.8; ) 6.5 6.7 6.9 7.1 7.2 7.3 7.4 7.5 7.6 7.6 7.7 7.7 7.8 280 6.2 6.3 6.5 6.7 6.7 6.9 7.1 7.2 7.3 7.3 7.3 7.3 7.4 300 6.1 6.0I 6.2 6.4 6.4 6.5 6.6 6.7 6.9 6.9 6.9 7.1 7.1 3o20 6.0 6.0 6.0 6.0 6.1 6.2 6.3 6.5 6.5 6.6 6.6 6.8 340 6.3 6.2! 6.0 6.0 6.0 6.0 6.1 6.1 6.2 6.2 6.3 6.3 6.4 360G 6.7 6.60 6.4 6.1 6.0 5.9 6. 5.9 5 5.9 6.0 6.1 6.2 3809 7.2 7.1 6.8 6.6, 6.4 6.2 6.1 5.9 5. 5. 6 5. 59 o00 7.8 7.7 74 7. 6.8 6.6 6.4 6.1 6.0 5.8 5.6 5.6 420 8.4 8.2 iS.0 7.8 7.5 7.s 6.81 6.5 6.2 6.0 5.7 5.5 5.4 440 8.9 8.8 8.7 8.4, 8.2 7.5 7.1 6.6 6.2 6.0 5.7 5.6 460 9.2 9.2 9.2 9.0 8.8 8.5 8.2 7.9) 7.5 9.9 6.3 6.0 480 9.3 9.5 9.6 9.4 9.2 9.1 8.6 8.3 7.8 7.2 6.9 6.5 500 1 9.4 9.6t 9.8 110.0 99 9.8I 9.6 9.4 9.1 8.7 8.21 7.6 7.2 520 9.0 9.5 98 10.3 100 9 9 91 5 8.0 540 8.5 9.1 9 1.3 15 i 10.0 i 6 10.3 9. 8 9 8.5 9.0 560 7.8 8.5 9.0 9.5 9.9 10.4 10. 0.S 10.2 8 9.9 5801 6.91 7.6 8.3 9.0 9.7 0.0 10.4 10.7 11.1 10.2 t1.0 110.0 10.6 600!0 6.8 7.4 8.01 8.9 9.6 10.1 10.4 10.9 11.3 11.4 11.3 11.2 620 5.4 5.9 6.5 7.11 7.8 8.6 9.4 10.3 10.6 11.0 11.5 11.7 11.7 610 4.5 5. 5 I 6.2 6.S8 7.6 8.4 9.2 10.0 10.7 11.1 11.6 j11.8 660, 3.7 4.1 4.7 5.2 5.9 6.5.3 8.3 9.1 9.8 110.5 11.2 11.5 680 3.1 34 3.8 4.3 4.8 5.5 6.2 7.0 7.8 87. 9.6 16.2 11.0 700 2.9 2.8'3.0 3.4 3.9 4.51 5.2 6.0 6.7 7.5 8. 9.4 10.1 720 2. 2.6 2.5 2.7 3.1i 3.5 4.0 4.8 5.i 6.4 7.3 8.1 9.1 740 | 2.F5 2.4 2.4!4 2 5 2.7 3.1 3.6 4.5 5.2 6.1 6.9 7.8 7601 2.5 2.:3 2.2 11 2.11 2.31 2.4 2.8 3.2 4.1 4.7 5.7 6.6 780 2.7 2.3 2. 1 2 2.0 1.9 2.1 2.2 2.5 2.9 3.6 4.4 5.2 800 2.8 2.71 2.4 2.2''.0 1.8 1.8 1.8 20 2.3 2..1 3.2 4.0 20 3.3 3.0! 2.7 2.3 2.1 1.9 1.8 1.5 1.7 1.7 2.0 2.2 2.9 840 3.8 351' 03.0 2.6 2.3 2.1 1.9 1.6 1.5 1.5 1.6 1. 2.2 860 4.4 4.0; 3.5 3.21 2.8 2.3 1.9 1.7 1.4 1.3 1.2 1.4 1.6 880 4.7 4.4! 411 37 3.3 3.0 2.5 2.1 1.7 1.4 1.3 1.2 1.2 300 5.2 5.0 4.61 4.0 3.6 3.2 2.7 2.2 1.6 1.3 1.2 1.1 9201 5.7 5.31 5.01 4.6 4.2 3.8 3.4 2.9 2.3 1.9 1.3 1.1 9340 6.1 5. 56 5.4 5.2 4.8 4. 3.9 3.5 3.1 2.6 2.1 1.5 960 6.6 6.4 6.2 5. 5.41 5.4 4.7 4.3 3.7 3.2 2.8 2.3 1 980 7.2 6.9, 66 61.4 6.2 5.9 5.6 5.3 5.0 4.6 4.0 3.5 3.0 it(001 7.7 7.4 1 6.9 e 1 6.7 6.4 6.1 5.81 5.5 5.2 4.8 4.4 3.71 68 I9 i 90 700 710 1 20 1 730 1 740 750! 760 770 - 800i TAELE XXXI. Perturbations produced by Mars Arguments II. and IV. IV. I. 800 81 0 85 820 830 840 850 860 870 801890 900 910 0 0 3.71 3.2 2.6 2.1 1.7 1.3 0.9 0.7 0.7 1.0 1.2 1.6 221 20 4.7 4.2 3.6 3.1 2.4 1.9 1.5 [ 1.2 0.8 0.6 0.9 1.2 15 40 5.3. 4.9 4.5 3.8 3.3 2.7 2.0 1.7 1.4 1.0 0.8 0.9 1.0 60 6.01 5.7 5.2 4.7 4.1 3.61 3.1 2.6 2.0.5, 1.2 0.9 1.0 80 6.51 6.3 6.0 5.5 5.0 4.6 4.0 3.4 2.7 2.2 1.8 1.5 1.3 l00 I 7.0 6.7 6.5 6.3 5.9 5.3 4.9 4.4 3.7 3.1 2.5 2.1 1.7 120 7.5 7.3 7.0 6.81 6.5 6.2 5.7 5.1 4.7 4.1 3.51 2.9 24140 7.7 7.7 7.5 7.3 j 7.0 6.71 6.4 6.0 5.6 5.1 4.5 3.8 33 160 7.8 7.9 7.7 7.6 7.4 7.21 7.0 6.8 6.3 5.8 5.41 4.8 421 189 7.9 7.8 7.9 7.9 7.7 7. 7.5. 1 7.0 6.6 6.1 5.7 5.2 200 7.9 7.9 7.8 7.9 7.8 7.7 7.6 7.5 7. 7.1 6.8 6.3 6. 6.86 6.1.5. 220 8.0 7.9 7.8 7.8 7.8 7.8 7.81 7.8! 7.6 7.5 7.4 711 6.7 2401 7.8 7.7 7.7 7.7 7.7 7.7 7.8 7. 7.7 7.6 7.6 7.5 7.2 260 7.8 77 7.7 7.6 7.7 7.7' 7.7 7.7 7.7 7.7 7.8 7.8 7.6 28()1 7.4 7.4 7.5 7.5 7.5 7.51 7.5 7.5 7.5 7.6 7.6' 7.8 7.7 300i 7.1 7.2 7.3 7.3 7.3 7.3 7.3 7.4 7.5 714 7.5 7.5 7.7 3201 6 81 6.9 6.8 7.0 7.11 7.1 7.1 7.1 7.3 7.3 7.31 7.4 7.4 340 6 41 6.5, 6.6 6.6 6.7 6.7 6.8 6.9 7.01 7.1 7.2 7.2 7.2 360, 6.21 6.21 6.2 6.3 6.4 6.4 6.5 6.6 6.7 6.7 6.9 6.91 7.1 380 5 91 5.81 5.8 5.9 6.0 6.1 6.2 6.3 6.4 6.4 6.4 6.6 6.8 4001 5,6 5.6 5.6 5.7 5.7 5.7] 5.8 5.91 5.9 6.0 6.1 6.2 6.4 420 54 5.4 5.5 5.5 5.5 5.5 5.5 5.51 5.61 5.6 F.6 5.7 5.8 440 561 5.3 5.3 5.3 5.3 5.2 5.2 5.2 5.2 5.1 5.0 5.3 i.5 460 |G o0 5.6 5.4 5.3 5.2 5.2 5.1 5.0 5.1 5.2 5.21 5.21 5:3 480 65 6.0 5.7 5.4 5.2 5.2 5.1 4.9 4.9 4.9 4.91 5.01 50 500O 7.2 6.8 6.3 5.9 5.6 5.3 5.0 4.8 4.9 4.8 4.81 4.8 49 520 8.0 7.4 7.0 6.5 6.1 5.5 5.4 5.1 4.9 4.7 4.7 4.,7 4.8 540 9.01 8.4 7.8 I 7.3 6.7 6.3 5.8 5.4 |5.2 4.9 4.7 4.7 4' 560 9.91 9.5 8.8 8.2 7.7 7.1 65 6.0 5.7 5.3 5.0 711 4.8 4.6 580 10.61 10.2 98 9.3 8.8 8.1 7.2 6.8 6.4 6.0 5.61 5.11 4.91 600 11.2 11.0 10.7 10.3 9.6 9.1 8.5 7.7 7.1 6.4 6.1 5.61 5.3 620 11.71 11.5111.4 11.0 10.6 9.9 9.5 S.9 8.1 7.4 6.8 6.3 5.9 640 11.81 11.9 11.8 11.7 11.3 11.0 10.4 9.8 9.31 8.5 7.8 7.1 6.61 660 11.5 11.8 12.0 12.1 11.9 11.6 11.2 10.8 10.2 9.6 8.91 8.2 7.5 680 11.0! 11.6 i2.1 12.2 12.1 12.2 i 12.1 11.1 11.1 10.6 10.1 9.2 8.5 700. 10.1 10.9 11.6 12.1 12.4 12.3 12.3 12.3 11.9 11.4 10.8 10.4 9.7 720 9.1 10.0 10.6 111.4 11.9 124 12.6 12.5 12.4 12.0 111.6 11.2 10.8 740 7.8 8.81 9.7 10.5 11.3 11.8 12.3 12.8 12.6 12.6 12.3 11.9 11.5 760 6.6 7.6 8.5' 9.4 10.3 11.0 11.7 12.1 12.6112.8 12.7 12.5!12.1 7800 5.2 6.3 7.1 8.1. 9.2 10.1 10.7 11.6 12.0 I 12.4- 12.8 12.9 12.8 800 4.0i 4.8 5.7 6.7 7.7 8.7 9.7 10.5 11.3 11.9 12.3 12.5 12.9 820 2.9 3.6 4.4 5.4 6.4 7.3 8.4 9.5 10.3 11.0 11.7 12.1 12.5 840 2.2. 2.7 3.3 4.0 4.9 6.0 7.0 8.0 91 10.0 10.8 11.4 12.0 860 1.6 1.6 2.2 2.9 36 4.61 5.6 6.61 7.6 8.6 9.6 10.5 11.2 880 1.2 1.3 1.5 1.9 2.6 3.3 4.1 5.2 6.1 7.1 8.2 9.2 1.0. 1 900 1.1!.1 1.2 1.3, 1.7 2.2 2.9 3.8 4.8 5.7' 6.8 7.9 8.8 I 920 1.1! 1.0 1.0 1 1 1.1 1.4 1.9 2.61 3.4 4.4 5.3 6.31 7.4 940 1.5 1.1 0.8 1.0 1.1 1.3 2.3 3.1 3.9 5.0 5.9 I960 23 1.7 1.3 0.9 0.7 0.8 0.9.2 1.4 2.0 2.8 3.5 4.61'St' 30 2.5 1.9 1.4 1.2 1.0 0.8 0.9 1L.2 1.4 1.7; 2.4 3.3! IO ~O 37 3.2 2.6 2.1 1.7 1.3' 0.9 0.7 o.r 1.0 1.2! 1.6 2.2 S 00 _ 810 890 830 840 1 850 860 870 880 890 900 910 1 20 40 TABLE XXXI. TABLE XXXII. Perturbations by Mars. Pert's. by Jupiter Arguments II. and IV. Arg's. II. and V. IV. V. I. s9z0 1 930 940 950 960 970 980 990 10001 0 ~ 10 i 20, 30 0 2.2 3.0 3.8 4.8 5.8 6.9 7.8 8.4 9.5 15.3 15.1 15.0 15.0 20 1.5 2.1 2.6 3.4 4.4 5.5 6.5 7.6 8.7; 14.9 14.9 14.7 14.8 40 1.0 1.4 1.8 2.5 3.2 4.0 5.2 6.0 7.1 14.7 14.6 14.6 14.5 60 1.0I 1.1 13 1.8 2.3 3.0 7 4.8 5.8 14.4 14.4! 14.4 14.4: 80 1.31 1.1 1.2 1.4 1.6 2.2 2.7 3.6 4.513.4 13.9 14.0 14.2 100 1.7 1.3 1.2 1.2 1.3 1.6 2 2.6 3.3 13.2 13.4 13.6 13.7 12.0 2.4 2.0 1.5 1.4 1.4 1.4 1.7 1.9 2.4 12.3 12.7 13.0 13.3 140 3.3 2.8 2.3 2.0 1.7 1.5 1.5 1.8 2.1 11.3 11.8 12.1 12.5 160 4.2 3.6 3.1 2.6 2.1 2.0 1.7 1.7 1.9 10.2 10.7 11.2 11.7 180 5.2 4.6 4.0 3 5 3.1 2.5 2.0 2.0 1.9 9.1 9.6 10.1 10.7 200 6.1 5.5 5.0 4.4 3.9 3.5 2.8 2.7 2.9 7.8 8.3 8.9 9.5 220 6.7 6.3 5.8 5.4 4.9 4 4 3.9 3.2 3.0 6.8 7.2 7.7 8.3 240 7.2 6.9 6.6 6.1 5.6 5 3 4.8 4.2 3.7 5.7 6.2 6.6 7.2 260 1 7.6 7.5 7.1 6.8 6.5 6.0 5.6 5.2 4.8! 4.8 5.2 5.6 6.1 280 7.7 7.7.5 3 7.1 6.7 6.3 5.9 5.5 3.9 4.1 4.7 5.2 300 7.71 7 7 7.7 7.7 7.4 7.2 7.0 6.6 6.11 3.4 3.5 3.9 4.3 320 7.4 7.4 7.6 7.7 7.6 7.6 7.3 7.1 6.9 3.2 3.1 3.4 3.6 340 7.2 7.2 7.3 7.5 7.7 7.6 7.6 76 7.7 3.2 3.0 3.0 3.1 360 7.1 7.1 7.1 7.2 7.2 7.6 7.6 7.6 7 5 3.5 3.2 2.9 2.9 3801 6.8 6.9 7.0 7.0 7.0 7.1 7.3 7.5 7.5 4.5 4.0 3.4 3.1 00~ 6.4 6.6 6.6 6.7 6.7 6.9 7.0 7.1 7.3 5.0 4.3 3.8 3.5 420 5.8 5.9 6.2 6.3 6.6 6.5 6.7 6.7 6.9 6.1 5.2 4.6 4.1 440 5.5 5.6 5.7 5.8 6.0 6.1 6.3 6.5 6.5 7.5 6. 6 58 4.9 460 5.3 5.3 5.4 5.7 5.7 5.7 5.9 6.1 6.2 9.0 7.91 7.0 6.3 480 5.0 5.0 501 51 53 5.4 5.5 5.6 5.83 10.5 9.5 8.5 7.6 500 4.9 4.9 5.0 s.o 5.0 5. 1 5.2 5.3 5.3: 12.3 11.3 10.0 9.1!520 4.8 4.8 4.8 4.8 4.8 4.7 4.9 5.0 5.1 14.0 12.7 11.7 10.7.540 4.7 4.7 4.6 4.6 4.6 4.5 4.6 4.6 4.7 15.6 14 5| 13.3 12.3 5601 4.6 4.5 4.5 4.4 4.5. 4.5 4.5 4.5 4.4 17.1 16.1 15.1 14.0 580 s 4.9 4.7 4.6 4.5 4.4 4.4 4.4 4.4 4.2 18.6 17.4 16.5 15.7 t000 5.3 4.9 4.8 4.7 4.5 4.4 4.4 4.3 4.1 19.8 19.0 17.9 17.0 620 5.9 5.5 5.1 4.8 4.6 4.5 4.4 43 4.2 20.8 20.1 19.2 18.4 840 6.6 6.1 5.6 5.0 4.7 4.6 4.5 4.3 21.6 20.9 20.2 19.5 |b60 7.5 6.8| 6.3 5.9 5.5 5.3 4.9 4.8 4.6 22.1 21.6 21.0 20.4 680 8.5 7.8 7.3 6.5 6.1 5.6 5.4 5.1 4.8 22.3 22.0 21.6 21.2 700 9.7 8.9 8.1 7.6 7.0 6.3 5.9 5.6 5.3 22.2 22.0 21.7 21.5 720 11.8 10.0 9.3 8.5 7.9 7.2 6. 6 6.1 5.8 22.0 21.9 21.7 21.6 740 11.5 11.0 10.21 9.7 8.9 8.2 7.6 6.9 6.5 21.6 21.6 21.5 21.5 760 112.1 11.8 11.3 10.5 10.0 9.3 8.5 7.9 7.3 21.2 21.1 21.1 21.2 780 12.8 12.3 11.9 11.4 10.9 10.2 9.6 9.0 8.2 20.4 20.5 20.6 20.7 800 12.9 12.9 12.5 12.1 11.7 11.2 10.5 9.8 9 2 19.6 19.8 19.9 20.1 820 12.5 12.7 12.8 12.7 12.2 11.9 11.2 10.7 10.1 18.8 19.0 19.2 19.4: 840 12.0 12.4 12.6 12.8 12.6 12.4 12.2 11.5 10.9 18.1 18.2 18.4 18.6 860 11.2 11.8 12.3 12.5 12.7 12.5 12.5 12.3 11.7 17.4 17.5 17.6 17.9 880 10.1 111.0 11.5 i 12.1 12.3 12.6 12.6 12.4 12.3 16.9 16.9 16.9 17.1 900 8.8 9.8 10.6 11.3 11.8 12.2 12.4 12.5 12.4 16.3 16.4 16.4 16.5 920 7.4 8.4 9.3 10.2 11.0 11.5 12.1 12.2 12.3 16.0 15.9 15.9 16.0 940 5.9 7.1 8.1 8.9 9.91 10.7 11.2 11.7 12.1 15.8 15.7 15.7 15.6 960 4.6 5.6 6.7 7.7 8.7 9.4 10.2 10.9 11.4 15.5 15.4 15.3 15.4 980 3.3 4.2 5.2 6.2 7.31 8.2 8.9 9.9 10.6 1 15.3 15.2 15.2 15.1 1000 2.2 3.0 3.8 4.8 5.8 6.9 7.8 8.7.5 115.3 15.1 15.0 15.0 920 930 340 950 960 i 970 1 980 990 11000o O 10 20 30 TABLE XXXII. 41 Perturibations p)'roduced by Jupiter. Arguments II. and V. V. 1l, 30 40 50 60 70 1 80 90 100 j 110 120 130 140 1501.... —,, _j._ -1- ---- --; 0 15.0 14.8 14.7 14.7 14.6 14.5 14.5 14.4 114.5 14.5 14.6 14.7 14.8' 20 14.8 14.7 14.6 14.4 14.4 14.2 14.2 14.1 114.1 14.1 14.1 14.1 14.2 40 14.5 14.4 14.4 14.3 14.2 14.1 13.9 13.8 13.8 13.8 13.8 13.8 13.7 60 14.4 14.3 14.3 i4.2 14.1 13.9 13.8 13.6 1? ]5 13.5 13.5 13.4 13.3 80 14.2 14.2 14.1 14.5 14.0 13.8 13.7 13.5 13.4 13.2 13.1 13.0 13.1 100 13.7 13.7 113.9 13.9 13.8 13.7 13.6 13.5 13.4 13.2 13.0 12.8 12.7 120 13.3 13.4 13.4 13.5 13.6 13.5 13.5 13.3 13.3 13.2 13.( 12.8 12.6 140 12.5 12.8 13.0 13.1 13.2 13.2 13.3 13.2 13.1 13.0 12.9 12.8 12.6 160 11.7 12.0 12.4 12.6 12.7 12.8 12.9112.9 13.0 12.9 12.8 12.7 112.5 180 10.7 11.1 11.6 11.9 12.2 12.3 112.5 12.5 12.6 I2.7 12.8 12.6 12.5 200 9.5 10.0 10.6 11.0 11.5 11.7 11.9 12.2 12.2 12.3 12.4 12.3 1 12.3 221) 8.3 8.8 9.5 9.9 10.4 10.8 11.3 11.5 11.8 11.9 12.0 12.0 12.0 240 7.2 7.7 8.2 8.9 9.41 9.8 10.3 10.6 11.0 11.3 11.5 11.7111.8 260 6.1 6.5 7.1 7.6 8.3 8.8 9.3 9.7 10.1 10.5 10. 1 1.0 11.2 280 5.2 5.5 6.0 6.5 7.1 7.6 8.2 8.7 9.2 9.6 10.0 10.4 10.6! 300 4.3 4.7 5.1 5.5 6.1 6.6 7.1 7.6 8.1 8.7 9.1 9.4 9.9 320 3.6 3.9 4.3 4.6 5.1 5.4 6.0 6.6 7.2 7.7 8.1 8.5 8.9 340 3.1 3.3 3.5 3.S 4.1 4.5 5.0 5.4 6.11 6.6 7.2 7.6 8.0 360 2.9 3.0 3.1 3.3 3.6 3.8 4.1 4.5 5.0 5.5 6.1 6.6 7.1 380( 3.1 2.81 2.81 2.7 2.8 2.9 3.0 3.2 3.5i 4.1 4.6 5.0 5.6 400:3 5 3.11 2.91 2.9 2.8 2.8 3.0 3.1 3.4 3.8 4.2 4.7 5.2 420 4.1:3.6 3.31 3.1 2.8 2.7 2.8 2.9 3.1 3.2 3.5 3.8 4.3 440 4.9 4.41 3.9 3 4 3.1 2.7 2.8 2.7 2.8 3.1 3.1 312 3.5 460( 6.3 5.41 4.8 4.3 3.7 3.2 2.9 2.8 2.8 2.7 2.7 2.8 3.2 480 7.6 6.71 5.9, 5.2 4.61 4.1 3.6 3.1 3.0 2.8 2.8 2.6 2.7.50 0 9.1 8.1 7.2 6.4 5.7 5.0 4.4 3.9 3.4 3.2 3.1 2.9 2.7 520( 10.7 9.5 8.7 7.7 6.9 6.1 5.5 4.8 4.2 3.8 3.5 3.2 3.1 5401 12.3 11.1 10.2 9.1 8.41 7.4 6.6 5.9 5.3 4. 7 4.1 3.8 3.5,5 60114.0 13.0 11.9110.8 9.9 8.7 7.9 7.1 6.4 5.81 5.21 4.5 4.1 580U 15.7 14.5 13.6 12.5 11.4 10.4 9.3 8.3 7.7 6.9 6.2 5.5 5.0 600 17.0 16.0 15.0 14.0 13.11 12.0 11.0 10.1 9.21 8.2 7.5 6.7 6.0 620 18.4 17.4 16.5 15.5 14.7 13.6 12.6 11.6 10.7 98 9.0 8.0 7.3 640 19.5 18.5 17.9 17.0 16.0 15.1 14.2 13.1 12.2 1.3 10.8 9.4 8.7 660 20.4 19.7 18.9 18.1 17.4 16.3 15.6 14.6 13.7 l1.8 11.9 11.0 10.1 680 21.2 20.5 19!.9 19.1 118.5 17.6 16.8 16.0 15.1 114.2 13.5 12.5111.6 700 21.5 21.0 20.6 1 20 0 19.3 18.7 18.0 17.1 16.5 15.6 14.7 13.8 13.0 720 21.6 21.2 21.0 20.5 20.0 19.3 18.9 18.3 17.5 16.8 16.1 15.1 14.3 740 21.5 21.2 21.1 20.8 120.5 20.0 19.4 18.9 18.4 17.7 17.2 116.8 15.7 760 21.2 21.0 21.0 20.8 20:7 20.3 120.0 19 4 19.0 118.6 17.9 17.4 16.7 7 20.7 20.7 20.7 20.6 20.6 20.3 20.2 19.8 19.4 19 1 189 7 18.1 17.6 8u0 20.1 120.2 120.3 120.3 20.41 20.3 120.1 19.9 19.7 119.3 19.1 18.7 18.2 820 19.4 19.5 19.7 19.8 19.9 19.9 199 19.8 19.8 19.6 19.2 18.9118 7 840 18.6 18.8 18.9 119.0 19.2 19.3 19 4 19.4 19.4 19.4 19.4 19.0 1 18.9 860 17.9 18.0 18.3 18.4 18.6 18.7 18.8 18.9 19.0 19.1 19.1 19.0 18.8 880 17.1 17.2 17.5 117.6 17.918.0 1.2 18.3 18.5 18.6 18.6 18.6 18.7 900 16.5 16.6 16.8 16.9 17.1 17.1 17.41 17.5 17.7 17.9 18.1 18.2 18.2 920 16.0 16.0 16.1 16.2 16.4 16.5 16.7 16.8 17.0, 17.2 17.4 17.5 17.7 940 15.6.5 15.5 6 1 15.6 15.7 15.8 16.0 16.1 16 3 16.5 16.8 16.8 17.1 960 15.4 15.3 115.3 15.2 15.2 15.2 15.3 15. 15.6 15.7 15 9 16.0 16.3 980 15.1 15.0 15.0 14.9 14.9 14.8 14.9 14.9 14.9 15.0 15.2 15.3 15.5 1000 15.0 14.8 [4.7 14.7 14.6 14.5 14.5 14.4 14.5 14.5 14.6 14.7 14.8 30 40 0 60 70 80 90 1 100 110 120 130 140 1501 42 T ABLE XXXII. Perturbations produced by Jupiter. Arguments II. anld V. V, 11. 150 160 170 180 190 200 1 210 220 230 240 250 260 270 _" i"!" lr0 ) 1, _ 2201230 24025026 __ 0 14.8 115.0 1 5.3 15.5 15.8 15.9 16.2 16.3 16.7 17.0 17.1 17.3 17.5 20 114.2 14.3 14.6 1 1. 14.9 15.2 5.5 15.7 15.9 16.2 16.6 16.8 17.1 40 113.7 13.7 13.9 14.1 14.3 14.5 1 14.8 15.0 15.3 15.5 15.8 16.2 16.41 60 113.3 13.2 13.4 113.5 13.6 13.8 14.1 14.3 14.61 1115.5 15.8 80 113.1 13.0 13.0 113.0 13.1 13.1 13.3 13.5 13.8 14.1 14.4 14.5 15.1 100 12.7 12.7 12.7.6 12.7 12.6 12.8 12.9 13.1 13.4 13.7 14.0 14.2 120 112.6 112.5 15 112. 112.3 12.2 12.3 12.3| 12.6 12.S 13.0 13.3 13.6 180 j 12.5 112.3 12.2 140 12.6 12.4 2.4 i 12.3 12.1 12.0112.0!2.0j 12.11 12.1 12.3 12.5 12.8 160112.5 12.31.2 12.1 12.1 11.9 11.8 11.8 11.8 11.8 11.9 12.0 12.2 18O 12.5 112.3 12,.2 12. 11.9 11.8 11.7 11.5 11.5 11.5 11.6 l11.7 11.8 200 112.3 1 12.2 2 12.2 12.0 11.9 11.7 11.7 11.5 11.4 11.3 11.2 11.3 11.5 220 12.0 12.01 12.1 12.0 111.8 11.6 11.6 11.5 11.4 11.3 11.2 11.1 11.1 240 11.8 111.8 111.9 11.9 11.8 11.6 i 11.5 11.4 11.3 11.2 11.1 11.1 11.0 260 11.2 11.5 11.6 11.6 11.6 11.5 1 11.3 11.3 11.3 11.2 11.1 1.1.0 10.9 280 10.6 10.8 11.1 11.2 11.2 11.2 11.31 11.3 11.2 11.2 11.1 11.0 10 9 300 9.9 10.1 10.5 10.8 10.9 11.0 111 11.0 11.0 11.0 11.0 11.1 16.9 320 8.9 9.4 9.7 10.1 10.4 10.5 10.7 10.8 10.8 10.8 10.8 10.8 10.9 340 8.0 8.5 9.1 9.31 9.61 9.9 10.2 3 10.5 10.6 10.6 10.7 10.7 360 7.1 7.5 8.0 8.4' 8.9 9.2 9.51 9.8 10.1 10.3 10.4 10.5 10.5 380 5.6 6.2 6.8 7.J1 7.8 8.3 8.9 9. 9.7 10.0 10.0 10.1 10.2 400 5.2 5.6 6.2 6.6 7.0 7.5 7.9 8.4 S.8 9.1 9.4 9.7 9.9 420 4.3 4.8 5.3 5.8 6.2 6.6 7.1 7.4 7.9 8.4 8.7 9.1 9.4 440 3.5 3.9 4.4 4.9 I 5.4 5.7 6.2i 6.7 7.1 7.6 7.9 8.4 8.7 460 3.2 3.3 3.8 41 4. 49 5.41 5.7 6.3 6.7 7.21 7.7 8.0 4801 2.71 2.9 3.2 3.6:3.9 4.31 4.7 5.0 5.01 5.9 6.3 6.8 7.3 500 2.7 2.7 2.9 3.1 3.4 3.6 4.0 4.4 14.8 4 5.2 5.7 5.9 6.4 520 3.1| 2.8 2.9 3.0 3.1 3.21 3.5 3.8 4.2 4.7 4.9 5.4 5.7 540, 3.5 3.1 3.1 3.0 3.0 3.0. 3.3 3.5 3. 7 4.1 4.3, 4.7 5.1 560 4.1 |3.8 3.61 3 3. 2 3.2 3.2 3.2 3.3 3.5 3.7 4.0 4.3 4.5 580 5.0 4.6! 4.2 4.0 3.6 3.5 3.3 3.21 3.41 35 3.7 4.0 4.2 600 6.01 5.41 5.1 4.6 4.3 3.9 3.7 3.5 3.5 3.6 3.7 3.8 4.0 620 7.3 6.6G 6.0 5.61 5.1 4.6 4.2 4.01 3.9 3.8 3.91 3.9 4.0 640 8.7 7.8' T.3 6.6 6.1 5.5 5.2 4.71 4.4 4.2 4.0 4.0 4.1 6600110.1 9.3 8.61 7.7 7.2 6.5 6 5.9 5.3 4.9 4.6 4.5 4.4 680 11.6 10.8 10.0 9.3 8.5 7.5 7.3 6.7 6. 3 5.8 5.5 5.2 4.9 /00 13.0112.1. 11.5 10.7 9.9 9.0 8.5 7.8 7.4 6.9 6.3 6.0 5.8 72( 14.3 13.5 12.8 12.1 11.3 10.6 9.8! 9.1 8.7 8.0 7.6; 7.0 6.6 740 15.7 14.9 14.2 13.4 12.7 12.0 11.2 10.5 9.7, 9.3 8.91 8.2 7.7 760 16.7 15.9 15.5, 14.7 13.9 13.3 12.6 11.811. 21 10.5 10.0 9.5 9.0 780 17.6 17.0 16.4 15.7 15.1 14.6 13.8 13.2 1.2.6 11.9 11.2 10.9 1 ).: 800 18.2 17.8 17.3 16.8 16.2 16.0 15.0114.3 13.7 13,1 12.6112.0 11.5 820 18.7 118.3 18.0 117.6 17.0 16.6 16.0 15.3 14.9 14.3 13.7 113.1 12.6 840 118.9 18.7 18.4 18.2 17.7 17.2 16.8 16.3 15.8 15.3 14.9 14.4 13. 8 860 18.8 18.7 18.6 18.4 18.3 17.9 1.7.4 17.1 16.7 16.3 15.9 15.41 15.0 80 118.7 18.5 18.6 18.5 118.3 18.2 18.0 17.7 17.4 17.1 16.6 |6.3 15.9 900 1 18.2 18.2 18.3 18.3 118 3 18.1 18.1 18.0 17.8 17.6 173 17.0 16.7 920 17.7 17.9, 18.0 1180 18.1 18.1 18.(0 18.0 18.0 17.8 17.7 17.6 17.3 940 i17 17. 1 17. 4 17. 17.6 17.7 17.8 17.8 17.9 18.0 17.8 17.8 i 17.7 960 1 163 16.5 1l. 16 9 117.1 17.2 17.4 17.5 17.6 117.8 17.9 18.0 1 7.9' 90 15.5 1 15.7 1.1 16.3 1 16.5 16.7 i6.8 17.01 17.2 17.3 17.617.7 17. 9 1000 4.8 15.0 15.3 1 5 1 15.8 15.9 16.2 1.3 16.7 17.0 17.11173 1 1.5 15 1 1) 170 181 11t I 2() 2 210 2`20 1 3o) 2) 1 25 260 I 27 TABLE XXXII. 42 Pertur-bations produced by Jupiter. Arguments II. and V V 1.] 279 280 290 300 310 320 330 340 350 1360 1370 380 390 0 17.5 117.5 17.7, 17.8 17.9 17.9 18.0 18.0 17.9 17.7 17.6 17.5 17.5 20 17.1 17.3 17.5 17.6 17.8 17.8 18.0 18.1 18.1 18.1 18.0 180 18.0 40 16.4 16.8 16.9117.2 117.6 17.7 17.9 18.1 18.3 18.3 18.4 18.4 18.6 60 15.8 16.0 16.4 16.7 16.9 17.3 17.6 17.9 18.2 18.3 i8.5 18.5 18.7 80 15.1 15.4 15.7 16.1 16.4 16.7 1]7.0 17.5 117.8 18.0 18.3 18.5 18.8 100 14.2 14.6 15.1115.0 15.8 16.1 16.5 17.0 17.2 17.5 17.9 18.3 18.7 120 13.6 13.7 14.2 114.5 15.0 15.4 15.8 16.2 16.7 17.1 17.3 17.9 18.3 140 12.8 13.1 13.3 13.7 14.2 14.4 15.1 15.5 15.9 16.3 16.8 17.3 17.7; 160 12.2 12.4 12.6 12.9 13.4 13.8 14.1 14.6 15.2 15.5 16,0 16.5 17.1 180 11.8 11.9 12.1 12.3 12.5 12.8 13.3 13.7 14.4 14.7 15.2 15.7 16.3 200 11.5 11.5 11.6 11.7 12.0 12.1 12.5 13.0 13.4 13.8 14.3 14.7 15.51 220 11.1 11.1 11.2 11.3 11.6 11.7 11.9 12.3 12.7 13.0 13.5 14.0 14.5 240 11.0 10.9 10.9 11.0 11.2 11.3 11.5 11.8 12.1 12.3 12.8 13.2 13.8 260 10.9 10.8 10.8 10.8 10.9 10.9 11.1 11.3 11.4 11.6 12.0 12.3 13.0 280 10.9 10.8 10.7 110.6 10.7 110.6 10.8 11.0 11.2 11.3 11.5 11.8 12.2 300 10.9 10.8 10.7110.6 10.6 10.5 10.6 10.7 10.8 10.9 11.1 11.4 11.8 320 10.9110.7 10.7 iO.6 10.6 10.5 10.5 10,6 10.7 10.6 10.7 11.0 11.2 340 10.7 1.7 10.6 10.5 10.5 10.4 10.5 10.5 10.6 10.5 10.6 10.7 10.8 360 10.5 107 5 10.5 10.5 10.5 10.4 10.4 10.4 10.4 10.3 10.5 10.6 10.8 380 10.2 10.3 110.3 110.3 10.4 10.3 10.4 10.4 10.4 10.3 10.3 10.4 10.6 400 9.9 10.0 10.0 10.2 10.353 10.2 10.52 10.3 10.4 10.3 10.3 10.3 10.5 420 9.4- 9.61 9.8 9.9 10.1 10.2 10.1 10.2 10.2 10.2 10.3 1 0 3 10.4 440 8.7 9.0 9.2} 9.4 9.7 9.8 10.0 10.1 10.2 10.1 10.41 0.2 10.4 460 8.0' 8.4 8.6 8.8 9.1 9.3 9.6i 9.9 10.1 10.0 10.0 10.2 1).3 480 7.3 7.6 7.9 8.4 8.7 8.9 9.1 9.4 9.6 9.7 9.8 10.0 10.1 500 6.4 6.9 7.2' 7.6 8.0 8.3 8.6 8.9 9.2 9.4 9.5 9'7 9.9 520 5.7 6.11 6.6 6.9 7.3 7.6 7.9 8.3! 8.6 8.9 9.1 9.4 9.7 540 5.1 5.41 5.8 6.2 6.7 7.0 7.4'7.7 8.0 83 8.6 8.9 9.2 560 4.5 4.9 5.1 5.5 6.Q 6.3 6.7 7.2 7.5i 1.7 8.0 8.: 8.7 580 4.2 4.4, 4.8 5.0.5.3 5.7 6.1 6.6 6.9 7.1 7.4i 7.7 8.1 600 4.0 4.2 4.3 4.7 4.9 5.2 5.6 6.0 6.3 6.5 6.8 7.2 7.6 6201 4.0 4.0 4.1 4.3 4.7 4.8 5.1 5.5 f.8 61 6.4 6.7 7.01 640 4.1 4.1 4.2 4.2 4.4 4.6 4.8 5.1 5.4..61 5.3 6.3 1 6.6 660 4.4 -.3 4.3 4.3 4.5 4.5 4.7 4.9 5.1 5.3 5.5 5.8.2 680 4.9 4.9 4.7 4.6 4.7 4.5 4.6 4.8 5.0.5.1 5.3 5-5 5 7001 5S.8 5.4 5.2 5.1 5.0 4.9 4.9 4.9 5.1 5.2 5.3 5.4 720 6.6 6.2 5.9 5.7~ 5.6 5.5 5.4 5.3 5.3 5.3 5.3I 5. 4 740 7-.7 7.2 6.8 6.5 6.4 6.1 6.0 5.9 5.8 5.7 5.6 5 5.7 760 9.0. 8.2 7.9 7.5 7.2 6.9 6.7 6.5 6.3 6.1 5.9 5.9 6)0 780 9.7 9.11 8.4 8.2 7.7 7.6 7.4 7.24 6.9 6.f 6. 5 6 5 800 11.5 11.0 10.4 9.8 9.4 8.7 8.5 8.3 8.0 7.7 7.6 7.3'.11 820 12.6 1 1.71111.7 11.2 10.6 10.1 9.7 9.2 9.1 8.6 8.3 1 79 840 13.8 13.2 i 12.8 112.3 111.9 11.3! 10.9 10.5h 10.2 9.6 9.4 9.1 S 9 860 15.0 11.4 1 13.8 13.5 113.1 12.6 12.1 11.7 11.2 10.7 10.4! 10.1 1) ) 880 115.9 15.4 15.0 114.4 114.2 13.7 13.4 12 9 12.51 12.0 11.5 11 11.I 900 16.7 j 16.4 15.9 15.5 15.2 14.8 14.4 14.1 13.7 13.2 12.8 12 4 12.2 1 920 17.3 17.1! 16.8 16.5 16.2 15.7 I 15.5 15.2 14.8 14 3 14.0113 6 | 13.3 940 17.7 17.5 17.3 17.1 16.9 16.6 16.3 16.1 16.0 15.5 15.0 14 7 14.5j 960 1 7.9 17.8 17.61 17.5 17.4 176.2 1508 1 5.6t 980 i.' 17. 17. 17.8 17.8 1 17.8 17.8 17.6 17.5 17.3 17.2 17.0 16.8 1661 000 117.5 17.7 9 17.7 9 17.8 117.9 | 17.9 1.0 79 17.7 176 27 280 290 003 310 30 0' 340 350 360 370:350 390 I8().7.... 44 TABLE XXXII. Perturbations piroduced by Jupiter. Arguments II. and V. V. l. 390 400 410 420 430 440 450 460 470 480 490 500 610 0 117.5 17.1 17.0 16.7 16.5 16.3 16.1 15.8 15.6 15.1 14.6 14.3 13.9 20 18.0 18.1 17.7 17.5 17.5 17.2 17.1 16.8 16.7 16.3 16.0 15.6 15.3 40 18.6 18.6 18.5 18.4 18.3 18.1 18.0 17.8 17.6 17.3 17.2 16.8 16.5 60 18.7 18.9 18.9 18.91 18.9 18.7 18.8 18.6 18.7 18.4 18.1 17.9 17.7 80 ]8.8 18.9 19.2 19.3 19.4 19.3 19.3,19.3 19.3 19.2 19.2 18.9 18.8 100 18.7 18.9 19.1 19.4 19.7 19.8 19.8 19.8 19.8 19.8 19.9 19.7 19.7 20o18.3 18.6 18.9 19.2 19.5 19.8 20.0 20.1 20.3 20.3 20 4 20.4 20.4 i40 17.7 18.2 18.6 18.9 19.2! 19.6 20.3 20.5 20.6 20.'1 20.8 21.0 160 | 17.1 17.G 17.9 18.5 10.0 19.3 19.8 20.2 20.5 20.6 20.9 21.1 21.2 180 16.3 16.8 17.3 17.9 18.3 18.8 19.3 19.8 20.3 20.6 20.9 21.1 21.4 200 15.5 16.0116.5 17.1 17.7 11.2 18.6 19.1 19.8 20.2 20.7 21.0 21.4 220 14.5 15.0 15.6 16.1 16.9 17.4 18.0 18.6 19.0 19.7 20.3 20.7 21.1 240 13.8 14.2 14.7 15.2115.9 16.5 17.1 17.7 18.4 18.9 19.5 20.1 20.7 260 13.0 113.4 14.4 15.0 15.5 16.3 16.9 17.5 18.0 18.6 19.3 20.0 280 12.2 12.7 13.0 13.5 14.2 14.7 15.3 15.9 16.7 17.2 17.8 18.4 19.1 300 11.8 11.9 12.4 12 8 13.3 13.8 14.4 14.9 15.7 16.3 17.0 17.6 18.2 320 11.2 11.5 11.8 12.2 12.7 13.0 13.6 14.1 14.7 15.3 16.0 16.6 17.4. 340 10.8 11.2! 11.4 11.6 12.1 12.4 12.9 13.4 13.9 14.4 15.1 15.7 16.4 360 10.8 110.8 11.0 11.2 11.6 11.9 12.3 12.6 13.2 13.6 14.2 14.8 15.5 380 10.6 10.6 10.7 10.9 11.2 11.4 11.9 12.2 12.6 12.9 13.5 13.9 14.5 400 10.5 10.5 10.6 10.6 10.9 11.1 11.4 11.8 12.2 12.5 12.9 13.31 13.8 420 10.4 10.4 10.5 10.6 10.7 10.9 11.2 11.3 l11.7 11.9 12.4 12.8 13.3 440 10.4 10.4 10.4 10.5 10.7 10.8 10.9 11.1 11.3 11.6 11.9 12.2 12.7 460 10.3 10.4 10.4 10.4 10.6 10.6 10.7 10.9 11.2111.3 11.7 11.9 12.2 480 10.1 10.2 10.3 10.4 10.6 10.6 10.7 10.8 11.0 11.2 11.4 11.7 12.0 500 9.9 10.0 10.1 10.l2 10.4 10.5 10.7 10.8 10.9 11.0 11.2 11.3 11.7 5201 9.7 9.8 9.8 10.0 10.2 10.3 10.5 10.6 10.9 10.8 11.1 11.3 11.5 540 9.2 9.4 9.6 9.8 10.0 10.2 10.3 10.4 10.6 10.7 10.9 11.1 11.4 560 8.7 8.9 9.1 9.3 9.7 9.8 10.1, 10.3 10.5 10.6 10.7 10.8 11.2 580 8.1 8.5 8.7 8.7 9.2 9.4 9.7 9.9 10.2 10.4 10.6 10.7 10.9 600. 7.6 7.9 8.2 8.5 8.8 9.0 9.3 9.5 9.8 10.0 10.3 10.5 10.7 620 7.0 7.31 7.6 7.9 8.2 8.5 8.8 9.0 9.4 9.6 10.0 10.1 10.4 6401 6.6 6.8 7.1 7.4 7.7 7.9 8.2 8.6 8.9 9.1 9.4 9.7 lo0., 660 6.21 6.4 6.6 6.9 7.3 7.6 7.9 8.1 8.3 8.6 8.9 9.2 9.5 680 5.8 6.1 6.2 6.5 6.8 7.0 7.4 7.6 7.9 8.1 8.4 8.7 9.0 700 5.6 5.8 6.01 6.2 6.4.6 6.9 7.1 7.4 7.6 7.9 8.2 8.5 7201 5.5 5.6 5.7 5.9 6.2 6.3 6.5 6.8 7.1 7.2 7.5 7.7 8.0 740 5.7 5.7 5.7 5.8 6.0 6.11 6.2 6.4 6.7 6.9 7.1 7.2 7.5 760l 6.0 6.0 6.01 6.0 6.0 6 1 6.2 6.3 6.4 6.5| 6.7 6.8 7.1 q80o 6.5 6.3 6.2 6.2 6.31 6.3 6.3 6.3 6.41 6.4 6.5 6.7 6.8 800 7.1 7.0 6.7 6.6 6.7 6.5 6.5 6.4 6.5 6.5 6.5 6.6 6.7 820 7.9 7.6 7.5 7.3 7.2 7.0 7.0 6.8 6.8 6.7 6.6 6.6 6.7 8401 8.9 8.6 8.3 8.1 7.8 7.7 7.6 7.4 7.3 7.1 7.0 6.8 6.8 860 10.0 9.7 9.3 9.0 8.7 8.4 8.2 8.1 7.9 7.71 7.6 7.3 7.2 880 11.1 10.5 10.4 10.0 9.7 9.5 9.2 8.9 8.7 8.4 8.2 7.9 7.7 900 12.2 11.8 11.5 11.0 10.8 10.5 10.3 9.9 9.7 9.4 9.0 8.8 8.5 1 90O 13.3 13.0112.612.3 12.1 11.5 11.3 11.0 10.6 10.2 10.1 9.7 9.4 9AO 14.5114.1 13.8 13.5 13.2 12.8 12.5 11.9 11.8 11.3 11.0 10.7 10.4 9('0 15.6' 15.3 14.9 14.6l 14.4 14.0 13.713.3 13.0 l2.5 12.1 11.8 11.5 980 16) t 16.3 16.0 15.715.6 15.2 11.9 14.6 14.2 133.8113.6 12.9 12.7 1000 z17..5 17.1 17 6716.5.16.3 16.1 15.8 15.6 15.1 14.6 14.3 i13.9 -/3901400 410 420 430 440 450140 4 470 480 490 500 510 -._ I TABLE XXXII. 4 Perturbations produced by Jupiter. Arguments II. and V. V. i1L. 510 1520 530 540 1 550 1 560 570 580 590 600 610 620 630: 0 13.9 13.4 13.1 12.7 12.1 11.8 11.3 10.8 10.2 9.9 9.4 8.9 8.4 20 16.3114.9 14.4 13.9 13.5 13.1 12.5 12.1 11.5 11.0 10.4,10.0 9.4 40 16.516.3 15.7 115.4 15.0 14.3 13.8 13.4 12.8 12.3 11.7 11.1 10.5 60 17.7 17.3 17.0 16.6 16.1 15.8 15.3 14.7 14.3 13.7 13.0 12.4 11.8 80 18.8 18.5 1 17.9 17.4 17.1 16.6 16.2 15.7 15.1 14.5 13.9 13.2 100 19.7 19.5 19.2 19.0 18.8 18.4 17.9 17.6 17.0 16.5 16.0 15.2 14.7 120 20.4 20.3 20.2 20.0 19.7119.5 19.1 18.8 18.4 18.0 17.3 16.8 16.2 140 21.0 21.1 21.0 20.8 20.7 20.4 20.2 19.9 19.6 19.3 18.8 18.3 17.7 160 21.2 21.5 21.51'216 21.5 2'1.3 21.2 21.0 20.6 20.4 20.1 19.6 19.1 180 21.1 21.6 21.8 22.0 22.0 22.1 21.9 21.8 21.6 21.4 21.1 20.7 20.3 200 21.4 21.7 21.9 22.1 22.3 22.5 22.5 22.5 22.4 22.3 22.1 21.8 21.5 220 1 21.1 21.5 21.8 22.2 22.5 22.8 j 23.1 23.1 22.9 22.8 22.9 22.6 22.51 240 20.7 21.1 21.5 21.9 22.3 22.7 23.0 23.3 23.4 23.5 23.4 23.3 23.2: 260 20.0 20.6 21.0 21.6 22.0 22.4 22.8 23.2 23.5 23.8 23.8 23.8 23.9. 2 80 19.1 19.9 20.4 20.9 21.5 22.0122.4 23.0 23.3 23.7 24.0 24.1 24.1 300 18.2 19.0 19.6 20.3 20.7 21.3121.8 22.3 23.0 23.4 23.8 24.1 24.3 320 17.4 18.9 18.7 19.4 20.0 20.6 21.1 21.8 22.3 22.9 23.3 23.7 24.2 340 16.4 17.0 17.6 18.5 19.2 19.9 20.4 21.1 21.6 22.2 22.8 23.3 23.7 360 115.5116.2 16.7 17.4 18.2 18.9 19.5 20.1 20.8 21.5 22.0 22.6 23.2 380 14.5 15.2 15.9 16.6 17.1 17.9 18.6 19.3 19.8 20.5 21.1 21.8 22.5 400 13.8 14.4 14.9 15.6 16.2 16.8 17.6 18.4 19.1 19.7 20.3 20.9 21.5 420 13.3 13.7 11.2 14.8 15.3 16.0 16.5 17.4 18.0 18.7 19.4 20.0 20.6 440 12.7 13.1 13.6 14.1 14.6 15.2 15.7 16.4 17.1 17.8 18.4 18.9 19.6 460 12.2 12.7 13.0 13.5 13.9 14.4 15.0 15.6 16.1 16.9 17.5 18.2 18.7j 480 12.0 12.2 12.5 13.0 13.4 13.9 14.3 14.8 15.3 15.9 16.6 17.3 17.9 500 11.7 12.0 12.2 12.6 12.9 13.3 13.8 14.3 14.7 15.2 15.7 16.4 16.9 520 11.5 11.9 12.0 12.3 12.6 13.0 13.2 13.8 14.2 14.7 15.1 15.5 16.2 540 11.4 11.6 11.9 12.2 12.4 12.7 12.9 13.3 13.7 14.2 14.6 15.0 15.4 560 11.2 11.4 11.5 11.9 12.1 12.41 12.7 13.1 13.4 13.8 14.1 14.5 14.91 580 10.9 11..2 11.4 11.6 11.9 12.2 12.4 12.8 13.1 13.5 13.8 14.2 14.5 600 10.7 10.8 11.1 11.5 11.7 12.0 12.2 12.5 12.8 13.1 1,3.4 13.8 14.2 620 10.4 10.7 10.7 11.1 11.4 11.6 12.0 12.3 12.5 12.9 13.1 13.4 13.8 640 10.1 10.4 10.6 10.7 11.0 11.3 11.6 12.0 12.3 12.6 12.9 13.2 13.5 660 9.5 9.9 10.2 10.5 10.6 11.0 11.3 11.6 11.9 12.3 12.6 12.9 13.2 680 9.0 9.3 9.6 10.0 10.3 10.5 10.8 11.3 111.5 11.9 12.2 12.4 12.8 700 8.5 8.9 9.1 9.5 9.8 10.1 10.3 10.7 11.1 11.4 11.8 12.1 12.4 720 8.0 8.3 8.5 9.0 9.2 9.6 9.9 10.2 10.5 10.9 11.3 11.7 12.0 740 7.5 7.8 8.0 8.3 8.6 9.0 9.3 9.7 9.9 10.4 10.8 11.1 11.5 760 7.1 7.3 7.5 7.9 8.1 8.4 8.6 9.1 9.4 9.7 10.1 10.5 10.9 780 6.8 7.0 7.1 7.3 7.6 7.9 8.1 8.5 8.8 9.2 9.4 9.8 10.2 800 6.7 6.8 6.8 7.0 7.1 7.3 7.5 7.8 8.2 8.5 8.8 9.1 9.5 820 6.7 6.8 6.6 6.8 6.9 7.0 7.1 7.4i 7.6 7.9 8.1 8.4 8.7 840 6.8 6.8 6.8 6.8 6.8 6.9 6.9 7.1 7.2 7.4| 7.6 7.9 8.1 s860 7.2 7.1 7.1 7.0 6.9 6.9 6.8 6.8 6.9 7.1 7.2 7.3 7.6 880 1 7' 7.5 7.4 7.3 7.1 7.0 6.8 6.8 6.7 6.8 6.8 7.0 7.2 900 8.5 8.2 7.9 7.7 7.5 7.3 7.2 7.1 6.9 6.9 6.8 6.8 6.8 920 9.4 9.2 8.7 8.4 8.1 7.9 7.6 7.4 7.1 7.0 6.9 6.8 6.7 940 10.4 10.0 9.7 9.4 8.9 8.6 8.3 8.1 7.7 7.4 7.1 6.9 6.7 96 111.5 11.2 10.7 10.4 9.8 9.5 9.1 8.8 8.5 8.1 7.7 7.4 7.1 980 12.7i 12.3 11.8 11.5 11.1 10.6 10.0 9.7 9.2 8.9J 8.51 8.1 7.7 1000 13.9j 13.4 13.1 12.7 12.1 11.8 11.3 10.8 10.2 9.9 9.4 8.9 8.4 510 590 530 540 550 560 570 580 590 600 610 6o20 630 146 TABLE XXXII. Perturbations produced by Jupiter. Arguments II. and V. V. i630640 650 660 670 680 690 700 710 720 730 740 750 f If If f If / I 0 8.4 8.0 7.7 7.3 6.9 6.7 6.5 6.5 6.3 6.2 6.2 6.4 6.5 20 9.4 9.0 8.4 8.0 7.5 7.1 6.9 6.7 6.4 6,3 6.0 6.1 6.1 40110.5 10.1 9.4 8.91 8.3 7.8 7.4 7.0 6.6 6.4 6.2 5.9 5.8 60 11.8 11.3 10.6 10.1 9.3 8.7 8.2 7.7 7.2 6.8 6.4 6.2 5.8 80 13.2 12.7 12.0 11.3 10.5 9.9 9.2 8.7 8.1 7.6 7.1 6.6 6.2 100 14.7 14.1 13.4 12.8 12.0 11.3 10.6 9.9 9.1 8.5 7.9 7.3 6.8 120 16.2 15.4 14.9 14.2 13.4 12.7 12.0 11.3 10.4 9.8 8.9 8.2 7.6 140 17.7 17.2 16.4 15.6 14.9 14.2 13.4 12.7 11.9 11.1 10.2 9.6 8.8 160i19.1 18.6 17.9 17.3 116.6 15.7 15.0 14.2 13.3 12.6 11.7 10.9 10.0 180 20.3 19.9 19.4 18.8 18.0 17.3 16.7 15.8 15.0 14.1 13.2 12.4 11.5 200o21.5 21.2 20.8 20.2 19.3 18.9 18.1 17.5 16.6 15.7 14.9 14.0 13.1 220 22.5 22.3 21.9 21.5 21.0 20.319.7 19.0 18.2 17.5 16.6 15.5 14.7 240 23.2 23.0 22.9 22.5 22.0 21.6 21.1 20.5 19.8 19.1 18.2/ 17.3 16.4 260 23.9 23.8 23.7 23.5 23.1 22.7 22.3 21.8 21.2 20.6 19.8 19.1 18.1 280 24.1 24.3 24.2 24.2 24.0 23.7 23.5 23.1 22.4 21.8 21.2 20.5 19.8 300 24.3 24.5 24.6 24.6 24.5 24.41 24.2 23.9 23.6 23.1 22.5 21.9 21.2 320 24.2 124.5 24.7 24.9 24.8 24.8 24.8 24.7 24.4 24.1 23.7 23.1 22.5 340 23.7 24.2 24.5 24.7 25.0 25.2 25.1 25.0 25.0 24.9 24.6 24.1 23.7 36023.221 3.7 24. 24.5 24.7 25.0 25.1 25.3 25.4 25.3 25.1 24.9 124.5 380 22.5 23.1 23.6 24.1 24.4 24.7 25.1 25.2 25.4 25.5 25.4 25.3 25.2 400 21c.5 22.3 22.8 23.4 23.9 24.3 24.7 205.1 25.2 25.4 25.6 25.6 25.5 420 20.6 21.3 22.0 22.6 4 23.1 23.61 24.1 245 25.0 2.2 25.4 "25.6 25.7 440 191.6 20.3 21.0 21.8 22.3 22.9 23.4 23.9 24.3 24.8 25.0 25.2 25.6 460118.7 19.4 201 207 21.31291.9 22.6 23.3 23.6 24.1 24.6 24.825.1 480 17.9 18.5 1931 1197 2013 21.0 21.6 22.2 22.8 23.3 23.8 24.3 24.6 500 16.9 17.6 18.2 18.8 19.3 1 19.9 20.7 21.4 21.9 22.5 22.9 23.4 23.9 I 520 16.2 16.8 17.3 17.9 18.41 19.0 19.7 20.4 21.0 21.6 21.1 22.6 23.0 540 1!5.4 16.1 16.6 17.2 17.5 18.1 18.7 19.3 19.9 20.5 1 2 227 22.2 I560i14.9115.4/16.0 6 2.15072. 56014.9 15.4 16.0116.5 16.91 17.3 17.9 18.4 18.9 19[6 20.1 20.7 21.3 580 14.5 15.0 15.3 15.9 16.3 16.7 17.1 17.6 18.1 18.7 19.3 19.8 20.3 600 14. 2114.6 14.9 15.3 15.8 16.3 16.6 17.0 17.4 17.9 18.3 18.9 19.4 620 13.8 14.2 14.6 14.9 105.11157 1652 16.6 16.9 17.3 17.6 118.0 18.5 640 13.5 14.0 14.2 14.6 14.8 15.1 1516 16.1 16.51 16. 17.1 17.5 17.9 660 13.2 13.5 13.9 114.3 14.6 14.9 15.2 15.6 15.9 16.4 16.6 17.0 17.3 680 12.8 113.2 13.5 13.9 14.2 14o5 14.9 15.2 15.6 16.0 16.2 1615 16.8 700 12.4 12.9 13.3 13.5 13.8 14.2 115 14.9 15.1 15.61 15.9 i16.2 16.4 720 12.0 12.4 12.8 13.2 13 5 13.8 14.2 14.5 14.8 15.1 15.5 15.8 1.1 1 740 11.5 11.9 12.2 12.6 12.9 13.3 13.8 14.2 14.5 14.8 15.1 15.4 15 7 760 10.9 11.4- 11.8 12.2 12.4 12.8 i3.2 13.7 14.1 14.5 14.7 15.0 15.4 780 10.2 10.6 11.2 11.6 111.9 12.4 12.8 13.2 13.5 13.9 14.3 14.6 14.9 800 9.5 10.0 10.3 10.9 11.3 11.6 12.1 12.6 12.9 13.4 13.8 14.2 1.5 820 8.7 9.3 9.7 10.0 10.5 10.9 11.4 11.9 12.3 12.8 13.2 13.6114.0 840 8.1 8.4 8.8 9.3 9.6 10.1 10.6 111.1 11.6 12.1 12.5 13.0 13.4 860 7.6 7.9 8.1 8.5 8.8 9.2 9.7 10.2 10.7 11.2 11.7 12.1 12.6 880 7.2 7.4 7.6 7.8 8.1 8.5 8.8 9.4 9.8 10.2 10.7 11.2 11.8 900 6.8 7.0 7.1 7.3 7.4 7.8 8.21 8.5 8.9 9.4 9.8 /10.3 10.8i 920 6.7 6.8 6.8 6.9 7.0 7.0 7.4 7.8 8.1 8.6 8.9 9.4 9.9 i 940 6.7 6.7 6.7 6.8 6.7 6.8 6.8 7.1 7.4 7.7 8. 1 8.4 8.9 960 7.1 7.0! 6.8 6.7 6.5 6.5 6.6 6.7 6.8 7.1 7.3 7.7 8.0 980 7.7 7.4 7.1 6.9 6.6 6.5 6.41 6.4 6.3 6.5 6.(' 6.9 7.3 1000 8.4 8.0 7.7 7.3 6.9 6.7 6.5 6.5 6.3 6. 6.26.4 6.5' 6 63) 640 650 660 670) 68 6!)0 70) 719 0- 7913 1 75-0i TABLE XXXII. 47 Perturbations produced by Jupiter. Arguments II. and V. V. II. 750 760 770 780 70 810 820 830 840 850 860 870 6. 5 17 6.77 7._I I.'If _ I.. i 0 6.5 6.8 7.2 7.5 8.0 8.4 8.8 9.5 10.1 10.5 11.0 11.6 12.4 20 6.1 6.2 6.5 6.7 7.0 7.4 7.9 8.4 9.0 9.5 10.0 10.6 11.1 40 5.8 5.9 5.9 6.2 6.4 6.6 6.9 7.4 7.8 8.2 8.8 9.5 10.0 60 5.8 5.7 5.7 5.7 5.9 6.1 6.2 6.5 6.9 7.2 7.7 8.3 8.8 80 6.2 5.8 5.7 5.6 5.4 5.6 5.7 5.9 6.1 6.3 6.7 7.3 7.8 100 6.8 6.3 5.9 5.6 5.5 5.3 5.3 5.4 5.4 5.6 5.9 6.3 6.8 120 7.6 7.4 6.5 6.0 5.7 5.5 5.1 5.2 5.1 5.1 5.2 5.5 5.8 140 8.8 8.1 7.4 6.8 6.2 5.8 5.4 5.2 5.0 4.9 4.8 5.0 5.1 160 10.0 9.3! 