Cornell University Library The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924004250597 QB 145 N^°8'"«"""'«"'«y Library * fliiMMuii?" ''^"■°"°'"y' spherical and p 3 1924 004 250 597 A TREATISE OS ASTRONOMY, SPHERICAL AND PHYSICAL; WITH A.STRONOMICAL PROBLEMS, SOLAR, LMAR, AND OTHER ASTRONOMICAL TABLES. FOR TUB USB Or COLLEGRS AND SCIKNTIFIO SCHOOLS. WILLIAM A. NOETON, M.A., PROFESSOR OF CIVIL ENGmEEEmO m TALE COLLE&B. FOUBTH EDITION. REVISED, REMODELLED, AND ENLARGED. NEW YORK: JOHN WILEY & SONS, 15 ASTOB PLACE. 1878. Entered according to Act of Congress, in the year 186T, Bt WILLIAM A. NORTON, la the Clerlis Office of the District Court of the United States for tlie Southern District of New Yatt, r^ CORNELLS UNIVERSITY LiBH, Tbow*s Printing and Bookbinding Co.) 205-213 /^ast isiA Si.f NEW YORK. PREFACE TO THE REVISED EDITION. In the preparation of tlie present edition the work has ceen 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 formulae, occurring 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 de- scription of astronomical facts and appearances, and a more com- plete 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 con- nection. 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 IV PREFACE. unceasingly execute His will. Some prominence is given to tlie author's theory of the variable intensity of the repulsive force of the Sun, acting on different portions of coraetic 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 ia 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 woik. The Astronomical 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 figm-es, and several plates of telescopic appearances, have been added. One of the most important of the special improvements intro- duced consists in the adoption of the new and more accurate determination of the Sun's parallax, and mean distance fi-om 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 il a more accurate determination of the distances of all the planets fi'om 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 methods used in the United States Coast Survey in deter- mining from astronomical observations the latitude and longitude of a place. These may be chai-acterized as the American PEEFACE. V methods, as they were devised and perfected by American as- tronomers and engineers; and are superior to all others that have 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 necessai-ily 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 peri- odicals have been consulted. Professor Chauvenet's admirable work on Spherical and Practical Astronomy, should be particu- larly mentioned as having been especially consulted in prepar- ing the. chapter on Instruments. In the mention of new dis- coveries and theoretical views, as well as of the signal advances vvhich modern Astronomy has made, the name of the discoverer, or author, is generally given. The history of Astronomy can- not 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 COlfTENTS. PAET I. SPHERICAL ASTRONOMY. CHAPTER I, Definitions and Fundamental Conceptions. — General Phenomena of the Heavens 1 CHAPTER II. Celestial and Terrestial Spheres 11 CHAPTER in. Astronomical Instraments. — ^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 rv. Corrections of Measured Angles . 54 Refraction 54 Parallax 60 Aberration 66 CHAPTER V. figure and Dimensions of the Earth. — ^Latitude and Longitude of a Place 72 Determination of the Latitude and Longitude of a Place . . 75 Vm CONTENTS. CHAPTER VL rAox Apparent Motion of the Sun in the Heavens ... .81 CHAPTER VII. Precession of the Equinoxes. — ^Nutation 86 Nutation 88 CHAPTER VIII. Measurement of Time "■'• Different kinds of Time *"• Conversion of One Species of Time into Another ... 92 Determination of Time, and Regulation of Clocks by Astrono- mical Observations "* The Calendar 98 CHAPTER IX Motions of the Sun, Moon and Planets in their Orbits . . . 102 Kepler's Laws »&• Definitions of Terms 106 Elements of the Orbit of a Planet 108 Determination of the Sun's Apparent Orbit, or the Earth's Real Orbit 109 Mean Motion ib. Semi-Major Axis ih. Eccentricity 110 Longitude and Epoch of the Perigee 112 Determination of the Elements of the Moon's Orbit . . 113 Longitude of the Node ib. Inclination of the Orbit 114 Mean Motion ib. Longitude of the Perigee, Eccentricity, and Semi-Major Axis . 115 Mean Longitude at an Assigned Epoch 116 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 Longitude and Latitude, and the Radius Vector, for a given time i5_ Longitude of the Perihelion, Eccentricity, and Semi-Major Axis 121 Epoch of the Perihelion Passage ...... 122 True and Mean Elements 123 CHAPTER X. Determination cf the Place of a Planet, or of the Sun or Moon, for a given time, by the EUiptic Theory. — Verification of Kepler's Laws 126 CONTENTS. ix Place 01 a Planet in its Orbit Heliocentric Place of a Planet Geocentric Place of a Planet Places of the Sun and Moon Verification of Kepler's Laws PAQfl 126 127' 128 ib. 129 CHAPTER XI. Inequalities of the Motions of the Planets and of the Moon. — Tables for Finding the Places of these Bodies 130 Tables of the Sun, Moon, and Planets 135 CHAPTER XII. Motions of the Comets 136 Halley's Comet 140 Encke's Comet 141 Biela's Comet 142 Fayes' Comet ib. Lexell's Comet of 1770 143 The Great Comet of 1843 144 Donati's Comet 145 Conspicuous Comets of the Present Century .... 146 CHAPTER XIII. Motions of the SateUites 147 CHAPTER XI-V. The Sun and the Phenomena Attending its Apparent Motions . 151 Inequality of Days ib. Twilight *. ... 156 The Seasons 159 Form and Dimensions of the Sun 162 Sun's Spots, and Rotation on its Axis. — Physical Constitution of the Sun 164 Zodiacal Light 175 CHAPTER XV. The Moon and its Phenomena 180 Phases of the Moon ib. Moon's Rising, Setting, and Passage Over the Meridian . . 182 Rotation and Librations of the Moon 184 Dimensions and Physical Constitution of the Moon . . . 186 ' Description of the Moon's Surface 188 CHAPTER XVI. Eclipses cf the Sun and Moon. — Occultations of the Fixed Stars . 190 CONTENTS. Eclipses of the Moon . . . Calculation of an BcHpse of the Moon Construction of an Eclipse of the Moon Eclipses of the Sun . Calculation of an Eclipse of the Sun Occultations ... Txa% 190 194 198 199 206 ib. 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 Mars 220 Jupiter and its Satellites 221 Saturn, with its Satellites and Ring 223 Uranus and its Satellites . 227 Neptune . ib. The Planetoids 228 CHAPTER XVni. Comets Their General Appearance. — Varieties of Appearance Form, Structure and Dimensions of Comets Physical Constitution of Comets .... Constitution and Mode of Formation of Tails of Comets Condition and Origin of the Nebulous Envelopes . 229 ib. 233 236 237 241 CHAPTER XIX. The Fixed Stars 245 Constellations. — Division into Magnitudes .... ib. Number and Distribution over the Heavens .... 247 Annual Parallax, and Distance of the Stars .... 250 Nature and Magnitude of the Stars 252 Variable Stars 253 Double Stars 256 Proper Motions of the Stars 259 Clusters of Stars 261 Nebulse j-j_ Distance and Magnitude of Nebulae 267 Number, Mutual Distance, and Comparative Brightness of the Component Stars of Clusters 269 CONTENTS. XI 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. PAET II. PHYSICAL ASTSONOMT. CHAPTER XXI. Principle of Universal Grravitation 278 CHAPTER XXII. Theory of the BUiptic Motion of the Planets 282 CHAPTER XXIII. Theory of the Perturbations of the Elliptic Motions of the Planets and the Moon 287 CHAPTER XXIV. Relative Masses and Densities of the Sun, Moon, 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 Rota- tion. — ^Precession of the Equinoxes, and Nutation . . . 299 CHAPTER XXVI. the Tides 302 Comparison of the Theory of the Tides vrith 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. Estabhshment of the Port. — Tide Tables »^ Xa CONTENTS. ' PAET in. ASTRONOMICAL PROBLEMS. rAOH Explanations of the Tables 317 Pkob. 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 Peoe. II. To take from a table the quantity corresponding to a given value of the argument, or to given values of the argu- ments of the table ... .... 321 Peob. III. To convert Degrees, Minutes, and Seconds of the Bqua- . tor into Hours, Minutes, &c., of Time 327 Peoe. IV. To convert Time into Degrees, Minutes, and Seconds . ib. Peob. V. The Longitudes of two Places, and the Time at one of them being given, to find the corresponding time at the other 328 Peob. VI. The Apparent Time being given, to find the correspond- ing Mean Time ; or, the Mean Time being given, to find the Apparent 329 Peob. VII. To correct the Observed Altitude of a Heavenly Body for Refraction 332 Peob. VIII. The Apparent Altitude of a Heavenly Body being given, to find its True Altitude 333 Peob. IX. To find the Sun's Longitude, Hourly Motion, and Semi- diameter, for a given Time, from the Tables .... 335 Peob. X. To find the Apparent Obliquity of the Ecliptic, for a given time, from the Tables 337 Peob. XL Given the Sun's Longitude and the Obliquity of the Ecliptic, to find his Right Ascension and Declination . . 338 Peob. XII. Given the Sun's Right Ascension and the Obliquity of the Ecliptic, to find his Longitude and Declination . . . 339 Peob. XIII. The Sun's Longitude and the ObUquity of the Ecliptic being given, to find the Angle of Position .... ib. Peob. XIV. To find from the Tables, the Moon's Longitude, Lati- tude, Equatorial Parallax, Semi-diameter, and Hourly Motions in Longitude and Latitude, for a given Time .... 340 Peob. XV. The Moon's Equatorial Parallax, and the Latitude of a Place, being given, to find the Reduced Parallax and Latitude 349 Peob. XVI. To find the Longitude and Altitude of the Nonagesi- mal Degree of the Ecliptic, for a given Time and Place . . ib. Peob. XVII. To find the Apparent Lorigitude and Latitude, as affected by Parallax, and the Augmented Semi-diaraeter of the Moon ; the Moon's True Longitude, Latitude, Horizontal Semi- diameter, and Equatorial Parallax, and the Longitude and Alti- tude of the Nonagesimal Degree of the Ecliptic, being given . 352 CONTENTS. . xm FAOI Pbob. XVIII. To find the Mean Right Ascension and Declination, or Longitude and Latitude of a Star, for a given Time, from the Tables 356 Peob. XIX. To find the Aberrations of a Star in Eight Ascension and Declination for a given Day 357 Pkob. 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 Declina- tion of a Star, for a given Day . • . . . 360 Pkob. 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 Heav- enly Body from its Right Ascension and Declination, the Obhquity of the Ecliptic being given 362 Peob. 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 Tear and Month 365 Pkob. 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 Pbob. XXXI. To find the Moon's Longitude, &c., from the Nau- tical Almanac 392 APPENDIX. Trigonometrical Formula 39!3 I. Relative to a Single Arc or angle a ih. II. Relative to Two Arcs a and h, of which o is supposed to be the greater H. III. Trigonometrical Series SO? IV. Differences of Trigonometrical Lines ib. V. Resolution of Right-angled Spherical Triangles . . . ib. VT. Resolution of ObUque-angled Spherical Triangles . . 399 XIV . CONTENTS. Tsat IlfVESTIGATION OF ASTKONOMIOAL FoKMtTL^ ..... 402 Formulae for the Parallax in Eight Ascension and Declination, and in Longitude and Latitude ib. Formulae for the Aberration in Longitude and Latitude, and in Right Ascension and DeoUnation . . . . • . 409 Formulae for the Nutation in Right Ascension and Declination 413 Formulas 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 L Construction of Tables 432 Note IL 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. SPHER-ICAL ASTRONOMY. CHAPTEK I. Definitions and Fundamental Conceptions; G-enerai. Phenomena of the Heavens. 1. The sun, moon, and stars — the luminous bodies dissemi- nated through the heavens, or indefinite space surrounding the earth — are called Heavenly Bodies. The heavenly bodies, consi- dered 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, mag- nitude, 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 uni- verse, 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, an-" astronomical observation ; practical determinations, as of tht latitude or longitude of a place, from instrumental observation ; and the solution of astronomical problems with the aid of tables. I FUNDAMENTAL CONCEPTIONS. 3. Form 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, tlien ot 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 por- tions 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. (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 observa- tion, and is on all sides equally depressed. To illustrate this proof, let BOA (Fig, 2) represent a portion of the earth's sur- Jkce supposed to be spherical, P the position of the eye of the observer, and DPC a horizontal line. If we conceive lines, such VISIBLE PORTION OP THE HEAVENS. 3 as PA and PB, 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, B, etc., which would be the visible horizon, is a circle. It is also manifest that the angles of depres- sion 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, in 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 this surface, which send suSicient light to the eye, are visible. That portion of the heavens which is above this surface, presents the appearance of a solid vault or canopy, resting upon the earth at the visible horizon, (see Fig. 2 ;) and to this vault the sun, moon, and stars seem to be attached. It is hardly necessary to remark that this is an optical illusion. It will be seen in the sequel that the heavenly bodies are distributed through space at various dis- tances from the earth, and that the distances of all of them 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 every variety of absolute direction in space. 6. The Sky. The earth is surrounded with a transparent gaseous medium, called the earths s atmosphere, estimated to be some 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 lumi- nous canopy over our heads by day and by night. This is called the Sky. 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 the Heavens. The most conspi- cuous 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 Must, being replaced by others that come into view, or rise; others, to descend towards the opposite quarter, the West, 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 de- scend 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 North, 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 Pok Star ; and the stars which revolve about it. and never set, are called Qircumpolar Stars. It should be remarked, however, that the pole star, when accurately observed by means of instruments, is found not to be strictly stationary, biit to de- scribe 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 simpli- city this conception is generally adopted. This motion, common to all the heavenly bodies, is called their Diurnal Motion. It is ascertained, by certain accurate methods of observation and com- putation, that the diurnal motion of the stars is strictly uniform and circular. 8. 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 eartKs surface, 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 Fixed 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 designation they are divided into the two classes of Planets, and Planetoids or Minor Planets. The former class comprises the planets Mercury, Venus, Mars, Jupiter, Saturn, Uranus, 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 sill been discovered since the year 1780. 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 Motions of the Planets. The apparent mo- O 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 Snn and moon. 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 motion 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) repre- Fig. 3. sents a section of the earth perpendicular to the axis of rotation of the imaginary sphere of the heavens, (8,*) conceived to pasa through the earth's centre ; the large circle H Z S a section of • Kumbers thus Inclosed in a paronthesia refer to a previous article. APPAEENT MOTION OP 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 tha 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, iu, uv, &c., are each equal to S r. Then, at the end of two or three weeks, an hour after gunset 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 posi- tions 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 di- urnal motion ; that is, towards the east. Another proof of tiie progressive motion of the sun among tha stars from west to east, is found in the fact that the same stars rise and set earlier each successive night, and week, and raontb during the year. At the end of six months the same stars risa 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 iu 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 from day to day by certain methods of observation, hereafter to be explained, it appears that his path is an exact circle, inclined about 28° to a circle running due east and west (8). IS. The Zodiac. The motions of the sun, moon, and pla- nets, are for the most part confined to a certain zone, of about 18° in breadth, extending around the heavens obliquely from west to east, which has received the name of the Zodiac. 14. Cometi. 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. O GENERAL PHENOMENA. These bodies are seen 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 Satellites. 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. Tlie Solar System. The sun and moon, the planets, (including the earth,) together with their satellites, and the comets, compose the Solar System. From the consideration of the apparent motions and other phenomena of the solar system, several theories have been form- ed in relation to the arrangement and actual motions in space of the bodies that compose it. The theory, or system, now univer- sally received, is, in its most prominent features, that which was taught by Copernicus in the sixteenth century, a.nd which is known by the name of the Gopernican System. It is as follows : The sun occupies a fixed centre, about which the planets (in- cluding the earth) revolve from west to east, in planes that are out slightly inclined to each other, and in the following order: Mercury, Venus, the Earth, Mars, the Planetoids, Jupiter, Sa- turn, Uranus, and Neptune. The earth rotates from west to east, about an axis inclined to the plane of its orbit about 66^°, 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 i~ight 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 himself in ima- gination at or near the centre of the system, and on the north side of the plane of the earth's orbit within which the planeta may all, for the present, be conceived to revolve. 17. Symbols. The principal planets, and the sun and moon, are often designated by the following conventional characters or symbols. The Sun, Mercury, . . . . ^ Saturn, . . . . ^ Venus, . . . . S Uranus, . . , . iji The Earth, . . . ® Neptune, . . . . f Mars, 5 The Moon, ...']) 18. Inferior, aud Superior Planets. The two planet^ y EFFECTS OF THE EARTH'S KOTATIOK 9 Mercury and Venus, whose orbits lie within the earth's orbit, are called Inferior Planets. The others are called Superior Planets. The terms inferior and superior as here used, have merely the signification of lower and higher in place, or in posi- tion with respect to the sun, as compared with the earth. 19. Vast Distance of tlie Fixed Stars. The angular dis- tance 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 Diurnal Motion of the Hea- vens. On the hypothesis of the earth's rotation, the diurnal motion of the heavens is a mere illusion occasioned by the rota- tion of the earth. To explain this, suppose the axis of the earth to be prolonged till it intej^sects the heavens considered as con- centric 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 sta- tionary, 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 men- tioned, 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 insensi- ble 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 Sun's apparent Motion. We 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 B, E' (Fig. 4) represent two positions of the earth in its orbit EE'B" about the sun S. When the earth i« 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 // and being in the mean time in- sensible 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 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 unOzr an angle of 66^° (16), the plane of the sun's apparent puth, which is the same as that of the earth's orbit, will be ini;lmed 23^° to a circle perpendicular to the earth's axis, or to u circle directed due east and west. CELESTIAL AKD TEEEESTIAL SPHERES. 11 CHAPTER II. Celestial and Terresteial Spheres. 22. Celestial Spliere. 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, hav- ing the station of the observer, or what comes to the very same thing, the centre of the earth, for its centre. This imaginary spheri- cal surface is called the Sphere of the Heavens, or the Celestial Sphere. 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 sphere, 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 celes- tial sphere, (19.) The other heavenly bodies are referred by observers at different stations to points somewhat different. Definitions. For the purposes of observation and compu- tation, 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, ia 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. Verliad Plane. Every plane that is perpendicular to the vertical line, is called a Horiaonial 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 from the invisible portion of the heavens, (*■) (3.) 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 Zefnitk 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 Ihe Heavens 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 per- formed. 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 Souih Pole of the heavens is the point in which the axis of the heavens meets the southern part of the celestial sphere. 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 be a point on the earth's 'sur- CELESTIAL SPHEEE. 13 S-S face, and OZ the vertical line at the station 0. — Then HOE will be the plane of the sensible horizon, HCE 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 OP 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 HOE, and HOE, in the same circle, passing through the points H and E. Unless we are seeking for the exact apparent place in the hea- vens of some other heavenly body than a fixed star, we may con- ceive the observer to be stationed at the earth's centre, in which case OP will become the same as CP, and HOE the same as HCE ; as represented in Fig. 6. In this diagram, the circle of the horizon being supposed to be viewed from a point above its plane, is repre- sented by the ellipse HAEa, Z 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 HAEa is pro- jected into its diameter HCE. Every different place on the surface ef the earth has a different zenith, and except in the case of diametrically opposite places, a fiG. T. different horizon. To illustrate this, let nesq (Fig. 7) represent the earth, and HZEP' the sphere of the heavens; then considering 14 CELESTIAL SPHERE. the four stations, e, 0, n, and q, the zenith and horizon of the firsl will be respectively E and PeP' ; of the second Z and HOR ; of the third P and QwE ; of the fourth Q and PqP. The diametri- cally opposite places O and 0' have the same rational honzon, viz. HOE. The same is true of the places n and s, and e and q. Their rational horizons are respectively QCB and PGP . (7.) Vertical Circles are great circles passing through the zenitti and nadir. They cut the horizon at right angles, and their planes are vertical. Thus ZSM (Fig. 6) represents a vertical circle passing through the star S, called the Vertical Circle of the Star. (8.) The Meridian of a place is that vertical circle which con- tains the north and south poles of the heavens. The plane of the meridian is called the Meridian Plane. Thus, PZRP' is the meridian of the station C. The half HZR, above the horizon, is termed the Superior Meridian, and the other half RNH, below the horizon, is termed the Inferior Meridian. The two points, as H and E, 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 Meridian Line, or the North and South Line. (9.) The Prime Vertical is the vertical circle which crosses the meridian at right angles. It cuts the horizon in two points, as e, w, called the Hast and West Points of the Horizon. (10.) The Altitude of any heavenly body is the arc 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 arc. Thus, SM or MCS is the altitude of the star S. (11.) The Zenith Distance of a heavenly body is the arc of a vertical circle, intercepted between its centre and the zenith ; or the distance of the centre of the body from the zenith as mea- isured by the arc 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 complements 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 arc of the horizon, intercepted between the meridian and the vertical circle passing through the centre of the body ; or the angle comprehended be- tween 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. HM or HCH represents the azimuth of the star S. The Azimuth and Altitude, or azhnuth and zenith distance of a heavenly body, ascertain its position with respect to the horizon and meridian, and therefore its place in the visible hemisphere. Thus, the azimuth HM determines the position of the vertical cir DEFINITIONS. 15 cle ZSM of the star S with respect to the meridian ZPH, and the altitude MS, or the zenith distance ZS, the position of the star in this circle. (13.) The Amplitude 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 ho- rizon 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 ampli- tude at rising, and wA its amplitude at setting. (14.) The Celestial Equator, 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 '&ivQ,e. This circle is also frequently called the Equator, 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 Diurnal Circles. (16.) Celestial Meridians, Hour Circles, and Declination Cir- cles, 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 from the meridian towards the west, are called Hour Angles, or Horary Angles. (17.) The Ecliptic is that great circle of the heavens which the Bun 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 23^°. (19.) The Equinoctial 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 Winter 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 Equinoctial Colure is the celestial meridian passing through the equinoctial points ; and the Solstitial Colure is the celestial meridian passing through the solstitial points. (22.) The Polar Circles are parallels of declination at a distance from the poles equal to the obliquity of the ecliptic. The one about the north pole is called the Arctic arch; the other, about the south pole, is called the Antarctic Circle. r • ■ 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 TVopic of Capricorn. The tropics touch the ecliptic at the solstitial points. Fio. 8. Let C (Fig. 8) represent the centre of the earth and sphere, POP' the axis of the heavens, EVQA 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; PKWK'T the solstitial colure', the angle TCQ, or its measure the arc TQ, the obliquity of the ecliptic ; KmU, K'm'U', the polar circles ; and TnZ, Wn'Z', the ti-opics. 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 Signs. Each sign contains 30°. The division com- mences 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 T, Taurus y, Gemini n, Cancer 25, Leo 5L, Virgo rrg, Libra d^, Scorpio fil, Sagittarius / , Capricornus V5, Aquarius ^, Pisces jf. The first six are called northern signs, beingnorth of the equinoctial. TheothersarecalledsoM^er-wst^r?^. The vernal equinox corresponds to the first point of Aries, and the autumnal equinox to the first point of Libra. The sum- mer 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 arc of the equator intercepted between the vernal equinox and the declina- tion circle which passes through the centre of the body, as reckoned from the vernal equinox towards the east. It mea- sures the inclination of the declination circle of the body to the equinoctial colure. Thus, PSK being the declination circle of the star S, and V the place of the vernal equinox, VE -is the right ascension of the star. It is the measure of the angle YPS. If PS'K' be the declination circle of another star S', SPS', or EE', will be their difference of right ascension. (27.) The Declination of a heavenly body is the arc of a circle of declination, intercepted between the centre of the body and the equator. It therefore expresses the distance of the body from the equator. Thus, ES is the declination of the star S. Declination is North or Scnith, according as the body is north or south of the equator. 2 18 CELESTIAL SPHERE. In the use of formulse, a south declination is regarded as negative. The right ascension and dedinalion of a heavenly hodi/ are two co-ordinates, which, taken together, fix its position in the sphere of the heavens: for they malte known its situation with respi^ct to two circles, the equinoctial colure, and the equator. Thus, VK and ES 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 arc ot a declination circle, intercepted between the centre of the body and the elevated pole. The polar distance is the complement of the declination, and, therefore, when either is known the other may be found. (29.) Oirchs of Latitude are great circles of the celestial sphere, ■which pass through the poles of the ecliptic, and tiicrulbre cut this circle at right angles. Thus, KSL represents a part of the circle of latitude of the star S. (30.) Tiie Longitude of a heavenly body is the "arc of the eclip- tic, interc(!pted between the vernal equinox and the circle of latitude which passes through the centre of the body, as reckoned from 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 South, according as the body is north or south of the ecliptic. In the use of formulse, 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 pass- ing through the vernal equinox, and the ecliptic. Thus, VL and LS fix the position of tlie star S, making known its situation with respect to the circles KVK'A and VTAW. (32.) Tlie Angle of Position of a star is the angle includeil at the star between the circles of latitude and di.'cli nation passing through it. PSK is the angle oC position of the star S. (33.) The Astronomical LaUlnde, or the I/atitude of a place, is the arc of the meridian intercepted between the zenitli 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 ; QUE or QCE being the equator. 23. Terremtrial Sphere. The earth's surface, considered as DEFINITIONS. 19 Bpherical (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 eor;- stractions appertain to the terrestrial sphere, as it is employed for the purposes of astronomy. It will be observed that they correspond to thos* of the celestial sphere above described, and are used for similar objects. DefinitioBs. (1.) The XoriJi and SotUh Poles of the Earth are the two points in which the axis of the heavens intersects tlie 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 pacing through the centre of the earth, and perpendicular to the axis of the heavens and earth, cuts the terrestnal 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 equa- tor divides the earth are called, respectively, the Northern Hemi- sphere and the Southern Hemisphere. (3.) Terrestrial Meridians are great circles of the terrestrial sphere, parsing 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 rneridian, 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^ and^' it= poles. Then, elq will represent the ferres/n'oZ ey^ator (ELQ representing the celestial equator.; and pep' and psp' terrestnal meridians (PEP' and PSP represer.ting celestial meridians). (i.) The Seduced Latitude of a place on the earth's surface is the arc of a terrestrial meridian, intercepted between the place and the equator, or the angle at the centre of the earth measured by this arc. Thus, oe, or the anjrle oOe, is the reduced latitude of the place o. latitude is Sorth or South, according as the place is north or south of the equator. The reduced latitude dif- fers somewhat from the astronomical latitude, by reason of the slight deviation of the earth from a spherical form. Their differ- ence is called the BeducfAon of Ziatiiiide. (5. 1 Parallels of Latitude are sma.l circles of the terrestrial sphere parallel to the equator. Every point of a parallel of lati- tude has the same latitude. The parallels of latitude which correspond in situation with the polar circles and tropics in the heavens, have received the same appellations as these circles. (See dels. 22, 23, p. 16.) (6.) The longitude of a place on the earth s surface is the inclination ot ts meridian to that of some particular station, fixed 20 TERBESTRIAL SPHERE. upon as a circle to reckon from, and called the First Meridian. 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 or to the west of the first meridian. Thus, if ^5;^' be supposed to represent the first meridian, the angle spy, or the arc 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 Green- wich. The longitude and latitude of a place designate its situation on the eartKs surface. They are precisely analogous to the right ascen- sion and declination of a Litar 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 circJes of the celestial sphere, represents the aspect of thia ALTITUDE OF THE POLE. 21 Pig. 10. = PB — ZP=90° — ZP. = ZB. Bpliere at a place at which the north pole of the heavens is some- where between the zenith and horizon. Such is the - -^ position of the north pole at all places situated be- tween the equator and the north pole of the earth. For, let (Fig. 10) repre- sent a place on the earth's surface, HOR the horizon, OZ the verticnl, HZR the me- ridian, and ZE the latitude. QOE will then represent the equinoctial, and P, 90° dis- tant from E and on the su- perior meridian, the elevated pole. Now we have HP = ZH — ZP = 90° — ZP; ZE = Whence, HP = Thus, the altitude of the pole is everywhere 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 declina- tion 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 appa- rition ; the line a& (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 oE, at a distance from the south pole equal to the latitude of the 22 CELESTIAL SPHERE. Fl8. 11. place, will lie entirely below the horizon, and jusi touch it at the south point. The parallel thus situated is called the cvrde of perpetual occuUation. ine stars comprehended between it Jf.^-4S .. and the south pole will neve) The celestial equator (which passes through the east and west points) will intersect the meridian at a point E, whose zenith distance ZE is equal to the latitude of the place (def. 33, Art. 22), and consequently, whose altitude RE is equal to the CO latitude of the place. There- fore, in the situation of the ob- server above supposed, the equator QOE, passing to the south of the zenith, will, to- gether with the diurnal circles nr, st, etc., which are all parallel to it, be obliquely 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, aH and oR, with the exception of the equator, will be divided unequally by the horizon. The greater parts of the circles nr, 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 hori- zon 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 equa- tor. Again, the stars will all culminate, 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 meri- dian to the western horizon. The circumpolar stars will pass the meridian twice in 24 hours; once above, and once below the pole. These meridian passages are called, respectively, ASPECTS OF THE CELESTIAL SPHERE. 23 Upper and Lower Gulminations, 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 con- stantly visible and invisible, are continually on the increase. It is evident, from what is stated in Art. 24, that the circum- stances 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. Pig. 12. 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 con- tinually visible, while the other hemisphere will be continually invisible. This is called a Parallel Sphere. It is represented in Fig. 13. 24 ASTRONOMICAL INSTEUMENTS. CHAPTER III. ASTEONOMICAL INSTRUMENTS. — ASTEONaMICAL 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 measure angles in the meridian plane, the axis of rotation is horizontal, and a single vertical limb is used. 29. The Reticle. 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 CoUimaiion 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 the axis of rotation of the telescope. These screws are shown at aa and bb, Fig. 14. They pass through narrow slits ip 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 pass- ing through the centre of the eye-glass and the centre of the object-glass, or the optical axis of the telescope, is perpendi- cular, 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 MICEOMETER WIRE. 25 Betide Tube. The reticle is placed in a tube, •which slides in the lower end of the principal tube of the telescope. The eyepiece is inserted in the outer end of this tube; and can be pressed in or drawn out until the wires of the reticle are dis- tinctly 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, Improved Form ot 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 com- monly by two very close parallel wires (Fig. 15). In meridian instruments, and those for measuring altitudes, tlie single wire, or the equiva- lent 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 two horizontal wires. The point of the middle transit wire that lies midway between the two horizontal ones, corre- sponds to the point of intersection of the two cross wires in Fig 14. ITie vnre-plaie 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 trans- verse frame, and is moved by means of a screw. By turning this screw the eye-piece may be brought into such positions that the star observed is kept in the middle of the field of view. 31. Movable Micrometer Wire. In the focus of the eye- glass there is often fastened to a transverse sliding plate, and movable with it, a wire at right angles to the direction in which 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 movahle micrometer-wire, and the whole apparatus, being especially designed for the measurement of small angular distances, is called a Micrometer. The same name a c 1 \ \ I 6 d Pie. 15. 26 ASTRONOMICAL INSTKUMENTS. 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 gradu- ated 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 Verai£r, and the Heading Microscope. 33. The Vernier 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 ver- nier, 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 coin- cides with a line of division on the limb. In Fig. 16, MN represents a portion of the limb of an instru- ment, divided into degrees and 10' spaces ; V the Vernier, ten 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 sup- posed 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, there- fore, 4' beyond the point 30° 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. To THE READING MICRUSCOPE. 27 ■show this, let x = the number of minutes iu a division of the vernier, then by what is stated above, 10 X = 9 X 10' ; whence x - 9', and 10' — a; = 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 iifty-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 microscope 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 iiidex, 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 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 microscope is set at the proper distance from the limb, it is coincident with the 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 cross- wires are attached ; Gr, the milled head of the screw ; EF, the graduated cylin- drical head, called the graduated head of the screw ; and i the fixed index. 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 micro- meter, 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 hh, 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 regular- ity to the movements of the frame under the action of the screw. The divisions of the limb are shown as short, heavy, equidis- i'lG. 17. 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 same 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 tlie angle the observer looks through the micro- scope 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 vernier- plate. 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 num- ber 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 readin" ASTRONOMICAL OBSERVAIIO.^S. 29 is thus measured at different points of the circle, and the mean of the 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 micro- scopes. 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 Mean Solar Time. Sidereul Time 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, commenc- ing 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 Ap- parent Solar Hours. Mean Solar Time is measured by the diurnal motion of an imaginary sun, called the Mean Sun, conceived to .move uni- formly 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 suns mean longitude. A Mean Solar Day com- mences when the mean sun is on the superior mei idian, and is divided into 24 Mean Solar Hours. Since the mean sun moves uniformly and directly towards the east, the length of the mean solar day must be invariable. The Astronomical Day commences at noon, and is divided into 24 hours; but the Cale^idar 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 import- ance are made out of this plane. The chief instruments em- ployed 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, in- asmuch as they serve, as will soon be explained, for the deter- mination of the places of the heavenly bodied, which are the fundamental data of astronomical science. The: principal iDfltrn- 80 ASTRONOMICAL INSTRUMENTS. ments used for making observations out of the meridian plane, are the Altitude and Azimuth Instrument^ the Equatorial, and the Sextant. THE TRANSIT INSTEUMBNT. 38. The Transit Instrument, or Transit, is an instrnnc ent 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 in- struments 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 instru- ment 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, M ; 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 iu their boxes to allow one of the pivots to ba 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, E, from one end of which depends a strong brass hook that supports the friction rollers X, under the end of the axis. A counterpoise, W, is adapted to the other end of the lever, which 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 — the 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. , THE TRANSIT INSTRUMENT. 81 Fia. 19. 82 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 tele- scope, and is reflected to the eye-glass; thus illuminating the field of view, and exhibiting the wires of the reticle, at w, 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 eye- glass, by two small lamps (omitted in the drawing) suspended upon the telescope, near the eye-piece, which throw their li^ht obliquely upon the wires, through openings in the eyeAnbe, 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 honzontal 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 Fio. 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 accom- plishing this, the handles, BB and D, are used. The handle D acts upon a clamp that fastens the rotation axis. When the telesa^pe has been depressed nearly to the required position, it ia 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 Beversing Apparatus, or Car, with which the instrument may be lifted from the Y's, and the rotation axis reversed, is shown at Hi. 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 friction rollers at XX. The car is then rolled out from between the piers, bearing 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 amch, 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 Telieving the observer of the fatigue of a constrained posi- tion when the telescope is directed upon stars at high altitudes. L is a striding level, which is used in levelling the rotation axis. 3ft. Adjustments of the Transit. To secure accurate obser- vations with the transit, three adjustments of the instrument are necessary : 1. The axis of rotation is to be brought into a horizontal po- sition. 2. The line of coUimation is to be made perpendicular to the axis of rotation. 3. The line of coUimation 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. 40. First Adjustment. The first adjustment is efiected by means of the striding level, L, which is applied to the pivots, W ; the feet of the level having the form of an inverted V for this pur- pose. 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 INSTRUMENTS. 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 en- tailed from this cause, upon the observation. 41. Second Adjustment. The second, or collimaiion adjv^t- meni, 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-defined point of a distant terrestrial object. Then, by means of the reversing apparatus raise the telescope from the T's, and replace it with the ends of the axis reversed.- Brmg 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, and the remaining distance by the screws that give a lateral motion to one end of the rotation axis. By one or more repeti- tions of this process, the desired adjustment may be effected. The better plan, and the one ordinarily adopted by astronomi- cal 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 mov- able micrometer-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, con- vert 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 collimation. The opportunity of reversing the instrument also enables the observer to determine the correction for ineqitaUty of the pivots ; that is, the inclination of the mathematical axis of rotation to the horizon that may result from any such inequality. This correc- tion is equal to one quarter of the difference between the inclina- tions of the line on which the feet of the level virtually rest, as determined by the level, in both positions of the telescope. Gollimating Eye-Piece. The most convenient method of determining the error of collimation fZ ~^^L ^^ ^^ making a certain observation with what is •f— — Hff called the collimating eye-piece, substituted for I \ J.f t^^ ordinary eye-piece of the telescope (Fig. &f i/'M'fS ^■'"^' '^^^^ differs from the common eye-piece ' ^'Jm ^^ having an opening in one side of the tube, J . ,*ij(ji and a metallic reflector, of the form of an ellip- r „..^:L.ra a tic^l ring, set obliquely within the tube, to re- t . M fleet the light of a lamp upon the wires of the Pig. 21. micrometer. _ The observation to be made with it consists simply in looking vertically down- ward through the telescope at the image of the micrometer- THIRD ADJUSTMENT, 35 wires, reflected from a basin of mercury placed on an immo- vable stone slab under the telescope. If the axis has been truly levelled, the error of collimation will be half the dis- tance between the middle wire, as seen directly, and its reflected image. This distance can be measured by means of the mova- ble 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. Tliird Adjustment. 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 me- ridian. 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 s and ^, in the constellation of the Great Bear, is in a vertical position. The pole-star is also known to be on the meridian when it attains to its greatest, or least alti- tude. When the instrument has thus been approximately 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 telescope moved in azimuth, 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 from 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 mid- dle 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 stars remote from the zenith (the pole-star in preference). This pro- cess may be many times repeated, until the line of collimation of the transit telescope is brought, with all attainable accuracy, into the meridian plane. Or, the error of the adjustment may be cal- culated from the results of the observations upon the star near the zenith and the pole-star, and allowed for in subsequent obser- 36 ASTRONOMICAL INSTRUMENTS. vations. This method of adjustment is called the method of high and hw stars. The final result obtained by it may be tested by the method of circumpolar stars already alluded to ; which haa 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 ex- plained. 43. The Time of the Meridian 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 crossing 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 equi- distant. 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 equato- rial intervals, divided by the cosine of the declination of any star, will be the wire-intervals for that star. By means of these inter- vals 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 pas- sage of both the western and eastern limb across each of ihe 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. 44. Electro- Chronograph. Theaccuracy 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 manner 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. 403. 41s. 42s. 43s. 44s. 46s. 46s. Ha. 48s. ^ - 1^9. 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 Har- vard College. It has since been introduced at the Greenwich and other principal observatories. 45. To determine the Right Ascension of a Star. When a star is on the meridian, its declination circle (def. 16, p. 15) coincides with the meridian ; moreover, the arc of the equator which lies between the declination circles of two stars, measures their difference of right ascension. Thus, ER' (Fig. 8) is the 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 arc RE', 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 de- termined. 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, YR' may be found by adding VR and RR'. The manner of determining the position of the vernal equinox, or the value of VE, will be explained in the chapter on the Apparent Motion of the Sun, Bight ascensions are commonly expressed in time. 38 asteonomical uststeuments. ASTRONOMICAL CLOCK. 46. The Astronomical Clock is provided with a pendulum so constructed that its length is unaffected by changes of tempera- ture. 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 discon- nected from the floor of the observatory. It keeps sidereal time. 47. To Regnlate 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 obser- Tatories, 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 determin- ing 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. 48. A Meau 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 Measurement of Time. UEBIDIAN CIRCLZ. 89 MERIDIAN CIRCLE. 49. The Meridian Circle is an instrument nsed 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. The graduated circle is firmly attached at right angles to the horizontal axis of rotation, and turns with it. The angle is read from 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 ate used, at equally distant points of the limb. The degrees, minutes, and seconds, are read from 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 from imperfect graduation, inaccurate centring, and unequal expansion of the limb, may be materially lessened. 50. Fig. 23 represents a meridian circle manufactured by Eepsold, a celebrated German instrument-maker, and mounted in 1852 in the observatory of the United States Naval Academy. It has two graduated circles, CO 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 EE. 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 hori- zontal 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, far 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 Uvel^ LL. A horizontal arm, FG, seen to the right of the telescope in the figure, extends out from 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 isi clamped in the collar ; and then a screw (not seen in the drawing), connected with the arm FG, and acting horizontally upon the 40 ASTEONOMICAL INSTBUMEUTS FlO. 23. MERIDIAN CIRCLE. 41 vertical arm, gives a fine motion to the telescope. FG turna 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, pre- paratory 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 regu- lated by revolving discs with eccentric apertures, at the extremi- ties 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 micrometer- screw. 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 measure- ment of small differences of declination. 51. Mural Circle. This is another form of the meridian circle that has been much used in large observatories. The gra- duated limb of the mural circle is secured to one end of a hori- zontal 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 determina- tions, to the form of meridian circle just described. Mural circles have been constructed as large as eight feet in diameter. 52. Adjustments. 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 liirib. 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. To determine the Horizontal Point of the Limb. 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 mer- cury, and note the angle as before. By a fundamental law of reflection the angle of depression of this image will be equal to the angle of elevation of the star. Accordingly the arc 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 ASTEONOMICAL INSTRUMENTS. This metliod 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 tha 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 anothei observation taken upon its image, as it is orossmg the last transit wire. The instrument is first set to the altitude of the star, as nearly known, and the cor- rection 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 measure tlie Altitude of a Heavenly Body. (1). Of a fioced 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 r&sults. This must be corrected for the deviation of the horizontal point from the zero of the limb, and all the detected errors that result from 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 alti- tude 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 establishing 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 Declination of a Heavenly Body. 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 (0), (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 Bs'=ZE— Zs', (6) ; and if its point of transit be some point s" to the south of the ALTITUDE ANB AZIMUTH INSTRUMENT. 43 equator (E), we shall have Es"=Zs"— ZB, and— Es"=ZE— Zs", (c). The three formulae {a), (J), and (c), may all be compre- hended in one, viz. : Declination =:latitude+meridian zenith distance, ... (1) If we adopt the following conventional rules : (1) north lati- tude is -f-, 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 determiiie the Kiongitiide and Liatittide of a Body. When the right ascension and declination of a heavenly body have been obtained from observation, with a transit instru- ment and circle (45, 55), its longitude and latitude may be com- puted. 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 SE. the declination, and the angle KPS = ER=:BV+VE=90° + right ascension; and if we sup- pose 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 = 180°— WKS = 180°— (WV+ VL)=:180° —(90° -I- 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 iJSD AZIMUTH INSTRUMENT. 5''. This instrument consists essentially of a telescope 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 instru- ment 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 INSTBUMENTS. be the index error. This having been determined, if the tele- scope 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 ihe ver- tical circle serves to measure the altitude. The altitude and azimuth instrument is sometimes called the Altazimuth; also the Asironomical Theodolite. 58. The nieridiau I.iiie (def. 8, p. 14) at a place may easily be determined with the altitude and azimuth instrument, by a method called the Method of Equal Altitudes. Let (Fig. 24) represent the place of ob- servation, NPZ the meridian, and S, S' two positions of the same star, at which the altitude is the same. Now, the spheri- cal triangles ZPS and ZPS' have the side ZP common, ZS= ZS', and (allowing the stars to moveinc«Vcfes)PS=PS'. Hence Fia. 24. they are equal, and consequently the angle PZS=PZS'; that is, equal altitudes of a star correspond to equal azimuths. 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 vici- nity 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 ad.aptation to the telescope of a mi- crometer with a movable horizontal wire. If such a micrometer be adapted to a transit instrument, that EQUATOEIAL. 45 instrument may 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 gra- duated circular limb at right angles to the axis, and therefore parallel to the plane of the equator ; from 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, aud 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. L'ses of the Equatorial. A telescope thus equatorially mounted, and provided with a movable micrometer-wire, is espe- cially adapted to the measurement of the apparent diameter of a heavenly body, the angular distance between stars in close prox- imity, 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 satisfactorUy shown by Professor 0. S. Lyman, of Tale Collega (see American Journal of Science, Vol. XXX., p. 52). The zenith telescope is essentially the invention of Oapt. Au(Jr6W 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 Taloott's method (Chaurenet's Spherical aud Practical Astrcmomy). 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 VnQ Position Filar Micrometer ; with which the measure- ments in question can be made with great accuracy. Differences of right ascension and declination can also be mea- sured with the equatorial, by means of the hour and declination circles, but with much less accuracy than with the transit instru- ment and meridian circle. 62. PosUion Filar Micrometer. This piece of apparatus serves at the same time to measure small angular distances, and the angle included between the line oonaeet- ing two stars in close proximity and the celestial meridian. This angle is called the angle of position of one of tlie stars with respect to the other. It is estimated from the S. round by the W. to 360°. The Filair 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). 'Eia. 26. Fm. 25. Jt consists of two forks of brass, U'l, cc'c, sUding within a reetanguLir 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- thev !il!="°\\ ' ^^\^^^ parallel wires which they carry, by micrometer screw^ passmg through the ends of the box, and attached to the forks. A third ani stationary wire ^ perpendicular to the other two, is attached to a diaphragm dis- connected from the forks. The heads of the screws are not shown in the^u?^ but they may be seen m Fig. 26, in which 6 is the micromet«r-box. The evel piece IS screwed mto the micrometer-box, as shown m Fig. 26 The eradu'.tpd screw-heads are connected with nuts wliich turn, without advancinff won the screws that are festened to the forks. AccordmgTy by turning Z71, X forks may be moved either forwards or backwards. A stationary comb-scale on one side of the box, Indicates the number of reTol* EQUATORIAL. 47 ttons of either screw, answering to any distance that the wires may be separated from each other ; and the fractional part of a revolution is ahown by the gradu- ated head of the screw. The value of one revolution of the micrometer-screw may be found 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 inten'al between them. The interval of time thus obtained, converted into the equivalent angular space by allowing 15" to 1', will be the number of seconds of arc answering to the assumed number of revolutions of the screw. To adapt the filar micrometer to the measurement of cmgles of position, the mi- crometer-box, with its attached eye-piece, is so mounted as to admit of a rotation around the centre of a graduated circle (Pig. 26). The circle is fastened at the end of the reticle-tube, and in a plane perpendioulai: to the optical axis of the telescope. The revolving motion is produced by a milled-head screw «, 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 ob- servatory instruments, the chief design of whose construction is to furnish the places of the 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 astro- nomical 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 portable than any of these, and has not to be set up and adjust- ed 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. €4. Construction : — Principle of Construction, 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 quan- tities 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 BG (Fig. 27), comprising about 60 degrees of the entire circle, which is attached to a triangular frame BAG ; two mirrors, of which one (A) called the Index 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 AG drawn 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 principle of the amstruction and use of the sextant may be understood from 48 ASTEONOMICAL 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 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 CAGr measured upon the limb of the in- strument (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, (=ADB ;) atid the observer, looking through the telescope, will see the same ob- ject S' both by direct and reflected light. Now, if the index- glass be moved from this position through any angle, GAG, 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 — CAG. Therefore the angle made with the index-glass by the new incident ray SA, which after reflection now pursues the same course ADB, and reaches the eye at E, as it is always equal to that made by the re- flected 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 reflee- THE SEXTANT. 49 tion from the two mirrors, is equal to twice the angle CAGr that the index-glass is moved from the position (AC) of parallelism to the horizon-glass. Hence the limb is divided into 120 equal parts, which are calhd 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 appar- ent semi-diameters of the two bodies, or bring the farthest limba into contact, and subtract this sum. 65. Tbe Detail of the Construction of the Sextant is shown in Fig. 28. The limb, and the triangular frame to which it ia attached, are of hammered brass, and strengthened by cross-platea. The graduation is upon silver inlaid in the brass. Bach degree i.s 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 sss, and movable with it about the centre C of the graduated arc. These two mirrors are of plate-glass silvered. The upper half -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, K, 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, mov- able 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 B, under the index-bar, is a screw for clamping it to the limb ; and G is a tangent screw for giving the bar, with the index-glass, a small motion, in securing the accurate contact or coincidence of the images. H is a wooden handle at the back of the sextant, by which it is held when an observa- tion is taken. At E and F are colored glasses of different shades, to diminish the intensity of the light when the sun is observed. Those at F are interposed between the index-gla.ss 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 sight of the telescope parallel to the plane of the graduated arc, and in determining the index efiror. 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 continu- ous arc. If the reflected image does not appear to form a trne continuation of the arc, the index-glass is not perpendicular to the plane of the sextant. It may be corrected by loosening the screws sss, 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 mov- ing the index until the direct and reflected image of the star pass each other. If, in passing, 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, 100° to 120° 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 near- est the plane of the sextant : if then, on moving the instrument, liie contact re- mains 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 re- qnired. It can be made, when necessary, by means of two small screws in the ring which supports the telescope. To find the index 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 arc. This reading will be the index error, and may be either positive or negative. 67. Taking an Angle. 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 band until the reflected image of this object is brought, at the centre of the field of the telescope, int(~ apparent contact with the object seen directly • THE SEXTANT. 51 the contact being finally effected by the use of the tangent screw. The angle is then read from the limb and vernier, with the mi- croscope. "When the sextant is employed to take the altitude of a heavenly body, a horizontal reflector, called an Artificial Horizon, is placed in 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 body, and from the result is subtracted the depression of the visible horizon below the horizontal line, called the Dip of the Horizon. 68. Hadley's Quadrant. Hadley'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 120°; 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 Eeflecting Circle is but an en- larged sextant. Its limb is a full circle, and the index-arm is prolonged ia 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 sex- tant, recently introduced. It takes its name from the fact that a reflecting prism is used in place of the ordinary horizon-glass. This prism also occupies a different position with respect to the index-glass. The graduated limb extends 120°. The prismatic sextant can be used to measure aa angular distance of 180°, and an altitude of 90°. It is also superior to the ordinary sextant in certain other peculiarities of construction. Pri^natic Reflecting Circles are also constructed which possess similar advantages over the ordinary reflecting circle. ERRORS OP INSTRUMENTAL ADMEASUREMENT. 71. Whatever precautions may be taken, the results of instru-" mental admeasurement will never be wholly free from errors. Errors that arise from inaccuracy in the workmanship or ad justment of the instrument, may be detected and allowed for. But errors of observation are, obviously, undiscoverable. Since, aowever, the chances are, that an error committed at one obaer- 52 ASTBONOMICAL INSTRUMENTS. vation, will be compensated by an opposite error at another, it la to be expected that a more accurate result will be obtained if a great number of observations, under varied circumstances, ba made, instead of one, and the mean of the whole taken for the element sought. And accordingly, it is the uniform practice of astronomical observers to multiply observations as much as is practicable. 73. Instrumental x;rrors may be divided into three classes ; viz. errors of construction, errors oj adjustment, and incidentai errors. Errors of construction, in the best instruments, result chiefly from imperfect graduation, an eccentricity of the limb, an inequality or an eUipticity of thepivots, and an imperfect rigidity of the telescope or axis. The efiect of eccentricity and of the elliplicity 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 tailing the mean of all the readings. When the construction of the instrument is such that the principle of repetition 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 mto 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 temper- ature, especially an unequal expansion of diiferent parts of the Urab, and a derange- ment of the microscopes ; from flexure produced by weight ; and also from vibra- tions produced by passing vehicles, and other derangements from extraneous mechanical causes. All such errors may be mostly neutralized by making nume- rous measurements, under a great variety of circumstances. THE TELESCOPE. 7 3 . An observatory is not completely furnished unless it is supplied with a largo 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 occa- sional celestial phenomena, as eclipses of the sun and moon, occultations of the stars, etc. Telescopes are divided into the two classes of Eeflecting and Refracting Tele- scopes. In the former class, the image of the object is formed by a concave specu- lum, 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 Hersehelian, 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. Magnifying power— illuminating powei-— space-penetrating power. The magni- fying power of a telescope is to be carefully, distinguished 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, nebulas, etc., by intercepting and converging to a point a much larger beam of rays. The magnifying power is measured by the ratio of tJie focal length' of the object-glass, or speculum, to th.it 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 diS'erent 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 THE 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 lighc from an increase of distance, will be just suppliei if the aperture of the telescope exceed in its diameter that of the pupil of the eye 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 eye (0.2 in.). In the larger reflecting telescopes, the space-penetrat ing power, calculated by this rule, requires to be diminished about one-fiith, in consequence of the loss of hght 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 tran- quillity 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 eye- piece. 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 cer- tain 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. With the best telescopes, a magnifying power of four or five hundred is the highest that can be apphed 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 defin- ing 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 pohsh of the lenses, their freedom from chromatic and spherical aberrations, and other niceties of construction. T6, The fidd of view of telescopes diminishes in proportion as the magnifying power increases. It is stated that with a magnifymg 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 fihed 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. Lwrge Telescopes. The largest reflecting telescope that has yet been con- structed and directed to the heavens, is the great Eosse Telescope, devised and con- structed by Lord Eosse, of Ireland. It has a focal length of 53 feet, and an aper- ture 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 Olarlt Telescope, made by Clark & Sons, Cambridgeport, Mass., and recently set up in the Chicago Observatory. It has a clear aperture of 1 &i inches, and a focal length of 23 feet. It has, by the adaptation of different eye-pieces, different magnifying powers, varj'ing from 70 to about 2,000. The great telescope of the Observa- tory of Harvard College has an aperture of 1 5 inches, and a focal length of 22-J feet. Its highest magnifying power is 2,000. The refractor of the observatory at Pul kova, in Russia, is but shghtly inferior to this in its dimensions and capabilities Refracting telescopes of large dimensions and great excellence, are mounted equa- toriaUy in all the prominent observatories m the United States and in Europe. C-i C0EBECTI0N3 OF MEASUHEL ANGLES. CHAPTER IV. Corrections of Measured Angles. TS. 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 Plxice of a heavenly body is its real place in the heav- ens, 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 apparent co-ordinates, and those of its true place its trae co-ordi- nates. 79. Corrections. The apparent co-ordinates are reduced to the true, by the application of certain corrections, called Refrac- tion, Parallax, and Aberration. Kefraction and aberration are cor- rections 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 observa- tions from the surface to the centre of the earth, is to render ob- servations made at different places on the earth's surface directly comparable 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 observa- tions cannot, therefore, be compared together, unless they be re- duced to the same point, and the centre of the earth is the most convenient point of reference that can be chosen. EEFEACTION. §0. Atmospheric Refraction. We learn from the princi- ples of Pneumatics, as well as by experiments with the barome- ter, that the atmosphere gradually decreases in density from the earth's surface upwards. We learn also from the same sources, that it may be conceived to be made up of an infinite number of strata of decreasing density, concentric with the earth's sur- face. 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 KEFEACTION. 55 that its actual altitude is not far from 50 miles. Now, it ia 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 tl-^ surface at the point of incidence. It follows, therefore, that tl « light which comes from a star, in passing into the earth's atmi • sphere, or in passing from one stratum of atmosphere into anothe ■, is refracted towards the radius drawn from the centre of tl i earth to the point of incidence. Path of a ray of light. Let MmnN, NwoO, OojQ, (Fig. 29 i represent successive strata of the atmosphere. Any ray, Sp, will, then, instead of pursuing a straight course, Sprc, follow the broken line pahc; being bent downwards at the points p, a, 6, 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, hay, &c., as well as the lengths of the lines pa, ah, &c., are infinitely small ; and therefore jjaJc, 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 re- presented in Fig. 30. Moreover, it lies in the vertical plane con- taining the original direction of the ray ; for this plane is per- pendicular to all the strata of the atmosphere, and therefore the ray will continue in it in passing from one to the other. 81. Astronomical Refraction. The lineOS' (Fig. 30) drawn tangent to paO, the curvilinear path of the light, at its lowest jMiint, will represent the direction' in which the light enters the oye, and therefore the apparent line of direction of tlie star. If, tiien, OS be the true direction of the star, the angle SOS' will be the displacement of the star produced by Atmospheric Befrac- lion. This angle is called ih.Q Astronomical Refraction, or simply the Befraction of the star. Since i'aO is concave towards the earth, OS' will lie above 56 CORRECTIONS OF OBSERVATIONS. OS ; consequently, refraction makes the apparent altitude of a star greater than its true altitude, 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 ol the observer upon the earth's surface.) Thus, to obtain the true altitude irom the apparent, we mast siibtract the refraction; and to obtain the true zenith distance from the apparent, we must add the refraction. As refraction takes effect wholly in a verti- cal plane (80), it does not alter the azimuth of a star. The amount of the refraction varies with the apparent zenith dis- tance. In the zenith it is zero, since the light passes perpendi- cularly 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 find the Amount of tlie Refraction for a given Zenith Distance or Altitude. Let us first show a method of resolving this problem by the general theor3' of refraction. Ac- cording to this theory, the amount of the refraction, 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 the density which obtains at the earth'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 hah (Fig. al) represent its surface ; also let represent the station of the observer upon the earth's surface, and Sa a ray incident upon the atmosphere at a. Denote the angle of refraction OaG by p, and the refraction Oax by r. The angle of incidence REFRACTION. 57 Z'aS = Z'aS'+ S'aS = OaC + Oax =j) + 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 = m sin OaC, or sin (p + r) = m sin p; Fia. 31. developing, (App. For. 15,) sin p cos r + cos ^ 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.) Whence, l+cot^./'sin l"=m, or r"=^ — p7X — — =Atang^; m — 1 putting A = ■ -,„ • Let ZCa = C ; and ZOa = Z. OaC = % sin i. ZOa — ZCa, or ^ = 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 sup- position 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 refrac- tion, except in the vicinity of the horizon, varies nearly as the tan- gent of the apparent zenith distance. It has been ascertained by experiment that m, the index of refraction (the barometer being = 29.6 inches, and the ther- mometer — 50°), = 1.0002803. Substituting in equation (3), after having restored the value of A, and reducing, there results r = 57".8 tang Z (4). 83. Formnlae of Refraction. With the aid of this for- mula, 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. For, let P (Fig. 32) be the elevated pole, Z the zenith, PZB the meridian, HOR the horizon, S the true place of a star, and S' its apparent place. Suppose the apparent zenith distance ZS' to have heen mea- sured. Now, m the triangle ZPS, ZP the co-latitude and PS the polar distance are known by hypothesis, and the angle. P is the sidereal time which has elapsed since the star's last meridian transit, (or, if the star he to the east of the meridian, the difference between this interval and 24 sidereal hours,) converted into degrees by allowing 15° to the hour. Therefore we may compute the true zenith distance ZS, and subtracting from it the apparent zenith distance ZS', we shall have the re- fraction. For the solution of this prob- lem, the polar distance may be found by taking the complement of the declination computed from an bhserved 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 tak- ing the half sum of the greatest and least zenith distances of the pole-star. But it is obvious that neitlier 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 90", the corresponding refrao. tions may be at first ascertained with considerable accuracy by means of equation (4). When more correct formulas have been obtained by this or any other jro- cess, the latitude and polar distance, and therefore the refraction answering to the measured zenith distance, will become more accurately known. The various formulae of refraction having been tested by nu- merous observations, it is found that they are all, though in dif- ferent degrees, liable to material errors when the zenith distance exceeds 80°, or thereabouts. At greater zenith distances than REFRACTION. 59 this tke refraction is irregular^ or is frequently different in amouiit ■when the circumstances on which it is supposed to depend are the same. §4. JHean Refractions. — Corrections for the varying 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 refractions. The refrac- tions corresponding to any other height of the barometer or thermometer, are obtained by seeking the requisite corrections to be applied to the mean refractions in consequence of the differ- ence 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 zenith distances, or al- titudes, are computed from the formula, as also the corrections for various heights of the barometer and thermometer, and in- serted 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. Otiier Effects of Atmospheric Refraction. Atmo- spheric refraction makes the apparent distance of anj' two heav- enly 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 i. 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 accele- rates 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 33'; 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 iti The 60 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 ap- plied, the zenith distance of a body is reduced from the surface of the earth to its centre, by means of a correction called Par- allax. §6. Definitions. Parallax is, in its most general sense, the angle made by the lines of direction, or the arc of the celestial sphere comprised between tlie 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) re- present 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 ce- lestial sphere, and at B to the point s'; the angle BSA, or the arc ss', is the paral- lax. The arc ss' is taken as the measure of the angle BSA, on the principle that the celestial sphere is a sphere of an indefi- 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 be- tween the lines of direction of a heavenly body, as seen from a point on the earth's surface and from its centre ; or the angle sub- tended at a heavenly body by a radius of the earth. If C (Fig. 34) is the centre of the earth, 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 paral- lax, 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 Hori- zontal 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 hori- zontal 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 surface are inclined to each other ; aa Fi(J. 34. represented in Fig. 35, where OC is the radius, and OC the ver- tical. The points Z and s, in which the vertical and radius z z Ro. 35. pierce the celestial sphere, are called, respectively, the Apparent Zenith and the True, or Central Zenith. In perfect strictness, the horizrntal parallax is the parallax at the time asOS, the apparent 63 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. ST. True Zenith Distance. Let the apparent zenith dis- tance ZOS = Z, (Fig. 34,) the true zenith distance ZCS = z, and the parallax 080=^?. Since the angle ZOS is the ex- terior angle of the triangle OSC, we have ZOS = ZCS + OSC, and hence also ZCS = ZOS — OSC ; or, Z = z + p, and s = Z — ^ .... (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. §8. To find an Expression for the Parallax in Alti- tnde, 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 ^ K, CS = distance of the body S == D, and COS = 180°— ZOS = 180°— apparent zenith distance = 180° — Z ; and we have by Trigonometry the proportion sin OSC : sin COS :: CO : CS ; whence, and or. sin p : sin (180°— Z) :: E : D ; D sin ^ ^ E sin Z ; sin ^ = — 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 «i'iH=| (7); substituting, we have sin^ = sin H sin Z . . . . (8). This equation may be somewhat simplified. The distances of the heavenly bodies are so great, that p and H are always very small angles ; even for the moon, which is much the near- est, the value of H does not at any time exceed 62'. We may PARALLAX. 63 therefore, witliout material error, replace sin 'p and sin H by ^J and H. This being done, there results, j3 = H sin Z (9). "Wherefore, the parallax in altitude equals the product ofthehorv zontal parallax 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 H the horizontal paral- lax 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 H, the horizontal parallax. 89. To find an Expression for tbc Horizontal Paral- lax, in terms of measurable quantities. Let 0, 0' (Fig. 35) repre- sent two stations upon the same terrestrial meridian OEO', and remote from each other, Z, Z' their apparent zeniths, and s, z' their true zeniths, QCE the equator, and.S the body (supposed to be in the meridian) the parallax of which is to be found. Let the angle 030'= A, zOS = Z, z'O'S = Z'; also let CO = R, CO' = R', CS = D, the parallax in altitude OSC — p, and the par- allax in altitude O'SO = p'. Now, by equation (6), replacing the sine of the parallax by the parallax itself, (88,) M = ?L sin Z, and »' = 5-' sin Z'; whence , '^ . ry ^' ■ rr, R siu Z + E' siu Z' ,^ ^^ p + /= ^sm Z + jQ sm Z'= ^ ; . (10); and, (equ. 7,) R ^ R H = D ' °^ ^ = H' Substituting this value of D, and deducing the value of H, we have T, _ njp+p') Rx A ,^^^ Rsin Z + R' sin Z' ~ R sin Z + R' sin Z' ' ' ' ^ '' It remains now to find an , expression for A in terms of mea- surable quantities. Let Os and O's (Fig. 35) be the directions, at and &, 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 zenith dis- tances 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 repre- sented by d '. Now, OSO' = OLO' — SO'5 = SOs— SO's, or A = dr-d . 64 COERECTIONS OF OBSERVATIONS. If the body be seen on different sides of the star by the two observers, we shall have Substituting in equation (11), there results, H= ^(^^'^O . . .(12). EsinZ+R'sinZ' ^ ^ If we regard the earth as a sphere, R=Il', and dividing by R, we have H=-,_^' .... (13). sinZ+sinZ' 90. To Determine the Horizontal Parallax of a body, f roni Observation ; by means 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 followipg 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 suc- cessive transits : diff. of long., expressed in time, {{) :: varia- tion 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 obser- vation 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 having recourse to methods of the greatest possible accuracy, in the case of the sun and planets, will appear in the sequel. 91. Horizontal Parallax in Different £iatitudes. In consequence of the spheroidal form of the earth, the horizontal Earallax 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. g5 sinH = ^, andsinH'=5^; , . ^ . „, E E' whence, sin H : sin H':: ^ : "k " E : E'. Also, as H and H' are small, we have very nearly, H:H'::E:E'. 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 than at the poles. In the case of the sun, or of any planet, the difference is in every instance less than k". 92, Equatorial Parallax. The horizontal parallax of a body, for a station on the equator, is called its equatorial hori- zontal 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 -J". The horizontal parallaxes of the planets, at their varying dis- tances from the earth, are comprised between the limits Si" and 0."8. 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 Eng- lish 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 for which it is calculated. 93. Parallax in Riglit Asceusioai and in Declination. 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, respec- tively, 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 in latitude. 6 66 CORRECTIONS OF OBSERVATIONS. Formulae for calculating the parallax in right ascension, and in declination, as well as in longitude and latitude, are investi- gated in the Appendix. ABERRATION. 94. The celebrated English astronomer, Dr. Bradley, com- menced in the year 1725 a series of accurate observations, with the view of ascertaining whether the apparent _ placets 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 dis- covering their true explanation. His 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 Liglit. Kthe 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 displace- ment of the stars, first discovered by Dr. Bradley. These facta show that the aberration of light is the true cause of these pheno- mena, 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 dis- covered by Eoemer, a Danish astronomer, in 1675, from a com- parison 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 18.1883 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 determina- tion, in which the mean distance 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. 102. 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 fake 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 cor- responds with that of the sea. 103. method of deteriniiiing tlie Form of a. Terres- trial lUeridiaii. 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 arc. It is also called a Degree of Latitude. 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 aGb between the vertical lines Z'aG, Z6C, is 1°. s s Fig. 39. The length of a degree at any place will serve as a mea- Bure of the curvature of the meridian at that place ; for it ia 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 ah (Fig. 39) may, with but little if any error, be considered as an arc of i° of a circle which has its centre at C, the point of intersection of the verticals Ca, Ci, 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 propor- tion 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 duterniine the Liengtta of a Degree of a Terres- trial iUeridiaii. 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 sight- ing 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 adjust- ments may be corrected from time to time by astronomical obser- vations (42, 58). (2.) To find the length of the arc passed over. When the ground is level, the length of the arc 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 bv 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 0, 0' (Fig. 39) 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'lS = 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 degree, having been found by the foregoing operations, thence to compute 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 FIGUKE AND DIME>'SIONS OF THE EAKTH. N: A::l° whence x =^ l°x A N ■ ,..(17). 105. Re§nlts of the ITIea§urcnieiits of Degrees. Degreea 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 proceed from the equator towards either pole. It follows, there- fore (103), that the curvature of a meridian is greatest at the equator, and diminishes as the latitude increases ; and conse quently, that the earth is flattened at the poles. The fact of the decrease of the curvature of a terrestrial meri- dian 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. Ana- lytical investigations, founded on the lengths of a degree in dif- ferent 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 spheroid. The same inves- tigations 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 Oblateness of the earth. FlO. 40. The form of the earth has also been determined by othei methods, which cannot here be explained. All the results of measurements, taken together, indicate an oblateness of 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 45° latitude 3 956.20 " Mean length of a degree of meridian . . . 69.048 " The fourth part of a meridian 6,2 14.33 " LATITUDE OF A PLACE. 75 106. Inclination of Radius to Vertical Line. Owing to the elliptical form of a terrestrial meridian, the radius and ver- tical line at a place do not coincide. Let ENQS (Fig. 40) repre- sent a terrestrial meridian.' For any point situated on this meridian, CO will be the radius, and the normal line ZOB the ver- tical. 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 -±5°, and is there equal to about 11^'. DETBEMINATION OP THE LATITUDE AND LONGITUDE OP A PLACE. lOT. The latitude and longitude of a place ascertain its situa- tion upon the earth's surface, and are essential elements in many astronomical investigations. 108. To find the Lratitude 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 haa 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- 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 = HS + PS, and HP = HS' — PS' = HS'— PS ; whence, 2HP = HS+HS', or, HP = ^S + HS^ _ _ ^^^g^^ In like manner we obtain, ZP = 5^i-?-....(19). 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 pre- ferred to all others. (2.) By a single meridian altitude or zenith distance. Let s, s', s" (Fio-. 10, p. 21) be the points of meridian passage of three diffei ent 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, ZB = s'B f Zs', ZB = 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 differ- ent 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 refrac- tion, 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 Hie zenith, on opposite sides. This is TalcotCs Method 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 ^ its declination ; z' and 5' the true zenith distance and declination of the other star ; and I the lati- tude of the station : we then have I — S+z, and I = §' — ' z', and the>rcfbre, l = i{8+8')+i{^—z')....{a). Also, let Z denote the apparent zenith distance of the star that passo-o to the south of the zenith, r its refraction, and Z', r' tho co»^e»ponding quantities for the other star ; then, LATITUDE OP A PLACE. 77 2! = Z+r, anda' = Z'+r'; and substituting in equation (a) we obtain, I = ^(5 + 5')+^(Z — Z')+K^ — r'). . . .(5). As we may suppose the declitiations of the two stars to ba 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 ob- tained 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 — - /, very minute, and lialDle to no appreciable uncer- tainty. If m and m' denote the micrometer readings in observ- ing the two stars, converted into their equivalent angular values, equation {b) becomes, l=l{^ + ^')+:^{m—m')+\{r — r') (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 hori- zontal level, attached to the instrument in a position perpendi- cular 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. 32), sets the telescope to the mean of the zenith distances of the selected pair of stars, and when the preceding star has entered the field fol- lows 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 tele- scope, 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 readings 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".l, which answers to a distance on the meridian of only ten feet. Seduced Latitude. The astronomical latitude being known, the reduced latitude (p. 19, def. 4) may be obtained by subtract- ing from it the reduction of latitude. For if 00 (Fig. 40) repre- sents the radius, and OB the vertical, at any place O, 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, OBQ = OCQ + COB, and OCQ = OBQ — COB .... (20). (For the practical method of resolving this problem, see Prob- lem XV.) 109. EiOngitnde of a Place : — Oeiieral 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 timea {whether sidereal or solar), at any place and on the first meridian, is the longitude of the place expressed in time ; and consequently, also, the difference 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 arc of the celes- tial 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 honv 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 be/ure 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 timea and of the longitudes will subsist equally if sidereal instead of solar time be used. LONGITUDE OF A PLACE. 79 110. To find tlie liOugitude 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 phenome- non 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 ' difference 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 satel- lites of "Jupiter and 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.) Transport a chronometer which has been carefully adjiisted to the local time at Greenwich, to the place whose longitude is sought, and compare the time given by the chronometer with 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 and 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 chro- nometers 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 som,e celestial 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, occul- tations 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 longitude is estimated from Washington, the Washing- ton 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 approxi- mately 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, bas recently been introduced and perfected by American astronomers. It consists in the use of the electric telegraph for the transmission of signals from one station to the other, and the introduction of the electro-chrono- graph into the 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 break- circuit 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, from the corresponding break obtained by the other set; then con- vert this into the equivalent interval of time, and take the mean of all the intervals. This will be his determination of the dif- ference 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 determina- tions. For greater accuracy a number of selected stars should be observed. The observations should also be many times repeat- ed ; 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 aSect the mtervals 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 thia method. OBLIQUITY OF THE ECLIPTIC. 81 CHAPTER VI. Apparent Motion of the Sun in the Heavens. in. 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 from day to day, is the JEkliptic. 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 ; EGA the ecliptic ; and 00, OQ, lines drawn through 0, the centre of the earth, and perpendicu- lar to the line of the equinoxes, AOE : then the angle COQ will be the obliquity of the ecliptic. This angle has for its measure the arc CQ, and therefore the ohliquity of the ecliptk 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 ob- served 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 obli- quity. A formula has been investigated, which gives in terma 6 82 APPARENT MOTION OF THE SUN. of determinable quantities the difiference between any of tba greater declinations and tbe maximum declination. 'By reducing, by means of this formula, a number of the greater declinations to the maximum declination, and taking the mean of the indivi- dual results, a very accurate value of the obliquity may be found. The obliquity of the ecliptic changes slightly from year to year. It is also subject to a slight diminution from century to centurv. Its mean value at the present date (Jan., 1867) is 23° 27' 24". 113. To find tlie Position of the Vernal or Antnmnal Equinox, (1.) On inspecting the observed declinations of the sun, it is seen that about the 21st of March the declination changes in the interval of two successive noons from south to north. The ver- nal equinox occurs at some moment of this interval. Let RS, E'S' (Fig. 43) represent tbe declinations at the noons between Pie. 43. which the equinox occurs : as one is north and tbe 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 tha equinox and the instant of the equinox, state the proportion S:RS:: T:x = T^i^; S ' on the principle that the declination changes, for a day or more, proportionally to the time. Next, take the daily change in right ascension (ER') on the day of the equinox and compute the value of RE, by the proportion T:ir, or^g^::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 tha POSITION OF THE 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 may be found by a similar process, the only difference in the circumstances being that the declination clianges from 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 the day preceding the equinox, the result will be the time of the equinox. (2.) In the triangle EES (Fig. 42) we have the angle EES = a the obliquity of the ecliptic, and KS = 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. EES) tan ES = cot o tan D (21), whence we can find ER. And by taking the sum or difference of ER and MR, according as the star observed is on the opposite side of the sun from the equinox or the same side, we obtain ME 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 results be taken, a more exact value of ME will be obtained. MB 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 obser- vations upon the sun in different positions, S, S', S", etc., esta- blishes the truth of this hypothesis. It is also confirmed by the fact that the right ascensions 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. L.ongitude of the Sun. The longitude of the sun may be expressed in terms of the obliquity of the ecliptic and the right ascension or declination. In the triangle ERS (Fig. 42), ES ( = L) represents the longitude of the sun supposed to be at S, BR ( = R) its right ascension, and RS ( = D) its declina- tion. Now, by Napier's first rule, cos EBS=tan BE cotES, orcotES=52?45§=cosEEScotER; ' tan ER thus, cot L = cos o cot R, or tan L = (22). cos w Also (Napier's second rule, Appendix), 84 APPARENT MOTION OF THE SUN". sin ES = COS (co. EES) cos (co. ES) ; whence, sin ES = - — =-p^ ; sin KhjO or, . -p sin D /oo\ sin L = . . . (23). sin cj "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 E = tan L cos o ; sin D = sin L sin o (24). These formulae will make known the right ascension and decli- nation 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. 1 15. Tropical Tear. 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 Mean Tropical Year. As the changes in the length of the true tropical year are very minute, the length of the mean tropi- cal 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 Xiongitude. 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. : Id. :: 360 : a; = 59' 8".33. If from the right ascension or declination of the sun, found on two successive days, the corresponding longitudes be de- duced (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 same throughout t/ie year, but, on the contrary, is continually varying. It gradu- ally increases during one-half of a revolution, and gradually de- creases during the other half, and at the end of the year hag recovered its original value. Thus, the greatest and least daily motions occur at opposite points of the ecliptic. They are, re- spectively, 61' 10" and 57' 12". The exact law of the sun^s unequable motion, can only be ob- tained by taking into account the variation of his distance from, the earth ; for the two are essentially 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 PEECESSION OF THE EQUOOXES. CHAPTER VII. Precession of the Equinoxes. — Nutation. 117. Proof of an Annual Prece§sion of the Equinoxes. The determination of the position of the vernal equinox, consists in deducing from the results of certain observations tlie 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 VE in Fig. 8. We have seen (45) that virhen this has become known, the absolute right ascensions of all the stars may 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 num- ber of years, it is found that the value of ME has materially in- creased, 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 retro- grade motion, or towards the west, from year to year. Again, if the longitudes and latitudes of the same fixed stars, obtained as above, at ditferent 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 tiiat sL 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 lias been ascertained that the autumnal is always at the distance of 180° from the vernal equi- nox, it must have the same motion. This retrograde motion of the equinoctial points is called the Precession of the Equinoxes. 11§. Ecliptic Stationary. As the latitude of a star is its angular distance from the ecliptic, it follows from the circum- stance 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 tha MOTION OF THE POLE OF THE HEAVENS. 87 ecliptic, remains in the course of this motion very nearly the same. 119. Progressive ITEotion 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 ita stationary pole ; P the position of the north pole of the equator, or of the heavens, at any given time, and VEA the correspond- ing 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 and 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 retro- grade 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 pok-siar, so called, will not always be nearer to the poh 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 1^° becomes reduced to less than J°, 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 Hipparchus, 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 ascensions 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. Jjffect of Precession on tUe 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' 8".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 £cliptic. The ecliptic, although very nearly stationary, as stated in Art. 118, is not strictly so. By comparing the values of 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 from 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 eclip- tic, 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 fai overlooked. NUTATION". 122. Discovery of IVutatiou. Dr. Bradley, in observing the polar distance of a certain star (/ 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 from the results of previous observations, by allowing for the change due to the precession in the interval ; the proper correc- tions for re'^-action and aberration having been applied in both ELLIPSE OF NUTATION. 89 cases. On continuing his observations he found that the polai 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 IVutation. 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 sup- posed 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, pgfg' represents the ellipse of nutation, and P the Fig. 45. mean pole ; the direction of the motion of revolution being from p towards/ The major axis (/g'' lies in the solstitial colure KPL, and is equal to 19" ; and the minor axis ff is equal to 14". "While the true pole revolves in its ellipse about the mean pole P, the mean pole haa a uniform retrograde movement in a cir- cle 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 eclip- tic. Its actual motion must therefore be in a slightly waving curve, passing alter- FiG. 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 i.s also greatly exaggerated in Fig. 45. 124, Effects of 'Mutation. As the equator must move with the axis of the earth or heavens, nviotaon must change the position of the equinox and IM obliquity of the ecliptic. It is plain that its efl'ect 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 wlU be to make it alternately greater and less than the obliquity corresponding to the mean pole. The position of the equinox which cor- responds to the mean pole is called the Mean Equinox; and the obliquity corres- ponding to the mean pole is called the Mean Obliquity. Mean Equator has a like signification. The real equinox and the real equator are called, respectively, tho True Equinox and the True Equator: The actual obliquity of the ecliptic is termed the Appa/rent 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 dedination, etc. ; to distinguish them from the corresponding elements referred to the true equator and true equinox. The distance of the true from the mean equi- Qoz in longitude, is called the eqitation of the equinoxes in longitude. DIFPEEENT KINDS OF TIME. 91 CHAPTEE VIII. Measurement of Time, different kinds of time. 125. In Astronomy, as we have already stated, three kinds of time are used — Sidereal, True or Apparent Solar, and Mean 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 Mean 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 length of the true solar day 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 arc, and therefore the true solar day, varies £rom two causes, viz. : (1.) The inequality of the Sim's daily motion in longitude. (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 arc of longitude to the equator is subject to a variation ; and this, it is plain (see Fig. 42), will be attended with a varia- tion 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. lar. Mean Solar Time, As the length of the apparent solar day 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 meaa 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 there- fore 2-4h. 3m. 56.555s. of sidereal time. Hence, a clock regulated to sidereal time will gain 3m. 56.555s. in a mean solar day. To find the expression for the sidereal day in mean solar tinte, we must uss the proportion 24h. 3m. 56.555s. : 24h.::24h. : x = 23h. 56m. 4.092s. The difference between this and 24 hours is 3m. 55.908s. ; and therefore, a mean solar clock will lose with respect to a sidereal clock, or with respect to the fixed stars, 8m. 55.908s. in a side- real 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 sola/r time, we must sub- 3m 55 91s tract for each hour — : — '. 1= 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 3m 56 55s time in sidereal time, we must add for each hour : '. l-= 9.86s., and for miu- 24 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. CONTERSION OF ONE SPECIES OF TIME INTO ANOTHER, 128. The difference between the apparent and mean time ia 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 Time. 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 1.5, 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, Fia. 47. MS VM — VS appar. time = :i^ = 1ft iP. ^^ 15 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' VM VS' mean time = _ = 13 1^: 15 15 thus, equa. of time = mean time — ap. time = — . . . (25); or, (he equation of iivie 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 TSD ought properly to be considered as the moan 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 bo corrected by the arc TT', 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. The equation of time is zero, or mean and true time are the same four times in the year, viz. about the loth of April, the 15th of June, the 1st of September, and the 24th of December. 94 MEASUREMENT OF TIME. Its greatest additive value (to apparent time) is about 14J minutes, and occurs about the 11th of February ; and its greatest subtractive value is about 16j: minutes, and occurs about the 3d of November. 130. To convert Sidereal Time into mean Time, and vice versa. — Making use of Pig. 47, already employed, the arc TM, called the EigM Ascension of Mid- Heaven, 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 TS', converted into time, be denoted respectively by S, M, and L. Now, VM = MS' + VS'; or, S = M + L..(26); and M = S — L. .(2f). If M + Ii 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 sub- traction possible. This problem may in practice be solved most easily by means of an ephemeris of the Sim (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 Bolar 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 REGULATION OF CLOOKh BY ASTRONOMICAL OBSERVATIONS. 131. The regulation of a clock consists in finding its errm and its rate. 132. mean 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 clock- time 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 sum, 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, 93 method of calculating such apparent places is given in Prob. XXI. 134. Tinie bf Ob§erTation$ out of the ITIeridian. In default of a transit instrument, the time may 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 Single Altitudes, and the method of Double' 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 declination, and also the latitude of the place being given. Let us first suppose that the altitude of the sun is taken ; cor- rect 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 elevated pole, and S the sun ; in the triangle ZPS we shall know ZP= co-latitude, PS = co-declination, and ZS = co-alti- tude, frona which we may com- pute the angle ZPS (= P), which is the angular distance of the sun from the meridian. or, if expressed in time, the time of the observation from apparent noon; by the following equations (App., Eesolution of oblique-angled spherical triangles, Case 1), 2 k— ZP+PS + ZS = co-lat. -I- co-dec. + co-alt (28) ; sin(^ — ZP) sin(^ — PS) ,„„. Fia. 48. or, sm' ^P = siaHP = sin ZP sin PS sin {k — co-lat.) sin ik ■ — co-dec.) .(30). sin (co-lat.) sin (co-dec.) The value of P being derived from these equations and con- verted into time (see Prob. HI-), 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. A more 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 wiU be sufficiently exact if the measurements be all made within the space of 10 or II minutes, and wJien the sun is near the prime verUcaU. 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 lu8 ap- 96 MEASUKEMENT OF TIME. parent aomi-diaraeter. In practice, the declination of the sun may he taken fol tho aolutioa ol' this problem from an ophemeris of the sun. For tills purpose, the time of the observation and the longitude of the place must be approximately known. Example. On March 20, 1867, the following double altitudes of the sun were taken with a prismatic sextant, at New Haven ; upper limb, 6i° 12' 0", 64° 21' 35", 64° 33' 0",— lower limb, 63° 3»' 50", 63° 51' 0", 63° 58' 5" ; the correspondi-ng times of obser. vation, noted by a watch, were 9h. 6m. 49s. a.m., 9h. 7m. 20.5s., 9h. 7m. 56s., 9h. 8ra. 29.5s., 9h. 9m. 7.7s., 9h. 9m. 31s. ; the barometer stood at 30.47in., and the thermometer at 34°. 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 K P. dist 90 11 37 Mean of measured double altitudes 64° 5' 45" Index error — 1 3 2)64 4 42 Appar. alt. of sun's centre 32° 2' 21" Sefraction (Tables VIII., and IX.) — 1 37.3 True alt. of sun's centre 32 43.7 Lat. of station.. 41° 18' 37" Co-lat 48 41 23 ar. co. sin. 0.124276 Co-dec 90 11 37 ar. co. sin. 0.000003 Co-alt 57 59 16.3 2)196 52 16.3 k 98 26 8 i — co-dec 8 14 31 sin. 9.156408 4 — co-lat 49 44 45 sin. 9.882630 2 ) 19.163317 iP = 22° 26' 7".8 9.581658 P = 44 52 15 .6 .4 179m. 29s. 2'" 2h. 59m. 29.03s. 12 TIME BY OBSERVATIONS OUT OF THE MERIDIAN. 97 9h. Om. 30.97s. A. M. Equa. oftime. +7 41.37 M. 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. 64° 11' 45" 9h. 10m. 15s. A. M. L. L. -( 64 18 35 9 10 36 64 25 20 9 10 58 65 41 40 9 11 37 U. L. ^ 65 49 50 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 -i- 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 observa- tion 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 obser- vations. 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 Alti- tudes. 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; 7 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 meridian, 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. IVatnral Periods of 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 con- spicuous 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, comprising 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 four 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 lunar 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. 48m. 46s. The lunar month consists of about 29^ days. The week, con- sisting 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 admit. Different nations have, in general, had calendars more or less different : and the proper adjustment or regulation of the calendar by astronomical observation has in all ages, and with all nations, been an object of the highest importance. We pro- pose, in what follows, to explain only the Julian and Gregorian Calendars. 