8.5 7.8 7.2 6.5 5.9 5.5 5.1 5.9 4.7 4.7 4.7 180 11.5 10.6 9.7 9.0 8.2 7.5 6.9 6.3 5.8 5.2 4.8 4.7 4.5 200 13.1 12.2 11.2 10.4 9.5 8.8 7.9 7.1 6.5 5.9 5.3 5.0 4.7 220 14.7 13.8 12.9 12.0 11.1 10.2 9.3 8.4 7.5 6.7 6.1 5.5 5.2 240 16.4 15.3 14.5 13.6 12.6 11.7 10.7 9.8 8.8 7.9 7.0 6.5 5.9 260 18.1 17.2 16.3 15.3 14.3 13.3 12.2 11.4 10.4 9.4 8.3 7.7 6.9 280 19.8 18.9 17.9 17.0 16.1 15.0 14.0 13.0 11.9 10.9 9.9 8.9 8.0 300 21.2 20.4 19.6 18.7 17.7 16.8 15.8 14.7 13.7 12.6 11.5 10.5 9.4 320 22.5 21.9 21.2 20.4 19.4 18.5 17.4 16.5 15.5 14.2 13.2 12.3 11.2 340 23.7 23.0 22.4 21.8 21.1 20.2 19.2 18 3 17.1 16.1 15.0 13.9 12.9 360 24.5 24.0 23.6 2 3.0 | 22.4 21.6 20.8 19.9 18.9 17.9 16.8 15.9 14.7 380 25.2 24.9 24.5 24.0 23.5 22.8 22.1 21.4 20.5 19.5 18.5 17.6 16.5 400 25.5 25.4 25.1 24.8 24.5 23.9 23.4 22.7 21.9 21.0 20.1 19.2 18.2 420 25.7 25.6 25.5 25.3 25.0 24.5 24.2 23.7 23.2 22.3 21.5 20.7 19 8 440) 25.6 25.6 25.7 25.7 25.5 25.3 24.9 24.6 24.1 23.4 22.7 22.0 21.2 460 25.1 25.3 25.5 25.6 25.8 25.7 25.4 25.2 24.8 24.3 23.7 23.1 22..5 480 24.6 24.9 25.2 25.4 125.6 25.6 25.5 25.4 25.2 24.9 24.5 24.1 23.5 500CO 23.9 24.2 24.7 25.0 25.3 25.1 25.5 25.5 25.4 25.2 24.9 24.7 24.3 520 23.0 23.6 23.9 24.3 24.7 24.9 25.2 25.4 25.4 25.3 25.2 25.1 24.8 540 22.2 22.6 23.2 23.6 124.0 24.4 204.6 24.9 215.1 25.0 25.1 25.1 25 0 560 21.3 21.7 22.2 22.8 23.2 23.7 24.0 24.3 24.6 24.7 24.8 24.9 24.9 580 20.3 20.8 21.3 21.8 22.3 22.7 23.2 23.7 23.9 24.1 24.4 24.6 24.7 600 19.4 19.9 20.4 20.8 21.4 21.9 22.2 22.7 23.1 23.4 23.7 24.1 24.3 620 18.5 19.0 19.5 20.1 20.5 20.9 21.4 21.8 22.2 22.6 22.9 23.3 23.6 640 17.9 18.3 18.7 19.2 19.7 20.1 20.5 22.0 21.3 21.7 22.1 22.5 22.8 660 17.3 17.6 18.1 18.5 18.9 19.4 19.6 20.1 20.5 20.7 21.2 21.7 22.0 680 16.8 17.1 17.4 17.8 18.2 18.6 18.9 19.4 19.7 20.1 20.4 20 7 21.2 700 16.4 16.7 16.9 17.3 17.7 18.0 18.3 18.7 18.9 19.2 19.6 20.0 20.3 720 16.1 1 6.3 16.5 16.9 17.2 17.6 17.8 18.0 18.3 18.5 18.7 19 2 19.5 740 15.7 16.0 16.2 16.5 16.7 17.0 17.3 17.6 17.8 17.9 18.1 18 5 18.8 760 15.4 15.7 16.0 16.1 16.4 16.6 16.7 17.2 17.4 17.4 17.8 18 0 18.2 780 14.9 15.3 15.6 15.9 16.1 16.3 16.5 16.7 16.9 17.1 17.3 17.6 17.7 800 14.5 14.7 15.2 15.5 15.8 15.9 16.2 16.5 16.6 16.8 16.9 17.1 17.3 820 14.0 14.4 14.7 15.1 15.4 15.7 15.8 16.1 16.3 16.4 16.6 16.9 17.0 840 13.4 13.7 14.1 14.5 15.1 15.4 15.4 15.8 15.9 16.1 16.2 16.6 16.7 860 12.6 13.1 13.5 13.9 14.3 14.8 15.2 15.5 15.6 15.8 16.0 116.3 16.4 880 11.8 12.3 12.8 13.3 13.7 14.1 14.5 15.0 15.3 15.4 15.6 15.9 116.1 900 10.8 11.3 11.9 1 2.4 13.0 13.4 13.7 14.2 14.7 15.0 15.2 15.5 15.7 920 9.9 10.3 10.8 11.4 12.0 12.5 12.9 113.4 14.0 14.3 14.7 15.0 15.31 940 8.9 9.4 9.9 10.4 11.0 11.6 12.1 12.5 13.0 13.6 13.9 14.4 14.7' 960 8.0 o 8.3 8.8 9., 10.0 10.6 11.1 11.7 12.2 12.5 13.1 13.7 14.1 980 7.3 7.6 7.9 8.4 8.91 9.5 9.9 10.5 11.1 11.0 12.1 12.8 13.3 1000 6.5 6.8 7.2 7.5 8.01 8.4 8.8 9.5 10.0 10.5 11.0 11.6 12.4 - 750 760 77 780 790 810 -810 i 820 S30 ]840 85. 860 1 870 48 TABLE XXXII. Perturbations produced by Jupiter. Arguments II. and V. V. II. 870 880 890 900 910 920 930 940 950 960 970 980 990 100 - i _..... 0 1 12.4 12.9 13.2 13.6 13.9 14.2 14.4 14.8 15.0 15.1 15.1 15.2 15.2 15.3 20 11.1 11.7 12.2 12.7 13.2 13.6 13.8 14.1 14.4 14.7 14.8 15.0 14.9 14.93 40 10.0 10.5 11.1 11.7 12.3 12.6 13.0 13.4113.7 14.1 14.3 14.6 14.7 14.7 60 8.8 9.4 9.9 10.6 11.2 11.8 12.1 12.6 12.9 13.3 13.6 13.9 14.2 14.4 80 7.8 8.3 8.7 9.3 10.0 10.5 11.1 11.6 112.1 12.5 12.8 13.2 13.5 13.8 100, 6.8 7.2 7.6 8.1 8.6 9.4 9.9 10'5 10.9 11.4 12.0 12.4 12.8 13.2 120 5.8 6.1 6.6 7.1 7.6 8.1 8.7 9.4 9.9 10.4 10.8 11.4 11.8 12.3 140 5.1 5.3 5.6 6.0 6.5 7.0 7.5 8.2 8.7 9.3 9.7 10.3 10.8 11.3 160 4.7 4.8 4.8 5.2 5.6 5.9 6.3 6.8 7.4 8.0 8.6 9.2 9.7 10.2 180 4.5 4.5 4.4 4.5 4.8 5.1 5.4 5.8 6.2 6.9 7.4 8.0'8.4 9.1 200 4.7 4.5 4.2 4.2 4.2 4.4 4.6 5.0 5.3 5.7 6.3 6.9 7.4 7.8 220 5.2 4.7 4.3 4.2 4.1 4.1 4.0 4.3 4.5 4.8 5.1 5.7 6.2 6.8 240 5.9 5.3 4.7 4.3 4.11 4.0 3.8 3.9 4.0 4.2 4.3 4.7 5.2 5.7 260 6.9 6.1 i.4 4.9 4.41 4.1 3.8 3.7 3.6, 3.7 3.8 4.1 4.3 4.9 280 8.0 7.2 6.3 5.7 5.21 4.6 4.1 3.8 3.5 3.5 3.5 3.6 3.7 3.9 300 9.4 8.5 7.5 6.8 6.1 5.4 4.7 4.3 3.9 3.6 3.3 3.3 3.3 3.4 320 11.2 10.1 9.1 8.1 7.3 6.5 5.7 5.0 4.4 4.0 3.6 3.4 3.2 3.2 340 12.9 11.8 10.7 9.6 8.7 7.7 6.8 6.0 5.2 4.6 4.1 3.7 3.4 3.2 360 14.7 13.4 12.3 11.1 10.1 9.2 8.3 7.4 6.4 5.7 4.9 4.3 3.8 3.5 380 16i.5 15.4 14.2 13.0 11.8 10.8 9.7 8.7 7.8 6.9 6.1 5.41 4.6 4.1 400 18.2 17.2 16.0 14.9 13.8 12.4 11.4 10.4 9.3 8.3 7.3 6.4 5.6 5.0 420 19.8 18.8 17.7 16.7 15.5 14.4 13.1 11.9 10.9 9.8 8.8 8.0 6.9 6.1 440 21.2 20.3 19.3 18.3 17.3 16.2 14.9 13.8 12.7 11.5 10.5 9.5 8.4 7.5 460 22.5 21.6 20.6 19.7 18.9 17.9 16.7 15.6 14.3 13.3 12.2 10.9 110.0 9.0 480 23.5 22.7 22.0 21.1 20.2 19.3 18.2 17.3 16.2 15.0 13.8 128 111.6 10.51 500 24.3 23.s 23.0 22.3 21.6 20.7 19.7 18.8 17.8 16.7| 15.4 14.5 13.4 112.3 520 24.8 24.3 23.7 23.2 22.7 21.9 21.1 10.2 19.2 18.3 17.2 116.1 15.0 14.0 540 25.0 24.8 24.3 23.9 23.4 22.8 22.1 21.3 20.6 19.7 18.7 117.6 16.6 15.6 560 24.9 24.8 24.7 244i24.0 23.6 22.9 22.4 21.6 20.8 20.0 19.1 18.2 17.l1 580 24.7 24.7 24.6 24.5 24.3 23.9 23.5 23.1 22.5 21.9 21.1 120.3 19.5 18.6 600 24.3 24.3 24.3!24.3 24.3 24.1 23.8 23.5 23.0 22.5 122.01 21.4 20.6 19.8 620 23.6 23.7 23.9 24.0 24.1 24.1 23.9 23.7 23.4 23.1 22.6 22.1 21.4 20.81 640 22.8 -23.1 23.2 23.4 23.6 23.7 23.8 23.7 23.5 23.2 22.9 22.6 22.1 21.6 660 22.0122.3 22..5 22.8 23.0 23.2 23.2 23.3 23.2 23.1 23.0 22.8 22.5 22.1 680 21.2 21.5 21.7 22.0 22.3 22.5 22.6 22.8 22.9 22.9 22.8!22.7 22.7'22.3 700 20.3 1 20.7 20.9 21.2 1 21.5 1 21.71 21.9 22.2 22.3 22.5 22.5 22.5 22.4 22.2 i 720 19.5 19.8 20.1 20.4 20.8 21.1 21.2 21.4 21.6 21.8 21.9 22.0 22.0 22.0 740 18.83 19.0 19.2 19.6 19.9 20.2 120.5 20.7 20.9 21.1 21.2 21.5 21.5 21.6 760 18.2 18.5 18.4 18.8 19.1 1.9.4 19.6 19.9 20.1 20.,3 20.5 20.8 21.0 21.2 780 17.7, 17.8 18.0 18.1 118.4 18.7 18.8 19.1 19.3 19.5 19.7 20.0 20.2 20.4 800 17.3 17.4j17.4 17.7 117.9 1 18.0 118.1 118.4118.6118.91 18.9 119.1 119.4119.6 820 17.0 17.2 17.2 17.2 17.4 17.4 17.6 17.8 17.8 18.1 18.3 18.5 18.6 18.8{ 840 16.7 16.8 16.8 16.9 17.2 17.2 17.1 17.1 17.3 17.4 17.5 17.8 17.9 18 1 860 16.4 16.5 16.5 16.6 16.6 16.7 16.8 16.9 16.9 17.0 17.0 17.1 17.2 17'.4 880 16.1 16.3 16.3 16.5 16.5 16.5 16.6 16.6 16.6 16.6 16.6 16.7 16.7 16.9 900 15.7 15.9 16.1 16.2 16.3 16.4 16.3 16.3 16.2 16.2 16.2 16.3 16.3 16.3 920 15.3 15.5 15.6 15.9 16.0 16.1 16.1 16.1 16.0 16.1 16.1 16.1 16.0 16.0 940 14.7 15.9 15.2 15.4 15.7 15.8 15.8 16.0 15.9 15.9 15.9 15.8 115.7 15.8 i 960 14.1 14.3 14.5 14.8 15.2 15.5 15.5 15.7 15.7 15.7 15.6 15.6 15.5 15.5 980 13.3 12.7 13.9 14.2 14.5 14.8 15.1 15.3 15.4 15.5 15.4 15.4115.4 15.3 1000 12.4 12.9 13.2 13.6 13.9 14.2 14.4 14.8 15.0 15.1 15.1 15.2 115.2 15 3 870 880 890 900 9101 920 930 940 950 960 970 980 990 1000 TABLE XXXIII. 49 Perturbations p? oduced by Saturn. Arguments II and VII. VII. II 0 100 200 300 400 500 600 700 800 900 1 000 0 1.2 1.5 1.4 1.0 0.7 0 6 0.5 0.5 0.4 0.8 1.2 100 0.9 1.2 1.3 1.1 0.9 0.8 0.7 0.7 0.6 0.7 0.9 200 0.7 0.9 1.0 1.1 1.0 0.9 0.8 0.8 0.9 0.8 0.7 300 0.9 0.8 0.7 0.8 0.9 1.0 1.0 1.0 1.0 1.0 0.9 400 1.0 0.9 0.6 0.4 0.6 0.9 1.0 1.1 1.1 1.1 1.0 500 1.1 1.0 0.8 0.4 0.2 0.5 1.0 1.3 1.3 1.2 1.1 600 1.2 1.1 0.9 0.6 0.2 0.2 0.5 1.1 1.5 1.5 1.2 700 1.4 1.1 1.0 0.8 0.4 0.1 0.3 0.8 1.4 1.7 1.4 800 1.6 1.3 1.0 0.8 0.6 0.4 0.1 0.3 1.0 1.6 1.G 900 1.5 1.4 1.1 0.9 0.7 0.6 0.3 0.2 0.6 1.2 1.5 1000 1.2 1.5 1.4 1.0 0.7 0.6 0.5 0.5 0.4 0.8 1.2 Constant, 1." 0 TABLE XXXIV. Variable Part of Sun's Aberration. Argument, Sun's Mean Anomaly. Os Is II IIIls IVs Vs o 0 0 0.0 0.0 0.1 0.3 0.5 0.6 30 3 0.0 0.0 0.2 0.3 0.5 0.6 27 6 0.0 0.0 0.2 0.3 0.5 0.6 24 9 0.0 0.0 0.2 0.3 0.5 0.6 21 12 0.0 0.1 0.2 0.4 0 5 0.6 18 15 0.0 0.1 0.2 0.4 0.5 0.6 15 18 0.0 0.1 0.2 0.4 0.5 0.6 12 21 0.0 0.1 0.3 0.4 0.6 0.6 9 24 0.0 0.1 0.3 0.4 0.6 0.6 6 27 0.0 0.1 0.3 0.4 0.6 0.6 3 30 0.0 0.1 0.3 0.5 0.6 0.6 0 XIs Xs IXs VILIs VIIs VIs Constant, 0." 3 50 TABLE XXXV. Moon's ]pochs. YEARS. 1 2 3 4 5 6 T 8 9 10 11 12 13 1830 00174 4541 4461 4638 9885 0635 5979 9921 7623 219 226 458 468 1831 00103 1 749 4127 9381 2357 6432 7040 2378 6487 82,5 587 177 94() 1832 B 00032 8957 3793 4125 4829 2229 8100 4835 5351 432 948 897 413 1833 00235 6816 4499 9156 7636 8399 9219 7683 4239 108 340 687 920 1834 00164 4024 4164 3900 0107 4196 0279 0140 3103 715 701 406 393 1835 00093 12'32 3830 8644 2579 9993 1340 2598 19G7 321 061 125 866 1836 B 00022 8441 3496 3388 5051 5791 2400 5055 0831 928 422 845 339 1837 00224 6299 4202 8419 7858 1960 3518 7903 9719 605 814 635 846 1838 00153 3508 3868 3163 0329 7757 4579 0360 8583 211 175 1354 319 1899 00082 0716 3534 71907 2801 3555 5639 2818 7447 818 536 074 792 1840 B 00011 7925 3199 2651 527 39352 6700 5275 6310 424 896 7993 265 1841 00213 5783 3906 7682 8080 5522 7818 8123 5199 101 288 583: 772 1842 00142 2991 3571 2425 0551 1319 8879 0580 4062 707 649 302 245 1843,00071 o200 3237 7169 3023 711 89939 3038 2926 314 010 022 718 1844 B 00000 7408 2903 1913 5495 2914 1000 5495 1790 920 371 741 191 1845 00203 5266 3609 6944 8302 9083 2118 8343 0678 597 763 531 698 1846 00132 2475 3275 1688 0774 48801 3179 0800 9542203 123 250 171 1847 00061 9683 2941 6432 3245 0678 4239 3:257 8406 810 484 970[ 644 1848 B 99990 6892 2606 11 7 615717 6475 5300 5715 7270 416 845 689 117 1849 00192 4750 3312 6207 8524 2644 6418 8563 6i158 093'237 479 624 1850 00121 1958 29781 0951 0995 8442 7479 1020 5022 700 597 199 097 1851 00050 9167 2644 5695 3467 4239 8539 3477 3885 306 958 918 570 1852 B 99979 6375 2310 10439 5939 0036 9600 5935 2749 913 [319 637 043 1853 00181 4233 3016 5469 8746 6206 0718 8782 1637 589 711 4271 50 1854 00110 1442 2681 0213 1217 2003 1778 1240 0501 196 072 147 023 1855 00039 8650 2347 4957 3689 7801 2839 3697 9365 802 432 8661 496 18.56 B 99968 5859 2013 9701 6160 3598 3899 6155 8229 409 793 586 969 1857 00171 3717 2719 4732 8968 9767 5018 9002 7 117 086 185 375 476 1858 00100 0925 2385 9476 1439 5565 6078 1460 5981. 692 5 4 6 0(95 949 1859 00029 8134 2051 4220 3911 1362 7139 3917 4845 299 91()7 814 422 1860 B 99958 5342 1716 8964 6383 7159 8199 6374 3709 905 26T7 534 895 1861 00160 3200 2423 3995 9190 3329 9317 9222 2597 581 659 123a 402 1862 00089 0409 2086 8739 1661 9)126 (0378 16t79 1461 188 020 043 875 1.863 00018 7617 1754 348. 41.33 4923 1438 41 37 [0)3,24 795 381 76zl 348 1864 B 99947 4826 1420 8227 6605 10721 2499 6594 9188 401 742 4182 821 1865 00149 26S4 2126 3257 9412 6890 3617 9442 8()7 G 078 / 134 272 328 1866 00078 9893 1792 8001 1883 2687 4678 1899 6940 G685 494 991 801 18;67 00007 7101 1457 2745 4355 8485 5738 4357 580(4 291. 1 55 [711 274 1868 B 99936 4309 1123 7489 1827 4282 679!9. 6814 4668 898 26 1431 747 1869 00138 2168 1829 2520 9634 0452 7917 9662 3556 574 608 2200 254 1870 00067 9376 1495 7264 2105 6249 8978 2119 2420 181 968 940t 727 1 871 99996 6585 1161 2008 4577 2046 0038 4576 1283 787 329 1659 200 187 2 B 99925 3793 o0827 6752 17049 7843 1099 7034 0147 394 690 378 6773 I 873 001'27 1651 1533 1782 9856 4013 12217 988119035 070 082 168 189 1874 00056 9860 1198 6526 2327 981( 3277 2339 17899 677 443 888 653 1875 99985 6068 0864 1270 4799 5608 4338 4796 6763 283 803 607 126 1876 B 99914 32771 0530 6014 7280 1405 5398 7254 5627 890 164 327 599 1877 00117 1 155 1 236 1045 0078 7574 6517 0101 4515 567 556 11 16 106 1878 00046 8343 0902 578'9 2549 3372 7577 2559 13379- 17: 191'[836 579 1879 99975 5552 0568 0()533 5021 9169 863S8 5016 2243 780 278 5515 052 1880 B 99904 2760 0233 52777 7493 4966 9698 7473 1107 386 638 275 525 18S1 10106 0618 0940 0308 0300 1136 0816 0321 9995 062 030 064 032 1882 00035 17827 0605 5052 27'1 16933 1877 2798 8859 669 391 784 505 1883 99964 5035 0271 9796 5243 2730 2937 5236 7722 276 75252 513 978 1884 B 99893 2244 9937 4540 7715 8528 3998 769316586 8821113 223 451 1885 00095 0042 0643 9570 0522 4697 5116 0541 5474 5591505101.3 957 TABLE XXXV. Moon's Epochs. Years. 14 15 16f 17 18 19 20 21 22 23 24 25 26 271 28 29 30 31 1830 921 3921230 588 462 523 536 52 60 44 94 51 47 98 99 99 89 52 1831 115 5321589 940 937 296 703 30 70 41 65 53 94 48 24 24 51 44 1832 B. 309 673 949 293.412 070 870 07 81138 36 55 42 97 48 49 14 35 1833 602 844 345 6881913 845 037 85 92 15 07 61 92.53 77 7 7 7 7 27 1834 796 984 704 040 -38S 619 2013 62 03 42 77 63 40 03 101 01 39 18 1835 989 124 063 3931863 392 370 39 13 38 48 65 87 51 26 26 02 10 1836 B. 183 265 423 745,338 166 537 17 24 35 19 67 34 01 50 51 64 01 18 37 476 436 819 140)840 942 704 94 35 42 90 173 85 58 79 79 27 93 1838 670 576 178 492 31.5 7 15 870 72 46 38 60 75 32 07 04 04 89 84 1839 864 716 537 845 790 489 037 49 56 35 31177 80 56 28 28 52 76 1840 B. 058 857 897 1 97 265 262 204 26 67 32 02 79 27 06 53 53 14 67 1841 351 028 293 592 766 138 371 04 78 39 73 85 77 62 81 81 77 59 1842 544 168 652 944 241 811 537 81 89 35 43 8 7 25 12 06 06 40 51 1843 738 308 012 297 716 585 704- 58 99 32 14 89 72 161 30 31 02 42 1844 B. 932 449371 6 19 1 358 871 3 6101298591119 10155555165134 1845 2251 62( 767 044 692 134!038 1131 1136156197 70 67 84 83 27 26 1846 419 760 126 396 167 9.07 204 91 32 3226 99 17 16 0808 90 17 1847 618 91.148t1 749,643 681 37 168142129197 01 65 65133133 52 09 1848 B. 806 )G041 1845101 18 454 58 45 53 26 68 03 12 15 57 58 15 00 1849 099 212 2141 4961619 23() 705 23 64 33 39 ()9 63 71 86 86 77 92 1 850 293 352 600 848 094 (10)3871 00 75 29 09 1010120 10 10 40183 1851 487 493 960 2015,069 777 038 78 85 2, 80 12 57 70 5 35 02 75 1852 B. 681 63313195553 4 a50 2151 55196 23'51 14104 19159 60 65 66 1853 974 804 715 948 S4.5 3'26 372 33 07 30 22 20 55176 88 88 28158 18.54 16819441074}300 020 099 5 3 10 18 126932 0312 51212 1 190150 1855 361 085 434 653 495 8731705 87 28 23 63 24 501 71 37 37 53141 1856 B. 555 225 793 005 970 646W872 65 39 20 34 26 97 23 61 62 15 33 1857 848 1396 189 400 471 4221039 42 50 27 105 32 48 30 9090'781 24 1858 042 537 548 752 947 1958 206 20 61 24|76 34|95 29 15 1540 16 1859 236 677 908 105 422 969 3 7 2 97 711 20 46 36 42 79 39 40 03 07 1860 B. 430 8171267 457 897 742 a539 74 82 11717 38 89 28 64 64165 99 1861 7231988 663 852 398 518 706!52193124,88 4441 84 92 192128 91 1862 916 129 022 204 873 291 873 29, 04 20 60 46 88 34 17 17 91 82 1863 110 2691382 557 348 065 039 0(6 14 17 29 48 35 8241 42 53 7.4 1864 B. 304 409 741 991 823 838 2006 84 25 14 00 50 82 32 66 66 16 65 1] 865 597 580 137 304 324 61437 3 61 36 21 71 56 33 89 95 94 78 57 1866 791 721 496 657 799 387 540 39 47 17 42 58 80 38 19 19 411 49 1867 9851861 1856 009 274 161 707 16571414 2160 28 8714444103l)40 1868 B. 17810011215 362 7491934 8731 93168 11 83 62 75137 68 69161632 1869 471 172611 756 251 l71Ol 040 71179 18 54 68 26 93 97 97 28 213 1870 665 313 970 109 7')6 483,207 48 901 15 261691731 43 21 21 1191115 1871 8591454 330 462 201 257 374 426 00 12 97171 12019314646153 107 1872 B. 053 594 689 814 67 6 030 541 03 11 109 68 73 67 42 70 71 16 98 1 873 346 765 0185 209 177 806l 708 181 22116139179 1899 919919979190 1874 54()1905 484 561 6521579 875 58133 1 21 081166148S23 23 41 82 1875 733 046 804 914 127 353 41 35143109 1801831 13197 148 48 10473 1876 B. 927 186 163 266 602 126 208 13154106[51 85160 46172t73'66165 1877 2201357 5591 661 103 902:3751 590 165131 22 91 11 03 01 ()1129 56 1878 414 4198.918 113 579 675 5412 68 76 10193193 58a52, 2612519148S 1879 608 6388278 366 54449 708 1145 86106 63 95 05 02 50.1 154 39 1880 B. 80I2 778 637 718.529 222 875 22 97 03 34 97a 5251 75 75 16 31 1881 095 949 0331 113 30 1981042 10008 10 05 03 04107 03 03 79 23 1882 288 90 3921465 505 771 2091 7719 06 l77( 05 51 57 28 2842 14 1883 4821 230 752 818 980 545 375 154 29103 46 07 98 05 52 153 41406 1884 B. 676 370111 1170 455 318 542 132140 10 17 09145155"77 177 67 197 1885 969 541 507 565 9561 941 709 09 51 07 88 15 96 12 06 05 2!9891 TABLE XXXV. Moon's Epochs. Years. Evection. Anomaly. Variation. Longitude. S 0 1 It t 0 I 1 0 t 1830 5 17 412 11 24 31 4.5 2 13 2 39 11 22 55 3T7.7 1831 11 7 35 41 223 14 24.6 6 22 40 4 4 2 1842.8 1832 B 4 28 17 11 5 2157 44.4 11 2 17 28 8 114148.0 1833 10 29 57 40 9 344 5S.5 3 24 6 21 1 4 15 28.4 1834 4 20 29 11 0 2 28 18.5 8 3 43 45 5 ] 3 38 33.6 1835 10 11 0 40 3 1 11 38.6 0 13 21 10 9 23 1 38.8 1836 B 4 1 32 9 5 29 54 58.7 4 22 58 34 2 2 24 44.0 1837 10 3 22 39 9 11 42 12.8 9 14 47 27 6 24 58 24.5 1838 3 23 54 9 0 10 25 32.9 1 24 24 651 11 4 2129.8 1839 9 14 25 38 3 9 8 53.1 6 4 2 16 3 13 44 35.0 1840 B 3 4 57 8 6 7 52 13.2 10 13 39 42 7 23 7 40.4 1841 9 6 47 37 9 19 39 27.5.3 5 28 33 0 15 41 20.9 1842 2 27 19 7 0 18 22 47.6 7 15 5 58 4 25 4 26.2 1843 8 17 50 37 3 17 6 7.9 11 24 43 23 9 4 27 31.6 1844 B 2 8 22 7 6 15 49 28.1 4 4 20 48 1 13 50 37.0 1845 8 10 12 36 927 36 42.5 8 26 9 40 6 6 24 17.5 1846 2 044 6 026 20 2.8 1 547 5 10 154723.0 1847 7 21 15 35 3 25 3 23.2 5 15 24 30 2 25 10 28.3 1848 B 11 47 5 6 23 46 43.5 9 25 1 55 7 4 33 33.7 1849 7 13 37 35 10 5 33 57.9 2 16 50 47 11 27 7 14.5 1850 1 4 9 4 1 4 17 18.3 6 26 28 12 4 6 30 19.9 1851 6 24 40 35 4 3 0 38.6 11 6 5 37 8 15 653 25.4 1852 B 0 15 12 5 7 1 43 59.2 3 15 43 3 0 25 16 31.0 1858 6 17 2 34 10 13 31 13.7 8 7 31 54 5 17 50 11.6 1854 0 7 34 4 1 12 14 34.1 0 17 9 20 9 27 13 17.2 1855 5 28 5 33 410 57 54.7 4 26 46 44 2 6 36 22.7 1856 B 1118 37 3 7 9 4115.2 9 6 2410 6 15 59 28.2 1857 5 20 27 33 10 21 28 29.8 1 28 13 2 11 8 33 9.1 1858 1110 59 2 1 20 11 50.3 6 750 27 3 17 56 14.6 1859 5 1 30 33 4 18 55 10.9 10 17 27 53 7 27 19 20.1 1860 B 10 22 2 3 7 17 38 31.4 2 27 5 18 0 6 42 25.8 1861 4 23 52 32 10 29 25 46.1 7 18 54 10 4 29 16 6.6 1862 1014 24 2 128 9 6.6 1128 3135 9 8 39 12.2 1863 4 4 55 32 4 26 52 27.3 4 8 9 1 1 18 2 17.9 1864 B 9 25 27 2 7 25 35 48.0 8 17 46 25 5 27 25 23.5 1865 3 27 17 31 11 723 2.7 1 9 35 18 10 19 59 4.3 1866 9 17 49 2 2 6 6 23.3 5 19 12 43 2 29 22 10.1 1867 3 8 20 31 5 44944.0 9 28 50 9 7 8 45 15.7 1868 B 8 28 52 2 8 3 33 4.7 2 8 27 34 1118 8 21.4 1869 3 0 42 33 11 15 20 19.6 7 0 16 26 410 42 2.3 1870 82114 2 2 14 340.3 11 95351 820 5 8.0 1871 2 11 45 33 5 12 47 0.6 3 19 3116 0 29 28 13.5 1872 B 8 2 17 3 8 1130 21.2 7 29 8 42 5 8 57 19.1 1873 2 4 7 32 11 23 17 35.7 0 20 57 33 10 1 24 59.7 1874 7 24 39 2 3 22 0 56.1 4 20 34 59 2 10 48 5.3 1875 1 15 10 31 5 20 44 16.7 9 10 12 23 6 20 11 10.8 1876 B 7 5 42 1 8 19 27 37.2 1 19 49 49 10 29 34 16.3 1877 1 7 3231 0 1 14 51.8 6 1138 41 322 7 57.2 1878 6 28 4 0 2 29 58 12.3 10 21 16 6 8 131 2.7 1879 0 18 35 31 5 28 41 32.9 3 0 53 32 0 10 54 8.2 1880 B 6 9 7 1 8 27 24 53.4 7 10 30 57 16 20 17 13.9 1881 0 10 57 30 0 9 12 8.1 0 2 19 49 9 12 50 54.7 1882 6 129 0 3 7 55 28.6 41157 14 122 14 0.3 1883 1122 0 30 6 6 38 49.3 8 21 34 40 6 1 37 6.8 1884 B 5 12 32 0 9 56 22 10.0 1 1 12 4 10 11 0 11.6 1885 11 14 22 29 0 17 9 24.7 5 23 0 57 3 3 33 52.4 TABLE XXXV. 53 Mboon's Epochs. YEARS. Supp. of Node. II V VI VII VIII IX X Xi XII s " sI ~ o. 1830 6 7 711.0 102446 498 502 900 904 427 062 025 433 1831 626 26 53.3 2 15 18 912 914 208 210 506 001 211 710 1832 B 715 46 35.51 6 5 50 326 327 516 516 586 940 397 986 1833 8 5 9 28.4 10 7 31 774 779 852 856 702 885 624 297 1834 824 29 10.7 1 28 3 187 191 159 163 782 825 810 573 135 9 13 48 53.0 5 18 35 601 6o3 467 469 861 764 996 850 1836 B 10 3 8 35.2 9 9 8 015 016 775 775 941 703 182 127 1837 10 22 3128.1 11049 463 468 111 116 057 648 409 437 1838 1111 5110.4 5 1 21 876 880 419 423 137 588 59.)5 714 1839 0 1 10 52.6 8 21 53 290 292 726 729 217 527 781 991 1840 B 0 20 30 34.9 0 12 25 704 705 034 035 296 466 967 268 1841 1 9 53 27.7 4 14 6 152 157 370 375 412 411 194 578 1842 1 29 13 10.0 8 438 566 569 678 682 4921 350 380 855 1843 2 18 32 52.2 11 25 10 980 980 986 988 572 290 566 131 1844 B 3 7 52 34.5 3 15 42 393 394 293 294 651 229 752 408 1845 3 27 15 27.4 7 17 23 840 846 629 634 767 174 979 718 1846 416 35 9.6 11 7 55 254 258 937 941 847 113 i65 995 1847 5 55451.8 2 2827 668 670 245 247 927 05'3 351 272 1848 B 5 2. 14 34.1 6 18 59 082 083 553 553 006 992 537 549 1849 6 14 37 27.0 10 20 40 531 535 889 893 122 937 764 859 1850 7 357 9.2 2 11 12 944 947 196 200 202 876 950 136 1851 723 76 51.5 6 144 3.58 359 504 506 282 816 136 413 1852 B 8 12 36 33.6 9 22 17 772 772 812 812 362 755 322 689 1853 9 1 59 26;.5 1 23 58 220) 223 148 152 477 700 549 000 1854 9 21 19 8.8 5 1430 634 636 456 459 557 639 735 276 1855 10 10 38 51.1. 9 5 2 0470448 763 765 637 579 921 553 1856 B 10 29 58 33.3 0 25 34 461 461 071 071 717 518 107 830 1857 11 19 2126.2 4 27 15 909 912 407 411 832 46:3 334 140 18,58 0 841 8.4 8 17 47 323 325 715 718 912 402 520 417 1859 0 28 0 50.7 0 8 19 736 787 023 024 992 9 342 706 694 1860 B 1 17 20 32.9 3 28 51 150 150;s130 330 072 281 892S 971 1861 2 6 43 25.8 8 0 32 598 601 666 670 187 226 119 281 1862 2 26 3 8.0 11 21 4 012 014 974 977 267 165 305 558 1863 3 15 22 50.1 3 11 36 426 426 282 283 347 105 491 834 1864 B 4 442 32.3 7 2 8 839 839 590 589 427 044 677 111 1865 424 525.2 11 349 287 291 926 929 542 989 904 422 1866 513 25 7.3 2 24 21 701 703 233 236 622 928 090 698 1867 6 24449.5 61453 115 115 541 542 702 868 276 975 1868 B 6 22 431.7 10 5 26 529 528 849 848 782 807 462 252 1869 71127 24.6 2 7 7 977 980 185 188 897 752 689 562 1870 8 0 47 6.7 5 27 39 390 392 493 495 977 691 875 83,9 1871 8'20 649.0 9 18 11 804 804 801 801 057 631 061 116 1872 B 9 9 26 31.1 1 844 218 217 109 107 137 570 247 392 1873 9 28 49 24.0 5 10 25 666 668 445 447 252 515 474 703 1874 10 18 9 6.3 9 057 080 081 753 754 332 454 660 979) 1875 11 7 28 48.6 2129 493 493 060 060 412 394 846 256 1876 B 11 26 48 30.8 412 1 907 906 368 366 492 333 032 533 1877 016 11 23.7 8 13 42 355 357 704 706 607 278 259 843 1878 1 6 31 5.9 0 414 769 770 012 013 687 217 445 120 1879 124 50 48.2 3 24 46 182 182 320 319 767 157 631 397 1880 B 214 10 30.4 7 15 18 596 595 627 623 847 096 817 674 1881 3 3 33 23.3 11 16 59 044 046 913 9.65 962 041 044 984 1882 3 22 53 5.5 3 7 31 458 459 271 272 042 980 230 261 1883 4 12 12 47.6 6 28 3 872 871 579 578 122 920 416 537 1884 B 5 1 32 29.8 10 18 35 2o5 284 887 884 202 859 602 814 1885 5 20 55 23.0 2 20 16 733 736 223 224 317 804 829 125 32 54 TABLE XXXVI. Moon's Mllotiors for MZlonths. Months. 1 2 31 4 5 6 7 8 9 10 11 12 13 January 00000 0000 000010000 0000 0000 0000 0000 0000 000 000000 000 February 08487 0146 2246 8896 0402 1533 1789 2099 0753 175 965i 184 059 March Com.- 16153 8343 1371 6931 9797 1951 3404 3027 1433 139 836 157 016 Bis. 16427 8993 2411 7218 013212323 3462 3418 1457 209 868 228 050 April { Com. 24640 8490 3616 5827 0199 3484 5193 5126 2186 314 801 3421076 April Bs. 24914914014657 61140534 3856 5251 5517 2210 384 832o 412 11 0 Com. 32853 79864822 4436 026514646 6924 6835 2914 419 735 456 101 May Bis. 33127 8636jI586i24723 060015018 6982 722612938 489 766 526 135 Tu Con. 41340 8133 7067 3332 0666 6179 8713 8934 3667 593 700 640 160 une s. 41614 783 8107 3619'1002 6551 8771 9325 3691 663 731 710 194 July Corm. 49554 7629 827311942 0732 7341 0444 0643 4396 698 634 754 185 Bis. 49828 8279 9313 2228 1068 771310502 1034 4420 768 665 824 219 Com. i58041 7776 0518 0838 1134 88741 2233 2742 5148 873 59919381245 Bis. 58315 8426 155811125 1470j9246 2290 3133 5173 943 630 009 279 Com. 66528 792212764 9734 1536'0408 4021 4842,5901 048 5631231304 Sept Bis. 66802 8572 380410021 187110780 4079 5232 5925 118 595 193 338 Oct. Com.174741 7419 396918343 1602 15695752 655016630 152 49712371329 Oct.- { Bi.s. 75015 8069 5009 8630 1938 1941 5810 6941 6654 222 528 307 363 Nyov. S Com. 83228 7565 6215 7239 2004 3102 7541 8649 7382 327 462 421 388 Bis. 83502 8215 7255 17526 2339 3475 7599 9040174071397 493 492 423 Com. 91442 7062 7420 5848 2070 4264 9272 0358 8111432 396 535 414 Bis. 91716 7712 8460 613512405 4636 9330 0749,813,5502 427 606448 TABLE XXXVI. Moon's Motions for Months. Months. Evection. Anomaly. Variation. Longitude. 8 ~ t 8 0 s tt e 0' 0 t s ~' "t January 0 0 00 000 0. 0000 0 0 0 0.0 February 11 20 48 42 1 15 0 53.1 017 54 48 1 18 28 5.8 March Coll. 10 7 40 26 12 050 4.2 11 29 115 1 27 24 26.6 RBs. 10 18 59 26 2 3 53 58.2 011 26 42 2 10 35 1.6 April Com. 9 28 29 8 3 5 50 57.3 017 10 3 3 15 52 32.5 p Bis. 10 9 48 8 3 18 54 51.2 029 21 29 3 29 3 7.5 May Com. 9 7 58 51 4 7 47 56.4 022 53 24 4 21 10 3.3 Ma Bis. 9 19 17 50 4 20 51 50.3 1 5 4 50 5 4 20.383 Je ( Com. 8 28 47 33 5 22 48 49.4 110 48 11 6 9 38 9.1 Jun Bis. 9 10 6 33 6 5 52 43.4 122 59 38 6 22 48 44.1 Jly Comr. 8 8 17 16 6 24 45 48.5 116 31 302 7 14 55 39.9 u Bis. 8 19 36 15 7 7 49 42.5 128 42 59 7 28 6 15.0 Aug. Comrn. 7 29 5 59 8 9 46 41.6 2 4 26 20 9 3 23 45.8 Aug Bis. 8 10 24 58 8 22 50 35.5 216 37 47 9 16 34 20.8 Com. 7 19 54 41 9 24 47 34.6 222 21 7 10 21 51 51.6 Sept. TBis. 8 1 13 40 10 7 51 28.6 3 4 32 34 11 5 2 26.7 Com. 6 29 24 24 10 26 44 33.7 228 4 28 11 27 9 22.4 Oct. Bis. 7 10 43 23 11 9 48 27.7 310 15 55 0 10 19 57.5 Nov. Com. 6 20 13 6 0 11 45 26.8 315 59 16 1 15 37 28.3 Bis. 7 1 32 5 0 24 49 20.7 328 10 43 1 28 48 3.3 Dec. Co. 5 29 42 49 1 13 42 25.9 321 42 37 2 20 54 59.1 Bis. 6 11 1 48 1 26 46 19.8 4 3 54 4 3 4 5 34.1 TABLE XXXVI. 55 Moon's Motions for Months. Months. 14 15 16 17 18 19 20 21122 23 24 25 26 27128 29 30 31 January 000 000 000 000 00010000 00 00 00100 00 00 0000 00100 00 February 074 946 135 3041805 066,014 24126 14!82 28 14 17 29 96 05i07 March 5 Com. 851 801 159 482 532 125 027 45150 98 57 43 18 12146 82 10 15 Bis. 950 831 196 524 5581127 027 46 51 08 59 47 21 19151 85 10 15 April Corn. 925 747 294 78613361191 041 68 77 12 3 9 70 32 2976 77 15 23 Bis. 024 778 331 828 3621193!042 69 77 22 42 74 36 36 80 80 16|23 M Com. 899 663 392 047 115!254 055 91 02 15 19 94 43 38[01 70 21130 Y { Bis. 999 693 429 089 141 256 055 92 03 26 22 98 47 45 05 73 21330 June { Com'. 973 609 527 351 920 320 069 15 28 29 01 21 57 55 31 65 26 38 e Bis. 073 639 563 3931946 322 069 15t29 40 04 25161 62 35 68 26'38 July Corn. 948 525 625 613 699 384*083 37154 33 81 45168 64 56 58 31145 Bis. 047 555 661 655 725'3861083 38 55 43 8449i72 71 60 6131146 Au. i Corn. 022 471 759 917 503 449 097 61 80 47 64 72 82 81 85 53 36153 Bis. 121 501 796 959 529 451 097 62 81 57 66 77186 88 90 56 36 53 St Corn. 096 417 894 221 308 515 111'85107 61 46 0097 97 15 49 42:61 P Bis. 195 447 931 263 334 517 111'85108 71 49 04101 04/19 5242 61 O Corn. 071 333 992 483 (87 578 125 071 32 65 26 23 08 0740 41 4768 O Bis. 1701363 0291525 113 581 126:08133 75 28 28 11 14!44 44 47 69 Nov. N Com. 145 279 1271787 892 644 139 131 59 79 08 51 22 23 70 37 52.76 Bis. 244 309 163 1829918 646 1401,32 60 89 11 55 26 30 74 40152 76 IDec. Corn. 120 194 225 0490670 708 1535418583 88174 33 33 95 29157 84 * Bis. 219 225 261 091,696 710 153154 86 93907937 40 99 325784 TABLE XXXVI. Moon's Motions for Months. Months. Supp. of Node. II V VI VII VIII IX X XI XII January 0 0 0 0.0 0 0 0 000 000 000 000 000 000 000 000 February 0 1 38 29.7 11 15 43 054 224 875 045 111 165 290 043 March Com. 0 3 7 27.5 9 27 59 007 330 666 989 114 313 455 984 Bis. 0 3 10 38.2 10 9 8'0411369 694 023 150 I319 496 018 April Com. 0 4 45 57.3 9 13 42 0611 554 542 034 225 478 745 027 Bis. 0 4 49 7.9 9 24 51 1095 593 570 068 261 484 787 061 Ma Com. 0 6 21 16.4 8 18 15 1081 738 3S9 046 300 638 993 036 Y Bis. 0 6 24 27.0 8 29 25 1115 778 417 080 336 643 034 070 June Com. 0 7 59 46.1 8 3 58 1361962 264 091 411 s2 282 079 Bis. 0 8 2 56.7 8 15 8 170 002 293 124,4471808 324 113 Com. 0 9 35 5.21 7 8 32156147 112 1031486 962 531 088 u Bs. 0 9 38 15.9 7 19 41 1901 186 140 136 522 967 572 122 Aug. Com. 0 11 13 35.0 6 24 15 210 371 987 147 597 126 820 131 Bis. 0 11 16 45.6 7 5 24 244411 015 182633 132 862 164 Sept Corn. 0 12 52 4.7 6 9 58 265 595 862 193 708 291 110 173 Bis. 0 12 55 15.4 6 21 7 299 635 891 227 744 296 1521207' oct. Corn. 0 14 27 23.8 5 14 32 285 780 710 204 783 451 358 182 Bis. 0 14 30 34.4 5 25 41 319 819 738 238 819 456, 400 216 ov. Corn. 0 16 5 53.5 5 0 15 339 004 585 250 894 615 1648 225 {B's. 0 16 9 4.2 5 11 24 3731 043 613 283 930 621 1690 259 Dec. Com. 0 17 41 12.6 4 4 49 359 188 432 261 969 775 1896 234 D Bis. 0 17 44 23.31 4 15 58 393 228 461 295 0051 780 938 1268 56 TABLE XXXVII. Moon's Motions for Days. D. 1 2 3 4 5 6 7 8 9 1lO 11 l2 13 1 ooooo0 00o0 0000 0000 0000 0000 1oo 0000 0000 000 000 000 oo i 2 00274 0650 1040 0287 0336 0372 0058 0390 0024 070 0311 070 034 3 00548 1300 2080 0574 0671 0744 0115 0781 0049 140 062 141 068 4 00821 1950 3121 0861 1007 1116 0173 1171 0073 210 093 211 103 5 01095 2600 4161 1148 1342 1488 0231 1561 0097 281 125 282 137 6 01369 3249 5201 1435 1678 1860 0289 1952 0121 351 156 352 171 7 01643 3899 6241 1722 2013 2232 0346 2342 0146 421 187 423 205 8 01916 4549 7281 2009 2349 2604 0404 2732 0170 491 218 493 239 9 02190 5199 8321 2296 2684 2976 0462 3122 0194 561 249 564 273 10 02464 5849 9362 2583 3020 3348 0519 3513 0219 631 280 634 308 11 02738 6499 0402 2870 3355 3720 0577 3903 0243 702 311 705 342 12 03012 7149 1442 3157 3691 4093 0635 4293 0267 772 342 775 376 13 03285 7799 2482 3444 4026 4465 0692 4684 0291 842 374 845 410 14 03559 8449 3522 3731 4362 4837 0750 5074 0316 912 405 916 444 15 03833 9098 4563 4018 4698 5209 0808 5464 0340 982 436 986 478 16 04107 9748 5603 4305 5033 5581 0866 5854 0364 052 467 057 513 17 04380 0398 6643 4592 5369 5953 0923 6245 0389 1122 498 127 547 18 04654 1048 7683 4878 5704 6325 0981 6635 0413 193 529 198 581 19 04928 1698 8723 5165 6040 6697 1039 7025 0437 263 560 268 315 20 05202 2348 9763 5452 6375 7069 1096 7416 0461 333 591 339 649 21 05476 2998 0804 5739 6711 7441 1154 7806 0486 403 623 409 683 22 05749 3648 1844 6026 7046 7813 1212 8196 0510 473 654 480 718 23 06023 4298 2884 6313 7382 8185 1269 8586 0534 543 685 550 752 24 06297 4947 3924 6600 7717 8557 1327 8977 0559 614 716 621 786 25 06571 5597 4964 6887 8053 8929 1385 9367 0583 684 747 691 820 26 06844 6247 6005 7174 8389 9301 1443 9757 0607 754 778 762 854 27 07118 6897 7045 7461 8724 9673 1500 0148 0631 824 809 832 888 28 07392 7547 8085 7748 9060 0045 1558 0538 0656 894 140 903 923 29 07666 8197 9125 8035 9395 0417 1616 0928 0680 964 872 973 957 30 07940 8847 0165 8322 9731 0789 1673 1I'9 0704 034 903 043 991 3'. 08213 9497 1205 8609 0066 11161 1731 1709 0729 105 934 114 025! TABLE XXXVII 57 Moon's Motion for Days. D 14 15 16 17 18 19 20 21 22 23 24 25 26 i27 28129 30 31 1 ooo00oo0 00oooo ooo000 000 000 000 00 00 00 00 00 00 00 00 0oo0 00oo 00oo 2 099 031 037 042 026 002 000 101 01 10 03 04 04 07 04 03 00 00 3 198 061 073 084 052 004 001 0202 20 05 08 07 14 08 0 s06 00 00 4 297 092 110 126 078 006 001 02 03 30 08 12 11 21 13 09 01 01 5 397 122 146 168 104 008 002 03 03 41 11 16 15 28 17 12 01 01 6 496 153 183 210 130 011 002 04 04 51 13 21 18 35 21 15 01 01 7 595 183 220o252 156 013 003 05 05 61 16 25 22 42 25 18 01 01 8 694 214 256 294 182 015 003 05 06 71 19 29 26 49 29 22 01 02 9 793 244 293 336 208 017 004 06 07 81 21 33 30 56 33 25 01 02 10 892 275 329 379 234 019 004 07 08 91 24 37 33 63 38 28 02 02 11 992 305 366 421 260 021 005 08 09 01 27 41 37 70 42 31 02 02 12 091 336 403 463 286 023 005 08 09 11 29 45 41 77 46 34 02 03 13 190 366 439 505 312 025 005 109 10 22 32 49 44 84 10 37 02 03 14 289 397 476 547 337 028 006 10 11 32 34 53 48 91 54140 02 03 15 388 427 512 589 363 030 006 11 12 42 37 58 52 98 58 43 02 03 16!487 458 549 631 389 032 007 11 13 52 40 62 55 05 63146 03 04 171587 488 586 673 415 034 007 12 14 62 42 66 59 12 67 49 03 04 18 686 519 622 715 441 036 008 13 14172 45 70 63 19 71152 03 01 19 785 549 659 757 467 038 008 114 15182 48 74 66 26 75 55 03 04 20 884 580 695'799 493 040 009 14 16192 50 78 70 33 79 59 03 05 21 983 611 7321841 519 0421009 15 17 03 53182 74 40 184 62 03 05 22 082 641 769 883 545 044 010 16 18 13 56 86 77 47 88 65 04 05 23 182 672 805 925 571 047,010 117 19 23 58 90 81 54192 68 04 05 24 281 702 842 96'7 597 09011 l 17 20 33 61 95 85 61 96 71 04 06 25 380 733 878 009 623 051011 18 12043 64 99 89 68100 74104 06 26 479 763 915 052649 064953101119 2153 66 03 92 75104177 04 06 27 578 794 952 094 675055! 0122 20 22 63 69 07 96 82 09 80104 06 28 677 824 988 136 701 057i012 20 23 73 72 11 00 89 13 83 05 06 29 777 855 0251178 727 059 0131 21 24 84 74 15 03 96 17 86 05 07 301876 885 061 220 753 061 013 22{25 94 77 19 07 03 21 89 05 07 31 975 916 098 262 27790641423 26104 80 23 111025920 5 07 58 TABLE XXXVVI. Moon's Motions for Days. D. Evection. Anomaly. Variation. M. Longitude. $S 0 f s 0 0 1 0000 0000 0 000 0 0 00 2 0 11 18 59 0 13 354.0 0 12 1127 013 10 35.0 3 0 22 37 59 0 26 747.9 0 24 22 53 026 21 10.1 4 1 3 6 58 1 9 11 41.9 1 6 34 20 1 931 45.1 5 1 15 15 58 1 22 15 35.9 1 18 45 47 122 42 20.1 6 1 26 34 57 2 5 19 29.8 2 0 57 13 2 552 55.1 7 2 753 57 2 18 23 23.8 2 13 840 219 3 30.2 8 2 19 12 56 3 1 27 17.8 2 25 20 7 3 2 14.2 9 3 031 55 3 14 31 11.7 3 7 31 34 315 24 40.2 10 3 1150 55 3 27 35 5.7 3 19 43 0 328 35 15.2 11 3 23 9 54 4 10 38 59.7 4 1 54 27 411 45 50.3 12 4 428 54 4 23 42 53.7 4 14 554 424 56 25.3 13 4 15 47 53 5 6 46 47.6 4 26 17 20 5 8 7 0.3 14 4 27 6 53 5 19 50 41.G 5 8 28 47 521 17 35.4 15 5 825 52 6 2 54 35.6 5 20 40 14 6 428 10.-1 16 5 19 44 51 6 15 58 29.5 6 2 51 40 617 38 45.4 17 6 1 3 51 6 29 2 23.5 6 15 3 7 7 0 49 20.4 18 6 12 22 50 7 12 6 17.5 6 27 14 34 713 59 55.5 19 6 23 41 50 7 25 10 11.4 7 9 26 1 727 10 30.5 20 7 5 0 49 8 8 14 5.4 7 21 37 27 810 21 5.5 21 7 16 19 49 8 21 17 59.4 8 3 48 54 823 31 40.5 22 7 27 38 48 9 4 21 53.4 8 16 021 9 642 15.6 23 8 857 47 9 17 25 47.3 8 28 11 47 919 52 50.6 24 8 20 16 47 10 0 29 41.3 9 10 23 14 10 3 3 25.6 25 9 135 46 10 13 33 35.3 9 22 34 41 10 16 14 0.7 26 9 12 54 46 10 26 37 29.2 10 4 46 7 10 29 24 35.7 27 9 24 13 45 11 9 41 23.2 10 16 57 34 11 12 35 10.7 28 10 532 15 11 22 45 17.2 10 29 9 1 11 25 45 45.7 29 10 165144 0 5 49 11.1 II 11 20 28 0 856 20.8 30 10 28 10 43 0 18 53 5.1 11 23 31 54 022 6 55.8 31 11 929 43 1 1 56 59.1 0 5 43 21 1 5 17 30.8 TABLE. XXXVII. 59 Moon's Motions for Days.!).iSupp. of Node. II V VI VII VIII IX X XI| XIf o.o o o o o oo ooo ooo Il0 0 0 0.0 0 0 0 000 000 000 000 000 000 000 000 20 0 3 10.6 0 11 9 034 039 028 034 036 005 042 034 3 0 0 6 21,3 0 22 18 068.079 056 067 072 011 083 067 400 9 31.9 1 3 27 102 118 085 101 108 016 125 101 5 0 0 12 42.5 1 14 37 136 158 113 135 143 021 166 135 6 0 0 15 53.2 1 25 46 170 197 141 169 179 027 208 168 7 0 0 19 3.8 2 6 55 204 237 169 202 215 032 250 202 8 0 0 22 14.5 2 18 4 238 276 198 236 251 037 291 235 9 0 0 25 25.1 2 29 13 272 316 226 270 287 043 333 269 10 0 0 28 35.7 3 10 22 306 355 254 303 323 048 374 303 11 0 0 31 46.4 3 21 31 340 395 282 337 358 053 416 336 12 0 0 34 57.0 4 2 40 374 434 311 371 394 058 458 370 13 0 0 38 7.6 4 13 50 408 474 339 405 430 064 499 404 14 0 0 41 18.3 4 24 59 442 513 367 438 466 069 541 437 15 0 0 44 28.9 5 6 8 476 553 395 472 502 074 583 471 16 0 0 47 39.5 5 17 17 510 1 592 424 506 538 080 624 505 170 0 50 50.2 5 28 26 544 632 452 539 573 085 666 538 18 0 0 54 0.8 6 9 35 578 671 480 573 609 090 707 572 19 0 0 57 11.5 6 20 44 612 711 508 607 645 096 749 605 20 0 1 0 22.1 7 1 53 646 750 537 641 681 101 791 639.21 0 1 3 32.7 7 13 3 680 790 565 674 717 106 832 673 22 0 1 6 43.4 7 24 12 714 829 593 708 753 112 874 706 23 0 1 954.0 8 521 748 1 869 621 742 788 117 I 915 740 24 0 1 13 4.6 8 16 30 782 908 650 775 824 122 957 774 25 0 1 16 15.3 8 27 39 816 948 678 809 860 128 999 807 26 0 1 19 25.9 9 8 48 850 987 706 843 896 133 040 841 27 0 1 22 36.5 9 19 57 884 027 734 877 932 138 082 875 2810 1 25 47.2 10 1 6 918 066 762 910 968 143 123 908 29 0 1 28 57.8 10 12 16 952 106 791 944 003 149 165 942 3010 1 32 8.5 10 23 2 986 145 819 978 039 154 207 975 3110 1 35 19.1 11 4 34 020 185 847 011 075 ] 159 l48 _ 009 60 TABLE XXXVIII. Moon's Motions for Hours 1 2 3 4 5 6 7 8 9 10 11 1 13 1 11 27 43 12 14 16 2 16 1 3 1 3 1 2 23 54 87 24 28 31 5 33 2 6 3 6 3 3 34 81 130 36 42 47 7 49 3 9 4 9 4 4 46 108 173 48 56 62 10 65 4 12 5 12 6 5 57 135 217 60 70 78 12 81 5 15 6 15 7 6 68 162 260 72 84 93 14 98 6 18 8 18 9 7 80 190 303 84 98 109 17 114 7 20 9 20 10 3 91 217 347 96 112 124 19 130 8 23 10 23 11 9; 103 244 390 108 126 140 22 146 9 26 12 26 13 10 114 271 433 120 140 15.5 24 163 10 29 13 29 14 11 12.5 298 477 131 1514 171 26 179 11 32 11 32 16 12' 137 325 520 143 168 18S 29 195 12 35 16 35 17 13 148 352 563 155 182 202 31 211 13 38 17 38 18 I41 160 379 607 167 196 217 34 228 14 41 18 41 20 15 171 406 650 179 210 233 36 244 15 44 19 44 21 16 182 433 693 191 224 248 38 260 1G 47 21 47 23 17 194 460 737 203 238 264 41 276 17 50 22 50 24 i8 205 487 780 215 252 279 43 293 18 53 23 53 25 19 217 515 823 227 266 295 46 309 19 56 25 56 27 20 228 5412 867 239 280 310 418 325 20 58 26 58 28 21 239 569 910 251 294 326 50 311 21 61 27 61 30 22 201 596 953 263 308 311 53 358 22 64 28 64 31 23 262 623 997 275 322 35'7 55 374 23 67 30 67 33 241, 274 650 1040 237 336 372 58 390 24 70 31 70 34 Hours. Evection. Anomaly. Variation. Longitude. 1 0 28 17 0 32 39.7 0 30 29 0 32 56.5 2 0 56 35 1 519.5 1 057 1 5 52.9 3 1 24 52 1 37 59.2 1 31 26 1 38 49.4 4 1 53 10 2 10 39.0 2 154 2 11 45.8 5 2 21 27 2 43 18.7 2 32 23 2 44 42.3 6 2 49 45 3 15 58.5 3 252 3 17 38.8 7 3 18 2 3 48 38.2 3 33 20 3 50 35.2 8 3 46 20 4 21 18.0 4 349 4 23 31.7 9 4 14 37 4 53 57.7 4 34 17 4 56 28.1 10 4 42 55 5 26 37.5 5 446 5 29 24.6 11 5 11 12 5 59 17.2 5 35 15 6 2 21.0 12 5 39 30 6 31 57.0 6 543 6 35 17.5 13 6 7 47 7 436.7 6 36 12 7 8 14.0 14 6 36 5 7 37 16.5 7 640 7 41 10.4 15 7 4 22 8 956.2 7 37 9 8 14 6.9 16 7 32 40 8 42 36.0 8 738 8 47 3.4 17 8 0 57 9 1515.7 838 6 91959.8 18 8 29 15 9 47 55.5 9 835 9 52 56.3 19 8 57 32 10 20 35.2 9 39 3 10 25 52.7 20 9 25 50 10 53 15.0 10 9 32 10 58 49.2 21 9 54 7 11 2554.7 1040 1 11 31 45.6 22 10 22 24 11 58 34.5 11 10 29 12 4 42.1 23 10 50 42 12 31 14.2 11 40 58 12 37 38.6 24 11 18 59 13 3 54.0 12 11 27 13 10 35.0 TABLE. XXXVIII. 61 Mloon's Motions for Hours. H. 14 15 16 17- 18 -19 20 21 22 23 24 25 26 27 28 29 1 4 1 2 2 1 0 0 0 0 0 0 0 0 0 0 0 2. 8 3 3 4 2 0 0 0 10 1 0 0 3 12 4 5 5 3 0 0 0 0 1 0 1 0 1 1 0 4 16 5 6 7 4 0 0 0 0 2 0 1 1 1 1 1 5 21 6 8 9 5 0 0 0 0 2 1 1 1 1 1 1 6 25 8 9 11 6 0 0 0 0 3 1 1 1 2 1 1 7 29 9 11 12 8 1 0 0 0 3 1 1 1 2 1 1 8 33 10 12 14 9 1 0 0 0 3 1 1 1 2 1 1 9 37 11 14 16 10 1 0 0 0 4 1 2 1 3 1 1 10 41 13 15 18 11 1 0 0 0 4 1 2 2 3 2 1 11 45 14 17 19 12 I0 0 0 5 1 2 2 3 2 1 12 49 1518 21 13 1 0 0 0 5 1 2 2 3 2 2 13 54 16 20 23 14 1 0 0 0 5 1 2 2 4 2 2 14 58 18 21 25 15 1 0 0 0 6 2 2 2 4 2 2 15 62 19 23 26 16I 1 0 0 0 G6 2 3 2 4 3 2 16 66 20 25 28 17 11 0 1 1 7 2 3 2 5 3 2 17 70 21 26 30 18 1 0 1 1 7 2 3 3 5 3 2 18 74 23 28 32 19 2 0 1 8 2 3 3 5 3 2 19 78 24 29 33 21 2 0 1 1 8 2 3 3 6 3 3 20 83 25 31 35 22 2 0 1 1 8 2 3 3 6 3 3 21 87 26 32 37 23 1 2 0 1 1 92 4 3 6 4 3 22 91 8 34 3 9 24 1 21 0 1 1 9 2 4 3 6 4 3 23 95 29 35 40 25 H20 1 1 10 3 4 4 7 4 3 24 99 31 37 42 26 2 0 1 1 10 3 4 4 7 4 3 H. ISup. of Nod, II V VI VII VIII IX X XI XII 0 - 1 0 7.9 0 28 1 2 1 1 1 0 2 1 2 0 15.9 0 56 3 3 2 3 3 0 3 3 3 0 23.8 1 24 4 5 4 4 4 1 5 4 4 0 31.8 1 52 6 7 5 6 6 1 7 6 5 0 39.7 2 19 7 8 6 7 7 1 9 7 6 0 47.7 2 47 9 10 7 9 9 1 10 9 7 0 55.6 3 15 10 12 8 10 10 2 12 10 8 1 3.6 3 43 11 13 9 11 12 2 14 11 9 1 11.5 4 11 13 15 11 13 13 2 15 13 10 1 19.4 4 39 14 16 12 14 15 2 17 14 11 1 2'7.4 5 7 16 18 13 15 16 2 19 15 12 1 35.3 5 35 17 20 14 1 18 3 21 1 7 1 3 1 43.3 6 2 18 21 15 18 19 3 23 18 1 4 1 51.2 6 30 20 23 16 19 21 3 24 19 1 5 1 59.2 6 58 21 25 18 21 22 3 26 21 1 6 2 7.1 7 26 23 26 1 9 22 24 4 28 22 117 2 15.0 7 54 24 28 20 24 25 4 29 24 1 8 2 23.0 8 22 26 29 21 25 27 4 31 25 19 1 2 30.9 8 50 27 31 3 22 27 28 4 33 27 20 2 38.9 9 18 28 32 24 28 30 4 35 28 21 46.8 9 45 30 34 25 29 31 5 37 29 22 2 54.8 10 13 31 36 26 31 33 5 38 31 23 3 2.7 10 41 33 38 27 32 34 5 40 32 21 3 10.6 11 9 34 39 28 34 36 5 42 34 062 TABLE XXXIX. Moon's Motions for Minutes. 1i 23 4156 7 8 9 10ll11 12 13 14 15 16 17 18 311 6 14 22 6 7 8 j 1 8 0 1 1 1 1 2 1 1 1 1 32 6 14 23 6 7 8 1 9 1 2 1 2 1 2 1 1 1 1 33 6 15 24 7 8 9 1 9 1 2 1 2 1 2 1 1 1 1 34 6 15 25 7 8 9 1 9 1 2 1 2 1 2 1 1 1 I 35 7 1G 25 7 8 9 i 10 1 2 1 2 1 2 1 1 1 1 36! 7 16 26 7 8 9 1 10 1 2 1 2 1 3 1 1 1 1 37 7 17 27 7 9 10 1 10 1 2 1 2 1 3 1 1 1 1 38 7 17 27 8 9 10 2 10 1 2 1 2 1 3 1 1 1 1 39 7 18 28 8 9 10 2 11 1 21 2 1 3 1 1 1 1 40! 8 18 29 8 9 10 2 11 1 2 1 2 1 3 1 1 1 1 411 8 19 30 8 10 11 2 11 1 2 1 2 1 3 1 1 1 1 42- 8 19 30 8 10 11 2 11 1 2 1 2 1 3 1 1 1 1 43 8 19 31 9 10 11 2 12 1 2 1 2 1 3 1 1 1 1 44 8 20 32 9 10 11 2 12 1 2 1 2 1 3 1 1 1 1 45! 9 20 32 9 10 12 2 12 1 2 1 2 1 3 1 1 1 1 46 9 21 33 9 11 12 2 12 1 2 1 2 1 3 1 1 1 1 47/ 9 21 34 9 11 12 2 13 1 2 1 2 1 3 1 1 1 1 48! 9 22 35 10 11 12 2 13 1 2 1 2 1 3 1 1 1 1 49 9 22 35 10 11 3 2 13 1 2 1 2 1 3 1 1 1 1 50; 9 23 36 10 11 13 2 13 1 2 1 2 1 3 1 1 1 1 51 10 23 37 10 12 13 2 14 1 2 1 2 1 4 1 1 1 1 52 10 24!38!10 12 13 2 14 1 3 1 3 1 4 1 1 1 1 53 10 24 38 111 12 14 2 141 1 3 1 4 1 1 1 1 54 10 24 39111 12 1142 14 1 3 1 3 1 4 1 1 2 1 55 10 25 40l11 13 14 2 15 1 3 1 3 1 4 1 1 2 1 56 11 25 40 11 13 14 2 15 1 3 1 3 1. 4 1 1 2 1 5711 26 141 11 13 15 2 15 1 3 1 3/1 4 1 1 2 1 5S811 26 42. 12 13 15 2 16 1 3 1 3 1 4 1 2 2 II 59 11 27 43i12114 15 2 16 1 3 1 3 1 4 1 2 2 1 60 11 27143 12 114 15 2 16 1 3 1 3 1 4 1 2 2 I TABLE XXXIX. 63 Moon's Motions for Minutes. Sup. Min. Evec. Anom. Varia. Long. Nod. II V VI VII VII IX XI XII, 1 0 28 0 32.7 0 30 0 32.9 0.1 0 0 0 0 0 0 0 0 2 0 57 1 5.3 1 1 1 5.9 0.3 1 0 0 0 0 0 0 0 3 1 25 1 38.0 1 31 1 38.8 0.4 1 0 0 0 0 0 0 0 4 1 53 2 10.6 2 2 2 11.8 0.5 2 0 0 0 0 0 0 0 5 2 21 2 43.3 2 32 2 44.7 0.7 2 0 0 0 0 0 0 0 6 2 50 3 16.0 3 3 3 17.6 0.8 3 0 0 0 0 0 0 0 7 3 18 3 48.6 3 33 50.6 0.9 30 0 0 0 0 0 0 8 3 46 4 21.3 4 4 4 23.5 1.1 4 0 0 0 0 0 0 0 9 4 15 454.0 434 4 56.5 1.2 4 0 0 0 0 0 0 0 10 4 43 5 26.6 5 5 5 29.4 1.3 5 0 0 0 0 0 0 0 11 5 11 5 59.3 5 35 6 2.4 1.5 5 0 0 0 0 0 0 0 12 5 40 6 31.9 6 6 6 35.3 1.6 6 0 0 0 0 0 0 0 13 6 8 7 4.6 6 36 7 8.2 1.7 6 0 0 0 0 0 0 0 14 6 36 7 37.3 7 7 7 41.2 1.9 7 0 0 0 0 0 0 0 15 r 4 8 9.9 7 37 8 14.1 2.0 7 0 0 0 0 0 00 16 7 33 8 42.6 8 8 8 47.1 2.1 7 0 0 0 0 0 0 0 17 8 1 9 13.3 8 38 9 20.0 2.3 80 0 0 0 0 0 0 18 8 29 9 47.9 9 9 9 52.9 2.4 8 0 0 0 0 0 1 0 19 8 58 1020.6 9 39 10 25.9 2.5 9 0 0 0 0 0 1 0 20 9 26 10 53.2 10 10 10 58.8 2.6 9 0 1 0 0 0 1 0 21 9 54 1125.9 10 40 11 31.8 2.8 10 0 1 0 0 0 1 0 22 10 22 1158.6 1111 11 12 4.7 2.9 10 1 1 0 0 1 1 O0 23 10 51 1231.2 11 41 12 37.6 3.0 11 1 1 0 0 1 1 0 24 11 19 13 3.9 12 12 13 10.6 3.2 11 1 1 O 1 1 1 1 25 11 47 1336.6 12 42 13 43.5 3.3 12 1 1 O 1 1 1 1 26 2 16 14 9.2 13 13 14 16.5 3.4 12 1 1 1 1 1 1 1 27 12 44 14 41.9 13 43 14 49.4 3.6 13 1 1 1 1 1 1 1 28 13 12 15 14.6 14 13 15 22.3 3.. 13 1 1 1 1 1 1 1 29 13 40 15 47.2 14 44 15 55.3 3.8 13 1 1 1 1 1 1 1 30 14 9 16 19.9 15 14 16 28.2 4.0 14 1 1 1 1 1 1 _-~~ ~, oo ~ 64 TABLE XXXIX. Moon's Motions for Minutes. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 1 O O O O O O O O O O O O I2 O 1 1 O O 1 O 1 O O O O O O O O O O ~o3 1 1 2 1 1 1 o oo o 4 1 2 3 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 51 24 1 1 1 1 0 000000000 61 1 3 4 1 1 2 o 2 o o o o 1 0 0 0o 7 1 3 5 1 2 2 0 2 0 0 0 0 0 0 0 0 0 8 2 4 6 2 2 2 0 2 0 0 0 0 0 9 2 4 6 2 2 2 o 2 o o 0 o o 1 o o o o 10 2 5 7 2 2 3 0 3 oo o0 0 0 1 o o o o 11 2 5 8 2 3 3 0 3 0 1 0 1 0 1 0 0 0 0 12 2 5 9 2 3 3 0 3 0 1 0 1 0 1 0 0 0 0 13 2 6 9 3 3 3 1 4 O 1 0 1 o 1 0 0 0 0 14 3 6 10 3 3 4 1 4 0 1 0 1 0 1 0 0 0 0 15 3 7 11 3 3 4 1 4 0 1 0 1 0 1 0 0 0 0 16 3 7 12 3 4 4 1 4 0 1 0 1 0 1 0 0 0 0 17 3 8 12 3 4 4 1 1 0 1 0 1 0 1 0 0 0 0 18 3 8 13 4 4 5 1 5 0 1 0 1 0 1 0 0 1 0 19 4 9 14 4 4 5 1 5 0 1 0 1 0 1 0 0 1 0 20 4 9 14 4 5 5 1 5 0 1 0 1 0 1 0 1 1 0 214 10 154 5 5 1 6 0 1 0 1 0 1 0 1 1 O 22 4 10 16 4 5 6 1 6 0 1 0 1 1 2 0 1 1 0.23 4 10 17 5 5 6 1 6 0 1 0 1 1 2 0 1 1 0 24 5 11 17 5 6 6 1 7 0 1 1 1 1 2 1 1 1 0 25 5 11(185 6 6 0 1 1 1 2 1 1 1 0 26 5 12 19 5 6 7 1 7 0 1 1 1 1 2 1 1 1 0 i27 5 12 19 5 6 7 1 7 0 1 1 1 1 2 1 1 1 0 28 5 13 120 6 7 7 1 8 0 1 1 1 1 2 1 1 1 0 29s 6 1 321 6 7 7 1 80 1 1 1 2 1 1 1 0 30 6 14 22 6 8 1 8 0 1 I 2 1 I 1 0 _2 1 [1[~ TABLE XXXIX. 65 Moon's Motions for Minutes..Sup.. mNaria.dong. upII V VI VII VIH IX XI 1 31143716 5 154517 1.2 4.1 14 1 1 1 1 1 1 1 3215 17 2 16151734.14.215 1 1 1 1 1 1 33 341757.916 46 7.14.41 1 11 1 1 1 1 3416 830.517161840.04.5 16 1 1 11 1 351630193.2 17471912.9 4.7 16 1 1 1 1 1 1 1 3616581935.8 18171945.94.8 17 1 1 1 1 1 1 37175 184820 18.8 4.9 17 1 1 1 1 1 3817552041.219182051.85.0 18 1 1 1 1 1 1 3918232113.8 19492124.7 5.2 18 1 1 1 1 1 1 1 4018522146.5 20 19 2157.6 5.3 19 1 1 1 1 1 1 4119202.2 20502230.6 5.4 19 1 1 1 1 11 421948225.8 212023 3.5 5.6 20 1 1 1 1 1 1 43 2016 2324.5 21 1 2336.5 5.7 20 1 1 1 1 11 1 452 11324~29.8 22 52 24 42.3 6.0 21 1 11 1 1 1 1 4621 25.523222515.36.1 21 1 1 1 1 1 1 1 47 2210 2535.1 2353 25 48.2 6.2 22 1 1 1 1 1 1 1 48 22 38 26 7.8 24 23 26 21.2 6.4 22 1 1 1 1 1 l 1 492362640.5 245426 54.1 6.5 23 1 1 1 1 1 1 1 5023342713.1 252427 27.0 6.6 23 1 1 1 11 1 51243274.8 255528 0.0 6.8 24 1 1 1 1 1 522431281. 5 262 8 32.9 6.9 24 1 1 1 1 1 532459285.1 2 5629 5.9 7.0 25 1 1 1 1 1 1 5425282923.8 27262938.8 7.1 25 1 1 1 1 1 2 1 55 2556 2956.4 2756 3011.8 7.3 26 1 11 1 1 2 1 56262430291 28273044.7 7.4 26 1 1 1 1 1 21 57 26 52 31 1.8 28 57 31 17.6 7.5 27 1 2 1 1 1 2 1 58 27 2 1 3 1 34.4 29 28 31 50.6 7.7 27 1 2 1 1 1 2 1 [-60 2-8 17 32 39.8- $0'l29 32 56.5 7.9 281,21 1, i2, I 66 TABLE XL. Moon's AMltons for Seconds. |Sec. Evec. Anom. Var. Long.I Sec. Evec. Anom. Var. Long. 1 0 0.5 1 0.5 31 15 16.9 16 17.0 2 1 1.1 1 1.1 32 15 17.4 16 17.6 3 1 1.6 2 1.0 33 16 18.0 17 18.1 4 2 2.2 2 2.2 34 16 18.5 17 18.7 5 2 2.7 3 2.7 35 17 19.1 18 19.2 6 3 3.3 3 3.3 36 17 19.6 18 19.8 7 3 3.8 4 3.8 i 37 18 20.1 19 20.3 8 4 4.3 4 4.4 38 18 20.7 19 20.9 9 4 4.9 5 4.9 39 18 21.2 20 21.4 10 5 5.4 5 5.5 40 19 21.8 20 22.0 11 5 6.0 6 6.0 41 19 22.3 21 22.5 12 6 6.5 i 6 6.6 42 20 22.9 21 23.1 13 6 7.1 7 7.1 43 20 23.4 22 23.6 14 7 7.6 7 7.7 44 21 24.0 22 24.2 15 7 8.2 8 8.2 45 21 24.5 23 24.7 16 8 8.7 8 8.8 46 22 25.0 2'3 25.3 17 8 9.2 9 9.3 47 22 25.6 24 25.8 18 9 9.8 9 9.9 48 23 26.1 24 26.4 19 9 10.3 10 10.4 49 23 26.7 25 26.9 20 9 10.9 10 11.0 50 24 27.2 25 27.4 21 10 11.4 11 11.5 51 24 27.8 26 28.0 22 10 12.0 11 12.1 52 25 28.3 26 28.5 23 11 12.5 12 12.6 53 25 28.9 27 29.1 24 11 13.1 12 13.2 54 26 29.4 27 29.6 25 12 13.6 13 13.7 55 26 29.9 28 30.2 26 12 141.1 13 14.3 56 26 30.5 28 30.7 27 13 14.7 14 14.8 57 27 31.0 29 31.3 28 13 15.2 14 15 4 58 27 31.6 29 31.8 29 14 15.8 15 15.9 5" 28 32.1 30 32.4 30 1 14 16.3 15 16.5 60 28,32.7 30 32.9 TABoff XLI. 67 First Equation of Moon's Longitude.-Argument 1. Diff. Diff. Diff. Diff. Arg. 1 for 10 Arg. 1 for 10 Arg. i for 10 Arg. for 10 i r11 for 10 Arg. 1 _I........,., 01240.0 4 2500 1 407 500012 40.0 50023 39.3 002 4 24 0.16 4.06004750022 50 12 18.8 4.22 2550 1 41.5 0.1 5050 13 0.3.0 7550;23 39.4 4.22 0.28 4.04 0.10~~0. I0011 57.7 2600 1 42.9 5100[13 205 760023 38.9 4.22 60.42 4.04 0.24 15011 36.6 2650 1 45.0 5150 13 40.7 4 650 23 37.7 4.20 0.54 4.04 0.38 4 47650i2 20011 15.6.2700 1 47.7 5200 14 0.9 770023 35.8 250j1 r4.18 0.66 4.00 0.5038 2510 54.7 2750 1 51.0 5250 14 20.9 7750123 33.3 4.16 0.80 4.00 0.62 300 10 33.9 414 2800 1 55.0 0.92 5300 14 40.9 7800 23 30.2 350 10 13.2 4 2850 1 59.6 5350 0.8 7850 23 26.4 4.12 1.04 7532 0.87 400 9 52.6 2900 2 4.8 5400 15 20.5 7900 23 22.0 0.88 4.06 1.18 3.92 1.02 450 9 32.3 2950 2 10.7 5450 15 3.92 404202 5450 40.1 3 7950 23 169.9 1.1402 17.8505563.000 500 9 12.1 400 3000 2 550015 59.684 8000 23 11.2 1.14 4.00 3.84 ~~~~~~~~~~~1.26 550 8 52.1 3050 2 24.2 154 5550 16 18.8 8050 23 4.9 3.94 1.54 3.80 81000 600 8 32.4 088 3100 2 31.9 164 5600 16 37.8 8100122 57.9 1.40 0.88 [1.64 3 815 650 8 13. 3150 2 40.1 565016 56.7 37 8150 22 50.3 1.52 700 7 53.8' 3200 2 48.9 1.76 0017 15.3 8200 22 42.0 3.78 70.31 366 1.76 750 7 34.9 3.0 3250 2 583 1.8 5750 17 33.6 360 825022 332 1.90 7~~~~~~~50bir3. 8250 252 33.2 1.98 1.90 800 7 16.4 364 3300 3 8.2 580017 51.6 8300 22 23.7 850 6 58.2 3350 3 18.7 2.20 5850 18 9.4 8350 22 13.7 2.12 585 2.2 3.58' 900 6 40.3 3400 3 29.7 590018I 26.9 8400i22 3.1 3250 2232 3. 80 3.1 950 6 22.8 32 3450 3 413 595018 44.0 8450 21 51.9 2.24 3.4 2.42.6001 1000 6 5.7 343500 3 5314 2.42 6000 19 0.8 3.36 8500121 401 2.36 2.50 3.28 246( 1050 5 49.0 34 3550 4 5.9 5 6050 19 17.2 8550121 27.8 3.24[32 25 11 3 24 2.6236001 6100119 33.3 860021 15.0 3.16 3 49012.70 3.14 2 68 150 5 17.0 3650 4 352. 0 615019 49.0 86531 1 26 8 3.OS 2.80 04 8 2 2.78 i200 5 1.6 38 700 4 46.5 6200 20 4.2 870020 477 2.98 2.88 2.98 2.88 1.250 4 46.7 3750 5 0.9 6250 20 19.1 8750 20 33.398 2.881 2.28 1300 4 32.3 278 3800 5 15.8 3.04 6300 20 33.5 8800.20 18.4 130 1.4 850 800 1.O30 1350 4 18.4 2 50 5 31.0 31635020 47.5 2. 8850 20 3.0 Q900 5 46.7 6409 0 1.0 90 13~~~~~~ ~0021 1438 1404502.56',3.22 2.62 19 72 3.24: 14501 3 52.2 2.0 3950 6 2.8 6450121 147.1 85019 3.03.,.468~38 250 13 1500 3 39.9 236 1000 6 19.2 336 6500 21 26.642 9000l9 14. 42 1550 3 ~28.1 4-050 6 36.0 6550 2L1 38.7 9050118 5,7.2 2.,,4 3.42 2.32 3.50 1600 3 16.9 4100 6 53.1 6600121 50.3 910018 39. 2.12 350 2120 3.58 1650 3 6.3 200 4150 7 10.6 356 6650 22 1.3 915018 21. 6 2.00 3.56 2.10 i 3.64 17001 2 56.3 4420 7 28.4 11.8 9200:18 3.6 1.90, 360,702 1.98 3.70 1750 2 46.8 4250 46.4 675022 217 8 9250117 45.13 166 430, 8 47 3721800:2 1850 2 29.7 1 4350 8 23.3 685i22 39.9 935017 7.0 6.52 378 1.64 3.88 1900 2 22.1 140 4400, 8 42.2 690022 48.1 1 9400 16 47.6 1950 2 15.1 4450 9 1.2 3.80 950122 55.8 1 9450 16 27.9 1.26 3.84 2 142 1 4.00 20001 2 8.8 114 500 9 3.90 1.28 9500116 4.04 2050 2 3.1 4550 9 39.9 705023 93 955015 47.7 2200 1 498 4700110 39.1 38 00023 250 0 014 46.1 0.62 5 4.001 0.80 4.016 2250 1 46 7r 4750 10 59.1 725023 29.0 9750 14 25.31 2 1 47 ol750 0 4.00 4 0.66 2300 1 44.2 4800 11 19.1 7300 03 32.3 9800 14 4.4 23501 1 42 3 0.38 4850'11 39.3 4.04 735023 3510 054 5013 43. 20 024 404 042 98 3 43.4 4122 2400 1 41.1 0.24 594.04 740023 37.1 0.28 990013 22.3 422 490011 9800I 24501 4(0.610- 4950 12 19.746 745023 38.5 6 131.21 i250o 8 9 40 95017.02 4.412 42 2 1 500012 7 0. 5 700023 39.3 1.14~t~ 68 TABLE XLII. Equations 2 to 7 oj Moon's Longitude. Arguments 2 to 7 Arg. 2 diff 3 diff| 4 diffj 5 diff 6 diff 7 liff Arg. 2500 4 57.3 003 01 6 30.3 04 339.4 0.2 0 6.2 02 0 0.8 0 12500 2600 4 57.0 0.9 0 2.4 6 29.9 1.1 3 39.2. 0 6.41o 0 0.91 4 1200 2700 456.1 0 2.8 5 628.8 1338.5 0 6.9 0 1.310 5 2300' 1.4 0.5..' 1.0' l' 0.' 1' uo 2800 4 54.7 0 3.3 6 269 3 375. 0 7 70.8 0' 2200 0 " 0.8 2.6 1.5 11 019 2900 4 52.7 0 4.1 6 24.3 3 36.0 0 8.8 0 2.7 2100 2.6 1.0 3g3 1.9 1.5 1.0 2100 3000 4 5i0.1 10 5.1 6 21.0 3 34.1 0 10.315 0 3.71 2000 3.1 1.3 4.1 2.4 1.8 1.3 3100 447.0 3.7 0 6.4.4 6 16.9 4.7 331..7 0 12.1 2.1 0 5.0 1.4 1900 -53 14. 24 1800 320014 43.3 4.2 0 7.8 16 612.2 54 3 29.0 3 0 14.2 0 6.4 1800 3300 439.1 47 0 9.4 1.9 6 6.8 611325.9'35 016.6 2'6 0 8.1 19 1700 3400 4 34.4 0 11.3 20 6 0.7 673 22.4 3 0 19.2 3. 0 10.0. 1600 3500 4 29.2 0 13.3 5 54.0 3 18.5 0 22.2 3. 0 12.1 2.15()0 5.7 2.2 7.4 4.2 3.2. 2.3 36004 23 5 61 0 15.524 546.6 7.9 314.34 0 25.4 35 0 14.4 24 1400 37004 17.466 0 17.926 5 38.7 3 9.7 4 0 28.9 38 016.8 2 1300 3800 4 10.8 69 0 20.5 27 5 30.3 3 4.9 2 0 32.7 039 19.5 28 1200 3900 4 3.9 0 23.2 9 5 21.3 92 597 0 36.6 41 l 22.3 29 1100 4000 3 56.6 0 26.1 5 11.9 2 54 3 0 40.7 0 25.2 1000 7.7 3.0 9.9 9 5.7 4.4 3.1 410013 48.9 7.9 0 29.1 331 5 2.0110.3 2 48.6 5.9 0 45.1 4.5 0 28.3 3.2 900 42003 41.0 8.3 032.2 4 51.7 242 42. 6.1 0 49.614'7 0 31.5 800 4300 3 32.7 8.5 0 35.4 3.4 1 41.0 107 2 36.fi6 3 0 54.3 4.9 0 34.8 3. 700 3.7' 34 10.9 6.3 ~. 3.4 440013 24.2 8.7 0 38.8 3.4 4 30. 113 2 30.3 0 59.214 9 38.2 35 600 450013 15.5 8.9 042.2 35 4 18.811'5 2 23.8 6"6 1 4.1 1041.7 36 50( 460013 6o.6190 0 45.7 3.5 4 7.311.6 2117.2 6.7 1 9 5.1 0 45.3 3.6 400 470012 576 9.1 0 49.236 355.7 11.8 2 10.568 14 3 5.2 0 48.9 3.7 300 480012 48.59.3 0 52.8 36 4.9 2 3.76.8 1 1 59.552 5.63.37 200 4900239.2920 56.43 3 31.9 9 1 56.9 1 24.7 056.3 317 1001 50002 30.0 1 0.0 320.0 119 1 50.0 9 1 30.0 5.31 0.0 5100| 220.8 9 1 3.6136 3 8.1 12.0 1431168 1 35.315' 12 3.7 7 9900i 5200 2 11.51i 1 7.23 2 56.1 136.3 1140.5 2 1 7.43 9800 53002 2.4. 1 10.8;3:5 244.3 118 1 29.5 7 145.7 5. 111.1 9700 5400 1 53.4 1 14.35 232.7 1 22.86 14 5. 114 6 196001 ~ 9 3.5 11.5 6.6 5.1 3'6 550011 44.51 1117.81 221.2 116.2 1 55.9t 118 3 95001 8.7 3.5 4 11.3 6.5 4.9 3.5 1550011 35.8 i. 2.9 56001 35. 8S5 1 21.234 2 9.9 10.9 1 9.762 0 8 49 1 21.319400 5700 1 27.383 1124.6 2 1 59.0 107 11 3.41 61 2 5.74.7 1125.2 3. 9300 5800 1 19.0 1 27.81 1 48.3 1 0 57.3. 210.445 1 28.5 32 9200 59001111.1 130.9 310 1 38.01 9 90 51.45.7 12 14.9 4 1 31. i3'1 9100 000 1 34 133.9 1 28.1 4 045.7154 2 19 4. 1 34 1001056.169 136.8 27 118.7 90 040.35 223.439 1137.728 8900 6200'049.2 1 39.5 1 9.71 0 35.1 42 27.3 1 40.5 28800 630010 42.6 6 142.1 1 1.3 0 30.3 2 31.1 1 43.2 640010 36.5 1 44.5 22 0 53.4 0 25.7 4 2 34.6 32 1 45.6 2 86001 6500 0 30.8 146.7 0 46.0 021.5 2 37.8 1 47.9 8500 660010 5.6 1 48.79 0 39.3 610 17.63 2 40.8 1 50.0 8400 67001 020.9 2 1 50.6 10 33.2 0 14.1 3 l 2 243.4 24 1 51.9 8300 6800|0 16 7 1 52.2 0 27.8' 11.0 02' 458 1 153.6 8200' 3.7 1.4 4.7 2.7 2.1 1. 8100I 6900 0 13.0 1 53.61 0 23.11 0 8.34 2 47.9 1 55.0 8100 7000~0 9911 154.9 0 19.0 0 5.9 2049.711 56.3 8000 2.6 1.0 3.3 1.9 1.5 1.0 7100 0 7.320 155.9 015.7 2.60 4.0 i 251.2 157.309 7900 720010 5.3 14156.70' 0 13.1 0 2.5 2o 52.3' 158.2 7800 73000 3.9 157.20.5 011.2 1 l.0 1 53.1 158.7 7700 0.9 0.4 1.1'S7 0.5 0.4 74000 3.0 1 57.6 010.1 O 0.8 1 253.6' 59.1 760 0.3 01 0.8 2 532 0.2 0.1 750010 2.7j 1 57.7 0 9.7 0 06.2 53.81 j 1 9.2 7500' o.~ o ~ o ~ o ~ to _ o TABLE XLIII. TABLE XLIV'. 69 Equations 8 and 9. Equations 10 and 11. Arg. 8 9 Arg. 8 9 lArg. 10 11 lArg. l10 1 0 I 20.01 200 O 000 120.0 1 20.0 0 10.0 10.0 o500 10.0 10o.o 100 115.5 1 287 5100 1 24.4 1 25.8 10 9.3 11.1 510 9.6110.8 200 111.111 37.3 5200 128.8 1 31.4 20 8.6 12.11 520 9.2111.5 300 1 6.7 145.7 5300 133.1 1 36.9 30 8.0 13.1, 530 8.9112.3 400 1 2.3 1 53.7 5400 1 37.4 1 42.0 40 7.4 14.1 540 8.5 12.9 500 0 58.0 2 1.3 5500 1 41.6 1 46.8 50 6.8 15.0 550 8.2 13.6 600 0 53.8 2 8.3 5600 1 45.8 1 51.0 60 6.2 15.8 560 7.9 14.2 700 0 49.7 2 14.7 5700 1 49.8 1 54.6 70 5.7 16.6 570 7.7 14.6 800 0 45.7 220.2 5800 1 53.8 1 57.6 80 5.3 17.3 580 7.5 15.0 900 0 41.9 2 25.0 5900 1 57.6 1 59.8 90 4.9 17.9 590 7.4 15.4 1000 0 38.2 2 28.9 6000 2 1.2 21. 100 4.6 18.3 600 7.3 15.6 1100 0 34.7 2 31.9 6100 2 4.7 2 1.9 110 4.3 18.6 610 7.2 15.7 1200 0 31.4 2 33.9 6200 2 8.0 2 1.7 120 4.1 18.9 620 7.3 15.7 1300 0 28.2 2 34.9 6300 2 11.2 2 0.7 130 4.0 19.01 630 7.4 15.6 1400 0 25.3 2 35.0 6400 2 14.1 1 58.8 140 4.0 18.91 640 7.5 15.4 1500 0 22.6 2 34,1 6500 2 16.8 1 56.1 150 4.0 18.8 650 7.8 15.1 1600 O 20.1 2 32 2 6600 2 19.3 1 52.5 160 4.2 18.6l 660 8.1 14.7 1700 0 17.9 2 29.5 6700 2 21.6 1 48.3 170 4.4 18.2 670 8.4 14.2 1800 0 15.9 2 25.9[ 6800 2 23.7 1 43.4 180 4.6 17.7, 680 8.7 13.5 1900 0 14.2 2 21.5 6900 2 25.4 1 37.8 190 4.9 17.1 690 9.2 12.8 2000 0 12.7 2 16.4: 7000 2 27.0 1 31.7 200 5.3 16'.5 700 9.7 12.1 2100 0 11.5 2 10.7[ 7100 2 28.2 1 25.1 210 5.7 15.7 710 10.2 11.3 2200 0 10.5 2 4.4 7200 2 29.2 1 18.2 220 6.2 14.9 1 720 10.7 10.4 2300 0 9.9 1 57.7 7300 2 30.0 1 11.1 230 6.7 14.1 1 730 11.2 9.5 2400 0 9.5 1 50.7 7400 2 30.4 1 3.8 240 7.2 13.2 740 11.7 8.6 2500 0 9.4 1 43.5 7500 2 30.6 0 56.5 250 7.7 12.3 7 750 12.3 7.7 2600 0 9.6 1 36.2. 7600 2 30.5 49.3 260 8.3 1 1.4 760 182.8 6.8 2700 010.1 1 28.9 77002 30.1 0 42.3 1270 8.8 10.5 770 13.3 5.9 2800 0 1.1 21.8 7800 2 29.5 0 35.6 280 9.3 9.6 r780 13.8 5.1 2900 0 11.8 1 14.9 7900 228.5 0 29.3 290 9.8 8.7 790 14.3 4.3 300010 13.0 1 8.3 8000 2 27.310 23.6 300 10.3! 7.9 800 14.7 3.5 3100 0 14.6 1 2.2 8100 2 25.8 01.8.5 310 10.8 7.2 810 15.1 2.9 3200 0 16.3 056.6 82000 224.1 0 14.1 320111.3 6.5 S20 15.4 2.3 3300 0 18.410.51.7 8300 2 22.1 0 10.5 330 11.6 5.8 830 15.6 1.8 3400 0 20.7l0 47.51 8400 2 19.9 0 7.8 310 11.9 5.3 840 15.8 1.4 3500 0 23.2 0 43.9 8500 2 17.4 0 5.9 350 12.2 4 9 1 850 16.0 1.2 3600 0 25.9 0 41.2 8600 2 14.7 0 5.0 360 12.5 4.6 860 16.0 1.1 3700 10 8.8 0 39.3 8700 2 11.8 0 5.1 370 12.6 4.4 870 16.0 1.0 3800 0 32.0 0 38 3. 8800 2 8.6 0 6.1 3S0 12.7 4.3 880 15.9 1.1 3900 0 35.3 0 38.1 8900 2 5.30 8.1 390 12.8 4.3 890 15.7 1.4 4000 1 0 38.8 1 0 38.7 9000 2 1.8 0 11.1 400 12.7 4.4 i 900 15.4 1.7 4100 10O 42.410 40.2 9100 158.1 0 15.0 410 12.6 4.61 910 15.1 2.1 4200 0 46.2 0 42.4 9200 1 54.3 0 19.8 420 12.5 5.0 920 14.7 2. 4300 0 50.2 0 45.4 9300 1 50.3 0 25.3 430 1.3 5.4 1 930 14.3 3.4 4400 0 54.2 0 49.0 9400 1 46.2 0 31.7 440 12.1 5.8 940 13.8 4.2 4500 0 58.4 0 53.2 9500 1 42.0 0 38.7 450 11.8 6.4 950 13.2 5.0 4600 1 2.6 0 58.0 9600 137.70 46.3 460 11.35 7.1 960 112.61 5. 4700 1 6.9 1 3.1 9700 1 33.3 0 54.3 470 11.1 7.7 970 12.01 fi.9 4800 111.2 1 8.6 9800 1 28.9 12.7 480 110.8 8.6 980 11.4 7.9 4900 1 15.6 114.2 9900 124.5 111.3 10.4 9.i 90 10.7 8.91 i 000 1 20.0 120.0 100001 20.011 20.0 500 1 10.0 1 10.0 10( ) I 0 10.0 33 7 0 TABLE XLV. TABLE XLVI. Equations 12 to 19. Equlation 20. Arg. 12 13 14 15 16 17 18 19 Arg. Arg. 20 Arg. 250 2.3 1.6 7.8 0.0 33.7 3.4 16.7 0.4 250 0 10.0 500 260 2.3 1.6 7.8 0.0 33.7 3.4 16.7 0.4 240 10 10.9 510 270 2.4 1.7 7.9 0.1 33.6 3.5 16.6 0.4 230 20 11.8 520 280 2.6 1.9 8.0 0.2 33.5 3.5 16.6 0.5 220 30 12.7 530 290 2.9 2.2 8.2 0.3 33.2 3.6 16.5 0.5 210 40 13.5 540 300 3.2 2.5 8.4 0.5 33.0 3.7 16.4 0.6 200 50 14'I 550 O1() 3.5 2.9 8.7 0.7 32.7 3.9 16.2 0.7 190 60 15.0 560 320 4.0 3.4 9.0 1.0 32.4 4.0 16.1 0.8 180 70 15.7 570 33.0) 4.5 3.9 9.3 1.2 32.0 4.2115.9 1.0 170 80 16.2 580 34) 0 5.1 4.4 9.7 1.6 31.6 4-4 15.7 1.1 160 90) 16.7 590 350 5.7 5.1 10.11 1.9 31.1 4.7 15.4 1.3 150 100 17.0 600 3630 6.4 5.8 10.6 2.3 30.6 4.9 15.2 1.5 140 110 17.2 610 370 7.1 6.6 11.1 2.7 30.1 5.2 14.9 ].7 130 120 17.4 620 380(1 7.9 7.4 11.7 3.2 29.4 5.5 14.6 1.9 120 130 17.4 630 390) 8.7 8.3 12.2 3.6 2S.7 5.8 14.3 2.1 110 140 17.2 640 400 9.6 9.2 112.8 4.1 28.0 6.1 13.9 2.3 100 150 17.0 650 410 10.5 10.1 13.5 4.6 27.3 6.5 13.6 2.5 90 160 16.7 660 420 11.5 11.1 14.1 5.2 26.6 6.8 13.2 2.8 80 170 16.2 670 430 12.5 12.2 14.8 5.7 25.8 7.2 12.9 3.1 70 180 15.7 680 440 13.5 13.2 15.5 6.3 25.0 7.6 12.5 3.3 60 190 15.0 690 450 14.5 14.3 16.2 6.9 24.2 8.0 12.1 3.6 50 200 14.3 700 460 15.6 15.4 17.0 7.5 23.4 8.4 11.7 3.9 40 210 13.5 710 470 16.7 16.5 17.7 8.1 22.G 8.8 11.3 4.1 30 220 12.7 720 480 17.8 17.7 18.5 8.7 21.7 9.2 10.8 4.4 20 230 11.8 7.3, 490 18.9 18.8 19.2 9.4 20.9 9.6 10.4 4.7 10 240 10.9 7i40 500 20.0 20.0 20.0 10.0 20.0 10.0 10.0 5.0 0 250 10.0 750 510 21.1 21.2 20.8 10.6 19.1 10.4 9.6 5.3 990 260 9.1 760 520 22.2 22.3 21.5 11.3 18.3 10.8 9.2 15.6 980 2701 8.2 770 530 23.3 23.5 22.3 11.9 17.4 11.2 8.7 5.9 970 280] 7.3 780 540 24.4 24.61 23.0 12.5 16.6 11.6 8.3 16.1 960 290 6.5 790 550 25.5 25.7 23.8 13.1 15.8 12.0 7.9 6.4 950 300 5.7 800 5 60 26.5 26.8 24.5 13.7 15.0 12.4 7.5 6.7 910 310 5.0 810 I 570 1 27.5 27.8 25.2 14.3 14.2 12.8 7.1 6.9 930 320 4.3 820 5S0 28.5 28.9 25.9 14.8 13.4 13.2 6.8 7.2 920 330 3.8 830 590 29.5 29.9 26.5 15.4 12.7 13.5 6.4 7.5 910 340 3.3 840 600 30.4 30.8 27.2 15.9 12.0 13.9 6.1 7. 900 350 3.0 850 610 31.3 31.7 27.8 16.4 11.3 14.2 5.7 71.9 890 360 2.8 860 620 32.1 32.6 28.3 16.8 10.6 14.5 5.4 8.1 880 370 2.6 870 630 32.9 33.4 28.9 17.3 9.9 14.8 5.1 8.3 8701 380 2.6 880 640 33.6 34.2 29.4 17.7 9.4 15.11 4.8 8.5 860 390 2.8 890 650 34.3 34.9 129.9 18.1 8.9 15.3 4.6 8.7 850 400 3.0 900 660 34.9 35.6 30.3 18.4 8.4 15.6 4.3 8.9 840 410 3.3 910 670 35.5 36.1 30.7 18.8 8.0 15.8 4.1 9.0 830 420 3.8 920 680 36.0 36.6 31.0 19.0 7.6 16.0 3.9 9.2 820 430 4.3 930 690 36.5 37.1 31.3 19.3 7.3 16.11 3.8 9.3 810 440 15.0 940 700 36.8 37.5 31.6 19.5 7.0 16.3; 3.6 9.4 800 450 5.7 950 710137.1 37.8 31.8 19.7 6.8 116.4 3.5 9.5 790 460 6.5 960 720 37.4 38.1 32.0 19.8 6.5 16.5 3.41 9.5 780 470 7.3 970 730 37.6 38.3 32.1 19.9 6.4 116.5 3.4 9.6 770 480 8.2 980 740 37.7 38.4 32.2 20.0 6.3 16.6 3.3 9.6 760 490 9.1 990 750) 37.7 38.4 32.2 20.0 6.3 16.6 3.3 9.6 750 j0 10.0 1000l TABLE XLVII. TABLE XLVIII. 71 Equations 21 to 29. Equations 30 and 31 |21 22 23 24 25 26 27128 29 *...9.... Arg.- 30 31 25 7.8 3.2 7.1 6.1 5.9 4.1 58 4.3 5.7 25 27 7.8 3.2 7.1 6.1 5.9 4.1 5.8 4.3 5.7 23 5 29 7.7 3.3 7.0 6.1 5.9 1.1 5.8 4.3 5.7 21 2 5.0 5.0 31 7.6 [3.3 7.0 6.0 5.8 4.2 5.7 4.3 5.7 19 6 4.9 5.1 33 7.5 3.4 6.8 6.0 5.8 4.2 5.7 4.4 5.6 17 6 4.9 5.2 35 7.3 3.5 6 7 5.9 5.7 4.3 5.6 4.4 5.6 15 37 7.0 3.7 6.5 5.8 5.7 4.3 5.6 4.5 5.5 13 10 4.8 5.2 39 6.8 3.9 6.3 5.7 5.6 4.4 5.5 4.6 5.4 11 12 4.7 5.3 41 6.5 4.0 6.1 5.6 5.5 4.5 5.4 4.6 5.4 09 14 4.6 5.4 43 6.2 4.2 5.9 5.5 5.4 4.6 5.3 4.7 5.3 07 16 4.5 5.5 18 4.4 5.5 45 fi.9 4.4 5.6 5.3 5.3 4.7 5.2 4.8 5.2 05 18 4.4 5.5 471 5.5 4.7 5.4 5.2 5.2 4.8 5.1 4.9 5.1 03 20 4.2 5.6 49 j 5.2 4.9 5.1 5.1 5.1 4.9 5.0 5.0 5.0 01 22 4.1 5.7 51 4.8 5.1 4.9 4.9 4.9 5.1 i5.0 5.0 5.0 99 24 4.0 5.8 53 4.5 5.3 4.6 4.8 4.8 5.2 4.9 5.1 4.9 97 26 3.9 5.8 55 4.1 5.6 4.4 4.7 4.7 5.3 4.8 5.2 4.8 95 28 3.8 5.9 5' 13.8 5.8 4.1 14.5 4.6 5.4 4.7 5.3 4.7 93 30 3.7 5.9 59 3.5 6.0 3.9 14.4 4.5 5.5 4.6 5.4 4.6 91 32 3.7 5.9 61 3.2 6.1 3.7 4.3 4.4 5.6 4.5 5.4 4.6 89 34 3.7 5.9 63 3.0 6.3 3.5 14.2 14.3 5.7 4.4 5.5 i4.5 87 36 3.7 5.9 65 2.7 6.5 3.3 4.1 4.3 5.7 4.4 5.6 4.4 85 38 3.8 5.8 672.56.6 3.2 4.0 4.2 5.8 4.3 5.6 4.4 83.240 3.9 5.7 69 2.4 6.7 3.0 4.0 4.2 5.8 4.3 5.7 4.3 81 42 4.1 5.6 71 2.3 6.7 3.0 3.9 4.1 5.9 4.2 5.743 9 4 4.3 5.5 73 2.2 6.8 2.9 3.9 4.1 5.9 4.2 5.7 4.3 77 46 4.5 5.3 75 2.216.8 2.9 3.9 4.1 5.9 4.2 5.7 4.3 75 4 4.8 5.2 50 5.0 5.0 TABLE XLIX. 52 5.2 4.8 54 5.5 4.7 Equation 32. Argument, Supp. of Node. 56 5.7 4.5 58 5.9 4.4 -__ -__ __________ - __60 6.1 4.3 | I Ills TVs Vs| VIS| VITS| VIIIS 62 6.2 4.2 64 6.3 4.1 o " " " " " ~ 66 6.3 4.1 0 3.1 4.0 6.5 10.0 13.5 16.0 30 68 6.3 4.1 2 3.1 4.2 6.8 10.2 13.7 1 16.1 28 4 3.1 4.3 7.0 10.5 13.8 1.6.2 26 70 6.3 4.1 6 3.1 4.4 7.2 10.7 14.0 16.3 24 72 6.2 4.1 8 3.2 4.6 7.4 11.0 14.2 16.4 22 74 6.2 4.2 10 3.2 4.7 7., 11.2 14.4 16.5 20 76 6.0 4.2 12 3.3 4.9 7.9 11.4 14.6 16.6 1.8 78 9 14 3.3'5.0 8.1 11.7 14.8 16.6 16 80 5.8 4.4 16 3.4 5.2 8.3 11.9 15.0 16.7 14 82 5.7 4.5 18 3.4 5.4 8.6 12.1 15.1 16.7 12 84 5.5 4.d 20 3.5 5.6 8.8 12.4 15.3 16.8 10 86 5.4 4.6 22 3.6 5.8 9.0 12.6 15.4 16.8 8 88 5.3 4.7 24 3.7 6.0 9.3 12.8 15.6 16.9 6 90 5.2 4.8 26 3.8 6.2 9.5 13.0 15.7 16.9 4 92 5.1 48 28 3.9 6.3 9.8 13.2 15.8 16.9 2 94 5.1 49 30 4.0 1 6.5 10.0 13.5 16.0 16.9 0 96 5.0 4.9 -2.. I -i'O.(1- [ 1-. 98 5.0 5.0 5. 1s Is Os XIs Xs JXs 100 5.0 5.0 Constant 55" 71~2 ~TABLE L. Evection. Argument. Evection, corrected. i-os 1~ Ills I 1~ iii IvslV-S o Diff.o20 Diffi2s Diff. 20 Dff 20 Diff20 Diff. 0 30 00.0 10 43.5 40 97 50 25o5 39 8.3 9 42.0 85.5 73 2 o 40.9 2.0 43.4 72.7 1 31 21s5.5 11 56.7 7.'40 50.6.950 23.52. 38 24.9 8 29.3 2 32 50.9 3 9.0 41 301.5 201 37 404 75 16.0 85.4 71.6 142 8.3 38.2 4.9 45.8 74.0 3134 16.3 14 206 83 50 15.2 36 54.6 6 2.074.0 341.8.53,70.7;'" 5. 6.4 47.0'74.6 4 35 41.6 15 313' 4245.1 50 88 36 4.6 447.4 85.1 69.8 ~~~~ ~ ~~~~ ~~~~~~~~~~~~35.5 7.8 48.1 75.2 5 37 6.7 16 41.1 43 20.6 50 1.0 35 19.5 332.2 85.1 69.0 34.1 9.3 49.3 75.9 3831.884 17 50.1 6 43 34. 4951.7 34 30.2 2 16.35 /QO~~~~' 1o~~~82 3339.7 ____ 74. 84 9 68.1 32.7 10.7 50.5 99 6.4'i 39 56.71 18 58.2 44 27.4 49 411.0 33 39.7 i. 41 21.4 20 5.3 66 44 58.8 31.4 4 28.8 12.2 32 48.1 51.6 59 430 76.9 8141 21.4 84.4 66.rt I f29.9 13.r 527. 77.4 91 42 45.8 2111.5 45 28.7 299 49 15.1 13.7 31 554 57 58 25.6 78. 9142 45 84.3 65.2' 428.6 14.9 53.8 78.0 10 44 10.1 22 16.7, 45 57.3 49 0.2 31 1.6 57 7.6 83.9 64.3 27.2 16.7 54.9 758.4 114534. 7 23 21.0632 i 46245 48 43.5 30 6.7 505 49.2 78.91 11567~' I 491.0 33.7. 3 79 5.7 1 8324 24.2 46 50.2 25.7 48 25.6 17.9 29 10.7 50 54 30.3 12946 57.7183 4 62.2 24.3 19.3 57.0 79.3 14 1 5 26.4 4714.5 8 6.3 28 13.7 5311.079 7 13/48 21.1 830 26.61.2 2. 120.8 8 7. 14 49 44.1 26 27.6 47 37.422.9 47 45.5 27 1517 5 51 51.3 1551 86.7 27 27.6 60107 58.821. 47 23.32 26 16.6 50 31.2 82.2 6.7659.0 20.0 23.5 60.0 80.5 16 52 28.9 81 8 28 26f 58.0 48 18.8 46 59.8 25.0 25 16.6 49 10. 37.8 58.0 156 241S6 61.0 47.9 1753 50.7813 2924.6 48 37.41 4634.8 24 15.6 47 49.91 81.3 56.5 17.1 26.3 6~~~~~~~ _/a 11 o 18 5512.0 30 21.4 4. 8 54.5 46 8'5 23 13.6 246 28.8183 4 809 41' 5. 26.6' 477.4 6f.2294 55:~' 2 575 19 56 32.9 31170 556 49 10.1156 45 40.727.8 2210.769 45 781. ii8O.3.54.5 14.3 29.1 63 9 4 581.7. 20 57 53.2' 32 11.5 49 24.4 45 11.6 21 6.8 4345.8 79.8 53.3 12.7 30.4 64.7 81.9 2513.0179333 4.85 49 37.1 44 41.2 20 2.1 42 23.9 21169~~ 25.0 24. 152.6 12 3178 22 032.3 787 33 57.0 5 49 483 11.244 9.5 7 18 56.4 65 41 1.8 82.3 22 —-3.3787 50.9 9.8 33.1 66 5 82.3 23 1 51.0 781 3447.9 8 49 58.1. 43 3614 33.1 17 49'9 6 39 82' ~ 4. 81.3/ /44.5/!7'1~4 8.5i'/31.l 4, 24 3 9177435 37.71 50 6.48 43 1.9 16 42.6 6 38 17.0 9. 7.4 48 5 6.9 35.7 68 2 82j.6 25 4 26.5 36 26.21 50 13.3 69 226.2 15 4.4 36 54.4 76.8 47.2 5.4 37.0 68.9' 82.7 26 5 43.3 3713.4' 50.187 41492 1425.5 3531.7 /46.0 3.9 38.4 69.8 82.9 27 659.4 37 59.4 50 22.6 41 10.8 13 15.7 34 82.9 75.539.5 0..3 8 149 38 44.2 5025.0 44.39.6 12 52 711. 32 45.9 82.9 28384.43 1.0 440.8 1040 3 230 29 929.6 39 27.6 50 26.0 9 50.4 10 54.0 7 23. 51.0782.1[ 8.133. 4'5~4.916' 38 1.08. 30 10 43.5 40 9.7 50 25.5 0 39 8.3 42.1 9 42.0 72.0 0 83.0 520 20 10 3 54.4 TABLE L. 73 Evection. Argument. Evection, corrected. |v | VI | viiis| Ixs Xs j XIs | | |'10 Diff. 00 Diff. 00 Diff00 if o Diff. 0o Diff. 0 19 49 16. 5, 01,30 00 50 18.0 20 51.7 9 34.5 19 50.3 49 16.5 i'049 60 20 419641 14215 128 37.0 8 49 6.0 20 9.640 9 34.0 20 32.4 5) 30.4 82.9 71.2 40.8 1.0 43.4 74.7 227 14.1 8 47 54.8 19 28.8 9 35.0 21 15.8 51 45.1.75 9 5 2.9 0.5 39.6 2.4 644.8 75.5 3125 51.2 46 44.3 18 49.2 937' 22 0.6 63 0.6 3o182'9 69.8 384 3.9' 46.0 76.1 41,4 28.3 827 45 8.9 37.4.0 54 54 181. 