136. Tlie Julian Calendar divides the year into 12 months, containing in all 365 days. ISTow, it is desirable that the calen- dar should always denote the same parts of the same season by the same days of the same months : that, for instance, the sum- mer 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 equi- nox, the natural commencement of spring, should, if once, be always on the '20th of March. For thus the labors of agriculture, whicti 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 comple- tions 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 astronomical year. At the completion of a fourth civil year the sun would be more distant than on the two preceding ones from the equinoc- tial point. In order to rejoin it, and to complete the astronomi- cal 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 March 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 8 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 neces- sary 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. By this intercalation, what would have been March 21st is called March 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 intro- duced 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 TIME. 137. Reformation of the Calendar.— Wregorian Calen- dar. The astronomical year being equal to b65d. 5h. 48m, 46.1s, it is less than the mean Julian by 11m. 13.9s., or 0.007800d. The Julian correction, therefore, itself needs correction. The calendar regulated by it would, in process of time, become erro- neous, 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 March IQth. 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 con- tinue 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 equi- nox 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 5lh, should be called the 15th : thus suppress- ing 10 days, which, in the interval between the years 32.5 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 inter- calary 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 centurial year thai is divisible by 400 is a Bissextile or Leap 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. Conse- quently 0.1200d., or 2h. 52m. 48s. in 400 years, or 1 day in 3333 THE CALENDAR. 101 years, is the measure of the degree of inaccuracy of the Gregorian correction. The Oregorian Calendar was adopted immediately on its pro- mulgation, in all Catholic countries, but in those where the Pro- testant 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 oi Neio Style (the next day) was called the 14th, instead of the 3d. The same legislative enactment which established the Gregorian Calendar in England, changed the time of the begin- ning of the year from the 25th of March 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. Eussia 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 tlie pro- duct as many days as there are Bissextile years in the period. 102 MOTIONS OF THE SUN, MOON, AND PLANETS. CHAPTEK IX. Motions of the Sun, Moon, and Planets, in theib Orbits. KEPLER'S LAWS. liiS. The celebrated astronomer, Kepler, by examining the observations upon the planets that had been made by the re- nowned Danish observer, Tycho Brahe, discovered, early in the seventeenth century, that the motions of these bodies were in con- formity with the following laws : (1.) The areas described by the radius-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 an 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 cubes of their mean distances from the sun, or of the semi-major axes of their orbits. These laws are known by the denomination of Kepler'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 from them. These laws being established, the third is obtained by simply comparing the known major axes and periods of revolution. 139. ITIotioii of the Sun in its Apparent 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 from 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. 108 position of the earth in its orbit, the corresponding apparent posi- tion 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 the sun (or a line drawn from the sun to the earth) in any inter- val 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 arc EB' 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 verifiesd 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 Higher Apsis. The line join- ing 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. 142. Law of the Angular Motion of a Planet. The law of the angular motion of a planet about the sun may be deduced from Kepler's first law. Let PpA^" (Fig. 50) repre- sent 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 Inl', and with the radius Sa, equal to unity, describe the arc ah. It is plain that the Fig. 6( two positions p, p' may be taken so u<.ar to each other, that the area Spp' will be sensibly equal to the circular sector ^11'. If we suppose this to be the case, as the measure of the sector is ^InV X Sn = ^ah X Sra (substituting for InV its value, ah x Sn), we shall have area ^pp' ^ \ah x Sn . When the planet is at any other part of its orbit, as n', if S>p"p"' be an area described in the same interval of time aa before, we shall have area Sp"p"' = ^a'b' x S^''. But these areas are equal according to Kepler's first law : hence, iah x Sn' = ia'h' X S^'\ . . .(31) ; and ah : a'V :: Sn' : Sn ; that is, the angular motion of a planet about the sun for a short interval of time, is inversely 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 correspond- ing 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. JTIean Place. — True 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 angular motion of a Elanet, and to set out at the same instant with it from the peri- elion. Let PMAM' (Fig. 51) represent the elliptic orbit of the planet, and PBaB' the circle described by the body. The posi- tion B of this fictitious body at any time, will be the meanplact of the planet as seen from the sun. The two bodies will accoiiX' MEAN AND TEUE PLACE. 103 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), but it continually decreases until the planet reaches the aphelion, ■while the motion of the body remains constantly equal to the mean motion. The planet will therefore take the lead, and its angular distance f'SB from the body will increase until its mo- tion 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 velo- city 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 re- main unaltered: thus, the body will now get in advance of the planet, and their angular distance ^'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 ths perihelion to the aphelion the true place is in advance of the mean place ; and that from the aphelion to the perihelion, on the contrary, the mean place is in advance 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 Centre. Thus, ^SB is the equation of the cen- tre corresponding to the particular position^ 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 Equation of the Centre. As the laws of the motion of the moon (140), and of the appa- rent motion of the sun (139), are the same as those of a planet, the principles established in the two preceding articles are aa applicable to these bodies in their revolution around the earth, as to a planet in its revolution around the sun. 106 MOTIONS OF THE SUN, MOON, AND PLANETS. DEFINITIONS OF TEEMS. 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 Longitude and Latitude appertain to the geo- centric place, and Heliocentric Longitude and Latitude 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 iv. 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 op- posite side of the sun. In the former situation they are said to be in Lnferior Conjunction, and in the latter in Superior Conjunction. (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 con- junctions. (6.) The Periodic Time of a planet is the period of time in which it accomplishes a revolution around the sun. (7.) The Xodes 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 Q,. The other is called the Descending Node, and is marked y. {\) The Eccentndty 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. 14.5. To illustrate these Definitions, let EE'E" (Fig. 52) represent the orbit of the earth ; CDC the orbit of Venus, or Mercury, which we will suppose, for the sake of simplicity, to lie in the plane of the ecliptic or of the earth's orbit ; LXP a part of the orbit of Mars, or of any other planet more distant from the sun S than the earth is ; and ANB a part of the projec- tion of this orbit on the plane of the ecliptic. N or ft will re- present the ascending node of the orbit; and the descending node will be diametri'ially opposite to this in the direction Sw'. DEFINITIONS OF TEEMS. 107 Also let SV be the direction of the vernal equinox, as *eeii 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 ba the projection of that position on the plane of the ecliptic. The heliocentriG longitude and latitude of Mars, in the position P, are respectively VSjo and PSj? ; and if the earth be at E, hia geocenmG longitude and latitude are respectively VE;? 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 opposition. Again, if we suppose the earth to be at E, and Venus at C, she will be in superior coiyunction ; but if we suppose that Venus is at C at the time that the earth is at E, she will be in inferior conjunction. The term inferior is used here in the sense of lower in place, or nearer the earth ; and superioi' in the sense of higher in place, or farther from 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 r&volution, is, for the same reason, longer than the periodic time, or time of passing around from one point of the orbit to the same again. 108 MOTIONS OF THE SUN, MOON, AND PLANETS. ELEMENTS OP THE OKBIT OP 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 infor- mation are subdivided into seven distinct elements, called the Elements of the Orbit of a Planet, 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 semi- major 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 ih.e positiim, of the plane of the planet's orbit ; the third and fourth, the dimensions of the orbit ; the fifth, \he position of the orbit in its plane ; the sixth, the place of the planet at a given epoch; and the seventh, its mean rate of motion. The elements of the earth's orbit, or of the sun's apparent orbit, are but fve in number ; the first two of the above-men- tioned elements being wanting, as the plane of the orbit is coin- cident with the plane of the ecliptic. The elements 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 geocentric longitude of the perigee and the geocentric longitude of the node of the moon's orbit answer, respectively, to the helio- centric longitude of the perihelion and the heliocentric longitude of the node of a planet's orbit. 147. Tlie r,iiiear ITiiit adopted, in terms of which the semi- major axes and radius-vectors of the planetary orbits are ex- pressed, 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 distauce from the earth, as unity. MEAN DISTANCE OF THE SUN; 109 DETEEMINATION OF THE ELEMENTS OF THE SUN'S APPA- RENT ORBIT, OR OF THE EARTH'S REAL ORBIT. MEAN MOTION. 148. The sun's mean daily motion in longitude results from the length of the mean tropical year obtained from observa- tion (115). SEMI-MAJOR AXIS. 14». 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 Horizontal Parallax of a body being given, to find its Distance from tlie Eartli. We have (equation 7, Art. 88) sin H where H represents the horizontal parallax of the body, D its distance from the centre of the earth, and E the radius of the earth. The parallax of all the heavenly bodies, with the excep- tion of the moon, is so small, that it may, without material error, be taken in this equation in place of its sine. Thus, D=^\=Ex 1;....(32). sm H i± Again, since 6.2831853 is the length of the circumference of a circle of which the radius is l,and 1296000 is the number of secondsin 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, D^E?^|1::806___ (33)_ 150. Determination of the Snn'§ Mean 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 one- fourth of the whole parallax, may be committed, so that the dis- tance 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 accu- rate methods of obtaining the sun's parallax. By one method, which will be noticed in another connection, the equatorial paral- lax 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 impor- tant element, by independent astronomical methods. The differ- ent 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 Mars 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 defini- tively 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. 151. Calculation of Sun's Mean Distance. We have, then, for the sun's mean distance from the earthy or the semi- major axis of its orbit, D ^b 206264". 806 ^ 23046.347 E = 91,328,064 miles; H taking for E the equatorial radius of the earth, 8962.80 miles. ECCENTEICITT. 152. First Method. 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 peri- gean distance of the sun ; and r' the greatest or apogean dis- tance ; and we shall have, by the principle of Art. 142, r :r':: y/v' : V v; whence, r' + r : r' — r \\ ij v ■\- -J v' '. -y/ v — V u', r' + r , ^ V + J v' , — ,— ; or, — - — : r — r : : -i — Z_L_1 ; V v — y/ v i but, r' + r : semi-major axis = 1 ; and r' — r = 2(eccentricity) = 2 e ; ECCENTRICirY OF SUN'S APPARENT ORBIT. Ill thus, 1 : 2e : : ±^L±dl : ^T- V^ and e = ^ZZ^....{S4:). 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 and 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 dia- meters (116). 153. Second Method. By {he greatest equation of (he centre. (1.) To iind the greatest equation of the centre. Let L ^ the true lon^tude, 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 lougi- tude, when the motions are equal between the apogee and perigee ; and E = the greatest equation of the centre. Then ( 1 43) L = M + E, and L' = M' — E ; whence, L' — L = M' — M — 2E, and ^ = 'E=}^l±=R....i^i). 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 apsides, 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' — M = 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 intervS by 69' 8".330, the mean daUy motion in longitude. For example : from observations upon the sun, made by Dr. Maskelyne, in the year \115, it is ascertained in the manner just explained, that the sun was near its greatest equation at noon, or at Oh. 3m. 353. mean solar time, on the 2d April, and at noon on the 31st, or at 231i. 49m. 35s. mean solar time, on the 30th of Septem- ber. The observed longitudes were, at the first period 12° 33' 39".06, and at the second 188° 5' 44".45. The interval of time Detween the two epochs is 182d — 14m. Mean motion in 182d. — 14m 1'79» 22' 41".56 Diflerenoe of two longitudes 175 32 5.39 Difference 2) 3 60 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, Delambre, the greatest equation was in 1775, 1° 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 K' 3.2' 3.5.2" ■&c....(36), ji which K stands for the expression _ _ (E being the greatest equation of the centre). In the case of the sun's orbit, K being a small fraction, all ita powers beyond the first may be omitted. Thus, retaining only the first term of the series, and taking E = I'' 55' 31".66 the greatest equation in 1775, we have e = ^ = ^SilH^^i- = .016803. 2 2 X 57''.29o7795 154. Equation of Centre depends on Eccentricity. It appears from the law of the angular velocity of a revolving body, investigated in Art. 14:2, that the amount of the propor tional variation of this velocity, which obtains in the course of a revolution, depends altogether upon the amount of the propor- tional variation of distance, or, in other words, upon the eccen- tricity of the orbit (def 8, 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 formuIiB 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. LOXaiTUDE AND EPOCH OF THE PERIGEE. 165. Methods of Determination. As the sun's angular velocity is the greatest at the perigee, the longitude of the sua 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). The viore accurate method 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 orbit whose longitudes differ by 1S0°, in passing from one to the other of which the sun employs half a year. This principle may be inferred from Kep- ler's law of areas, for it is a well known property 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 tbe law in question, equal areas correspond to equal times. 156. Progressive ITIotioii of the Perigee. Bj a compari- son of the results of observations made at distant epochs, it ia discovered that the longitude of the perigee is continually increas ing at a mean rate of 6l".7 per year. As the equinox retro- grades 50".2 in a year, the perigee must then have a direct angu- lar 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 tune employed by the sun in moving through an arc of 5". 7, the sidereal motion of the apogee and perigee in half a year. According to the most exact determinations, the mean longi- tude of the perigee of the sun's orbit at the beginning of the year 1800, was 279° 29' 56". It is now 280f°. 137. Tlie Heliocentric Longitude of the 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 (3!&c — 180° = geocentric longi- tude 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 liongitnde of the Sun, at any assumed epoch, may be obtained by means of the mean motion in longi- tude (116), the epoch and mean longitude of the perigee of the sun's orbit having once been found. DETEEMINATION OP THE ELEMENTS OF THE MOON'S OEBIT. 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 ana- logous 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 equi- nox, IS ascertained (113). 8 114 MOTION OP THE MOON IN SPACE. INCLINATION OF THE OBBIT. 160. Among the latitudes computed from the moon's observed right ascensions and declinations, the greatest measures the incli- nation of the orbit. It is found to be about 5° ; sometimes a little greater, and at other times a little less. MEAN MOTION. 161. Tropical Revolntiou. With the longitudes of the moon, found from day to day, it is easy to obtain the interval from the time at which the moon has any given longitude till it returns to the same longitude again. This interval is called a Tropical 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 longi- tude 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 longitude at the same time is derived from it by correcting for the equation of the centre 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 determi- nation of the longitude of the moon at an ancient epoch. 162. inean Daily motion iii liongitude. The mean tropi- cal revolution of the moon is found to be 27.321582d. or 27d. 7h. 43m. 4.7s. (5s. nearly). Hence, 27.321582d. : Id. : : 360° : 13°.17639 = 13° 10' 35".0 =t moon's mean daily motion in longitude. LONGITUDE OF THE PERIGEE. 115 163. Sidereal Revolution. Since the equinox has a retro- grade motion, the sidereal revolution of the moon m-ust 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°.17G : Id. : : 3".752 : 6.8s. = emcess. "Wherefore, the mean sidereal revolution of the moon is 27d. 7h. 43m. 11.5s. 164. Secular Acceleration of Kioon's motion. 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 wUl not, how- ever, 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. ] 66. Orbit liongitndes. 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 eclip- tic, in the case of the moon corresponding an- gles are laid off in the plane of its orbit. These angles are reckoned from a line drawn making an angle with the line of nodes equal to the ' longitude of the ascending node, and are called OrW, Lcmgitvdes. The orbit longitude is equal to the moon's angular distance from the ascend- ing node plus the longitude of the ascending node. Thus, let VNO (Fig. 53) represent the plane of the ecliptic, and V'NM a portion of the moon's orbit; N being the ascending node; also let ET be the direction of the vernal equi- nox, and let EV be drawn in the plane of the moon's orbit, making an angle V'EN with the line of the nodes equal to VEN, the longitude of the asoendmg node N. The orbit longitudes lie in the plane of the moon's orbit, and are estimated from this line, while the ecliptic lon- gitudes lie in the plane of the ecliptic, and are estimated from the line EV. Thus, T'BM, or its measure V'NM, is the orbit longitude of the moon in the position M; and T Em is the eclip- tic longitude ; that is, the longitude as it has been hitherto coasidered. V'NM = V'N -1- NM = VN +NM ; that is, orbit longi = long, of S + Q)'s distance from S. 116 MOTIONS 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 orfm 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 soon as MN, the moon's distance from the node, becomes known : now, by Napier's first rule we have cos MNot = 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 moon; 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, The Eccentricity of the moon's orbit is more than three times as great as that of the sun's apparent orbit. Its great- est equation exceeds 6° (154). MEAN LONGITUDE AT AN ASSIGNED EPOCH. IVO. We have already explained (161) the principle of the determination of the mean longitude of the moon from an ob- served 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 Liongitttde and Radins-Tector of the Earth. The methods of determining the elements of the planetary orbits, suppose the possibility of finding -the heliocen- tric 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 com- puted by means of Kepler's first law ; and the radius-vector from the polar equation of the orbit, as given in treatises on Analyti- cal 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 HELIOCENTEIC LONGITUDE OF THE ASCENDING NODE. 172. First nietliod. 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 ia passing through it. Now, if the right ascension and declination of the planet b» observed from day to day, about the time it ia passing from one side of the ecliptic to the other, and converted into longitude and lati- tude, the time at which the latitude is zero, and the longitude at that time, may be ob- tained by a proportion. When the planet is C' again in the same node, the geocentric longi- ^, tude 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 Imtgitudes of the same node be found, its heliocentric hngitude may be computed. Let S (Fig. 64) be the sun, N the node, and E one of the positions of the ( \Z /A L earth for which the geocentric longitude of I ^I^tCT / / 'j' the node (VEN) is known- Denote this V I / /gf—/- /v angle by G, the sun's longitude TES by S, \,^ 1 / ^ ' and the radius-vector SE by r. Also, let E' ^^^— t«rC^.... / be the other position of the earth, and de- ^ y^ note the corresponding quantities for this position, TE N, TB'S, and SE', respectively, ^'^- ^^ 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. Th3 triangle SNE gives sin SNE : sin SEN : : SE : SN; but SEN = TES — VEN = S — 6, and SNE=VAN — TSN = VEN — TSN = G— X; hence, sin (G — X) : sin (S — G) : : r : T, or, rBin(S — G) = Vsin(6 — -X) (37). In like manner, r sin (S' — G') = T sin (G' — X). Dividing, J^sinJS-Gl ^ sin_(G^^), ^' r'sin(S' — G') sin (G— X)' rsinfS — G) sin G cos X — sin X cos G sin G — cos G tan X. or, i —^ • =^ 1 r' sin (S' — G') sin G' cos X — sin X cos G' sin G — cos G' tan X whence, ■y — *■ s™ (S — G) sm G' — r' si n (S' — G') sin G ^gv ~rsin(S — 6)cosG' — r' sin(S' — G) cos G Equation (37) gives T = ZI%i^-.^. . . .(39). sm (tr — X) 173. Second 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 pro- cess 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 ques- tion, or will differ 180° from it. This will be seen on inspecting IV] MOTIONS 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 VBS, the longitude of the sun, will be equal to VSN, the heliocentric longitude of the node. If the earth should be Fio. 65. at E" and the planet in opposition, the longitude of the sun would be VE"S = VE"N + 180^ ^ VSN + 180° = 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 opposi- tion 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 OEBIT. 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 geocen- tric longitude and latitude. Let ENj) (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 2, denote the geocentric latitude PEp ; E the arc Np = Yp — VIST = geo. long, of planet — long, of node ; and I the inclination PNp. The right-angled triangle PNp gi^'cs PERIODIC TIME. 119 sin Np = tan Pp cot PJTp = tan ;^ cot I ; . T sin E J X T tan Jt Dcnce, cot I = , and tan I = __I^ ; tan ?j sin E or, tan inclination = -. _tan_lat. ^^q, sin (long. — long; of node) It win be understood, that to obtain an exact result, we must compute the pre- ease time of day at which the longitude of the sun is the same ae that of th6 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 thug ascertained and the time of the observation above mentioned. PEEIODIC TIME. 175. The interval from the time the planet is in one of its nodes till its return to the same, gives the periodic time or side- real revolution. Another atid 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 ob- served 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. Mean Daily Motion. 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 EADIUS-VECTOE, FOR A GIVEN TIME. 177. Oeueral Problem. The earth being in constant motion in- its orbit, and being thus at different times very diiferently situated with regard to the other planets, as well in respect to distance as direction, it is necessary for the purpose of compar- ing 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 accor- dingly it is to the sun that the observations are in reality refer- red. The reduction of observations from the earth to the sun, as it is actually performed, consists in the deduction of the helio- centric longitude and latitude from the geocentric longitude and latitude ; these being calculated from the observed right ascen- sion and declination. The requisite formulae for effecting this reduction are investi- gated in the Appendix. 120 MOTIONS OF THE PLANETS IN SPACE. ITS. Special Cases. The heliocentric longitude, or radius vector of a planet, may be more readily obtained if the observa tions be made upon it when it is in certain favorable positions. Case 1. When the planet is in conjunction or opposition, its helio- centric longitude will then, either be equal to the geocentric longitude, or differ 180° from it. When the heliocentric longitude is thus found, the latitude for the same time may be obtained by solv- ing the triangle PNp (Fig. 56). For, by Napier's first rule, sin Np = cot PlSTp tan Vp, or tan Pp ^ sin !Np tan PNp ; where Vp is the latitude sought, PNp the known inclination of the orbit, and 'Np — Y'Sp — VIST = long, of planet — long, of node, both Fi8. 56. of which may be considered as known. The radius-vector may be computed for the same time from the triangle ESP ; for the side SB, 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° — heliocen- tric latitude. Case II, When an infei-ior 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 ascer- taining from instrumental observations the places of the planet and sun in the heavens, and connecting these by an arc 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 dis- Pie. 51. tances PS and Vp, which are the complements of the observed declinations, and the angle SPp the difference of their observed right ascensions, from which the angular distance S^ between the two bodies may be calculated. The maximum elongation being LONGITUDE OF THE PERIHELION. 121 then supposed to be known, let NPP' (Fig. 58) represent the orbit of the inferior pla- net. The line EP drawn from the earth to the planet will, at the time of maxi- mum elongation, be perpen- dicular to SP, the radius- vector of the planet; and thus we shall have in the right-angled triangle EPS, the line ES, and the angle SEP, from which the radius- vector SP may be computed. As the earth and planet are in motion, the greatest elongation will occur at dif- ferent points of the planet's orbit, and therefore we may find by the foregoing process different radius- vectors. 179. The Orbit liongitude of a Planet may be derived froni the ecliptic longitude in the same manner that the orbit longitude of the moon is calculated from its ecliptic longitude (166). 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. LONGITrCE OP THE PERIHELION", BCCBNTRIOITY, AND SEMI-MAJOR AXIS. l§0. These elements may be calculated from the heliocentric orbit longitude and radius-vector, found for three different times. Let SP, SP', SP" (Fig. 69), be the three given radius- vectors : V'SP, V'SP', V'SP", the three given longitudes; and AB the line of apsides of the planet's orbit. Let the angles PSP', PSP", which are known, be represented 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 semi-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 furnishea for their determination the three equations, 122 MOTIONS OF THE PLANETS IN SPACE. 1 + e cos x' 1 + e cos (x -t- m)' 1 + e cos (x + n)' Tlie process of solution is given in the Appendix. When x has been found, by subtracting it from V'SP we obtain V'SB, the longitude of the perihelion. ISl. Other iTIethodt of Determining tbe Semi-major 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 Kepler's third law has been established by investi- gations in Physical Astronomy, it furnishes the most accurate method of finding 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.)' : V" : : V : a' ; ^^^■^^^^ '^ = (365:2T6359d:)'--"^*^)- 182. liongitude of Perihelion, and Eccentricity, by Approximate ITIethods. 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 longi- tude 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 difference between the greatest and least cal- culated radii to the mean of the whole, will be the approximate value of the eccentricity. EPOCH OP THE PERIHELION PASSAGE. 183. 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 pro- portion. The Mean Longitude at an assigned epoch is obtained on the same principles as the mean longitude of the sun or moon (158, 170). EEMAEKS. 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 TRUE AND MEAN ELEMENTS. 123 the planet Uranus was discovered bj Sir William Herschel, La- place contrived methods by which the elements of its elliptic orbit were determined from four observations within little mora than a year from its first discovery by Herschel. After the dis- covery of Ceres, Gauss invented another general method of cal- culating 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 (keir variations. The elements of the planetary orbits, ootained 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 cen- tury, and that the changes experienced during equal long periods of time are very nearly the same. The amount of the variation of an element iu a period of 100 years is called its Secular Va/rialion, Upon reducing the elements, found at differ- ent 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. Mean Elements and {heir Secular Variations. The mean elements at, an assigned epoch may be had by finding the true elements at various times, and re- ducing 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 par- ticular 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. 187. 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 wUl 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 com- paring the computed with the observed places of the planet; and for this purpose many thousands of observations have been made. ISO. Tables II. and III. contain the elements of the orbita 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. 12 i MOTIONS OF THE PLANETS IN SPACE. If an element be desired for anytime 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. Secular Tariations. 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. Eccentricities 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, Xeptune, 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. The 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 f °, of the former 7°. 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. Tbe ITIean Distances of tiie Planets from tiie i»nn, expressed in miles, are in round numbers as follows: Mercury 35 millions, Venus 66 millions, the earth 91 millions. Mars 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. The Approximate Periods of Revolution of the planets are : of Mercury 3 months, Venus 7^ months, Mars IJ DIMENSIONS OF THE SOLAE SYSTEM. 125 years, Juno 4| years, Jupiter a little less than 12 years, Saturn 29^ years, Uranus 84 years, Neptune 164J- years. Tiie periods and mean distances are more exactly given in Table II. (For the planetoids, see Table II. (a) ). 194. Boimeiisioii§ of the Solar System. A better idea of the dimensions of the solar system than is conveyed by the state- ment 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 Mercury from 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 If miles, that of Uranus by 3| miles, and that of Neptune by 5|- miles. On the same scale, the distance of the moon from the earth would be only 2^ feet. 126 DETERMINATION OF THE PLACE OF A PLANET. CHAPTER X. Determination op the Place of a Planet, or of the Sum OR Moon, for a Given Time, by the Elliptic Theort.— Verification of Kepler's Laws. PLACE OP A PLANET IN ITS ORBIT. 196. True and Mean Anomaly. The angle contained between the line of apsides of a planet's orbit and the radius- vector, as reckoned from the perihelion towards the east, is called the True Anomaly. Thus, let BPAP' (Fig. 60) represent the Fl8. 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 Mean 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 difference 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 B^A on the line of apsides as a diameter ; through P draw^PD perpendicular to the line of apsides, and join p and C ; the angle BCp, which the line thus deter- mined 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 Anomalistic Beijoluiion. 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 to = the sidereal motion of the apsis answering to the periodic time, and M = the mean daily motion of the planet ; then, M : Id. : : TO : X = diff. 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. 19S. Calculation of ]nean Anomaly. From the anoma- listic revolution, and the epoch of 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° -^. . . .(42). 199. Tlie Place of a Body in its Elliptic Orbit is ascertained by finding its true anomaly. The problem which has for its object the determination of the true anomaly from the mean, was first resolved by Kepler, and is called KepWs Prob- lem. Another and more convenient method of obtaining the true anomaly, is to compute the equation of the centre from 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. The Place of a Planet in the Plane of its Orbit is designated by its orbit longitude (166) and radius-vector. To find the orbit longitude we have the equation V'SP = VSB + BSP(seeFig. 60); or, long. = long, of perihelion -f true anomaly. The orbit longitude may also be deduced from the mean lon- gitude, by adding or subtracting the equation of the centre ; for, V'SP = V'SM4- MSP, 128 DETERMINATION OF THE PLACE OF A PLANET. or true long. = mean long. + equa. of centre : also, V'SP' = V'SM' — M'SP', or, true long. = mean long. — equa. of centre. The radius-vector results from the polar equation of the ellip* tic orbit, viz. : V = A(l^Zfl....(43). 1 + e cos X ' in which x denotes the true anomaly, e the eccentricity, and a the semi-major axis. 201. To find the Heliocentric Liongitude and Liatitnde, which ascertain the position of the planet with respect to the ecliptic, the triangle NPp (Fig. 56, p. 120) gives sin Pp = sin NP sin PNp ; 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, or, tan (long. — long, of node) = tan (orbit long. — long, of node) X cos (inclination) .... (45). GEOCENTRIC PLACE OP A PLANET. 202. The theoretical determination of the place of a planet, as it would be seen from the centre of the earth, consists in deduc- ing its geocentric longitude and latitude, and its distance from the earth, from its heliocentric longitude and latitude and radius- vector; the latter having been calculated by the methods just explained. (For the detail of the solution of this problem see Appendix.) PLACES OP THE SUN AND MOON. 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 + long, of perigee, and the polar equation of the orbit. 204. Tile Orbit Liongitude and the Radins-vector of the noon, are found by the same process as the longitude and radius-vector of the sun. The orbit longitude being known. VEEIFICATION 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 longi- tude and latitude of a planet are found (201). VERIFICATION OP 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 Snn or Jnoon, at different times, result for this purpose, from measurements oi the apparent diameter, upon the principle that any two distances are inversely proportional to the corresponding apparent diame- ters. Let A = semi-diameter corresponding to the mean dis- tance, and 5 = semi-diameter corresponding to any distance D : then ;5 : A : : 1 : D; whence, D =-^ (46); an equation which, when A has been found, will make known the distance corresponding to any observed semi-diameter 5, in terms of the mean distance as a unit. Now, to find A, denote the greatest and least semi-diameters, respectively, by 8', 8", and the corresponding distances by D' and D", and we have A A ^ - S" S'" and thence, ^(D'-H D") = i(-| + ^,), or, 1 = i(|;-f-|r,); whence, A = w;^^ (*7). 130 INEQUALITIES OF PLANETARY MOTIONS. CHAPTER XL Inequalities of the Motions of the Planets and of the Moon ; Tables for finding the Places of these Bodies. 207. Oravitatioii. 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 Grarnta- iion. This force of attraction, common to all tbe bodies of the Solar System, is the general physical cause of their motions. The 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 NewtonJs Theory of Universal Gravitation is established, appertains to Physical Astronomy, and will be presented in Part II. aOS. 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 com- puted 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. Di!sturt>iitg Force. In treating of the perturbations in the motions of one planet, resulting from 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 iittraction of M for the planet. Decompose PD into two forces, COMPONENTS OF DISTUEBING FORCE. 131 PE and PP, one of which, PE, is equal and parallel to SG, the attraction of M for the sun ; the other, PF, will be itnown in position and intensity. The two forces, PE and SG, being equal and parallel, they cannot alter the relative motion of the sun 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 bodies for the sun and planet, when they 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. Components of Disturbing Force ;— their Effects. It is plain that the disturbing force 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, and of radius vector. 211. Deterini nation of Inequalities. Let us now c )n- sider how these inequalities may be determined. In the farst 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 the 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 ex- pressions for the perturbations in longitude, latitude, and radius- vector, involving variables and constants. 212. TQuations of Specific Ini^qiialities of L.oiigi- tucie. The general expression for the whole perturbation in longitude, due to the action of anyone disturbing body, is of the form C sin A H- C sin A' -1- C" sin A", etc., in which 0, 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 coefBcient is equal to the great- est value of the inequality. 213. Calculation of In«qiialities. 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 making repeated determina- tions of the difference between the observed and computed longi- tude 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 unjjnown quantities, 0, 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. mu***r '"•'^("^••t'es of L.atitU4le and Radius-vector. ihe theory of these mequalities, and of their computations, is similar to that of the inequalities of longitude just explained.' 215. Inequalities are Periodic. "We have seen that the arguments of the inequalities are angles depending on the con- figurations of the disturbing and disturbed planets with respect to each other and the sun, or with respect to the nodes or peri- helia 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 through all its gradations of positive and negative value," is called the Period 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. Inequalities of Elliptic Elements. 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 centurj' 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 longi- tude of the perihelion), but are really periodic inequalities whose periods comprise many centuries. Prom 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 furnishes expressions cslXeA &cular Equa- tions, which give the value of an element at any assumed time. 217. The Inequalities of the Moon's ITlotion 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 134 INEQUALITIES OF THE MOON'S MOTION. 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 num- ber and magnitude those of any single planet. There are three lunar inequalities of longitude which are promi- nent above the rest, and were early discovered by observation. The most considerable is called the JEvection, and was dis- covered 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' 30". 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 Annual Equation, from the cir- cumstance 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 ex- ception of one inequality of latitude^, is due to Physical Astro- nomy. 21 §. Calculation of £xact Heliocentric Place of a Planet, To present now at one view the entire process of cal- culating 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 ele- ments, 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-veotor, by the elliptic theory (200, 201). (3). Compute the values of the inequalities in longitude, lati- tude, 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 appa- rent 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'fi motion are then to be applied to these as corrections. ASTRONOMICAL TABLES. 135 TABLES OF THE 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 con- struction of tables adapted to this purpose are explained in Part III. We will only remark here that the tables save the neces- sity 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 argu- ments are also readily obtained from tables especially designed for this purpose. Table? of the sun, moon, -and of eacli of the principal planets, have been calculated by different astronomers, and are now in general use. 220. Eptaemeris. With the aid of these tables an ephemeris of each body is computed, and published for each year in ad- vance, in the American and English Nautical Almanacs. \ii Ephemeris of a heavenly body is a collection of tables exhibitiDi^ the longitude, latitude, right ascension, declination, parallnw, semi-diameter, etc., of the bodj^, at stated periods of time, a' at noon of each day throughout the year. J36 MOTIONS OF THE COMETS. CHAPTEE XII. Motions of the Comets. 221. Apparent ^notions. When first seen, a comet is ordi- narily at some distance from the sun in the heavens, and moving towards it. After this, it continues to approach the suti, 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 Febru- ary, 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 Comets. Sir Isaac Newton, from observa- tions 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 radius- vector 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 form of the orbit will depend upon the projectile velocity alone. With 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 velo- city from which a parabolic orbit will result, and as any comet, which may have originally moved in a hyperbola, must hav9 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. Elements 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 as- cending 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. Entire Elliptic Orbitis.— Periods of Revolution. Astronomers do not in general seek to deduce, from the obser- vations 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, inas- much as in the interval during which they are made, the comet describes 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 succes- sive perihelion passages. A comet cannot be recognized at a second appearance by its aspect; for this is liable to great altera- tions. But it may be identified by means of the elements of its parabolic orbit (223), as it is extremely improbable that the ele- ments of the orbits of two different comets will agree throughout. This method of identifying a comet may sometimes fail of appli- cation, inasmuch as the orbit of a comet may experience great alterations from the attractions of the planets. 225. Comets of Known 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 cer- tainty. These have all reappeared, and in some instances repeat- edly, 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 Honey's, Enck^s, Bkla's, Faye's, De Vice's. Urorson's, D' Arrest's, and Winnecke's. The last seven are known as Comets of Short Period ; their periodic times being comprised within the limits of 3.3 years and 7^ years. Their mean dis- tances from the sun are less than that of Jupiter, and they re- volve within the orbit of Saturn. Halley's comet, in its recess from 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. Pio. 62. 226. Comets ivliose Periods have 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, respeotiyely. Five form, with Halley's comet, a distinct 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 degree 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 Cornels of Comparatively Sliort 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 strik- ing 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 com- paratively 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. 22§. Comets of L.oiig 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. Comets of Small Perihelion Dis^tauce. 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 reniarkable 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 accom- plished a semi-revolution around the sun (from n to n', Fig. 63) in the astonishingly short interval of 2 hours. 230. Number of Comets, The number of recorded appear- ances 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 mven- tion of the telescope, only those comets which were very con- Bpicuous 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 appear- ance every year, of ■s^hicli the great majority were telescopic. The periods of these, as well as of the others, are in general of Buch 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 ap- pears, 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 Halley, 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 comet had made its appear- ance in these several years, and predicted that it would 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. Accord- ingly 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 pre- dicted 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. l^l suoTi increased accuracy that the error in the prediction of ita 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 particu- larly conspicuous in the years 1066 and 1456. ENCKB'S COMET. 232. This comet is remarkable for its short period of revolu- tion, 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 1866 ; the next will be in September, 1868. 23^. Disturbing Effects of a Kesistiiig Medinm. 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 dis- turbing actions of the other bodies of the system. Professor Encke found that after allowance had been made for all the per- turbations produced by the planets, the actual time of each peri- helion passage anticipated the time calculated from the duration of the previous revolution about 2| hours; and that the comet now arrives at its perihelion about 2|- days sooner than it would if 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 there- fore diminished, and its period shortened. A similar pheno- menon to this is presented in the oscillations of a pendulum freely. suspended. It is well known that the arc of vibration of 142 MOTIONS OF THE COMETS. the pendulum shortens, and consequently its rapidity of oscilla- tion increases, under the influence of the resistance of the air. BIBLA'S COMET. 234. In February, 1826, M. Biela, of Josephstadt, in Bohemia, detected a telescopic comet in the constellation Aries ; and subse- quently 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 in- specting 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 in- clined 12^° 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 coinci- dence, 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 8,000,000 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 es- caped detection. The next return will be in September, 1872. FATE'S COMET. 835. 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 accele- rated, 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. 23G. 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 elon- gated 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 fur- nished an illustration of this general fact. 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 explana- tion 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 COMET OF 1843. 837. This comet has already been alluded 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 Fia. 63. comet at several different dates, with the corresponding positions of the earth, are also indicated ; n is the ascending and n' the descending node. The perihelion is within 500,000 miles of the sun's centre, and nearly midway between n and n'. The incli- nation of the orbit is 36°. The comet passed its perihelion on February 27, at about 5 P.M. (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 ofif 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 from 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 Douati 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 Fio. 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 oa the north side of the plane of the earth's orbit. When first seen, on June 2, the cornet was 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 Jeter- mined ; 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 (Halley'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. EEVOLUTION OF THE SATELLITES. 147 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 Nep- tune's one. 241. Tbe Satellites of Jupiter are perceptible with a tele- scope 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 First 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 Neptune cannot be seen, except through excellent telescopes. They experience changes of apparent position, similar to those of Jupiter's satellites. 242. The Satellites Revolve around the Planet. The apparent motion 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 fre- quently 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 satel- lites are opake bodies illuminated by the sun, and that the satel- lites revolve around the planet from west to east. On this hypo- thesis, 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 au eclipse, occasioned by its entering the shadow of the planet. As the transit of the shadow occurs during the passage of the satellit') from the eastern to the western side of the planet, and 118 MOTIONS OF THE SATELLITES. the eclipse of the satellite daring 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. Eclip$e§. — Transits of Shadows. Let us now exa- mine 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 Fig. 65. earth, PPT" the orbit of Jupiter, and ss's" that of one of its satellites, supposed to lie in the plane of Jupiter's orbit. Sup- pose that E is the position of the earth, and P that of the planet, and conceive two lines, aa', b¥, to be drawn tangent to the sun and planet : then, while the satellite is moving from s to s' it will be eclipsed; and, while it is moving from ftof,' its shadow will 149 fall upon the planet. Again, if Ee, Ee' represent two lines drawn from tlie 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 A', be behind the planet, and thus concealed from 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 disappear- ances or immersions of the satellite will take place on the western side of the planet ; and that the emersions, 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 visi- ble 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, from which cause the fourth satellite sometimes escapes an eclipse. 244. JPeriods.— Mean inotions.— iTIeaii 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 from 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 inter- val 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 peri- odic 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 dis- tances from the primary. 150 motiojSts of the satellites. The same law also has place with the satellites of Saturn and Uranus. 245. The Conipntation of tlie 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 satel- lites occasion sensible perturbations of their motions, of which account must be taken when it is desired to determine their places with accuracy. 24C. Relations of Mean 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 con- sequently that they can never be all eclipsed at once. INEQUALITY OF DATS. 151 CHAPTER XIV. The Sun, and the Phenomena attending its Apparent Motions. mEQUALITT OP DAYS.* 247. Sun's inotion relative to the Equator. We will first give a detailed description of the sun's apparent motion with respect to the equator, the phenomenon upon which the ine- quality of days (as well as the change of seasons, soon to be treated of) immediately depends. Let VBAQ (Fig. 66) represent the equator ; VTAW (inclined to VEAQ, under the angle TOE, measured by the arc TB, equal to 23i°), the ecliptic ; TwX and Wn'X', the two tropics; POP', the axis of the heavens ; and PEP'Q the meridian, andHVEA the horizon, in one of their vari- ous positions with respect to the other circles. About the 21st of March the sun is in the ver- nal equinox Y, 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 it crosses the meridian at the Fio. 66. point E, south of the zenith a distance ZE equal to the latitude of the place. Advancmg 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. 152 THE SUN AND ATTENDANT PHENOMENA. 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, TnX ; 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 turns about and goes towards the south. f 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. Dur- ing 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 AW, 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 WV, crossing the meri- dian at the points s", s', etc. ; so that on the 21st of March it is again at the vernal equinox. 248. Explanation of Ineqnality of Days. The pheno- menon of the inequality of days obtains at all places on the earth situated north or south of the equator. At all such places, the observer is in an oblique sphere ; that is, the celestial equator and the 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 hori- zon, equator, ecliptic, and parallels of declination, which are represented as ellipses in Fig. 66, are in Fig. 11 projected into right lines upon the plane of the meridian. Since the centres of the parallels of declination are situated upon the axis of the 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 dis- parity between the parts will be greater in proportion as the parallel is more distant from the equator ; also, that to the north * Prom Sol, the sun, and sto, to stand. f From rpsn-u, to turn. INEQUALITY OF DATS. 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 equa- tor, 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 sum- mer solstice T to the winter solstice W; whence it appears that the day will increase in length from the winter to the summer solstice, and diminish in length from the summer to the winter solstice. 249. L.eiigtli of Day. As the equator is bisected -by the aorizon at the equinoxes, the day and night must be each twelve Qours long. But, when the. sun is north of the equator, the .greater part of its diurnal circle lies above the horizon, in north- ern 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 correspond- ing periods, be of ttie same length as the days in the former. 250. Effects of Iiicrea§e of L.atitude. 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. li, p. 22, where HOR, H'OR', represent the positions of the horizons of two diiferent places with respect to these circles; H'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 northward,' during the period that the sun is north of the equinoctial, and the shorter, during the period that he is south of this circle. 251. l.o>ige$t 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 the arctic circle the day will be 21: hours long at the time of the summer solstice ; for the polar distance of the sun will then be 66|-°, which is the same as the latitude of the arctic cir- cle; 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 hand, when the sun is at the winter solstice, the night will be 24 hours long on the arctic circle. 