472 512;3 5.6 i44 25.6.17 33.8 9 46.7'23 3.8 55 33.5 82.6 68 2 35.7 6.9 48.5 77.4 62143.0825 43 17.467.3 16 58.1 9 53.6 3 24 223,1 56 50.9 2o 082.5 67.3 36 821.3 49.8 78.17 0 20..5 2 42 10.1 16 23.6 10 1.9' 25 12.11.58 9.()7 82.3 66.or 33.1 9.8 50o. 9 787 18 58.2 1821. 65.7 1550. 31,7 10 11.712 26 3.0522 79.3 9 17 36.1 8 139 57.9 c 15 18. 10 22.9 1 26 55.2 5 47.079' 81.9 64. 3. 12.7 53.3 79.8 10 16 14.2.38 53.2 14 48.4 10 35.6 27 48.5' 2 6.8 81.7 63.9 29.1 14.3 54.5 80.3 1114 52.5813 37 49.3 9 14 19.3 8 10 49.9 2S 43.0 327.1 1213 31.281 36 46.42 13 51.5 11 15 29 38.6 4 48.0 81.1 62.0 26.3 7. 1 56.8 81.3 13t12 10.1 8 135 44A 61 0 13 5.2 250 11 22.6 i186 30 35.4 58 6 9.3 81.3 1,1049.38 3' 13 02 11 41.2 31 3358.0 7 31.181 151 9 28.8 8(. 33 13.4 1600 12 36.7 23.5 12 12 20.0 32 32.4 59.0 8 533 82.2 80.1 59.1 22.2 21.4 60.0 82.6 161 8 8.7 7 32 44.3 129 14.5 12 22.6 33 32.4 10 15.9 13.0 79.7 5a.0, 0.8 22.9 61.2 83.0 17 6 49.0 931 46.3570 11 53.7'12 45. 3 33.6 11 38.9 ~79.3 57.0( ~ 19.3 /~,3 62.2 83.4 5 29. 78 309 53.3 11 34.4 13 9. 35 5. 13 2 3 83 19 4 10.8 78.4 29 53.3 11 165 1 3 35.5 36 39.0 64 14 26.0 83. 20 2 52.4 28 58.4 10 59.8 14 2.7 37 43.3 15 49.981 0i 78.0 53.8 14.9 28.6 65.2 184.3 21 3 1 34 28 4.6 10 44.9 1431.3 3848.5 17142 1.477.4 521 2.7 15 1229.9 66.2 8144. 22 17.0 27 11.9 10 31.2 15 1.2 39 54.7 18 38.6 2 0776'9 51.6 122 41 11 203.8 259 0. 76 4 26 20.3 5o5 10 19.0 107 15 32.63 41 1.8 1 8. 24 5743.7 759 25 29.8 10 8.3 16 5.3 42 919 O21 28.28 125 56 27.8 24 40.5 9 59.0 16 39.4 43 18.9 22 53.31 75.2 48.1 7.8 35.5 69.81 85.1 26 55 12.6746 23 52.470 951.2 64 1714.96 8 4428.7 24184 27 53 58.0 740 23 5 415 9 44.8 17 51.7 45 394 25 43.7 28 52 44.0 22 19.6 9399 18 29'9 1651.0 27 9 8 29 513.07 73.3 21 35.1 441 9 36.5 19 9 48 372.3 85.4 72.7 20 51.7 73.4 50 180 2 20 51.7 9 34.5 19 50.3 49 16.5 30 0. 10! 00.0 00 - 00 -10 - J-;.... 74 TABLE LI. Equation of Moon's Centre. Argument. Anomaly corrected. | | Os __ - | sIs - IIs I IIIs | ] | Vs | f70 Diff 1 Diff I120 Diff 130 Diff Diff Diff forlO 100 f forlO' forl0 forl0 for for1 f. _ 20579 _ __. 0 0.070 2057.9 59.2 3843.6 301 1735.2 48 116 20.8 32 5828.9 30 332.6 7092355.6 589 4014.0. 1720.9 11435.335 5543.855 1 0 7 5.2 0 26 522 58 41 4. 29.0 17 4.8 591 2 48.5 36 52 58.0 5 301037.8708 29 477 58 43 9.6 2 16 47.1 11 04 5011.655 0S,4 5834 1 ] 28.4. 2 7.1[ 6.5 1 364 55.70. 2 0o1410.3 70 8 3242.0 577 4434.9. 1627.6 7.0 911.1369 4724.5 55.9 30 17 42.7 35 35.2 4558.4 16 6.5 720.5 4 36.8 301 5 4'' 1519.2 23 35.6 38 59.5 70.8 57.3 27.3 7.6 37.3 56.1 3 0 2 19.4 70.8 3 7.1 472.2 26. 15 43.7 528.7 3 41 48.5 56. 7076 561 25.5 9.1453 3 3 38. 56. 30 31 51.2 46 56.0 51 15.3 14 2.2 59 45.8 33519.81 70.6 55.76 557 98 389 56.9 5 0 35 23.0 4943.2 555230.2 25. 1355.8 5'749.1 30 29.1 70.5 55.3 24.4 10.4 39.3 57.1 303854.5 704 52 29.1 549 53 43 3 13 24.7 19 55 51.1 397 7 37.8 6 0 42 25.S 704 ~3355 13.8 55 547 22 1 51.9 1 5 552.0 24 45.9 30 45 56.970 57572 0 56 4. 1217.412.0 51 51.7 2153.5 70.3 540 26 405 5i.0 7 0 49277 039. 1 57 12.3 11 41.4254 432 40 19 0 57.8 305135 70.2 3. 53.6 22.1 16.6 1 8.1 10 V2 10. ~ 2f656 0' 51 15.43,'~ 1417.3 137 154.8 1 1.' 5 70.1 3 153.2 1 121.5 1 13.1 41.3 58.0 8 0, 5628.5.05 5975 5922.9 1024.3136 45 43.8417 1313.1582 30159 58.4 70.0.855'74 52.4 0 68. 37.9 0025.6 209 43.4 43 38.9 10 18.6 9 0 32.0 11 14852.3 1265 20.3 9 0.8 1 41 328 4 23.6 30 657.269.7 13 50.351.8 225 1 8 16.6 9 251 5127 424 28.19 5 69.6 51.4 19.1 6 15.3 39 25.62i 428.14 58.6 10 0 1026.0 161624.5 543 23.0 730.8 3717.3 1 32.2 69.5 50.9 18.6 15.8 43.1 58.8 3011354.5 18657'3 47187 6 43.41635 7.94 58535.8 11 01722569.321 50.5 1729..8 9 2 01722.5 28.8 5 12.5 11 54.44 3257.4 55 38.9 30120 50.1 23 58.80 6 4.6'1 5 3.916.8 3045.8 52 41.759.1 12 02417.13 26 27.5 6 54.9 4 111 7 2S3.1 41 434.3.30,27744.068.9 2854.7 743.5i16.2 3 18.0 1719 26 19.444.46 45.8(59.4 68.7 48.6 15.6 18.4 44.9 159.5 13 0,31 102 3120.54 8 890.3 2 22.7 24 436 43 47.3 3013435.8 68.5 481 5415.0 15.819.0 21488 40 48.459.6 1068.4 4 13 200 459. 6 - 9.9, 14 0 38 1.0 36 7.91 958. 4 02714- 19.5 19 31.9 37 49159.81; I~.699 1148.2 147.2' 13898 20.0 45.9 59.9 30o41 256.6'. 38 29.44 10 40.1 53 9 27 51. 4 14. 3. 15 04449.680140 546.6 3 20.5 2 3149 46 180 169'50 120 _'9 s TABLE LI. 75 Equation of Moon's Centre. Argument. Anomaly corrected. YVIPS VII ~ VIIIS IXs XS ~tjt Diff Diff' Diff Diffl Dil0 bil r0 40 10~~~~~ 00 10 tl forlOf for40 forll forlo forlO3 I 0 ________ ____ - - __ __ _ __ _ _ __ _ _ - - —.... —o o 0 0.061 8 5.4392 42204.8 42 21 1.437 T9. 0 O 0 U.O!~~ 1131 1 43 39.2 423. 164 39 2 1. 618 _...8 34.7 4.2 30.7 30 - -~~41 55.0 4 It 2 8 41 0-.5L9. 3056 54.6 61'8 5846.7546 4 42 12.1 36 22 48.5312'o60.0 1 0 5349.2 6156 3.04 40 12.0' 42 1.2 24422.2 41 61../54 3''338 3.1; 30 5043.916 1'5320.0'5413830.5I3' 4152.0 25 57.7 4 1.76 305 3961.8 541 3-.5 32.4 160.7 2 0 47 38.6617 50 37.7 53.8 3650.3 330 41 44.4 119 27 34.8 3310 5 7 61.0 61~~~~~~~~~~~.~2535 3. 4, 1.7!60' 3014433'4. 47 56'2 3511.3 3.41 38.7'429 13.7 54 6.7 61.8 53.6 32.5 1.4 33.5 61.3 3 0/41 28.1 6 45 15.4 33 33.7 44134.6 7 30 54.2 57 0.761.61.7.53.4 32.413. 0.8 ~ 340 X-617r 303823.0 42 35.31 3157.5 1. 41 32.2 03236.3 0 i5. 6 161.7 53.1 01.6 0.2 3'.6 62.0 4 0135 18.0 39 56.0 o,30 213.6 41,31.6 14 420.2 31 321.8 1617 52.9 31.2 0.4 35.1 6,. 30]32 13.0 3717.4 2849.0 41'32.7 136 5.65 6.8627 61 6 52 b 30. ~.0 u5r G. 019 8.1 61.6 3439.6 52.6 27 16.8 30. 41 35.6 ~3752.8 9 36.8 61.6 52.3 30.2 1.5 36.2 63.0 30[26 3.4 32 2.7 2546.1 1245 61.5 2 52.1 29.8 24114.4 3941.5 1 63.3 6 012258.8 6 29 26.5 2416.7 4146.4 41 32.07 1555.5 ]' 1 5'51.8 -~'298' 2.7 7 3011954.3 61 2651.1 152 48.7 2.34154.5.34324.0 19 6.23 I'61.4'515 - 5.'. z~. ";5; 7 011650.0 614 24 16.6 512 21 22.1 28. 42 4.3 45 17.7 22 17.8 6 61.4 51.2 28.4 3'9 38.A 7. 64.2 30 13 45.8 21 42.9 19 56.9 42 15.9 47 12.9 38.4025 60.3 61.3 51.0 27.9 4.4 39.0 6.5 8 011041.9 19100 0.7 1833.1 27.44229.2 o 49 98 3.28 3.7 ~61.3'507r' 74 o i.' 3. 5o''/ 47 30 7 38.0 6 1633.0 1710.8 42 44.2, 1 8.31 0 7.s31 57 0'61.2,' 50.4'270 ~'' 6 o t. q 9 0 434.46. 1.14 6.9 50.4 1549.8 27.043 1.1. 53.40 3512.9163 6101 I 0. 35.6.7!65.3l 3ol 131.0 1 1136.6 50.1 1430.4265 43 19 6 55 10.1 8 28 3 61.1 1 49.8 26.0 68 41.1 65' 61.0 I 49.5 25.5 7.4 41.6 1 65.8:30 5524.9 638.9.1 1155.9 5..144 2.0 598 18.2 45 2.6 t0.9 4. 2.5,0 8o 4,'.1 66.0 11 0 52222 60. 411.3 1040.9 4425.9 815 124.5426 663 r'60.8'489'24.5'8.5 4 30 49 19 7 60 144 7 9 27.3 4 51.5 3 32.4 51 39.6 12'61715 59 18.9 8 152 4. 5 188 541.9 559. 60.6 482 2345 97 43.667 30143 15.6 565422 7 4.6 4548.0 7 5249 58 19.3 60.5 47.9 23.1 10.3 44.2 1 67.0 13 0 4014.0 0 54304 5 5554 2 46 189 19 10 55 7 14013 30 3712.6 60.'52 7.547 447.82. 4651.5 1219.51 5 1.9 14 03411.6 60.3 4945 4 34167 22.0.47260 11.5 11435. i 4..2.1676 ~ 60 2' 47 0'37.1121.234 72. 67.' 3031 10.9 60.2 47247 47.0 23721.5 12.1 16 52.' 1146.9' 60.1'46.6 21.0 12.6 2 _ 16.2 67.1 15 0 28 10.6 45 4.8 1 34.1 48 40.1'1 19 10.74 5 1 04 - 0 I 0 10 00 20 190. 20- ~ - - 76 TABLE Li. Equatzon of Mloon's Centre Argument. Anomaly corrected. Os I Is II IIIs IV Vs OS _- l Ill s ] V _ _I is0o Diff o Diff 13 Diff Diff Diff 8~ Diff i forlO 11 folO forlO 13 forlO forlO for 10 1.,,,,,,,,, f,, I.,, 15 0 44 49.6 40 49.3 11 19.9 58 25.9 21 55.2 466 31 4 1 30 48 13.1 67643 7.9 4 11 57. 12 57 22.9 21 12 35.3470 28 491 66.7.6 4.7 19.7 23.5 4.3 60.6 16 01 59 541. 431.2 142 34. 56 1S.3 10 14.4 18.4 67.4. 11.5 22.0 47.3 0 6.3 30 54 58.1 47 4405 15. 57 4 55 10 2 275 52.57 22 747.41 60 291 671.2 246.7 1 0.9 1 4 22. 5 5 5.7 630.5 05 17 4 49 14 113 41.1 4.5 429.6' 19 46.0O,30 75. 0 7 4. 1 9 5 263.1 3:.3 47.3 16 44.4 60.5 6 1 5.7 16. 43.7 9. 23.5 48.3 5 160.6!8 O! 5 0.9 i 65 18.1 14 4.2 51 441 0 41.1 3 42.5 / 66.1 5 43.2 0., 2 1.1 4.6 60.1 30 5 8 20.4 56 27.6 2 8.5 50 32.7 50 15.3 10 40.3 19 0 3169.3 7 3 3.5 15 1.1 49 19.1 4'5. 7. 6 66.0 42.1 08.0 44.0 49.2 60.9 65.38 34216 416 1 4 3.91 38 173 15.0 4 161.4 65.5 341.1 6. 26.0 49.8 61.1 130 2.2 6399 1 44.3 167 4359. 2354.4 54 21.0 21 47!65.3 6,51.6 40.6 16 58.9 62 IO." 23 1 145 53.1 54 1 55 2561.1 304 5.' 2.2 851.7 175 3..2 43 22.0 5.1 64.7 3 19.5 5.0 27.4 30.7 61.3 2'4014 4. 18489 181.3 61.03 25 341 239. 126 47.2 162 1 4.1 40 3.1 12 47. 1 46 14. 1 62.2 35.4 3.9 51.2 61.. 23 037 42.2 14 42.3 1 57.7 0 37.9 35 43.4 7410. 611 30 40 53. 0 3916 236. 0 2 7 5.5 37 11.3 33 8.9 51.5 40 57 61 _ 63.6 47.3 2.7 29.91 6 4.5 18 28.0 3 8. 13.6 0. 5 43.3 30 33.5. 7 1.2 27 29.825.4 26.3 14447 18318 30 47 1.3 0 18.5 18 19.9 3 27 57.3 33 5 61.40 3.0 1.9 33.0 523. 61.6 26 0350 23.2 22732 IS 21.4 32 43.252 25 640.4 30 51.9 162.. 1.0 314 0.8 42. 52.6 61.8:3.2 62.3 23 54.4 IS 27.3 0.431 11.0 32 542.900 61.a 26 0151) 3 25 39.4 IS 2s.4 3 47.4 20 4.0 2442.0 3OK7 9960103i. 4 [1 3.. 1 o 1.50.2 31 5.6 5 61.8 23 0 8 56.3 207 25.2 3 3 7 175.2.2 2309. 3 922.3 0 1 621.4 8 011 5 1135 37.8 130 55.8 19 8.0 6 10.8 42 31.2 9 [44.3 54.6 8 17 59.2 37 11.5 17 47.9 is 5.0 1/13.30 3 5.4 2579 343631735.2 16,0.'2.9 044- 4.5[63'- [120 {12'- 270 TABLE LI 77 Equation of Aloon's Centre. A rgument. Anomaly corrected. I S| VIP VIIIs IXs xis | xs Is o Dilffo Diff1 Diff 0 Diff Diff Diff 50 forlO for101 forlO 0~ for]0 2 forlO forl1 o -,1' —15 028 10.6 4.8 4 1341 48 40.11 19 10.7466 15 10.4 30 25 10.5. 4245.9' 32.6 0,9 19.9 21 30.64 834.468. 16 O.2 10.9 5940281 56 5932.6'195 50 1.4 1.4 23 52.1 21 59.068 16 9 ~ 21 1 59n 27 428.6 4tt375 a1 tl. 163;t4475 79 32.6 47.7 68.4 30 1911.6 59 38 112 0.6150 5. 1 68 1 59.6 42 5.3 19.1 451', 168.5 17 0 16 12.7 56 374.318.4 5.129.7 15.6 28 9486 2 9.8 68.7 301 3 141 5. i. 5216.5 31 5.. 3 3216.0 594 44.6 17.9 16.'2! 49.1 68.9 18 0'10 16.15 31 26.9 55 48.3 33 32 357 1 30 7 18.31 718.23 3 1456. 1174 6 5.51,4 636 0..54 39 9.9 195 04 2914.2 9 6.8 174 50.0. 69.2 19 O1 4 291.1 291 2.6 47 5.6 5445 38 31.2)0 42 37.5 69.3 30 124.2 35.1 163. 1.6 41 50.946 69.5 20 0 527 58. 22 42. 56 37.0 43 35. 5 49 4.0 (2 58. 6 42.8 15.3 19.1 5l4 69.6 3055 31.9 20 34.442.451 43.414 57 34.3 46 9. 53 2.8 69.7 21 052 36.4 53.4. 3 13 3049 155.4 1621.4210 14.72 137 51 69.9 212 0112~ 35.51 15) 4827.92 1 25. 1 149'6 97 8.9 i 2230 9 41.49 58.2 1416 416 40 50 16.6 5116 0 20 51 02. 1 52 0 1.6 70.0 58.0 413 3 037.3321 5 4 03532 331.57.1 3043 529 1212.4 4856.3 141.5 56 399 7 1.8 157.8 40.9 12.6 22.1 53.6 70. 23 0 40 59.4 10 9.7 48 18.6 2 47.7 59 20.1 10 3.3 3 040j3S 6.59 7.6 40.5 12.0 22.6 -540 70.3 303 6.5 8 8.3 47 42.6 355.6 2 8 28 14 3. 21 0 3514.1 57.5 40. 1 11.5 23.2 2 4.5 70.4 24 3514.1573 6 8.0 47 8.1 109 5.3 446.2 17 34.2 70.4 3032 22.2 4 8.9 46 35.3 16 16.7 7 30.9 21 551705 25 0> 2,930.9 2 10.9 46 4. 2 7 29.8 10 16.81 24 37.0 156.9 38.9 9.8 25.0 55.7 70.6 30 26 40.2 0 14.2 45 34.8 8 44.72 13 4.0 28 8.8 03-. 56.7 85 9.3 25' 56.1 70.6 26 0 23 50.0 7 45 6.9 10 13.'4 15 52.451 3140.7 40.9 38.1 28.6 22.1 55 30,21 0.5 56524.4 I4440.8. 11197 18242.01 3512.8707,1( 1 15 6.3 j 13 37-7 841.-.2 3.2 70.7 27 (11811.5 54~313 44 16.3 1239.8 2132.9 38045.1 30 15 23.2 52 39.5 43 53.5 14 1.6 42 17.3 70.8 i 55.9 36.9 7.o0 2"I ~7.8 1.157.6.4 494. 011235.5 50 48.9 43 32.42718.01 4549.7 55. 0836.443.4 6. 2.4 i1 4as.1.708 30 948.4 48 59.6 143 12.9 59. 0 1 49 22.2 129 0 7 2.0 47 11.5 42 55.2 1817.3' 33 7.8' 5 70.9 55.3 35.6 5.1 1 29.6 0 8.9 54 27.4709 45 4735.2 4.8 30.1 59.2 70.9 30 0 131.1 5.14339.2 4224.8 21164 39 2.1 0 0.0 240., I 0 14 2 0 8 44.7I 1 11 8.3-70 10' 43'70. 7 78 TABLE LII. Variation. Argument. Variation, corrected. OsIs 111 HIS IiVa1 V1sI 00 Diff. 10 Diff. 0 Diff. Oo Diff.0 lDif. oo Diff. Z i 0 38 0.0 8 1.5 6 57.9 35 54.4 5 29.5i 6 1.6 1 3.3 34.0'39.9 74.0 35.3 40.0 1 39 1.3.3.3 8 3555 6 18.0.34 44 54. 641.6 240 2 9 735.9 33 26.6 7 4 1. 7 523.944 773.0 9 9.3 A14.21 73.6 30 307 8 4 3141 39. 1 3426.9 26 4 517 3 2 13.07 3 506 8 8 5 4142 52~21727 29 46.2 30 735 4 28.3 46.6 44254 4 5 30 59.6 3 22.3 8 55.0 54z4 4 5 72.3 10 27.9 24.5 3 17.31 29 46.7 2 2 56.5. 9 43.7.71.9 22.0 150.1 72.4 i 23.4 50.8 6 45 16.4 10 49.9 1 2 27.21 28 34.3 2 33.1 10 34.5 8 71.3 19.5'51.9 1 204 71.9 3.0 53.8 7 46 27.7 11 9.4 1 35.3 27 22.4 2 12.1 11 27.35 8 47 8.4 9 12644 0 41.653.26 112. 153.7 12 28. 4 699 14.5 55.5 70.5 15.9 56.6 9148 48.369 11 40.9 19 46.1571 25 0.7 696 137.8 13 1]8.6 69'1 1,7i 5. 69.6 13.3 58.3 10 49 57.4 11 52.9 58 49.01 23 51.1 1 24.5 14 16.9 68.2 9.3 158.8 68.8 10.8 60.1 11 51 5.6 12 2.2 57 50.2 22423'1137 15170 12 52 12.8 12 9.0 6.56 50.0 16. 2134.5 67. 1 5 8.25 16 61.7 0 661 4.2 61 6.76 5.5 63.3 1; 53 18.9 12 13.2 55 4.3 120 27.9 0.0 30 1722.0 64.9 36 63.16. 7 66.6 0 3.0 64.9 4154 23.863 12 14.8 1.6 54 45.2 63 19 223'0 57.0 18 26.9 63.7 0.9 8.,3 618 00 64 3 0.3 66.2 15 55 27.5 1 2 1 0.9 53 42096.0 563.7 19 33.1 6' (2.3 3.6 i 65.6 i 63.0 2.3 67.6 116 56 29.8609 12 10.3 61 152 35.36 17 150 60 590 9 20 40.7689 60.9 6.' 66'8 61,6 4.9 68.9 175730.7 9 2 4.1 151 2 9.5 16 13.4 1.1 39 7 149.6 1858 30.1 11 55.5 11. 20.7 15 13.2 111.5 22 59.61 19159 2.0 11 44.211 149 11.91 14 14.65 101.9.8 56.2 11 19697 1 57 1 128 72.1 0 24.2 11 30.5 148 2.21 1317.5 1 34.4 229 54.5 16.4 05 553 154 73~0 121 1 18.7 11 14.1;46 51.7, 1" lr 2 2 2!rI i 149 8 26 3591 52.7 188 112 9.8 5 180 73.9 22 2 11.4 10 55 2 115 O.5 11 28 5r 81 5 27 49.8 13 509 J,',~ Ji3.21 8.31;7.9 FI 20.5 1 74. 23 33 2 13 10 34.0 442 16, "o 3'.7 43 S 2 98.3 029 45752 21'48.9 13 8 f7 49 9 231 63. 75.2 6 34 51.2j O 012.0' ll'.il 2. 30 19.7 25 4 38.2' 9 440' 4-.3.72 58.8 36. 31 5.96 144-.9t | |S.6 i 73.3 i'46. i I S3.1 7.6.3! 26. 523.1 9 154 4049.9! 8 12.7 3' 3.61519 3~.9::0.7J~;Oa' " "3.7 1!.9 76. 27 6 6.0 7 38 44.5 39 3, 7 28.71 4 15 6.934 86 28 6467 467 8112 3 82`. 6 46.8s 448 25 71 29 725285 735.7355 13 84740 6 7.1137 3763,.1 29.2; 7 35.7 37 8.4;7 675 230.9:7.6 70.0 36.3 37.8'74.0 5237.6 37 4/7.3 301 1.' 18 1 6 5. 5544 1 29.5 I6 1.65'38 o' I 1 - 1 -- - I00! | 11~ I 11~ I lo 10 10. " I 2.0 _.',il4.j!~19! 46 [ 16 28 TABLE LII. 7 Variation. Argument. Variation corrected. i VII VIIIs IXs X X18 a!0~ 1Diff 0 Diff. 1i0 Diff. 0o Diff. 10 Diff. 100 Diff. 038 0.0 9 58.4 10 30.5 40 5.6 9 2.1 758.5 tt. o 7. at. u t,-J. 3 at 36.3 139 17.3' 10 36. 1 9 52.9 37.6 38 51.6 74.08 24.3 37.8 34.8 36 23 77.1 35.4 39.7 740 35.5 38.5 24034 31.4 1177.0 11.5 9 13.2 37376 8 748.8 9 13.3 40.7 rr.0 32.9' 4 1.9~'73.8'33.31 40.7 34151.4 66 7l 44.4. 831.3 44.0 3623.81 737 715.5,,' 9 54.0 43 8.1 7. 1215.0 747.3 3510.1 6 44.6.6 1036.9 76.3 28.1 4ot 1 429 5144 24.4 76312 43. 1 7 1.2 3356.8 736 16.0 28.6 11 21.8 75.9 25.5 48.0 72.9 / 26.2 47.0 6 4540.3,13 8.61 613.2 49.9 3,2 43.9 7 549.823,8 12 8.8 714655.5 7. 133172 3. 523.3,31 31.4 71.5 26.021.3 12 57.7 489 431.5 51.8 195 4'7. 50.9 8'48 1072 13 52.2 31.5 30 19.5 5 4. 1 48.6 949 4 73.9 14 10.2 37 29 71.2 45.9 1441.3 73.0 15.4 55.3 44.5 16.4 54.5 105037.1 1425.6 242.5 2757.8 4 7 29.5 15435. 72.1 12.8 57.1 69.7 13.7 56.2 1115149.2 1438.4 1 45.4. 26 48.1 6 4 15.8, 16 32.0579 12153 0.47 1448.51 0 546. 6 2539.36.4 4.5' 1729.9 tu. t. o 6...mr J ~.o 8.t 59.4 13154 10.47.0 1456.1 7.6 68602 2431.5 67.8 3o55.8. 1829.3 068.9 4.9 5946.6 61.6 668 6 2.1 60 14 55 19.3 15 1.0 61.6'845016 j 23 4.7 3 49.7 19 30. 60.9 67.6 23 5845 65.6 3.6 62.3 15 56 26.9 15 3.3 57 42.0 2219.1 346.1 20 32.5 167366.2 0.3 164.3 64.3 0.9 63.7 1657 33.6. 4 1 3.0 56 37.7 21 14.8 3 45.2 21 36.2 o64.9 3.0 65.6 63.1 1.6 64.9 17 58 38.0 0611 5 32.1 20' 3 46.8 122 41.1 63; 5 066... 6137 4. 2 66.1 185941.3 14 54.5,5425.5 6 1910.0 3 51'0,2347.2 ~ —1317 8.D 67. Dr. ~ D.x~ D.i 19-04306 14 46.3 12 53 17.7 67.8 18 9.8 60.2 3 57.8 2454.4 67.2 0 3060.1 10 8 68.8 58.8 9.3 2 446. 20 143.1 1435.5 852 8.9 17.11.0.84 7.1 26 2.6 58.3 13.3 69.6 57.1 12.0 69.1 21 241.4 1422.2 50 59.3 16 13.9 54 19 1 27 11.7699 56.6 15.9.98. 70.5 55.5 14.5 69.9 22 338.0 14 63 1. 49488 7. 1518.4 4336 1. 2821.6707 1317918.4 487 1.2 1424W 32.7.3 23 432.7 71' 4 50.6 1.29 32.3 24525..~. 32921.0 475 1.9 51.9 19.5 71.3 24 525.5 13 26. 47 25.7 13 32 5 10.1 30 43.6 50.8 13 23.4 72.4 128 50.1 1122.0 3 5 71.9 25 6 16.3 13 3.5 46 1313 1242.7 5 32.1 31 55.5 48.7 25.8 72.9 48.2 24.5 72.3 26 7 5.0. 1237. 2 45 04.4 115455 556.6.33 7.8 72.7 2 46.6 28.3 4334 4' 46.2 26.9' 27 27 751644.5 30.7 43. 73.6 44.2 6 23.5293 35 20.5 73.0 28 836.1 11138.7 42 33.4 1024.11 6 52.8 3533513'42.3 3.'39I' 38'42.1'31.7'73.2 129 918.4 42111 5.s 41 19.6 9 42.0 7 24.5 36 46.7 30 958.4 10 30.5 40 5.6 7 9 2.1 758.5 38 0.03'K- 7-' —-' -- -0 — __107-1-10_-_-107- K~ IO 00 00 00 00 i JO ~ 80'1 ABLE LIII. Reduction. Argument. Supplement of Node+Moon's Orbit Longitude. Os VIs Diff. Is VIIIs DifF Ils Vls Duff. Ills IXs Diff. IVs Xs Diff. Vs XIs Diff.,I _ _ I __ __ _.- _ 07 0.0 14.4 1 3.070 1 3.0 74 7 0.0144 12 57.0 12 57.0 74 6 45.6. 0 56.0 1 10.4 7 14.4 13 4.0 12 49.6 14.4 6.5 7.9 14. 65 2 4.6 7 216 311.21 0 49.5'1 18.3 7 28.8 13 105 12 41.7 14.3 611 8.2 14.3, 61 SAI 3,6 16.9 0 43.4'.1 26.5 7 43.1 13 16.6 12 33.5 4j6 2.6 14.3 37.8 5.6 1 352 8.7 7 57.4 14.3 13 22256 12 248 8.7 5 5 48.4 140 32.7 5.1 1 44.2 9.0 8 11.6 14.2 13 27.35.1 12 15.8 9.0 114. 5 9.5 14.1 4.5 9.. 65 34.311 0 28.2 1 53.7 8 25.7 13 31.8 1i2 6.3 91 75 203 14.0 43 9.8 14.0113 36.3 11 56 7 5 20.31 0 23.9' 2 3.5 8 39.7 11 56.5 8 13.91 3'9 10.2 13.9 13.9 10.2 8 5 6.4 200 2 13.7.8 53.6 13 40.0 11 46.3 1398 8 5313.8 32 10 916.8 224 10.519 7.4 13 43O.2 11 35.8108 13.6 2 2. 108 13.6 2.7.31oo 10 4 39.0 - 14.1127 2 3.0 9 21.0 13 45.92 11 25.011. 13.4 2.3 11.2 13.4 2.3 11.2 114 25.613 11 46. 9 34.4 1348.2 1.7 11 13.8 1 a3 3 12.6 0 7803 3 33.9 1 0 26.11 126 13 52.203 110 26.11 12.31 0.3 12.6 12.3 0.3 12.6 163 21.61.1 0 S 3 465 10. 13 51.9 1 0 13. 5 173 9.5' 0 8801 3 59.13 12.10505 113 512 L0 102. 11.8 4 6.3. 94 1.3 1.3 18 2 577 11 10 01 741 34 3 1 3 111 2. 113 49.91 7 9 47.71 19 2 46.112 118 234 25.6 11 138.0 1 13 45.9' 9 10 13.4 20 2 35.0 1. 0 14.1 234 39.0 13 4 11 25.0111.' 2.3 9 21.0 1 108 2.7 136 1 0. 1 2 452, 2.7 13.6 21'2 24.2 0 16.8 4 52.6 1135.8 13 43.2 9 713.8'2.5 13' 1..5 20 32 18 1413. 222 13.71 0 20.0 5 6.4 11 46 13 40. 8 53.6 232 3.51 10 23.9 5 20. 19 1 56.5 13 36.91 8 39.7 24 4.1' 22.6 I 1.0 18 82. 5 14.0 241 53.7 28.2 34.3 12 6.3 13 31. 8 25.7 25'1i5 3391'0 25 1 44.2 950 32.71 5 48.4 1 12 15.8 13 27.3 I 8 11.6 9.0 5.1 14.21 9.0 5.1 14.2 1 i5.1 142.3 261, 35.218 0 37]8 6 2.6 1312 24.38 3 22.2 7 57414. 27 1 20.5 8 43.4 6 16.9 12 30.5 23 166 7 431 28 1 3.5 1 0 6 6 3 18.1 2 13 10.51 7 288.38' 10.4'0 56.0 6 456 12 496 13 41 6 7 144144 741 56.017.0 6 14.4 7.0 ~~ 57.0 14.4 3130 70.01 3.01 * 301 712057'13 1.257 7.0 7 ~0O 1 TABLE LIV. Lunar Nutation ti Longitude. Argument. Supplement of the Node. Os Is s1| IlTs | Il | Vs| I0 o0.0' 8.5 14.8 17.3 15.2 8.8 430, 2 0.6 9.0' 15.1 17.2 14.9 8.1 28 4 1.2 9.4 15.4 17.2 14.5 87.7 526 6 1.7 10.0 15.6 17.2 14.2 7.2 24 8 2.3 10.4 15.9 17.2 13.8 6.5 22 10 2.9 10.9 16.4 117.1 13.5 6.1 0 12 3.5 11.4 16.3 17.0 13.0 5.4 18 14 4.1 1 1.8 16.5 16.9 12.6 4.8 16 16 4.6 12.2 16.7 16.7 12.2 4.3 14 18 3 5.2 12.6 16.8 16.5 1 1.8 3.7 12 20 5.8 13.1 16.9 16.4 11.3 3.0 10 22 6.2 13.4 17.1 16.2 10.9 2.4 8 24 6.9 13.8 17.1 15.9 10.4 1.8 6 26 7.4 14.1 17.2 15.7 9.8 1.3 4 28 7.8 14.5 17.2 15.4 9.4 0.6 2 30 8.5 14.8 17.3 15.2 8.8 0.0 0 XIs X- IXs VIIs. VIIs V.2 TABLE LV. 81 Aoon's Distance from the North Pole of the Eclptic Argument. Supplement of Node+Moon's Orbit Longitude. IIIs IVs Vs V Is JIIs VIIIs 40 Diff. 87 ff. Dif: Diff. 84~ 85~ ffo 870 for lO 890 for 10 920 for 10 940 0 Q 0 39 16.0 20 42.7 13 46.6 4648 0.0 22 13.4 6 15 17.3 30 0 30 39 16.7'22 4.2 27.2 16 6.9. 5041.4 3.8 2433.1[4664 16f37.7 30 27.6 47.0 53.8 0 39 18.S 23 27.0 2.618 278 53 22.9 26 52.2' 17 56.8 29 0 8.3918. 0 7.2 53.8 46.0 30 39! 22.4124 51.0.4 20 49.5 47 6 4.3.29 10.2;' 19 14.6 30 2 0 39 27.3 126 16.2 88 23 11.847 58 45.7 538 31 27.514586 20 31.3 28 0 It 0 139 {r.-ZifI, 28 8 47.7 ]3.8 30 39 33.7 127 42.6 25 34.8 127.0 33 44.2 21 46.7 30 29.2 47.9 53.8 45.3 3() 39 41.5 29 10.1 2758.548 4 8.3 36 0.2 23 0.8 270 30 39 50.6 30 38.9 30.0 30 2.8148 6 49.5 7 38 15.3 44 24 13.7 0 3 8300'48.3 53.7 44.8 4 0 40 1.2 32 8.8 30.4 32 47.748.5 9 30.6 40 29.7 25 25.3 26 0 30 40 13.2 33 39.930 35 13.2 7 12 11.6 6 42 43.3443 26 35.7 30 5 0 40 26.7 35 12.2 37 39.3 14 52.5 44 56.2 2744.8 25 0 31.1 48.9 53.6 44.0 30 40 41.5 36 45.6 315 40 6.1 491 1733.3 53647 8.1438 28 52.6 30 6 0 40 57.7 38 20.11 42 33.4493 20 14.0 49 19.4434 29 59.0 24 0 30 41 15.4 39455.8 45 1.2 49. 22 54.4'35 51 29.7 43 31 4.3 30 3441 32.3 49.5 535 43.2 7 0 41 31.4 41 32.73 47 29.6 25 34.8 - 53 39.349 32 8.2 230 30 41 54.8 43 10.6 32.6 49 58.6 28 14.9.. 55 48.0 2. 33 10.9 30 33.0 49.8 53.3 42.6 8 0 42 16.744 49.7 334 52 28.1 500 30 54.9 53 7 55.8 34 12.2 22 0 30 42 39.9 46 29.9 33 54 58.2 033 34.7 0 2.8 4210 35 12.2 30 9 40 3 4.6 48 11.2 34.1 5728.7 4 36 14.3 2 8.9 41736 10.9 21 30 4330.6 504 53.1 41.7 30 43 30.649 53.534 59 59.8 5 53.7 0 414.1 41 37 8.3 30 10 0 43 58.1151 37.0 231.3 4132.8 618.4 38 4.4 20 0 34.9 50.7 53.0 41.1 30 44 26.953 21.6 2 3.3508 44 1l.752 8 21.8408 38 59.1 30 11 0 |4457.1|55 7.1 357 735.8 1 0 4650.4 81024.3405 3952.5| 19 0 30 4528.8 5653.83. 10 8.8 51 49 28.7 71225.9402 4044.6 30 12 0 46 1.8 58 41.61 912421.1 52 6.852 14 26.6 4135.31 180 1 851.3 52.6 39.9 30 14636.1 0036.21516.0 3 54 4452.6 1626339 4224.7 30 1 36.6 51.4 152.5 39.6 13 014711.9j 220.137 17 50.21 15722.1 5 1825.0393 4312.7 17 0 37.3 51.7- 52.3 380 14 0 48 27.5 6 2.9376 22 59.9 51.82 36.2522 22 19.71386 4444.7 16 0 30 49 7.4 7 55.738 0 25 35.3 5 12.7 52. 24 15.5 383 45 28.7 30 15 0 14948.7 949.6 28 11.1 1 748.9 26 10.4 46 11.3 15 0 J8413 1860 1 88 910 93 J 940 jIls I IO X l~ I40 i HS is 08 ~ ~~~~' -142.6'.. 4 135. 82 TABLE LT. Mloon's Distance from the North Pole of the Ecliptic. Argument. Supplement of Node + Moon's Orbit Longitude. | H IVsr -VI VIIs VIIJs 84o 86~ Diff. 8 Diff. o Diff 93 Diff 94 for 10 fr 10 for 10 for 10 — o, —-,, _ o1 15 0 49 48.7 9 49.6383 28 11.1 52 7 48.9 5 26 104 46 11.315 0 38.3 52.1 51.9 38.0 30 50 31.311 44.5386 30 47.3 522 10 24.7 518 28 4.3 376 46 52.6 30 6 0 51 15.313 40.3 390 33 23.8 52.3 13 0.1 517 29 57.1 3 47 32.514 0 30 52 0.615 6 52.4 15 35.1 5 31 9.0 48 11.0 30 39.3 ID 52.4 51 6 37.0 17 0 52 47.3 17 3o.0 96 38 37.9 52.18 9.8 33 39.9 6 48 48.113 30 53 35.3 19 33.7 41 15.4 5 20 44.0 1. 35 29.7 49 23.9 30 39.9 52.61 51.3 36.2 18 0 54 24.721 33.4 2 43 53.2 52.7 23 17.9 37 18.4 9 49 58.2 12 0 30 55 15.4 23 34.1 405 46 31.3 25 51.2 39 6.2 350 31.2 30 40.5 52.8 51.0 35.6 19 056 7.512535.7 49 9.6 2824.2 40 52.9 352 51 2.911 0 40.8 52.9 50.8 35.2 30 57 0.9127 38.2 41.1 51 48.3 53.0 30 56.7 50.7 42 38.4 34 9 51 33.1 30 20 0 57 55.6 29 41.6 54 27.2 33 28.7 44 23.0 52 1.9 10 0 It 4. 41.4.53.0 50.5 34.5 30 58 51.7 31 45.9 57 6.3 53 36 0.2 46 6.5 34 1 52 29.4 30 21 0 59 49.1 33 51.1 4259 45.7 38 31.3 4748.8 5255.4 9 0 042.0 53.2 50.2 33.8 30 0 47.835 57.2 42.3 2 25.3 3.3 41 1.8 50. 49 30.1 33.4 53 20.1 30 22 0' 147.8138 4.249 5 51 5 4331.9 51 10.3 3.0 5343.3 8 0 4.2 42.6 5 5.1153.3 49.8 30 2 49.1 40 12.0 745.1 46 1.4 49.8 52 49.4 54 5.2 30 1 42.9 53.4 49.7 32.6 23 0 3 51.8 42 20.7432 10 25.2 48 30.449 54 27.3323 5425.6 7 0 30 4 55.7 144 30.3 43. 13 5.6 5 5 50 58.8 49. 56 4.2 319 54 44.6 30 24 0 6 1.0146 40.6 43 15 46.0 53 53 26.6 49 57 39.9 315 55 2.3 6 0 30 7 7.4 148 51.9 440 18 26.7 55 53.9 489 59 14.4 3151 55 18.5 30 25 0 8 15.2151 3.8. 21 7.5 3 58 20.7 4 0847.8 55 33.3 5 0 44.3 53.6 48.7 0 30.8 30 9 24.3153 16.7 23 48.4 0 46.8 201 30 4 55 46.8 30 44.5 53.7 48.5 300 5. 26 0 10 34.7 55 30.3 448 26 29.4 3 12.3 51.230 5 58.8 4 0 30 11 46.3 j57 44.7 450 29 10.5 53. 37.2 39. 30 27 0112 59.2 59 59.8 31 51.71 *7 8 1.5 649.9 292 5618.5 30 30 1413.3 215.8 34 33.0' 10 25.2 8 17.4 5626.3 30 1 l 45.6 53.8 47.7 28.8 28 0 15 28.7 1 4 32.5 37 14.3 12 48.2 9 43.8 5F6 32.7 2 O 30 16 45.4 6 49.8 39 55.7 15 10.5 11 9.0 56 37.6 30 6 29 0 18 3.2 9 7.8 42 37.1 53.817 32.2 4712 33.0. 56412 1 0 4637 7.4 53.8 9 47.0 13 55.27.6 543.3 30 30 19 22.3 111 26.946 45 18.6 53 8 119 53.1 46 7 1. 30 i25 0 81542.2l513 36.8 146.6 30 0o 2042.7 13 46.6 48 0.0 12213.4 1517.31 5644.0 0 0 850 870 890 920 940 9 0 11 4Os I X4- X 1- IX --- | Ite 1 I8 1 OS i | XI8 1 I X.s I2. |, TABLE LVI.'3 Equation II of the Moon's Polar Distance. -Argument II, corrected. r IIIs diff. IVs dif. Vs diff VIs diff. VIls di VIIIs diff.I 0D 13.8 1 124.4 46 4 36.9 80 9 0.0 92 13 23.1 16 35.6 30 1 0 13.9 129.0 4 44.9 9 9.2 13 31.0 -16 40.2 4'6 29 01391' 29 0 7'.8 46 20 14.1 2 1 33.8 4.84 53 0 8.1 9 18.4 9.2 13 38.8 78 16 44.6 28 20 04. 494 84 1 91.886.6 30 14.5 4 138.7 4.19 5 1.1 829 27.5 9.1 13 46.6 76 16 48.9 27 Og6 5 1 8.2 t' 4.1 4 15.1 3. 58 5 9.3 3 9 36.7 913 54.2 76 16 53.0 9 26 5 0158 149.0 5 17.6 9 458..9 14 1.8 16 56.9 25 0.9 5.3 8.4 9.1 7.5 3.8 6 0 16.7 10 154.3 55 5 26.0 8.4 955.0 9 14 9.3 74 17 0.7 37 24 70 17.7 1598 534.4 10 4.1 14 16_, 171 4.4 23 81018.! 1.4 2 5.6 8.5 9.1 43.5 8018.9 2 5.4 542.9 1013.2 14240!7 17 7.9 22 9 0 20.31 2 11.1 5 51.4 10 223 14 31.2 7 17 11.3 2 21 0 1 5 5.8 311 9.1 83.2 10 0 21.8 2 16.9 56 0.0 10 31.4 9 14 38.2, 17 14.5 20 1.7 6.0 8.7 9.0 17.0 3.0 11 023.5 1.8 222.9 16 8.787 1040.4 901445.2 69 1717.5 29 19 0291 23 1 4 635.0 7.3 15 5.52 6 1 911 93.'7 25.8, 16 213 6 4 8.8 8.0 136 2.5 15 0 31.7 1 247.9 6143.8 111 6.2 15 12.1 17 28.3 15 251. 6.6 8.9 8.8 6.4 2.3 160 34.2 212 54.5 6 652.7 89 125.0 8815 118.5 6 17 30.6.1 |14 17036.8 283 1.1 7 1.6 190 1133.8 815 24.8 67 1732.7 2.0 13 18 39.6 3 7.9 7 10.6 90 11 42.6 15 31.0 1 11 134 7 1 12 93 69 116. 12 0 3 2.8 728.6 12 o.o 15 43.1.9 17 38.2 10 1 3.2 7.0 19.1/'5.1 1.6 5.8 1.5 35 7..3' 858.85 5 123. 223 01 525. 1 43 6 5891 12 25.6 15 7.5 l4 2. 1i 3.7 s7. 9.1 1 1. 174 1.0 12131 38 3.1 358.2 75814.1919 1 242.48.416 11.5 17 44.2 10 51 3.91 7.6 9.2 8.3 5.2 0.7 261 7.0 4 14 5.8 768123.39,211250.7,8 116162 5 1174491664 4 27 1 0 13.4 8'3259 1.2 58.9 16 21.31 18 4 43 718 9.1 S. I 4.1 1 4-5 79 2115.41 421.2 84161 13 78.1 1626.2 481745.9 29 1 19.84 4 29.0 7.8 8 50.89 13 15 8.1 16 31.014617461 0 2 1 198'2 850.0 9.2 231 01 6-4 3 2 9.1 x~ I 5s Os I XIs Xsl IXs.4.... 1 o I I 16.0 14.9 12.0 84.0 4.0 1.1 30 6 16.0 14.5 11.3 7.21 33 0.7 24 12 15.86 13.9 10.5 6.3 2.6 0.4 18 18 15.6 13.4 9.7 5.5 2.1 0.2 12 24 15.3 12.7 8.8 1 4.7 1. 0.0 16 30 14.9 12.0 8.0 4.0 1.1 0.0!23 |IIs Is Os XIs XI s IX 84TABLE LVIII. TABLE LIX. Equcations of Moon's Polar Distance. To convert Degrees Arguments, Arg. 20 of Long.; V to IX and Minutes into corrected; X not corrected; and XI I)ecimal Parts. and XII corrected. |&Min. Dect. Arg 20 V VI VII VIII IX X X I Arg Arg rg ]&Min.'parts. jain~r ~arts ~ sI J I-I I I —XI-I Arg. 1 5 003 250 0.3 55.9 6.1 2.625.1 3.0 0.7 0.9 1250 0 4.0 500 1 26 4 2C0 0.3 55.8 6.2 2.7,25.1 3.11 0.7 0.9 240 10 3.7 510 1 48 5 270 0.4 55.7 6.3 2.8 25.0 3.2 0.8 1.0 230 20 3.4 520 210 6 280 0.6 55.4 6.5 3.024.9 3.5 1.0 1.0 220 30 3.1 530 2 31 7 290 0.8 55.1 6.9 3.3 24.8 3.8i 1.2 1.1 210 40 2.8 540 2 53 8 300 1.0 54.6 7.3 3.7'24.7 4.31 1.5 1.2 200 50 2.5 55(. 3 14 9 310 1.3 54.1 7.8 4.224.4 4.9! 1.8 1.3 190 60 2.3 56( 3 36 to 320 1.7 53.4 8.4 4.724. 1 5.6 2.2 1.4 180 702.1 570( 358 11 330 2.1 52.7 9.1 5.423.8 6.4! 2.7 1.5 170 8011.9 58) 4 19 1 2 3'40 2.6 j51.9 9.8 6.1 23.5 7.21 3.2 1.7 160 90 1.7 591 4 41 13 350 3.1 51.0 10.7 6.9 23.2 8.2! 3.8 1.9 150 10011.6 600 5 2 14 360 3.7 50.0111.6 7.7,22.8 9.2 4.4 2.1 140 110 1.5 610 5 24 15 370 4.3 148.9112.6 8.7 22.4 10.3 5.1 2.3 130 1201.5 620 546 16 380 4.9 147.7113.6 9.7 21.9 11.5 5.8 2.51 120 130/1.5 630 6 7 17 390 5.6 46.5 14.8 10.7 21.4 12.8 6.6 2.8 110 140 1.5 640 6 29 18 400 6.4 145.2 16.0 11.8 20.9 14.1 7.4 3.0 100 150 1.6 650 6 50 19 4:10 7.1 413.9 17.2 13.0 20.4 15.5 8.31 3.3 90 160 1.7 660 7 12 20 420 7.9142.5 18.5 14.219.9 17.0 9.1 3.5 80 170 1.9 670 7 34 21 430 8.8 141.0 19.81 15.51 19.3118.5 10.1 3.8 70 18012.1 680 7 55 22 44|0 9.6 39.5 21.2 16.8M 18.7 20.1 11.0 4.1 60 1'3012.3 690 8 17 23 450 10.5 38.0 22.6118.1 18.1 21.7 12.0 4.4 50 20012.5 700 8 38 24 460 11.3 36.4 24.1119.417.5 23.3 12.9 4.7 40 21)2.8 710 9 0 25 470 12.2 34.9 25.5 20.8 16.9 24.9 13.9 5.0 30 22G3. I 720 9 22 26 480 13.2 33.227.0 22.216.3 26.6 15.0/ 5.4 20 230,3.4 73( 9 43 27 490 14.1 31.6 28. 23..6 15.6128.3 16.01 5.7 10 24013.7 740 10 5 28 500 15 0 30.0 30.0125 0 15.0130.0 17.0 6.0 0 25014.0 750 10 26' 29 510 1.5 9 28.4 31.5126,4 14.4 31.7 18.01 6.3 990 26041.3 760 10 48 30 520 116.8 26.8133.0 27.8 13.733.4 19.0 6.61980 12704.6 770 11 101 31 1530 17.8 25.1134.5129.2 13.1135.1 20.1 7.0 970 28014 9 |780 11 31 32 540 18.7 23.6 35.9 30).6 1.5136.7 21.1 7.31 960 2905.2 90 11 53 133 550 119.5 22.0137.4.31.9 11.9 38.3,22.0 7.61 950 1300 5.5 S(0O 1214 34 560 20.4 20.5/38.8!33.211.3!39.9.23.0 7.9 940 31015.7 810 12 36 35 570 121.2 19.0 40.2134.5 10.7141.5 23.9 8.21930 32015.91 820 12 58 36 580122.1 17.5 41.5135.8 10.1143.0124.9 8.51920 330 6.1 830 13 19 37 1590 22.9 16.1142.837.01 9.6 44.5 25.7 8.7 910 340 6.3 840 13 41 3 8 600 23.6 14.8 44.0 38.21 9.1 45.9126.6 9.0 900 350 6.4 850 14 2 39 610 24.4 13.5 45.2 39.31 8.6 47.2527.4i 9.2 890 360 6.5 860 14 24 40 1520 25.1 112.3 46.4 40.3 8.1 48.5 28.2 9.51880 370 6.5 870 1446 4:1 630 25.7 111.1147.4 41.3 7.6 49.7 28.91 9.7 870 380 6.5 880 15 7 42 640 126.3 10.0 48.4 42.3 7.2 50.8 29.6 9.91860 390 6.5 890 1529 43 650 26.9 9.0 49.3i43.1 6.851.830.2 10.1 850 400 6.4 900 15 50 441 660 27.4 8.1 50.2 43.9 6.5 52.830.8 10.3 840 41016.3 910 16 12 45 670 27.9 7.3 50.944.6 6.2 53.6231.3.10.5 830 42016.1 920 1634 46 680 28.3 6.6151.fi645.3 5.9 54.431.8 10.6820 4305.9 930 16 55 j 47 690 28.7 5.9 52.2 45.8 5.6155.1132.2 10.7 810 4-40 5.7 940 i17 17 48 700 29.0 5.4 52.7 46.3 5.3'55.7 32.5110.8 800 460 5.5'350 117 38 49 710 29.2 4.9 53.1 46.7 5.21 56.2 32.8 10.9 790 46015.2 60 18 0 50 720129.4 4.6153.547.0 5.1156.533.0 11.0 780 470l4.9 970 18 22 51 730;29.6 4.3 53.7 47.2 5.0156.8 33.2,11.0 770 48014.61 980 18 43 52 740 129.7 4.2 53.8 47.3 4.9 56.9 33.3 11.1 760 490 4.3 1 990 19 5i 53 750 129.7 4.1153.9 47.4 4.9157.0'33.3 11.11_750 1500 4.0 1000 Cornstant 10" TABLE LX. TABLE LXI. 85 Small Equations of Mloon's Parallax. Moon's Equatorial Parallax. Args., 1, 2, 4, 5, 6, 8, 9, 12, 13, of Long. Argument. Arg. of Evection. A. 1 2| 4 5 6 8 9 1213 A O Is IIs IVsW Vs O o.0 1.6 0.6 1.6 1.9 0.0 3.6 1.4 2.0 100 0 1 20.8,1 15.6 1 1.5 42.6 24.1 10.8 30 3!0.0 1.6 0.6 1.6 1.9 0.0 3.5 1.4 2.0 97 1 1 20.8 1 15.2 1 0.9 41.9 23.6 10.5129 6!0.0 1.5i 0.6 1.5 1.8 0.0 3.1 1.4 1.9 94 2 1 20.811 14.9 1 0.3 41.3 23.0 10.2128 29Th1 1.5 0.6 1.5 1.8 0.1 2.6 1.331.8 91 1 20.7 1 14.5 59.7 40.6 22.5 9.9 27 12 0.1 1.4 0.511.4 1.7 0.2 1.9 1.2 1.7 88 4 120.7;114.2 59.2 40.0 21.9 9.626 15 0.1 1.3 0.5 113 16 0.2 1.3 1. 1 5 5 1 20.6 113.8 58.6 39.4 21.4 9.4 25 1 01. 05. 1.602.3I.68 6 1 20.6 1113.4 57.9 38.7 20.9 9.1 241 180.21.1 0.4 1.1 1.4 0.3 0.7 1.0 1.4 82 7 1 20.51 13.0 57.3 38.1 20.4 8.8231 21 Ol.1.0 0.4 1.0 1.3 0.5 0.2 0.9 1.2 8 1 20.4'1 12.6.7 37.4 19.9 8.6 22 24 04 0.9 0.3 0.9 1.2 0.6 0.0 0.7 1.0 76 8.421 91 20.311 2.2 56.1936.. 19.4 8 270.5 0.7 0.3 0.71.0 0.7 0.1 0.6 0.9 73 101120.21 11.7 55.5 36.1118.9 8.220 30 0.5 0.6 0.2 0.6 0.9 0.8 0.4 0.5 0.7 70 111 20 1 113 54 35 184 8 19 ] 111120.1111.3 54.9 35.5184 8019 33 0.6 0.4 0.2 0.4 0.7 0.9 0. 8 0.4 0.5 6 7 12 119.9 110.8 54.2 34.9 17.9 7.82 0;3610.7 0.3 0.1 0.3 10.6 1.0 1.5 0.3 0.4 64 1311 19.8 1 10.4 53.6 34.2 17.5 7.6 17 39 0.7 0.2 0.1 0.2 0.5 1.1 2.1;.2 0.2 61 1411 19.6 1 9.9 53.0 33.6 17.0 7.4116. 42 0.8 0.1 0.010.1 0.4 1.1 2.8 l 0.110.1 58 1511 19.5 1 9.4 52.3 33.0 16.6 7.2115 45 0.8 10.0 0.0 0.0 1 0.3 1.2 3.2 0.0 0.0 55 161 19.3 1 9.0 51.7 32.4 11.1 7.1114 48 ().8 0.0 0.0 0.0 0.3 1.2 3.5 0.0 0.6 52 171119.1 1 8.5 51.1 |1.7 15.7 6.9113 50 0.8s0.0 o 1 0.0 10.3 1.2 3.6!0.0 0.0o j 50 1811 18.9 1 8.0 50.4131.1 15.2 6.8 12 Consttnt 7" - 1911 18.7 1 7.5 49.8 30.;14.8 6.7 11 IConstnt 7" 201 18.4 1 7.0 49.1 29.5 14.4 6.5 10 The first two figures only of the Arguments 211 18.2 1 6.5 48.5 29.3 14.0 6.4 Q Iare talen. I2211 18.0 1 5.9 47.8128.7 113.6 6.3 8 231117.71 5.4 47.2 128.1 1\3.2 6.3 7 241 17.4 1 4.8 46.5127.5 12.9 6.2 6 251 17.11il 4.3 45.9 26.9 12.5 6.1 5 261 16.911 3.8 45.2126.3 12.1 6.1 4 27o 116.6 1 3.2 44.6125.8 11.8 6.1 3 281 16.2 1 2.6 43.9125.2 11.5 6.0 2 29 1 15.9 1 2.1 43.324.7 11.1 60 1 301 15.611 1.5 42.6124. 10.81 6.01 0 XIs Xs j IX I 4V I.5sVI.IsViaJ 34 86 TABLE LX!I. Moon's Equatorial Parallax. Argument. Anomaly. ( I diff IIs diff Ifs diff HIs diff Ws duff Vs diff O 5857.700 5827.0 57 57.9 31 529.8 54 1.9 5 53 3.2i "4 30 1 5857.7' 58 25 2.0 257 48 55 26.6 2 53 59.4 53 1.8 29 2 15857.6 0.1 58 230.0 57 1.6 55 23.4 53 56.9 53 05 1. 28 2 0 22 1 3526 3.2 25 4 12 3 5 57.4.2 58 20.9 1 56 58.4 2 55 20.2. 53 54.5 2. 52 59.3 1.2 27 0.3 2.2 3.2 3.2 2.4 1.2 4 58 57.1 0.3 58 18.7 2.2 56 55.2 55 17.0 3.2 53 52.1 2. 52 58.1 1 26 5 58 56.8 58 16.522 56 52.0 3.55 13.8 53 49.'7 52 57.01 25 0.4 2.2 3.2 3.2 2.3 1.2 6 5856.44 58 14.3 56 48.8 3 55 10.6 53 47.4 52 55.8 24 7 5856.006 58 12.0 2 56 45.5 3.55 7.5 3.1 53 45.1 2.52 54.81.0 23 8 5855.46 58 9.6 2.56 42.3 5 4.4 53 42.9 2.2 52 53.8 22 9 5854.8 0 58 7.2 56 39.0 3 55 1.3 3.1 53 40.6 52 52.8 21 0.6 284 06 3.1 21 05 9 10 58 54.2. 58 4.8, 2 56 35.7 54 58.2. 53 38.5 52 51.9 20 0.8 2.5 3.3 3.1 2.2 0.9 11 5853.4 58 2.3 56 32.4 5455.1 3 53 36.3 52 51.0 19 12 58 52.6 0.8 5759.8 2. 5629.1. 5452.1 3'0 53 34.2 211 5250.1 0 1 8 0.8 75982.6 3.3 3.0 2.1 0.8 13 58 51.8 0 57575.2' 26 56 25.8 3.3 449.1 310 53 32.1 20 52 49.30 17 14 58 50.8 10 57 54.612 7 56 22.5 3.3 5446.1 30 53 30.12'0 52 48.6 16 15 5849.8 57 51.9.56 19.2 54 43.1.53 28.1. 52 47.9. 15 1.1 2.7 3.3 2.9 1.9 0.7 16 5848.7 1.1 57 49.2.8 56 15.9 33 5440.2 2.9 53 26.2 1.9 52 47.2 0.6 14 17 5847.6 1.2 57 46.4 756 12.6 3,3 54 37.3 27 9 53 24.3 1.9 52 46.60 6 13 18 5846.4 57 4372.7 56 9.3 54 34.9 53 22.4 52 46.0 12 19 5845.1 13 5740.8,2 9 56 6.0 33 5431.5 2.9 5320.6 1.8 5245.5 11 20 5843.8 13 57 38.0 156 2.7. 54 28.7 2.8 53 18.8 52 45.0 0.5 10 1.4 2.9 3.4 2.8 1.8 0.4 21 5842.4 5 5735 1 25559.3 3 5425.9 253 17.0 5244.6 4 9 15 7322.9 5 3535 2.8'317524 04 22 58 40.91',57 32.22 55 56.0 54 23.1 53 15.3 52 44.2 8 23 58 39.4 57 29.3 55 52.7 54 20.3 253 13.7 152 43. 7 24 58 37.8 1'6 57 26.3 3.055 49.4 154 17.6 27 53 12.0 1. 52 435 0.3 6 1.6 3.0 3.3 ).7 1i6 0.2 25 58 36.2 57 23.3 55 46.1 54 14.9 2 53 10.4 52 43.3 5 1.8 1 13.0 3.3 2.7 1.5 0.2 26 5834.417 57 20.2 55 42.8 54 12.2 26 53 8.9 15 52 43.1 4 27 83.1.7 3.0 39.2.'S 02 27 58 32.71 85717.2 55 39.6 54 9.62 53 7.45 52 42.9 0 3 29 5829.9 57 11. 1 55 36.4 54 7.0 53 5.9 52142.8 2 1.9 3.1 3.3 2.6 1t4' 0.1 5827.0 5711.01' 55 33.15 3o4 4.4{25153 4.5 35242.7 0 30 58 27.0o 57 T.9' 155 29.8 54 1.9 153 3.2 152 42.7 0 I I_ -1I VIII - 8I. I I&- - ~______________________________________ ]_______- TABLE LXIII. 87 Moon's Equatorial Parallax. Argument. Argument of the Variation.... - IIs II- IVHV8 W 0 55.6 42.3 16.0 3.7 17.6 44.0 30 1 55.6 41.5 15.3 3.8 18.5 44.8 29 2 55.5 40.7 14.5 3.8 19.3 45.6 28 3 55.5 39.8 13.8 3.9 20.1 46.3 27 4 55.3 39.0 13.1 4.1 21.0 47.0 26 5 55.2 38.1 12.4 4.3 21.9 47.7 25 6 55.0 37.2 11.7 4.5 22.7 48.4 24 7 54.8 36.3 11.1 4.7 23.6 49.1 23 8 54.6 35.5 10.4 5.0 24.5 49.7 22 9 54.3 34.6 9.8 5.3 25.4 50.3 21 10 54.0 33.7 9.2 5.6 26.3 50.9 20 11 53.7 32.7 8.7 6.0 27.2 51.5 19 12 53.3 31.8 8.2 6.3 28.2 52.1 18 13 52.9 30.9 7.7 6.8 29.1 52.6 17 14 52.5 30.0 7.2 7.2 30.0 53.1 16 15 52.0 29.1 6.7 7.7 30.9 53.5 15 16 51.5 28.2 6.3 8.2 31.8 54.0 14 17 51.0 27.2 5.9 8.7 32.8 54.4 13 18 50.5 26.3 5.6 9.3 33.7 54.8 12 19 49.9 25.4 5.3 9.8 34.6 55.1 11 20 49.4 24.5 5.0 10.5 35.5 55.4 10 21 48.8 23.6 4.7 11.1 36.4 55.7 9 22 48.1 22.7 4.5 11.7 37.3 56.0 8 23 47.4 21.9 4.3 12.4 38.2 56.2 7 24 46.8 21.0 4.1 13.1 39.0 56.4 6 25 46.1 20.1 3.9 13.8 39.9 56.6 5 26 45.4 19.3 3.8 14.5 40.8 56.8 4 27 44.6 18.5 3.7 15.3 41.6 56.9 3 28 43.9 17.6 3.7 16.1 42.4 56.9 2 29 43.1 16.8 3.7 16.8 43.2 57.0 1 30 42.3 16.0 3.7 17.6 44.0 57.0 0 XIs Xe IXs VIIIa IIs I VIi 88 TABLE LXIV. TABLE LXV. Reduction of the Parallax, Moon's Semi-diameter. and also of the Latitude. Argument. Latitude. Argument. Equatorial Parallax. atRed. Red. of Eq.Par Semidia. Eq.Par'Semidia. Eq.Par Semidia.! sec Pro. L at._ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ P ar. 0o,,,,, 53 0'14 26.5 56 0 15 15.6 59 0 16 4.6 1 0.8 0 0.0 0 0.0 53 10 14 29.3 56 10 15 18.3 59 10 16 7.4 2 0.5 3 0.0 1 11.8 53 20114 3".0 56 20 15 21.0 59 20 16 10.1 3 0.8 6 0.1 2 22.7 53 30 14 34.7 56 30 15 23.8 59 30 16 12.8 4 1.1 9 0.3 3 32.1 53 40 14 37.4 56 40 115 26.5 59 40 16 15.6 5 1.4 12 0.5 4: 39.3 12 0.5 4 393 4 53 50 14 40.2 56 50 15 29.2 59 50 16 18.3 6 1.6 15 0.7 543.4 54 0 14 42.9 57 0 15 31.9 60 0 16 21.0 7 1.9 18 1.0 6 43.7 54 10 14 45.6 57 10 15 34.7 60 10 16 23.7 8 2.2 21 1.4 7 39.7 54 20 14 48.3 57 20 15 37.4 60 20 16 26.4 9 2.4 24 1.8 8 30.7 54 30 14 51.1 57 30 15 40.1 60 30 16 29.2 10 2.7 32 2.7 9 55.1 54 40 14 53.8 57 40 15 42.8 60 40 16 31.9 30 2.7 9 55.4 54 50 14 56.5 57 50 15 45.6 60 50 16 34.6 33 3.3 10 28.3 55 0 14 59.2 58 0 15 48.3 61 0 16 37.3 36 3.8 10 54.3 55 10 15 2.0 58 10 15 51.0 61 10 16 40.1 39 4.4 11 13.2 55 20 15 4.7 58 20 15 53. r 61 20 16 42.8 42 4.9 11 24.7/ 45 5.5 11 28.7 55 30 15 7.4 58 30 15 56.5 61 30 16 45.5 55 40 15 10.1 58 40 15 59.2 61 40 16 48.2 48 6.1 11 25.2 55 50 15 12.9 58 50 16 1.9 61 50 16 51.0 51 6.7 11 14.1 56 0 15 15.6 59 0 16 4.6.62 0 16 53.7 54 7.2 10 55.7 57 7.8 10 30.0 60 8.3 9 57.4 63 8.8 9 18.3 66 9.2 8 32.9 69 9.7 7 42.0 TABLE LXVI. 72 10.0 6 45.9 75 10.3 5 45.4 Augmentation of Moon's Semi-diameter. 78 10.6 4 41.0 81 10.8 3 33.5 84 11.0 2 23.7 Horizon. Semi-diameter. Horizon. Semi-diameter. 87 11.1 112.3 Alt. 1Alt. 8907\ 11.1 0 10.30 14'30"1 15' 16' 17 14' 30'" 15' 1 16' 17 90 10.1 0 0.0 Subsidiary Table. 2 0.6 0.6 0.7 0.8 42 9.2 9.8 11.2 12.6 Lat. +3' 1-3' 4 1.0 1.1 1.3 1.5 45 9.7 10.4 11.8 13.3.-.. 6 1.5 1.6 1.9 2.1 48 10.2 10.9 12.4 14.0 0o ",, 1 8 2.0 2.1 2.4 2.7 51 10.6 11.4 13.0 14.7 0 +0.0 -0.0 10. 2.4 2.6 3.0 3.4 54 11.1 11.8 13.5 15.2 6 0.0 0.0 612 0.0 0.0 12 2.9 3.1 3.6 4.0 57 11.5 12.3 14.0 15.8 15 0.0 0.0 14 3.4 3.6 4.1 4.7[ 60 118 12.7 14.4 16.3 15 0.0 0.1 0.1 01 16 3.8 4.1 4.7 5.3 a 63 12.2 13.0 14.9 16.8 24l 0.1 l0.1 18 4.3 4.6 5.2 5.9 B 66 12.5 13.4 15.2 17.2 21 4.9 5.3 6.0 6.8 69 12.8 13.7 15.6 17.6 30 0.1 0.1 360 0.21 0-.2 24 5.6 6.0 6.8 7.7 72 13.0 13.9 15.9 17.9 342 0.21 0.2 127 6.2 6.7 7.6 8.6 75 13.2 14.1 16.1 18.2 482 0.23 0.3 3(0 6.9 7.3 8.4 9.5 ] 78 13.4 14.3 16.3 18.4 48 03 03 33 7.5 8.0 9.1 10.3 81o 13.5 14.4i16.5 18.6 36 8.1 8.6 9.8 11.1 84 13.6 14.5 16.6 18.7 60 0.4 0.4 319 8.6 9.2 10.5 111.9 190 13.7 14.6 16.7 18.8 72 0.5 0.5 78 0.6 0.6 84 0.6 0.6 90 - 0.6 -0.6 TABLE LXVII. 89 Moon's Horary Motion in Longitude. Arguments. 1 to 18 of Longitude. Arg. 2 3 4 5 6 1 7 8 9 Arg. 0 5.0 0.0 2.9 1.9 0.0 0.00 0.00 0.00 0.16 100 2 5.0 0.0 2.8 1.9 0.0 0.00 0.00 0.00 0.15 98 4 4.9 0.0 2.8 1.9 0.0 0.01 0.00 0.02 0.15 96 6 4.8 0.1 2.8 1.9 0.1 0.03 0.01 0.05 0.14 94 8 4.7 0.2 2.7 1.8 0.1 0.06 0.01 0.09 0.12 92 10 4.5 0.3 2.6 1.7 0.2 0.09 0.02 0.14 0.10 90 12 4.3 0.4 2.5 1.7 0.2 0.13 0.02 0.19 0.09 88 14 4.1 0.6 2.3 1.6 0.3 0.18 j 0.03 0.26 0.07 86 16 3.8 0.7 2.2 1.5 0.4 0.23 10.04 0.33 0.05 84 18 3.6 0.9 2.0 1.4 0.5 0.28 0.05 0.41 0.03 82 20 3.3 1.1 1.9 1.3 0.6 0.34 0.06 0.50 0.02 80 22 3.0 1.3 1.7 1.1 0.7 0.40 0.07 0.58 0.01 78 24 2.7 1.5 1.5 1.0 0.8 0.46 0 08 0.67 0.00 76 26 2.3 1.7 1.3 0.9 0.9 0.52 0.10 0.77 O 00 74 28 2.0 1.9 1.2 0.8 1.0 0.58 0.11 0.86 0.00 72 30 1.7 2.1 1.0 0.7 1.1 0.63 0.12 0.94 0.01 70 32 1.4 2.2 0.8 0.5 1.2 0.69 0.13 1.03 0.01 68 34 1.2 2.4 0.7 0.4 1.3 0.74 0.14 1.11 0.03 66 36 0.9 2.6 0.5 0.3 1.3 0.78 0.15 1.18 0.05 64 38 0.7 2.7 0.4 0.3 1.4 0.82 0.16 1.25 0.06 62 40 0.5 2.8 0.3 0.2 1.5 0.86 0.16 1.30 0.08 60 42 0.3 2.9 0.2 0.1 1.5 0.89 0.17 1.35 0.10 58 44 0.2 30 0.1 0.1 1.6 0.91 0.17 1.39 0.11 56 46 0.1 3.1 0.0 0.0 1.6 0.93 0.18 1.42 0.12 54 48 0.0 3.1 0.0 0.0 1.6 0.94 0.18 1.44 0.13 52 50 0:0 3.1 0.0 0.0 1.6 10.94 0.18 1.44 0.13 50 Arg. 10 11 12 13 14 15 16 17 18 Aig. 0 0.00 0.26 0.00 0.00 0.00 0.00 0.26 0.00 0.21 100 2 0.00 0.25 0.00 0.00 0.00 0.00 0.26 0.00 0.20 98 4 0.02 0.24 0.01 0.00 0.01 0.00 0.16 0.00 0.20 96 6 0.04 0.22 0.03 0.01 O.C2 0.01 0.25 0.00 0.20 94 8 0.08 0.20 0.04 0.02 0.04 0.01 0.25 0.1O 0.20 92 10 0.12 0.17 0.07 0.03 0.06 0.02 0.24 0.01 0.20 90 12 0.16 0.14 0.09 0.04 0.09 0.0210.22 0.02 0.19 88 14 0.20 0.11 0.12 0.06 0.12 0.03;0.21 0.02 0.19 86 16 0.24 0.08 0.16 0.07 0.15 0.0410.20 0.03 0.18 84 18 0.28 0.05 0.19 0.09 0.19 0.05 0.19 0.04 0.18 82 20 0.31 0.03 0.23 0.11 0.22 0.06 0.17 0.05 0.17 80 22 0.34 0.01 0.27 0.13 10.26 0.07 10.15 0.06 10.17 78 24 0.35 0.00 0.31 0.15 0.30 0.08 0.14 0.07 0.16 76 26 0.36 0.00 0'35 0.17 0.34 0.08 0.12 0.07 0.16 74 28 0.35 0.01 0.39 0.19 0.38 0.09 0.11 0.08 0.15 72 30 0.34 0.02 0.43 0.21 0.42 0.10 0.09 0.09 0.15 70 32 0.32 0.04 0.47 0.23 0.45 0.11 0.07 0.10 0.14 68 34 0.29 0.06 0.50 0.25 0.49 0.12 0.0610.11 0.14 66 36 0.26 0.09 0.54 0.26 0.52 0.13 0.05 0.12 0.13 64 38 0.22 0.11 0.57 0.28 0.55 0.14 0.04 0.12 0.13 62 40 0.18 0.14 0.59 0.29 0.58 0.14 0.02 0.13 0.12 60 42 0.15 0.16 0.62 0.30 0.60 0.15 0.01 0.13 0.12 58 44 0.12 0.19 0.63 0.31 0.62 0.15 0.01 0.14 0]12 56 46 0.10 0.21 0.65 0.32 0.63 0.16 0.00 0.14 0.12 541 48 0.09 0.22 0.66 0.32!0.64 0.16 0.00 0.141012 52 50 0.08 0.22 0.66 0.32 0.64 0.16 0.00 10.14i0.11 50 TABLE LXVIII. Moon's Horary Alotion in Longitude. Argument. Argument of the Evection. | Os Is Is IIIs IVs Vs 0 80.3 74.7 59.6 39.4 19.8. c 30 1 80.3 74.3 58.9 38.7 19.3 5.6 29 2 80.3 73.9 58.3 38.0 18.7 5.3 28 3 80.2 73.5 57.7 -37.3 18.1 5.0 27 4 80.2 73.1 57.1 36.6 17.6 4.7 26 5 80.1 72.7 56.4 36.0 17.0 4.4 25 6 80.1 72.3 55.8 35.3 16.5 4.1 24 7 80.0 71.9 55.1 34.6 15.9 3.8 23 8 79.9 71.4 54.5 33.9 15.4 3.6 22 9 79.8 71.0 53.8 33.2 14.9 3.4 21 10 79.7 70.5 53.1 32.5 14.4 3.1 20 11 79.5 70.1 52.5 31.9 13.9 2.9 19 12 79.4 69.6 51.8 31.2 13.4 2.7 18 13 79.2 69.1 51.1 30.5 12.9 2.5 17 14 79.1 68.6 50.5 29.9 12.4 2.3 16 15 78.9 68.1 49.8 29.2 11.9 2.1 15 16 78.7 67.6 49.1 28.6 11.4 2.0 14 17 78.5 67.0 48.4 27.9 11.0 1.8 13 18 78.2 66.5 47.7 27.2 10.5 1.7 12 19 78.0 66.0 47.0 26.6 10.1 1.6 11 20 77.8 65.4 46.4 26.0 9.7 1.4 10 21 77.5 64.9 45.7 25.3 9.3 1.3 9 22 77.2 64.3 45.0 24.7 8.8 1.2 8 23 77.0 63.7 44.3 24.1 8.4 1.2 7 24 76;.7 63.2 43.6 23.5 8.0 1.1 6 25 76.4 62.6 42.9 22.8 7.7 1.0 5 26 76.1 62.0 42.2 22.2 7.3 1.0 4 27 75.7 61.4 41.5 21.6 6.9 0.9 3 28 75.4 60.8 40.8 21.0 6.6 0.9 2 29 75.0 60.2 40.1 20.4 6.2 0.9 1 30 74.7 59.6 39.4 19.8 5.9 0.9 0 XIs i Xs IXs VIIIs VIIs VIs TABLE LXIX. Moon's Horary Motion zn Longitude. krguments. Sum of Equations, 2, 3, &c., and Evection werrccted, o0" I 10"J 20" - s o'__-s 0 0 0 0 0 0.2 0.5 XII 0 I 0 0.0 0.2 0.4 XI 0 II 0 0.1 0.2 0.3 X 0 III 0 0.2 0.2 0.2 IX 0 IV 0 0.3 0.2 0.1 VIII 0 V 0 0.4 0.2 0.0 VIi 0 I 0 0.5 0.2 0.0 VI O -I Q" I 0"10" l 0"l TABLE LXX. 91 Moon's Horary Motion in Longitude. Arguments. Sumn of preceding equations, and Anomaly corrected of 0" 10" 20" 30/ 40" 50" 60" 70" 80, 90" 100" O 0 4.1 5.3 6.5 7.6 8.8 10.0 11.2 12.4 13.5 14.7 15.9 XII 0 5 4.1 5.3 6.5 7.7 8.8 10.0 11.2 12.3 13.5 14.7 15.9 25 10 4.2 5.4 6.5 7.7 8.8 10.0 11.2 12.3 13.5 14.6 15.8 20 15 4.3 5.5 6.6 7.7 8.9 10.0 11.1 12.3 13.4 14.5 15.7 15 20 4.5 5.6 6.7 7.8 8.9 10.0 11.1 12.2 13.3 14.4 15.5 10 25 4.8 5.8 6.9 7.9 9.0 10.0 11.0 12.1 13.1 14.2 15.2 5 I 0 5.1 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 14.9 XI 0 5 5.4 6.3 7.2 8.2 9.1 10.0 10.9 11.8 12.8 13.7 14.6 25 10 5.7 6.6 7.4 8.3 9.2 10.0 10.8 11.7 12.6 13.4 14.3 20 15 6.1 6.9 7.7 8.5 9.2 10.0 10.8 11.5 12.3 13.1 13.9 15 20 6.6 7.2 7.9 8.6 9.3 10.0 10.7 11.4 12.1 12.8 13.4 10 25 7.0 7.6 8.2 8.8 9.4 10.0 10.6 11.2 11.8 12.4 13.0} 5 II 0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0 12.5 X 0 5 7.9 8.4 8.8 9.2 9.6 10.0 10.4 10.8 11.2 11.6 12.1 25 10 8.4 8.7 9.1 9.4 9.7 10.0 10.3 10.6 10.9 11.3 11.6 20 15 8.9 9.1 9.4 9.6 9.8 10.0 10.2 10.4 10.6 10.9 11.1 15 20 9.4 9.5 9.7 9.8 9.9 10.0 10.1 10.2 10.3 10.5 10.6 10 25 9.9 9.9 9.9 10.0 10.0 10.0 10.0 10.0 10.1 10.1 10.1 5 III 0 10.4 10.3 10.2 10.1 10.1 10.0 9.9 9.9 9.8 9.7 9.6 IX 0 5 10.8 10.7 10.5 10.3 10.2 10.0 9.8 9.7 9.5 9.3 9.2,5 10 11.3 11.0 10.8 10.5 10.3 10.0 9.7 9.5 9.2 9.0 8.7 20 15 11.7 11.4 11.0 10.7 10.3 10.0 9.7 9.3 9.0 8.6 8.3 15 20 12.1 11.7 11.3 10.9 10.4 10.0 9.6 9.1 8.7 8.3 7.9 10 25 12.5 12.0 11.5 11.0 10.5 10.0 9.5 9.0 8.5 8.0 7.5 5 IV 0 12.9 12.3 11.7 11.2 10.6 10.0 9.4 8.8 8.3 7.7 7.1 VIII 0 5 13.3 12.6 11.9 11.3 10.6 10.0 9.4 8.7 8.1 7.4 6.7 25 10 13.6 12.9 12.1 11.4 10.7 10.0 9.3 8.6 7.9 7.1 6.4 20 15 13.9 13.1 12.3 11.5 10.8 10.0 9.2 8.5 7.7 6.9 6.1 15 20 14.1 13.3 12.5 11.6 10.8 10.0 9.2 8.4 7.5 6.7 5.9 10 25 14.4 13.5 12.6 11.7 10.9 10.0 9.1 8.3 7.4 6.5 5.6 5 V 0 14.6 13.7 12.7 11.8 10.9 10.0 9.1 8.2 7.3 6.3 5.4 VII 0 5 14.7 13.8 12.8 1.19 10.9 10.0 9.1 8.1 7.2 6.2 5.3 25 10 14.9 13.9 12.9 12.0 11.0 10.0 9.0 8.0 7.1 6.1 5.1 20 15 15.0 14.0 13.0 12.0 11.0 10.0 9.0 8.0 7.0 6.0 5.0 15 20 15.1 14.1 13.0 12.0 11.0 10.0 9.0 8.0 7.0 5.9 4.9 10 25 15.1 14.1 13.1 12.0 11.0 10.0 9.0 8.0 6.9 5.9 4.9 5 VI 0 15.1 14.1 13.1 12.1 11.0 10.0 9.0 8.0 6.9 5.9 4.9 VI 0 o" 10"' 20" 1 30" 1 40' 50" 60"' 70" 80" 90"' 100" 92 TABLE LXXI. IMoon's Horary Motzon in Longitude. Argument. Anomaly corrected. Os diff. Is diff. IIs Idiff IIIs diff. IVs diff. Vs diff. 0o _ _ _,., I 0 0 441.5 404.1 5 309.3 195.3 95.8 30.6 30 1 441.5 0. 401.6 2 305.6 7 191.6 7 93.0 8 29.2 4 29 0.1 2 4 37i 7 3.7 2.8 1.4 2 441.3 399.2 2'4 301.9' 187.9 90.2 2 27.8 28 3 441.1 0.2 396.6 6 1298.1 38 184.3 3.6 87.6 2o 26.4 1*3 27 4 440.8 0.3 394.0 2.6 294.4 3 8 180.6 3.6 84.9 2.6 25.1 1.3 26 5 440.4 0.4 391.3 2.7 290.6 3. 177.0 3. 82.3. 23.8. 25 0.5 2.7 3.8 3.6 2.6 1.2 6 439.9 388.6 286.8 173.4 79. 22.6 26 24 7 439 0.5 238.8 283.0 3.8 169. 77.1 2 21.4 1. 23 8 438.7 07 383.0 2.8 279.2 166.3 74.6 20.3 1.1 22 9 438.0 08 380.1 30 275.4 3.8 162.8 72.1 2.5 19.2 21 10 437.2 377.1 271.5 159.3 69.7 18.2 1.0 20 0.9 3.0 3.8 3.5 2.4 1.0 11 436.3 374.1 267.7 3 155.8 3 67.3 2 17.2 09 19 1.0 3.0. 3. 9 2.3 0.9 12 435.3 371.1 263.8 15.4 65.0 16.3 9 18 1.1 3.1 3.8 3.5 2.3 0.9 13 434.2 1.1 368.0 3.1 260.0 3.8 148.9 4 62.7 2 15.4 8 17 14 433.1 1.364.8.2 256.2 145.5 60.4 14.6 16 15 431.8 1.3 361.6 252.3 142.2 5s.2 13.8 15 1.3 3.2 3.8 3.3 2.1 0.7 16 430.5 1 28.4 248.5 138. 3 56.1 2 13.1 14 17 429.1 4 355.1 244.6 391135.6 53.9 12.4 13 15 4 35 3 38 3.3 2.0 0.6 18 427.6 1.5 351.8 3.3 240.8' 132.3 3.2 51.9 2 11.8 0.6 12 19 426.1 1.5 348.4 3.4 236.9 3 129.1. 49.8 11.2 11 20 42415 1.6 345.0 3.4 233.1 3.8 125.9 3.2 47.9 10.7 0.5 1.1.7 3.4 3.8 3.2 2.0 0.5 21 422.7 341.6 229.3 3.9 122.7 31 45.9 1 9 10.2 0A 9 22 421.0 338.1 225.4 119.6 44.0 9.8 8 23 419.1 1 334.6 221.6 38 16 3.51 42.2 1.8 9.4 3 24 417.2 9 331.1 217.8 113.4 1 40.4 1 9.1 03 2,5 415.2 327.5 214.0 3.8 110.4 38.7 8.8 5 2.1 3.5 3.7 3.0 1.7 0.2 26 413.1 324.0 210.33 8 1107.42 370 7 86 02 4 27 410.9 320.3 31206.5 I104.5 35.3 8.4 " 28 408.7 2.2 316.7 202.8 3.7 1 1016 29 133. 6 8.1 2.3 3.7 2.9 1 6 8. 2 29 406.4 2.3 313.0 37199.0 3.78 98.7 32.931 8.20. 1 30 404.1 2.3 309.3 195.3 95.8 30.6 8.2 0 Xs l I s IX VIIIs IIs I IS VIs.~~~~~~~~~~~~ __ i TABLE LXXII. 93 Moon's Horary Mlotion in Longitude. Arguments. Sum of preceding Equations, and Arg. of Variation. 0 50 100 150 200 250 300 350 400 450 500 550 600 0 0 4.5 5.5 6.5 7.6 8.6 9.6 10.6 11.6 12.6 13.7 14.7 15.7116.7 XII 0 5 4.6 5.6 6.6 7.6 8.6 9.6 10.6 11.6j12.6 13.6 14.6 15.6 16.61 25 10 4.8 5.8 6.8 7.7 8.7 9.6 10.6 11.5 12.5 13.4 14.4 15.3 16.3 20 151 5.3 6.1 7.0 7.9 8.8 9.7 10.55 11.4 12.3 13.1 14.0 14.9 15.81 15 20 5.8 6.6 7.4 8.2 8.9 9.7 10.5 11.2 12.0 12.8 13.5 14.3 15.1 10 25 6.6 7.2 7.8 8.5 9.1 9.7 10.4 11.0 11.7 12.3 12.9 13.fi6 14.2 5. 0 o 7.4 7.8 8.3 8.8 9.3 9.8 10.3 10.8 11.3 11.8 12.3 12.7 13.2 XI o 5 8.3 8.6 8.9 9.2 9.5 9.9 10.2 10.5 10.8 1 1.2 11.5 111.8 12.1 25 10 9.2 9.3 9.5 9.6 9.8 9.9 10.1 10.2 10.4 10.5 10.7 10.8 11.0 20 1.5 10.2 10.1 10.1 10.1 10.0 10.0 10.0 10.0 9.9 9.9 9.9 9.8 9.8 15 20 11.1 10.9 10.7 10.5 10.3 10.1 9.9 9.7 9.5 9[2 9.0 8.8 8.6 1i5 25 12.1 11.7 11.3 10.9 10.5 10.2 9.8 9.4 9.0 8.6 8.3 7.9 7.5 5, If 0 12.9 12.4 11.8 11.3 10.8 10.2 9.7 9.1 8.6 8.1 7.5 7.0 6.4 X 0 5 13.7 13.0 12.3 11.6 11.0 10.3 9.6 8.9 8.2 7.5 6.9 6.2 5.5 25 10 14.3 13.5 12.7 11.9 11.1 10.3 9.5 8.7 7.9 7.1 6.3 5.5 4.7 20 15 14.9 14.0 13.1 12.2 11.3 10.4 9.5 8.6 7.7 6.8 5.8 4.9 4.0 15 20115.3 14.3 13.3 12.3 111.4 10.4 9.4 8.4 7.5 6.5 5.5 4.51 3.6 101 25 15.5 14.5 13.5 12.4 11.4 10.4 9.4 8.4 7.4 6.3 5.3 4.3 3.3 5 III 0 15.6 |14.5 13.5 12.5 11.4 10.4 9.4 8.4 7.3 6.3 5.3 4.2 3.2 IX 0 5115.4 14.4 13.4 12.4 11.4 10.4 9.4 8.4 7.4 6.4 5.4 4.4 3.3 25 10 15.2 14.2 13.3 12.3 11.3 10.4 9.4 8.5 7.5 6.5 5.61 4.6 3.6 20 15 14.8 13.9 13.0 12.1 11.2 10.4 9.5 8.6 7.7 6.8 5.9 5.1 4.21 15 20 14.2113.4 12.6 11.9 11.1 10.3 9.5 8.8 8.0 7.2 6.4 5.6 4.9 101 25 13.5 12.9 12.2 11.6 10.9 10.3 9.6 9.0 8.4 7.6 7.0 6.3 5.7 5 IV 0 12.7 12.2 11.7 11.2 10.7 10.2 9.7 9.2 8.7 8.2 7.7 7.2 6.71 Vfi10 511.9 11.5 11.2 10.8 10.5 10.1'9.8 9.5 9.1 8.8 8.4 8.1 7.7 25 10.10.9 10.7 10.6 10.4 10.2 10.1 9.9 9.7 9.6 9.4 9.2 9.1 8.9 20 15 9.9 9.9 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.1. 10.1 10. 1 15 20 8.9 9.1 9.3 9.5 9.7. 9.9 10.1 10.3 10.5110.7 10.9 11.1 11.3 10 25 8.0 8.4 8.7 9.1 9.5 9.9 10.2 10.6 11.0 11.3 11.7 12.1 12.T- 51 V 0 7.1 7.6 8.2 8.7 9.2 9.8 10.3 10.9 11.4 11.9 12.5 13.0 13.6 VII 0 5 6.3 7.0 7.6 8.3 9.0 9.7 10.4 11.1 11.8 12.5 13.2 13.9 14.6' 25 10 5.6 6.4 7.2 8.0 8.8 9.7 10.5 11.3,12.1 13.0 13.8 14.6 15.4 201 15 5.0 5.9 6.8 7.8 8.7 9.6 10.6 i11.5 12.4 13.3 14.3115.2 16.1 151 20 4.6 5.6 6.6 7.6 8.6 9.6 110.6 i11.6112.6 13.6 14.6 15.7 16.7 10 25 4.3 5.4 6.4 7.5 8.5 9.6 10.611.7 12.713.8 14.9 15.9 17.0 5 VI 0 4.2 5.3 6.4 7.4 8.5 9.6 10.6 11.7 12.8 13.9 14.9 16.0 17.1 VI ~| 0 50 100 150 200250 1300 350 400 450 500 550 600 TABLE LXXIII. Moon's Horary Motion in Longl:ude. Argument. Argument of the Variation. Os IIs IIIs IVs Vs 0 — o. -;- 1 —— I- - 0 77.2 57.8 20.3 2.4 21.5 59.7 30 1 77.2 56.7 19.2 2.5 22.7 60.9 29 2 77.1 55.5 18.1 2.6 23.8 62.0 28 3 77.0 54.3 17.0 2.7 25.0 63.1 27 4 76.8 53.1 16.0 2.9 26.2 64.2 26 5 76.6 51.8 15.0 3.1 27.5 65.3 25 6 76.4 50.5 14.1 3.3 28.7 66.3 24 7 76.1 49.3 13.2 3.7 30.0 67.3 23 8 75.7 48.0 12.3 4.0 31.3 68.3 22 9 75.3 46.7 11.4 4.4 32.6 69.2 21 10 74.9 45.4 10.6 4.9 33.9 70.1 20 11 74.4 44.1 9.8 5.3 35.2 70.9 19 12 73.9 42.8 9.0 5.9 36.5 71.7 18 13 73.3 41.5 8.3 6.4 37.8 72.5 17 14 72.7 40.2 7.6 7.0 39.2 73.3 16 15 72.0 38.9 7.0 7.7 40.5 74.0 15 16 71.3 37.5 6.4 8.3 41.8 74.7 14 17 70.6 36.2 5.8 9.1 43.2 75.3 13 18 69.8 34.9 5i.3 9.8 44.5 75.8 12 19 69.0 33.6 4.8 10.6 45.8 76.4 11 20 68.1 32.3 4.4 11.5 47.2 76.9 10 21 67.2 31.1 4.0 12.3 48.5 77.3 9 22 66.3 29.8 3.7 13.2 49.8 77.7 8 23 65.3 28.6 3.3 14.2 51.1 78.1 7 24 64.4 27.3 3.1 15. l1 52.4 78.4 6 25 63.4 26.1 2.9 16.1 53.6 78.6 5 26 62.3 24.9 2.7 17.1 54.9 78.9 4 27 61.2 23.7 2.5 18.2 56.1 79.0 3 28 60.1 22.5 2.5 19.3 57.3 79.2 2 29 59.0 21.4 2.4 20.4 58.5 79.2 1 30 57.8 20.3 2.4 21.5 59.7 79.2 0 - XIS Xs _8vI VIhIs VPIs\_s xI~ —-"~ —-,!~/ —vr-m-~ {~~~.m { v.,! — TABLE LXXIV. 95 Mlloon's Horary Motion in Longitude. Arguments. Arg. ot Reduction and Sum of preceding Equations 0 I 50 I 0 15C 200 2.50 300 3.50 400,450 500 550 6001650 -0 - ]Q I t l / t [. 0 2 i.1 3.9 12.7 2.5 2.3.1 1.9 1.7 1.511.3 1.1 0.910.7 [II 0 i 3.3 3. 2. 2.5 2.3 2.1 1.9 11.7 1.5 1.3 l.l 0.9 0.7 lO 13.2 3.0 2.8 2.6 2.4 2.3 2.1 1.9 1.711.5 1.3 1.1 1.0 0.8 20 15 13.1 2.9 2.8 2.6 2.4 2.2 2.1 1.9 1.7 1.5 1.4 1.211.010.9 15 20 3.0 2_8 2.7 2.5 2.4 2.2 2.1 1.9 1.8 1.6 1.5 1.3 1.1 1.0 O10 25 2.8 2.7 2.6 2.4 2.3 2.2 2.1 1.9 1.8 1.7 1. 1.4 1.3 1.2 5 I 0 2.6 2.5 2.4 2.3 2.1 2.0 2.0. 1 1.9 1. 1.8.7.41.3 X1 0 5 2.4 2.4 2.3 2.2 2.2 2.1 2.0 2.0 1.9 1.8 1.8 1.7 1.6 1.6 25 1 2.2 2.2 2.2 2.1 2.1 2.0 2.0 12.0 1.9 1.9 1.9 1.8 1.8 l.8 20 15 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0.0 2.0 2.0 2.0 2.0 2.0 15 20 1.8 1.8 1.8 1.9 1.9 1.9 2.0 2.0 2.1 2.1'2.1 2.2 2.2 2.2 10 25 1.6 1.6 1.7 1.8 1.8 1.9 2.0 2.0 2.1 2.2 2.2 2.3 2.4 2.4 5 II 0 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 X 0 5 1.2 1.3 1.4 1.6 1.7 1.8 1.9 2.1 2.2 2.3 2.5 2.6 2.7 2.8 25 10 1.0 1.2 1.3 1.5 1.6 1.8 1.9 2.1 2.2 2.4 2.5 2.7 2.9 3.0 20 15 0.9 1.1 1.2 1.4 1.6 1.8 1.9 2.1 2.3 2.5 2.6 2.8 3.0 3.1 15 20 0.8 1.0 1.2 1.4 1.6 1.7 1.9 2.1 2.3 %.5 2.7 2.9 3.0 3.2 10 25 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9 3.1 3.3 5 III 0 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9 3.1 3.3 IX 0 5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9 3.1 3.3 25 10 0.8 1.0 1.2 1.4 1.6 1.7 1.9 2.1 2.3 2.5 2.7 2.9 3.0 13.2 20 15 0.9 1.1 1.2 1.4 1.6 1.8 1.9 2.1 2.3 2.5 2.6 2.8 3.0 3.1 15 |IV 0 114 1.5 1.6 1.7 1.8 1.9 2.0 2.1 12.2 213 214 2.5 2.6 2.7 VIll 0 5 1.6 1.6 1.7 1.8 1.8 1.9 2.0 2.0 2.1 2.2 2.2 2.3 2.4 2.4 25 10 1.8 1.8 1.8 1.9 1.9 1.9 2.0 2.0 2.1 2.1 2.1 2.2 12.2 2.2 20 15 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 1 2.0 2.0 2.0 15 20 2.2 2.2 2.2 2.1 2.1 2.0 2.0 2.0 1.9 1.9 1.9 1.8 1.8 1.8 10 25 2.4 2.4 2.3 2.2 2.2 2.1 2.0 2.0 1.9 1.8 1.8 1.7 1.6 1.6 5 V 0 2.6 2.5 2.4 2.3 2.2 2.1 2.0 1.9 1.8 1.7 1.6 1.5 1.4 11.3 Vii 0 1.2 i 5 3.1 2.9 2.8 2.6 2.4 2.2 2.1 1.9 1.7 1.5 1.4 1.42 1.0.9 25 20 3.2 3.0 2.8 2.6 2.4 2.3 2.1 1.9 1.7 1. 1.3 1.1 1.0 0.8 10 25 3.3 3.1 2.9 2.7 2. 2.3 2.1 1.9 1.7 1.5 j.3 1.1 0.9 0.7 5 I 0 3.3 3.1 2.9 2.7 2.5 2.3 2.1 1.9 1.7 1.5 11.3 1.1 0.9 0.7 VI 0 0 50 1100 1501200.2501300j350J400 450,500 550 6i00 650 96 TABLE LXXV. TABLE LXXVI. Moon's Horryorary Motion in Long. I'Ioon's Horary Motion in Loang. (Equation of the second order.) Arg. Arg. of Reduction. Arguments. Arg's.of Table LXX Os VIs Is V Ls Is VlIs _- - _ Arg. 0 50 100 o,,,,,, o _.. I 0 2.1 6.0 14.0 30 [r ~ 1 2.1 6.3 14.2 29 O 0 0.05 0.05 0.05 2 2.1 6.5 14.4 28 I 0 0.08 0.05 0.02 3 2.1 6.8 14.7 27 II 0 0.10 0.05 0.00 4 2.2 7.0 14.9 26 III 0 0.10 0.05 0.00 5 2.2 7.3 15.1 25 IV 0 0.09 0.05 0.01 6 ~2.2 7.5 24 V 0 0.07 0.05 0.03 6 } 2.2 7 7.5 15.3 24 7 2.3 7.8 [ 15.5 23 VI 0 0.05 0.05 0.05 8 2.4 8.1 15.7 22 VII 0 0.03 0.05 0.07 9 2 5 8.4 15.9 21 VIII 0 0.01 0.05 0.09 10 2.5 8.6 16.1 20 IX 0 0.00 0.05 0.10 X 0 0.00 0.05 0.10 11 2.6 8.9 16.2 19 XI 0 0.02 0.05 0.08 12 2.7 9.2 16.4 18 XI 0 0.05 0.05 0.05 13 2.9 9.4 16.6 17 _ _-. o.. 14 3.0 9.7 16.7 16 16.. 15 3.1 10.0 16.9 15 0 50 100 16 3.3 10.3 17.0 14 17 3.4 10.6 17.1 1 3 18 3.6 10.8 17.3 12 19 3.8 11.1 17.4 11 20 3.9 11.4 17.5 10 21 4.1 11.6 17.5 9 22 4.3 11.9 17.6 8 23 4.5 12.2 17.7 7 T4 4.7 12.5 17.8 6 25 4.9 12.7 17.8 5 26 5.1 13.0 17.8 4 27 5.3 13.2 17.9 3 28 5.6 13.5 17.9 2 29 5.8 13.7 17.9 1 30 6.0 14.0 17.9 O s VsX s I Vsj IXs Ills ionstant to be added 27'24".0. TABLE LXXVII. Moon's Horary Motion in Longitude. (Equations of the second order.) Arguments. Arguments of Tables LXXII and LXXIV. Variation. Reduction. 0 100 200 300 400 500 600 0 600 O. VI. 0 0.14 0.14 0.14 0.14 0.14 0.14 0 14 0 03 0.03 I. VII. 0 0.22 0.19 0.16 0.13 0.10 0.06 0.02 0.01 0.05 I. VII. 15 0.23 0.20 0.17 0.13 0.10 0.05 0.01 0.01 0.06 II. VIII. 0 0.22 0.19 0.16 0.1,3 0.10 0.07 0.03 0.01 0.05 III. IX. 0 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.03 0.03 IV. X. 0 0.06 0.09 0.12 0.15 0.18 0.21 0.26 0.05 0 01 IV. X. 15 0.05 0.08 0.11 0.15 10.18 10.23 0.28 0.05 0.00 V. XI. 0 0.06 0.09 0.12 0.15 10.18 6.22 0.26 0.05 0.01 VI. XII. 0 0.14 0.14 ]0.14 [0.14 0.14 C,.14 0.14 0.03 0.0:3 _~~~~~~~~.o [ TABLE LXXVIII. 97 Moon's Horary Motion in Longitude (Equations of the second order.) Arguments. Args. of Evection, Anomaly, Variation, Reduction. Evec. Anom. Var. Red. {Evec. Anom.j Var. Red. I _o. _ _ I l. 0 0 0.16 1.05 0.34 0.08 0.16 1.05 0.34 0.08 XII 0 5 0.15 0.93 0.28 0.09 0.18 1.17 0.40 0.06 25 10 0.13 0.81 0.22 0.10 0.19 1.28 0.46 0.05 20 15 0.12 0.70 0.17 0.11 0.21 1.40 0.51 0.04 15 20 0.10 0.59 0.12 0.12 0.22 1.50 0.56 0.03 10 25 0.09 0.49 0.08 0.13 0.24 1.60 0.60 0.02 5 0 0.08 0.40 0.05 0.14 0.25 1.70 0.63 0.01 XI 0 5 0.07 0.31 0.02 0.15 0.26 1.78 0.66 0.01 25 10 0.05 0.24 0.01 0.15 0.27 1.86 0.67 0.00 20 15 0.04 0.17 0.01 0.15 0.28 1.92 0.67 0.00 15 20 0.03 0.12 0.01 0.15 0.29 1.98 0.67 0.00 10 25 0.03 0.07 0.03 0.15 0.30 2.02 0.65 0.01 5 II 0 0.02 0.04 0.06 0.14 0.31 2.05 0.62 0.01 X 0 5 0.01 0.02 0.09 0.13 0.32 2.08 0.59 0.02 25 10 0.01 0.00 0.13 0.12 0.32 2.09 0.54 0.03 20 15 0.00 0.00 0.18 0.11 0.32 2.10 0.50 0.04 15 20 0.00 0.00 0.24 0.10 0.33 2.09 0.44 0.05 10 25 0.00 0.02 0.29 0.09 0.33 2.08 0.39 0.06 5 III 0 0.00 0.04 0.35 0.08 0.33 2.06 0.33 0.08 IX 0 5 0.00 0.07 0.40 0.06 0.33 2.03 0.27 0.09 25 10 0.01 0.10 0.46 0.05 0.32 2.00 0.22 0.1o 1 20 15 0.01 0.14 0.51 0.04 0.32 1.96 0.17 0.11 15 20 0'01 0.18 0.56 0.03 0.31 1.91 0.12 0.12 10 25 0.02 0.23 0.60 0.02 0.31 1.87 0.08 0.13 5 IV 0 0.03 0.28 0.63 0.01 0.30 1.82 0.05 0.14 VIII 0 5 0.03 0.34 0.66 0.01 0.29 1.76 0.02 0.15 25 10 0.04 0.39 0.67 0.00 0.28 1.70 0.01 0.15 20 15 0.05 0.45 0.68 j 0.00 0.27 1.64 0.00 0.15 15 20 0.06 0.52 0.67 0.00 0.26 1.58 0.00 0.15 10 25 0.08 0.58 0.66 0.01,0.25 1.52 0.02 0.15 5 V 0 0.09 0.64 0.64 0.01 0.24 1.45 0.04 0.14 VII 0 5 0.10 0.71 0.60 0.02 0.23 1.39 0.08 0.13 25 10 0.11 0.78 0.56 0.03 0.22 1.32 0.12 0.12 20 15 0.12 0.84 0.51 0.04 0.20 1.25 0.16 0.11 1 2O 0.14 0.91 0.46 0.05 0.19 1.18 0.22 0.10 10 25 0.15 0.98 0.40 0.06 0.18 1.12 0.28 0.09 5 VI 0 0.16 1.05 0.34 0.08 0.16 1.05 0.34 0.08 VI 0 98 TABLE LXXIX. Moon's Horary Motion in Latitude. Argument. Arg. I of Latitude. Is 1Is IIs' IV6 v -d 0 378.0 354.3 289.2 200.0 110.8 45.7 30 1 378.0 352.7 286.5 196.9 108.1 44.2 29 2 377.9 351.1 283.8 193.8 105.4 42.7 28 3 377 8 349.4 281.0 190.7 102.8 41.3 27 4 377.6 347.7 278.3 187.5 100.2 39.9 26 5 377.3 346.0 275.5 184.4 97.7 38.6 25 6 377.0 344.2 272.6 181.3 95.1 37.3 24 7 376.7 342.3 269.8 178.2 92.6 36.1 23 S 376.3 340.5 266.9 175.1 90.2 34.9 22 9 375.8 338.5 264.0 172.1 87.7 33.8 21 10 375.3 336.6 261.1 169.0 85.3 32.7 20 11 374.7 334.5 258.1 165.9 83.0 31.6 19 12 374.1 332.5 255.2 162.9 80.7 30.7 18 13 373.5 330.4 252.2 159.8 78.1 29.7 17 14 372.7 328.3 249.2 156.8 76.1 28.9 16 15 372.0 326.1 246.2 153.8 73.9 28.0 15 16 371.1 323.9 243.2 150.8 71.7 27.3 14 17 370.3 321.9 240.2 147.8 69.6 26.5 13 18 369.3 319.3 237.1 144.8 67.5 25.9 12 19 368.4 317.0 234.1 141.9 65.5 25.3 11 20 367.3 314.7 231.0 138.9 63.4 24.7 10 21 366.2 312.3 227.9 136.0 61.5 24.2 9 22 365.1 309.8 224.9 133.1 59.5 23.7 8 23 363.9 307.4 221.8 130.2 57.7 23.3 7 24 362.7 304.9 218.7 127.4 55.8 23.0 6 25 361.4 302.3 215.6 124.5 54.0 22.7 5 26 360.1 299.8 212.5 121.7 52.3 22.4 4 27 358.7 297.2 209.3 119.( 50.6 22.2 3 28 357.3 294.6 206.2 116.2 48.9 22.1 2 29 355.8 291.9 203.1 113.5 47.3 22.0 1 30 354.3 289.2 200.0 110.8 45.7 22.0 0 XIs Xs IXs VIIIs VIIs VIs TABLE LXXX. Moon's Horary Motion in Latitude. Arguments. Args. V, VI, VII, VIII, IX, X, XI, and XII, of Latitude. Arg| V VI VI VIVII IX X XI XII Arg. 0 0.00 0.50 0.34 0.00 0.50 0.04 0.12 0.08 1000 50 0.01 0.49 0.33 0.00 0.49 0.04 0.12 0.07 950 100 0.04 0.45 0.30 0.02 0.45 0.04 0.11'0.05 900 150 0.09 0.401 0.27 0.04 0.40 0.03 0.10 0.03 850 200 0.16 0.33 0.22 0.061 0.33 0.03 0.08 0.01 800 250 0.23 0.25 0.17 0.09 0.251 0.02 0.06 0.00 750 300 0.30 0 17 0.12 0.12 0.17] 0.01 0.04 0.01 700 350 0.37 0.10 0.07 0.14 0.10 0.01 0.02 0.03 650 400 0.42 0.05 0.04 0.16 0.05 0.00 0.01'0.05 600 450 0.45 0.01 0.01 0.18 0.01 0.00 0.000.07 550 500 0.46 0.00' 0,00 0.18, 0.00 0.0010.00!0.08 500 --— I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ TABLE LJXXXI. Moon's Horary Alotion in Latitude. 99 Arguments. Precedirlg equation, and Sum c,f equations of Horary Motion in Lorngitude, except the last two. 0") " 50" 100" 150" 200" 250" 300" 350" 400".50o" 500' 0"550"'600"1650"i Pr.' _- _- --- --- --- --—: —- --—.. eq. 1".6 1".4 1."1 0'.9 0".6 0".4 0".1 0".2 0". 0".7 0".9 ".2 1 ".41 1".7 D11. 2059.0 54.5 50.0 45.4 40.9 36.4 31.8 27.3122.8 18.2 13.7 9.1 4.6 0.1 4.5 30 57.4 53.1 48.9 44.6 40.3 36.0 3,1.7 27.4 23.2 18.9 14.6 10.3 6.0 1.7 4.3 40 55.8151.8 47.7 43.7 39.7 35.6 31.6 27.6 23.6 19.5 15.5 11.5 7.4 3..r 4.0 50154.2 50.4 46.6 42.9 39.1 35.3 31.5 27.7 24.0 20.2 16.4 12.6 8.8, 5.1 3.8 60 52.6149.1 45.5 42.0 38.5 34.9 31.4 27.9 24.4 20.8 17.3 13.8 10.21 6.7 3.5 70 51.0 47.7 44.4 41.1 37.9 34.6 31.3 28.0 24.8 21.5 18.2 14.9 11.7 8.4 3.3 80 49.3.46.3 43.3 40.3 37.3 34.2 31.2 28.2 25.2 22.1 19.1 16i.l 13.1 10.0 3.0 90 47.7 45.0 42.2 39.41 36.7 33.9 31.1 28.3 25.6 22.8 20.0 17.3 14.5 1i.7 2.8 100 46.1 143.6 41.1 38.6 36.0 33.5 31.0 28.5 26.0 23.4 20.9 18.4 15.9 13.4 2.5 110 44.5 42.2'40.0 37.7 35.4 33.2 30.9 28.6 26.4 24.1 21.8 19.6 17.31 15.0 2.3 120 42.9 40.91 38.9 36.9 34.8 32.8 30.8 28.8 26.8 21.8 22.7 20.7 18.71 16.7 2.0 130 41.3i39.5 37.8 36.0 34.2 32.5 30.7 28.9 27.2 25.4 23.7[ 21.9 20.11 18.4 1.-8 140139.7 38.2 36.7 35.1 33.6 32.1 30.6 29.1 27.6 26.1 24.6 23.0 21.5 20.0 1.5 150 38.1 36.8 35.51 34.3/ 33.0 31.8 30.5 29.2 28.0 26.7 25.5 24.2 23.01 21.7 1.3 160 36.5 35.4 34.4 33.4 32.4 31.4 30.41 29.4 28.4 27.4 26.41 25.41 24.4 23.31 1.0 170 34.8 34.1 33.3 32.6 31.8 31.1 30.31 29.5 28.8 28.0 27.31 26.5 25.8i 25.01 0.8 180133.2 32.7 32.2 31.7 31.2 30.7 30.21 29.7 29.2 28.7 28.21 27.7 27.21 26.7 0.5 190 31.6 31.4 31.1 30.9 30.6 30.4 30.11 29.8 29.6 29.3 29.1 28.81 28.6 28.3 0.3 200 30 0130.0 30.0 30.01 30.0 30.0 30.0 30.0 30.01 0.0 30.0 30.0 30.01 30.01 0.0 210o28.4128.6 28.9| 29.1 29.4 29.6 29.9 30.2 3(.4.30.7 30.9 31.2 31.41 31.7 0.3 220126.8 27.3 27.8S 28.3 28.8 29.3 29.8 30.3 30.8 31.3 31.81 32.3 32.8 ~33.3 0.5 230125.2 25.91 26.71 27.41 28.2 28.9 29.7 30.5 31.2 32.0 32.71 33.51 34.2 35.010.8 240 23.5 24.6 25.61 26.67 27.6 28.6 29.6 30.6 31.6 32.6 33.6 34.6 35.61 36.7 1.0 250 21.9 23.2i 24.5 25.7 27.0 28.2 29.5 30.8 32.0 33.3 34.5 35.8 37.1 38.3 1.3 260120.3 21.81 23.3 24.91 26.4 27.9 29.41 30.9 32.4 33.9 35.4 37.01 38.51 40.0 1.5 270 18.7120.5fi1 22.2 24.0 25.8 27.5 29.81 31.1 32.8 34.6 36.3 38.11 39.9 41.6 1.8 280 17.1 19.11 21.1 23.1 25.2 27.2 29.2 31.2 33.2 35.2 37.3 39.3 41.3: 43.3 2.0 290 15.5 17.8 20.0o 22.3 24.6 26.8 29.1 31.4 33.61 35.9 38.2 40.41 42.7 45. 1 2.3 300 13.9 16.4 18.9 21.4 24.0 26..5 29.0 31.5 34.01 36.6 39.1 41.61 44.1 46.6 2.5 310 12.3115.0 17.8 20.6 23.3 26.1 28.9 31.7 34.4 37.2 40.0 42.7 45.5 48.31 2. 320 10.7 13.71 16.71 19.7 22.7 25.8 28.8 31.8 34.8 37.9 40.91 43.91 46.9 50.0 3.0 330 9.0112.3 15.6 18.9 22.1 25.41 28.7 32.0 35.2 38.5 41.8 45.11 48.31 51.61 3 1 340 7.4j 10.9 14.51 18.0 21.5 25.1 28.61 32.1! 35.61 39.2 42.71 46.2 49.81 53.3 35'3501 5.8 9.6 13.41 17.1 20.9 24.7 28.51 32.3 36.0' 39.8 43.61 47.4 51.2 54.9; 3 8 360 4.2 8.2 12.31 16.3 20.3 24.41 23.4 32.4; 36.41 40.5 44.51 48.51 52.61 56.6 40 370 2.6 6.9 11.11 15.4 19.7 24.01 28.3 32.61 36.8 41.1 45.41 49.71 54.01 58.31 4 3 380 1.0 5.5 10.01 14.61 19.1 23.6 28.2 32.71 37.2 41.8 46.3 50.9 55.41 59.91 4..5 I —; _.4 - 0" 50"1)o111i,150Tj200 250",300" 3F40''1400" 1450" 50 o0 550"ir 600 65 0 TABLE LXXXII. Moon's Horary lnotion in Latitude. Argument. A rg I11. of Latitude. Os I s Ils IlIls IUs Vs j 0 9.3 871 7.1 5.0 2.9 1.3 30 3 9.3 8.6 6.9 4.8 2.7 1.2 127 1 6 19.2 8.5 1 6.7 4.6 2.5 1.1 24 9 9.2 8.3 6.5 4.3 2.3 1.0 21 12 9.2 8.2 6.3 4.1 2.1 0.9 18 15 9.1 8.0 6.1 3.9 2.0 0.9 15 18 9. 1 7.9 5.9 3.7 1.8 0.8 1 12 21 9.0 7.7 5.7 3.5 1.7 0.8 9 24 8.9 7.5 5.4 3.3 1.5 0.8 6 27 F 8 7.3 5.2 3.1 1.4 0.7 3 30 97 7. 1 5.0 2.9 1.3 0.7 0 XIs i X I Xs VI IIIsVi i[ Vls 100 TABLT LXXXIII. TABLE LXXXIV. Moon's Horary Motion in Latitude. Moon's Hor. NIotion in Lai Arguments. Preceding equation, and Sum (Equa. of second order.) of equations of Horary Motion in Longi- Argument. Arg. I of Lat. tude, except the last two. Prec. "' I equ. 0 100 200 300 400 500 600 700 -' —;Y,, I,,,,:I,,,,,, I,,,,, O 0 0.90 0.90 XII 0 0 2.1 1.8 1.5 1.2 0.9 0.6 0.3 0.0 5 0.83 0.97 25 1 1.9 1.6 1.4 1.1 0.9 0.7 0.4 0.2 10 0.75 1.05 20 2 1.7 1.5 1.3 1.1 1.0 0.8 0.6 0.3 15 0.68 1.12 15 3 1.5 1.4 1.2 1.1 1.0 0.9 0.8 0.6 20 0.61 1.19 10 4 1.3 1.2 1.2 1.1 1.1 1.0 0.9 0.9 25 0.54 11.26 5 5 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 I 0 0.47 1.33 XI 0 6 0.9 1.0 1.0 1.1 1.1 1.2 1.3 1.3 5 0.41 1.39 25 7 0.7 0.8 1.0 1.1 1.2 1.3 1.4 1.6 10 0.35 1.45 20 8 0.5 0.7 0.9 1.1 1.2 1.4 1.6 1.9 15 0.29 1.51 15 9 0.3 0.6 0.8 1.1 1.3 1.5 1.8 2.0 20 0.24 1.56 10 10 0.1 0.4 0.7 1.0 1.3 1.6 1.9 2.2 25 0.20 1.60 5 _ | t -| -,,-, II 0 0.16 1.64,X 0 0 100 200 300 400 500 1600 700 5 0.12 1.68 25 - o bst10 0.09 1.71 20 Constant to be subtracted 237".2. 