154 THE SUN AND ATTENDANT PHENOMENA. To the north of the arctic circle, the sun will remain continually above the horizon during the period, before and after the sum- mer 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. lit the Southern Hemisphere, 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. Problena I. The latitude of the place and the declination of tlie sun being given, to find the times of the sun's rising and setting and the length of tlie day. . Let HPE (Fig. 67) be the me- ridian, HMR the horizon, and BsD the diurnal circle described by the sun. The hour angle EP<, or its measure E<, which, convert- ed into time, expresses the inter- val between the rising or setting of the sun and his passage over the meridian, is called the Semi- diurnal Arc. Now, Ei! = EM + MC = 90° + Mt, Fig. which gives cos Et : : — sin Mi : and we have, by Napier's first rule, sin Mt = cot tMs tan ts = tan PMH tan EB = tan PH tan EB: whence, cos Ei = — tan PH tan EB, 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 appa- rent 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") 9.68740 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. What are the lengths of the longest and shortest days at Boston ; the latitude of that place being 42° 21' 15" N ? Ans. 15h. 6m. 26s., and 8h. 53m. 35s. Exam. 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, 6h. 19m. 48s. ; time of setting, 6h. 40m. 12s. 254. Problem II. To find the time of the sun's apparent ris- ing 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 of 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 -f PS -I- ZS _ co-lat. + co-dec. -t- (90° + E) k - -2 - 2 • • • -^^^^ . . , j^ sin (^ — ZP) sin {k — PS) and sm'iP = ^^ — ^^—55 , sm ZP sm PS ' . ,„ sin (^ — co-lat.) sin (yfc — co-dec.) ,_^. or, s n'iP = — — 7— TTV-^-T — i — T — ' (50). ' ^ sm (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 appa- rent 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" -f 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, 156 THE SUN AND ATTENDANT PHENOMENA. TWILIGHT. 255. Bxplaiiatioii. 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 re- flected down to the earth, so as to occasion a certain degree of light ; which gradually diminishes as the sun descends farther below the horizon, and the portion of air posited above the hori- zon, 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 Gre- pvisculum or Twilight. The explanation of twilight will be better understood on examining Fig. 68, where AON represents apor- ^ Fig. 68. tion of the earth's surface, H/cR the surface of the atmosphere above it, and hn^ a line drawn touching the earth and passing through the sun. The unshaded portion, Z;cR, of the body of air which lies above the plane of the horizon, HOR, is still illiimi- iiated by the sun, and shines down, by reflection, upon the station of the observer at 0. As the sun descends, this will decrease, until finally, when the sun is in the direction RNS', it will illumi- nate 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 which 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 marked 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 dis- appear. 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. Approixmate Deteriiiiuatioii of Height of Atmo- spbere. As we have just seen, at the end of evenincr twilii/ht the angle TKS' (Fig. 68) is equal to 18°; WcU being the hmit ot that portion of the atmosphere which is capable of reflectino a sensible amount of light to the eye, in the direction RO. Now^ if the vertical lines at 0, m, and N, be produced to the centre of the earth, 0, we shall have the angle OCN" equal to TJRS', or 18°, and therefore OCR equal to 9°. If, then, we denote the radius of the earth Cin by R, we shall have, height of atmos.=mR=CR— Cm=::Rsec 9°— R=R (sec 9° — 1) Making the calculation, we obtain for the height of the atmo- sphere, 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 kB,, and yet not have suffi- cient 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. Problem. The latitude of the place and the sun's declina- nation being given, to find the time of the beginning or end of twi- light. The zenith distance of the sun, at the beginning of morning or end of evening twilight, is 90° + 18° ; 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 sub- tracted from the time of sunrise, the remainder will be the dura- tion of twilight. 258. Variattle Duration of Xwiliglit. 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 cor- responding declinations of the sun. These facts are consequences of the different situations with respect to the hori- zon of tlie 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 con- ceive a circle to be traced in the heavens parallel to the horizon, and at the dis- tance of 18° 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 suu, comprised between the horizon and the crepusculum circle, which, for the sake of brevity, we wiU 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 uorth of the equator ; and let hr, si, and feV, 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 tha equator, the are of twilight is hs, and when he is on the parallel of declination h'k'r' it is lis'. Draw the chords hs, h's', mm, and the radii, cs, cs', cr', en, cp. The angle r'h's' is the half of r'cs', and the angle pmn is the half oi pen ; but r'cs' is lesa than pen, and therefore r'Ks' is less than pmn. Again, chs is the half of res, and It THE SUN AND ATTENDANT PHENOMENA. Fig. 69. thereibre greater than pmn, the half of the less angle pan. Whence it appears that the chord ft'»' is more oblique to the horizon, and therefore greater than the chord mn, and this more oWique and greater than the chord hs. It follows, there- fore, that the arc ftV is greater, and contains a greater number of degrees than the arc pm, and that this arc is greater than hs. Thus, as the sun re- cedes- from the equator towards the nort", the arc of twilight, and there- fore the duration of twilight, increases from two causes, viz. : 1 st. The increase in the distance of the line of intersec- tion of the liorizon with the diurnal circle from the centre of the drcle; and, 2d. The diminution in the size of the circle. The change will manifestly be greater in proportion as the latitude is greater. When the sun is south of the equa- tor, twilight will, for the same declina- tion, be shorter than when lie is north of the equator, because, although the diurnal circle will be of the same size, and its intersection with the horizon at the same distance from its centre, on the opposite side, the intersection with the erepusculum circle will now fall between the intersection with the horizon and the centre, and therefore, by what has just been demonstrated, the arc of twilight wUl be shorter. The shortest twilight occurs when the sun is somewhat to the south of the equator, because the arc of twilight, for a time, decreases by reason of the diminu- tion of its obUquitj' to the horizon more than it increases in consequence of the decrease in the size of the diurnal circle. That the obliquity of the arc of twilight, or rather of the chord of the arc, to the horizon diminishes, for a time, when the snn gets to the south of the equator, will appear from this, viz., that the chord is perpendicular to the horizon when the centre of the diurnal circle is midway be- tween the horizon and the erepusculum circle ; which will happen when the sun is a certain distance south of the equator, varying with the inclination of the axis of the heavens to the plane of the horizon, and therefore with the latitude of the place. The difference in the length of the summer and winter twilights, resulting from the causes above specified, is augmented by the inequality in the height of the atmosphere. Twilight also increases in length with the obliquity of the sphere. 259. Twiligbt in £.ow and jniddle L.atitudes. At the equator, the shortest astronomical twilight occurs at the equi- noxes, and is Ih. 12m. in duration. At latitude 41°, it occurs when the sua is about 6° south of the equator, and continues about 1\ hours. At the polar circle it happens when the sun is b^° south of the equator, and continues over 3 hours. The longest twilight at the equator is Ih. 19m. ; and at latitude il°, is 2h. 3m. in duration. 260, Twilight in High Latitudes. At the latitude 49°, the sun at the time of the summer solstice is only 18° below the horizon at midnight ; for the altitude of the pole, on the parallel of 4'.:*°, differs only 18° from the polar distance of the sun, at this epoch. This may be illustrated by Fig. 66, p. 151, taking X as the point of passage of the sun across the inferior meridian, and supposing I'H to be equal to 49°. At the summer solstice, PX = 07°; and thus the distance of the sun below the horizon THE SEASONS. 159 at midnight = HX = PX — PH = 67° — 49° = 18°. At this latitude, therefore, evening and morning twilight will each con- tinue half the night, at the summer solstice, and therefore nearly 4 hours. At higher latitudes than 49°, twilight (evening and morning) will continue all night for a certain period of time before and after the summer solstice, during whicn the polar distance of the sun is less than the latitude augmented by 18°. At the polar circle, this will be the case for 2 1 months before and 2^ months after the summer solstice. To the north of the arctic circle, as far as 84° of latitude, during the long night that prevails before and after the winter solstice, there should be more or less of a twilight over the southern hori- zon, about the hour of noon of every day of i!4 hours. At either pole twilight commences about a month and a half before the sun appears above the horizon, and lasts about a month and a half after he has disappeared. For, since the horizon at the pole is identical with the celestial equator, the twilight which precedes the long day of six months will begin when the sun in approach- ing the equator, upon the other side, attains to a declination of 18°; and this will be about 50 days before he reaches the equa- tor, and rises at the pole. The evening twilight will continue, in like manner, until the sun has descended 18° below the equator. It should be observed, with reference to the above results, that the assumed limiting angle of depression of the sun below the horizon, 18°, having been determined from observations in the middle latitudes, is probably too great for high latitudes ; and also that the astronomical twilight above considered, is much longer than what is ordinarily regarded as the period of twilight, THE SEASONS. 261. General Explanation of Change of Seasons. The amount of heat received, at any place on the earth, directly from the sun, in the course of 24 hours, depends upon two operative causes ; the length of time that the sun remains above the hori- zon, and the obliquity of its rays at noon. By reason of the obliquity of the ecliptic, both of these general circumstances vary materially in the course of the year; whence arises a variation of temperature, or a change of seasons. Since the obliquity of the ecliptic is a consequence of the inclination of the earth's axis to the perpendicular to the plane of the orbit, the inclined posi- tion of the axis is the primary cause of the change of seasons. _ 202. Climatic Zones. The tropics and polar circles divide the earth into five parts, called Zones, throughout each of which the yearly change of temperature is occasioned by a similar change in the circumstances of the sun's thermal action. The ICiO THE SUN AND ATTENDANT PHENOMENA. part contained between the two tropics is called the Torrid Zone, the two parts between the tropics and polar circles are called the Temperate Zones ; and the other two parts, within the polar cir- cles, are called Frigid Zones. At all places in the north temperate zone, the sun will always pass the meridian to the south of the zenith ; for the latitudes of such places exceed 23^°, the greatest declination of the sun (see Fig. 6H). The meridian zenith distance will be greatest at the winter solstice, when the sun has its greatest southern decli- nation ; and it will vary continually between the values which obtain at the solstices. The day will be longest at the summer solstice, and shortest at the winter solstice, and will vary in length progressivel}' from the one date to the other. We infer, therefore, that throughout the zone in question the greatest amount of heat will be received from the sun at the summer solstice, and the least at the winter solstice ; and that the amount received will gradually increase, or decrease, from one of these epochs to the other. The solstices are not, however, the epochs of ma.Kiraum and minimum temperature, but are found from observation to precede these by about a month. The reason of this circumstance is, that the earth continues for a month, or thereabouts, after the summer solstice to receive dur- ing the day more heat than it loses during the night, and for about the same length of time after the winter solstice continues to lose during the night more heat than it receives during the day. Within the torrid zone, the length of the day varies after the same manner as in the temperate zone, though in a less degree ; but the motion of the sun with respect to the zenith is different. At all places in the torrid zone the sun passes the meridian daring a certain portion of the year to the south of the zenith, and during the remaining portion to the north of it; for all places so situated have their zeniths between the tropics in the heavens, and the sun moves from one tropic to the other, and back again to its original position, in a tropical year. Through- out the torrid zone, therefore, the sun will be in the zenith twice in the course of the year, and will be at its maximum distance from it on the one side and the other at the solstices. An inhabitant of the equator, or its vicinity, will have summer at the two periods when the sun is in the zenith, and winter (or a period of minimum temperature) both at the summer and win- ter solstice. Near the tropic, there will be but little variation in the daily amount of heat received, during the period that the sun is north of the zenith. At tlie frigid zone, a new cause of a change of temperature exists; the sun remains continually above the horizon for a greater or less number of days about the summer solstice, and continually below it for the same number of days about the win- ter solstice. THE SEASONS. 161 263. General Fffccts of Increase of Liatitude. The amount of the yearly variation of temperature increases with the latitude of the place; for the greater is the latitude the greater will be the variation in the length of the day. Also, the mean yearly temperature is lower as we recede from the equator and ap- proach the poles ; for since the sun is, in the course of the year, the same length of time above the horizon, at all places, the mean yearly temperature must depend altogether upon the mean obli- quity of the sun's rays at noon, and this increases with the latitude. 264. Special Canses of Cliang^e of Climate. It is im- portant to observe, that although, in the main, climate varies with the latitude, after the manner explained in the foregoing articles, it is still dependent, more or less, upon local circum- stances, such as the vicinity of lakes, seas, or mountains, prevail- ing winds of some particular direction, etc. As the result of the operation of such special causes two .places may be situated on the same parallel of latitude, and yet have climates quite diflerent. Such differences of thermal condition are very marked on the oppo- site coasts of the Atlantic Ocean, in the middle and high latitudes. 265. Seasons Astronomically Defined : — Comparative L.engtiis. In the north temperate zone. Spring, Summer, Au- tumn, and Winter, the four seasons into which the year is divided, are considered as respectively commencing at the times of the Vernal Equinox, Summer Solstice, A utumnal Equinox, and Winter Solstice. Let V (Fig. 70) represent the vernal, and A the autumnal ■--41 Fio. TO. equinox ; S the summer, and W the winter solstice. The perigee of the sun's apparent orbit is at present 10° 40' to the east of the winter solstice. Let P denote its position. The lengths of the seasons are, agreeably to Kepler's law of areas, 11 162 THE SUN AND ATTENDANT PHENOMENA. respectively proportional to tbe areas VES, SEA, AEW, and WEV. Thus, the winter is the shortest season, and the summer the longest; and spring is longer than autumn. Spring and summer, taken together, are about eight days longer than autumn and winter united. 266. Secular Variation of Lieiigth of Seasons. Since the perigee of tlie sun's orbit has a progressive motion, the relative lengths of the seasons above defined must be subject to a continual variation. At the beginning of the year I811O, the longitude of the sun's perigee was 279° 29' 56". If from this we take 180°, the longitude of the autumnal equinox, the remainder, 99° 29' 56", is the distance of the perigee from the autumnal equinox at that epoch. The motion of the perigee in longitude is, at the present date, at the rate of 61'.70 per year. Dividing 99'' 29' 56" by 61".70, the quotient is 5,805 years. Hence it ap- pears that if the annual motion of the perigee had been constantly equal to 61''.7, about 5,800 years anterior to 1800 the perigee would have coincided with the au- tumnal equinox. But the motion of the perigee has in fact been very different in different centuries; and it appears from the calculations of Leverrier that 10,000 years before the beginning of the present century, the perigee was still 78° to the east of the autumnal equinox, and that the two points were in approximate coinci- dence 20,000 years earlier. 267. Secular Variiilions of Temperature. The eccentricity of the earth's orbit is so small (0.0L7) that the. present annual change in the sun's distance from the earth has but little effect in producing a variation of tempera- ture upon the earth's surface. The annual change of its heating power from this cause amounts to no more than one-fifteenth. So far as this cause operates, it makes the winters warmer and the summers colder in the northern hemisphere. But tlie eccentricity has not always had its present small value ; it has been for ages slowly diminishing from a certain maximum value. Recent calculations made by Leverrier and other eminent computers, have made known its value at inter- vals of 10,000 years, or 50,000 years, for a period of 1,000,000 years previous to the beginning of the present century. From these results, it appears that it has increased and decreased during alternate periods comprising, in general, about 50,000 years; and that its recurring maximum value has fluctuated generally be- tween the limits .05 and .07 5, while its minimum value has been about .01. The highest maximum value occurred 850,000 years since. The most recent maximum occurred 20,000 years ago, and was only ,019. At the epoch of the highest maximum, the earth reached its perihelion during the summer (civil reckoning) in the northern hemisphere, and its aphelion during the winter. At that epoch the heating power of the sun was, by reason of the eccentricity of the earth's orbit, about one-fourth greater at the beginning of summer than at the beginning of winter ; and the midwinter temperature, owing to the greater distance of the Bun, was much lower than at present. In an article in tlie Philosophical Maga- zine for February, 1867, by James Croll, it is computed that the midwinter temperature of Scotland was not less than 45° P. lower than at present; and that, at the same epoch, the midsummer temperature was correspondingly liigher. It is also maintained that, by a diversiou of the gulf-stream, the midwinter tem- perature may have been reduced many degrees lower ; and that this incidental effect of the great eccentricity of the earth's orbit, at that remote period, may have been the determining cause of the glacial epoch of the earth's geological history. FORM AND DIMENSIONS OF THE SUN. 268. The sun presents the appearance of a luminous circular disc ; but it does not follow from this that its surface must be really flat, for such is the appearance of all globular bodies when viewed at a great distance. It is ascertained from observations with the telescope that the sun has a rotatorj^ motion ; this being the fact, its surface must in reality be of a spherical form ; for DIMENSIONS OF SUN. 163 otlierM'ise it would not, in presenting all its sides, always appear under the form of a circle. No sensible difference between the equatorial and polar diame- ters of the sun can be detected by the nicest raicrometrical mea- surements. 269. Dimensions of Sun. The sun's real diameter is calculated from his apparent diameter and horizontal parallax. Let ACB (Fig. 71) represent the sun, or other heavenly body, and B the place of the earth ; and let g = AEB, the sun's apparent diameter ; d = 2 AS, its real diameter; D = ES, its distance from the earth ; Fia. 71. and E = the radius of the earth. We have, from the triangle AES, AS = ES sin iAEB, or 2AS = 2BS sin ^AEB ; and thus d — 2T) sin ^5 : but (equa. 7), D —— — , sin H whence, c? = 2R -?!^ = 2E ^ = 2R — (nearly) .... (51). sm H H 2H The apparent diameter of the sun at the mean distance is 32' 0", and the corresponding equatorial horizontal parallax is 8".95. Accordingly we have, for the real diameter of the sun (by equa. 51, whether the sines be taken or the arcs) d = 2R X 107.263 = 7925'".60 x 107.263 = 850,123 miles. The mean diameter of the earth is 7,912.40 miles ; the diame- ter of the sun exceeds this in the ratio of 107.442 to 1. The volume of the sun then exceeds that of the earth in the propor- tion of (107.442)' to 1', or 1,240,286 to 1. The surface of the sun bears to that of the earth the ratio of (107.442)' to 1", or 11,544 to 1. If models were constructed to show the comparative dimen- sions of the sun and earth, and the earth were represented by a ball one inch in diameter, the sun would be represented by a globe nine feet in diameter. Perhaps a juster conception of the enormous bulk of the sun may be obtained from the considera- tion, that, if the centre of the sun were coincident with the centre of the earth, its mass would extend nearly 200,000 miles beyond the orbit of the moon. 270. General Principle. From equation (51) we may de- rive the proportion d-.'m-.-.h: 2H. Thus, the real diameter of a heavenly body is to the diameter of the i6i THE SON AND ATTENDANT PHENOMENA. earth, as the apparent diameter of the body is to duuble its horizontal parallax. SUN'S SPOTS, AND ROTATION ON ITS AXIS.— PHYSICAL CONSTITUTION OF THE SUN. 271. When the sun is viewed with a good telescope, provided with colored glasses to protect the eye, black spots, or maculxB, of an irregular form, surrounded by a dark border of a nearly uniform shade, called a. penumbra, are often seen on its disc (Fig. 72). Sometimes several spots are included within the same V «!. Fi&. 72. penumbra. On the other hand, a large penumbra has occasion- ally been seen without any central black spot. The spots usually appear in clusters, composed of various numbers, from two to sixty or seventy. It is even said, that as many as 200 have been counted, in one instance, in a single group. In most of the individual spots, the central spot, or umbra, is not perfectly black ; but a black nucleus is observed in the ma- jority of the large and symmetrical spots, to occupy some part of the umbra, generally the centre. This distinction of shade in the umbra has been overlooked by most observers. The penumbra has almost always a perceptibly darker shade at its outer edge than in any other part ; and its light generally increases somewhat to its inner edge. 272. inagnitude of the Spots. The absolute magnitude of the solar spots is often very great. Spots are not unfrequently seen that subtend an angle of 1', or 60". Now the apparent diameter of the earth, as viewed at the distance of the sun, is equal to double the sun's horizontal parallax, or IS" ; the breadth of such spots must therefore exceed three times the diameter of the earth, or 24,000 miles. Spots have been observed whose sun's spots. 165 linear diameter ,was more than 45,000 miles, and which were therefore, in area, eight times as large as the entire surface of the earth. Some spots have attained to even a greater size than this, and become visible to the naked eje. A spot was seen in June, 1843, that continued visible to the naked eye for a. whole week, and which, according to the measurements of M. Schwabe, of Dessau, had a breadth of 74,000 miles. A group of spots, with the penumbra surrounding it, will frequently cover a still larger portion of the sun's disc. One noticed in April, 1845, had a linear extent of 147,000 miles. 273. Variability ot tiie Spots. The form and size of the spots are subject to rapid and almost incessant variations. When watched from day to day, or even from hour to hour, they are seen to enlarge or contract, and at the same time to change their form. They sometimes vanish in an incredibly short space of time, while others make their appearance as suddenly. Some spots disappear almost immediately after they become visible; others remain for weeks or even months. When a spot disap- pears, it usually contracts into a point, and vanishes before the penumbra, which gradually closes in upon it. When a new spot is developed, it is not till it has attained some measurable size that a penumbra begins to be perceived distinct from the umbra. The black nucleus within the umbra makes its appearance still later. The spot usually grows very rapidl}', and often attains its full size in less than a day. During the period of increase, and while it remains without material change of size, its edges are sharply de- fined, and the penumbra exhibits a general uniformity of shade. Several observers have, however, distinctly noticed a radiated appearance in the peuumbral fringes, as if they were traversed by bright veins diverging from the central spot. In the act of decreasing, the edges of the spot are less strongly defined, being apparently seen through a thin, luminous veil, which gradually extends over the spot. The process sometimes eventuates in the sudden appearance of a luminous line traversing the dark inter- val, which is then rapidly followed by the filling up and disap- pearance of the spot. The velocity of the inward movement of the penumbral edges is found to have exceeded, in some of the larger spots, 44 miles per hour. 274. Periodicity of the Spots. It has been ascertained, by systematic observations upon the spots, that their number varies considerably in different years. It will sometimes happen that, on every clear day during a particular year, the sun's disc always contains one or more of them, while, in another year, for weeks or even months together, no spots of any kind can be per- ceived. After twenty-five years of continued observations, M. Schwabe discovered that there was a regular alternate increase and decrease in the varying numbers and sizes of the spots ob- 166 THE SUN AND ATTENDANT PHENOMENA. served during successive years; the period from one maximum or one minimum to another being about ten years. More recently, Prof. Wolf, of Zurich, by a careful discussion of the observations of the solar spots made during the last one hundred years, has determined that the period of the spots has varied, during this interval, from 8 to 16 years, and that its mean value has been 11.11 years. 1860 was the last year of maximum. Agreeably to the mean period, the year 1866 should have been a year of minimum spots. 275, Faculae. Curved or branching streaks more luminous than the general body of the sun, are frequently perceived upon parts of his disc, especially in the region of large spots, or of extensive groups of spots, or in localities where dark spots subsequently make their appearance. These are called Faculm. They are chiefly to be seen near the margin of the disc. Adja- cent bright spaces are also an invariable accompaniment of the spots. These, in the majority of instances, are most conspicuous behind the spots, or in a direction opposite to that of the sun's rotation. It has recently been established by an observation made by Dawes, a distinguished English astronomer, that the facuke. are ridges or masses of luminous matter, elevated above the general level of the sun's surface. In 1859, he observed a bright streak at the very edge of the disc, which projected irregularly beyond the circular contour of the edge, like a low range of hills. For such elevations to have been distinctly perceptible, their actual height could not have been less than 500 miles, and was probably two or three times as great as this. 276. Oeneral Telescopic Appearance of $tin'« Disc. The part of the sun's disc not occupied by spots is far from being uniformly bright. Inequalities of brightness prevail in all parts of the disc, which give it a coarsely mottled appearance. When more attentively scrutinized, its grou-nd is seen to be finely mot- tled with minute dark dots or pores, which often appear to be in a state of change. It is also observed that the general luminous surface of the sun presents the appearance of bright granules scattered irregularly over it, and that, on the darker spaces be- tween the granulated portions, the minute dark pores are espe- cially prevalent. It is not yet decided whether these bright granules are to be regarded as distinct masses of greater bright- ness, or ai merely different conditions of the luminous cloudy surface, diversified by elevated ridges or waves. This general granulation of the surface is entirely wanting on the faculse, and on the luminous border of the penumbra of each of the dark spots. But lines of distinct, elongated, and comparatively bright masses are often seen projected on the penumbra, directed toward the centre of the spot, and even extending irregularly into the umbra,' or central black spot. PERIOD OF ROTATION. 167 277. motions of the Spots :— Rotation of tlic Sun. When the positions of the spots on the disc are obseived from day to day, it is perceived that they all have a common motion in a direction from east to west. Some of the spots close up and vanish before they reach the western limb; others disappear at the western limb, and are never afterwards seen ; a few, after becoming visible at the eastern limb, have been seen to pass entirely across the disc, disappear from view at the western limb, and reappear again at the eastern limb. The time employed by a spot in traversing the sun's disc is about 14 days. About the same time is occupied in passing from the western to the eastern limb, while it is invisible. The motions of the spots are account- ed for, in all their circumstances, by supposing that the sun has a motion of rotation from west to east, around an axis nearly perpendicular to the plane of the ecliptic; and that the spots are portions of the solid body of the sun. The truth of this explana- tion of the apparent motions of the sun's spots, is confirmed by the changes which are observed to take place in the magnitude and form of the more permanent spots during their passage across the disc. When they first come into view at the eastern limb, they appear as a narrow dark streak. As they advance towards the middle of the disc, they gradually open out and increase in magnitude ; and after they have passed the middle of the disc, contract by the same degrees until they are again seen as a mere dark line upon the western limb. A spot returns to the same position on the disc in about 27.^ days. This is not, however, the precise period of the sun's rota- tion ; for during this interval the sun has apparently moved forward nearly a sign in the ecliptic ; the spot will theretbre have accomplished that much more than a complete revolution, when it is again seen by an observer on the earth in the same position on the disc. 278. Period of Rotation. The apparent position of a spot with respect to the sun's centre may be accurately determined, from day to day, by observing, when the sun is crossing the meridian, the right ascension and declination both of the spot and centre. From three or more observations of this kind the period of the sun's rotation and the position of his equator may be ascertained. The period of the sun's rotation, as determined from observa- tions upon the spots, is found to increase with the latitude of the spot; from which it is to be inferred that the spots are not sta- tionary, and have different rates of motion along the surface, in a direction parallel to the equator. If, as seems most probable, the general direction of motion is opposite to that of the rotation, then the spots nearest the equator have the slowest motion ; and the period of rotation deduced from observations upon these spots ap- proximates most nearly to the actual period of rotation of the body 16S THE SUN AND ATTENDANT PHENOMENA. of the sun. The period in question is 25 days. Spots observed in the latitude 34° give a period of nearly 27 days. The incli- nation of the sun's equator to the ecliptic is about 7|-° ; and the heliocentric longitude of the ascending node of the equator is about 74°. 279. Kegions of the §pots. The solar spots are mostly confined to two zones parallel to the equator, and extending from 5° to 35° of latitude. Beyond 85° they are rarely seen, and in the polar regions never. The actual equator is also sel- dom, if ever, visited by spots. They are most abundant toward the middle of the spot-belts, and prevail more in the northern than in the southern hemisphere. It is observed that the spots have a tendency to form groups lying in lines or belts parallel to the equator. This is apparently the result of a tendency of new spots to break out behind the old ones. 280. JVatiii-e of tlie Spots. The dark spots on the sun are depressions below the luminous surface. This important fact was first established by Dr. Wilson, of Glasgow. He noticed that as a large spot, which was seen in November, 1769, came near the western limb, T.he penumbra on the side toward the centre of the disc contracted and disappeared, and that afterwards the luminous matter on that side seemed to encroach upon the central black spot, while in other parts the penumbra underwent but little change. On the reappearance of the spot at the eastern limb, he found that the penumbra was again wanting on the side toward the centre of the disc ; and that when this part made its appear- ance, after the spot had advanced a short distance upon the disc, it was much narniwer than the opposite part. These various appearances of the spot in question are represented in Fig. 73. Pro. 73. They show conclusively that both the black central spot and the penumbra were below the luminous surface of the sun. Dr. Wilson estimated the depth of the spot to be nearl3'' 4,000 miles. It has since been observed that similar changes of appearance are experienced by the spots in general, in their passage across the disc. 281. Theories of Phfsical Constitution of Sun : and of Foi'inalion of Spots. Wihon^ Theory. Dr. Wilson drew from the various appear- ances of the spot observed by him in 1769 the natural conclusion PHOTOSPHERE OF THE SUN. 169 that the solar spots were the dark body of the sun, seen through excavations made in the luminous matter at the surface. The luminous matter he conceived to have the consistence of a fog or cloud, rather than of a liquid; and suggested that openings might be made in it by the working of some sort of elastic vapor generated within the dark globe. The penumbra surrounding each black spot he conjectured to be the sloping sides of the opening in the stratum of luminous clouds. HerscheVs Theory. Sir "William llerschel, after an assiduous study of the aspects and phenomena of the sun's spots, adopted substantially Dr. Wilson's views, but conceived it to be necessary in order to explain the uniform shade of the penumbra, to sup- pose the existence of an opake, non-luminous, cloudy stratum, posited between the luminous medium and the dark solid globe. On this hypothesis, the spots are accounted for by supposing that openings occasionally take place in both the luminous and non-luminous envelopes, through which the dark body of the sun is seen. The penumbra is the portion of the obscure enve- lope situated immediately around the opening made in it, and shining by reflected light only. Herschel supposed the openings to be made by the exertion of some sort of explosive energy from beneath ; and that the same upheaving agency, when not of sufBcient intensity to rend the luminous envelope, forced it up into masses or waves of hundreds of miles in height. The ridges of these waves he conceived to be the faculse, which are distinctly seen only when near the margin of the disc, because the waves there appear in profile, and when near the middle of the disc are seen in front, or foreshortened. Sir John Herschel, who has also been an attentive observer of the sun's spots, has advanced the opinion that the agency by which .the spots are formed, is exerted from above downwards, instead of from below upwards. Recent observations made by Dawes indicate the existence of a second non-luminous envelope, posited below that which is seen in the penumbra of a spot ; tbat this is seen in the umbra, and that the black nucleus often observed near the centre of the umbra is an opening made in this lower stratum. 282. Photosphere of the Suu. It is the received opinion among astronomers of tlie present day, that the sun, as main- tained by Wilson and Herschel, consists of a comparatively dark globe, either solid or liquid, surrounded by one or more lumi- nous envelopes, in a vaporous or nebulous condition, and some thousands of miles in total height. This exterior region per- vaded by the medium which is the great source of the sun's light, is called iha photosphere of the sun. That this luminous me- dium is really in the aeriform state, or in the condition of cloudy masses floating in a gaseous medium, may be inferred from its great mobility. The velocity of expansion and contraction 170 THE SUN AND ATTENDANT PHENOMENA. of the spots, which often exceeds 40 miles per hour, is incom' patible with the supposition of a liquid condition. It does not necessarily follow from the fact that the solid or liquid globe appears perfectly dark, that it has no degree of luminosity; for it has been observed that intensely ignited solids appear only as black spots on the disc of the sun, when held between the sun and the eye. Professor Henry, and more recently Professor Secchi, of Rome, has established by experi- ment that the dark spots emit less heat than the luminous sur- face. But it is to be observed that the result of the experiments does not give any decisive indication as to the comparative tem- peratures of the photosphere and dark body of the sun, since a considerable fraction of the heat radiated from the latter is no doubt intercepted by the sun's atmosphere, below the level of the luminous surface. Depth of photosphere. Secchi has succeeded, from measure- ments made upon several spots at the time of the disappear- ance of the penumbra on the side toward the centre of the disc, in effecting an approximate determination of the depth of the photosphere, on the supposition that the penumbra is made up of the sloping sides of an opening in a single luminous enve- lope (281). He estimates the depth to be about one-third of the radius of the earth, or 1,300 miles. M. Faye has undertaken to determine the depth of the photosphere by a different method, and makes it about 4,000 miles. !i§S. Liiiniiiious Appearances exterior to tlie Photo- sphere. Wlienever the sun becomes totally eclipsed by the moon interposed between it and the eye of the observer, the region exterior to the photosphere is seen to be pervaded, for a considerable distance, by luminous matter, which offers a variety of remarkable appearances. The principal of these are the corona, luminous streamers, or jets of light, and rose-colored pro- tuberances. 2S4. The Corona is a ring or halo of white light encircling the sun, which becomes visible when the body of the sun is con- cealed from view. It is brightest next the dark limb of the moon, where it has a rosy tint, and gradually decreases in lustre until it becomes undistinguishable from the general light of tlie sky (See Plate II.) It has presented to observers more or less of a radiated appearance, as if made up of luminous radiations, or traversed by them. This appearance has been less distinct near the moon where the corona is brightest than at the more distant and fainter portions. In the total eclipse of July 18, 1860, the interruptions of continuity became distinctly percep- tible at a distance from the photosphere of the sun equal to the sun's semi-diameter, or more than 400,000 miles. Beyond this, ■the, corona was distinctly radiated. Its extreme breadth, both HEIGHT AND EXTENT OF PROTUBERANCES. 171 in that eclipse and the eclipse of September 7, 1858, exceeded the diameter of the sun, or 850,000 miles. The extreme outline of the corona is perceptibly elliptical in its form ; the major axis lying in the plane of the sun's equator. 285. The L,uiiiiiioiis Streanier§, whether forming part of the corona, or distinct from it, have been seen to extend, at par- ticular points, far beyond the general outline of the corona. In the eclipse of 1860, some of them were traced to a distance from the sun's photosphere equal to twice the diameter of the sun, or 1,700,000 miles (Plate II.). They present, in general, the appear- ance of radiations of luminous matter, in directions perpendicular to the sun's surface. Observations made with the polariscope have established that the light of the corona and streamers is in part reflected light. 28e. The Rose-colored Protuberances are regarded by observers as the most remarkable and beautiful phenomena wit- nessed in total solar eclipses. They consist of apparent cloudy masses, more or less tinged with red light, and of various forms and sizes, noticed just without the dark limb of the moon (Plate II.). In some instances, they have been seen entirely detached from the moon's limb. They are seen at various points of the limb, and in every variety of position with respect to the equator of the sun. The latter circumstance shows that they have no connection with the sun's spots, since these do not occur in high latitudes. It has been repeatedly observed that, in the progress of a total eclipse of the sun, the protuberances which become visible at the eastern limb of the moon continually decrease in their apparent dimensions, as if the moon were screening more and more of them from view ; while those seen at the western limb continually in- crease in their dimensions, as if they were more and more un- covered by the moon in its advance. These facts indicate that the protuberances in question are luminous masses connected with the sun, and elevated above the photosphere. Careful measurements have fully established this conclusion. 2S7. Height and Extent of the Protuberances. Some of the protuberances observed in the eclipse of 1860, had an appa- rent height of nearly 70,000 miles. The breadth of single pro- tuberances is but a tew minutes of arc, but they sometimes extend in a continuous chain for many degrees. Near the end of the eclipse of 1860, one chain of low elevations was observed by Secchi, just without that part of the moon's contour at which the sun was about to make its appearance, which extended 60°. About the middle of the eclipse, no less than ten distinct protu- berances were counted, which were about regularly distributed around the disc. In view of these facts, it seems highly probable, as intimated by Secchi, that the rose-colored protuberances ob- Berved in that eclipse, were but the higher portions of cloudy 172 THE SUN AND ATTENDANT PHENOMENA. masses that formed at a lower level, one continuous reddish enve lope surrounding the sun. This envelope must have extended upwards from the sun's photosphere to the height of thousands of miles, and risen at some points into cloudy peaks of tens of thousands of miles in height. 288. JValiire of tlie Corona. The corona is supposed by Sir John Herschel. and other astronomers, to be a gaseous solar atmosphere, extending above the sun's photosphere; but its vast extent (28i) seems to be fatal to this explanation. Upon no reasonable supposition that can be made with regard to the effect of the sun's heat in expanding such an atmosphere, can it be supposed to have sufficient density to reflect a sensible amount of light, beyond a few thousand miles from the body of the sun. The natural indications of the phenomena are that the corona consists of luminous matter streaming off from the sun into space ; and at the same time that the appearance of distinct radi- ations arises from an inequality of emanation from different parts of the sun's surface. The elliptical form of the corona (284) in- dicates that the emission of luminous matter is most abundant from the equatorial regions. It has been suggested that the radiated appearance of the corona may result from a partial interception of the sun's light by clouds floating in the sun's atmosphere ; but the course of the rays which pass unobstructedly through the spaces between the clouds, could not be recognized unless they encounter matter of sufficient density to reflect a sensible quantity of light to the eye, and such matter cannot extend to a distance of more than a million of miles from the sun (285), unless' there are material emanations proceeding from the sun. We can only avoid this conclusion by assuming that there is a dense mass of meteoric bodies, or of cosmical matter, revolving around the sun within this distance. ; It is maintained by some astronomers that the corona and streamers are piienomena of diflfraction, resulting from the pas- sage of the sun's light near the borders of the moon. But upon this explanation there ought to be certain attendant phenomena of variegated colors which are not in reality seen. 289 Temperature of the Sun's Surface. To an observer at the luminous surface of the sun, its disc would appear to cover an entire hemisphere in the heavens, and therefore the heat that falls upon a small area at the surface of the sun should exceed that received upon the same area at the distance of the earth, in the proportion that a hemisphere of the heavens exceeds the area occupied by the sun's disc as seen from the earth, or nearly in the ratio of 100,000 to 1. A heat many times less intense than this suffices to dissipate the most refractory metals in vapor. That the calorific rays emitted from the sun have a far higher intensity than those which proceed from the hottest furnaces, or ORIGIN OF THE SUN'S HEAT. 173 result from the most vivid ignition obtained by chemical or gal vanic processes, may be inferred from the fact that they penetrate glass with far greater facility. Inequalities of temperature. Secchi has made a series of obser- vations upon the comparative amounts of heat received_ from different parts of the sun's surface. He finds that the polar emit less heat than the equatorial regions ; and that the two hemi- spheres separated by the equator have not exactly the same tem- perature. It also appears from his observations that the breaking out of a spot at any point of the sun's disc, occasions a fall of temperature there and at surrounding points ; and that the faculte do not sensibly augment the temperature of the points where they make their appearance. Also, the calorific rays proceeding from the centre of the sun's disc have a higher intensity than those proceeding from the bor- ders. The same is true of the luminous rays. From this fact it is inferred that the sun is surrounded by an atmosphere extend- ing far above its photosphere, or by some form of matter, in a condition to intercept a large amount of light and heat, 290. Intensity of SiinS L,ight. The most intense artificial lights are the Drummond Light, produced by the flame of the oxyhydrogen lamp directed against a surface of chalk, and the Electric Light, generated by the passage of a galvanic current between two charcoal points. Fizeau and Foucault found, by ingenious and carefully conducted experiments, that the light of the sun's disc exceeded in intensity the Drummond light, in the ratio of 146 to 1 ; and that it exceeded the electric light from forty large plates of a Bunsen's battery in the ratio 2^ to 1. It appears, therefore, that the electric light is the only artifi- cial light that approximates in intensity to the light of the sun. 291. Origin of the Sun's Heat. There would seem to be but two possible physical causes in operation that might be ade- quate to the development and maintenance of the high tempera- ture of the sun. These are — (1.) The contraction of the body of the sun from an original va)*)rous state to its present size and density. (2.) The fall of meteoric masses into the photosphere of the sun. Upon the hypothesis, hereafter considered, that comets and meteors were originally discharged from the surface of the suu during the successive stages of vaporous diffusion through which the sun's mass is supposed to have passed, these two causes are, physically speaking, essentially the same; since the heat ulti- mately developed must be the same, whether the subsidence of the matter is by gradual contraction, or by gravitation. Yet the fall of the revolving meteors one after another into the photo- sphere of the sun might now determine a much higher tempera- 174: THK SUN AND ATTENDANT PHENOMENA. ture tlian would have resulted from contraction alone. The heat due to their fall has been, as it were, stored up in these meteoric bodies, to be suddenly developed, instead of being gradually dissipated during the ages in which the sun has been going through its process of formation. 292. Results of Recent Investigations, concerning the sun's spots and physical constitution. The careful scrutiny and assiduous study, by several astronomers, of all the phenomena observable on the surface of the sun for a number of years past, has led to the following important discovery : The sun's spots are for the most part developed hy, or in some way connected with, the operation of a physical agency exerted hy the planets upon the photosphere. This remarkable fact has been conclusively established by the observations of Schwabe, Car- rington, Secchi, and others ; and especially by the detailed dis- cussion to which all the reliable observations upon the spots, made during the last 100 years, have been subjected by Professor Wolf, of Zurich. The planets which exercise the greatest influ- ence are Jupiter and Venus. The planetary agency is directly recognized in the origination of spots on the parts of the sun's surface brought by the rotation into favorable positions, and in the subsequent changes experienced by the spots while subject to the direct action of the planet. It is also shown by the dependence of the epochs of the maximum and minimum of spots upon the positions of the planets, especially of Jupiter and Ven us. Effects of so marked a character, exerted by the planets upon the photosphere of the sun, cannot reasonably be attributed to their natural attractive action, and must apparently result from a repulsive or impalsive action exerted upon the photospherio matter. Rotation of Spots. Some spots have been observed to have a motion of rotation around their centres ; but according to Dawes, who has been a diligent observer of solar phenomena for many years, this is a phenomenon of exceedingly rare occurrence in the case of well-developed spots. 293. Theory of the Origin of the Sun's Spots. The foUow- Ing ia a brief outliue of a theory of the development of the sun's spots, based upon the principle of planetary action above stated. (1.) The matter of the sun's photosphere, and for a certain distance beyond the luminous surface of the photosphere, is, either wholly or in part, In a magnetized state, and arranged in columns, or lines of magnetic polarization, like the auroral matter in the upper atmosphere of the earth. (2.) A repulsive or impulsive action exerted by the planets upon the molecules of these columns, tends to disturb their electric and magnetic equilibrium, and induce electric discharges along certain of the upper columns, by which they are widely dispersed, and the mechanical equilibrium of the portions below disturbed, lu this way, a vast column of expanding and ascending matter is originated in the photosphere, which in the process becomes more or less dissipated, and may reveal the body of the sua to view. ZODIACAL LIGHT. 175 (3.) The matter dispersed, from a loss of magnetic intensity, or by the electrir discharges, and certain portions of the vaporous matter of the column, as theyrisb above the photosphere, are brought into that subtle or nebulous condition observed in the matter of comets, in which it becomes subject to an effective repulsion from the sun, and so is expelled indefinitely into space. Other portions may be- come condensed above the photosphere, and subside into it. (4.) The planets may bo conceived to operate in two ways, to initiate the process i ol dispersion of the tops of the photosplierio columns, and so develop spots on the sun ; viz., by originating in the upper photosphere electric currents radiating in all directions from the region exposed to most direct action, or by developing electro-magnetic currents running in a direction opposite to that of the rotation. Such radial electric currents would be attended with an exaltation of the statical electric condition of the region exposed to planetary action ; and such magnetic currents would tend to demagnetize, or magnetize, the upper photospherio columns, according as the upper or lower currents prevail. It appears, from the results of observation, that the planets operate unequally in different parts of the ecliptic, and in different relative positions ; and their effects are apparently modified, in certain positions, by the electro-magnetic cur- rents developed iu the sun's photosphere by the motion of the solar system through space. (5.) The spots are more likely to occur in low than in high latitudes, because the induced magnetism of the photospherio columns has a lower intensity in proportion as the magnetic latitude is less ; and spots do not make their appearance on the equator, nor in its immediate vicinity, because the columns of magnetic matter are there parallel to the surface of the sun. (6.) In the supposed electric discharges along the magnetic columns, with the attendant accumulation of luminous matter in certain localities above the ordinary surface of the photosphere, we have at the same time an adequate explanation of the faculce, and of the rose-colored protuberances at a still higher level. When, in special localities, the discharge has attained to a sufficient intensity, or continued for a sufficient length of time, openings are made through the whole depth of the photosphere, and spots are seen in the region where the faculss were before observed. It may be added, in this connection, that the supposition of the distribution of the photospherio matter in separate columnar masses, accords with the granulated appearance presented by the sun's disc (2'76). (7.) The photospherio matter dispersed by reason of the varying action of the planets, sufficiently to become subject to a repulsive action from the sun as it flows away into space, forms the corona, with its accompanying radiations and streamers, visible in total eclipses. (8.) A portion of the attenuated matter thus expelled to an indefinite distance from the sun, is received into the upper atmosphere of the earth, and, by develop- ing electric currents there, becomes one of the operating causes of the disturbances of the magnetic needle on the earth's surface; which are observed to increase and decrease, pari pasm with the sun's spots. The impulses attendant upon the electric discharges occurring in the sun's photosphere, and propagated indefinitely into space, constitute another cause of magnetic disturbance upon the earth. The escaping solar matter received into the earth's upper atmosphere, supplies the matter of terrestrial auroras, which also have the same periods as the sun's spots. (For a more complete exposition of the author's theoretical views, see Am. Journal of Science, Vol XLL, Nos. 121 and 122. See also Note in Appendix,) ZODIACAL LIGHT. 294. At certain periods of the year a luminous appearance is observed in connection with the sun, extending upwards from the western horizon after evening twilight, and from tlie eastern horizon before daybreak, which is called the Zodiacal Light, from the circumstance of its being mostly comprehended within ihe ^1 f) THE SUN AND ATTENDANT PHENOMENA. Via. 74. zodiac. Its color is white, with a decided tinge of yellow at the lower altitudes. When most con- spicuous, it has a striking bril- liancy near the horizon, and fades upwards by imperceptible de- grees. Its apparent form is near- ly triangular, the base resting on the horizon, from which it tapers upwards to an indistinct vertex. The axis, or central line, lies near- ly in the ecliptic. Its length varies with the season of the year, and the state of the atmo- sphere. As estimated from the sun, it is sometimes more than 100°, but ordinarily not more than 40° or 50°. Its breadth near the horizon varies from 8° to 30° or 40°. It is nowhere abruptly terminated, but gradu- ally merges into the general light of the sky (Fig. 74). 