15 0.07 1.73 15 TABLE LXXXV. 20 0.05 1.756 10 Moon's Horary Motion in Latitude. III 0 0.04 1.76 IX (Equations of second order.) 5 0.04 1.76 25 Arguments. Preceding equation, and Sum 10 10.05 11.75 20 of equations of Horary M.otion in Longi- 15 0.07 1.73 15 tude, except the last two. 20.0 09 1.71 10 25 iO. 12 1.68 5 Prec, I IV 0 10.16 1.64 VIII 0 equ. 0 100 200 300 400 500 600 700 5 0.20 1 25 10'0.24 1.56 20 0.00 0.65 0.57 0.48 0.39 0.31 0.21 0.12 0.00 15 10.29 1 1.51 15 0.10 0.62 0.55 0.47 O. s9 0.31 0.23 0.15 10.04 20.35 145 10 0.20 0.69 0.53 0.46 10.39 10.32 /0.25 0.18 0.09 25 0.41 1.39 5 0.30 0.66 0.51 0.45 0.39 0.33 0.270.210.13 V 0 7 1.33 VII 0 0.40 0.63 0.48 0.44 0.39 0.34 0.29 0.24 0.17 5 0.54 11.26 25 0.50 0.50 0.46 0.43 10.38 0.35 0.30 0.27 0.21 10 0.61 20 I0'50'' 10 0.61 1.19 20 0.60 0.47 0.44 0.42 0.38 0.36 0.32 10.29 0.25 15 0.68 1.12 15 0.70 0.44 0.42 0.40 0.38 10.36 {0.34 [0.32 0.30 20 0.75 1.05 10 0.80 0.41 0.40 0.39 0.38 10.37 0.36 10.35 10.34 25'O.83 097 5 0.90 {0.38 0.38 0.38 10.38 10.38 0.38 10.38 0.38 VI 0 190 0190 Vi 1.00 10.35 0.36 0.37 0.38 0.39 10.40 10.41 0.42 1.10 10.32 10.34 A0.36 [0.38 10.40 0.42 10.44 0.46 1.20 0.29 o0.32!0.34 10.38 01.40 0.44 0.47 0.51 1.30 0.26 10.30 0.33 0.38 0.41 0.46 10.49 0.55 1.40 10.23 0.28 10.32 0.37 10.42 10.47 10.52 0.59 1.50 10.20 i0.25 10.31 0.37 0.43 10.49 10.55 10.63 1.60 {0.17 0.23 0.30 0.37 0.44 0.51 10.58 0.67 1.70 10.14 0.21 10.29 10.37 0.45 10.53 10.61'0.72 1.80 10.11 0.19 }0.28 0.37 0.45 {0.55 0.64 0.76 0 1)30 200 300 400 500 600 700t TABLE LXXXVI. 101 2lean New Moons and Arguments, in January. Mean New Years. AMoon hi I. II IV. N. January. d. h. r. 1836 B 17 10 32 0469 1246 1 7 08 669 1837 5 19 20 0171 9852 00 97 692 1838 24 16 53 0681 9175 99 85 799 1839 14 1 42 0383 7780 82 74 822 1840 B 3 10 30 0085 6386 65 63 844 1841 21 8 3 0595 5709 63 51 951 1842 10 16 51 0297 4314 46 40 974 1843 29 14 24 0807 3637 44 28 081 1844 B 18 23 13 0509 2243 28 17 104 1845 7 8 1 0211 0848 11 06 126 1846 26 5 34 0721 0171 09 94 234 1847 15 14 22 0423 8777 92 84 256 1848 B 4 23 ]1 0125 7382 75 73 278 1849 22 20 43 0635 6705 73 61 386 1850 12 5 32 0337 5311 56 50 408 1851 1 14 21 0()38 3916 40 39 431 1852 B 20 1 53 0549 3239 38 27 538 1853 8 20 42 0251 1845 21 16 560 1854 27 18 14 0761 1168 19 04 668 1855 17 3 3 0463 9773 02 93 690 1856 B 6 11 51 0164 8379 85 82 713 1857 24 9 24 0675 7702 84 70 820 1858 13 18 13 0376 6307 67 59 843 1859 - 3 3 1 0078 4913 50 48 865 1860 B 22 0 34 0588 4236 48 36 972 1861 10 9 23 290 2842 31 25 994 1862 29 6 55 800 2164 29 13 102 1863 18 15 44 502 770 12 2 124 1864 3 8 0 32 204 9376 95 91 146 1865 25 22 5 714 8699 94 79 254 1866 15 6 54 416 7304 77 68 276 1867 4 15 42 117 5910 60 57 299 1868 B 23 13 15 628 5234 58 46 406 1869 11 22 3 329 3838 41 35 428 1870 30 19 36 i840 3161 40 23 536 1871 20 4 25 541 1767 23 12 558 1872 B 9 13 13 243 372 6 1 581 1873 27 10 46 753 9695 4 89 688 1874 16 19 34 455 8301 87 78 710 1875 6 4 23 157 69(6 70 67 733 1876 B 25 1 55 667 6229 69 55 840 1877 13 10 44 369 4835 52 44 862 1878 2 19 33 71 3441 35 33 885 1879 21 17 5 581 2763 33 21 993 1880 B 11 1 54 283 1369 16 10 15 1881 28 23 27 793 692 14 99 123 1882 18 815 495 9297 98 88 145 1883 7 17 4 197 79(~3 81 77 167 1884 B 26 14 36 707 7226 79 65 275 1885 14 23 25 409 5832 62 54 297 35 102 TABLE LXXX II AMean Lunations and Changes of the Argumezs. Num Lunations. I.III.I N.V d. h m 2 14 18 22 404 5359 58 50 43 1 29 12 44 808 717 15 99 85 2 59 1 28 1617 1434 31 98 170 3 88 14 12 2425 2151 46 97 256 4 118 2 56 3234 2869 61 96 341 5 147 15 40 4042 3586 76 95 426 6 177 4 24 4851 4303 92 95 511 7 206 17 8 5659 5020 7 94 596 8 236 5 52 6468 5737 22 93 682 9 265 18 36 7276 6454 37 92 767 10 295 7 20 8085 7171 53 91 852 11 324 20 5 8893 7889 68 90 937 12 354 8 49 9702 8606 83 89 22 13 383 21 33 510 9323 98 88 108 TABLE LXXXVIII. Number of Days from the commencement of the wea to thefirst of each month. Months. Com. Bis. I Days. Days, January.... 0 0 February.. 31 31 March.... 59 60 April.... 90 91 May.... 120 121 June.... 151 152 Ju.... 181 182 August. 212 213 September.. 243 244 October... 273 274 November.. 304 305 December.. 334 33fi L _.~~~~~~~~~3, TABLE LXXXIX, 103 Equations for New and Full Moon. Arg. I I Arg. I II Arg [II IVT Arg h m i h rL h m h m mn n 0 4 20 10 10 5000 420 10 10 25 3 31 25 I00 I 4 36 9 36 5100 4 5 10 50 26 3 31 24 200 4 52 9 2 5200 349 11 30 27 3 30 23 300 5 8 828 5300 334 12 9 128 3 30 22 400 5 24 7 55.5400 3 19 12 48 29 3 30 21 5005 40 7,.500 3 4 13 261130; 3 30 20 60015 55 6 49 5600 2 49 14 3131 3 30 19 700 6 10 6 17 5700 2 35 14 39 132 4 30 18 800 6 24 5 46 5800 2 21 15 13 133 4 29 17 900,6 38. 15 5900 2 8 15 46 134 4 29 16 1000 6 51 446 6000 155 16 18 35 4 29 15 1100 7 4 417 6100 1 42 16 48 36 5 28 14 120017 15 3 50 6200 1 31 17 161 37 5 28 13 1300 727 324j 6300 119 17 42 38 5 27 12 1400 737 2 59 6400 1 9 18 639 5 27 11 1500 747 235 6500 0 59 18 28 40 6 26 10 1600 7 55 2 14 6600 0 50 18 48 41 6 26 9 1700 3 1 53 6700 0 42 19 6 42 7 25 8 1800 810 1 35 6800 0 34 19 21 43 7 25 7 1900 816 1 18 6900 0 28 19 33 44 7 24 6 2000 82 1 181 3 7000 0 22 19 44 45 8 23 5 2100 825 0 700 17 1952 46 8 23 4 2200 8 29 0 40 7200 014 19 57 147 9 22 3 23008 31 0 o311 73001 01 20 11 48 921 1 2 2400 832 025 7400 0 9 20 1149 10 21 1 2.500 8 32 0 21 7500 0 8 199 590 10 20 0 2600 831 119 76 8 55 51 101 9 99 2700 829 0 20 7700/0 9 19 48 52 11 19498 2800 8 26 0 23 7800 0 11 19 40 53 11 18 97 2900 823 028 7900 015 19 29 54 12 17 96 3000 8 18 0 36 8000 0 19 19 17 55 12 17 95 3100 8 12 0 47 8100 0 24 19 2 156 13 16 94 32008 6 0 59 8200 0 0 S 1845 57 13 15 93 3300 7 58 1 14 8300 037 18 27 58 13 15 92 3400 7 50 1 32 8400 0 45 18 6 59 14 14 91 3500 7 41 1 52 850 0.53 17 45 60 14 14 190 3600 7 31 2 14 8600 1 3 17 21 161 15 13 89 3700 7 21 238 8700 113 16 5662 15 13 88 38007 9 3 4 8800 1 25 16 301163 15 12 87 3900 6 58 3 32 8900 1 36 16 3 64 15 12 86 4000 6 45 4 21 900011 49 15 34 65 16 11 85 4100 6 32 4 341 91001 22 15 5 66 16 11 84 4260 6 19 5 71 9200 2 16 14 34 67 16 11 83 4300 6 541 9300 2 30 14 368 16 10 82 4400 5 51 6 17[ 9400 2 45 13 31 69 17 10 81 4500 5 36 6 54 9500 3 0 12 58 70 17 10 80 4600 5 21 732'" 9600 3 16 12 25 71 17 10 79 47005 6 8 11 970013 32 11 52 72 17 10 78 4800 4 511 8 50 9800 348 11 18 73 17 10 77 4900 435 930 9900 4 10 44 74 17 9 76 5000 4 20 10 10 110000 4 20 10 10 75 17 9 75 104 TAELE XC. Mern Right Ascensions and Declinations of 50 principal Fixed Stars, for the beginning of 1840. Stars' Name. Ma~g Right Ascen. AnnumlalVar. Declination. Ann. Var. h rn s n s o,, 1 Algenmb 2.3 0 5 0.31 + 3.0775 14 17 38.82N -+ 20.051 2 / Andromedae 2 1 0 46.7 3.309 34 46 17.2 N 19.35 3 Polaris 2.3 1 2 10.38 16.1962 88 27 21.96N 19.339 4 Achernar 1 1 31 44.88 2.2351 58 3 5.13 S -18.473 5 aArietis 3 1 58 9.94 3.3457 22 42 11.81 N - 17.4Fr5 6 a Ceti 2.3 2 53 55.34 + 3.1257 3 27 30.09N + 14.561 7 aPersei 2.3 3 12 55.97 4.2280 49 17 8.74N 13.371 8 Aldebaran 1 4 26 44.77 3.4264 16 10 56.82N 7.949 9 Capella 1 5 4 52.67 4.4066 45 49 42.81N 4.793 10 Rigel 1 5 6 51.09 2.8783 8 23 29.29S - 4.620 11 dTauri 2 5 16 10.96 + 3.7820 28 27 58.20 N + 3.825 12 yOrionis 2 5 16 33.1 3.210 6 11 55.3 N + 3.82 13 a Columbae 2 5 33 51.52 2.1688 34 9 47.41 S 2.291 14 aOrionis 1 5 46 30.71 3.2430 7 22 17.14N + 1.191 15 Canopus 1 6 20 24.18 1.3278 52 36 38.42 S 1.778 16 Sirius 1 6 38 5.76 + 2.6458 16 30 4.79 S + 4.449 1'7 Castor 3 7 24 23.06 3.8572 32 13 58.89N - 7.206 18 Procyon 1.2 7 30 55.53 3.1448 5 37 48.92N 8.720 19 Pollux 2 7 35 31.07 3.6840 28 24 25.57N 8.107 20 aHydrae 2 9 19 43.57 2.9500 7 58 4.83S + 15.341 21 Regulus 1 9 59 50.93 + 3.2220 12 44 49.70N -17.356 22 aUrsae Majoris 1.2 10 53 47.98 3.8077 62 36 48.93 N 19.221 23 / Leonis 2.3 11 40 53.69 3.0660 15 28 1.16N 19.985 24 lVirginis 3.4 11 42 21.4 3.124 2 40 26 N 19.98 25 yUrsae Majoris 2 11 45 22.93 3.1914.54 35 4.67N 20.014 26 a 2 Crucis 2 12 17 43.7 + 3.258 62 12 47. 9S + 19.99 27 Spica 1 13 16 46.36 3.1502 10 19 24.39 S 18.945 28 0Centauri 2 13 57 18.0 3.491 35 34 41.9 S 17.499 29 aDraconis 3.4 14 0 2.8 1.625 65 8 32.1 N -17.37 30 Arcturus 1 14 8 21.96 2.7335 20 1 7.67N 18.956 31a2Centauri 1 14 28 47.84 ~ 4.0086 60 10 6.24S +15.152 32a2Librae 3 14 42 2.44 3.3088 15 22 18.25$ i5.256 33 /$Ursae Minoris 3 14 51 14.66 - 0.2787 74 48 34.18 N -14.712 34y2Ursae Minoris 3.4 15 21 1.3 - 0.179 72 24 14.1 N 12.81 35 a CoronaeBorealis 2 15 27 54.87 + 2.5277 27 15 27.71N 12.361 36 aSerpentis 2.3 15 36 23.43 + 2.9386 6 56 2.80N -11.770 37 flScorpii 2 15 56 8.68 3.4729 19 21 38.82S + 10.330 38 Antares 1 16 19 36.49 3.6625 26 4 13.13 S 8.519j d9 aHerculis 3.4 17 7 21.30 2.7317 14 34 41.43N - 4.576 40 aOphiuchi 2 17 27 30.56 2.7724 12 40 58.65N 2.844 41 6Ursae Minoris 3 18 23 56.48 -19.';072 86 35 28.89N + 2.161 42 Vega 1 18 31 31.19 + 2 0116 38 38 16.85N 2.742 43 Altair 1 19 42 58.61 2.9255 8 27 0.21 N 8.701'44a2Capricorni 3 20 9 10.34 3.3323 13 2 5.57 S — 10.705.45 aCygni 1 20 35 58.80 2.0416 44 42 41.38N + 12.614'46 aAquaril 3 21 57 33.93 + 3.0835 1 5 38.00S - 17.256 47 Fomalhaut 1 22 48 47.67 3.3114 30 28 4.91 S 19.092 4 8 iPegasi 2 22 56 1.1 2.878 27 13 1.7 N + 19.255 49 Markab 2 22 56 47.75 2.9771 14 20 46.92N 19.295 50 a Andromedae 1 24 0 7.72 3.0704 28 12 27.06N 20.05Uj TABLE XCI. 105 Constants for the Aberration and Nutatzon in Rtqght Ascension and Declination of the Stars in the preceding Catalogue Aberration. Nutaticn. M 08 N 2.' M' 0 I N' ^- F ~ -- S ~, | S S ~' - 1 8 28 47 0.1087, 7 27 12 0.9657 i 6 8 24 0.0300 5 28 30 0.8381 2 8 13 39 0.1830 6 19 12 1.07401 6 19 53 0.0838 5 10 8 0.8496 3 8 13 51 1.6526 5 16 57 05 8 16 1.3427 510 22 0.8493 4 8 5 20 0.3801 10 26 46 1.2798 4 10 12 0.0775. 5 0 31 0.8629 5 7 28 26 0.1397 7 0 2 0.8972 6 11 1 0.0695 4 22 53 0.8765 6 7 14 11 0.1149 8 23 s 0.8678 6 1 26 0.0322 4 8 16 0.9078 7 7 9 30 0.3020i 5 3 5 1.0630 6 18 13 0.1849 4 3 47 0.9179 8 6 21 43 0.1447i 7 23 12 0.5760 6 3 27 0.0726 3 1754 0.9502 9 6 12 51 0.2875 3 25 37 0.9112 6 5 46 0.1830 3 10 29 0.9605 10 6 12 20 0.1355; 9 342 1.0300 528 47 1.9966 3 10 4 0.9608 1 1 610 13 0.1873 4 19 21 0.3917 6 2 52 0.1008 3 8 19 0.9626 12 6 10 6 0.1340 8 26 4 0.7851 6 0 40,0.0441 3 8 14 0.9626 13 6 6 5 0.2145 9 4 24 1.2348.5 26 18 1.8750 3 4 57 0.9648 14 6 3 13 0.1361 828 2310.7521 6 0 15 0.0481 3 2 37 0.9657 15 525 22 0.3491 825 53 1.2960 6 8 46 1.6679 226 15 0.96571 16 521 21 0.15011 8 25 51 1.1152 6 1 51 1.9658 2 22 58 0.9636 17 510 4010.2010 1 2 17 0.66201l 5 24 2 0.1257[ 2 14 610.9535 18 5 9 6 0.1297 9 6 54 0.8071 528 47 0.04141 2 12 4710.9513 19 5 8 2 0.1829 0 14 32 0.6052 5 24 20.111411 2 11 53 0.94993 20 4 12 39 0.1158, 8 17 31 0.9967 6 3 41 0.0081 18 37 0.9007 21 4 2 22 0.1162 10 3 471 0.84.57 5 23 47 0.04.80 I 1 7 59 0.87S2 22 3 18 7 0.4366 0 3 28 1.2394 4 18 58 0.24071 0 21 57 0.8520 23 3 5 21 0.11171 10 6 20 0.9621 5 20 56 0.0344! 0 6 35 0.8393i 24 3 4 57 0.0958 9 6 51 0.9075 528 25 0.0253 0 65 0.8390 25 3 4 8 0.3229 11 17 28 1.2298 4 21 46 0.1465 0 5 5 O.S3S8 26 2 25 19 0.4261' 6 8 5 1.2585 7 16 2 0.2089, 11 24 14 0.83930 27 2 9 22 0.10661 8 3 31 0.8862 6 5 51 0.0154411 5 6 O0.S 5591 28 1 28 40 0.1942 6 7 12 1.01761 6 17 31 01.10621 10 23 8 0.8760' 29 1 27 53 10.4824 10 23 28 1.2995 3 25 50 0.10()90 10 22 16 0.8777 30 1 25 46 0.1336 928 18 1.0974 5 18 49 1.9937 10 20 1 0.822 31 1 20 32 0.4123 5 7 54 1.1201 6 29 6 0.2460 110 11 36 0.89371 32 1 17 26 0.1273 7 18 24 1 0.6923 6 6 29 0.0593 11011 28 0.90)06 33 1 14 42 0.6961 10 15 51.3087 2 26 45 0.2235 10 8 47 0.9066 34 1 7 20 0.6386 10 7 33 1.3087 2 27 7!0.0960 10 1 45 0.9'225 35 1 5 45 0.1704 9 22 28 1.1785 5 17 18 1.9510 10 0 180.9257( 36 1 3 43 0.1237| 9 8 22 0.9994 5 27 30 0.00~58 9 28 261 0.9298 37 0 28 58 0.14851 4 40.6237 6 520 0.07951 924 120.9386 38 023 24 0.1728 5 27 59 0.5816 6 5 49 0.1029 9 19 21 0.9478 39 0 12 13 0.1451| 9 5 25 1.0962 5 27 45 1.9742 9 9 58 0.9610 40 0 7 34 0.1427 9 3 4 1.0786 5 28 48 1.9803 9 6 9 0.9642 41 1 11 23 47 11.3571 8 22 49 1.2821 1 119 31 0. 8S57 8 24 57 0.9650 42 11 22 50 0. 2393 8 24 29 1.2545 6 5 31 1.8436 8 24 10 10.964,143 11 6 15 0.1309 8 22 59 1.023711 6 2 16 1.99sS 8 10 21 0.9472 44 11 0 2 0.1341 9 29 33 0.69611 5 26 12 0.06)9 8 4 55 0.93681 45 10 23 29 0.2668 8 0 39 11.26341 6 28 32 1.90421l 7 29 0 10.9242 46 1 0 2 57 9 2 31 0.8988 1 5 29.6 60.0264 7 8 37 0.8794 47 9 19 26 1 0.1638 11 1 734 11.02711 5 13 8 0.076.5 6 23 30 10.8540 48 917 29 0.14911 717 0 1.11715 617 2 0.0162 621 1310.8511 49 9 17 1710.11201 8 2 51L.0138 6 8 23 0.0157 6 20 581 0.8508 501 9 0 6 0.1495 7 6 4211.078511 617 20 0.044411 6 0 810.8380 106 TABLE XCII. IAean Longitudes and Latitudes of some of the principal Fixed Stars for the beginning of 1840, with their Annual Variations Stars' Nane, Mag Longitude. Varl Latitude. Amnal Var. Var. a Arietis 3 1 5 25 27.6 50.277 9 57 40.9 N + 0.161 Aldebaran. 2 7 33 5.9 50.210 5 28 38.0; - 0.335 Capella 1 2 19 37 17.8 50.302 22 51 44.4 N - 0.052 Polaris 2.3 2 26 19 20.1 47.959 66 4 59.5 IN + 0.552 Sirius 1 3 11 52 32.9 49.488 39 34 4.3 S + 0.319 Canopus 1 3 12 44 59.6 49.366 75 50 57.6 S + 0.459 Pollux 2 3 21 0 22.0 49.502 6 40 20.2 N + 0.255 Regulus 1 4 27 36 13.2 49.946 0 27 38.3 N + 0.220 Spica 1 6 21 36 29.2 50.085 2 2 29.7 5 + 0.171 Arcturus 1 6 22 0 4.7 50.711 30 51 17.5 N + 0.214 Antares 1 8 7 31 45.2 50.120 4 32 51.6 S -- 0.424 Altair 1.2 9 29 31 5.9 50.795 29 18 37.3 N + 0.080 Fomaihaut 1 1I 1 36 22.0 50.595 21 6 49.7 S + 0.213 Achernar 1 11 13 2 5.3 50.346 17 6 17.3 S 0.083 a Pegasi 2 11 21 15 24.7 1 50.112 19 24 40.9 N + 0.098! TABLE added to TABLE XC. WIean Right Ascensions and Declinations of Polaris and J Ursae Minoris for 1830, 1840, 1850, and 1860. Stars. Years Right Asc. Ann. Var. Declination. Ann. Var. 1830 0 59 30.76 + 15.478 88 24 8.82 + 19.371 1840 1 2 10.32 16.470 88 27 22.43 19.309 Polaris 1850 1 5 0.29 17.567 88 30 35.40 19.240 1860 1 8 1.79 18.784 88 33 47.64 19.163 1830 18 27 5.13 -19.167 86 35 5.70 + 2.363 UrsaetMinoris 1840 18 23 53.03 19.241 86 35 27.93 2.085 1850 18 20 40.21 19.305 86 35 47.36 1.805 1860 1 1 117 26.77 19.360 86 36 3.97 1.523 TABLE XC. (a). 107 Mean Places of 50 Principal Fixed Stars. For January Od., 1870. Star's Name. Mag. Right Aseen. Annual Var. Declination. Annual Var, h. m. u. s. aAndrormede. 2 0 1 40.238 + 3.0864 N. 280 22' 21".62 +19."899 y Pegasi (Algenib).. 3.2 0 6 32.548 3.0811 N. 14 27 38.40 20.027 a Urs. Min. (Polaris) 2 1 11 16990 20.1966 N. 88 36 58.74 19.091 a Eridani(Achernar) I 1 32 52.026 2.2349 S. 57 53 51.51 18.419 a Arietis..2....... 1 59 50.914 3.3665 N. 22 50 46.85 17.224 a Ceti..... 2.3 2 55 29.039 + 3.1273 N. 3 34 40.00 ~14.344 a Persei.......... 2 3 15 3.176 4.2481 N. 49 23 44.88 13.169 a Tauri(Aldebaran). 1 4 28 27.782 3.4353 N. 16 14 44.14 7.622 a Aurigte (Capella). 1 57 5.338 4.4217 N. 45 51 44.60 4.154 2 Orionis (Rigel).... 1 5 8 17.411 2.8799 S. 8 21 15.13 4.464 2 Tauri........... 2 5 18 4.487 + 3.7873 N. 28 29 40.42 + 3.446 6 Orionis.......... 2 5 25 21.975 3.0641 S. 0 23 52 63 2.980 a Columbe........ 2 6 34 56.721 2.1778 S. 34 8 40.13 2.187 a Orionis.......... Var. 5 48 8.026 3.2462 N. 7 22 48.64 + 1 015 a Argfs (Canopus).. 1 6 21 6.083 1.3303 S. 52 37 32.06- 1.842 a Canis Maj. (Sirius) 1 6 39 25 283 + 2.6452 S. 16 32 24.53 - 6.657 a' Geminor (Castor). 2.1 7 26 18.152 3.8417 N. 32 10 14.97 7.651 a CanisMin.(Procyon) 1 7 32 29.688 3.1446 N. 5 33 22.44 8.908 2 Geminor (Pollu). 1.2 7 37 21.465 3.6812 N. 28 20 15.56 8.324 a Hydre... 2... 2 9 21 11.889 2.9485 S. 8 5 47.54 15.397 a Leonis (Reguluzs). 1.2 10 1 26.176 + S3.2023 N. 12 36 5.31 -17.423 a Urss Majoris.....2 10 55 41.171 3.7653 N. 62 27 7.09 19.360 2Leonis.2..... 2 11 42 25.582 3.0648 N. 15 17 55.39 20.099 y Ursm Majoris..e 2.3 11 46 58.914 3.1887 N. 54 25 2.81 20.027 i Virginis........ 3.4 12 13 15.2441 3.0650 N. 0 3 21.65 20.054 a' Crucis..........1 12 19 22.701 + 3.2650 S. 62 22 38.17 -19.932 a Virginis (Spica).. 1 13 18 20.737 3.1506 S. 10 28 55.45 18.932 (Virginis........ 3.4 13 28 4.232 3.0523 N. 0 4 11.64 18.527 a Bootis (Arclurus). 1 14 9 43.897 2.73:38 N. 19 51 37.41 18.903 al Centauri...... 1 14 30 48.325 4.0367 S. 60 17 39.13 15.022 E Bootis........ 2.3 14 39 18.503 + 2.6194 N. 27 37 24.36 -15.395 a'lLibroa........... 2.3 14:43 41.335 + 3.3058 S. 15 29 59.61 15.21! 2 Ursme Minoris.. 2 14 51 6.857 - 0.2489 N. 74 41 11.24 14.7,57 3 Librae,.. 2.. 2 15 10 0.758 + 3.2188 S. 8 54 6.16 13.562 a Coronm Borealis.. 2 15 29 10.990 2.5377 N. 27 9 13.63 12.335 aSerpentis.2... 2.3 15 37 51.859 + 2.9492 N. 6 50 11.26-11.598 /' Scorpii.... 2 15 57 52.802 3.4776 S. 19 26 50.27 10.206 a Scorpii (Antares).. 1.2 16 21 26.333 3.6668 S. 26 8 27.62 8.387 a Herculis....... Var. 17 8 43.136 2.7322 -N. 14 32 26.06 4.404 a Ophiuchi........ 17 28 53.949 + 2.7808 N. 12 39 24.42 - 2.921 6 Urse Minoris.. 4.5 18 14 16.673 -19.3995 N. 86 36 21.06 + 1.260 a Lyrm (Vega)....1 18 32 32.155 + 2.0304 N. 38 39 51.25 3.126 a Aquilem (Altair)... 1.2 19 44 26.344 2.9272 N. 8 31 37.06 9.210 a2 Capricorni......3.4 20 10 50.295 3.3324 S. 12 56 45.08 10.840 a Cygni........... 2.1 20 36 59.960 2.0430 N. 44 49 0.98 13.690 611 Cygni.... 5.6 21 1 3.949 + 2.6737 N. 38 6 41.36 +17.495 a Aquarii......... 3 21 59 6.275 3.0823 S. 0 57 1.81 17.317 aPis.Aus(Flomalhaut) 1.2 22 50 27.642 3.3288 S. 30 18 38.37 18.966 a Pegasi (Marckab) 2 22 58 17.136 2.9831 N. 14 30 23.04 19.312 y Cephei.......... 3.4 23 34 1.860 2.4018 N. 76 54 24.86 20.077 108 TABLE XCIII. Second Differences. HIlurs &Minutes. 1' 2' 3' 4' 5' 6' 7 8' 9 10' 1 h mo o o m o o o o o0 0 12 0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0 10 11 50 0.4 0.8 1.2 1.6 2.0 2.4 2.9 3.3 3.7 4.1 4.5 0 20 11 40 0.8 1.6 2.4 3.2 4.1 4.9 5.7 6.5 7.3 8 1 8.9 0 30 11 30 1.2 2.4 3.6 4.8 6.0 7.2 8.4 9.6 10.8 12.0 13.2 0 40 11 20 1.6 3.1 4.7 6.3 7.9 9.4 11.0 12.6 14.2 15.7 17.3 0 50 11 10 1.9 3.9 5.8 7.8 9.7 11.6 13.6 15.5 17.4 19.4 21.4 1 6 11 0 2.3 4.6 6.9 9.2 11.5 13.8 16.0 18.3 20.6 22.9 25.2 1 10 10 50 2.6 5.3 7.9 10.5 13.2 15.8 18.4 21.1 23.7 26.3 29.0 1 20 10 40 3.0 5.9 8.9 11.9 14.8 17.8 20.7 23.7 26.7 29.6 32.6 1 30 10 30 3.3 6.6 9.8 13.1 16.4 19.7 23.0 26.3 29.5 32.8 36.1 1 40 10 20 3.6 7.2 10.8 14.4 17.9 21.5 225.1 28.7 32.3 35.9 39.5 1 50 10 10 3.9 7.8 11.6 15.5 19.4 23.3 27.2 31.0 34.9 38.8 42.7 2 0 10 0 4.2 8.3 12.5 16.7 20.8 25.0 29.2 33.3 37.5 41.7 45.8 2 10 9 50 4.4 8.9 13.3 17.8 22.2 26.6131.1 35.5 40.0 44.4 48.8 2 20 9 40 4.7 9.4 14.1 18.8 23.5 28.2 32.9 37.6 42.o 47.0 51.7 2 30 9 30 4.9 9.9 14.8 19.8 24.7 29.7 34.6 39.6 44.5 49.5 54.4 2 40 9 20 5.2 10.4 15.6 20.7 25.9 31.1 36.3 41.5 146.7 51. 57.0 2 50 9 10 5.4 10.8 16.2 21.6 27.1 32.5 37.9 43.3 48.7 54.1 59.5 3 0 9 0 5.6 11.3 16.9 22.5 28.1 33.8139.4 45.0 150.6 56.3 61.9 3 10 8 50 5.8 11.7 17.5 23.3 29.1 35.0 40.8 46.6 52.4 58.3 64.1 3 20 8 40 6.0 12.0 18.1 24.1 30.1 36.1 42.1 48.1 54.2 60.2 66.2 3 30 8 30 6.2 12.4 18.6 24.8 31.0 37.2 43.4 49.6 155.8 62.0 68.2 3 40 8 20 6.4 12.7 19.1 25.5 31.8 38.2 44.6 50.9 57.3 63.7 70.0 3 50 8 10 6.5 13.0 19.6 26.1 32.6 39.1 45.7 52.2 58.7 65.2 71.7 4 0 8 0 6.7 13.3 20.0 26.7 33.3 40.0 46.7 153.3 60.0 66.7 73.3 i 4 10 7 50 6.8 13.6 20.4 27.2 34.0 40.8 47.6 54.4 61.2 68.0 74.8 4 20 7 40 6.9 13.8 20.8 27.7 34.6 41.5 4814 55.4 62.3 69.2 76.1 4 30 7 30 7.0 14.1 21.1 28.1 35.2 42.2 49.2 56.2 63.3 70.3 77.3 4 40 7 20 7.1 14.3 21.4 28.5 35.6 42.8 49.9 157.0 64.2 71.3 78.4 4 50 7 10 7.2 14.4 21.6 28.9 1 36.1 43.3 50.5 57.7 64.9 72.2 1 79.4 5 0 7 0 7.3 14.6 21.9 29.2 36.6 43.8 51.0 58.3 65.6 72.9 80.2 5 10 6 50 7.4 14.7 22.1 29.4 36.8 44.1 51.5 58.8 i66.2 73.6 80.9 5 20 6 40 7.4 14.8 22.2 29.6 37.0 44.4151.9 159.3'66.7 74.1 81.5 5 30 6 30 7.4 14.9 22.3 29.8 37.2 44.7 52.1 59.6 167.0 74.5 81.9 5 40 6 20 7.5 15.0 22.4 29.9 37.4 44.9 52.3 59.8 67.3 74.8 82.2 5 50 6 10 7.5 15.0 22.5 30.0 3715 45.0 52.5 60.0 67.4 74.9 82.4 6 0 6 0 7.5 15.0 22.5 30.0 37.5 45.0 52.5 60.0 67.5 75.0 82.5 _~~~~~~~~~256. 6.4_ TABLE XCIII. 100 Second Digerences. Hours& Min.l10" 20" 30" 40"50" 1 2" 3" 4' 5" 6" 7" 8 9 i- -- -—, — - - -- _ —........- 6 —--- hmlhm 0 0 112 0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0O.O 0 10 11 50 0.1 0.1 0.2 0.3 0.3 0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.1 0 20 11 40 0.1 0.3 0.4 05 0.7 0.0 0.0 0.0 0.1 0.1 0.1 0.1 0.1 0 30 11 30 0.2 0.4 0.6 0.8 1.0 0.0 0.0 0.1 0.1 0.1 0.1 0.1 0.2 0.2 0 40 11 20 0.3 0.5 0.8 1.0 1.30 0.0 0.1 0.1 0.1 0.1 0.2 0.2 0.2 0.2 0 50 11 10 0.3 0.6 1.0 1.3 1.610.0 0.1 0.1 0.1 0.2 0.2 0.2 0.310.3 I 0 11 0 0.4 0.8 1.1 1.5 1.9 10.0 0.1 0.1 0.2 0.2 0.2 0.3 0.3 0.3 1 10 10 50 0.4 0.9 1.3 1.8 2.2 0.0 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 1 20 10 40 0.5 1.0 i 1.5 2.0 2.5'0.0 0.1 0.1 0.2 0.2 0.3 0.3 t0.4 0.41 130 10 30 0.5 1.1 1.6 2.2 2.7 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 1 40 10 20 0.6 1.2 1.8 2.4 3.0 0.1 0.1 0.2 0.2 0.3 0.4 0.4 0.5 0.5 1 50 10 10 0.6 1.3 1.9 2.6 3.21 o0.1 0.1 0.2 0.3 0.3 0.4 0.5 0.5 0.6 2 0 10 0 0.7 1.4 2.1 2.8 3.5 10.1 0.1 0.2 0.3 0.3 0.4 0.5 0.6 0.6 2 10 9 50 0.7 1.5 2.2 3.0 13.7 0.1 0.1 0.2 0.3 0.4 0.4 0.5 0.6 0.7 2 20 9 40 0.811.6 2.3 3.1 i 3.9 0.1 0.2 0.2 0.3 0.4 0.5 0.5 10.6 10.7 2 30 9 30 0.8 1.6 2.5 3.3 4.1 0.1 0.2 0.2 0.3 0.4 0.5 0.6 0.7 0.7 2 40 9 20 0.911.7 2.6 13.5 4.3 0.1 0.2 0.3 0.3 0.4 0.5 0.6 0.7 0.8 2 501 9 10 0.9 1.8 2.7 3.6 4.5 10.1 0.2 0.3 0.4 0.5 0.5 0.6 0.7 0.8 3 0 9 0 0.9 1.9 2.8 3.8 4.7 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.7 0.8 3 10 8 50 1.0 1.9 2.9 3.91 4.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 3 20 8 40 1.0 2.0 3.0 4.0 5.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9' 3 30 8 30 1.0 2.113.1 4.1 5.2 0.1 0.2 0.3 0.4 0.5i 0.6 0.7 0.8 0.9 3 40 8 20 1.1 2.1 3.2 4.2 5.3 0.1 0.2 0.3 0.5 0.6 0.7 0.7 0.8 1.1 3 50 8 10 1.1 2.2 3.3 4.3 5.49i0.1 0.2 0.3 0.5 0.5 0.7 0.8 10.9 1.1 4 0 8 0 101 2.2 3.3 4.4 5.6 0.1 0.2 0.3 0.4 0.6 0.7 0.8 0.9 1.0 4 10 7 50 1.2 2.3 3.4 4.5 5.7 0.1 0.2 0.3 0.5 0.6i.7 0.8 0.9 1.0 4 20 7 40 1.212.3 3.5 4.6 5.8 0.1 0.2 0.3 0.5 0.6 0.7 0.9 0.9 1. 4 30 7 30 1.2 2.3 13.5 4.7 5.9 20.1 0.2 0.4 0.5 0.6 0.7 0.8 1.9 1.1 4 40 7 2011.2 2.4 3.6 4.8 5.9 10.1 0.2 0.4 0.5 0.6 0.7 0.8 1.0 1.1 4 50 7 1011.2 2.4 3.6 4.8 6.0 10.1 0.2 0.4 0.5 0.6 0.7 0.8I1.0 1.1 5 0 7 0 1.2 2.4 3.6 4.9 6.11 0.1 0.2 0.4 0.5 0.61 0.7 0.9 1.0l 1.1 5 10 6 50 1.2 2.5 3.7 4.9 6.1;0.1 0.2 0.4 0.5 0.6 O0. 0.9 1.0 1.1 5 20 6 40 1.2 2.5 3.7 4.9 6.1 10.1 0.2 0.4 0.5 0.61O.7 0.9 1.0 1.1 5 301 6 30 1.2 2.5 3.7 5.0 6.2 i0.1 0.2 0.4 0.5 0.61.7 0.9 1.0 1.1 5 40 6 20 1.2 2.5 3.7 5.0 6.2 0.1 0.2 0.4 0.5 0.6 0.7 0.9 1.01. 5 50 6 10 1.2 2.5 3.7 5.0 6.2 0.1 0.2 0.4 0.5 0.6 0.7 0.9 1.0 1.1 6 0 6 0 1.3!2.6 3.8 5.0 6.31 0.10.2 0. 05 0.61 0.7 0.91._1< 2.,.1o2o41.1..1. 110 TABLE XCIV. Third Differences. Time after Time after noowi or 10" 20" 30" 40" 50" 1' 2' 3' 4' 5 noon or midnight. midnight. Oh. Om. 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 12h. Om. 0 30 0.0 0.1 0.1 0.1 0.2 0.2 0.4 0.5 0.7 0.9 11 30 1 0 0.1 0.1 0.2 0.2 0.3 0.3 0.6 1.0 1.3 1.5 11 0 1 30 0.1 0.1 0.2 0.3 0.3 0.4 0.8 1.2 1.6 2.1 10 30 2 0 0.1 0.2 0.2 0.3 0.4 0.5 0.9 1.4 1.9 2.3 10 0 2 30 0.1 0.2 0.2 0.3 0.4 0.5 1.0 1.4 1.9 2.4 9 30 3 0 0.1 0.2 0.2 0.3 0.4 0.5 0.9 1.4 1.9 2.3 9 0 3 30 0.1 0.1 0.2 0.3 0.4 0.4 0.9 1.3 1.7 2.2 8 30 4 0 0.1 0.1 0.2 0.2 0.3 0.4 0.7 1.1 1.5 1.9 8 0 4 30 0.0 0.1 0.1 0.2 0.2 0.3 0.6 0.9 1.2 1.5 7 30 5 0 0.0 0.1 0.1 0.1 0.2 0.2 0.4 0.6 08 1.0 7 0 5 30 0.0 0.0 0.1 0.1 0.1 0.1 0.2 0.3 ).4 0.5 6 30 6 0.0 i0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 6 0 TABLE XCV. Fourth Differences. 3Time after 1 | Time after noon or 10" 20" 30" 40" 50" l 1' 2' 3' noon or midnight. I midnight. |h. m.,, " I, I,, I\ | I | I | h. in O O 0.0 0.0 0.0 00 0.0 0.0 0.0 0.0 12 0 0 30 0.0 0.1 0.1 0.1 0.2 l0.2 0.4 0.6 11 30 1 O 0.1 0.1 0.2 0.3 0.3 0.4 0.8 1.2 11 0 1 30 0.1 0.2 0.3 0.4 0.5 0.61.2 1.7 10 30 2 0 0.1 0.2 0.4 0.5 0.6 110.7 1.5 2.2 10 0 2 30 0.1 0.3 0.4 0.6 0.7 0.9 1.8 2.7 9 30 3 0 0.2 0.3 0.5 0.7 0.9 1.0 2.1 3.1 9 0 3 30 0.2 0.4 0.6 0.8 0.9 1.1 2.3 3.4 8 30 4 0 0.2 0.4 0.6 0.8 1.0 1.2 2.5 3.7 18 4 30 0.2 0.4 0.7 0.9 1.1 1.3 2.6 3.9 7 30 5 0 0.2 0.5 0.7 0.9 1.1 1.4 2.7 4.1 7 0 5 30 0.2 0.5 0.7 0.9 1.2 1.4 2.8 4.2 6 30 8 O 0.2 0.5 0.7 I 0.91 1.2 1.4 2.8 4.2 6 0 TABLE XCVI. Logistical Logarithms. 111 0i i1 2 i 3 I 4 5 6 7 8 1Kj 60 120 180 240 300 360 420 480 540 - -1.77821477 1.3010 l.1761 1.0792 1.0000 9331 8751 8239 1 3.5563 1.77101 1.4735 1.2986 1.1743 1.0777 9988 9320 8742 8231 2 3.2553 1.7639 1.4699 1.2962 1.1725 1.0763 95J 3 9310 8733 8223 3 3.0792 1.7570 11.4664 1.2939 1.1707 1.0749 9964 9300 8724,8215 4 2.9542 1.7501 1.4629 1.2915 1.1689 1.0734 9952 9289 8715 8207 5 2.8573 1.7434 1.4594 1.2891 1.1671 1.0720 9940 9279 8706 8199 612.7782 1.7368 1.4559 1.2868 1.1654 1.0706 9928 9269 8697 8191 7 2.7112 1.7302 1.4525 1.2845 1.1636 1.0692 9916 9259 8688 8183 8 2.6532 1.7238 1.4491 1.2821 1.1619 1.0678 9905 9249 8679 8175 9 2.6021 1.7175 1.4457 1.2798 1.1601 1.0663 9893 9238 8670 8167 10 2.5563 1.7112 1.4424 1 2775 1.1584 1.0649 9881 9228 8661 8159 11 2.5149 1.7050 1.4390 1.2753 1.1566 1.0635 9869 921818652 8152 12 2.4771 1.6990 1.4357 1 2730 1.1549 1.0621 9858 9208 8643 8144 13 2.4424 1.6930 1.4325 1.2707 1.1532 1.0608 984'6 9198 8635 8136 14 2.4102 1.6871 1.4292 1.2685 1.1515 1.0594 9834 9188 8626 8128 15 2.3802 1.6812 1.4260 1.2663 1.1498 1.0580 9823 9178 8617 8120 16 2.3522 1.6755 1.4228 1.2640 1.1481 1.0566 9811 9168 8608 8112 17 2.3259 1.6698 1.4196 1.2618 1.1464 1.0552 9800 915818599 8104 18 2.3010 1.6642 1.4165 1.2596 1.1447 1.0539 9788 9148 8591 8097 19 2.2775. 1.6587 1.4133 1.2574 1.1430 1.0525 9777 9138 8582 8089 20 2.2553 1.6532 1.4102 1.2553 1.1413 1.0512 9765 9128 8573 8081 21 2.2341 1.6478 1.4071 1.2531 1.1397 1.0498 9754 9119 8565 8073 2212.2139 1.6425 1.4040 1.2510 1.1380 1.04841 9742 9109 8556 8066 23 2.1946 1.6372 1.4010 1.2488 1.1363 1.0471 9731 90991854718058 2412.1761 1.6320 1.3979 1.2467 1.1347 1.0458 9720 9089 8539 8050 25 2.15S4 i.6269 1.3949 1.2445 1.1331 1.0444 9708 9079 8530 8043 26 2.1413 1.6218 1.3919 1.2424 1.1314 1.0431 9697 9070i8522 8035 27 2.1249 1.6168 1.3890 1.2403 1.1298 1.0418 9686 9060 8513 8027 28 2.1091 1.6118 1.3860 1.2382 1.1282 1.0404 967.5 9050 18504 8020 29 2.0939 1.6069 1.3831 1.2362 1.1266 1.0391 9664 9041 18496 8012 30 2.0792 1.6021 1.3802 1.2341 1.1249 1.0378 9652 9031 8487 8004 31 2.0649 1.5973 1.3773 1.2320 1.1233 1.0365 9641 902118479 7997 32 2.0512 1.5925 1.3745 1.2300 1.1217 1.0352 9630 9012 8470 7989 33 2.0378 1.5878 1.3716 1.22799 1.1201 1.0339 9619 90021 8462 7981 34 2.0248 1.5832 1.3688 1.2259 i.. 186 1.0326 9608 8992 8453 79741 35 2.0122 1.5786 1.3660 1.2239 1.1170 1.0313 9597 8983 8445 7966 36 2.0000 1.5740 1.3632 1.2218 1.1154 1.0300 9586 897318437 7959 37 1.9881 1.5695 1.3604 1.2198 1.1138 1.0287 9575 8964 1 8428 7951 38 1.9765 1.5651 1.3576 1.2178 1.1123 1.0274 9564 8954 8420 7944 39 1.9652 1.5607 1,3549 1.2159 1.1107 1.0261 9553 8945 8411 7936 40 1.9542 1.5563 1.3522 1.2139 1.1091 1.0248 9542 8935 8403:7929 41 1.9435 1.5520 1.3495 1.2119 1.1076 1.0235 9532 8926 8395 7921 42 1.9331 1.5477 1.3468 1.2099 1.1061 1.0223 9521 8917 18386 7914 43 1.9228 1.5435 1.34411 1.2080 1.1045 1.0210 9510 8907 8378 7906 44 1.9128 1.5393 1.3415 1.2061 1.1030 1.0197 9499 8898 8370 7899 45 1.9031 1.5351 1.3388 1.2041 1.1015 1.0185 9488 8888 8361 7891 46 1.8935 1.5310 1.3362 1.2022 1.0999 1.0172 9478 8879 8353 1 7884 47 1.8842 1.5269 1.3336 1.2003 1.0984 1.0160 9467 8870 8345 7877 48 1.8751 1.5229 1.3310 1.1984 1.0969 1.0147 9456 8861 8337 7869 49 1.8661 1.5189 1.3284 1.1965 1.0954i 1.0135 9446 8851 8328 7862 50 1.8573 1.5149 1.3259 1.1946 1.093911.0122 9435 8842 8320 78551 51 1.8487 1.5110 I1.3233 1.1927 1.0924 1.0110, 9425 883318312 78471 62 1.8403 1.50711 1.3208 1.1908 1.0909 1.0098 9414 8824 8304 7840, 53 1.8320: 1.5032 1.3183 1.1889 1.0894 1.0085 9404 8814 8296 7832 5fi4 1.8239 11.4994! 1.3158 1.1871 1.0880 1.00731 9393 8805 8288 7825j 551.8159 1.4956 1.3133 1.1852 1.0865 1.0061 9383 8796 8279 78189 56 1.8081 1.4918 1.3108 1.1834 1.0850 1.0049 9372 8787 8271 7811 57 1.8004 1.4881 1.3083 1.1816 1.0835 1.0036 9362 8778 8263 i 7803 58 1.7929 1.4844 1.3059 1.1797 I.0821 1.0024 9351 8769 8255 7796 |59 1.7855 1.4808 1.3034 1.1779 1.0806 1.0012 9341 8760 8247 7789 6011.7782 1.4771 1.3010 1 17611 1.0792 1.0000 9331 875118239 7782 112 TABLE XCVI. Logistical Logarithms.' 10 1 12 13 14 15 16 17 18 19 120 1 600 360 720 780 840 900 960 1020 1080 1i140 120)0 12601 0 7782 7368 6990 6642 6320 601 5740 5477 5229 4994 4771 4559 1 7774 7361 6984 6637 6315 6016 5736 5473 5225 4990 4768 4556 2 7767 7354 6978 6631 6310 6011 5731 5469 5221 4986 4764 4552 3 7760 7348 6972 6625 6305 6006 5727 5464 5217 4983 4760 4549 4 7753 7341 6966 6620 6300 6001 5722 5460 5213 4979 4757 4546 5 7745 7335 6960 6614 6294 5997 5718 5456 5209 4975 4753 4542 6 7738 7328 6954 6609 6289 5992 5713 5452 5205 4971 4750 4539 7 7731 7322 6948 6603 6284 5987 5709 5447 5201 4967 4746 4535 8 7724 7315 6942 6598 6279 5982 5704 5443 5197 4964 4742 4532 9 7717 7309 6936 6592 6274 5977 5700 5439 5193 4960 4739 4528 10 7710 7302 6930 6587 6269 15973 5695 5435 5189 4956 4735 4525 11 7703 /296 6924 6:581 6264 5968 5691 5430 5185 4952 4732 4522 12 1 769F) 7289 6918 6576 6259 5963 5686 5426 5181 4949 4728 4518 13 7688 7283 6912 6570 6254: 5958 5682 5422 5177 4945 47241 4515 14 768i 7276 6906 6565 6248 5954 5677,5418,5173 4941 4721 4511 15 1767-4 7270 6900 6559 6243 5949 5673154114 51694937147174503 16 7667'964 6894 6554 6238 5944 5669 5409 5165 14933 4714 4505 17 7660 7257 6888 6548 623315939 5664 5405 5161 4930 4710 45!01 18 7653 7251 6882 6543 62r28 5935 5660 5401 5157 4926 4707 4498 19 7646 7244 6877 6538 6223 5930.5655 5397 5153 14922 4703 4494 20 7639 7238 6871 6532 6218 5925 5651 5393 5149 4918 4699 4491 21 7632 7232 6S65 6527 621315920 5646 5389 5145 4915 469614488 2217625 7225 6859 6521 6208 15916 5642 5384 514114911 4692 4484 2317618 7219 6853 6516 6203 5911 5637 5380 513714907 4689 4481 24 7611 7212 65847 6510 6198 5906 5633 5376 513314903 4685 4477 25 7604 7206 68541 6505 6193 5902 5629 5372 5129 14900 4682 4474 26 7.597 7200 6836 6500 6188 5897 F5624 5368 5i25 4896 4678 4471 27 7590 7193 6830 6494 6183 5892 5620 5364 5122148S924675 4467 2817583 7187 6824 64'89 61785888 5615 5359 5118 4889 4671 4464 2917577 7181' 6818 6484 6173:5883 5611 5355 5114 14885 4-668 4460 3017570 71';5 6812 6478 6168 15878 5607 5351 5110 4881 4664 44i57 31 7563 71686807 6473 6163 5874 5602 5347 510614877 4660 4454 32 7556 7162 66801 6467 6158 5869 5598 5343 5102 48'74 4657 4450 33 75491 7156 679 6462 6153 1 5864 5594 5339 5098 4870 4653 44147 3417542 7149 6789 64571 614815860 5589 5335 5094 4866 4'650 4444 35 7535 7143 784 6451 6143 5855 5585 5331.5090 4863 4646 4440 36 7528 7137 6778 6446 613815355 55580 5326 5086 4859'4643 4437 37 7522 7131 6772 6441 16133 5846 5576 5322 5082 4855 4639443131 38 17515 7124 6766 6435 6128 5841 5572 5318 5079 4852 46361 4430 39 7508 7118 16761 6430 612315836 5567 5314 5075 4948 4632 44127 40 7501 7112 6755- 6425 6118 5832 5563 5310 5071 48414 4629 44:,i 41 7494 7106 6749 6420 6113 5827 5559 5306 5067 4841 4625 442,0 42 17488 7100 6743 6414 6108 5823 5554 5302 506 3 4837 462' 41i l17 4317481 7093 6738 6409 6103 5818 5550 5298 5059 4833 4618S 4414 44 7474 7087 6732 6404 609915813 5546 5294 5055 4830 1615 110 4537467 7081 6726 6398 6094 5809 5541 5290 505i1 4826 4611 4407 4617461 7075 6721 6393 16089 5804 5537 5285 5048 148221460814404 47 7454 7069 6715 6388 6084 5800 5533 5281 5044 4819 14604 4400 48 7447 7063 6709 6383 6079 5795 5528 5277 5040 4815 4601 4397 49 7441 7057 6704 6377 6074 5790 5524 5273 5036 4811 4597 4394 5017434 7050 6698 6372 16069 5786 5520 5269 5032 480814594 4390 51 7427 7044 6692 6367 6064 5781 5516 5265 5028 4804 4590 4387 5217421 7038 6687 6362 6059 5777 5511 5261 502514800 4587 4384 53 7414 7032 6681 6357 6055 5772 5507 5 5021 4797 4584 43804 54 7407 7026 6676 6351 16050 15768 1 5503 1 5253 5017 4793 4580 1 4377 55 7401 7020 6670 6346 16045 5763 549815249 5013 4789 4577 4374 56 7394 7014 6664 634116040 5758 5494 5245 5009 476 14573 4370 57 7387 7008 6659 6336 6035 5754 54190 5241 5005 47R2 4570 42fi7 58 7381 7002 6653 6331 6030 5749 154 -Si527 5002 4778 456i6 1431644 591 7374 6996t; I i6648 6325 60325. 575 5.18 5233 4 499( | 4775. 4563 43661 60 7368 1 6990 i 642 6320) fi6(.57410) 5477 1 5229 49(94! 4771 4 559 4357 TABLE XCVI. Logistical Logarithms. 113'i 22 1 23 124 25 1 26j 27 28 29 30 31 1 32 33 1320 1380 1440 1500 1560 1620 1680 1740 18o0 1860 192() 180S 0 4357 4164 3979 3802 3632 3468 3310 3158 3010 2868 2730 2596 1 41354 4161 3976 3799 3629 3465 3307 3155 3008 286612728 2594 2 14351 4158 3973 3796 3626 3463 3305 3153 3005 2863 12725 2592 3! 4347 4155 3970 3793 3623 3460 3302 3150 3003 2861 2723 2590 4143444 i4152 3967 3791 3621 3457 3300 3148 3001 285912721 2588 5 4341 4149 3964 3788 3618 3454 3297 3145 2998 2856 2719 2585 - 6 433814145 3961 3785 3615 3452 3294 3143 2996 2854 2716 2583 714334 14142 3958 3782 3612 3449 3292 3140 2993 2.52 2714 2581 814331 4139 3955 3779 3610 -3446 3289 3138 2991 28,'9 2712 2579 9 432814136 3952 3776 3607 3444 3287 3135 2989 2847 2710 2577 101432514133 3949 3773 3604 3441 3284 3133 2986 2845 2707 2574 11 4'321 4130 3946 3770 3601 3438 3282 3130 2984 2842 27,05 2572 12 4318 4127 3943 3768 3598 3436 3279 3128 2981 2840 2703 2570 1314315 4124 3940 3765 3596 3433 3276 3125 2979 2838 2701 2568 14l4311 4120 3937 3762 3593 3431 3274 3123 2977 2835 2698 2566 1514308 14117 3934 3759 3590 3428 3271 3120 2974 2833 2696 2564 1614305 14114 3931 3756 3587 3425 3269 3118 2972 2831 2694 2561 147 43024111 3928 3753 3585 3423 3266 3115 2969 2828 2692 2559 1814298 4108 3925 3750 3582 3420 3264 3113 2967 2826 2689 2557 1914295 4105 3922 3747 3579 3417 3261 3110 2965 2824 2687 2555 20 4292 4102 3919 3745 3576 341~ 3259 3108 2962 2821 2685 2553 21 4289 4099 3917 3742 3574 3412 3256 3105 2960 2819 2683 2551 2214285 409613914 3739 3571 3409 3253 3103 2958 2817 2681 2548 2314282 4092 3911 373613568 340713251 3101 2955 2815 2678 2546 2414279 4089 3908 3733 3565 3404 3248 3098 2953 2812 2676 2544 2' 14276 4086 3905 3730 3563 3401 3246 3096 2950 2810 2674 2542 26 42073 4083 3902 3727 3560 339913243 3093 2948 2808 2672 2540 2714269 4080 3899 3725 3557 3396 3241 3091 2946 2805 2669 2538 28 4266 4077 3896 3722 3555 3393 3238 3088 2943 2803 2667 2535 2914263 4074 3893 3719 3552 3391 3236 3086 2941 2801 2665 2533 I30 4260 4071 3890 3716 3549 3388 3233 3083 2939 2798 26631 2531 3114256 4068 3887 3713 3546 3386 3231 3081 2936 2796 2660 2529 32142153 4065 3884 3710 3544 3383 13228 3078 2934 12794 2658 2527 33 4250 4062 3881 3708 3541 3380 3225 3076 2931 2792 2656 2525 3414247 405913878 3705 3538 3378 3223 3073 2929 2789 2654 2522 3514244 4055 3875 3702 3535 3375 3220 3071 2927 2787 2652 2520 3614240 4052 3872 3699 3533 3372 3218 3069 2924 2785 2649 2518 37 4237 4049 3869 3696 3530 3370 3215 3066 2922 2782 2647 2516 3814234 4046 3866 3693 3527 3367 3213 3064 2920 2780 2645 2514 39 4231 4043 3863 3691 3525 3365 3210 3061 2917 2778 2643 2512 40j4228 404013860 3688 3522 3362 3208 3059 2915 2775 2640 210 41 4224 4037 3857 3685 3519 335913205 3056 2912 2773 2638 2507 42 4221 4034 3855 3682 3516 335713203 3054 2910 2771 2636 2505 4314218 4031 3852 3679 3514 3354 3200 3052 2908 2769 2634 2503 4414215 4028 3849 3677 3511 3351 3198 3049 2905 2766 2632 2501 45 4212 4025 3846 3674 3508 3349 13195 3047 2903 2764 2629 2499 46 4209 4022 3843 3671 3506 3346 3193 3044 2901 2762 2627 2497 47 4205 4019 3840 3668 3503 3344 3190 3042 2898 2760 2625 2494 48 4202 4016 3837 3665 3500 3341 3188 3039 2896 12757 2623 2492 4914199 4013 3834 3663 3497 3338 3185 3037 2894 2755 2621 2490 150 4196 4010 3831 3660 3495 3336 3183 3034 2891 2753 2618 2488 51 4193 4007 3828 3657 3492 3333 13180 3032 2889 12750 26161 2486 52 4189 4004 3825 3654 3489 3331 3178 3030 2887 2748 2614 2484 5314S86 4001 3822 3651 3487 3328 3175 3027 2884 2746 2612 2482 4183 3998 3820 3649:3484 3325 3173 3025 2882 2744 2610 2480 55 4180 3995 3817 36461 3481 3323 3170 3022 2880 2741 2607 2477 56 14177 399113814 3643 13479 3320 3168 3020 2877 2739 2605 2475 5714174 39881 3811 3640 3476 3318 31.65 3018 2875 2737 2603 2473 58 4171 3985 3808 3637 13473 3315 3163 301F1 2873 12735 2601 2471 59 4167 3982 3805 3635 3471 3313 13160 3013 2870 2732 2599 2469 60 4164 3979 3802 3632 3468 3310 3158 301012868 2730 259612467 114 TABLE XCVJ..Logistical Lolarithms. 34 35 36 37 38 39 40 141 42 431 44 1 45, 2100 2160; 2220 2280 2340 2400 2460 2520 2580 2640 2700 0 2467 2341 2218 2099 1984 1871 1761 1654 1549 1447 -f1347 1249 1 2465 2339 2216 2098 1982 1869 1759 1652 1547 1445 1345 1248 2 2462 2337 2214 2096 1980 1867 1757 1650 1546 1443 1344 1246 3 2460 2335 2212 2094 1978 1865 1755 1648 1544 1442 1342 1245 4 2458 2333 2210 2092 1976 1863 1754 1647 1542 1440 1340 1243 5 2456 2331 2208 2090 1974 1862 1752 1645 1540 1438 1339 1241 6 2454 2328 2206 2088 1972 1860 1750 1643 1539 1437 1337 1240 7 2452 2326 2204 2086 1970 1858 174811641 1537 1435 1335 1238 8 2450 2324 2202 2084 1968 1856 174611640 1535 1433 1334 1237 9 2448 2322 2200 2082 1967 1854 174511638 1534 1432 1332 1235 10 2445 2320 2198 2080 1965 1852 1743 1636 1532 1430 1331 1233 11 2443 2318 2196 2078 1963 1850 1741 1634 1530 1428 1329 1232 12 2441 2316 2194 2076 1961 1849 1739 1633 1528 1427 1327 1230 13 2439 2314 2192 2074 1959 1847 1737 1631 1527 1425 1326 1229 14 2437 2312 2190 2072 1957:845 1736 1629 1525 1423 1324 1227 15;2435,2310 2188 2070 1955 1843 1734 1627 1523 1422 1322 1225 16 2433 2308 2186 2068 1953 1841 1732 1626 1522 1420 1321 1224 17 2431 2306 2184 2066 1951 1839 1730 1624 1520 1418 1319 1222 18 2429 2304 2182 2064 1950 1838 1728 1622 1518 1417 1317 1221 19 2426 2302 2180 2062 1948 1836 1727 1620 1516 1415 1316 1219 20 2424 2300 2178 2061 1946 1834 1725 1619 1515 1413 1314 1217 21 2422 2298 2176 2059 1944 1832 1723 1617 1513 1412 1313 1216 262 2420 2296 2174 2057 1942 1830 1721 1615 151111410 1311 1214 23 2418 2294 2172 2055 1940 1828 1719 1613 1510 1408 1309 1213 24 2416 2291 2170 2053 1938 1827 1718 1612 1508 1407 1308 1211 25 2414 2289 2169 2051 1936 1825 1716 1610 1506 1405 1306 1209 26 2412 2287 2167 2049 1934 1823 171411608 1504 1403 1304 1208 27 2410 2285 2165 2047 1933 1821 1712 1606 1503 1402 1303 1206' 28 2408 2283 2163 2045 1931 1819 1711 1605 1501 1400 1301 1205 29 2405 2281 2161 2043 1929 1817 1709 1603 1499 1398 1300 1203 30 2403 2279 2159 2041 1927 1816 1707 1601 1498 1397 1298 1201 31 2401 2277 2157 2039 1925 1814 1705 1599 1496 1395 1296 1200 32 2399 2275 2155 2037 1923 1812 1703 11598 1494 1393 1295 1198 33 2397 2273 2153 2035 1921 1810 1702 1596 1493 1392 1293 1197 34 2395 2271 2151,2033 1919 1808 1700 1594 1491 1390 1291 1195 35 2393 2269 2149 2032 1918 1806 1698 1592 1489 1388 1290 1193 36 2391 2267 2147 2030 1916 1805 1696 1591 1487 1387 1288 1192 37 2389 2265 2145 2028 1914 1803 1694 1589 1486 1385 1287 1190 38 2387 2263 2143 2026 1912 1801 1693 1587 1484 1383 1285 1189 39 2384 2261 2141 2024 1910 1799 1691 1585 1482 1382 1283 1187 40 2382 2259, 2139 2022 1908 1797 1689 1584 1481 1380 1282 1186 41 2380 2257 2137 2020 1906 1795 1687 1582 1479 1378 1280 1184 42 2378 2255 2135 2018 1904 1794 1686 1580 1477 1377 1278 1182 43 2376 2253 2133 2016 1903 1792 1684 1578 1476 1375 1277 1181 44 2374 2251 2131 2014 1901 1790 1682 1577 1474 1373 1275 1179 145 2372 2249 2129 2012 1899 1788 1680 1575 1472 1372 1274 1178 546 2370 2247 2127 2010 1897 1786 1678 1573 1470 1370 1272 1176 47 2368 2245 2125 2009 1895 1785 1677 1571 1469 1368 1270 1174 48 2366 2243 2123 2007 1893 1783 1675 1570 1467 1367 1269 1173 49 2364 2241 2121 2005 1891 1781 1673 1568 1465 1365 1267 1171 50 2362 2239 2119j2003 1889 1779 1671 1566 1464 1363 1266 1170 51 2359 2237 2117 2001 1888 1777 1670 1565 1462 1362 1264 1168 52 2357 2235 2115 1999 1886 1775 1668 1563 1460 1360 1262 1167 53 2355 2233 2113 1997 1884 17741166611561 1459 1359 1261 1165 54 2353 2231 2111 1995 1882 1772 1664 1559 1457 1357 1259 11163 155 2351 2229 2109 1993 1880 1770 1663 15583'145h5 1355 1257 i1162 56 2349 2227 2107 1991 1878 1768 1661 1556 1454 1354 1256 1160 57 2347 2225 2105 1989 1876 1766 1659 1554 1452 1352 1254 1159 58 12345 2223 2103 1987 1875 1765 1657 1552 1450 1350 1253 1157 59 2343 |2220 2101 1986 1873 1763 1655 1551 1449 1 34 1251 1156 60 2341 2218. 2099i1984 1871 1761 1654 154-9 1447 11347 1249 1 1 54 TABLE XCVI. Logizstical Logartthms. 115!t 46 47 48 49 I 50 1 51 52 1 53 54 55 56 57 1_58 I 59' 2760 2820 2880 2940 3000 3060 3120 3180 3240 3300 3360 3420 3480 35401 0 1154 1061 0969 0880 0792 0706 0621 0539 0458 0378 0300 0223 0147 0073 1 1152 1059 0968 0878 0790 0704 0620 0537 0456 0377 0298 0221 01456 0072 2 1151 1057 0966 0877 0789 0703 0619 0536 0455 0375 0297 0220 0145 0071 3 1149 1056 0965 0875 0787 0702 0617 0535 0454 0374 0296 0219 0143 0069 4 1148 1054 0963 0874 0786 0700 0616 0533 0452 0373 0294 0218 0142 0068 5 1146 1053 0962 0872 0785 0699 099 0615 0532 0451!0371 0293 0216 0141 0067 6 114.5 1051 0960 0871 0783 0697 0613 0531 0450 0370 0292 0215 0140 00661 7 1143 1050 0959 0869 0782 0696 0612 0529 0448 0369 0291 0214 0139 00641 8 1141 1048 0957 0868 0780 0694 0610 0528 0447 0367 0289 0213 0137 0063 911140 1047 0956 0866 0779 0693 0609 0526 0446 0366 0288 0211 0136 0062 1011138 1045 0954 0865 0777 0692 0608 0525 0444 03651 0287 0210 0135 0061 11 1137 1044 0953 0863 0776 0690 0606 0524 0443 0363 0285 0209 0134 0060 12 1135 1042 0951 0862 0774 0680689 0605052 0442 0362 0284 0208 0132 0058 13 1134 1041 0950 0860 0773 0687 0603 0521 0440 0361 0283 0206 0131 0057 14 1132 1039 0948 0859 0772 0686 0602 0520 0439 0359 0282 0205 0130 0056 15 1130 1037 0947 0857 0770 0685 0601 0518 0438 0358 0280 0204 0129 0055 16 1129 1036 0945 0856 0769 0683 0599 051I 0436 0357 0279 0202 0127 0053 17 1127 1034 0944 085510767 0682 0598 0516 0435 0356 0278 0201 0126 0052 118 1126 1033 09421085310766 0680 0596 0514 0434 0354 0276 0200 012510051 19 1124 1031 0941 085210764 0679 0595 0513 0432 0353 0275 0199 012410050 20 1123 1030 0939 0850 0763 0678 0i594 0512 0431 0352 0274 0197 0122 0049 21 1121 1028 0938 0849 0762 0676 0592 0510 0430 0350 0273 0196 0121 0047 22 1119 1027 0936 0847 0760 0675 0591 0509 0428 0349 0271 0195 01201 0046 23 1118 1025 0935 0846 0759 0673 0590 0507 0427 0348 0270 0194 0119 0045 24 1116 1024 0933 0844 0757 0672 0588 0506 0426 0346 0269 0192 0117 0044 25 1115 1022 0932 0843 0756 0670 0587 0505 0424 0345 0267 0191 0116 0042 26 1113 1021 0930 0841 0754 0669 0585 0503 0423 0344 0266 0190 0115 0041 27 1112 1019 0929 0840 0753 0668 0584 0502 0422 0342 0265 0189 0114 00401,28 1110 1018 0927 0838 0751 0666 0583 0501 0420 0341 0264 0187 0112 0039 29 1109 1016 0926 0837 0750 0665 0581 0499 0419 0340 0262 0186 0111 0038 30 1107 1015 0924 0835 0749 0663 0580 0498 0418 0339 0261 0185 0110 00361 31 1105 1013 0923 0834 0747 0662 0579 0497 0416 0337 0260 0184 0109 0035 32 1104 1012 0921 0833 0746 0661 0577 0495 0415 0336 0258 0182 010710034 33 1102 1010 0920 0831 0744 0659 0576 0494 0414 0335 0257 0181 0106 0033.34 1101 1008 0918 0830 0743 0658 0574 0493 0412 0333 0256 0180 0105 0031 35 1099 1007 0917 0828 0741 0656 0573 0491 0411 0332 0255 0179 0104 0030 36 1098 1005 0915 0827 0740 0655 0572 0490 0410 0331 0253 0177 010310029 37 1096 1004 0914 0825 0739 0654 0570 0489 0408 0329 0252 0176 0101 0028 38 1095 1002 0912 0824 0737 0652 0569 0487 0407 0328 0251 0175 010010027 39 1093 1001 0911 0822 0736 0651 0568 0486 0406 0327 0250 0174 0099 0025 40 1091 0999 0909 0821 0734 0649 0566 0484 0404 0326 0248 0172 0098 0024 41 1090 0998 0908 0819 0733 0648 0565 0483 0 083 0324 0247 01 00961 0023 42 1088 0996 0906 0818 0731 0647 0563 0482 0402 0323 0246 0170 0095 0022 43 1087 0995 0905 0816 0730 0645 0562 0480 040010322 0244 0169 0094 0021 44 1085 0993 0903 0815 0729 0644 0561 0479 0399 10320 0243 01671 0093 0019 45 108410992 0902 0814 0727 0642 0559 0478 0398 0319 0242 0166 0091 0018 46 1082 0990 0900 P912 0726 0641 0558 0476 0396 0318 0241 0165 0090 0017 47 1081 0989 0899 0811 0724 0640 0557 0475 0395 0316 0239 0163 008910016 48 1079 0987 0897 0809 0723 063810555 0474 0394 0315 0238 0162 00880015 4911078 0986 0896 0808 0721 063710554 0472 0392 0314 0237 0161 0087100131 50 1076 098410894 0806 0720 06350552 0471 0391 0313 0235 0160 0085 0012 51 1074 0983 0893 0805 0719 0634 0551 0470 0390 0311 0234 0158 0084 0011 52! 1073 0981 0891 0803 0717 0633 0550 0468 0388 0310 0233 0157 0083 0010 53 1071 0980 0890 0802 0716 0631 0548 0467 0387 0309 0232 0156 0082 0008 541 1070 09780888 080110714 0630 0547 0466 0386 0307 0230 0155 0080 00071 55 1068 0977i0887 07991 0713 i 0628 0546 0464 03841030610229 0153 0079j 0006 56 1067 0975 j0885 0798 0711 0627 0544 0463 0383 0305 02,28 0152 0078 0005 57 1065 0974 088410796 10710 0626 0543 0462 0382 0304 0227 0151 0077 0004 58 1064 0972 0883 0795 0709 0624 0541 0460 0381 03021 0225 0150 0075 0002 159 1062 0971 0881 0793 0707 0623 0540 0459 0379 0301 0224 0148 100740001 60, 1061 0969 0880 0792 0706 0621 10539 0458 037S 8 0300 J 0223 0147 0073 00001.~~ ~ _... ~ l,lll PLATE I. —EQUATORIAL TELESCOPE OF THE OBSERVATORY OF HARVARD COLLEGE, PLATE II. —TOTAL ECLIPSE OF THE SUN, OF JULY 18, 1860, AS OBSERVED BY I DR FEILITZSCH, AT CASTELLON DE LA PLANA. PL),ATE Il, —-])ONATI'$f COMFT. 1ATsH IYV. —.OLU'ST:RS AND NN'UI,,M. *~-Y PJ^'ck: V.-NJuuk.s H FIG. 5. 13z FIG. 6. -- E j \g El~ ~ ~ FI \ *~~~~~~~:'.8 5 ~~ ~~ j......... i?.......~/....~~~~n z~~~~~~~~~.~......'"\..'....~~~] ~./ I~ "~ -. "~~~~'".' L R~~~~~~~~~~~~~~~~~~~~~~ W~~~~~~ a _ I I TR ~ I ~B~~~~~~~~~~~~~~~~~~~c - - B -1 (i I~~~~~~~~~~~~ _,, C pFIGe. 19. -' ]lB.~ ~~ 3 FaG. FIG. 23. I?- ~ FiLO 52. Po o ~ T FIG. 28. p fl A FiG. 6 2. M FIG. 61. G D: FIG. 65. FIG. 77. AFIG. r r FIG. $2. \........... —-—; —--------- F Mf' FIG. 84. t FIG98 I FIG. 98. 2180 F-g. 101. R/ FIG. 110. N FIG. 112. I -- N E N d o ih.1 FAG. 118. New York, August, 1873. JOHN WILEY & SON'S LIST OF PUBLICATIONS, 15 ASTOR PLACE, Under the Mereantile Library and Trade &Suerooms. AGRICULTURE. DOWNING. FRUITS AND FRUIT-TREES OF AMERICA; or the Culture, Propagation, and iManagement in the Garden and Orchard, of Fruit-trees generally, with descriptions of all the finest varieties of Fruit, Native and Foreign, cultivated in this country. By A. J. Downing. Second revision and correction, with large additions. By Chas. Downing. 1 vol. 8;o, over 1100 pages, with several hundred outline engravings. Price, with Supplement for 1872........................... $5 00 " As a work of reference it has no equal in this cllntry, and deserves a place in the Library of every Pornologist in America."-Jfarsshall P. TWilder. 66 ENCYCLOPEDIA CE' FRUITS; or, Fruits and FruitTrees of America. Part 1.-APPLES. With an Appendix containing many new varieties, and brought down to 1872. By Chas. Downing. With numerous outline engravings. 8vo, full cloth......................................... $2 50 ENCYCLOPEDIA OF FRUITS; or, Fruits and FruitTrees of America. Part 2.-CHEIIRIES, GRAPES, PEACIIES, PEARS, &C. With an Appendix containing many new varieties, and brought down to 1872. By Chas. Downing. With numerous outline engravings. 8vo, full cloth......... $2 50 FRUITS AND FRUIT-TREES OF AME:5RICA. By A. J. Downing. First revised edition. By Chas. Downing 12nmo, cloth............................................ $2 00 as 683ELECTED F'RUITS. From Downing's Fruits and FruitTrees of America. With some new varieties, including their Culture, Propagation, and Management in the Garden and Orchard, with a Guide to the selection of Fruits, with reference to the Time of Ripening. By Chas. Downing. Illustrated with upwards of four hundred outlines of Apples, Cherries, Grapes, Plums, Pears, &c. 1 vol., 12mo.... $2 50'* fALOUDON'S GARDENING FOR LADIES, AND COMPANION TO THE FLOWER-GARDEN. Second American from third London edition. Edited by A. J. Downing. 1 vol., 12tno.................$......... 2 00 DOWNING & THE THEORY OF HORTICULTURE, By J. Lindley. LINDLEY. With additions by A. J. Downing. 12mo, cloth.......$2 00 DOWNING. COTTAGE RESIDENCES. A Series of Designs for Rural Cottages and Cottage Villas, with Garden Grounds. By A. J. Downing. Containing a revised List of Trees, Shrubs, and Plants, and the most recent and best selected Fruit, with some account of the newe tsoyle of Gardens. By Henry Winthrop Sargent and Charles Downing. With many new designs in Rural Architecture. By George E Harney, Architect. 1 vol. 4to........................... $6 00 92 JOHN WILEY & SON'S LIST OF PUBLICATIONS. DOWNING & HINTS TO PERSONS ABOUT BUILDING IN THEI WIGHTWICK. COUNTRY. By A. J. Downing. And HINTS TO YOUNG ARCHITECTS, calculated to facilitate their practical operations. By George Wightwick, Architect. Wood engravings. 8vo, cloth....................... $2 00 KEMP. LANDSCAPE GARDENING; or, How to Lay Out a Garden. Intended as a general guide in choosing, forming, or improving an estate (from a quarter of an acre to a hundred acres in extent), with reference to both design and execution. With numerous fine wood engravings. By Edward Kemp. 1 vol. 12mo, cloth.........................$2 50 LIEBIG CHEMISTRY IN ITS APPLICATION TO AGRICULTURE, &c. By Justus Von Liebig. 12mo, cloth.... $1 00 LETTERS ON MODERN AGRICULTURE. By Baron Von Liebig. Edited by John Blyth, MI.D. With addenda by a practical Agriculturist, embracing valuable suggestions adapted to the wants of American Farmers. 1 vol. 12mo, cloth................................... $1 00 9' PPRINCIPLES OF AGRICULTURAL CHEMISTRY, with special reference to the late researches made in England. By Justus Von Liebig. 1 vol. 12mo................. 75 cents. PARSONS. HISTORY AND CULTURE OF THE ROSE. By S. B. Parsons. 1 vol. 12mo............................ $1 25 ARCH ITECTURE. DOWNING. COTTAGE RESIDENCES; or, a Series of Designs for Rural Cottages and Cottage Villas and their Gardens and Grounds, adapted to North America. By A J. Downing. Containing a revised List of Trees, Shrubs, Plants, and the most recent and best selected Fruits. With some account of the newer style of Gardens, by Henry Wentworth Sargent and Charles Downing. With many new designs in Rural Architecture by George E. Harney, Architect.........................$6 00 DOWNING & HINTS TO PERSON3 ABOUT 1BUILEDING IN THE WIGHTWICK. COUNTRY. By A. J. Downing. And HINTS TO YOUNG ARCHITECTS, calculated to facilitate their practical operations. By George Wightwick, Architect. With many wood-cuts. 8vo, cloth..$................$2 00 iHATFIELD. THE AMERICAN HOUSE CARP?; ITER. A Treatise upon Architecture, Cornices. and Mouldings, Framing, Doors, Windows, and Stairs; together with the most important principles of Practical Geometry. New, thoroughly revised, and improved edition, with about 150 additional pages, and numerous additional plates. By R. G. Hatfield. 1 vol. vo............................................... $3 50 NOTICES OF TIIE WORK. "The clearest and most thoroughly practical work on t-he subject." "This work is a most excellent one, very comprehensive, and lucidly arranged." "This work commends itself by its plractical excellence." " It is a valuable addition to the library of the architect, and almost indlspensabie to every scientific master-mechalnic.'-R. R1. Journal. HOLLY CARPENTERS' AND JOINERS' HEAND-BOOK, containing a Treatise on Framing, Roofs, etc., and useful Rules and Tables. By H. W. Holly. 1 vol. 18tmo, cloth........$0 75 THE ART OF SAW-FILING SCIENTIFICALLY TREATED AND EXPLAINED. With Directions for putting in order all kinds of Saws. By H. W. Holly. 18mo, cloth..............................................$0 75 RUSKIN SEVEN LAMPS OF ARCHITECTURE. 1 vol. 12mo, cloth, plates..................................... $1 75 JOHN WILEY & SON S LIST OF PUBLICATIONS. 93 RUSKIN. LECTURES ON ARCHITECTURE AND PAINTING. 1 vol. 12mo, cloth, plates...........................$1 560 LECTURE BEPFORE SOCIETY OF ARCHITECTS. 0 15 WOOD. A TREATISE ON THE RESISTANCE OF' MA. TERTALS, and an Appendix on the Preservation of Timber. By De Volson Wood, Prof. of Engineering, University of Michigan. 1 vol. 8vo, cloth........................$2 50 This work is used as a Text-Book in Iowa University, Iowa Agricultural College, Illinois Industrial University, Sheffield Scientific School, New HIaven. Cooloer Institute, New York. Polytechnic College, Brooldyn, University of Michigan, and other institutions. A TREATISE ON BRIDGES. Designed as a Text-book and for Practical Use. By De Volson Wood. 1 vol. 8vo, numerous illustrations,....................... $3 00 ASSAYINGC-ASTRONOMY. BODEMANN. A TREATISES ON THE ASSAYING OF LEAD, SILVER, COPPER, GOLD, AND MERCURY. By Bodemann and Kerl. Translated by W. A. Goodyear. 1 vol. 12nao, cloth............................................. 50 MITCHELL. A MANUAL OF PRACTICAL ASSAYING. By John 3Mitchell. Third edition, edited by William Crookes. 1 vol. thick 8vo, cloth...........................$10 00 NORTON. A TREATISE ON ASTRONOMY, SPHERICAL AND PHYSICAL, with Astronomical Problems and Solar, Lunar, and other Astronomical Tables for the use of Colleges and Scientific Schools. By William A. Norton. Fourth edition, revised, remodelled, and enlarged. Numerous plates. Svo, cloth........................................... $3 53 BIBLES, &c. BAGSTER. THE COMMENTARY WHOLLY BIBLICAL. Contents: -The Commentary: an Exposition of the Old and New Testaments in the very words of Scripture. 2264 pp. II. An outline of the Geography and History of the Nations mentioned in Scripture. III. Tables of AMeasures, Weights, and C ins. IV, An Itinerary of the Children of Israel froim Egypt to the Promised Land. V. A Chronological conmparative Table of the Kings and Prophets of Israel and Judah. VI. A Chart of the World's History from Adam to the Third Century, A. D. VII. A complete Series of Illustrative iIMaps. IX. A Chronological Arrangement of the Old and New Testaments. X. An Index to Doctrines and Subjects, with numerous Selected Passages, quoted in full. XI. An Index to the Names of Persons mentioned in Scripture. XII. An Index to the Names of Places found in Scripture. XIII. The Names, Titles, and Characters of Jesus Christ our Lord, as revealed in the Scriptures, methodically arranged. 2 volumes 4to, cloth................................ $19 50 2 volumes 4to, half morocco, gilt edges............. 26 00 2 volumes 4to, morocco, gilt edges................. 35 00 3 volumes 4to, cloth.......................:....... 20 00 3 volumes 4to, half morocco, gilt edges............. 33 00 3 volumes 4to, morocco, gilt edges.................. 40 00 BLANK-PAGED TNiE HOLY SCRIPTURES OF THE OLD AND NEW BIBLE. TESTAMENTS; with copious references to parallel and illustrative passages, and the altemnate pages ruled for MS. notes. This edition of the Scriptulres contains the Authorized Version, illustrated by thle references of " Bagster's Polyglot Bible," and enriched with accurate maps, useful tables, and an Index of Subjects. 1 vol. 8vo, half morocco........................$.. 9 00 1 vol. 8vo, morocco extra...........................10 50 1 vol. 8vo. full morocco............................12 CO0 94 JOHN WILEY & SON'S LIST O0 PUBLICATIONS. THE TREASURY Containing the authorized English version of the Holy Scriptures, BIBLE. interleaved with a Treasury of more than 500,000 Parallel Passages from Canue, Brown, Blayney, Scott, and others. With numerous illustrative notes. I vol., half bound.................................$7 50 1 vol., morocco....................................10 00 COMMON PRAYER, 48mo Size. (Done in Londont expresslyfor us.) COMMON No. 1. Gilt and red edges, imitation morocco...........$0 624 PRAYER. No. 2. Gilt and red edges, rims....................... 7 No. 3. Gilt and red edges, best morocco and calf....... 1 25 No. 4. Gilt and red edges, best morocco and calf, rims.. 1 50 BOOK-KEEPING. JONES. BOOKKEEPING AND ACCOUNTANTSHIP. Elementary and Practical. In two parts, with a Key for Teachers. By Thomas Jones, Accountant and Teacher. 1 volume Svo; cloth.............................................$2 50 64 3Z 00aBOOKKEEPING AND ACCOUNTANTSHIP. School Edition. By Thomas Jones. 1 vol. 8vo, half roan....... $1 50 t6 BOOKKEEPING AND ACCOUNTANTSHIP. Set of Blanks. In 6 parts. By Thomas Jones.............. $1 50 BOOKKEEPING AND ACCOUNTANTSHIP. Double Entry; Results obtained from Single Entry; Equation of Payments, etc. By Thomas Jones. 1 vol. thin 8vo... $0 75 CHEMISTRY. CRAFTS. A SHORT COURSE IN QUALITATIVE ANALYSISI with the new notation. By Prof. J. M. Crafts. Second edition. 1 vol. 12mo, cloth........................ $1 50 JOHNSON'S A MANUAL OF QUALITATIVE CHEMICAL ANALYFRESENIUS. SIS. By C. R. Fresenius. Edited by S. W. Johnson, Professor in Sheffield Scientific School, Yale College. With Chemical Notation and Nomenclature, old and new. 1 vol. 8vo, cloth..$..........................$4 50 A SYSTEM OF INSTRUCTION IN QUANTITATIVE CHEMICAL ANALYSIS. By C. R. Fresenius. From latest editions, edited, with additions, by Prof. S. W. Johnson. With Chemical Notation and Nomenclature, old and new..$...........................................$6 00 KIRKWOOD OOLLECTION OF REPORTS (CONDENSED) AND OPINIONS OF CHEMISTS IN REGARD TO THE USE OF LEAD PIPE FOR SERVICE IPIPE, in the Distribution of Water for the Supply of Cities. By Jas. P. Kirkwood. 8vo, cloth.............................$1 50 MILLER. ELEMENTS OF CHEMISTRY, THEORETICAL AND PRACTICAL. By Wm. Allen Miller. 3 vols. 8vo.. $1, 0(1 44 Part I.-CHEMICAL PHYSICS. 1 vol. 8vo...........$4 00 4' Part II.-INORGANIC CHEMISTRY. I vol. 8vo..... 6 0f *6 Part III.-ORGANIC CHEMISTRY. 1 vol. 8vo....... 1) 0C "Dr. Miller's Chemistry is a work of which the author has every reason to fee. proud. It is now by far the largest and most accurately written Treatise on Chemistry in the English language," etc.-Dublin M~ed. Journal. 66 ~ IbZMAGNETISM AND ELECTRICITY. By Wm. Allen Millei. 1 ol. 8vo........................................$2 50 JOHN WILEY & SON'S LIST OF PUBLICATIONS. 95 ifUSPRATT. CHEMIISTRY - THEORETICAL, PRACTICAL, AND ANALYTICAL-as applied and relating to the Arts and Manufactures. By Dr. Sheridan Muspratt. 2 vols. 8vo, cloth, $18.00; half russia........................... $24 00 NOAD. A MdANUAL OF QUALITATIVE AND QUANTITATIVE CHEIMICAL ANALYSIS. For the use of Students. By H. M. Noad, author of "Manual of Electricity." 1 vol. 12mo. (London.) Complete.......................$6 00 c BJQUANTITATIVE ANALYSIS. 1 vol. cloth.......... 4 00 PERKINS. AN ELEMENTARY MANUAL OF QUALITATIVE CHEM1VICAL ANALYSIS. By Maurice Perkins. 12mo, cloth..............................................$1 00 DRAWING AND PAINTING. BOUVIER HANDBOOK ON OIL PAINTING. Handbook of Young AND OTHERS. Artists and Amateurs in Oil Painting; being chiefly a condensed compilation from the celebrated Manual of Bouvier, with additional matter selected from the labors of Merriwell, De Montalbert, and other distinguished Continental writers on the art. In 7 parts. Adapted for a Text-Book in Academies of both sexes, as well as for self-instruction. Appended, a new Explanatory and Critical Vocabulary. By an American Artist. 12mo, cloth....................$2 00 COE. PROGIRESSIVE DRAWING BOOK. By Benj. H. Coe. One vol., cloth.............................$3 50 DRAV ING FOR LITTLE FOLiS; or, First Lessons for the Nursery. 30 drawings. Neat cover.,.......... $0 20 FIRST STUDIES IN DRAWING. Containing Elementary Exercises, Drawings from Objects, Animals, and Rustic Figures. Complete in tree number's of 18 studies each, in neat covers. Each........................0......... 020 $4 COTTAGES. An Introduction to Landscape Drawing. Contaibdiel 72 St7lies. Complete in four numbers of 18 studies each, in neat covers. Each................... $0.20 $4 EASY LESSONS IN LANDSCAPE. Complete in four numbers of 10 Studies each. In neat 8vo cover. Each, $0 20 a4 - HlEADS, ANIMALS, AND FIGURES. Adapted to Pencil Drawing. Complete in three numbers of 10 Studies each. In neat 8vo covers. Each.......................,$0 20 COPY BOOK, WITH INSTRUCTIONS.............$0 37i RUSKIN. THE ELEMENTS OF DRAWING. In Three Letters to Beginners. By John Ruskin. 1 vol. 12ino........... $1 00 "I THE ELEMENTS OF PERSPECTIVE. Arranged for the use of Schools. By John Ruskin................ $1 00 SMITH. A MANUAL OF TOPOGRAPHICAL DRAWING. By Prof. R. S. Smith. Second edition. 1 vol. 8vo, cloth, plates.................................$2 00 MANUAL OF LINEAR PERSPECTIVE. Form, Shade, Shadow, and Reflection. By Prof. R. S. Smith. 1 vol. 8vo, plates, cloth...... 00 WARREN. CONSTRUCTIVE GEOMETRY AND INDUSTRIAL DRAWING. By S. Edward Warren, Professor in the Massachusetts Institute of Technology, Boston:I. ELEMENTARY WOR S. 1. ELEMENTARY FREE-HAND GEOMETRICAL DRAWING. A series of progressive exercises on regular lines and forms, including systematic instruction in lettering; a training of the eve and hand for all who are learning to draw. 12mo, cloth, many cuts...................... 75 cts. Vols. 1 and 3, bound in 1 vol........................$1 75 96 JOHN WILEY & iSON'S LIST OF PUBLICATIONS. ELEMENTARY WORKS. -Continued. WARREN 2. PLANE PROBLEMS IN ELEMENTARY GEOMETRY. Wit% numerous wood-cuts. 12mo, cloth.................. $1 25 3. DRAFTING INSTRUMENTS AND OPERATIONS. Containing full information about all the instruments and materials used by the draftsmen, with full directions for their use. With plates and wood-cuts. One vol. 12mo, cloth, $1 25 4. ELEMENTARY PROJECTION DRAWING. Revised and enlarged edition. In five divisions. This and the last volume are favorite text-books, especially valuable to all Mechanical Artisans, and are particularly recommended for the use of all higher public and private schools. New revised and enlarged edition, with numerous wood-cuts and plates. (1872.) 12mo, cloth............................................. $1 50 5. ELEMENTARY LINEAR PERSPECTIVE OF FORMS AND SHADOWS. Part I.-Primitive Methods, with an Introduction. Part II.-Derivative Methods, with Notes on Aerial Perspective. and many Practical Examples. Numerous woodcuts. 1 vol. 12mo, cloth..........................$1 00 II. HIGHER WORKS. These are designed principally for Schools of Engineering and Architecture, and for the members generally of those professions; and the first three are also designed for use in those colleges which provide courses of study adapted to the preliminary general training of candidates for the scientific professions, as well as for those technical schools which undertake that training themselves. 1. GENERAL PROBLEMS OF ORTHOGRAPHIC PROJECTIONS. The foundation course for the subsequent theoretical and practical works. A new edition of this work will soon appear. 2. GENERAL PROBLEMS OF SHADES AND SHADOWS. A wider range of problems than can elsewhere be found in English, and the principles of shading. 1 vol. 8vo, with numerous plates. Cloth........................... $3 50 3. HIGHER LINEAR PERSPECTIVE. Distimguishedc by its concise summary of various methods of perspective construction; a full set of standard problems, and a careful discussion of special higher ones. With numerous large plates. Svo, cloth............................................. 4 00 2. ELEMIENTS OF 3MACHINE CONSTRUCTION AND DRAW~ING; or, Machine Drawings. With some elements of descriptive and rational cinematics. A Text-Book for Schools of Civil and Mechanical Engineering, and for the use of Me'chanical Establishments, Artisans, and Inventors. Containing the principles of gearings, screw propellers,.valve motions, and governors, and many standard and novel examples, mostly from present American practice. By S. Edward Warren. 2 vols. 8vo. 1 vol. text and cuts, and 1 vol. large plates..,,.$7 50 A FEW FRO1M MANY TESTIMONIALS. "It seems to me that your Works only need a thorough examination to be introduced and permanently used in all the Scientific and Engineering Schools." -Prof. J. G. FOX. Collegiate and Enigieering Iistitute, New lork2: Citl/. "I have used several of your Elementary Works, and believe them to be better adapted to the purposes of instruction than any others with which I am acquainted."