295. It varies in Distinctness. Tne Zodiacal Light is seen most distinctly, in our northern latitudes, in February and March after sunset, and in October and November before sunrise. During the month of March it may be seen directed towards the star Aldebaran. In December, though fainter, it may often be seen both in the morning and evening. Also, towards the sum- mer solstice, it is discernible, in a very pure state of the atmo- sphere, both in the morning and evening. The reason of the variations in the distinctness of the zodiacal light, from one sea- son to another, is found in the change of its inclination to the horizon at the time of sunset or sunrise, together with the varia- tion that occurs in the duration of twilight. As its length lies nearly in the plane of the ecliptic, its inclination to the horizon will be different, like that of this plane, according to the differ- ent positions of the sun in the ecliptic. At sunset, the zodiacal light will be most inclined to the horizon, and therefore extend farthest up' in the heavens, towards the vernal equinox, when the ecliptic is at sunset most nearly perpendicular to the horizon ; and at sunrise it will be most inclined to the horizon towards the autumnal equinox, when the inclination of the ecliptic to the horizon at sunrise is the greatest. The zodiacal light is much brighter and more frequently observed between the tropics than in these latitudes ; because the ecliptic, in general, makes there a larger angle with the horizon, and twilight is of shorter dura- tion. According to Arago it appears, from the entire series of obser- ZODIACAL LIGHT. 177 vations at Paris and Geneva, that the Zodiacal Light varies con- siderably from one year to another, and that the observed variations cannot result entirely from changes in the transpa- rency of the atmosphere. Extraordinary changes of brightness and form, in the course of a single evening, have also been noticed by several observers,- which were regarded as decided indications of a change in the intrinsic lustre or density of the substance of the Zodiacal Light. But all such abrupt changes may possibly be purely of atmospheric origin. 29(i. Recent Observations :— Important Results. The most valuable series of observations extant on the zodiacal light are those which were made in the years 1853-4—5, at various latitudes, from 41° 49' N to 53° 28' S, by Chaplain Jones, of the U. S. Navy; and by the same observer, in the years 1856-7, from the elevated station of Quito, very near the equator. The discussion of these observations has furnished the fol- lowing important results : 1. When the observer was in a position on either side of the plane of the ecliptic, the main body of the Zodiacal Light was on the same side of the ecliptic in the heavens ; and when he was in the plane of the ecliptic, this light was equally divided by its circle in the heavens. 2. When the observer was carried by the earth's rotation rapidly towards or from the plane of the ecliptic, the change of the apex of the light, and of the direction of its boundary lines, was equally great, and corresponded to the change of place. 3. As the ecliptic changed its position with respect to the horizon, the entire shape of the Zodiacal Light became changed. 4. The entire luminous appearance consisted of a, stronger light at the central part, and a much broader diffuse light extending beyond this on either side, and to a greater height. The stronger passed by degrees into the diffuse light, and the latter also gradu- ally faded away. Yet there was a discernible line of greater suddenness of transition, that could be taken for the boundary of the former. 5. The light was visible, with more or less distinctness, on every favorable night during the entire period of the observations. The position of the observer, generally at sea, or in the lower latitudes when on land, was more favorable than that which most observers have had. 6. On favorable nights, when the ecliptic was nearly perpen- dicular to the horizon, at the observer's station, the Zodiacal Light was visible at midnight, over both the western and eastern horizons. This singular phenomenon was observed at sea, at certain stations within the tropics. At Quito, the light was seen every favorable night, and at all hours, to extend as a broad Imminous Arch, entirely from one horizon to the other. At midnight it had a pale and nearly uniform white lustre, from 12 % •178 THE SUN AND ATTENDANT PHENOMENA. one horizon to the other. The breadth was then nearly uniform, and about 30°. 297. Explanation. Ko generally received explanation of this singular phenomenon has yet been given. It was at one time supposed to be the atmosphere of the sun, but Laplace has shown that this explanation is at variance with the theory of gravitation. He found that at the distance of about sixteen mil- lions of miles from the centre of the sun, the centrifugal force due to the sun's rotation balanced the gravity, and therefore that the solar atmosphere could not extend beyond this; but this distance is less than half the distance of Mercury from the sun, whereas the substance of the Zodiacal Light extends beyond the orbit of Venus, and even beyond the earth's orbit. The most plausible theory of the Zodiacal Light that has been advanced is that propounded by Laplace, that it consists of a broad ring, or lenticular mass of nebulous matter, encircling the sun in the plane of bis equator. He supposed it to be maintained in a per- manent condition by the revolution of its particles around the sun. But, as intimated in former editions of tliis work, another conception of the me- chanical condition of such a mass of nebulous matter may be formed, that accords equally well with the phenomena of the zodiacal light We may regard the whole mass as made up of the streams of particles which wo have recognised as continu- ally in tlie act of flowing away from the sun (288 and 293), under the operation of a force of solar repulsion ; or, in other words, that it is the indefinite continuation of the corona observed in total eclipses of the sun, with its attendant streamers. Upon this view it should appear elongated, like the faint outer boundary of the corona (284), in the plane of the sun's equator ; and this elongation may be attri- buted to a more copious discharge of photospherio matter from the equatorial than from the polar regions of the sun. Or we may conceive, in accordance with the theoretical views of the probable condition of the sun's photosphere that have been presented (293), that the discharges may take place from tho tops of the photo- spheric columns, in the direction of their prolongation. All such discharged parti- cles would thus receive a projectile velocity oblique to the sun's surface, and toward the plane of the equator, and, being subsequently repelled by the sun, would move away mto space iu hyperbolic orbits, convex toward the sun. As 2 necessary consequence, there would be an augmentation of the quantity of escap- ing matter in the plane of the sun's equator, and an elongation of its visible por tion in this plane. The light may vary in brightness from one year to another, with the varying activity of discharge from the sun's surface. The appearance and phenomena of the Zodiacal Light indicate that the principal portion of the light received experiences specuior reflection from the particles of its substance. In the report of the observations referred to In the last article, this idea is presented and advocated. Upon this view, the amount of hghl reflected to the eye from different directions will increase with the angle included between the directions of the incident and reflected rays, and with the density of the substance. The lateral shiftings of position of the light, as the distance of the zenith from the ecliptic varies, may be satisfactorily explained by means of the following genera^ considerations : 1. By reason of the small dimensions of the earth, as compared with the vas* extent of the entire collection of matter flowing away from the sun to an indefinite distance, the density of this nebulous matter must be sensibly the same for consi- derable distances from the earth, in all directions. But beyond a certain distance, which we will call D, the density must begin sensibly to decrease as the line' of sight makes a greater angle with the plane of tho ecliptic (which is nearly coinci- dent with the plane of the sun's equator, the supposed plane of greatest density of the solar emanations). This decrease of density will be more rapid as the portion ZODIACAL LIGHT, 179 of nebiilous matter conaidered is more remote from the earth. It is plain, then, tha* the light reflected to the earth, from aU portions of this matter situated at distances from the earth greater than D, will decrease in intensity from the ecliptic, in both directions. The brightness of the light, at its different points, will also augment as the aagular distance from the sun diminishes. The entire result from the light thus reflected should then be a luminous appearance similar to the observed Zodia- cal Light, with its axis or line of greatest brightness lying in the ecliptic. If wo now take account of the absorptive action of the atmosphere upon the transmitted lighf, which increases in proportion as the direction of the ray makes a less angle with the plane of the horizon we perceive that the axis wiU be thrown to that side of the ecliptic on which the zenith lies, whenever the inclination of the ecliptic to the horizon is less than 90°. The light received from all portions of the nebulous matter which are at a less distance than D from the earth, will produce a different result. Its intensity will be the same for all points at the same angular distance from the sun ; that is, if we disregard atmospheric absorption. If this be taken into account, it will be seen that the actual appearance will be a diffuse luminosity, decreasing in both direc- tions from the vertical circle passing through the sun below the horizon. From the station of the observer, within the shadow of the earth, the nearer portions of the shining nebulous matter will lie in the direction of this vertical circle. This cir- cumstance wUl tend to augment the brightness along this circle, and in its vicinity, as compared with points at a distance from it. The Zodiacal Light observed is the result of the combination of this light, which has Its axis in the vertical circle through the sun, in its position below the horizon, with the stronger light reflected from the more remote regions, whose axis lies in the ecliptic. 2. The axes of these two different luminosities coincide whenever the ecliptic i& perpendicular to the horizon; but when these circles are inclined to each other, a greater portion of the compound light falls on the side of the ecliptic on which the zenith and the vertical circle of the sun lie, than on the opposite side. When the inclination of the circles increases, the disparity between the portions of the light that lie on opposite sides of the ecliptic becomes greater, and the axis and the whole luminous appearance are displaced in tlie direction of the vertical circle. This displacement should certainly take place up to a certain limit of increase in the angle, which should be greatest at the lower altitudes, at which the observed displacement is greatest. It is also to be noticed that there is a secondary cause in operation, tending to augment thi3 lateral displacement; which consists in the fact, that as the ecliptic becomes more oblique to the horizon, the sun when at the same distance as before, along the ecliptic from the horizon, will be nearer the horizon in the direction of the vertical circle, and therefore at equal heights above the horizon, the luminosity which has its axis in the vertical circle, will be increased in brightness, and so have a greater displacing effect on the boundary of the ediptic Ught. It will also readily be perceived that up to a certain amount of deviation of the ecliptic from the vertical circle, the unequal atmospheric absorp- tion of the light will operate to increase the displacement. The Diffuse LigU noticed in the last article, probably has its origin iu the portions of the nebulous solar emanation that lie immediately beyond the distance D, the density of which will decrease from the ecliptic more slowly than that of the more remote portions. The Imrmams Arch seen at midnight in tropical regions (296) must be attributed, from the present point of view, to the radiant reflection, and feeble specular reflec- tion at angles of incidence less than 45°, of the sun's rays, from the portion of the solar matter that extends indefinitely beyond the earth's orbit. The variations in the amount of light reflected from regions at different angular distances from the sun, in the density of the reflecting substance, and in the effects of atmospheric ab- sorption, tend to equalize the light received from different directions lying in the plane of the ecliptic. The extent of the conical shadow oast by the earth is presumably small in comparison with that of the nebulous substance from wliich the light is received. it wUl be perceived that atmospheric aljsorption plays the prominent part in the phenomenon of the lateral displacement of the Zodiacal Light, operating both directly and indirectly. 18C THE MOON AND ITS PHENOMENA. CHAPTER XV. The Moon and Its Phenomena. PHASES OP THE MOOK 298. The most conspicuous of the phenomena exhibited by the moon, is the periodical change that is observed to take place in the form and size of its disc. The different appearances which the disc presents are called the Phases of the moon. The phenomenon in question is a simple consequence of the revolution of the moon around the earth. Let E (Fig. 75) represent the position of the earth, ABO the orbit of the C _Sx_^ Fia. 75. moon, which we will suppose for the present to lie in the p'ane of the ecliptic, and ES the direction of the sun. As the distance of the sun from the earth is about 400 times the distance of the moon, lines drawn from the sun to the different parts of the moon's orbit, may be considered, without . material error, as par- allel to each other. If we regard the moon as an opake non- luminous body, of a spherical form, that hemisphere which is turned towards the sun will be continually illuminated, and the other will be in the dark. Now, by virtue of the moon's motion, the enlightened hemisphere is presented to the earth under every variety of aspect in the course of a synodic revolu- tion of the moon. Thus, when the moon is in conjunction, as at A, this hemisphere is turned entirely away from the earth, and PHASES OF THE MOON. 181 it is invisible. Soon after conjunction, a portion of it on the right begins to be seen, and as this is comprised between the right half of the circle which limits the vision, and the right half of the circle which separates the enlightened and dark hemi- spheres of the moon, called the Circle o/" Illumination, it will ob- viously present the appearance of a crescent, with the horns turned from the sun, as represented at B. As the moon advances, more and more of the enlightened half becomes visible, and thus the crescent enlarges, and the eastern limb becomes less concave. At the point C, 90° distant from the sun, one-half of it is seen, arid the disc is a semi-circle, the eastern limb being a right line. Beyond this point, more than half becomes visible ; the nearer half of the circle of illumination falls to the left of the moon's centre, as seen from the earth, and thus becomes convex out- ward. This phase of the moon is represented at D. When the moon appears under this shape, it is said to be Gibbous. In ad- vancing towards opposition, the disc will enlarge, and the eastern limb become continually more convex ; and finally at opposition, where the whole illuminated face is seen from the earth, it will become a full circle. From opposition to conjunction, the nearer half of the circle of illumination will form the right or western limb, and this limb will pass in the inverse order through the same variety of forms as the eastern limb in the interval between conjunction and opposition. The different phases are delineated in the figure. The moon's orbit is, in fact, somewhat inclined to the plane of the ecliptic, instead of lying in it, as we have supposed ; but, it is plain that its inclination cannot change the order, nor the period of the phases, and that it can have no other effect than to alter somewhat the size of the disc, at particular angular distances from the sun. In consequence of the smallness of the inclination, this alteration is too slight to be noticed. 299, Definitions. When the moon is in conjunction, it is said to be JVew Moon ; and when in opposition. Full Moon. At the time between new and full moon, when the difference of the longitudes of the moon and sun is 90°, it is said to be the First Quarter. And at the corresponding time between full and new rhoon, it is said to be the Last Quarter. In both these positions the moon appears as a semi-circle, and is said to be dichotomized. The two positions of conjunction and opposition are called Syzi- gies; and those of the first and last quarter, Quadratures. The four points midway be: ween the syzigies and quadratures are called Octants. 300. Liunar Month. The interval from new moon to new moon again, is called a, Lunar Month, and sometimes a Lunation. The mean daily motion of the sun in longitude is 59' 8".83, and that of the moon 13° 10' 35".03; wherefore the moon sepa- ■ates from the sun at the mean rate of 12° 11' 26".70 per day, 182 THE MOON AND ITS PHENOMENA. and hence, to find the mean length of a lunar month, we have the proportion 12° 11' 26".70 : Id. : : 360° : x = 29d. 12h. 44m. 2.7s. 301. To Determine the Time of ITIeaii New, or Full Moon, in any Oiven mouth. Let the mean longitude of the sun, and also the mean longitude of the moon, at the beginning of the year, be found, and let the former be subtracted from the latter (addmg360° if necessary); the remainder, which call K, will be the mean distance of the moon to the east of the sun, at the beginning of the year. As the moon separates from the sun at the mean rate of.l2° 11' 26".70 per day, ~ will express the num- f •'' 12° 11' 26 '.70 ber of days and fractions of a day, which at this epoch have elapsed since the last new moon. This interval is called the Astronomical Epaxit. If we subtract it from 29d. 12h. 44m. 2 7s. we shall have the time of mean new moon in January. This being known, the time of mean new moon in any other month of the year results very readily from the known length of a lunar month. The time of mean new moon in any month being known, the time of mean full moon in the same month is obtained by the addition or subtraction, as the case may be, of half a lunar month. This problem is in practice most easily resolved with the aid ' of tables. (See Problem XXVII.) The time of true new moon differs from the time of mean new moon, for the same reasons that the true longitudes of the sun and moon differ from the mean. The same is true of the time of true full moon. For the mode of computing the time of true new or full moon from that of mean new or full moon, see Problem XXVII. 302. The Eartli goes through the same Phases, as viewed from the moon, in the course of a lunar month that the moon does to an inhabitant of the earth. But, at any given time, the phase of the earth is just the opposite to the phase of the moon. About the time of new moon, the earth, then near its full, reflects so much light to the moon as to reader the ob- scure part visible. (See Fig. 75.) MOON'S RISING, SETTING, AND PASSAGE OVER THE MERIDIAN. 303. To find the Time of tiie Meridian Passage of the Moon on a Given Day. Let S and M denote, respectively, the right ascension of the sun and the right ascension of the moon, at noon on the given day, and m, s, the hourly variations of the right ascension of the PHASES OF THE MOON. 183 sun and moon ; also let t = the required time of the meridian passage. At the time t the right ascensions will be, for the moon M. + (m, for the sun S + is; and, as the moon is on the meridian, the difference of these area will be equal to the hour angle t ; whence, <=M — S + t{m — s); or, if all the quantities be expressed in seconds, 3600 ^ ^ Thus, we find for the time of the meridian passage, 3600 (M-S) ,-.. ^ = 360o^;;r::^--"^^^^- The quantities M, S, m, s, are, in practice, to be taken from ephemerides of the sun and moon. Example. What was the time of the passage of the moon's centre over the meri- dian of New York on the 1st of August, 1837? Whoa it is noon at New York, it ia 4h. 56m. 4s. at Greenwich. Now, hy the English Nautical Almanat;, Aug. 1st, at 4h. 3's E. Asoen 8h. 58m. 36. '7s. " at5h. " " 9 38.3 Ih. : 56m. 4s. : : 2m. 1.6s. : Im. 53.68. Aug. 1st, at 4h. 3's B. Ascen 8h. 58m. 36. Ys. Variation of E. Aaceu. in 56m. 4s . . 1 53.6 3*3 R- Ascen. at M. Noon at N. York 9 30.3 Aug. 1st. 0's hourly Variation of R. Ascen 9.7048. Ih. : 4h. 56m. 4s. : : 9.704s. : 47.8s. Aug. 1st, M. Noon at Greenw., 0's R. Asc. ... 8h. 45m. 31.5s. Variation of R. Ascen. in 4h. 56m. 4s 47.8 0's E. Ascen. at M. Noon at N. York 8 46 19.3 Aug. 1st, M. Noon at Greenw., 3's E. Ase 8h. 50m. 27.7s. Aug. 2d, " " " 9 38 18.7 24)47 51.0 Aug. 1st, f5)'s mean hourly Varia. of R. Aso... 1 59.6 (m) " cb's " " "... 9.7 (s) m — s = 1 49.9 = 109.98. By Nautical Almanac, equation of time = 5m. 59s. Ih. : 5m. 59s. : : Im. 59.6s. : 11.9s. S)'s E. Ascen. at M. Noon at N. York 9h. Om. 30.3s. Correttion for equation of time — 11.9 n)'s E Ascen. at apparent Noon at N. York. . . 9 18.4 (M) e'8 " " " " ... 8 46 18.3 (S) M — S = 14 0.1 = 840.19, 184 THE MOOK AND ITS PHENOMENA. 3600 log. 3.55630 M — S = 840.1 log. 2.92433 3600 — (m — «l = 3490.1 ar. CO. log. 6.45716 Apparent time of meridian passage, 14m. 26.53. = 866.5s. . log. 2.93'?'79 Equa. of time at merid. passage, 5 58 Mean time of meridian passage, Oh. 20m. 243. The Nautical Almanac gives the time of the moon's passage over the meridian of Greenwich for every day of the year. From this, the time of the passage across the meridian of any other place may easily be determined, as follows : subtract the time of the meridian passage at Greenwich on the given day, from that on the fol- lowing day, and say, as 24h. : the diflerence : ; the longitude of the place : a fourth term. The fourth term, added to the time of the meridian passage at Greenwich on the given day, will give the time of the meridian passage on the same day at the given place. 304. Moon's Rising and Setting. Since the moon has a motion with respect to the sun, the time of its rising and setting must vary from day to day. When first seen after conjunction, it will set soon after the sun. After this it will set (at a mean) about 50m. later every succeeding night. At the first quarter, it will set about midnight; and at full moon, will set about sun- rise, and rise about sunset. During this interval it will rise in the daytime, and all along from sunrise to sunset. From full to new moon, it will rise at night and set during the day; and the time of the rising and setting will be about 50m. later on every succeeding night and day ; thus, at the last quarter, it will rise about midnight, and set about midday. 305. ]>ailf Retardation of Moon's Rising. The daily retardation of the time of the moon's rising is, as just stated, at a mean, about 50 minutes ; but it varies in the course of a revolu- tion from less than half an hour to one hour, in these latitudes. The retardation of the moon's rising at the time of full moon, varies from one full moon to another, in the course of the year, between the same limits. The reason of these variations is found in the fact, that the arc of the ecliptic (12° 11') through which the moon moves away from the sun in a day, is variously inclined to the horizon, according to its situation in the ecliptic, and therefore employs different intervals of time in rising above the horizon. This fact may be very distinctly shown by means of a celestial globe. It will be seen that the arc in question will be most oblique to the horizon, and rise in the shortest time, in the signs Pisces and Aries. Accordingly, the full moons which occur in these signs will rise with the smallest retarda- tion from day to day. These full moons occur when the sun is in the opposite signs, Virgo and Libra, that is, in September and October. They are called, the first the Harvest Moon, and the second the Hunter's Moon. The time of the moon's rising at these full moons will, for two or three days, be only about half an hour later than on the preceding day. 306. To find the time of the mooti's rising or setting on any given day. Compute EOTATION AKD LIBBATIONS OF THE MOON. 185 Ihe moon's semi-diurnal arc from equation (60), or (48), according as it is tie time of the apparent riaiog or setting, or the time of the true rising or setting, that is desired. Correct it for the moon's change of right ascension in the interval between the moon's passage over the meridian and setting, by the rollowing pro- portion, 24h : 24h + m — s :: semi-diurnal arc : corrected semi-diurnal arc ; and add it to tho time of the moon's meridian passage, found as explained in Art. 303. The result will be the time of the moon's setting ; and if this be subtracted from 24 hours, the remainder will be the time of the moon's rising. In consequence of the change of the moon's declination in the interval between its rising and setting, it would be more accurate to compute the semi-diurnal arc separately for the moon's rising, In computing the Semi-diurnal arc by equation (48), the decUnation 6 hours before or after the meridian passage may be used at first; and afterwards, if a more accurate result be desired, the calculation may be repeated with the declination found for the computed approximate time. In equation (49), R = refraction — parallax = 34' 54'' — 51' 3'' (at a mean) = — 22' 9". ROTATION AND LIBEATIONS OF THE MOON. 307. The moon presents continually nearly the same face towards the earth ; for the same spots are always seen in nearly the same position upon the disc. It follows, therefore, that it rotates on its axis in the same direction, and with the same angu- lar velocity, or nearly so, that it revolves in its orbit, and thua completes one rotation in the same period of time in which it accomplishes a revolution in its orbit. 308. Libratious of tbe moon. The spots on the moon's disc, although they constantly preserve very nearly the same situations, are not, however, strictly stationary. When carefully observed, they are seen alternately to approach and recede from the edge. Those that are very near the edge successively disap- pear and again become visible. This vibratory motion of the moon's spots is called lAhration. There are three librations of the moon, that is, a vibratory motion of its spots from three distinct causes. (1.) The moon's motion of rotation being uniform, small por- tions on its east and west sides alternately come into sight and disappear, in consequence of its unequal motion in its orbit. The periodical oscillations of the spots in an easterly and westerly direction irom this cause, are called the Libration in Longitude. (2.) The lunar spots have also a small alternate motion from north to south. This is called the Libration in Latitude, and ia accounted for by supposing that the moon's axis is not exactly perpendicular to the plane of its orbit, and that it remains con- tinually parallel to itself On this supposition, we ought some- times to see beyond the north pole of the moon, and sometimes beyond the south poie. (3.) Parallax is the cause of a third libration of the moon. The spectator upon tbe earth's surface being removed from its centre, the point towards which the moon continually presents the same hemisphere, he will see portions of the moon a little diiferent 186 THE MOON AND ITS PHENOMENA. according to its different positions above the horizon. The diurnal motion of the spots resalting from the parallax, is called the Diurrial or Parallactic Lihratwn. 309. Equator of tlie JTIoon. The exact position of the moon's equator, like that of the sun's, is derived from accurate observations of the situations of the spots upon the disc. From calculations founded upon such observations, it has been ascer- tained that the plane of the moon's equator is constantly inclined to the plane of the ecliptic under an angle of 1° 32', and inter- sects it in a line which is always parallel to the line of the nodes. It follows {rom the last mentioned circumstance, that if a plane be supposed to pass through the centre of the moon, parallel to the ecliptic, it will intersect the plane of the moon's equator and that of its orbit in the same line in which these planes intersect each other. The plane in question will lie between the plane of the equator and that of the orbit. It will make with the first an angle of 1° 32', and with the second an angle of 5° 9'. DIMENSIONS AND PHYSICAL CONSTITUTION OP THE MOON. 310. Diameter;— Surface;— Tolnine. The phases of the moon indicate that it is an opake spherical body. Its diameter is found by means of equ. (51), viz. : d = 2R A, 2H' in which d will denote the diameter sought, R the radius of the earth, S the apparent diameter of the moon at a given distance, and H its horizontal parallax at the same distance. The equatorial horizontal parallax of the moon, at the mean distance, is 57' 2".7, and its corresponding apparent diameter is 31' 7".0: thus we have d = 2B, J1L112 = 7925.6m. x J^!^ = 2161.6 miles. 114' 5".4 6845"4: The ratio of the diameter of the moon to the mean diameter of the earth (7912.4m.) is 0.27319. This is very nearly equal to ^, or a little more than i. The surface of the moon is therefore to the surface of the earth nearly as 3" to 11", or as 1 to 13J; and the volume of the moon is to the volume of the earth nearly as 3' to IP, or as 1 to 49. 311. Telescopic Appearances: — Inferences. When the moon is viewed with a telescope, the edge of the disc which bor- ders upon the dark portion of the face, is seen to be very irre- gular and serrated (see Fig. 76). It is hence inferred that the surface of the moon is diversified by mountains and valleys. The truth of this inference is confirmed by the fact that bright PHYSICAL CONSTITUTION OF THE MOON. 187 insulated spots are frequently seen on the dark part of the face, near the edge of the disc, which gradually enlarge until they become united to the disc. These bright spots are doubtless the tops of mountains illuminated by the sun, while the sur- Fia. 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. 312. Liunar Moniitaius. The number of the lunar moun- tains is very great. Many of them, by their form and grouping, furnish decided indications of a volcanic origin. From measurements made 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. Ac- cording to Herschel, the altitude of the highest is only about If English miles. But Schroeter, of Lilienthal, a distinguished Selenographist, makes the elevation of some of the lunar moun- tains to exceed 5 miles; and the more recent measurements of MM. Baer and Madler, of Berlin, lead to similar results. 313. There are 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 Heveliua Mare Imbriuw, Mare Orisium, 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 them. 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 in- ferred from the fact that the moon's face is never obscured by clouds or mists. 314, liUnar Atmosphere. It has long been a question among Astrono- mers, 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 dimin- ish or refract the light of a star seen in contact with the moon's limb ; for when a star experiencas 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 atmo- sphere. 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 whidi have been observed afford in- dubitable evidence of the presence of a certain limited body of air upon the moon's surface. Thus, the celebrated Schroeter, in the course of some delicate observa- tions 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. Tliis he conceived to be the moon's twilight, and hence inferred the existence of a lunar atmosphere. Prom the measurements which he 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 SrAt 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 comple- tion of the ring, at the point where the junction took place. DESCRIPTION OF THE MOON'S SUEPACE. 315. Oeiteral Topographical Features. The surface of the moon, like that of the earth, presents the two general varieties of level and moun- tainous 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 com- paratively 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 hmar plains vary in extent from 40 or 50 miles to '700 mUes in diameter. 316. The ITIouiitainoiis Forinatious of the other parts of the surface offer three marked varieties, viz. : (1.) Insviaied Mountains, which rise from plains nearly level, and which may be supposed to present an appearance somewhat similar to Mount 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 wheu 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 Centrai Mountains. (2.) Ranges of Mountains, extending in length two or three hundred mUes. These ranges bear a distinct resemblance to our AipSj 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. 1S9 surface. Some of them appear very rugged and preoipitoua, 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.) Oircula/r FormaMons. 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. : Walled PlaiTis, whose diameter varies from one hundred and twenty to forty or fifty miles , Hing Mountains, the diameter of which descends to ten miles ; and Craters^ which are still smaller. The term crater is sometimes extended to aU 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 in- ferior scale every attribute of the large one.f 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 tlian 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 mfle 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-flfths of the moon's visible surface. * Dick's Celestial Scenery, p. 257. f Niohol's Phenomena of the Solar System, p. 167. 190 ECLIPSES OP THE SUN AND MOON. CHAPTEK XVI. Eclipses of the Sun and Moon. — Occultations of the Fixed Stabs. 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 eclipses of the sun, of the moon, and of the satellites ; and, with respect to cir- cumstances, into tot(xl, partial, an- nular, and central. A total eclipse is one in which the whole disc of the luminary is darkened; & par- tial one is when only a part of the disc is darkened. In an annular eclipse the whole is darkened, ex- cept a ring or annulus, which appears round the dark part like an illuminated border; the defini- tion of a central eclipse will be given in another place. ECLIPSES OP THE MOON. 318. An eclipse of the moon is occasioned by an interposition of the body of the earth directly between the sun and moon, and thus intercepting the light of the sun ; or the moon is eclipsed when it passes through part of the sha- dow of the earth, as projected from the sun. Hence it is obvious that lunar eclipses can happen only at Fig. M. the time of full moon, for it is then only that the earth can be between the moon and the sun. 319. Earth's Shadow. Since the sun is much larger than CIRCUMSTANCES UNDER WHICH AN ECLIPSE OCCURS. 191 the earth, the shadow of the earth must have the form of a cone, the length of which will depend on the relative magnitudes of the two bodies and their distance from each other. Let the circles AGE, agb (Fig. 77), be sections of the sun and earth by a plane passing through their centres S and E ; Aa, B6. tangents to these circles on the same side, and Ad, Be, tangents on differ- ent sides. The triangular space aCh will be a section of the earth's shadow or Umbra, as it is sometimes called. The line EC is called the Axis of the Shadow. If we suppose the line cp to revolve about EC, and form the surface of the frustum of a cone, of which pcdq is a section, the space included within that surface and exterior to the umbra, is called the Penumbra. It is plain that points situated within' the umbra will receive no light from the sun ; and that points situated within the penum- bra will receive light from a portion of the sun's disc, and from a greater portion the more distant they are from the umbra. 320. To find the Length of the Earth's »ihadow. Let L = the length of the shadow ; E = the radius of the earth ; 3 = the sun's apparent semi-diameter, and p = sun's parallax. The right-angled triangle BaC (Fig. 77) gives EC = ^?_. sin ECa Ea = R; and ECa = SEA — E AC = 8 — p ; whence, L = -—^ (54). As the angle (5 — p) is only about 16', it will differ but little from its sine, and therefore, L = R -J (nearly) ; or, if 8 and p be expressed in seconds, L = R?^^-:::?(nearly)....(55). h—p The shadow will obviously be the shortest when the sun is nearest to the earth. We then have 5 = 16' 18", and p = 9", which gives L = 213 K The greatest distance of the moon is 65R. It appears, then, that the earth's shadow always extends to more than three times the distance of the moon. 321. Circumstances under which an Eclipse occurs. Let TMh be a circular arc, described about B the centre of the earth, and with a radius equal to the distance between the cen- tres of the earth and moon at the time of opposition. The angle MEm, 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-diameter of the Earth's Shadow. And the angle MBA, the apparent semi-diameter of a section of the penumbra, at the same distance, is called the Semi-diameter of the Penumbra. 192 ECLIPSES OF THE SUN AND MOON. Fw. 18. 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 orbit, or so near it that the moon's latitude does not exceed the sum of 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'O represents a portion of the ecliptic, and N'M a portion of the moon's orbit, N' the descending node, E the earth, ES, ES', ES", three dif- ferent directions of the sun, s, s', s", sections of the earth's shadow in the three several positions correspond- ing to these directions of the sun, and m, m', m", the moon in opposi- tion. It will be seen that the moon 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. U22. Calculation of Semi-diameter of Sbadow. 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, and let P = the moon's parallax. MEm = 'Ema — EOm (Fig. 77) ; but Ema = P and ECm = S—p(320); therefore, MEm = A = 'P + p — 8.. . .(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 S its maximum value. In these positions of the moon and sun, P = 52' 40", S = 16' 18", and p = 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 great- est apparent diameter of the moon is 33' 32". "Whence it ap- pears that the diameter of the earth) s shadow is always more than twice the diameter of the moon. The means of the greatest and leabt values of P and 5 are, re- spectively, 57' 11" and 16' 2" ; which gives for the mean semi- diameter of the earth's shadow, 41' 18". 323. L,nnar Ecliptic Liimits. If to P -f--^ — 5, the semi- diameter 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 oi 7 + p + d — 5, and a given inclina- tion 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 oi P + p + d — S, 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 13° 21' the hap- pening of the eclipse is doubtful. These numbers are called the Lunar Ecliptic Limits. To determine at what full momis in the course of any one year there will he 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 differ- ence 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 semi- diameter 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 3f h. 325. Parlicniar 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, and thus the apparent diameter of the earth's shadow, is found by observation to be somewhat greater than would result from this supposition. This circumstance is accounted for by supposing 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 larger. 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, ex- periences 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 pre- cision, and in consequence astronomers differ as to the amount of the increase in the size of the earth's shadow from the cause above mentioned. It is the practice, however, in computing an eclipse of the moon, to increase the semi-diameter of the shadow by a ^ part ; or, which amounts to the same, to add as many seconds as the semi-diameter 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 atmosphere, though of a totally different nature from the preced- ing. 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 taint light, in- terior 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 showa that the case is different with eclipses of the sun. CALCULATION OF AN ECLIPSE OP 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. We may also consider the apparent motion of the sun in longitude, and the motions of the moon in longitude and latitude, as uniform during the eclipse. These suppositions being made, the calculation of the circum- stances of an eclipse of the moon is very simple. 827. Relative Orbit. Let NF (Fig. 79) be a part of the ecliptic, 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 CO', which is the latitude of the moon in opposition, is the distance of the centres CALCULATION OF AN ECLIPSE OP THE MOON. 195 of tbe 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 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 cir- cumstances of the eclipse depend altogether upon the relative riiotion of the moon and shadow, this mode of proceeding is obvi- ously 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 oppo- sition, bv laying off Cm equal to the difference of the motions of the sun and moon in longitude in this interval, through m draw- ing mM perpendicular to KF, 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 CW (= 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 fol- lows, therefore, that the relative orbit of the moon N'C'M is ti 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 . ,. Mm' moon's motion in latitude tan inolma. = - _— - = —-. — , --:— -| m moon s mot. m long. — sun's mot. in long. 196 ECLIPSES OF THE SUN AND MOON. 328. Requisite Data. The following data are requisite in the ealoulation of die oiroumstanoes of a luuar eclipse : T = time of opposition. M = moon's hourly motion in longitude. « = moon's hourly motion in latitude. TO = sun's hourly motion in longitude. A = moon's latitude at opposition. d — moon's semi-diameter. i = 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 orhit. T, M, -rt, m, X, d, (i, P, and p, are derived from Tables of the sun and moon. (See Pioblems IX and XIV.) The quantities s, I, and ft, may be determined from these : s = P + p — ci-t-^(P +p — l){322 and 325).. ..(67); tangi = — -— (327). . . .(58). M — m The triangle O'Mm' gives 0'M = - O'ot' ,or,A = ^-^....(59). cos I COS MO'm' S29. Process of Calculation. The above quantities being supposed to be known, let N'CF (Pig. 80) represent the ecliptic, and C the stationary centre of the earth's shadow. Let CO' = A, and let N'O'L' represent the relative orbit of the _K.B FiS. 80. moon. We here suppose the moon to be north of the ecliptic at the time of oppo- sition and near its ascending node ; when it is south of the ecliptic A is to be laid off below N'GP, and when it is approaching either node, the relative orbit is in- cliaed to the right. Let the circle KPK'B, described about the centre C, repre- sent the section of the earth's shadow at the moon; and let/, /', 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. Gf=Gf':=s + d, and Og = Og'= s — d. Draw CM perpendicular to N'O'L', and M will represent the place of the moon's centre when nearest the centre of the shadow : it wiU also be its place at the middle of the eclipse ; for since C/= 0/', and CM is perpendicular to H'C/' kf= M/'. Middle 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 (as) employed by the moon in passing from M to C Now f CALCULATION OF AN ECLIPSE OP THE MOON. 197 (expressed in parts of an hour) x = — ; h uid in the right-angled triangle CC'M we have 00' = X, and < COM = < O'N'O = ], and therefore MO' =: X sin I ; whence, by substitution, X sin I X sin I X sin I cos I ^ = -ft- = M=^ (^i-^ 53) = -ii=-iir ! COS I or (expressed in seconds), x = '. — ?_ . X sin I . . (60), M — m Hence, if M := time of middle, we have M^T T :c = T T ?5^-^2!i.XsmI....(61). M TO It is obvious that the vpper 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, = CM = CO' cos COM = X cos I (62). Beginning and end of the edipse. 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 = Gt, the d.s- tance between the centres of the moon and shadow. In the interval i the moon's centre will pass over the distance Ml ; hence m Ml cos I t- h M — m but, m— \/'q^_ ilff = \/ jc' — k^ cos'I (equa. 62), and therefore t = ^— V A= _ /,> cos" I >' M — m " ,. j\ V 3600s. cos I , or (in seconds), « = ____ ^(j. ^ ^ ^^ j^ g^_^ ^^ j^ _ _ (g3^_ Let T' denote the time of the supposed phase of the eclipse, and M the time of the middle ; and we shall have T' = M + /, or, T' z= M — i, according as the phase follows or precedes the middle. Now, at the beginning and end of the eclipse, we have, k = Gf0TCf'-s + d: substituting in equation (63) we obtain ., 3600s. cos I , ~ M — m •♦/(s+d + X cosI)(s+d — xcosi) (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 iolcU eclipse. At the beginning and end of the total edipse, k — Gg = Gg' = s.^— d; whence, by equation (63), J,, 3600s. cos I , /gr\. t =. — ^ _^ |/(s— d + Xcos !)(« — d— X cos I) \^^)- and, denoting the time of the beginning by B' and the time of the end by E', we have B' = M—t",K=M + t". Quantity of the edipse. In a partial eclipse of the moon the magnitude or quan- tity of the eeUpse 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 SUK AND MOOIT. of a digit in the part immersed. "WTien the moon passes entirely within the sha- dow, as in a total eclipse, the quantity of the echpse is expressed by the uumbtr of digits contained in the part of the same diameter prolonged outward, which la comprised between the edge of the shadow and the inner edge of the moon. Thus the number of digits contained in SN (Pig. 80) expresses the quantity of the eclipse represented in the figure. Hence, if Q = the quantity of the eclipse, we shall have ^ _ NS _ 1 2NS ^ 12fflM + MS) _ 12 (KM + C S — CM) _ NV T^NV KT 12 (d + s- ■ > cos I) . 2d or, „ _ 6 (s + d — X cos I) y d . .(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 wiU be total COHrSTRUCTION OP AN ECLIPSE OP 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 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 EL' = X ± «, 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 right 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 inter- secting N'L' in/,/', and g, g' ; f and/' 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 M 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 7b. 23ra. 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 0' backward to the point 7, it will give the moon's place at that hour. This being found, take in the dividers the moon'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, 8, 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/y, 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 pointy falls between 9 and 10, and tiius 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 containsy 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 OP THE SUN. 331. Linniinous Frustum and Coue. 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 and earth by a plane passing through their centres S and E ; Aa, B&, tan- gents to the circles AGB and agb on the same side; and Ad, Be tangents to the same on opposite sides. The figure AaJB will be a section through the axis, of a frustum of a cone formed by rays tangent to the sun and earth on the same side, and the tri- angular space Ycd will be a section of a cone formed by raya 200 ECLIPSES OF THE SUN AND MOON. tangent on opposite sides. An eclipse of the sun will take place somewhere upon the' eaith's surface, whenever the moon comea within the frustum AabB, and a total or an annular eclipse when- ever it comes within the cone Fed. Fig. 82. and m'ES 332. Semi-diameters of Frii!«tnm and Cone. Let mm'M. (Fig 82) be a circular arc described about the centre B, 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 Tnoon. To find expressions for these semi-diameters in terms of determinate quantities, let the first be denoted by A, and the second bj A' ; and let P =^ the parallax of the moon, p = the parallax of the sun, and ^ = the semi-diameter of the sun. Then we have mES = A = toEA + AES = Ema — EAm -fAES; or, A = P— p + 5.-..((57}: wi'EB — BBS = Em'c — EBm' — BES ; A' = P— p — 3....(68). Taking the mean values of P, p, and 5 (322), we find for the mean value of A, 1° 13' 3" ; and for the mean value of A', 41' 1". 333. Circumstances of moon's Position in Solar Eclipses. 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 nodes of the moon's orbit, or so near it that the moon's latitude does not exceed the sum of the semi-diameters of the moon and luminous frustum at the moon's orbit. This may be illus- trated by means of Fig. 78, already used for a lunar eclipse, by supposing the sun to be in the directions Es, Es', E.s", and that s, «', s", are sections of the luminous frustum corresponding to these directions of the sun ; also that m, m\ m", represent the moon in the corresponding positions of conjunction. Thus, dei or, NUMBER 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 + 5 + cZ....(69). For a total eclipse, 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— i)— 5+(Z....(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 au annular eclipse is possible, we have -L"=V-p + h — d....{n). 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. Prediction of Eclipses :— Period. In order to dis- cover at w^hat 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 longi- tudes 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 29.5305887d. 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 commence- ment. The eclipses which occur during one such period being noted, subsequent eclipses are easily predicted. This period was known to the Clialdeans and Egyptians, by whom it was called Saros. 3.35. JViimber of Eclipses in a Year. As the solar eclip- tic limits are more extended than the lunar, eclipses of the sua must occur more frequently than eclipses of the moon. • More exactly, 18 years (of 365 days) plus 16d. 1h. 42m. 293. 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 year, 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 34:6 days. Now, if the sun be at a little less distance than 19° 44' from either node at the time of mean new moon, he may be eclipsed (333), and at the subsequent op- position 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 days 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 conse- quence 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 tbe inter- val the sun will move over an arc of 14° 32', whereas the great- est 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 the 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 hemisphere of the earth. To show this, suppose two lines to be drawn from 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 arc of more than 1 °. MOOlirS SHADOW CAST UPON THE EABTH. 203 336. moon's ShadoTir Cast upon the Earth. Instead of regarding an eclipse of tlie 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 moon's shadow falling upon the earth. Fig. 83 represents the moon'a shadow, as projected from the sun and covering a portion of the earth's surface. Wherever the umbra falls, there is total eclipse ; and wherever the penumbra falls, a partial eclipse. 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 tnoorHs shadow is about equal to the distance of the earth, being some- times 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 ■^, 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 32' 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 fiiid a General Expression for the r>engtb of the iTIoon's Shadow. Let AGB, a'g'h', and agh (Fig. 84) ba sections of the sun, moon, and earth, by a plane passing through Fia. 84. their centres S, M, and E, supposed to be in the same right line, and Aa', B6', tangents to the circles AGB, a'g'h' : then a'KJb' 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 i = the apparent semi-diameter of the sun. At K the vertex of the shadow, MKa' the apparent semi-diameter of the moon, will be equal to SKA 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 situ- ated at the centre of the earth. Now, since the apparent semi- diameter of the moon is inversely proportional to its distance, angle MKa' : c^ : : MB : MK; and thus, ^ : cZ : : ME : MK' : : D : L (nearly) : whence, L = D^ . . . . (72). K a more accurate result be desired, we have only to repeat the calculation, after having diminished S in the ratio of D' to (D' + L — D). 33§. To find the Breadth of the Penumbral Shadow cast upon the earth, let the lines Ad', Be' (Pig. 84) be drawn tangent to the circles AGB, a'g'V, on opposite sides, and prolonged to the earth. The space hc'dJk 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 gWi = 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 MEA gives angle MEA = S = MftZ — AME. The angle /iME is the moon's parallax in altitude at the station h, and MftZ is ts zenith distance at the same station. Denote the former by P' and the latter oy Z. Thus, ^-Z — V....{n). the triangle AMS gives ME = P' = MSft + MAS; LENGTH AND BREADTH OF MOON'S SHADOW. 205 MAS = d + S; and MSft is the sun's parallax in altitude at the station h: let it be denoted by p'. Wo have, then, 'P' — d + i+p' = d+6 (nearly) (74); and to find Z we have (equa. 9, p. 63), P' = P Bin Z, or sin Z = ?. (15). P' and Z being found by these equations, equa. (T3) 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: sinZ = ?i5Z....(76). sinP The penumbral shadow will obviously attain to its greatest breadth when the sun is at its least and the moon is at its great- est distance. The values of d, d, and P under these circumstances are respectively 14' 24", 16' 18", and 52' 60". Performing the calculations, we find that the breadth of the greatest portion of the earlKs surface ever covered by the penumbral shadow is 70° 17', or about 4,850 miles. 339. The Breadth of the ITinbra may be found in a simi- lar manner. The arc gh' (Pig. 84) represents one half of it : denote this arc or the equal angle pEA' by S'. MEA' = S' = MA'Z' — A'MB ; or, S' = Z — P'....('r7). A'MB = F = MSft' + MA'Si but MA'S = d — 6, and MSA' =p', the sun's parallax in altitude at A'; whence^ P' = d—i +p' = d — a (nearly) (78): and we have, as before, P' = P sin Z, or sin Z = (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 i = 15' 45", d=16' 46", P = 61' 82". Making use of these numbers, we deduce for the maximum Ireadih of the portion of the eartKs surface covered by the moon's shadow, 1° 54' ; or 130 miles. It should be observed that the deductions of the last two arti- cles 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. 340. Phases of Eclipse Different at each Place. Parallax not only causes the eclipse to be visible at some places and invisi- ble at others, as shown in Art. 335, but, by making the distance 206 ECLIPSES OF THE SUN AND MOON. between the centres of the sun and moon unequal, renders tbe cir- cumstances of the eclipse at those places where it is visible, differ- ent at eacli place. This may also be inferred from tbe 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 consid- ered in two points of view : 1st. With 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 hap- pen 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 laier at the eaatem. 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 set- ting 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 ecUpse of the sun begins on the western side and ends on the eastern. 5th. "When the straight line passing through the centres of the sun and moon passes also through the place of the spectator, the ecUpse 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 and a half minutes. 1th. 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 Uluminated part of the earth. CALCULATION 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 corre- sponding 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 sim- ple and direct, are given in tbe Appendix. OCCTJLTATIONS. 343. An occultation is an eclipse, or deprivation of the light of a star, resulting from 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 amount 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. 20T 344, Limits of Position of Stars Liiable to Occultation. The greatest value of the apparent semi-diameter of the moon la 16' 46", and that of its horizontal parallax is 61' 32". If we add the sura of these quantities to 5° 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° 38' 24" that can experience an occultation from the moon. In order that any of the stars situated within this distance from the ecliptic may sufier 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-dia- meter and horizontal parallax of the moon. 208 THE PLANETS, CHAPTER XVII. The Planets, and the Phenomena occasioned by their Motions in Space. APPARENT MOTIONS OF THE 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 inferior planets always accompany the sun, being seen alter- nately 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 from 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. ^316. Apparent OTotion of an Inferior Planet. Let CJiC'B (Fig, 85) represent the orbit of either one of the inferior MOTION OF A SUPERIOR PLANEl. 209 planets, Venus for example, and PKT the orbit of the earth — 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 being 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 Morhmg 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 alter- ing 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 elonga- tion of Venus varies from 45° to 47° 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 cir- cle 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° distantfrom the sun : and when it reaches B, the elongation is again 90°. ' At intermediate points, the elon- gation will have intermediate values. If, now, we restore to the 210 THE PLANETS. planet its orbital motion, we shall manifestly be conducted to the same results relative to the change 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 con- junction 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 conjunc- tion, 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. 34§. To find tbe Length of the ^Synodic Revolution of a Planet. Let us first take an inferior planet, Venus for instance. Suppose we assume, at a given instant, the sun, Venus, 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 (wi — M) ; at the end of s days, s {m — M). When the angle of separation amounts to 360°, that is, when s (m — M) = 360°, the sun, planet, and earth must be again in the same right line, and in that case . = _^....(80). m — M 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 dbiain from equation (80) another equation, in which • the synodic revolution is expressed in terms of the sidereal peri- ods of the earth and planet. Let P and^ denote the sidereal periods in question, then, since Id, : M° : : P : 360°, and 1 : m : : p : 360 ; M = , and m = ; substituting, P p 360° _ P^ ..(81). Equations (80), (81), although investigated for an inferior STATIONS AND EETEOGRADATIONS OF THE PLANETS. 211 planet, will answer equally well for a superior planet, provided we regard m as standing for the mean daily motion of tiie eartli, M for tiiat 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 ob- viously be a synodic revolution of the planet to an observer on the earth. 349. I,engtli8 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 revo- lution is, for an inferior 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 revolu- tions 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 approx- imately the time of its rising, setting, and meridian passage. 350. Planets as Evening or iTIorniug Stars. A planet will rise and set at the same hours at the end of a synodic revo- lution; 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 inter- val 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 9^ months ; Mars for 1 year and 1 month ; Jupiter for 6^ months ; Saturn and Uranus each a few days more than 6 months. STATIONS AND RETKOaRADATIONS 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 i-etrograde. 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 Betrogradations of the planets. We now propose to 212 THE PLANETS. inquire theoretically into the particulars of the motions in que* tion, and to show how the phenomena just mentioned result from the motions of the planets in connection with the motion of the earth. 352. Ca§e 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 ta a; that, while it was moving from A to B, that is, from greatest eastern to greatest western elongation, it would retro- grade in the heavens from atob; 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 Fio. 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 EETROGRADATIONS OP THB PLANETS. 213' ward in its orbit a certain distance A'D'; s6 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 con- tinue ; 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, m'p', at the begin- ning 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 dimin- ishing, the lines of direction of the planet at the beginning and end of the day will diverge, and therefore the motion will be re- trograde. 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 from 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 represent the positions of the planet and the earth at the time of the western station ; G'q = G'p, and Pn=Pm. The diminution of the elongation of the planet at its two sta- tions is not the only effect of the earth's motion in the case under consideration ; it also accelerates the direct, and retards the retro- grade motion of the planet, and gives to the planet along Vi^ith 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 ATB' 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 pG'q in the inferior part of its orbit, and a direct motion during the remainder of the synodic revolu- tion. 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 mo- ' tion during a small portion of its synodic revolution, about the time of opposition. (See Table V.) 2U THE PLANETS. PHASES OF THE INFEEIOR 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 posi- tions 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 con- junction. 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. 335. Explanation. The phases of Venus are easily account- ed for, by supposing it to be an opake spherical body, and to shine by reflecting the sun's light, and by taking into considera- tion its motion with respect to the sun and earth. The hemi- sphere turned towards the sun is illuminated and the other is in the dark, and as the planet revolves around the sun, various portions of the enlightened half are turned towards the earth ; in superior conjunction, the whole of it ; at the greatest elongations, one half; and near inferior conjunc- tion, but a small part. This will be abundantly evident on inspect- ing Fig. 87. The phases corres- ponding to the positions represent- ed are delineated in the figure. The phases of Mercury are ob- viously suceptible of a similar ex- planation. 356. 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 would not see the whole of the enlightened hemisphere, except in con- junction 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 Fm. 87. 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 OP 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. 358. Epochs of Transits: — Periods of Recurrence. The transits of Venus take place alternately at intervals of 8 and 105^ or 121^ 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 Venus. 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 finst, which will occur on the 4th, will be visible in this country. 359. A Transit is Calculated in a precisely similar man- ner with a solar eclipse ; the planet in the one calculation answering to the moon in the other. 360. A Transit is "an Important Pbenomenon in a practical point of view, as it furnishes an indirect but accurate method of ascertaining the sun's parallax. In order to under- stand how this phenomenon can be used for this purpose, wa have only to consider that, in consequence of the difference of Fnj. 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 accom- plished in unequal portions of time. This fact is represented tc the eye in Fig. 88. It is then to be expected, that, if the dura- tions 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 par- allaxes 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. 361. The Parallax of the Sun was quite accurately de- duced 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 com- puted 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 ap- parent 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 ap- parent 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 inagnltndes. The real dia- meter 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 princi- pal 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 Mars, 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 L.iglit 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 atmospheres. MBRCUET. 366. In 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 magni- tude. 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 ^ of that of the earth ; and its volume is about -^ of the earth's volume.* Mercury performs a complete rotation on its axis in 24h. 5|-m., 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. Schroter is the only observer who has supposed that he discerned distinct spots upon it. Later observ- ers 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 in- ferred that the planet is surrounded with a dense atmosphere loaded vrith clouds, that reflect a strong light, and, except when the atmosphere clears up in an un- usual degree, prevent the darker body of the planet from being seen. But the evidence in support of this conclusion needs confirmation. Schroter, in making observations upon Mercury at the time the disc had the "orm Of a crescent, discovered that one of the horns of the orescent became blunt at *The exact diameters, volumes, times of rotation, &c., of the different planets, aa .'ar 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 calculation, 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 thaa compensates for the diminished size of the disc. Venus attains to greater splendor in some revolutions thaa in others, in consequence of being nearer tile earth when in the favorable position just noticed. A com- 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 uniform brightness, and spots have very rarely been seen upon it. From the regular succession of phases through which the disc passes, as the planet changes its position with respect to the earth and sun, we infer that it is an opake spherical body, shin- ing 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 7,60(J miles ; and its volume ^ less than that of the earth. The period of its rotation is 23h. 2im. The inclination of its axis to the plane of its orbit is not exactly known, but is supposed to be not far from 18°. 369. Evidences of an Atmospbere surrounding Tenus. 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 Schroter. This Astronomer distinct- ly discerned, when the disc was seen as a nar- row 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 FlQ. 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 Ly. man, with the nine-inch equatorial of the Sheffield Observatory, Tale 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 momenta, 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 seeu also, by the same observer, with a five-foot telescope, so placed as to have the sun covered by a distant chimney. The maximum extent of the cres- cent observed at Dorpat, at the conjunction in 1849, was 240" j the planet being 3° 26' from the sun's centre. This remarkable prolongation of the cusps must be attributed mainly to the re- fraction of the sun's rays by the atmosphere of the planet. On this assumption, Madler, from the Dorpat observations of the extent of the cusps, made the hori- zontal atmospheric refraction of Venus 43'.7. The observations of Professor Lyman, at the late conjunction, givo 45'. 3. This is about i 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. Clouds in the Aimosphtre. Since the transparency of Venus's atmos- phere 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 aque- ous 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 some- times been noticed on the disc, is, that they are clouds, more highly reflective than the atmosphere, or than the clouds in general. 371. Inequaliiies on the Surface. There are great inequalities on the sur- face of Venus, and, it would seem, mountains, much higher than any upon our globe. Schroter detected these masses by several infallible marks. In the first place, the edge of the enlightened part of the planet is shaded, as seen ia Figs. 89, 90, 91, and as the moon appears when In crescent even to the naked eve. Fie. 90. Jio. 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 appears detached from the disc — the top of a high mountain, illuminated across a dark valley. Schroter found that these appearances recurred regularly, at equal intervals of about 23i hours ; the same period as that which Oassini had previously found fot the completion of a rotation, by observations upon the spots. 220 THE PLANETS. MARS. 378. 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 form 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 in- clined to the plane of the ecliptic in an angle of about 60°. It appears, from measurements made 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 deter- minations its oblateness (105) is -jL. 373. Telescopic Appearances : — Inferences. 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 coDJectured to be contineDts and seas. In fact, Sir J. P. W. Herschel has on several occasions, in exam- ining this planet with a good tele- scope, noticed that some of its spots are of a dull red color, wliile others have a greenish hue. Tho 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 bright spot at tfie top is at one of the poles of Mars. At other 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 Fi&. 92 are reduced in size, and some- times 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 Mars are seen with different degrees of distinctness at different times, and sometimes disappear, either partially or entire- ly ; parts of the disc also appear at limes 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 mountains have been detected upon Mars. 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 AND 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. 3T5. The ruddy color of the light of Mars has generally been attributed to its atmosphere, but Sir John Herschel finds a suflicient cause for this phenomenoB m the ochrey tinge of the general soil of the planet. JUPITER AND ITS SATELLITES. 376. Jupiter is the most brilliant of the planets, except Yenus, 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., o.H"^- The polar diameter is A- less than the equatorial. 377. Belts of Jupiter. When 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 appear- ance 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 situa- tion 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 William Herschel even saw them, on one or two occasions, broken up and distributed over the whole face of the planet; but this phenomenon is extremely rare. Branches runnmg out from the belts, and subdivisions, as represented in the figure, are by no means uncommon. Dark Spots 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, after the same manner as the sun's spots; which leaves no room to PlO. 93. 222 THE PLANETS. doubt ttat tbey ure on the body of the planet, and that this turns upon an axis. Bright Spots have also recently been detected upon the belts by two observers ; Dawes and Lassell. The belts generally retain pretty nearly the same appearance for several months together, but occasionally marked changes 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 Elanet seen through an atmosphere of variable transparency, but, in general, aving extensive tracts of comparatively dear sky in a direction parallel to the equator. These are supposed to be determined by currents analogous to our trade- winds, but of a much more steady and decided character, as would be the neces- sary consequence of the superior velocity of rotation of this planet. As remarked by Hersohel, 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 bolts 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 planet3: 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 reflect- ed 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 spot- belts of the sun, with 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 tem- perature 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. 3'78. The Satellites of Jupiter, as it has been already re- marked, 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 moon. The orbits of the satellites lie very nearly in the plane of Jupiter's equator. They are, there- fore, viewed nearly edgewise from the earth, and in consequence the satellites always appear nearly in a line with each other. Sir W. Herschel, in examiaing 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 tbe primary, and therefore, like the moon, presented continually the same Etce to the primary, SATURN, WITH ITS SATELLITES AND EING. 379. Saturn shines with a pale dull light. Its apparent diam- eter 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 diame- ter 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 per- formed in lOh. 29m. The inclination of its axis to the ecliptic is about 62°. Its oblateness is -jij-. 380. Belts of Saturn. The disc of Saturn, like that of Jupiter, is frequently crossed with dark bands, or belts, in a di- rection 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 Horschel's observations on the occultations of the satellites by the planet, indicate the existence of a dense atmosphere. 381. Saturn's Ring. The planet Saturn is distinguished 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°, and during the motion of Saturn in its orbit re- mains continually paral- lel to itself. The face of the ring is, therefore, never viewed perpendic- ularly from the earth, and for this reason nev- er appears circular, al though such is its actual form. Its apparent form is that of an ellipse, more or less eccentric, accord- Fic. 94. ing to the obliquity un- der 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 obliqui-" ty, 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 tele- scopes of high power. When it is approaching the latter state, it has the appearance of two handles or ansoe^ 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. When the plane of the ring passes through the centre of the sun, the illuminated edge can be seen only in telescopes of extraordi- nary power, and appears as a thread of light cutting the disc of the planet. 383. Circumstances of Disappearance 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 and 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 consider- able or great power. All this will perhaps be better understood on consulting Fig. 95, where efg represents the orbit of the earth. Pm. 95. The appearances of the ring in the different positions of the planet 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 ITO" and 350°. The ring will then disappear once in satuen's ring. 225 about fifteen years; but, owing to tbe different situations of the earth in its orbit, under varied circumstances: and the disap- pearance 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 80=, or 2f;0°. 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. 3§3. Rotation of Ring.— I>iineii8ion§. From observa- tions made upon bright spots seen on the face of the ring, Her- schel 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 lOh. 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 dis- tance 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. 384. Divisions of tbe Ring. What we have called Sa- turn'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 tbe 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 -^"^ 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 im- proved 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 Ring of Saturn, interior to the other two, was discovered by G. P. Bond, then assistant at the Observatory at Harvard College, on the 11 th of Novem- ber, 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 palo 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 now was con- nected with the old ring or not. The same appearances were noticed by Dawes, at his observatory near Maid- stone, England, on the 25th and 29th of November, and subsequently by Lassell, with his large reflector, at Starfleld, 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 ansse have not disappeared or reappeared at the same time. He has also found, from a discussion of the observatious which have been made upon the disappear, auces and reappearpnces 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 elUpse instead of a rectangle, and that it varies somewhat in sisM 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 microme- tric measurements of groat 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 shghtest 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 tht planet. 388. Physical Comtitution 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 untU the sources of disturbance are removed, when the parts thrown off would again reunite. Profes- for Pierce has followed up the speculations of Bond, by undertaking to demon- strate from purely mechanical considerations, that Saturn's ring cannot be soUd. 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 per- manently 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 equili- brium about the planet, solely by the attractive power of his satelUtes ; and that no planet can have a ring unless it is surrounded by a sufficient number of proper- ly arranged satellites. Upon the theory of the development of heat by collision with the ether of space (3T7), 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 act- ing forces. The latter supposition accords with Laplace's theory of the progres- sive 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. AH these but the 1st and 2d, are visible in a telescope of large aperture, with a magnifying 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 Joho Herschel, are given in Table VI. All of Saturn's satellites, with the exception of the Sth, re- volve very nearly in the plane of the ring, and of the equator of the primary. The orbit of the 8th is inclined under a con- siderable angle to this plane. The 6t,h satellite is much the larg- est, 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. TJEANUS AND ITS SATELLITES. 391. The planet now known by the name of Uranus, was dis- covered by Sir William Herschel. It is not visible to the nnked eye, except iu 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 com- parison 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 Her- schel, 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 out- etanding inequalities in the motion of this planet, which could not be referred to the action of the other planets, Le Verrier, the 228 THE PLANETS. eminent French astronomer, undertook in 1845 the problem of determining the orbit and mass of a planet capable of producing such inequalities. Tbe same problem was independently under- taken and successfully solved by Mr. Adams, of Cambridge, England. 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 tele- scope of tbe 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 36,000 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. la the tele- scope, 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 magni- tude. Pallas is the largest of this class of bodies. According to Lamont, Director of the Royal Observatory, Munich, its diame- ter is 670 miles. The diameter of Vesta is believed not to exceed- 300 miles ; and that of Ceres to be somewhat smaller. Juno ia the smallest of the four planetoids first discovered. All of the other minor planets are supposed to be less than 100 miles in diameter. COMETS. 229 CHAPTER XVm. Comets. THEIR GENERAL APPEARANCE :— VARIETIES OP ,': APPEARANCE. 394. The general appearance of comets is that of a mass of some luminous nebulous substance, to which the name CoTHia has been given, condensed towards its centre around a brilliant nucleus that is in most cases not very distinctl3' defined ; from which proceeds, in a direction opposite to the sun, a stream of Fig. 96. similar but less luminous matter, called the Tail or Train of the comet (Fig. 96). The nucleus, with the surrounding coma, forma 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 fadea away until it is lost in the general light of the sky. It is, in 230 COMETS. 1/ f /; genera], 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 dark 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 comet has just left. This curvature of the tail is most observable near its extremity. Tbe most remarkable example is that -of the comet 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 great circle passing through the sun and the head of the comet, but deviata'? more or less from this, the position of exact op- position to the sun in the heavens, 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. Variations of Length of Tail. The apparent length of the tail varies, from one comet to another, from zero to 100° and more; and ordinarily the tail of the same comet increases and diminishes very much in length during the period of its visibility. When a comet first ap- pears, in general no tail is perceptible, and its light is very faint. As it approaclies 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 grows less and less at the same time that, along with the head, it decreases in bright- ness, till at length the comet resumes nearly its first appearance, and finally disappears. (See Fig. 98.) It sometimes happens that, owing to Fis 97 peculiar circumstances, a comet does not make Great Comet of I8i3. ^^ appearance in the firmament until after it has passed the sun in the heavens, and not unt'l it has attained 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 APPEAEANCE. 231 396. Effects of the Position of tiie Earth, on tbe appar- ent size and brightness of a comet. The tail of a cornet is the longest, and the whole comet is intrinsically the most 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 Fio. 98. this, let ahcd (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 o or 6 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 M, while the earth is moving around from a to 5 or c; so as to be continually comparatively remote from the comet, and so that the comet will be in conjunction with tbe 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 bj' the great obli- quity of the tail, in some of its positions, to the line of sight. This seems to have been the case with tlie comet of 1843, on February 28 (see Fig, 63), and, it has been already intimated, 232 COMETS. Wiis 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 have 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 ap- pearance and disappearance of the comet, as well as its size and distinctness, may be very different 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. S9r. Tarieties of Asper-t. Individual comets offer consider- able varieties of aspect. Some comets have been seen which were wholly' destitute of a tail : such, among others, was the comet of 1682, which Cassini describes as being as round and as bright as Jupiter. Others have had more than one luminous train. The comet of 17-11: 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 tiie 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 1804. The comets that are visible only in telescopes, which are very numerous, have generally no distinct nucleus, and are often en- tirely destitute of every vestige of a tail. They have the appear- ance 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 99. — Encke's Comet removed from the centre of figure on the side towards the sun ; and sometimes also on the opposite side. ii9S. The Comets vrliicta have had the L.ongest 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° from the head; that of the comet of 1769, a distance of 97°; and that of the comet of 1618, 104". These are the apparent lengths as seen at certain places. By reason of the different de* grees 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 appear- ance, may be described as follows : The head of a comet consists of a central nucleus, or mass of matter brighter and denser than the other portions of the comet, enveloped on the side towards the sun, and ordinarily at a great distance from its suiface in comparison with its own dimensions, by a globular nebulous mass of great thickness, called the Nebulosity, or nebulous En- velope. This, it is said, never completely surrounils 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 be- tween 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, liowever, 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. Multiple Tails. In some instances the nucleus is fur- nished with several envelopes concentric with it: which are formed in succession as the comet approaches the sun, and then 234 C0MET3. recedes from it again. For example, the comet of 1744, eight days after the perihelion passage, had three envelopes. Some- times each of them is provided with a tail. Each of these sev- eral 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 tlie 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 princi- pal one, are by no means uncommon ; but they generallj'^ appear quite suddenly, and as suddenly disappear in a few days, as if the stock of materials from which' they were supplied had be- come exhausted. 401. The general Po§ition 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 re- gions 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 ex- ample, 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 11°- In the case of the comet of 1(364, the same angle about two weeks after the perihelion pas- sage was 43°, and was then decreasing at the rate of 8° per day. The comet of 1 823 might seem to present an exception to the general fact that the tail of a comet is nearly opposite to the sua; 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 ap- pear 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 volumi- nous bodies in the solar system. The tail of the great comet of 1680 was found by Newton to have been, when longest, no lesa than 128,000,OOU 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 from 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 3.')0,000 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. 40a. The rviiclei of comets, so far as they have been accu- rately determined, do not exceed a few hundred miles in diame- ter. For example, the great comet of 1811 had a nucleus of 428 miles, and that of 1798 one of 125 miles in diameter. In- stances 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. Tariation 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 16,000 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 somet 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 ITIass and Density. The quantity of mattel which enters into the constitution of a comet is exceedingly small. This is proved by the fact that comets have had no influ ence 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 prnx- imity to Jupiter's satellites, without deranging their motions in the least perceptible degree. Moreover, since this small quan- tity 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 comets 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 inconceiva- bly 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 nebulcjsity of the head, but is apparently of the same nature, and derived from the head. 406. IVucleus and IVebulosity. The nucleus is supposed by some astronomers to be, in some instances, a sohd, 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 Herschel, and it has been the more generally received notion since his time, that the nucleus of a comet is surrounded with a transpar- ent atmosphere of vast extent, within which the nebulous enve- lope 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 attrnction of the nucleus. 40r. Liuniinosity of Cometsi. Observations with the polari- scope 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 from the sun and earth. By this rule, the head of Donati's comet should have been 188 limes brighter on the 2d of October than on June 15th ; whereas it was actually 6,3ij0 times brighter. From which we may infer that the quantity of light emanating from it had increased in the proportion of 33 tc 1. This increase of actual light was confined chiefly to the nebulosity of the head, and is probably attributable, in a great degree, to an augmentation of the quantity of nebulous matter received from the nucleus. CONSTITUTION AND MODE OF FORMATION OP THE TAILS OF COMETS. 40§. Upon this topic we may lay down the following postu- lates: 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 bej'ond 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 in.stance, 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 tlie 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 ex- ceeded 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 in- versely as the square of the distance. This law of variation has in fact been established by the investigations of Bessel and Profes- sor Pierce, and confirmed by the author's determination of the form and dimensions of the tail of Donati's comet, upon th» 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 iiitnaiion and Curved Form ol Tail. Let PCA (Fig. 100) be a portion of a comet's orbit, the Fio. 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 SBB, 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 0, 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 Gba, 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 con- tinually emitted from the marginal parts of the head, we shall have the hollow conical tail actually observed. It is easy to see that Gba, the line of the tail, must be a curved line concave to- wards the regions of space which the comet has left. Supposing the arc 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 C6 parallel to BE ; whence RCa = CSA, and EC5 = CSB. Thus the line joining any particle with the nucleus always makes an angle with the prolongation of the radius vector, approximately equal * See Amerioau Journal of Science, Yol. xxix, pp. 79 and 383, and Vol. xsza p. 54, etc. FOKMATION 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 Ga and Cb. But in point of fact the velo- city 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 un- less the velocities of emission augment as rapidly as the velocity of revolution, the deviation in question will increase to the peri- helion, and afterwards decrease, as it is in fact known to do. 410. Dispersion of the Cometic Matter in the Plane of tlie Orbit. Observations made upon Donati's comet, have established that the nebulous matter was much more widely dis- persed in the direction of the plane of the comet's orbit than in the direction perpendicular to the orbit ; so that the transversa 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 phenome- non had its ojigin in the case of Donati's comet, in an inequality in the force of repulsion exerted by the sun upon different por- tions of nebulous matter expelled from the nucleus. The limits between which the repulsive force varied were and 1.21 (the intensity of the sun's ordinary force of attraction at the same dis- tance being the unit). It is shown also that nearly one-half 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 attract- ive force equal to -^-^ 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 rnna through the positions of the particles that left the head at several 240 COMETS. assumed dates, calculated for October, 5d. 7h. mean time at Greenwich, and is accordingly the outline of the train as theoreti- cally determined for that instant. The dotted curved line is the 2]8 ,---- '^. r 216' ,---''^^ ^* — .t — „---- — I » 211 • 0> "'^"'^ =m — ' FlO 101. outline of the actual train as observed \\ hours later, when its form and dimensions were sensibly the same as at 7h. The bro- ken 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 re- pulsion for the principal tail (1.21). 411. Columnar Structure of the Tail of Douati's Comet. The tail of the comet of Donati, was seen on certain occasions to be traversed, for a part of its length, by hands of unequal bright- ness, 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 nead 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. 2il 412. The Source of the Nebulous Stream, called the tail of the comet, has been generally supposed to be the envelope, or enve- lopes of the head ; but at the present day the preponderating weight Fro. 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 con- tinues 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 ap- pearance. 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 hy- pothesis 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. 4ia. 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 Jouraal of fcii noe, Vol. XXVIL, January, 1859.) H 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 par- ticles, and according to the other because the supply of outstream- ing 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 eometic particles which were distributed along the concave side of the tail of Donati'a comet were not repelled by the sun (410), we must hifer that they were not expelled from the nucleus by a force of repulsion, but were in all probability detached by some projectile force in opera- tion at or near its surface. On the other hand, the eometic 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 cousequence of the particles being more nearly in the condition of the ultimate molecules, iu which there is reason to believe that they become subject to both a molecular aud 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 iu 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 ; aud 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 aud the velocity of rotation.f 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. T/ie Projectile Force, whose existence we have here recognized, may have its ori- gin in electric discharges along magnetic vaporous columns, like the similar force supposed to be iu action upon the surface of the sun's photosphere (293). In sup- port of this view it may be urged that, if we assume the hypothesis that the nebu- lous matter at the nucleus of a comet is made up of particles susceptible of mag- netization, aud capable of being expelled by discharges along lines of magnetic polarization, we are enabled to give au adequate explanation of diverse luminous phenomena presented by comets, that are wholly inexplicable upon all previous hypotheses. 415. Theoretical Process of Hvolutlon of an Envelope. We would first remark that a rotatory motion of the uueleus, in conjunction with its orbital motion, should, by the collision of the molecules with the ether of space, bring ii 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 ex polled 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 wiU be nearly parallel to the surface of the nucleus near the magnetic equator, and that their angle of inoli- * See American Journal of Science, VoL xxxvm., p. 70. f See American Journal of Science, Vol. XLi., p. 62. ORIGIN OF THE NEBULOUS ENVELOPES. 24:3 nation to the surface will increase with the distance of the columns from the mag- netic equator, or approximately from the plane of the orbit. T}te envelope sJiovM therefore consist of two portions proceeding from parts of the nucleus that lie ou 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 lo its plane, or to the plane of the orbit nearly. If we conceive tl)e 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 paraboUc form, if the discharge extend be- yond 35° of magnetic latitude, and the expelling force and the solar repiilsion 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 pecuhar 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 dis- charge 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 enve- lope, frequently noticed, especially in the later stages of the envelope. This has been in some instances represented as a notch in the envelope. This deficiency ef light at the vertex in ob^riously what should result if the discharge should rela- tively 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 boUow. Upon the pre- sent theory, the brightest portions of the taU, 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 tbe axis (415). (4.) The great difference noticed in the brightness of the two branchy of the train, of Donaiis 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 tliat of the orbit." Theory, as we have seen, assigns it a position nearly per- pendicular to the plane of the orbit. (5.) The remarkable shifting of the superior brightness and eccentric position from me branch of the tail ofDojiati's comet, near the head, to the other, about October \f)th. 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 brauches of the train. Cotem- 244 COMETS. poraneously with these changes, the dark, axial stripe nearly disappeared, and reappeared at later dates. 417. Explanation oftlie Rise and Recess of Successive Envelopes. To understand how one envelope after another may rise and recede to a cer- tain 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 mag- netic 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 pro- cess 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 tlie Brlgbtness 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 hue of sight, of the separate streams of cometic particles, and to varjdng 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 enve- lope ; these may result from sudden variations in the intensity of the expelling force, or in the quantity of matter discharged. The da/rk spots sometimes seen are probably due to a deficiency of nebulous mat- ter 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 tiie envelope wiU rise and expand with the envelope. THE FIXED STABS. 245 CHAPTER XIX. THE FIXED STABS. 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 re- tained. The resemblance of the figure of a constellation to that of the animal or other object from which it is named, is in most instan- ces 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 LionJs Heart: and the brightest star in the constellation Taurus is situated in the eye of the Bull, and therefore sometimes called the BulVs Eye; while that con- spicuous 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 constella- tions are divided into three classes: Northern 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 FIXEU STARS. in the heavens. The signs of the zodiac coincided with the zo- diacal constellations of the same name, as now defined, about tha 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 con- stellation 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 begin- ning 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 ob- served that the constellations and signs of the zodiac have not precisely the same extent. 421. IfIodv§ of Designation of Individual Starts. The stars of a constellation are distinguished from each other by the letters of the Greek alphabet, and in addition to these, if necessary, the Eoman letters, and the numbers 1, 2, 3, &c. ; the characters, according to their order, denoting the relative magnitude of the stars. Thus » Arietis designates the largest star in the constella- tion Aries; ;3 Draconis, the second star of the Dragon, &c. Some of the fixed stars have particular names, as Siriiis, Aide- baran, Arcturus, He^idus, &c. 422. magnitudes. The stars are also divided into classes, or magmtudes, according to the degrees of their apparent bright- ness. The largest or brightest are said to be of the first magni- tude; 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 tlie telescope. The classification according to ap- parent 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 oc- cur in the heavens, viz. : Slrius or the Dog-star, Betelgeux, Rigel, Aldeharan, Capel'a, Procyon, Begidus. Denebola, Cor. Hydrm, Spica Virginis, Arcturus, Antares, Altair, Vega, JDeneb or Alpha Oygni, Dubhe or Alpha Ursce Majoris, Alpherat or Alpha Andro- mecke, Fomalhaut, Achernar, Canopus, Alpha Orucis, and Alpha Gentauri. It is the practice of Astronomers to mark more or less of these stars as intermediate between the first and the sec- ond magnitude ; and in some catalogues some of them are as- signed 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 tilobe. 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 6ther 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 con- cave surfiice 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 inter- section of the two circles will represent the equinoctial points. The point which represents the place of a star is found by mark- ing 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 char- acters, 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 en- tire 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-penetrat- ing 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 magni- tude, 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 iscale of brightness ; the whole number of stars already registered down to the seventh magnitude, inclusive, amounting to 12,000 or 16,000. The reason of this increase in the number of the stars, as we 248 THE FIXED STAKS. 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 mag- nitude, will be greater in proportion as this distance is greater, and thus as the apparent magnitude is lower. 426. Comparative Brightness. It is not to be understood that the classification of the stars into different magnitudes, ia made accftrding to any fixed definite proportion subsisting be- tween the degrees of apparent brightness of the stars belonging to dififcrent classes. Stars of almost every gradation of bright- ness, 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 cer- tainty to either magnitude. Thus, in the catalogue published by the Astronomical Society of London, 3 stars are marked aa 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 com- parisons of Sir Wm. Herschel, they are, from the first to the sixth magnitude, approximately in the ratio of the numbers, 10 ', 25, 12, 6, 2, 1. 427. Distribution ot the Stars. With the exception of the three or four brightest classes, the stars arc not distributed in- discriminately over the sphere of the heavens, but are accumu- lated 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 im- mense 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, ia a little over 2i», 000,000. 42S. stratum of the Milky Way. The great accumula- tion 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 AND DISTRIBUTION OVER THE HEAVENS. 249 Fig. 103. side of the point S, the stratum is certain distance into two laminae, as shown in the figure, which re- presents a section of the supposed stratum. This supposition is ne- cessary to account for the two branches, with a dark space be- tween them, into which the milky ■way is divided for about one- third of its course. Herschel undertook to gauge this stra^ turn in various directions, on tlie 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 actu- al form was very irregular ; its section, in- stead of being truly that of a segment of a sphere divided for a certain distance into two lanHuae, 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 be- tween the stars, and the breadth to be no- where 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 in- vestigations. 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 dross, and that the central parts are so vacant of stars that the whole approximates to the form of an annulus. upon this supposition the number of stars in the direction SO of the thick nessof th e stra tu m , will be less than in any other direction, and that the greatest number will lie in the direction of the breadth, as SB. On one supposed to be divided for a Fia. 101. 250 THE FIXED STARS. FlO. 105. ANNUAL PARALLAX AND DISTANCE OP 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 perpen- dicular 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 B the earth, in such a position that the radius -vector SE is perpendicular to Es the line of direction of the star, then the angle SsE is the annual parallax of the star s. 430. Lieast Distance of the Stars. If the annual par- allax of a star were known, we might easily find its dis- tance from the earth ; for in the right-angled triangle SE« 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 1;o 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 paral- lax. Now, such is the nicety 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 an- nvMl parallax of 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 tliat light, which traverses the distance between the sun and earth in 8m. 18s, and would perform the circuit of our globe in -^ of a Becond, employs 3J years in coming from the nearest fixed star to the earth. ANNUAL PAEALLAX AND DISTANCE OF THE STAES. 251 431. Determiiiation of tlie 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 observa- tions with a more efficient instrument by Maclear have furnished an angle of parallax differing but little from that obtained by Henderson. Its value is U".913, which answers to a distance about ^ less than the least limit of distance of the stars, just determined. The parallax and distance of Sirius and of the pole- star, have since been determined in a similar manner, but with less certainty. The result obtained for the parallax of the pole-star is 0".ll, and for that of Sirius an angle a little greater. A parallax of 0".ll answers to a distance that light would re- quire nearly 30 years to traverse. 432. Parallax or a Star found by tbe Differential method. The honor of being the first to determine with certainty the parallax and distance of a filed 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 henca inferred that they are bound together into one system by the principle of gravita- tion, 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 sup- posed to be at a much greater distance, seen in as nearly the same direction as pos- sible. 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 paraUas in the two stars, conceive two cones having the earth's orbit for a common base, and their vertices respec- 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 wiU be apparently described by the stars in the course of one revolution of the earth in its orbit. The two stars vriU always be sunilarly situated in their parallactic ellipses: thus, if one is at A the other wiU be at a ; and after the earth has made one-quarter of a revolution, they will be at B and i; and after another quarter of a revolution at and c, &o. Now it will be manifest, on inspecting the figure, the ellipses being of unequal size, Fia. 106. that the lines of the stars will be of un- equal lengths, and have different directions in the (Mferent situations of the stan 252 THE FIXED STARS. A much smaller angle of parallax may be found, with the same degree of certainty, hy this indirect method, than by the direct process explained in Art. 430; for since the two stars are seen in pretty nearly the game 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 tho determination of these several corrections, when it is attempted to find direutly the absolute parallax, by observing the right ascension and declination at oppo- site 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 tlie sake of greater certainty Bessel made the measurements of parallactic changes of relative situation between the star 61 Cygni and two small stars in- stead 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 dif- ference 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. Tbe 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, whicb 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 de- termined by Struve on the assumption that the stars are uni- formly 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 Herschelian telescope of 20 ft. focus are sep- arated from us by a distance that light takes 3,500 years to jour- ney 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. VAEIABLE STARS. 253 If it should be conjectured that some of the fainter stars might bo 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 dimen- sions 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. lUagiiitnde of the Stars. To be able to determine the magnitude of a star, we must know its distance, and also its ap- parent diameter. Now the distances of but few stars have aa yet been found; and the discs of all the stars, even in the most powerful telescopes, are altogether spurious ; so that in no in- stance 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 dif- ferent 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 » Lyrse 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 smalle? 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 STAES. 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 gradu- ally 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 331|- 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 grada- tions. It is even related by Hevelius, that in one instance it re- mained invisible for a period of four years. A similar phenome- non has been noticed in the case of the star x Cygni. According to the testimony of Cassini, it was scarcely visible throughout th3 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 re- sumes its regular variations, in the course of which it entirely disappears. The greater number of variable stars undergo a regular in- crease and diminution of lustre without ever becoming entirely invisible. Algol, or /3 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 dimin- ish in splendor, and in about 3^ hours is reduced to a star of the fourth magnitude. It then begins again to increase, and in 3^ 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, t Arietis, ^ 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 min- imum, as if some form of nebulous matter were interposed be- tween them and the eye. 438. 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 en- tirely 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 Cassio- peia, and was attentively observed by Tycho Brah^. 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 midday. 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, hke 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 tliis with any degree of certainty. Besides these three temporary stars, several others have made their appaarance, 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 aud 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 Algol the phenomena are precisely such as would result from the periodical interposition of an opake body. In those cases in which the period of ihe diminution of the hght is a large fraction 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 ©volution of light for a long period of time, and then burst forth anew with a sud- den 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 aa 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 Her- schel, 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 lattei hypothesis. 256 THE FIXED STARS. DOUBLE STARS. 440. Many of the stars which to the naked eye appeal 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 Multiple Stars. (See Fig. 107.) This class of bodies was first attentively observed by Sif William Herschel, who, in the years 1782 and 1785, published Castor. y Leonia. RigeL Pole-star. 11 Monoc. ^ Cancn. Flff. 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 Proximity. 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. 44*. 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 18^in. aperture. In some instances one of the constituents of a double star is itself double, e Lyrae offers the remarkable combination of a double-double star. 443. Diffenni Colors. It is a curious fact, that the two constituents of a dou- ble star in numerous instances shine with different colors ; and it is stUl inore 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 Herschel cites as probable examples of this effect tlie two stars I Cancri, and y Andromedse. But it is maintained by Nichol that Ibis expl& nation cannot be admitted ; for, if true, it ought to be universal, whereas there are many systems similar in relative magnitudes to the contrasted ones, is which DOUBLE STARS. 257 both stars are yellow, or otherwise belong to the red end of the apectrutn. Again, if the blue or violet color were the effect of contrast, it ought to disap- pear wlien the yellow star is hid from the eye ; which, however, it does not do. Thus, the star ff Cygni consists of two stars, of which one is yellow, and the other shines with an intensely blue light ; and when one of them ia concealed from view by an interposed slip of darkened copper, the other preserves its color unchanged. Tlie color, then, of neither of the stars can be accidental. It naay bo remarked in this connection, that the isolated stars also shine with various colors. For example, among stars of the tirst magnitude, Sirius, Vega, Altair, Spica are white, Aldebaran, Arcturus, Betelgeux red, Capella and Procy- on yellow. In smaller stars the same difference is seen, and with equal distinct- ness 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 unassooiated with a companion brighter than itself. 