-H. F. WALLING, Prof. of Civil and Topographical Ezgineering, Lafayette College, Lastosn. Pa. "Your Works appear to me to fill a very important gap in the literature of the subjects treated. Any effort to draw Artisans, etc., away from the'rule of thumb,' and give them an insight into principles, is in the righ-t direci;ion, and meets my heartiest approval. This is the distinguishing feature or rotu ]Elementary Works."-Prof. H. L. EUSTIS, Lawrence Sciestidflc Swchool, Caemb7ridcge, iass. #The author has happily divided the subjects into two great portions: the fol mer embracing those processes and problems proper to be taught to all students in Institutions of Elementary Instruction; the latter, those suited to advanced students preparing for technical purposes. The Elementary Books ought to be used in all High Schools and Academies; the Higher ones in schools of'Technology." —WM. W. FOLWELL, President of Univterity of Minneuota. JOHN WILEY & SON S LIST OF PUBLICATIONS. 97 DYEING, &c. IACFARLANE. A PRACTICAL TREATISE ON DYEING AND CIALICOPRINTING. Including the latest Inventions and Improvements. With an Appendix, comprising definitions of chemical terms, with tables of Weights, Measures, &c. By an experienced Dyer. With a supplement, containing the most recent discoveries in color chemistry. By Robert Macfarlane. 1 vol. Svo........................................ $5 00 REIMANN. A TREATIS TE ON THE MANUFACTURE OF ANILINE AND ANILINE COLORS. By M. Reimann. To which is added the Report on the Coloring Matters derived from Coal Tar, as shown at the French Exhibition, 1867. By Dr. Hofmann. Edited by Wm. Crookes. 1 vol. 8vo, cloth, $2 50 ", Dr. Reimann's portion of the Treatise, written in concise language, is profoundly practical, givingll toe minutest details of the processes for obtaining all the more important colors, with woodcuts of apparatus. Taken in conjunction with Hofmann's Iteport, we have now a complete history of Coal Tar Dyes, both theoretical and practical."-/zsenzist and Druzggist. ENGINEERING. AUSTIN. A PRACTICAL TIREATISE ON THE PREPARATION, COMBINATION, AND APPLICATION OF CALCAREOUS AND HYDRAULIC LIMES AND CEMENTS. To which is added many useful recipes for various scientific, mercantile, and domestic purposes. By James G. Austin. I vol. 12mo...................................... $2 00 COLBURN LOCOMOTIVE ENGINEERING AND THE MECHANISM OF RAILWAYS. A Treatise on the Principles and Construction of the Locomotive Engine, Railway Carriages, and Railway Plant, with examples. Illustrated by Sixty-four large engravings and two hundred and forty woodcuts. BY Zerah Colburn. Complete, 20 parts, $15.00; or 2 vols. cloth.................. $16 00 Or, half morocco, gilt top.........2..0......... 20 00 KNIGHT. THE MECHANICIAN AND CONSTRUCTO.R OR ENGINEERS. Comprising Forging, Planing, Lining, Slotting, Shaping, Turning, Screw-cutting, &c. Illustrated with ninety-six plates. By Cameron Knight. 1 vol. 4to, half morocco......................................... $15 00 MAHAN. AN ELEM3ENTARY COURSE OF1 CIVIL ENGINEERING, for the use of the Cadets of the U. S. Military Academy. By D. H. Mahan. 1 vol. 8vo, with numerous woodcuts. New edition. Edited by Prof. De Volson Wood. Full cloth..................$......................$5 00 "s BaDESCRIPTIVE GEOMETRY, as applied to the Drawing of Fortifications and Stone-Cutting. For the use of the Cadets of the U. S. Military Academy. By Prof. D. H. Mtahan. 1 vol. 8vo. Plates................................ $1 50 (u INDUSTRIAL DRAWING. Comprising the Description and Uses of Drawing Instruments, the Construction of Plane Figures, the Projections and Sections of Geometrical Solids, Architectural Elements, Mechanism, and Topographical Drawing. With remarks on the method of Teaching the subject. For the use of Academies and Common Schools. By Prof. D. H. Mahan. 1 vol. 8vo. Twenty steel plates. Full cloth............................$......... 3 00 A TREATISE ON FIELD FORTIFICATIONS. Containing instructions on the Methods of Laying Out, Constructing, Defending, and Attacking Entrenchments. With the General Outlines, also, of the Arrangement, the Attack, and Defence of Permanent Fortifications. By Prof. D. H. Ilahan. New edition, revised and enlarged. 1 vol. Svo, full cloth, with platies.. *$3 50 ELEMENTS OF PERMANENT FORTIFlICAIIONS. By Prof. D. H. Mahan. 1 vol. Svo, with numerous large plates. Cloth............................................. $6 50 98 JOHN WILEY & SON'S LIST OF PUBLICATIONS MAHAN. ADVANCED GUARD, OUT-POST, and Detachment Sewrcg of Troops, with the Essential Principles of Strategy and Grand Tactics. For the use of Officers of the Militia and Volunteers. Byv Prof. D. H. Mahan. New edition, with large additions and 12 plates. 1 vol. 18mo, cloth......$1 56 MAHAN MECHANTCAL PRINCIPLIES OF ENGINEERING & MOSELY. AND ARCHITECTURE. By Henry Mlosely, M.A., F.R.S. From last London edition, with considerable additions, by Prof. D. H. Mahan, LL.D. of the U. S. Military Academy. 1 vol. 8vo, 700 pages. With numerous cuts. Cloth... $5 00 MAHAN HYDRAULIC M1BOTORS. Translated from the French Courm & BRESSE. de Mecanique, appliqu6e par M. Bresse. By Lieut. F. A. Mahan, and revised by Prof. D. H. Mahan. 1 vol. 8vo, plates..................................... 50 WOOD. A TREATISE ON THE RESISTANCE OF MATERIALS, and an Appendix on the Preservation of Timber. By De Volson Wood, Professor of Engineering, University of Michigan. 1 vol. 8vo, cloth.........................$2 50 A TREATISE ON BRIDG]ES. Designed as a Text-book and for Practical Use. By De Volson Wood. 1 vol. 8vo, numerous illustrations, cloth.................3.......$3 00 CREEI(. BACSTER. GREEK TESTA!MENT, ETC. The Critical Greek and English New Testament in Parallel Columns, consisting of the Greek Text of Scholz, readings of Griesbach, etc., etc. 1 vol. 18mo, half morocco............................3 00'I ___ do. Full morocco, gilt edges................. 4 50 *(6 With Lexicon, by T S. Green. Half-bound........ 4 50 64 —- do. Full morocco, gilt edges.................. 6 00 _ —-- - do. With Concordance and Lexicon. Half mor., 6 00 66 ---- do. Limp morocco............................ 7 50 THE ANALYTICAL GREEK LEXICON TO THE NEW TESTAMENT. In which, by an alphabetical arrangement, is found every word in the Greek text in every form i~l whq7icli it appears-that is to say, every occurrent person, number, tense or mood of verbs, every case and number of nouns, pronouns. &c., is placed in its alphabetical order, fully explained by a careful grammatical analysis and referred to its root, so that no uncertainty as to the grammatical structure of any word can perplex the beginner, but, assured of the precise grammatical force of any word he may desire to interpret. he is able immediately to apply his knowledge of the English meaning of the root with accuracy and satisfaction. 1 vol. small 4to, half bound...............................$6 50 GREEK-ENGLISHP LEXICON TO TESTAMENT. By T. S. Green. Half morno....................... $1 50 HEBREW. GREEN. A GRAMMAR OF THE HEBREW LANGUAGE. With copious Appendixes. By W. H. Green, D.D., Professor in Princet,n Theological Seminary. 1 vol. 8vo, cloth....$3 50 AN ELEMENTARY HEBREW GPAM TIAR. With Tables. Reading Exercises, and Vocabulary. By Prof. W. 11. Green, D.D. 1 vol. 12mo, cloth................... $1 50 (6 HEi~.BREW CHRESTOMATHY; or, Lessons in Readbi:,- and Writing Hebrew. By Prof. W. H. Green, D. D. 1 vol. 8vo, cloth.............................................. S 00 LETTERISi A NEW AND BEAUTIFUL EDITION OF THIE HEBDRE W BIBLE. Revised and carefully examined by MlIyer Levi Letteris. 1 vol. 8vo, with key, marble edges..... $2 5( This edition has a large and much more legible type than the known one volunlme editions, and the print is excellent, while the name of LETTERIS is a sufficient guarantee for correctness." -'ev. Dr. J. M. WISE, Eclitor f tlhe ISRAELITE. JOHN WILEY & SON'S LIST OF PUBLICATIONS, 99 8AGSTER'S BAGSTER'S COMPLETE EIDITIO2N OP GESENIUS' GESENIUS. HEBREW AND CHIALDEE LEXICON. In large clear, and perfect type. Translated and edited with additions and corrections, by S. P. Tregelles, LL.D. In this edition great care has been taken to guard the student from Neologial tendencies by suitable remarks whenever needed. "The careful revisal to which the Lexicon has been subjected by a faithful and Orthodox translator exceedingly enhances the practical value of this edition." -Edinburgh Ecclesicastical Journal. Small 4to, half bound............................. $7 50 BAGSTER'S NOWe POCKET FHEBREW AND:ENGLISH LEXICON, The arrangement of this Manual Lexicon combines two things-the etymological order of roots and the alphabetical order of words. This arrangement tends to lead the learner onward; for, as he becomes more at home with roots and derivatives, he learns to turn at once to the root, without first searching for the particular word in its alphabetic order. 1 vol. 1Smo, cloth...........2.................. $2 00 "This is the most beautiful, and at the same time the most correct and perfect Manual He rew Lexicon we have ever used."-Eclectic Reviecw. IRON, METALLURGY, &c. BOEiEMANN. A TREATISE ON THE ASSAYING OF LEAD, SILVER, COPPER, GOLD, AND MERCURY. By Bodemann & Kerl. Translated by WV. A. Goodyear. 1 vol. 12mo, $2 50 CROOKES. A PRACTICAL TREATISE ON METALLURGY. Adapted from the last German edition of Prof. Kerl's Metallurgy. By William Crookes and Ernst Rohrig. In three vols. thick 8vo. Price........................... $30 00 Separately. Vol. 1. Lead, Silver, Zinc, Cadmium, Tin, Mlercury, Bismuth, Antimony, Nickel, Arsenic, Gold, Platinum, and Sulphur..................$1..................10 00 Vol. 2. Copper and Iron......................... 10 00 Vol. 3. Steel, Fuel, and Supplement................ 10 00 FAIRBAIRN. CAST AND WROUGHT IRON FOR BUILDING. By Win. Fairbairn. 8vo, cloth......................... $2 00 FRENCH. HIISTORY OF IRON TRADE, FROM 1621 TO 1857. By B. F. French. 8vo, cloth..........................2 00 KIRKWOOD COLLECTION OF REPORTS (CONDE NSED) AND OPINIONS OF CHEMIISTS IN REGARD TO THEE USE OF LEAD PIPE FOR SERVICOJ PIPE, in the Distribution of Water for the Supply of Cities. By I. P. Kirkwood, C.E. 8vo, cloth......................$1 50 LESLEY. THE IRON MANUFACTURER'S GUIDE TO THES FURN ACES, FORGES, AND ROLLING-MILLS O0 TViE UNITLD STATES. By J. P. Lesley. With maps and plates. 1 vol. 8vo, cloth........................$8 00 MACH I N ISTS-M ECHAN I CS. FITZGERALD. THE 3 BOSTON MACHINIST. A complete School for the Apprentice and Advanced Machinist. By W. Fitzgerald. 1 vol. 18mo, cloth............................... $0 75 HOLLY. SAW FiLING. The Art of Saw Filing Scientifically Treated and Explained. With Directions for putting in order all kinds of Saws, from a Jeweller's Saw to a Steam Saw-mill. Illustrated by forty-four engravings. Third edition. By H. W. HIolly. 1 vol. 18mo, cloth......$..5............ $0 7 KNIGHT. THE MbICHANISM AND ENGINEER INSTRUCTOR, Comprising Forging, Planing, Lining, Slotting, Shaping, Turning, Screw-Cutting, etc., etc. By Cameron Knight. 1 vol. 4to, half morocco......................... $15s jiO) JOHN WILEY & SON'S LIST OF PUBLICATIONS.'rURNING, &c. LATHE, THE, AND ITS USES, ETC.; or, Instruction in the Art of Turning Wood and Metal. Including a description of the most modern appliances for the ornamentation of plane and curved surfaces, with a description also of an entirely novel form of Lathe for Eccentric and Rose Engine Turning, a Lathe and Turning Machine combined, and other valuable matter relating to the Art. 1 vol. 8vo, copiously illustrated. Including Supplement. 8vo, cloth......$7 00 "The most complete work on the subject ever published." —Amerzcan Artisan. "Here is an invaluable book to the practical workman and amateur." —London, Weekly Times. TURNiNC, &c. SUPPLEMENT AND INDEX TO LATHE AND ITS USES. Large type. Paper, 8vo................... $0 90 WILLIS. PRINCIPLES OF MECHANISM. Designed for the use of Students in the Universities and for Engineering Students generally. By Rotbert Willis, M.D., F.R.S., President of the British Association for the Advancement of Science. &c., &c. Second edition, enlarged. 1 vol. 8vo, cloth........... $7 50 *** It ought to be in every large Machine Workshop Office, in every School of Mechanical Engineering at least, and in the hands of every Professor of Mechanics, &c.-Prof. S. EDWARD WARREN. MANUFACTURES. BOOTH. NEW AND COMPLETE CLOCK AND WATCH MAKERS' MANUAL. Comprising descriptions of the various gearings, escapements, and Compensations now in use in French. Swiss, and English clocks and watches, Patents, Tools, etc., with directions for cleaning and repairing. With numerous engravings. Compiled from the French, with an Appendix containing a History of Clock and Watch Making in America. By Mary L. Booth. With numerous plates. 1 vol. 12mo, cloth.................................. $2 00 GELDARD. HANDBOOK ON COTTON MiANUFACTURE; or, A Guide to Machine-Building, Spinning, and Weavin.g. With practical examples, all needful calculations, and many useful and important tables. The whole intended to be a complete yet compact authority for the manufacture of cotton. By James Geldard. With steel engravings. 1 vol. 1110, cloth....................................... $2 50 MEDICAL, &c. BULL HINTS TO MOTHIERS FOR TEO MANAGEMENT OF HEALTH DURING THE PE RIOD OF PREGNANCY, AND IN THE LYING-I'N ROOM. With an exposure of popular errors in connection with those subjects. By Thomas Bull, M.D. 1 vol. 12mo, cloth...........$1 00 FRANCKE. OUTLINES OF A NEW THEORY OF DISE.ASE: applied to Hydropathy, showing that water is the only true remedy. With observations on the errors committed in the piactlce of Hydropathy, notes on the cure of cholera by cold water. and a critique on Priessnitz's mode of treatment. Intended foi popular use. By the late H. Francke. Translated from the German by Robert Blakie, M.D. 1 vol. 12mo, cloth...$1 50 GREEN. A TREATISE ON DISEASES OF T-E:- AIR PASSAGES. Comprising an inquiry into the History, Pathology, Causes, and Treatment of those Affections of the Throat called Bron chitis, Chronic Laryngitis, Clergyman's Sore Throat, etc., etc. By Horace Greenl M.D. Fourth edition., revised and enlarged. 1 vol. 8vo, cloth................................... $3 00 A PRACTICAL TREATISE ON PULMONARY TUBERCULOSIS, embracing its History, Pathology, and Treat. ment. By Horace Green, HI.D. Colored platesa 1 vol. 8vo, cloth............................................. $5 OC JOHN WILEY & SON'S LIST OF PUBLICATIONS. 101 GREEN. OBSERVATIONS ON THE PATHOLOGY OF CROUP With Remarks on its Treatment by Topical Medications. By Horace Green, M.D. 1 vol. 8vo, cloth...... $12... 1 25 ON THE SURGICAL TREATMENT OF POLYPI OR THE LARYNX, AND CED:EMA OF THE GLOTTIS. By Horace Green, M.D. 1 vol. 8vo.................. $1 25 FAVORITE PRESCRIPTIONS OF LIVING PRACTITIONERS. With a Toxicological Table, exhibiting the Symptoms of Poisoning, the Antidotes for each Poison, and the Test proper for their detection. By Horace Green. 1 vol. 8vo, cloth..................................$2 50 TILT. ON THE PRESERVATION OF THE HEALTH OF WOMEN AT THE CRITICAL PERIODS OF LIFE. By E. G. Tilt, M.D. 1 vol. 18mo, cloth..............$0 50 VON )UBEN. GUSTAF VON DUBEN'S TREATISE ON MICROSCOPICAL DIAGNOSIS. With 71 engravings. Translated, with additions, by Prof. Louis Bauer, M.D. 1 vol. 8vo, cloth.............................................. $1 00 MINERALOGY. BRUSH. ON BLOW-PIPE ANALYSIS. By Prof. Geo. J. Brush. (In preparation.) DANA. DESCRIPTIVE MIINERALOGY. Comprising the most recent Discoveries. Fifth edition. Almost entirely re-written and greatly enlarged. Containing nearly 900 pages 8vo, and upwards of 600 wood engravings. By Prof. J. Dana. Cloth...................................... $10 00 "We have used a good many works on Mineralogy, but have met with none that begin to compare with this in fulness of plan, detail, and execution."Americant Journal of ifining. DANA& BRUSH. APPENDIX TO DANA'S MINERALOGY, bringing the work down to 1872. By Prof. G. J. Brush. Svo..... $0 50 DANA. DETERIMINATIVE MINERALOGY. 1 vol. (In preparation.) 66 ~ A TEXT-BOOK OF M!INERALOGY. 1 vol. (In preparation. ) MISCELLANEOUS. BAILEY. THE! NEVW, TALE OF A TUB. An adventure in verse. By F. W. N. Bailey. With illustrations. 1 vol. 8vo.....$0 75 CARLYLE. ON HEROES, HERO-WORSHIP, AND THE HEIROIC IN HISTORY. Six Lectures. Reported, with emendations and additions. By Thomas Carlyle. 1 vol. 1m1o, cloth... $0 75 CATLIN, THE BREATH OF LIFE; or, Mal-Respiration and its Effects upon the Enljoyments and Life of Ian. By Geo. Catlin. With numerous fwood engravings. 1 vol. 8vo, $0 75 CHEEVER. CAPITAL PUNISHMENT. A Defence of. By Rev. George B. Cheever, D.D. Cloth............................$0 50 "e 4i HILL DIFFICULTY, and other Mfliscellanies. By Rev. George B. Cheever, D.D. 1 vol. 12mo, cloth.........$1 00 6" JOURNAL OF THE PILGRIMS AT PLYMOUTH ROOC. By Geo. B. Cheever, D.D. 1 vol. 12mo, cloth........$1 00 G WANDERINGS OF A PILGRIM IN THE ALPS. By George B. Cheever, D.D. 1 vol. 12mo, cloth......... $1 00 "4 WANDERINGS OF THE RIVER OF THE VWATER OF LIFE. By Rev. Dr. George B. Cheever. 1 vol. 12mlo, cloth.............$1 00 CONYBEARE. ON INFIDELITY. 12mo, cloth.................... 1 00 CHI0LD'S BOOK OF FAVORITE STO!RIES. Large colored plates. 4to, cloth............................................. $1 50 102 JOHN WILEY & SON'S LIST OF PUBLICATIONS. EDWARDS. FRE.E TOWN LIBRARIES. The Formatf;on, ManaaelneLt and History in Britain, France, Germany, and America. Together with brief notices of book-collectors, and of the respective places of deposit of their surviving collections. By Edward Edwards. 1 vol. thick 8vo.............. $4 00 GREEN. THE PENTATEUCH VINDICATED PROM THE AS. PERSIONS OF BISHOP COL]ENSO. By Wm. Henry Green, Prof. Theological Seminary, Princeton, N. J. 1 vol. 12mo, cloth........................................$1 25 GOURAUD. PHRENO-MNEMOTECHNY; or, The Art of 1M5emory. The series of Lectures explanatory of the principles of the system. By Francis Fauvel-Gouraud. 1 vol. 8vo, cloth, $2 00 PHRENO-MNEM:OTECHNIC DICTIONARY. Being a Philosophical Classification of all the Homophonic Words of the English Language. To be used in the application of the Phreno-lMnemotechnic Principles. By Francis Fauvel-Gouraud. 1 voL 8vo, cloth............................$2 00 HEIGHCWAY. LEILA ADA. 12mo, cloth......................... 1 00 LEILA ADA'S RELATIVES. 12mo, cloth........... 1 00 KELLY. CATALOGUE OF AMERICAN BOOKS. The American Catalogue of Books, from January, 1861, to January, 1866. Compiled by James Kelly. 1 vol. 8vo, net cash.......$5 00 CATALOGUE OF AMERICAN BOOKS. The American Catalogue of Books from January, 1866, to January, 1871. Compiled by James Kelly. 1 vol. 8vo, net........... $7 50 AJIVER'S COLLECTION OF GENUI NE SCOTTISH MiVELODIES. For the Piano-Forte or Harmonium, in keys suitable for the voice. Harmonized by C. H. Morine. Edited by Geo. Alexander. 1 vol. 4to, half calf....................... $10 00 NOTLEY. A COMPARATIVE GRAMM A1R OF THE FRENCH, ITALIAN, SPANISH, AND PORTUGUESE LANGUAGES. By Edwin A. Notley. 1 vol., cloth......$5 00 PARKER. POLAR MAGNETISM. First and Second Lectures. By John A. Parker. Each............................ $0 25 NON-EXISTENCE OF PROJECTILE FORCES IN NATURE. By John A. Parker....................., 25 STORY OF A POCKET BIBLE. Illustrated. 12mo, cloth........ $1 00 TUPPER PROVERBIAL PHILOSOPHY. 12mo............... 1 00 WALTON -EPIE COMPLETE ANGLER; or, The Contemplative Man's & COTTON. Recreation, by Isaac Walton, and Instructions how to Angle for a Trout or Grayling in a Clear Stream, by Charles Cotton, with copious notes, for the most part original. A bibliographical preface, giving an account of fishing andl Fishing Books, from the earliest antiquity to the time ol Walton, and a notice of Cotton and his writings, by Rev. Dr. Bethune. To which is added an appendix, including the most complete catalogue of books in angling ever printed, &c. Also a general index to the whole work. 1 vol. 12mo, cl..................................... ~....... *: 00 cloth..... 00 WARREN. NOTES ON POLYTECHNIC OR SCIENTIFIC. SCHOOLS IN THE UNITED STATES. Their Nature, Position, Ainms, and Wants. By S. Edward Warren. Paper....$0 40 WILLIAMS. THE MiIDDLE KINGDOM. A Survey of the Geography, Gove'nmenlt, Education, Social Life, Arts, Religion, &Tc., of the Chinese Empire and its Inhabitants. With a new map of the Empire. By S. Wells Williams. Fourth edition, in 2 vol1s............................................. 4 00 JOHN WILEY & SON'S LIST OF1 PUBLICATIONS. 103 RUSKIN'S WORKS. Uniform in sizue and style. RUSKIN MODERN PAINTERS. 5 vols. tinted paper, bevelled toards, plates, in box.....................................$18 00 MODERN PAINTERS. 5 vols. half calf............ 27 00 (4' i" " without plates....... 12 00 6"'" " " half calf, 20 00 Vol. 1.-Part 1. General Principles. Part 2. Truth. Vol. 2.-Part 3. Of Ideas of Beauty. Vol. 3.-Part 4. Of Many Things. Vol. 4.-Part 5. Of Mountain Beauty. Vol. 5.-Part 6. Leaf Beauty. Part 7. Of Cloud Beauty. Part 8. Ideas of Relation of Invention, Formal. Part 9. Ideas of Relation of Invention, Spiritual. STONES OF VENICE. 3 vols., on tinted paper, bevelled boards, in box......... $7 00 STONES OF VENICE. 3 vols., on tinted paper, half calf.............................................. $12 0 STONES OF VENICE. 3 vols., cloth.............. 6 00 Vol. 1.-The Foundations. Vol. 2.-The Sea Stories. Vol. 3.-The Fall. SEVEN LAMPS OF ARCHITECTURE. With illustrations, drawn and etched by the authors. 1 vol. 12mo, cloth, $1 75 LECTURES ON ARCH~IITECTURE AND PAINTING. With illustrations drawn by the author. 1 vol. 12mo, cloth.............................................. $1 50 THE TWO PATHS. Being Lectures on Art, and its Application to Decoration and Manufacture. With plates and cuts. 1 vol. 12mo, cloth.......................... $1 25 -THSE ELEMENTS OF DRAWING. In Three Letters to Beginners. With illustrations drawn by the author. 1 vol. 12mo, cloth........................................ $1 00 THE ELEMENTS OF PERSPECTIVE. Arranged for the use of Schools. 1 vol. 12mo, cloth... $1 00 THE POLITICAL ECONOMY OF ART. 1 voL 12mo, cloth..............................................$1 00 PRE-RAPHAELITISM.. NOTES ON THE CONSTRUCTION OF 1 vo. 12mo, SHEEP]F'LDS. vi. 1 mol cloth, $1 00 KING OF THE GOLDEN RIVER; or, The Black Brothers. A Legend of Stiria. J RUSKIN SESAME AND LILIES. Three Lectures on liooks, Women, &c. 1. Of Kings' Treasuries. 2. Of Queens' Gardens. 3. Of the Mystery of Life. 1 vol. 12mo, cloth.......... $ 50 AN INQUI INQUIRY INTO SOME OF THE CONDITIONS AT PRESENT AFFECTING "THE STUDY OF ARCHITECTURE" IN OUR SCHOOLS. 1 vol. 12mo. paper............................................. $0 15 THE ETHICS OF THE DUST. Ten Lectures to Little Housewives, on the Elements of Crystallization. 1 vol. 12mo, cloth........................................$1 25 i "UNTO THIS LAST." Four Essays on the First Principles of Political Economy. 1 vol. 12mo, cloth............. $1 00 Os 0 JOHN WILEY & SON'S LIST OF PUBLICATIONS. RUSKIN THE CROWN OF WILD OLIEVE. Three Lectures on Works Traffic, and War. 1 vol. 12mo. cloth................ $1,e TIME AND TIDE BY WEARE! AND TYNE. Twentyfive Letters to a Workingman on the Laws of Work. 1 vol. 12mo, cloth......................................$1 00 "' THE QUEEN OF THE AIR. Being a Study of the Greek Myths of Cloud and Storm. 1 vol. 12mo, cloth...... $1 00 * LECTURES ON ART. 1 vol. 12mo, cloth............1 00 FORS CLAVIGERA. Letters to the Workmen and Labourers of Great Britain. Part 1. 1 vol. 12mo, cloth, plates, $1 00 lo FORS CLAVIGERA. Letters to the Workmen and Labourers of Great Britain. Part 2. 1 vol. 12mo, cloth, plates, $1 00 MUNERA PULVERIS. Six Essays on the Elements of Political Economy. 1 vol. 12mo, cloth.............$1 00 ARATRTA PENTELICI. Six Lectures on tho Elements of Sculpture, given before the University of Oxford. By John Ruskin. l12no, cloth, $1 50, or with plates....... $3 00 "r'I~CTHE EAGLE'S NEST. Ten Lectures on the relation of Natural Science to Art. 1 vol. 12mo...............$1 50 BEAUTIFUL PRESENTATION VOLUMES. -Printj4 op tlined caper, and elegantly bound in crape cloth extra, bevelled boards, gilt head. RUSKIN. THE TRUE AND THE BEAUTIFPUL IN NATURE, ART, MORALS, AND RELIGION. Selected from the Works of John Ruskin, A.M. With a notice of the author by Mrs. L. C. Tuthill. Portrait. 1 vol. ]2mo, cloth, plain, $2.00; cloth extra, gilt head....................2$9 50 ~6 ART CULTURE. Consisting of the Laws of Art selected from the Works of John Ruskin, and compiled by Rev. W. H. Platt, for the use of Schools and Colleges as well as the general public. A beautiful volume, with many illustrations. 1 vol. l2lno, cloth (shortly). " PRECIOUS THOUGHTS: Moral and Religious. Gathered from the Works of John Ruskin, A.M. By Mrs. L. C. Tuthill. 1 vol. 12mo, cloth, plain, $1.50. Extra cloth, gilt head.......................................... $2 00 " SELECTIONS FROM THE WRITINGS OF JOHiN RUSKIN. 1 vol. 12mo, cloth extra, gilt head........2 50 A6 SELECTIONS FROM THE WRITINGS OF JOHN RUSKIN. 1 vol. 12mo, plain cloth.................$2 00 4i SESAME AND LILIES. 1 val. 12mo................$1 715 ETHICS OF THE DUST. 12o..................... 1 75 r' O TCROWN OF WILD OLIVE. 12en..1 50 RUSKIN'S BEAUTIES. THE TRUE AND BEAUTIFUL 1 gilt head..........%;; 00 CHOICE SELECTIONS. do., half calf...10 t)0 RUSKIN'S POPULAR VOLUMES. CROWN OF WILD OLIVE. SESAME AND LILIES. QUEEN OF THE AIR. ETHICS OF THE DUSPT. 4 vols. in box, cloth extra, gilt head............. $6 00 RUSKIN'S WORKS. Revised edition. RUSKIN Vol. l.-SESAME AND LILIES. Three Lectures. By John Ruskin, LL.D. 1. Of King's Treasuries. 2. Of Qleens' Gardens. 3. Of the Mystery of Life. 1 vol. 8vo, cloth, $2.00. Large paper.............................. 2 5i JOHN WILEY & SON'S LIST OF PUBLICATIONS. 105 RUSKIN'S WORKS. Revised edition. 6" Vol. 2.-MUNERA PULVERIS. Six Essays on the Elementa of Political Economy. By John Ruskin. 1 volume 8vo, cloth............................................$2 00 Large paper....................................... 2 50 66 Vol. 3.-ARATRA PENTELICI. Six Lectures on the Ele ments of Sculpture, given before the University of Oxforu. By John Ruskin. 1 vol. 8vo.......................$4 00 Large paper....................................... 4 C THE POET RY OF ARCHITECTURE: Villa and Cottage. With numerous plates. By Kata Phusiri. 1 vol. 12mo, cloth................................ $1 50 Kata Phusin is the supposed Nom de Plume of John Ruskin.' FORS CLAVIGERA. Letters to the Workmen and Laborera ci great Britain. Part 3. 1 vol. 12mo, cloth........$1 50':USKIN-COMPLETE WORKS. Tnir COmLE ErE WORTis U TOIIN RusKIN. 27 vols., extra cloth, in a box..$40 00 Ditto 27 vols., extra cloth. Plates... 48 00 Ditto Bound in 17 vols., half calf. do.... 70 00 SHIP-BUILDING, &c. BOURNE. A TREATISE ON THE SCREWEV PROPELLER, SCREOW VESSELS,' AND SCREW ENGINES, as adapted for Purposes of Peace and War. Illustrated by numerous woodcnts and engravings. By John Bourne. New edition. 1867. 1 vol. 4to, cloth, $18.00; half russia................$24 00 WATTS. RANKINE (W. J,'M.) AND OTHERS. Ship-Building, Theoretical and Practical, consisting of the Hydraulics of ShipBuilding, or Buoyancy, Stability, Speed and Design —The Geometry of Ship-Building, or Modelling, Drawing, and Laying Off-Strength of Materials as applied to Ship-Building -Practical Ship-Building-Masts, Sails, and Rigging-Marine Steam Engineering-Ship-Building for Purposes of War. By Isaac Watts, C.B., W. J. M. Rankine, C.B., Frederick K. Barnes, James Robert Napier, etc. Illustrated with numerous fine engravings and woodcuts. Complete in 30 numbers, boards, $35.00; 1 vol. folio, cloth, $37.50; half russia, $40 00 WILSON (T. D.) SHIP-BUILDING, THEEORETICAL AND PRACTICAL. Ill Five Divisions. -Division I. Naval Architecture. II. Laying Down and Taking off Ships. III. Ship-Building IV. Masts and Spar Making. V. Vocabulary of Terms usedintended as a Text-Book and for Practical Use in Public and Private Ship-Yards. By Theo. D. Wilson, Assistant Naval Constructor, U. S. Navy; Instructor of Naval Construction, U. S. Naval Academy; Member of thc~ Institution of Naval Architects, England. With numerous plates, lithographic and wood. 1 vol. 8vo...................$7 50 SOAP. MORFIT. A PRACTICAL TREATISE ON THE MANUFACTURE OF SOAPS. With numerous wood-cuts and elaborate working drawings. By Campbell Morfit, M.D., F.C.S. 1 vol. 8vo..............................................$20 W, STEAM ENGINE. TROWBRIDGE. TABLES, WITH EXPLANATIONS, OF THE NONCONDENSING STATIONERY STATIONERY STEAM ENGINPe, and of High-Pressure Steam Boilers. By Prof. W. P. Tron bridge, of Yale College Scientific School. 1 vol. 4to plates............................................$2 5G TREATISE ON THE GENERATION AND UTILIZA TION OF HEAT THROUGH THE MEDIUM OF STEAMlS AND STEAM BOILERS. Designed as a TextBook and for Practical use. By Prof. W. P. Trowbridge Very fully illustrated, 1 vol. 8vo (shortly) L0G JOHN WILEY & SON'S LIST OF PUBLICATIONS. TURNING, &c. THE L.ATHE, AND ITS USES, ETC. On Instructions in the Art of Turnmm Wood and Metal. Including a description of the most modem appliances for the ornamentation of plane and curved surfaces. With a description, also, of an entirely novel form of Lathe for Eccentric and Rose Engine Turning, a Lathe and Turning Machine combined, and other valuable matter relating to the Art. 1 vol. 8vo, copiously illustrated, cloth.......... $7 00 SUPPLEMENT AND INDEX TO SAME. Paper...$O 90 VENTILATION. LEEDS (L. W.). A TREATISE ON VENTILATION. Comprising Seven Lectures delivered before the Franklin Institute, showing the great want of improved methods of Ventilation in our buildings, giving the chemical and physiological process of respiration, comparing the effects of the various methods of heating and lighting upon the ventilation, &c. Illustrated by many plans of all classes of public and private buildings, showing their present defects, and the best means of improving them. By Lewis W. Leeds. 1 vol. 8vo, with numerous wood-cuts and colored plates. Cloth........ $2 50 "It ought to be in the hands of every family in the country."-Technologist, "Nothing could be clearer than the author's exposition of the principles of the principles and practice of both good and bad ventilation."- Van Nostrand's Ensgineering MJlgazisne. "The work is every way worthy of the widest circulation."-Scfentflc Amerwcan. REID. VENTILATION IN AMERICAN DWELLINGS. With a series of diagrams presenting examples in different classes of habitations. By David Boswell Reid, M.D. To which is added an introductory outline of the progress of improvement. in ventilation. By Elisha Harris, lI.D. 1 vol. 12mo, $1 50 WEIGHTS, MEASURES, AND COINS. TABTLES OF WEIGaTITS, MEASURES, COINS, &c., OF THE UNITED STAT.ES AND ENGLAND, with their Equivalents in the French Decimal System. Arranged by T. T. Egleston, Professor of Mineralogy, School of Mines, Columbia Collego. 1 vol. 18mo..................... $0 75 "It is a most useful work for all chemists and others who have occasion to make the conversions from one system to another."-rAmsericanz CGhemist. "Every mechanic should have these tables at hand." —Amersican HorolZogoc Jouesssa. J. W. & SON are Agents for and keep in stock SAMUEL BAGSTER & SONS' PUBLICATIONS; LONDON TRACT SOCIETY PUBLICATIONS, COLLINS' SONS & CO.'S BIBLES, MURRAY'S TRAVELLER'S GUIDES, WEALE'S SCIENTIFIC SERaEL FPu Catalogues gratis on application. J. W. & SON import to order, for thie TRADE AND PUBLIC, BOOKS, P ERIO ICAL S, &c.. FROM,*' JOIIN WILEY & SON'S Complete Classified Catalogule of the most vala1 able and latest scientific publications supplied gratis to order.