444. Di§covery 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 con- sequence of the annual variation of tbe 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 aimual 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 regu- lar progressive changes which seemed to indicate a real motioa 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 ex- ist sidereal systems composed of two stars revolving about each other in regular orbits. These stars have received the appel- lation of Binary Stars, to distinguish .them from other double stars which are not thus physically connected, and wliose appar- ent 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. Similar stars, con- sisting of more than two constituents, are called Ternary, Quater- nary, &c. Since the time of Sir W. Herschel, the observations upon the binary stars have been continued hj Sir John Herschel, Sir James South, Struve, Bessel, Madler, and other astronomers. According to Madler the number of known binary and ternary Btars 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 sucb 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 I' of each other. This may be inferred from the fact, that the- number of such double and multiple stars actually ob- served, which amounts to over 3000, is at least ten times greater than the number of instances of fortuitous juxtaposition that would obtiiin on the supposition of a uniform distribution of the stars. Besides, there are a number of double stars not yet dis- covered to have a motion of revolution, which still give indica- tions of a physical connection. 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. 445. Period§ and Orbits of Biliary Stars. From the ob- servations made upon some of the binary stars, astronomers have been ennbled to deduce the form of their orbits, and approxi- mately the lengths of their periods. The orbits are ellipses of considerable eccentricity. The periods are of various lengths, as will be seen from the following enumeration of some of those considered tis b:st ascertained : f.' Bootis 650 years, y Virginis 171 years, ^ Ophiuchi 92 years, « Centauri 77 years, g'Cancri 58 years, i Herculis 36 years. Fig. 108 represents a portion of the Fig. 108. apparent orbit of the double star y Yirginis, and shows the rela- tive positions of the two members of the double star in various years. At tbe time of their nearest approach, in 1836, the inter- val between them was a fraction of 1", and thevvcould not be separated by the best telescopes, with a magnifying power of 1000. Since then their distance has been continually increasing. In 1844 it amounted to 2", and a power of from 200 to 300 was sufficient to separate them. 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. The actual distance hetvleen the members of a binary star has been found for 61 Cygni, and « 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 gravi- ty. 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 OP THE STARS. 446. It has already been stated that the fixed stal-s, so called, are not all of them rigorously stationary. By a careful compari- son of their places, found at different times with the accurate in- struments and refined processes of modern observation, it has been found that great numbers of them have a progressive mo- tion 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 Cyg- ni. During the last fifty years it has shifted its position in the heavens 4' 21"; the annual proper motion of each of the indi- vidual 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 and 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. motion of the Solur System througli 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 ex- amination of these motions, he conceived that the majority of them could be explained on the supposition of a general recess of the stars from a point near that occupied by the star a Hercu- lis towards a point diametrically opposite. Whence he inferred that the eun, 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 M. Argelander, of Abo. The investigations oi 260 THE FIXED STARS. Argelander, which were communicated to the Academy of St. Petersburgh in 183Y, have since been confirmed bj M. Otto Struve, of the Pulkowa Observatory, and other eminent ob- servers. 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: E. 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 Snn's motiou throngli Space. 0. 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".S39 (with a probable error of 0".(i25). So that, if we assume, according to the best determinations, 0".209 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. Vehcify of the proper motions of the stars. The above angle of 0".339is 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 moion (5".28) must result from an actual niotion in space. This is 14.37 times greater than the parallax of this star (0".35). Accordingly, if we sup- pose the direction of its motion to be perpendicular to its line 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 ascer- taining the actual direction of its motion, it is impossible to dis- cover how much the velocity exceeds this determination. 450. Sun's motwn 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 OP STARS. 451. In many parts of the heavens stars are seen crowded together 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 tele- scope 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 /Vcesepe, or the bee-hive, which a telescope of moderate power resolves entirely into stars. Within a space about i° 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 ■n and e. 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 u Cenlauri. The telescope shows it to consist of an immense mul- titude 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. NEBULA. 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 Nebulae. Some of these nebulous objects are partially visible to the naked eye, but the great majority of them cannot be discerned without the assis- tance of a good telescope, and very many are beyond the reach of any but the most powerful instruments. 453. JVumber and Distribution of ^Tebulae. The num- ber of nebulae hitherto discovered, is over 5,000. They are very unequally distributed over the heavens, especially in the north- ern hemisphere. They are most abundant in the constellationa Virgo, Leo, Coma 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 nebulae follow each other in rapid succession, from the diur- nal 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 de- tached spaces of considerable extent, visible to the naked eye, called the Magellanic Olouds, that shine with a nebulous light like the milky way, which are thickly sown with nebulae. 454. Diversity of Form and Appearnnce. As seen through telescopes of moderate power, the nebulae are, for the most part, round or oval in form ; but, when carefully examined with the larger telescopes, 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, charac- terized 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 tbe highest telescopic power is ap- plied to them. These Irresolvable Nebulae were supposed by Sir William Herschel to be masses of actual nebulous matter dis- seminated 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. 455. Classification of IVeitulae. The nebulae are classified according to their aspects and forms, as seen through the larger telescopes, as follows : (1) Globular Clusters, (2) Irregular Clus- ters, (3) Oval Nebulce, (4) Annular Nebulae, (6) Planetary Nebulae, (6) Stellar Nebulae, (7) Spiral Nebulae, (8) Irregular Nebulae, (9) Douhle Nebulae. 456. Olobular Clnsters take their name from their sup- posed 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, difiicult of resolution, and appear in telescopes of moderate power as small, round, nebu- lous specks, resembling a comet without a tail. Fig. 3, Plate IV., represents a globular cluster to be seen in the constellation Pegasus. 457. Irregfular Clnsters. These are more or less irregular and indefinite in their outline. They are generally less rich in stars,, and le^ condensed towards the centre than the globulai NEBULJS. 263 clusters. Fig 2, Plate IV., represents an irregular cluster situated in the constellation Capricornus. The Pleiades^ and Coma Bere- nices, 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 sur- rounding regions, to the perfectly defined globular cluster highly condensed at the centre. 458. Oval Nebiilic. Nebulae having a distinct elliptic out- line 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 and sudden varia- tions of brightness from one portion of their mass to another. This is very observable in Fig. 9, Plate V. Such nebulae, 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 stel- lar structure. In general the spheroidal clusters are far more difficult of resolution than globular clusters. 459, Dynamical Eqnilibriiini 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 in- stance of globular and spheroidal clusters, a general state of equilibrium would be possible only upon the supposition that the individual 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 subject at the same time to secular changes, from the varying combina- tions of individual orbital positions, and the disturbing actions of some of the component stars on one another. 4«0. Annular Mebulae. A very small number of observed nebulae have the annular form (Fig. J2, Plate V.). A conspicu- ous example of this singular class of nebulae may be seen with a telescope of moderate power, midway between the stars /3 and V Lyrse. The central vacuity is not perfectly dark, but filled with a faint nebulous light. The telescope of Lord Rosse, has resolved it into minute stars, and shown it to be fringed on its outer edge with filaments of stars (Fig. 11, Plate V.). Chacornac, of the Paris observatory, describes it as presenting, in Foucault's great telescope of plated glass, the appearance of a hollow cylin- drical 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 STABS. an equable light 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 ;3 Ursse 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 * 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 dimensions 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 dimension? more than 1,000 times greater than the above determination. One of the planetary nebulae has been resolved by Lord Eosse'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 holloM spherical shells. 462. Siellar IVebiilaB are those in which one or more stars are seen apparently connected with a nebulosity. This class of nebulae comprises several varieties, the most important of which is that of the Nebulous 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, i*late V., shows the appear- ance of a nebulous star in Gemini, as seen through Lord Rosse's telescope. The stars surrounded by these nebulous atmo- spheres 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 con- stellation Cygnus. It consists of a star of the 11th 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 link between planetary nebulae and nebulous stars. In the other varieties of stellar nebulae stars are seen occupy- ing various positions, in apparent connection 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 Nebulce, from their close resemblance to a comet with a spreading tail. Fig. 18, Plate v., represents a cometic nebula in the tail of Scorpio. 463. Spiral Nebulae. The great telescope of Lord Rosse has revealed the remarkable fact that some of the nebulae are made up of spiral convolutions proceeding from a common nucleus, or from two nuclei. The most conspicuous example of NEBULA. 265 this curious form is represented in Fig. IQ, Plate V. It ia situated near the star ri, at the extremity of the tail of the Great Bear. The spiral nebulous coils diverge from two bright cen- tres, 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 IVebnlae. Under this head are classed all the remaining single nebulae that, as seen through the best tele- scopes, 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, indicating a singular complexity of internal structure. Tlie Great Nebula in the sword handle of Orion is the most con- spicuous example of this class of nebulae. It consists of irregu- lar nebulous patches extending over a surface about 40' square, or about twice the size of the moon's disc. From its great mag- nitude and beauty, singularly grotesque form, and the wonderful variety of its light, it is the most remarkable of all the nebulae. One portion, near the trapezium or sextuple star ^, is uncom- monly 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 nebula remained enveloped in mystery from the time of its first dis- covery by Huyghens, in 1 656, until the telescope of Lord Rosse was directed upon it ; when the brighter portion near the tra- pezium 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. The Oreat Nebula in Andromeda is another remarkable irregu- lar 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 tele- scope, 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 elon- gated oval form, similar to Fig. 7, Plate IV. Tlie Crab Nebula. Fig. 4, Plate IV., represents the- appear- ance of this curious nebula as seen through Lord Rosses tele- 268 THE FIXED STABS. scope. It is described as studded with stare, mixed with a nebulosity probably consisting of stars too minute to be recog- nised, and exhibiting filaments extending out from the soutliern portion of the nebula. Tn ordinary telescopes these outlying branches, which have suggested the name of crab nebula, are invisible, and the part seen has an oval form. The Dumb-hell Nebula is so named from the fact that as seen through a telescope of moderate size, in which the brighter fortion alone is visible, it has the apparent form of a dumb-bell, n Lord Eosse'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 por- tions are included, its outer limit has an oval form (see Fig. 9, Plate v.), which shows the nebula as viewed through the smaller telescope of 3 feet aperture, constructed by Lord Eosse. 465. Double IVebulae. A considerable number of double nebulae occur in dififerent parts of the heavens. M. D' Arrest, of Copenhagen, enumerates fifty whose constituents are not over 5' apart, and estimates that there may be as many as 200 such double nebulae. The two constituents are most commonly circu- lar in their apparent form, and are probably real globular cluster. (Fig. 17, Plate V.) The individual members of most of these nebulae 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 Nebulae. Systematic observations have been made by Struve, D'Arrest, and other astronomers, with the view of ascertaining whether any of the nebulae 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 changes 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 detect- ed. It is an interesting fact that this diminution of brightness has proceeded pari passu with that of a small star which pre- sented 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 Orion 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 NEBUL-ffi. 267 DISTANCE AND MAGNITUDE OP NEBULA. 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 tele- scope was 2,080 times the mean distance of stars of the first magnitude. This should accordingly be the limiting distance of the resolved nebulse seen in Herschel's telescope. The corres- ponding limit for stars and nebulae, as seen in Lord Eosse's tele- scope, should be 3,120. But Struve, after determining the com- parative distances of stars of the different photometric magni- tudes, by comparing the actual number of stars of the different magnitudes, has been enabled to ascertain the actual space-pen- etrating power of any telescope in which all the stars up to any particular magnitude could be seen. According to his deter- minations, the actual space-penetrating power of Herschel's tele- scope 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".'il, and is traversed by light in 15.5 years. We may 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 3|- 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 Nebiilse. Herschel also undertook to determine the probable distance of the more remote irresolvable nebulae; He estimated that a certain cluster of stars (75 of Mes- sier'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 Eosse'a; telescope would be only a small fraction greater. 469. Extinction of the liight of the Stars, in its passage aba 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, con- ducted 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. He 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 ^Jr of its intens- ity. 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 Her- schel, 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 con- clude that stars can be discerned with the most powerful tele- scopes 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 100 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 dis- tance. 4T0. magnitude of IVebiilse. At the distance of 422 stellar intervals (the utmost actual reach of Lord Rosse's teles- cope) a linear extent of 10', in the heavens, answers to 1.23 times one of these intervals (467). Some of the planetary ne- bulae 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 direc- tion, 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 1^ units. We here suppose these two nebulae 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 BRIGHT- NESS OF THE COMPONENT STARS OF CLUSTERS. 471. Possible IVumbcr of Stars in a IVebula. We may obtain an approximate estimate of the number of stars that may be congregated together in a nebula that is completely re- solvable bj' a powerful telescope, by considering that if the tel- escope 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 /S, in the con- stellation of the Great Bear (461), has an apparent extent of 7 square minutes. If it were just resolvable it should then con- tain 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, -^V of 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. Liimit of Distance between Stars in a Resolved JVebula. 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, can- not exceed this amount. K 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. Explanation of Inequalities of Brightness in a [Vebuia. 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 nebulse 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 por- tions of such nebulae; or that instances of close juxtapositioQ more frequently occur, in groups of two or three, which appear united as one, as suggested by Sir John Herschel. STRUCTURE OP THE SIDEREAL UNIVERSE. 474. System of the milky Way. We have already seen (428) that Sir William Herschel made the grand discovery that tiie 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. He at first conceived that his telescope penetrated to the outer- most limits of the stratum, but later investigations, recently con- firmed 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- tum 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 de- creases, 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 increas- ing 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 intersec- tion of the galactic circle, or middle line of the milky way in the heavens, with the equator. These points lie in R. Asc. 6h. 40m., and R. Asc. 18h. 40m., between the constellations Orion and Canis Minor, and between Serpentarius and Antinous. Ac- cording to Struve the sun is on the north side of the plane of great STRUCTURE OF THE SIDEREAL UNIVEHbE. 271 BSt condensation, and at an estimated distance from it equal to the distance of « 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 way, passes through the points on the equator above-men- tioned, 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 Coma- Berenices and Canes Venatici. 475. The Galaxy, or Belt of the Milky Way. The lu- minous belt in the heavens called the milky way, as seen by the naked eye, 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 fainter 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 equa- tor, 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 10° 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 opposite sides of this broken plane. The line of greatest density before 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 con- tinuity 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 from it. 476. Relations of Clusters and IVebulae to the System of the 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 nebulae, some of which have been resolved, are, for the most part, either portions or outlying branches of the system. Some of tbose which have 272 THE FIXED STARS. not been resolved, may possibly be independent systems ezteriof to that of^he milky way. Oval nebulfe, and the irresolvable nebulae generally, do not hold the same relations to our starry firmament. They are mostly absent from that great belt in which the stars are so numerous and condensed, and the conspicuous clusters abound, and are congregated towards its poles. The region richest in nebulae lies around its north pole. They are more uniforrnly disseminated and more widely dispersed over the zone which surrounds its south pole; and are at the same time less numer- ous. But on the other hand, as already intimated, there are two luminous tracts of the southern heavens, called the Magellanic Clouds, in which they occur in large numbers. In these they are found associated with groups and clusters of stars of every 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 oral and irresolvable nebulae of the northern hemi- sphere, leads to the supposition that they may have originated in a different manner from the clusters and nehulse 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 iu some of the features of their physical constitu- tion. The latter supposition acquires additional probability from a recent discovery that the character of the light received from some of the nebulse is in certain re- spects different from that of the light received from the sun and the stars. A spectral smalysis of the light from some of these nebulije, by two eminent physi- cists, has disclosed the remarkable fact, that it is not made up of rays of widely different refrangibiUties, but is, the greater part of it, monodiromatic ; 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 ne- bulae in question can no longer bo regarded as clusters of suns, similar iu constitu- tion to the centre of our planetary system, but a^ objects having quite a different and peculiar composition ; and that instead of being considered as made up of bodies having a soUd nucleus, they must be regarded as enormous masses of lumi- nous 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 nebulce comea 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. 47§. General motiAii of Revolution of the Stars. Madler, after an elaborate discussion of the proper motions of a large number of stars, has arrived at the conclusion that the collective body of stars visible to us has, together with the sun, a common movement of revolution around a cen- tre 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 regarded as highly probable upon physical grounds, but it cannot be doubted DYNAMICAL CONDITION OF SIDEREAL SYSTEMS. 273 ttat 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 constella- tion Perseus, as suggested by Argelander. 4'!'9, Hypotheses respecting the ITIilky Way. Madler supposes that the stars of the milky way are arranged in seve- ral concentric rings of unequal thickness, and of varying dimen- sions in different directions, but lying nearlj' in the same plane. He conceives the sun to be eccentrically situated in the sys- tem, 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 cen- tral 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. 480. 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 Hues 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 Hypothesis. — This was proposed by Sir "William Her- schel. It accords with the different aspects presented by clus- ters 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 provi- sion 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 cen- tre 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 impossi- ble. In the history of such an organized system, from its be- ginning, there may be epochs of collision among its individual members, but when all such cases, inevitably resulting from cor- respondences of original position, have occurred, the motions which remain outstanding may ultimately tend to a' permanent stability. NEBULAE HYPOTHESIS. 275 CHAPTEE XX: Theories of the Evolution of Sidereal and Pi^ajs-etary Systems. NEBULAR HYPOTHESIS. 481. Primitive Tfebnlons Condition of all Systems. Although me telescope, by revenling the stellar constitution of many of the nebulae 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 pri- mitive nebulons condiiion of all worlds and systems of worlds. Numerous correspon- dences of structural and dynamical features, and intimations of a progressive crea- tion, 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 com- mon 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 aU revolve in the same direction around the sun; viz. : from west to east. 2. Their orbits he nearly in the plane of the sun's equator. 3. Their orbits are ellipses of small eccentricity. 4. The Sim 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, approximately 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 occnrs in the case of the satellites of Uranus, which have a common retrograde motion. Their orbits are also incUned to the plane of the echptic under a large angle (79°) ; but their conmion 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.). GThe 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 Hypothesis. 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 pei^ manently 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 the 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 oT cooling at its surface. This contraction of necessity accelerated the rotation of the body, and augmented the centrifugal force : until 276 EVOLUTIOK OF SIDEREAL AND PLANETARY SYSTEMS. finally the increasing centrifugal force at the equator balanced the gravity. Whoi. 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 concen tration at special points, eventually separated into fragments ; which, from the preponderating attraction of the larger fragment, or because of sMght 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 same condition of rotation as the original nebulous mass, might pass through a similar succession of changes, and thus a system of satelUtes 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 de- tached 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 othei systems, and so became permanently united with it by the bond of gravita- tion. But, with the evidence now afforded by accumulated facts, several con- siderations 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 httle distinguished, in their orbit- al motions, from the planetoids. They revolve in the same direction, and in orbits having about the same average inclination to the ecUptic, 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 eUiptie 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 echptic 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 mUes beyond the limits of the solar system, it appears th.it as many among them have a retrograde as a direct motion ; while the majority move in orbits incUned under large angles to the echptic. These exceptional facts do not necessarily imply that this class of comets have an origin extraneous to the sys- tem ; 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 attrac- tion 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 echptic. Besides, the masses which became detached at the equa- tor 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 an- swering to a motion of revolution m 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 m 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 I the surface, and assume it to have been of sufBoient 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 firma- ment, 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 rota/- tion 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 in in Appendix.) PAET 11. PHYSICAL ASTRONOMY. CHAPTER XXI. Peinciple of Univeksal Gravitatiok. 484. Force of Gravity. It is demonstrated in treatises on Mechanics, that if a body move iu 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 inces- sant 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 areas in equal times. Since the impulses are only communicated at the angular points A, B, C, D, &c., of the polygon, the motion will be uniform fUong each of the sides AB, BC, CD, &c. : and since we may suppose the times of describing these sides to be equal, we shall have the triangular area SAB equal to the triangular area SBC, and SBC equal to SCD, &c. Produce AB and make Be 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 fall the perpendiculars Am, era, Gr. Then, since AB = Be, Am = en ; and since the equivalent triangles SAB, SBC, have a common base SB, Am = Gr. It follows, therefore, that en = Gr, and consequently, that Cc is parallel to BS. The im- pulse which the body receives at B is therefore directed from B towards S. In the same manner it may be shown that the impulse which it receives ji jaq 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 radius- vectors 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 Bun. PRINCIPLE OF UNIVERSAL GRAVITATION. .r& 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. 4§5. Its liaw of Tariation. It is also proved by the prin- ciples 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 supposed elliptic orbit of the body, CA and CB its semi-axes, and S the focus towards wh oh the force is con- stantly directed. Also let P be one position of the body, PR a tangent to the orbit at P ; and draw RQ par- rallel to PS, Quu, HI, and CD, par- allel to PR, Qa; perpendicular to SP, and join S and Q. CP and CD are semi-conjugate diameters. Denote them, respectively, by" A' and B' ; and denote the semi-axes, GA and CB, by A and B Since HI is par- allel to PR, and, by a well-known property of the ellipse, the angle EPS is equal to the angle HPT, PH is equal to PI : and since HC is parallel to HI, E is the middle of SI. We have, therefore, FiS. 110. : SC, and CB PB=-P.S±^ PS-l-PH =CA = A. Now the force at P is measured by 2Pm ; and we may state the proportion Pu : Pi; :: PE : PC :: A : A' ; which gives Pi; = Pm:^. By the equation of the ellipse referred to its centre and conjugate diameters, pa and DL, If we regard Q as indefinitely near to P, then Qa =: Qu, and Qv — 2CP : and therefore Q^^=:|,'l (P«^'.3A')=^.2P«.. (a.) But Qa: Qa;::PE : PF::OA : PP: and, by Analytical Geometry, CD X PP=CA X OB, or, CA : PF :: CD : CB :: B' : B. :2A'i Hence Qa : Qa;::B' : B, Qa : Q» ::B'2 : B', and Qm Substituting in equation (a), Qa; — = — .2Pm; whence Qa; =— .2Pa. Qx 5 4fc Now triangular area SQP=4=SP x — ; whence Qa; = =. Substituting, there SP' lesnlts 4ft« B'' A 1 r=5= — .2Pm: or 2Pu = — .4i;^=j. S? A B^ SP (I). 280 PRINCIPLE OF UNIVEESAL GKAVITATION. To compare the intensities of the force at different points of the orbit, we must take the values of 2Pm, by which they are measured, for the same interval of time. On this supposition k is constaut, 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 pla.net towards the sun varies inversely as the square of the distance from the sun's centre. 486. It operates on all tlie 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 mat- ter of the sun for the matter of the planets, and is called the Solar Attraction. To deduce this consequence from Kepler's third law, let /, (', denote the perio- dic 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 irAB, irA'B'. Thus ki: k't :: AB: A'B', and k^t^ : k'^i' :: A*B" : A"'B'''. But, by Kepler's third law, t' :«'':: A' : A''. Dividing, and reducing, k^ :k' '::—-- : — - : A A that is, the squares of the areas described in equal times are as the parameters of the orbits. Now, let f, /, denote the forces soliciting the two planets. Then, bv eauation (I), Art. 485, ' ' ^ /=^-4*^-^>and/ = |;,'.«-.^; whence /:/:: 4*^ i- : 4. &-.Jl:: A l!.i. . ^ 51^ 1 ' From which it appears that the planets are solicited by a force of gravitation towards the sun, which varies from oneplanet to another according to the law of the inverse square of their distance. 4§T. 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 at^ tractive 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 plonets, may be shown upon this supposition to be neces- Newton's theoet of universal gravitation. 281 eary cousequences of the attractions of the planers 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§. The Constituent 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 attrac- tions 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. 4§9. Theory of Universal Gravitation. 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 inverse 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 New- ton, and is hence often called NewixnCs Theory of Universal Ora- vitation. 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 tha most important effects of the principle of gravitation thus ar- rived at by induction. The perfect accordance that will be ob- served 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 or THE ELLIPTIC MOTION OF THE PLANEIS. CHAPTER XXn. 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° : 1", which gives F=— j This, in the language of Dynamics, is the Accelerating Force of the planet, due to the attraction of the sun. M . . „ As —; 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 M be expressed by m — ;. 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 — . The sun will then tend towards 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, Li i. • Mm -KIT that IS, as — : — , or as M : 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 tUeir Common Centre of Oravity. 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 Principle of the Preservation of the Centre of Oravity. 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. Aa ■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. Ill) be the sun, and P any planet, supposed for the moment to be at rest. If neither of the two bodies should receive a velocity in a di- rection inclined to PS, the line of their cen- tres, they would move towards each other by virtue of their mutual attraction, and meet at C their common centre of gravity.* But, if the body P have a projectile velocity given to it in any direction Pi, 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. 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 pointy, the sun will have shifted its position to some point s on the line j)C pro- longed. Moreover, as the two bodies mutually gravitate towards each other, the path of each in space will be continually con- cave- towards the other body, and therefore also towards the cen- tre 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 SO 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. Por, draw Sp' parallel and equal to sp, and join Pp and Ss. Then, since sO : Cp :: SO : OP, Pp is parallel to Ss; and therefore Pp produced passes through p'. Whence, OP : Op :: SP : Sp'. Moreover, the angle PCp = P^'. 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 THE ELLIPTIC MOTION OF THE PLANETS. therefore, that the area PCp is similar to the area PSp' ; and thus that the oibit of P around C is similar to the apparent orbit of P around S. The latter is known from observation to be an ellipse. The former ia therefore also an eUipse. 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. 492. Entire Accelerating Force of Planet. If to both the sun and planet there should be applied a force equal to the accelerating force of the sun, !^, (49°), but in an opposite di- rection, the sun would be solicited by two forces that would destroy each other, but the planet would now be urged to- wards the sun remaining stationary, with the accelerating force — XJH:^ 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 libertj' to conceive the sun to be stationary, if we suppose the planet to be solicited by the accelerating force — IIl-!^. As the mass of the sun ia r" very much greater than that of any planet, but little error will be committed in neglecting the attraction of the planet, and tak- ing into account only the sun's action — 493. General Theoretical Re§ults. Analysis makes known the general laws of the motion of a body, when im- pelled by a projectile force, and afterwards continually attracted towards the sun's centre by a force varying inversely as the square of the distance. We learn by it that the body will neces- sarily describe some one 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 de- pendent, for any given distance of the planet, upon the intensity of the projectile force ; but that the position of the axis, and the tccentricity of the orbit, depend also upon the angle of projec- tion (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 sec- tions, a certain intensity of force will produce a parabola; any less intensity, an ellipse or circle ; and any greater, a hyperbola, 494. Tiieoretical Determination of Orbit of^Plaiiet 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 ita THEORETICAL DETERMUSTATION OF ORBIT. • 28o 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 d 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 mo- tion, 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. CeiitreofOravitjr of the Solar System. 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 quan- tity of matter in all the planets taken together is very small, compared with that in the sun (less than y^), 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 sentre 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 60 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 L.aw not rigorously true. It ap- pears from the physical investigation of the elliptic motion of the planets, that Kepler's third law is not strictly true. In con- • sequence of the actioQ of the planets upon the sun, the ratio of the periodic times of the different planets depends upon the mas- ses of the planets, as well as their distances from the sun. If p and p' be the periodic times of a.ny two of the planets, a and a' their mean distances from the sun's centre, and m and m' their quantities of matter, that of the sun being denoted by 1, then, disregarding the actions of the other planets, theory gives p" : p" 1+ m l-f-TTi'" As m and m' are very small fractions, the error resulting from their omission will be very small. If we omit them, wo shall have which is Kepler's third law. p-:p"::a':a" T'EETURBATIONS OF ELLIPTIC MOTION OF THE MOON. 287 CHAPTEK XXIII. ThEOET OF THE PERTURBATIONS OF THE ELLIPTIC MOTION OP 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 aotion 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, M 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' par- allel to ES, and the other from M towards E. If the component along MS' were equal to the 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 wiU be the disturbing force in the direction MS'. The component along ME is another disturbing force. It is called the Addititious Force, be- cause it tends to increase the gravity of the moon towards the earth. The disturbing force along MS' wUl generally be inclined to the plane of the orbit, and may be decomposed into three forces, one in the direction of the tangent, another in the direction of the radius- vector, and a third in the direction of the per- pendicular to the plane. The first mentioned component is called the Tangential Force; the second is called the Ablatitmts Force; and the third we shall call the Fei^endicular 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 Badial 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 (wfeioh for the present we shall suppose to be constant) be denoted by a, and the distances of the moon from the earth and Bun, 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 forca Fio. 112. 288 PERTURBATIONS OF ELLIPTIC MOTION OF THE MOOW. exerted by the sun upon the moon. Then, if we denote the mass of the earth by I, and take m to stand for the mass of the sun, we shaU have (490), y^ a' z" Let the force Q he represented by the line MS (Fig. 112); and let its oomponenl parallel to ES, or MS'=R, and its component along the radius-vector, or ME=;T Q: T:: MS: MB; or, - : T::z:y. Whence, addititious force T : In a similar manner we obtain .(82). ..(83). The disturbing force in the direction of the sun _ Tj p ma m '■(---\ W a') Now, let a, 0, Y, denote the angles made by the line MS' 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 ( — — _ "\ cos o (84) : \ z' o' / ablatitious force = ma( _ — _ ) cos /? (85); \z' a' / perpendicular force = mo ( _ — — 1 ( V 2' a' J cosy (86). Combining equ. (85) with equ. (82), we obtain for the radial force, radial force =mi/_ — ma ( _ — L N cos /?. z' \ z' a' ; The obliquity of the orbit of the moon to the plane of the ecliptic, affects but very shghtly the value of the tangential and radial forces. If we leave it out of account, or suppose the moon's orbit to he in the plane of the ecliptic, we shall have (Fig. 113) ,3 = S'ML = SEM, the elongation of the moon = , and z' = a' — ta'y cos # : neglecting the terms containing the higher -powsrs of y than the first, as they are very minute, y being only about j^u a. Thus, i = i 1 ^ 3y cos ^ . z' a' — Sa^yeos^ o' a* ' neglecting all the terms of the quotient that involve higher powers of y than the first. Substituting this value of— in equ. (87), we obtain, z' tangential force = ^JH^^^Lt^^ ■ a* or (App. For. 13), tangential force = ^Hj^^l' , . . .(go). 2i a Making the same substitution in equ. (88), and neglecting the term containing f*, there results, radial force = '!^=if?!!i); or (App. For. 9), radial force = ^mSUil^^J:) . . . .(91). 2 a' In equ. (89) we have to substitute, besides, the value of z, viz. a — y coa^; then dividing and neglecting as before, we have perpen. force = ^'"^ °°11 sin IT sin I. . . .(92). 501. Variations of disturbing forces. If the disturbing forces retained constantly the same intensity and direction, the result would be a continual progressive de- parture 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 re- sults 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 inclina- tion 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 19 290 PERTUKBATIONS OF ELLIPTIC MOTION OF THE MOON". the variations of the forces resulting from the first, and occasion each a distinct set of periodical perturbations. 603, Tangential Force. Let us now investigate, in succession, the effects of each of the'distarbing forces, commencing with the tangential force. The tan- gential 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 des- cription of areas. It also affects the radius-vector indirectly, by changing the cen- trifuo-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 be- come equal to zero' when sin 2(1=0, and will have its maximum value when sin 2ip=l. 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 Pig. 114, that it will be a retard- ing force in the first quadrant (AB). Ac- cordingly, 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 wiU also appear upon considering the direction of the 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 may regard it, instead, as urging the moon from the earth by the same force ; for the relative motion wiU be the same on this supposition. In the part of the orbit sup- posed, then, the disturbing force under consideration wiU be directed from the sun, as represented in Fig. 1 14. 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, 0, where the accelerat- ing 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 A3, 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 aa 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 wiU be in advance of its mean place, and will become more and more so tUl 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 wUl approach nearer to the true, till at the quadrature .they coincide. Beyond B, the true motion stUl 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 OP THE TANGENTIAL FORCE. 291 This is the inequality known in Spherical Astronomv by the name of Ywria- /ioTi (217). 503. Modifications of the effects of the tangential force, that result from the elliptic form of the sun's orbit. Suppose that at the moment when the moon sets out from conjunction the sun is in the apogee of its orbit : then it is plain that, during the whole revolution of the moou, the aun's disturbing force would be on the increase by reason of the diminution of the sun's distance, and that, in conseq\]ence, the retard- ation in the first quiidrant would be less than the acceleration in the second, aud the retardation in the third less than the acceleration in the fourth. So that, when the moou has again come round into conjunction, the acceleration will have over- coiopensated the retardation. This Icind of action would go on so long as the sun approaches the earth ; bnt when it has passed the perigee of its orbit, and begun to recede from the earth, the reverse effect would take place, and a retardation of the moon's 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 accelera- tion in each revolution of the moon during tbe passage of the sun from the apogee to the perigee of its orbit, would be compen.fatod for by an equivalent retardation in the revolution of the moon, answering to the same distance of the sun in its passage from the perigee to the apogee ; and tlie 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 accele- rations 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 half 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 performs 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 form of the moon's orbit. We remark, ir> the first place, that the orbit being an elhpse, the areas AEB, EEC, CED, and DBA (Pig. 114). will be unequal, and therefore, by the laws of elliptic motion, the arcs AB, BC, CD, and DA, will be described in unequal times. It follows from this, that the retar- dation in the first quadrant will not be exactly compensated by the acceleration in the second, and that the retardation in the third will not be exactly compensated by the acceleration 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 from one revolution to another, by reason of the ellipticity of its orbit. This varia- tion 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 compen- sated 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 moon's orbit. 505. Radial Force. We 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 dis- cuss the general analytical expression for it, already investigated. It is, ,. , . my (1 — 3 cos" a). radial force ——EJ^ II. a' We shall have radial force = 0, when 1 — 3 cos" ^ = 0, or when cos

rce from the earth. Therefore, as the sun approaches the perigee of its orbit, its di.'»tKnce from the earth diminishing, the mean diminution of the moon's gravity to the oarth will increase, and consequently the moon's distance from the earth will becom,-> ^;reater, and its motion slower, than it otherwise would be. The contrary will take place while the suu is moving from the perigee to the apogee. 510. I'erpendlcular Force. The disturbing force perpendicular to the piano of the moou'a 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= — ^ £ sm 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 aa the sun, and in the more remote hJlf, on the opposite side. We perceive also that it will be zero for every value of ^, or for every elongation of the moon, when tho 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 per- pendicular 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 the Nodes. Let NM'R (Fig. 116) represent the orbit 294 PERTURBATIONS OF ELLlPl.C 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 tha force, in the various points of the orbit, is in- dicated by the arrows drawn in tlie 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 thp disturbing force M'K', and compelled to move in such a line as M'i'. The node N' will there- fore retrograde to some point »'. When the moon is at any point M, moving from the as- cending node N towards L, its course will be changed to the lice M(, lying, like the line M'i', below the orbit, which being produced back- ward, meets the plane of the ecliptic in some point n behind N. The nodes, therefore, retro- grade in this position of the moon, as well as in the former. When the moon is in the half FlO. 116. K'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 same 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. Ac- cordingly, when tlie 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 consid- ered, as represented in Figs. 116 and 117. The effect of the disturbing force will be the samo as in the first situation from the quadrature } (Fig. 116) to the node N', and from the quadrature q' to the node N. But throughout the arcs Ng, TS'q, the direction of the force, and therefore the efiects, will be reversed The node will then retrograde, as before, while the moon moves over the arcs qS' and 12'N, and advance while it is in the arcs Ng, Wq. But as the force is greatest over the arcs jN', j'N, which contain the syzigies (510), and as these arcs are also longer than the arcs Ng, N'g', the node will, on the whole, retrograde each revolution. The velocity of retrogradation will, however, be less than when the nodes are in quadratures, and proportiouably less as the distance of the sun from this posltioi is greater. In the position represented in Fig. IIT, a direct motion will take place over the ar^s g'N' and 5N : but as Ng' and N'j, the arcs of retrograde motion, are of greater extent than g'N' and q'S, and moreover contain the syzigies, the retrograde motion in each revo- lution must exceed the direct, as before. If we suppose the sun to be situated on the other side of the line 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 ap- pears, then, that the line of the nodes has a retrograde motion in every possible position of the sun. 512. ^eci of Sun^s Motion. We have thus far supposed the sun to remainstation- ary in the various positions in which we have considered it, during the revolution of F18. 111. 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 ISq, TH'q' ■will increase, and the arcs gW and q''S diminish ; from ■« hieh it ap- pears, 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 dif- ference 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. OhoMge of the inclination of the orbit. If we refer to Pig. 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 in- crease whUe 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 indiuation will diminish, and while it is moving from L' to N the inclination will increase. The diminutions and incre- ments will compensate each other, and the original inclination wiU 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 incUnation will decrease from j to L and from g' to L', and increase from L to g' and from L' to q ; the effects being the same as when the nodes are in quadratures over the arcs gL and LN in Fig. 116, and NL and Lg' in Fig. Ill, and being reversed over FlQ. 116. Fia. 117. the arcs Ng and Wq' in Fig. 116, and qS and j'N' in Fig. 117. When tlie sun has the position represented in Fig. 116, the arcs of increase Lg' and L'g will be greater than the arcs of diminution gL and g'L'. The disturbing force wUl 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 Pig. 117, the arcs of diminution gL and g'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. S14. Consequences of the sun's motion during the revolution of the moon. As the sun moveE from S towards W (Fig. 1 16) the arcs Lg', L'g, over wMoh there is an increase of the inchnation, will increase ; and the arcs gL, g'L', over which there is a diminution, wiU diminish. The motion of the sun will, therefore, in ap- 296 PERTURBATIONS OF ELLIPTIC MOTION OF THE MOON. proaching the descending node, render the increase of the iiidination each revolu- tion 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 sta- tionary. The results wiU 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 dimi- nishes as the sun recedes from either node, and increases as it approadies eithor node, it wUl 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 considermg is modified by the retrograde motion of the node ; and thus, that, be- sides 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 com- pleting a revolution with respect to both the sun and moon. 516. Pertui-batlons Periodic. The perturbations of the elliptie motion of the moon, comprising ineqnaUties 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 wUl disappear when the configurations upon which they depend become the same. They are therefore periodical. 517. Xlie Perturbations of the ITIotions 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 are 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 dis- turbing 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 accompanyuig 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 com- prise many thousands of years, and on this account they are called Secula/r Mo- tions 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 pro- duced by a change of configurations are much more slowly effected, and thus the periods of the inequahties are much longer. 518. Acceleratlou of tlie IHoon. The motions of the moon would be subject to no secular variations if the apparent orbit of the sun were unchange- able ; but the secular variation of the eccentricity of the sun's orbit, which an- swers 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 inequahty 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: — Eelative Intensity of the Foece of Gea- VITY AT THEIK SuEFACE. 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 i-atio 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 ele- ment upon which it depends. This is one method of determining the masses of the planets. The masses of those planets which haye satellites may be found by another and simpler method, viz. : by comparing the attract- ive 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 in- versely as the squares of the distances of the satellite from the primary and of the primary from the sun. Thus calling the forces j^ F, the masses m, M, and the distances d, D, we have whence we obtain m : M.y./d' : FD^ 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, ill, and VI. The ratio of/ to F is equal to the ratio of the versed sines of the arcs actually de- scribed 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 lY 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. 520. Computation of the Densities of the Planets. 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 inspect- ing 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. The 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 suppos- ing 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. The 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 EAKTH. 29S CHAPTER XXV. Form and Density of the Earth : — Changes of its Period OF Eotation. — Precession of the Equinoxes, and Nutation. 522. We 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 Y^ff. 523. Density of the Eartli. From the amount and law of variation of the force of gravity upon the earth's surface, ascertained by observations upon tiie length of the seconds' pen- dulum, it is proved that the matter of the earth is not homo- geneous, 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 moun- tain. 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 moun- tain 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 deviation of the plumb-line from the true ver- tical 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 determin- ing the mean density of the earth by Dr. Maskelyne, in 1774, on the sides of the mountain Schehallien in Scotland. The ob- served 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 moun- tain 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 5.4-t. 524. Explanation of Spheroidal Form of £artfa. 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 animated 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 centrifu- gal 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 o^ 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 equili- brium, if it had the figure of an oblate spheroid, of which the axis was to the equatorial diameter as 229 to 2S0, or of which the oblateness was -jj^. 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. Po§$ibIe Changcii 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 distribu- tion of the matter of the earth, or in case of the impact of a foreign 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 alteratioa in the circumstances of the earth's rotation on its axis. earth's AXIS INVAEIABLB. 801 526. Earth's Dimensions and Axis Invariable. It in ascertained from direct observation, that there has in reality beea 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, d posteriori, that there has been no material change in the form and dimensions of the earth in this interval. Were the axis of the earth to experience any change of posi- tion with respect to the matter of the earth, the latitudes of places would be altered. A motion of 100 feet might increase or di- minish 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 haye 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 ar- ranged symmetrically with respect to the present axis of rotation, it is to be inferred that this axis is the same as that which ob- tained at the epoch when the matter of the earth changed from 'a fluid to a solid state. 5!27. Ptiysical Theory of Precession and IVntation. 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 theyare 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 ques- tion 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 thp 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. 528. 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 phe- nomenon known by the name of the Tides. The rise of the water is called the Flood Tide, and the fall the Ebh Tide. 529. Times of High and LiOtF 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, em- ploying 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 time of high water from one day to another is 50m., in its mean state. The time of high water is mainly dependent upon the position of the moon, being always, at any given place, about the same length of time after the moon's passage over the superior or in- ferior 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 dif- ferent at different places, and in some instances is of such length that the time of high water seems to precede the moon's passage. 530. 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 phases of the moon. It is greatest soon after thesyzigies; after which it diminishes and becomes the least soon after the quadratures. The tides which occur near the syzigies, are called the Spring Tides ; and those which occur near the quadratures are called the Neap 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 lilie 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 THE TIDES. 303 On the Atlantic coast of the United States the spring tides ex- ceed the neap tides in the proportion of 3 to 2. The tides are also affected hy 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 diflFer somewhat in height. At Brest, in the syzigies of the sum- mer 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 distance of the moon from the earth has also a sensible influ- ence upon 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 re-' tardation 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 quadratures, when the tidesare at their minimum. It also varies with the distances of the sun and moon from the earth, and with their declinations. 532. Physical 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 ia 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 conse- quence of the inequality and non-parallelism of the attractive forces exerted by the moon, as well as by the sun, upon the dif- ferent particles of the earth's mass. From this cause there re- sults 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 acid (Fig 118) represents the earti, and M tbe place of the moon ; then a will be the point of the earth's surface directly under the moon, h the point diametrically oppo- site to this, and the right line dc, per- pendicular to MO ,will represent the circle traced on the earth's surface 90° distant from a and i. Now, the at- traction of the moon for the general mass of the earth is the same as if the whole mass were concentrated at the centre 0. But the centre of the earth is more distant from the moon than the point a at tbe surface. It follows, therefore, that a particle of matter situ- ated at a will be drawn towards the moon with a proportionally greater force than the centre, or than the gen- eral mass of the earth. Its gravity or 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 point b. It is therefore attracted more strongly than a particle at h. The ex- cess will be a force tending to draw tbe 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 dimin- ished, as well as the particle at a. 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 ik parallel to OM, and partially in the direction of the tangent or horizontal line is. The component {is) 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 tk, 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, there- fore, that the component in question will be greater than the attraction for the centre, in the vicinity of the point a, and for a certain distance from it in all directions. The gravity of the par- ticles 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 h. Let us now FiS. 118. THEORY OF THE TIDES. 805 consider a particle at c, 90° from tbe points a and h. The at- traction of the moon for it will take effect in the two directions d 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 55° from a and b in all directions; and is augumented for a distance of 35° on each side of the circle dc, 90° distant from the points a and b. These distances are represented in the Figure. Tliis 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. —xy{l — 3 cos" ^). If we a' consider m as denoting the mass of the moon, a the moon's distance from the earth's centre, y tlie distance of a particle of matter at some point t of the earth's surface from the earth's centre, and

stream does not begin until half ebb-tide, and in New York Bay it begins at one- sixth of the ebb tide. Tidal currents owe their origin to the resistance opposed by shallow waters, and contracted channels, to the free propagation 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 Sound, and in Hell Gate. About the time of the turn of the tide, at the head of the Sound, there is a certain interval of slackwater there. After the tide- wave begins to move in the opposite direction, the aoonmulafve effect of TIDES OF THE PACIFIC COAST. 311 the resistances determines, in a certain interval of time, a sensible current, whioli shows itself first at the surface and in-shore, hut soon becomes general. In mid- channel, 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 Iriih Channel. The turu of the stream, whether flood or ebb, is simultaneous 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 considera- ble 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 aversige maximum drift of the cwreni 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 4J knots per hour ; and at Hell Gate nearly 6 knots. In New York har- bor 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 mean duration of the flood stream at difi'erent points of Long Island Sound varies between .4J^ hours and 7 J hours. The corresponding limits for the ebb stream are 5h. and S^h. The mean duration of slackwater varies between Om. and 45m. It is at most places less than 10m. The duration of the ebb or flood stream, differs as much as J of an hour in successive tides ; but commonly not more than 10m. The set of the currents is ordinarily nearly parallel to the shore. 343i Tides of Rivers. 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 Eichmond, the ebb continues 2^ hours longer than the flood tide. TIDES OF THE PACIFIC COAST. 544. Cotidal Liiiies. The cotidal lines of the Pacific coast of the United States are approximately paiallel to the coast. Thus, high tide occurs at about the same hour from San Fran- cisco to Vancouver's Island. South of San Francisco the tide- wave arrives at an earlier hour; at the southern extremity of California, about 2^ hours earlier. 545. Diurnal luequality. The tides of the Pacific coast 812 THE TIDES. are remarkable for the great inequality that prevails between the heights of tico 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 con- spicuous, but quite marked. The differences are greater for the ebb than for the flood tides. These diurnal inequalities increase with the moon's declination, north or south; and vanish en- tirely when the moon is in the equator. When the moon's de- clination 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 leet, and of the low- est 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 for 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 GULP OF MEXICO. 540. 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 ti^is tide is only from 1 foot to 1^ feet. The second tide is doubtless obliterated by the interference of the semi-diurnal flood-tide with a diurnal ebb-tide ; as happens ap- proximately on the Pacific coast (545). For some three to five days, 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 same places on the coast, the rise and fall 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 at- tributable 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. S13 TIDES OF THE MEDITEREANEAK 54T. The average height of the tide in the Mediterranean is said not to exceed l^ feet, though at some ports, as Tunis and Venice, it sometimes amounts to 3 or i feet. The Mediterranean is of sufficient extent for the sum 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 haa been detected at Chicago, on the southwestern shore of Lake Michigan. TIDES OP THE COAST OF EUROPE. 549. The tide-wave advancing from the south, makes a con- siderable angle with the coast of Europe, and thus the tide occurs continually later in following the coast from the Straits of Gibral- tar northward ; and along its entire extent from two to twelve hours later than the corresponding tides on the coast of the United States. Similar varieties of tidal phenomena occur on either coast. T/>e highest tides prevail in the Bristol Channel, and the Bay of St. Malo, on the northwest coast of France. At the head of the Bristol 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, the 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. From that point the height of the tide increases about equally in every direction, from to 15 feet on the opposite coast. The remarkably low tides at that locality result irom the fact that the tide stream is diverted by a promontory at the entrance of the channel to the opposite shore. ESTABLISHMENT OP 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 establishment of the port. The average of the intervals dur- ing a month's tides, is called the mean, or correct establishment. The mean establishment of Boston is llh.'ZTm. ; of New Haven llh. 16m.; of New York 8h. 13m.; of Charleston, S. C, 7h. and 26m. ; of San Francisco 12h. 12m. 551. Calculation of Time 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 da}"^, 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 establish- ment of the port. In the annual Coast Survey Reports a table is pub'ished, giving the interval be- tween the time of the moon's transit and the time of liigh water for different hours of transit, and for the principal ports on the U. S. coast. If the time of the moon's transit on any day be obtained from tlie Nautical Almanac, tlie interval correspond- ing to this time in the table, added to the time of transit, will give more aeeuralely the time of high water. 552. A tide table for the coast of the United States, is published in the same Reports, giving for numerous points of tlie coast the mean values of the interval be- tween 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 lime during which the sur- face of the water neither rises nor falls. A table is also given showing, for various ports, the rise and (all of tides corresponding to different hours of the moon's transit ; from which, by taking the time of transit for any day from the Almanac, the cor- responding rise and fall of the tide may be obtained for any of the ports mentioned in the tabia PAET III. ASTROlSrOMIOAL PROBLEMS. Explanations of the Tables. The Tables which form a part of this work, and which are em- ployed in the resolution of the following Problems, consist of Ta- nks 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, in- elusive, are, for the most part, abridged from Delambre's Solar Ta- bles. JThe mean longitudes of the sun and of his perigee for the beginnmg of each year, found in Table XVIII, have been com- puted from the formulae of Prof. Bessel, given in the Nautical Al- manac 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 Francoeur'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 frac- tion of a second of the result obtained immediately from De- lambre's Tables, as corrected by Bessel. The Tables of the Moon, which are from XXXIV to LXXXV, inclusive, are abridged and computed from Burckhardl's Tables of the Moon. To facilitate the determination of the hourly motions in longi- tude and latitude, the equations of the hourly motions have all been rendered positive, like those of the longitude. Some few new tables have been computed for the same purpose. The longitude and hourly motion in longitude will very rarely differ from the re suits of Burckhardt's Tables more than 0".5, and never as mucL 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 tablok is called the Argument of this quantity. 316 ASTRO^OMICAL PROBLEMS. Tlie angular arguments are expressed in some of the tables ac- cording 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., pun's. Tables are of Single or Double Entry, according as they con- tain one or tv/o arguments. The Epoch of a table is the instant of time for \viiich the quantities given by the table are computed. 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 ele- ments 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 II. (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 '1 .'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 from 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 pro- portional part of its variation in the interval between the giren date and the epoch of Table XII is easily derived. EXPLANATION OF THE TABLES. 817 Table XIV, contains certain other Corrections to be applied tc 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 uni- form degrees, in deducing then 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 arithmeti- cal operations Table XVIII, is a table of Epochs of the Sun's Mean Longi- tude, of the Longitude of the Perigee, and of the Arguments for finding the small equations of the Sun's place. They are all cal- culated 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 longi- tude of Venus ; Argument IV. is the heliocentric longitude of Mars ; Argument V. is the heliocentric longitude of Jupiter , Ar- gument VI. is the mean anomaly of the Moon ; Argument VII. is the heliocentric longitude of Saturn ; and Argument N is the sup- plement of the longitude of the Moon's Ascending Node. Argu- ment 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 re- maining 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 ; Argu- ments II. and V. for the equation depending on the action of Ju- piter ; and Arguments II. and VII. for the equation depending on the action of Saturn. Argument N is the argument for the Nuta- tion 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 irariations of the arguments from the begi» ning of any month to the beginning of any day of the month ; als 3 ihe 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 ASTRONOMICAL PROBLEMS. of each year contained in the table. It is found for any interme- diate lime 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 jn has the negative sign, its sup- plement 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'?. 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 Longi- tude, and of the Arguments of the equations used in determining the True Longitude and Latitude of the Moon. They are all cal- culated for the first of January of each year, at mean noon on the meridian of Greenwich. The Argument for the Evection is di- minished by 30'' ; the Anomaly by 2° ; the Argument for the Va- riation by 9°, and the mean longitude by 9° 45' ; and the Supple- ment 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. Tables 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 XL! 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, il EXPLANATION OP THE TABLES. 319 t;an be retained, and a cipher conceived to be written after the numbers in the cohimns of Arguments in the table. J n 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 Longitude. 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 determined by observation, to the corresponding latitude on the supposition of the earth being a sphere. The ellipticity to which the numbers in the table correspond is g-i^. Tables LXV and LXVI, Moon's Semi-diameter, and the Aug- mentation of the Semi-diameter depending on the altitude. Tables LXVII to LXXXV, inclusive, are for finding the Hourly 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. ; lOh. lOra. to equation 11. ; 10m. to equation III.; and 20m. to equation IV. Tables LXXXVII and LXXXVIII, are used with the preced- ing in finding the Approximate Time of Mean New or Full Moon in any given month of the year. Table LXXXIX furnishes the Equations for finding the Ap- proximate Time of New or Full Moon. Table XC contains the Mean Right Ascensions and Declina tions 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 Fourth Diflferences. These tables are given to facihtate the determina- tion, from the Nautical Almanac, of the moon's longitude or lati- tude for any time between noon and midnight. Table XCVI, Logistical Logarithms. This table is convenient in working proportions, when the terms are minutes and seconds, or degrees and minutes, or hours and minutes, — especially when the first term is Ih. or 60m 320 ASTRONOMICAL PROBLEMS. To find the logistical logarithm of a number composed ofmin utes and seconds, or degrees and minutes, of an arc ; or of min- utes 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 de- grees 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 com- plement 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 ob- tain 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, or 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 loga- rithms of the three given terms ; add together the logistical loga- rithms of the second and third terms and the arithmetical comple- ment 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 obtained 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 sec- onds, 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. Wlien the moon's hourly motion is 30' 12", what is its motion in 1 6m. 24s. ? As 60m. . . : 30' 12" . . . 2981 : : 16m. 24s. . . 5633 : 8' 15" .... 8614 2. If the moon's declination change 1° 81' in 12 hours, what wiU be thfe change in 7h. 42m. ? As 12h. . . . ar. co. 9.3010 : 1° 31' . . . 1.5973 •: 7h. 42m'. . . . 8917 : 0° 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. . . 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. lis. ? 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 dechnation change 54".7inone hour, what will be the change in 52m. 18s. ? Ans. 47".7. PROBLEM II. To take from a table the quantity corresponding to a given value of the argument, or to given values of the arguments of the table 21 S22 ASTRONOMICAL PROBLEMS. Case 1. When 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 esti- mated without the trouble of stating a proportion. Note 2. In some of the tables the degrees or signs of the quan- tity sought, are to be had by referring to the head or foot of the col- umn in which the minutes and seconds are found. (See Tables L, LI, LIT, and LV.) The degrees there found are to be taken^ if no horizontal mark intervenes ; otherwise, they are to be in- creased 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 consecu- tive 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, ac- cording 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 7'- 6° 24' 36". to find the corre spending quantity in Table L. 7^- 6° gives 0° 43' 17".4. The difference between 0° 43' 17". 4 and the next following quan- tity in the table is 1' 7".3. 60' : 1' 7".3 : : 24' 36" : 2r'.6.* * The student can work the proportion, either by the common method, or by Ic gistical logarithms, as he may prefer. In worliing this and all similar proportions hy 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 frac'iou to the nearcBl tenth 10 TAKE OUT A UUANTITY TROM A TABLE. 823 From 0° 43' 17" A Take 27 .6 42 49 .8 2. Given the argument 2'- 18° 41' 20", to find the corresponding quantity in Table XXV. 2'- 18° 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 1° 52' 32".5 Add 15 ,0 1 52 47 .5 3. Given the argument 9* 2° 13' 33", to find the correspond ing quantity in Table XII. 9»- 2° 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 5'- 8° 14' 52", to find the corresponding quantity in Table LII. Ans. 12' 36".0. 5. Given the ijgument 11'- 11° 23' 10", to find the correspond- ing quantity in Table LVI. Ans. 11' 48' .0. 6. Given the argument 0'- 26° 20', to find the corresponding quantity in Table XII. Ans. — 4^.0. Case >?. When the argument changes in the table by more or less than 1° ; or when it is given in lower denominations than signs. Take out of the table the quantity answering to the number in the column 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 tenn. This fourth term applied to the quantity first taken out, according to the rule given in the prece- ding 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 difterence of 10' in the argu ment. In obtaining quantities from these tables, it will be founa naore convenient to take for the first and se^nnd terms of the pro 324 ASTRONOMICAL PROBLEMS. portion, respectively, 10', and the difiference furnished by the table and work the proportion by the arithmetical method. (See note at bottom of page 268.) Exam. 1. Given the argument 0'- 24° 42' 15", to find the cor- responding quantity in Table LI. 0=- 24° 30' gives 9° 47' 14".3. The difference between 9° 47' 14". 3 and the next following quantity = 3 x 63".0 = 189".0. The argument changes by 30'. And the excess of 0'- 24° 42' 15" over O'*- 24° 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 9° 47' 14".3 Add 77 .2 9 48 31 .5 2. Given the argument 1° 12', to find the corresponding quar tity in Table VIII. 1° 10' gives 23' 13", and 5' : 33" : : 2' : 13" the correction. From 23' 13" Take 13 23 3. Given the argument 6'- 6° 7' 23", to find the corresponding quantity in Table LV. Ans. 90° 20' 53".5. 4. Given the argument 49° 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, mul tiply 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 „aken out of the table. The same rule will apply in taking quan- tities from tables in which the differences answering to a change of 10 in the argument are given, although the argument should actu- ally change by 50 or 100. If the argument changes by 100, mnl 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 com mon difference is 50, proceed as if it were 100, and double the result. TO TAKE OUT A QUANTITY FROM A TABLE. 325 ft Exam. 1. Given the argument 973, to find the corresponding quantity in Table XLV column headed 13. 970 gives 23".5. The diiference 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 quan- tity in Table XLI. Ans. 15' 37".7. 4. Given the argument 4225, to find the corresponding quan- tity in Table XLIII, column headed 8. Ans. 0' 47".2. Case 4. When the table is one of double entry, or quantities are 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 ex- cess 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 1. 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 quanti- ties correspondirig, respectively, to the value of the argument at the top next less than its given value and the other given argu- ment, 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- ments : 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- 326 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 al the top and 77 at the side to find the corresponding quantity in Table LaXXI. 50 and 70 give 47".7. The difference between 47".7 and the next quantity below it h I" A. The exce«s 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 -nl 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 .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 .9 Quantity corresponding to 220 and 448, 15 .1 The difference between 16".0 and the adjacent quantity in the next column on the right is 0".7. 0".7 3 To 15".l Corr for excess 3, .21 Add .2 15.3 TO CONVERT DEGREES, MINUTES, ETC., N TO TIME. 327 3. Given ihe argument 472 at the top and 786 at the side, to find the corresponding quantity in Table XXXI. Ans. 9".7, 4. Given the argument 620 at the top and 367 at the side, tt 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. Id" A. PROBLEM III. To convert Degrees, Minutes, and Seconds of the Equator into Hours, Minutes, <^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. 83° W 52" 4 gh. 32m. 47B. 28"' 2. Convert 34° 57' 46" into time. Ans. 2h. 19m. 5l8ec. 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 sail 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. yh. gn.. 34.. 60 4 ) 429 34 30" 3 desrees. i Ans. 171° 11' 15" 107° 23' 30" V. Convert llh. 24m. 45s. into degrees, &,c 328 ASTEONOMICAL PROBLEMS. PROBLEM V. The Longitudes of two Places, and the Time at one of them being given, to find the corresponding Time at the other. When the given time is in the morning, change it to astronomi cal 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 al- ready 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 longi- tude, 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 re- mainder 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, 24''- IS""- IS""- 22'- Diff. ofLong. ... 4 56 4 Time at New York . . 24 10 17 18 P. M. 2. When it is June 9th, 5h. 25m. lOsec. P. M. at Washington, what is the corresponding time at Greenwich 1 Time at Washington, June, 9''- 5'^- 25'"- 10'- Diff. of Long. ... 586 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 . . O''- g™- 21' .6 E. Do. of Philadelphia, . 5 39 .6 W. 5 10 1.2 15" .2h. 44m. 23* 5 10 1 Time at Paris, January, Diff. of Long. Time at Philadelphia, . 14 21 34 22 Or January 15th, 9h. 34m. 22sec. A. M. 4. When it is March 31st, 8h. 4m. 21sec. P. M. at New Haven, Rhat is the corresponding time at Berlin ? Ans. April 1st, Ih. 49m. 43sec. A. M. TO CONVERT APPARENT INTO MEAN TIME. 329 5. When it is August 10th, lOh. 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. 23(1, 7h. 3m. 55sec. A. M. PROBLEM VI. 77/e Apparent Time being given, to find 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, re- duce 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, lake out the motions in longitude for the given month, days, hours, and minutes, neg- lecting 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 the 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. When 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 Table XIII, and find by simple proportion the variation in the interval between the given year and 1840. The result, ap- plied 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 1 840, 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 withm a small fraction of a second is demanded, take the numbers in the columns entitled I, II, III, IV, V, N, in Tables, XVIIT, XIX, XX, answering respectively to the year, month, days, and hours, of the given time. With the respective sums of tlie numbers taken from each column, as arguments, enter Table XIV, and take out the corresponding quantities. These quantities added to the equatioi of time as given by Tables XII and XIII, and tne 330 ASTRONOMICAL PROBLEMS. constant 3.0s. subb'acted, will give the true Equation of Time, if tlie given time is Mean Time. When Apparent Time is given, i\ will be farther necessary to correct the equation of time as given by the tables, by stating the proportion, 24 hours : change of equation for 1° of longitude : : equation of time : correction. Note 2. The Equation of Time is given in the Nautical Alma- nac 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 sime place ? Time at New York, July, 1840, 16*- 9^- SS""- 22"- Diff. of Lonir. ... 4 56 4 Time at Greenwich, July, 1840, 16 14 31 26 1840 M. Long. 9'- 10° 12' 49' July 16d 5 29 23 16 14 47 5 14h 34 30 31m 1 16 M. Long 3 24 58 56 The equation of time in Table XII, corresponding to 3'' 24' 58 56", is + 5""- 44"- Mean Time at New York, July, 1840, 16''- 9^- SS"- 22"- 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, 8''- le"-- 15"'- 21»- Diff. of Long, ... 4 56 4 Time at Greenwich, 8 21 11 25 M. Long'. 1842 . 9"- 10° 43' 18" May 3 28 16 40 8d. 6 53 58 2lh. 51 45 11m. . 27 M. Long 1 16 46 8. Equa. of time, — 3m.45& TO CONVEUT APPARENT INTO MEAN TIME. 331 A.pparent Time at Greenwich, May, 1 842, Equation of Time, .... Mean Time at Greenwich, . DifF. of Lona; gd. 2ih. n™' 25'- -3 45 8 21 7 40 4 56 4 8 16 11 36 Mean Time at New York, . Or, May 9th, 4h. 11m. 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 1855 . . M. Long. I. n. m. IV. V. N. 9-- 10° 34' 30" 433 279 806 889 866 863 Feb. . . 1 33 18 47 85 138 45 7 5 3d. . . 1 58 17 68 5 9 3 2h. . . 4 56 3 43m. . . 1 46 10 13 12 47 551 369 953 937 873 868 Appar. Time at Greenwich, Feb., 1855, 3^- 2^- 43""' 36'- Equation of time by Table XII, . +14 8.6 lOOyrs. : 13s. (Sec. Var., Table XIII) :: 15yrs. : 1.9s. ... —1.9 Approx. Mean Time at Greenwich, 24h. : 6s. (change of equa. for 1° of long.) : : 14m. 11. III. . 11. IV. . II. V. I. . N. . Constant. . 0.1s. 3 2 57 42.7 + 0.1 0.8 0.4 1.0 0.3 0.1 —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 thn 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 1 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 froiK 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 suffi- ciently 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 50°. Add these to the mean re- fraction taken from Table VIII, if the barometer stands higher than 30in. and the thermometer lower than 50° ; but in the oppo- site 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 ba- rometer 29.5 inches, and the thermometer of Fahrenheit 70°, required the true altitude at the place of observation. The differ- ence between 29.5 inches and 30 inches is 0.5 inches, and the difference between 70° and 50° is 20°. Obs. alt. . 20°42'11".0 Refrac. (Table VIII), 2' 33".0; Bar.-l-lin.,5".12; ther.-l°,0".310 Corr.for— 0.5in.,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 11th of April, 1838, being 14° 17' 20", required the true altitude at the place of obser- vation. 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 ther- mometer 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 find 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 paral- lax in altitude by Table X, and add it to the observed altitude cor- rected for refraction. The result will be the true altitude sought. 2. If it is the altitude of the moon that is taken, and the hori- zontal 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 refrac tion. 3. If one of the planets is the body observed, the following for- mula will serve for the determination of the parallax in altitude when the horizontal parallax is 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 ob- tained 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 Al- manac. 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 ap- parent. 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 re- fraction 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 26° 40' 20", what is its true altitude ^ 334 ASTRONOMICAL PROBLEMS. Obs. alt. Refraction . True alt. at the station, Parallax in alt. (Table X), 26-' 40' 20" — 1 56 26 38 24 + 8 26 38 32 True altitude 2. Let the apparent altitude of the moon at New York on tha 17th of March, 1837, 8h. P. M., be 66° 10' 44" ; the barometei 30.4in. and the thermometer 62° ; required the true altitude. Appar. alt. . . 66° 10' 44" Meanrefrac. . 25.7 Corr. for + 0.4in., bar. + 0.3 Corr. for + 12°, ther. —0.6 Truerefrac. . . 25.4 logarithms cos. 9.60637 True alt. at N. York, 66 10 18.6 . 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 56° 54' 57", what was its apparent altitude at the time of the meridian passage, the horizontal pirallax 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. Refraction —1.0 + 37.9 log. 0.0158 Appar. alt. . . 56 55 34 4., What will be the true altitude of the sun on the 22d of Sep tember, 1840, at the time its apparent altitude is 39° 17' 50" ? Ans. 39° 16' 46". 5. Given 29° 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 the true altitude. Ans. 30° 22' 41". 6. Given 15° 24' 23" the true ahitude of Venus, and 8" its hori zontal parallax, to find the apparent altitude Ans. 1 5° 27' 41' . TO FIND THE SUn's r.ONGITDDE, ETC , FROM TABLES. 335 PROBLEM IX. To find the Sun's Longitude, Hourly Motion, and Semi-dtameter, for a given time, from the Tables. For the Longitude. When the given time is not for the meridian of Greenwich, re- duce 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 enti- tled M. Long, will be the tabular mean longitude of the sun ; the sum in the column entitled Long. Perigee will be the tabular lon- gitude 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 longi- tude. Subtract the longitude of the perigee from the sun's mean long-" tude, adding 12 signs when necessary to render the subtraction possible ; the remainder will be the sun's mean anomaly. W.th the mean anomaly take the equation of the sun's centre from Ta- ble 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 corre- sponding 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 Lu nar Nutation in Longitude from Table XXVII. Also take the So lar 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 Motion. With the sun's mean anomaly, take the hourly motion and semi- diameter from Tables XXIII and XXIV. * In adding quantities that are expressed in signs, degrees, &c., reject 12 or 24 eigns whenever the sura e.xceeds 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. 1 . 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 ex ceed .5. 2. The longitude of the sun, obtained by the foregoing rule, may differ about 3" from the same as derived from the most accu- rate 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 col- umns I, II, as arguments, take the corresponding equations from Tables XXlX and XXXIII. Also with the sun's mean anomaly take the equation for the variable part of the aberration from Ta- ble 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 rai-diameter, on the 25th October, 1837, a,t llh. 27m. 38s. A. M mean time at New York. Mean time at N. York, Oct. 1837, 24''- 23''- 27"'- 38'- Diff. of Long 4 56 4 Mean time at Greenwich, 25 4 23 42 1837 . October 25(1. . 4h. . 23m. . 42s. . Eq. Sun's Cent, I. II. III. II. IV. II. V. Const. . Lunar Nu.tation Solar Nutation San'struelong. M. Long. 9 M) 55 47.2 8 29 4 54.1 23 39 19.9 9 51.4 56.7 1.7 7 3 60 61.0 11 28 12 43.5 2.5 9.0 7. 19.3 3.0 7 2 4 16.0 — 6.3 — 1.2 7 2 4 8.5 Long. Perigee. I 9 10 II. 8 6'816 280 46'250j748 4810 66 6i 9 10 8 55 7 3 60 61 III, 549 215 107 1 IV. 321 397 35 882 94 872 753 416 939 348 63 5 N. 895 40 4 9 23 41 56 Mean Anomaly. Sun's Hourly Motion, . . 2' 29".7 Sun's Semi-diameter . 16' 17".8 2. Required the sun's longitude, hourly motion, and semi-diam etcr, on the 1 5th of July, 1837, at 8h, 20m. 40s. P. M mean time at Greenwich. TO FIND THE APPARENT OBLIQUITY OF THE ECLIPTIC. 337 1837 July 15d. 8h. 20m. 40s. . Eq. Sun's Cent. I. . . II. III. . . II. IV. . . II. V. . . I. VI. . II. VII. . . Aber. . . . Lunar Nutation Solar Nutation Sun's true long. IVI. Long. Long. Peri. 9 10 55 47.29 10 8 5 5 28 24 7.8 13 47 56.6 19 42.8 49.3 1.6 31 2 3 23 28 25.3 11 29 33 10.3 10.7 6.6 5.0 7.7 1.8 0.2 0.6 .9 10 8 38 3 23 28 25 6 13 19 47 Mean Anomaly. Sun's Hourly Motifin, . Sun's Semi-diameter, 3 23 2 8.2 — 7 + 0.8 3 23 2 1.2 816 129 473 11 429 n. Ill IV. V. 280549,321348 4968061263 41 815 418 20 604392 N. VL 895 787 27569 2.508 11 924 875 vn. 600 17 2 619 2' 23' 1 15' 45".4 3. Required the sun's longitude, hourly motion, and semi-diam- eter, on the 10th of June, 1838, at 9h. 45m. 26s. A. M. mean time at Philadelphia, (omitting the three smallest equations of longi- tude.) Ans. Sun's longitude, 2=- 19° 11' 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, 1 6' 14".7. PROBLEM X. To find the Apparent Ohliquity 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 echptic on the I5th of March, 1839. 22 338 ASTIIONOMICAL PROBLEMS. N. 1839, . 3 March, 9 15d. . 2 M. Obliquity, 23=27'36".9 14 . . . +9 .1 Solar Nutation for March 15th, +0 .5 Apparent Obliquity, . . 23 27 46 .5 2. Required the apparent obliquity of the ecliptic on t}ie 12th af July, 1845. Ans. 23° 27' 28".2. PROBLEM XI. Given the Sun's Longitude and the Obliquity of the Ecliptic, tc 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 u — 10, log. sin D = log. sin L + log. sin w — 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 me- ridian 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 ob- ■iquity of the ecliptic 23° 27' 36", to find his right ascension and decimation. 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— w= 23 27 36 . . . sin. 9.60000 D= 9 49 52S. . . . sin. 9.23235 — 2. The obliquity of the ecliptic being 23° 27' 30", required • The obliquity of the ecliptic at any given time for which the sun's longituda ffi known, is found by the foregoing Problem. TO FIND THE SUN S LONGITUDE AND DECLINATION. 389 ihe sun's right ascension and declination when his longitude is W 18' 25". Aus. Right ascension 41° 50' 30", and declination 16°8'40"N. PROBLEM XII. Given the Sun's Right Ascension and the Obliquity of the Eclip tic, tojind 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 u, 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 23° 27' 40" ? R = 142° 11' 34" . . . tan. 9.88979- u = 23 27 40 . . ar. co. cos. 0.03747 L = 139 46 30 . . . tan. 9.92726- R= 142 11 34 . . . sin. 9.78746 u= 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 obli- quity of the echptic 23° 27' 35", to find the longitude and declina- tion. Ans. Longitude 307° 59' 57", and declination 18° 17' 0" S. PROBLEM Xm. The Sun's Longitude and the Obliquity of the Ecliptic being given, to find the Angle of Position. Let p = angle of position ; w = obliquity of the ecliptic ; and L — sun's longitude. Then, log. tangp = log. cos L -}- log. tang u — 10. The angle of position is always less than 90°. The northern part of the circle of latitude will lie on the west or east side of the 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 obli- quity of tl" e ecliptic 23° 27' 32", required the angle of position. 340 ASTRONOMICAL PROBLEMS. L=24'' 15' 20" . . cos 9.93980 w = 23 27 32 . . tan. 9.63745 j9=21 35 10 . . tan. 9.59731 The northern part of the circle of latitude is to the wesl of the civcle of declination. 2. When the sun's longitude is 120° 18' 55", and the obliquity of the ecliptic 23° 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 find from the Tables, the Moon's Longitude, Latitude, Equa- torial Parallax, Semi-diameter, and Hourly Motion in Longi- tude and Latitude, for a given time. When the given time is not for the meridian of Greenwich, re- duce it to that meridian, and when it is apparent time convert it into mean time. Take from Table XXXV, and the following tables, the argu- ments 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 thousEinds 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, Va- riation, Longitude, Supplement of the Node, II, V, VI, VII, VIII, IX, and X ; and add the quantities in the column for the Supple- ment of the Node. For the Longitude. With the first twenty arguments of longitude, take from Tables XLI to XL VI, inclusive, the corresponding equations ; and with the Supplement of the Node for another argument, take the corre- sponding equation from Table XLIX. Place these twenty-one equations in a single column, entitled £^5. 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 Evectio/i 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 sum of the quantities in this column will be the corrected Anomaly. ro FIND THE MOON S LONGITUDE, ETC. 341 With the corrected Anomaly, take the Equation of the Centris 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 conslaats 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 equmox. 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 Ta- ble LIII, and add it to the Orbit Longitude. The sum will be the Longitude as reckoned from the mean equinox. With the Supple- ment 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. jpor the Latitude. The argument of the Reduction is also the 1st argument of Lat- itude. Place the sum of the first twenty-four equations of Longi- tude, taken to the nearest minute, in the column of Arg. IL 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 ar- gument 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 remain- ing 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 the Equatorial Parallax. With the corrected argiiments, Evection, Anomaly, and Varia- 842 ASTRONOMICAL PROBLEMS tion, take out the corresponding quantities from Tables LXl LXII, and LXIII. Their sum, increased by 7", will be the Equa corial 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 Table LXVII. Find the sum of these equations and the constant 3", and with this sum at the 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 sum just found, and with the result- ing sum at the top, and the corrected anomaly at the side, take the corresponding equation from Table LXX ; also with the corrected anomal}^ take ihe corresponding equation from Table LXXI. Add these two equations to the sum last found, and with the re- sulting 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 correspond ing equation from Table LXXIII. Add these two equations to the sum last found, and with the re suiting sum at the top, and the argument .of the Reduction at thij side, take the corresponding equation from Table LXXIV Also, with the argument of the Reduction take the corresponding equa- tion from Table LXXV. These two equations, added to the last sum, will give the sum of the 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. With the argument I of Latitude, take the corresponding equa tion from Table LXXIX. 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 corresponding equation from Ta- ble LXXXI. With the argument II of Latitude, take the corre- sponding 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 LXXXIII. Find the sum of iliose four equations and the TO FIND THE MOON's LONGITUDE, ETC. 343 constant!". 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 nep'ative 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 ar- guments, respectively, will be the arguments of eleven small addi- tional 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 va- riation 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 con- stant 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". The difference be- tween the sura 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 con- stant 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, take the cor- esponding equations fn ;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 precedent or follows the given time, with the arguments of Tables LXX, LXXII, and LXXIV, take the equations from Tables LXXVl 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 fol- lows the given time, add this sum to the hourly motion at the given time already found, and subtract 2".0 ; for the hour which pre- cedes, subtract it from the same quantity, and add 2".0. It will expedite the calculation to take the equations of the sec- ond 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 argu- ments 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 equatioH 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 wliich follows the given time, add this sum to the Hourly Motion in Latitude al- ready 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 lati- tude, on the 14th of October, 1838, at 6h. 54m. 34s. P. M. mear lime at New York. Mean time at New York, October, W- 6''- 54°'- 34'- Diff. of Long 4 56 4 i Mean time at Greenwich, October, 14 11 50 38 TO FIND THE MOON's LONGITUDE, ETC. 34S s o in coo 00 rH o 1— 1 o o a> CQ 00 W5 t' t^ * O — ( OS lO CO 1^ r- OD "^ ^ CM 00 Ir- o m CT Ol -rf to — 1 — (M -t -« CO CO -* en o ■ t- -* o o o f— 1 CO t^ c: r- r^ CO UO CO CO t^ ^ t- lO OO — ' r- m Oi O} Of ^ ^ CT o -^ in — CO '-0 CO — « o ^ -r CO in to ■* eo CO ^ r-i o yS -T CO CO ^ 1^ CO t- .^ CO OJ CO 00 CO CO 00 CT CT t- to to to Ci t- CO 00 CI in -^ 00 01 00 CTi CI 00 CO C — ' -r CJ CT in ^ Tf CT CO C- 00 g OS p^ CO — ' oi in OS m -^ in CT -^ r- uo 1— 1 ^ m 00 in GO 1838 . . October . 14d. . . llh. . . 50m. 38sec. >4 X 00 rH OS O^ O O 00 m to ' ^^ »n -^ o t* CO to to ^ o in CO GO to .-H CO CO '^ c CT o c OS o I 1-1 1— » t o o CO 00 rH o in 00 00 W w CO CO c- in to in (M to 1-H o in t- 00 "^ "* o -H -^ Tj< in 00 Tf '<* O O OS 00 OS OT in .-^ t-^ o o CI o* CO CT CT o» m CO -g •« — S " S 00 i? ^ — O 00 3 11 -^ ^ m CO CC CO OS OS c*i o hP=! S 346 ASTRONCUICAL PROBLEMS 1 •9 -. ooosm -#0 OOJ <'? .T « CO OT i-H .w -ot~: H m-* ^ oom s s a 7 CO o Ol • & • 6- • 1 « s„- (3 5 _w _w _ ■ ■ 1 ^^'^ o £; , S • 3 , o 13 o B p 3 o .£.h£§ ^% EC ^fi^t^p^O v3 •^ rt 3 ei O O O « L^ o 00 CM 00 ooto t- ^^05 TftO rt ^^ ^-S 3 t W5 P-H O O 1-H « d d r-i oi -; -; g==5 0(N -^ l^ 9S f ,7 rt > ^^_c ^.^ ^o. o o ^S o >> 1 a E 1 £ »4 n ^ ^ Ph O rf P5 O 3 ^ r-* (-4 • * CO lo m t* H a ^nco C "" 0(N <* » _0Q m lO ffl A s -1 • . • • (U .J iS a 1 < a (^ B J W M lill 1 § '? s rH o lo oi lo '?ps>^.ini^oiO'*(NO(Mr-mQOrtcoo m <£> 0^ IJ^ •* ■* <» - in sj so irf m tocri d Qodod "^oo TJ rH 00 -Jco ddo.iri t^ to CO c6 S" _BJ (N mcj « CT -■ ^ rt MCT rt rt rt .-1 lO -*CT r- ro ■^ in j A ^ m (Mooooo o>-( -^ r-l ■J to t m CT CO Tjt -- CM CO rf i- o o CT CO Oi ot (N H E >^ £ A E i rt-H«rt>H-Hrtr-lr-.rt01« § Sr'^ = 5 3 B 3 o (»C=3 32^ Ki> tc .%f>Vli^-.6 V ' TQ FIND THE MOON S LONGITUDE, ETC 347 S ISS§= "in ■n S IS5§= 'to 00 OS 35?^ 00 -=:** O -^ CT O ''t lO o CT o o o: 00 o to ?iE^'^ o OS o to ■<# ^ to CT CO 00 »0* CO -H OS 00 "OS _p°_ o o §! CO 1— 1 CT t^ o o t^ o ^ § o m to o C^ CT o 00 — o "lO 00 00 I-H ^ t^ CM t^ »n o Oi 00 »o (-- r- c^ -H -H 00 CO — t ro O CO (M in 1^ OJ CO r- CTi -H CTl 00 '^ -^ '* -r in 1 00 C^ CO t^ r-< t- CT t- -H ^ Ci rt to CO i-H CO CO r- '* --I t- CO Ol ^ lO CO CO >— 1 rM CO -«* O -^ c; O CO 1-- f- GO -^ to CTJ 00 o CO o:, as ■<* to ^ ^ c^J -^ — 1 CO CO -^ O 1— ( CM 1—1 lO CO ^ CO CO CM Oi in :Z m r- iO '^ r-t I— 31 O r^ (M OS o -H ir^ — 1 CO (M ^ Ci Acs o CO en coo to r^ ^ GO CO — < ^ LO to CO OD to O in in 00 O O ^ CD CO CO tra r- t>- i-H CO UOO r^ O to iO to CM t* Ol CM O CO CN C-- UO liO CT uo r~ t- '^ in o 2 r-H co t^ Oi t- — 1 CO vjO to CO C- 1— ' t- in 00 r-t C- r-^ -^ CO iO cncM (M -* w CM o CO m -^ ■X) in -^ to a> CO in -* CO CO ^ ^ o to -f CO CO ■-< r-H CO l~- >-l CO GO CO eo 00 OifM t- to to CD CM t^ CO CO CD in -* (M QO 01 ex- en oi 00 CO O -H Tt< C?S CM lO Tt -^ CM CO r- 00 __ _ CO r-t a> uo OT in -^ in CM -^ t-- in --H o -*co o i> o OS to OS - ' • ' * o t-H ■^ (M ^- in ^ o in 1-H GO CO fH CO CO i 1 C4H o )-H *n cx) — ( OS i-H o lO in >< 00 rt CTiOlOO 00 U) to w ^ o o rH !>■ CO to to — * o in CO 00 to rH CO 1-, 1--Tj< CJO 9 1-^ I-H > CO -^ 00 in rH o in (M o CO rH eo CO 1-H OS o !>• CO rH o in r^ rH to rH CO T}< c-co 1 > o o CO 00 rH o in CO o in ?5 ^ to m CM to rH o m t^ OO ^ rH CO 00 (M -^ in in to I-H ^ -.(Mast^coorH (M CO in (M -^ o rH '^ Tt in CM -HCM rH CO " in in -^ CO -5 0) -a o -^ 00 00 ■<* to o a o CO 00 1-- to d r-H CM rH (M in -* CO t-: (M CM =3 " incM ^ rH OS in o i CO O rH Tj< i-t 1-H l^ 1-H CM r^ OS •S° 1-t in to 00 -^-^ OO OJ OS ^to O 00 00 ■* 6 =: ostMinrHr-ldo CT CM CO (M CM CM m i^ in CM CO ffQ O O 1 1 " be J ^ rH OS C^ CT ^- O CM -H (M Tf OS r-l in ^ ^ 1 1 r-( O -^t- r^tO CM ° (MCM -H •-* I-H , >-* _ rH rH in * rH r-H liO in o .S 1 t rHQor-in'tpost-- m (M -^ rH (M rH -* " -«j* -^ 00 in in to (M CM COCM CO »n • in • " c o Tj* 00 oo in CM (MCM ^ » rH (M lO 2| rH X) <1> >3 OS l> to (M rH 1> O c^ 4 ^ (MCOrHt-^Cod-^ * COCO'^-HrHCM'-H s? ^ i ^ in -^ o OS t^ to (M Tf in in CM ^ -. o CO OS in CO O rH(M rH CO B o o in in ^ d .2 :z OSTt-COtM^OOt^ CM in r-H CO rH Tp - ■^ Tt to rH CO ■*# in (M rH CM CO S > o CO OS b- in CM CM CM to H - CO CO Tft n & W •1 • • -."s CO o . . J S p S C •-( '-I lo m !K 848 ASTRONOMICAL PROBLEMS. S oo ai Tt< om t* f^ 1^ be eo ■HH- « n a (M Of g 1 ^ mcocjt-ioooooooo C*3 -^ & & cqU I S 3 S 5 ■S SS iS Ed lA O t-H t~ o -J IN -HH- 120.2 2.6 4.1 05 o CO .«^ .w a c 3 O 18 si o o MCSBS XO tow coco ss o u 11 s s o o 11 H OrHOOOOCOOi-itOlricO S '2 oooooooo H S o rs o o rHC^'f^irStOOOCTiCTCO^Sj^ to Tt« O i^ CO CO t- o lo r-Soo ^o oooo oo o o o o o S ^ ©5 Pi^CT lO OJ -^ If'ooooooooooooooOoo — I t?l CO"^irtCOt-Q0050^HCTCO'tu^tOt-00 o m «-* -1 m t' CO Si CO mm O! 03 e •a .2 r. " " C 03 (u 5 > > " E ■ S ■ o = = c 03 ?^^^^^ o: o »-l CO o CO W5 on « QOO T-< 00 OS *i S9 1-1 E c a o s ^^i-|{»cococopccu^^^o^^■*c^oc^^^coco-^C4--ll:^Jr-^c3o^fl■T)^c;■^ ^■y^cicoij;?p^tocioG6oa6Tt'c6'^'r-;o6'--HCOoo --,_,30«OC0 OCO=OC05DTj• 4C"'- 22'- = 6382'- log. 3.8049S TO CALCULATE A SOLAR ECLii-SE 375 Middle, . 6 20 22'- = 6382'- 36 log. 3.80495 Beginning, . End, 4 34 8 6 14 P. M. 58 P. M. S-d+c S-d-c • • • • . 2357" . 1019 log. 3.37230 log. 3.00817 2 ) 6.38053 O"-- 46"" 6 20 9'- = 2769'- 36 3.19026 R 0.25206 Middle, '. log. 3.44232 Beg. of total eclipse, 5 34 27 P. M. End of total eclipse, 7 6 45 P. M S+d-c d Quantity, 973" 0.77815 log. 3.47202 ar. CO. log. 7.01189 18.3 digits, log. 1.26200 PROBLEM XXX. To calculate an Eclipse of the Sun, for a given Place. Having found by the rule given in the note to Problem XXVIII, that there is a probability that the eclipse will be visible at the given place, and calculated the approximate time of new moon by Problem XXVII, find from the tables, for this time or for the near- est whole or half hour, the sun's longitude, hourly motion, and semi-diameler ; and the moon's longitude, latitude, equatorial par- allax, semi-diameter, and hourly motions in longitude and latitude. Find also by Problem XVI, the longitude and altitude of the nonagesimal degree ; and thence compute by Problem XVII, the apparent longitude, latitude, and augmented semi-diameter of the moon, (using the relative horizontal parallax.) With these data compute the apparent distance of the centres of the sun and moon, at the time in question, by means of the following formulae ; log. tang ^ = log. X' -|- ar. co. log. a ; log. A = log. a. -(- ar. CO. log. cos & : 376 ASTRONOMICAL PROBLEMS. in which A =: appar. distance of centres ; X' = appar. Lat. of Moon ; o = Difi'. of appar. Long, of Moon and Sun = diff. of r.ppar. long, of Moon (found as above) and true long, of Sua 6 is an auxiliary arc. The value of 6 being derived from the first equation, the second will then make known the value of A. a and X' are in every instance to be affected with the positive sign.* For the Approximate Times of Beginning, Greatest Obscuration, and End. Let the time for which the above calculations are made, be de- noted by T. If the apparent distance of the centres of the sun and moon, found for the time T, is less than the siun of their ap- aarent semi-diameters, there is an eclipse at this time. But if ti is greater, either the eclipse has not yet commenced, or it has al- ready terminated. It has not commenced if the apparent longitude of the moon is less than the longitude of the sun ; and has termi- nated, if the apparent longitude of the moon is greater than the longitude of the sun. 1. If there should be an eclipse at the time T, from the sun's longitude and hourly motion in longitude, and the moon's longi- tude and latitude, and hourly motions in longitude and latitude, found for this time, calculate the longitudes and the moon's lati- tude for two instants respectively an hour before, and an hour after the time T. The semi-diameter of the sun, and the equatorial parallax and semi-diameter of the moon, may, in our present in- quiry, be regarded as remaining the same during the eclipse. Find the apparent longitude and latitude, and the augmented semi-diam- eter of the moon, (using in all cases the relative parallax,) and thence compute by the formulae already given, the apparent dis tance of the centres of the sun and moon at the two instants in question. Observe for each result, whether it is less or greater than the sum of the apparent semi-diameters of the two bodies. If the moon is apparently on the same side of the sun at the times T and T + Ih., take the difference of the distances of the two bodies in apparent longitude at these times, but, if it is on opposite sides, take their sum, and it will be the variation of this distance in the * A, the apparent distance of the centres, may be found without the aid of loga- rithms by means of the following equation : A = n/ a2 -I- X'2. Jf the logarithmic formuloB are used, it will be sufficient here to take out the angle 9 to the nearest minute. When we have occasion to obtain the distance of the centres exact to within a small fraction of a second, d must be taken to the nearest Una of seconds, if it exceeds 20° or SO*". TO CALCULATE A SOLAR ECLIPSE. 877 hour following T. Find in like manner the variation ol the dis- tance during the hour preceding T. Then, if the apparent distance of the centres at the times (T — Ih.), (T + Ih.) is less than the sum of the apparent semi-diameters, dednce from these results the variations of the distance in apparent longitude during the pre- ceding and following hours, allowing for the second difference, and observing whether the two bodies are approaching each other, or receding from each other. Thence, find the distance in apparent longitude at the times (T — 2h.), (T + 2h.) Find by the same method the apparent latitude of the moon at the instants (T — 2h.), (T + 2h.), observing that the variation of the apparent latitude in any given interval is the difference between the latitudes at the beginning and end of it, if they are both of the same name ; theii sum, if they are of opposite names. From these results derive the apparent distance of the centres of the sun and moon at the two instants in question. If there should still be an eclipse at the time (T + 2h.) or (T — 2h.), find by the same method the distance of the centres al the time (T + 3h.) or (T — 3h.) These calculations being effect- ed, the times of the beginning, greatest obscuration, and end of the echpse, will fall between some of the instants T,(T— lh.),(T-|- Ih.), &c., for which the apparent distance of the centres is computed. 2. If the eclipse occurs after the time T, the different phases will happen between the instants T, (T + Ih.), (T + 2h.), &c. Find the apparent distance of the centres of the sun and moon for the times (T + Ih.), (T -|- 2h.), by the same method as that by which it is found for the times (T -|- Ih.), (T — Ih.), in the case just considered. Then, if the eclipse has not terminated, deduce the distance of the moon from the sun in apparent longitude, and the moon's apparent latitude, for the time (T +3h.), from these distances and latitudes at the times T, (T-f Ih.), (T-|-2h.); as in the preceding case the distance and latitude for the time (T-H2h.) were deduced from the same at the times (T — Ih.), T, (T -\- Ih.) With the results obtained compute the apparent dis- tance of the centres of the two bodies at the time (T -|- 3h.) 3. In case the eclipse occurs before the time T, the apparent distance of the centres must be found by similar methods for the times (T - Ih.), (T - 2h.), &c. The calculation is to be continued until the distance, from being less, becomes greater than the sum of the semi-diameters. Now, let h = variation of apparent distance of centres in the interval of one hour comprised between the first two of the instants for which the distance is computed ; d = difference between the sum of the semi-diameters of the sun and moon and the apparent distance of their centres at the first instant ; and t = interval be- tween first instant and the time of the beginning of the eclipse Then, h:d:: 60""- • t (nearly.) 378 ASTRONOMICAL PROBLEMS. Find the value of t given by this proportion, and add it to llie time at the first instant, and the result will be a first apprnximation to the time of the beginning of the eclipse, which call i. Find, by interpolation,* the distance of the moon from the sun in appa rent longitude (a), and the moon's apparent latitude (XO, for this time, and thence compute the apparent distance of the centres. Take h = variation of apparent distance in the interval between the time b and the nearest of the two instants above mentioned, be- tween which the beginning falls, and d = difference between the apparent distance of the centres at the time b and the sum of the semi-disimeters, and compute again the value of t. Add this to the time b, or subtract it from it, according as b is before or after the beginning, and the result will be a second approximation to the time of the beginning, which call B. A result still more approxi- mate may be had, by taking h = variation of apparent distance of centres hi the interval B — 6, d = difference between apparent dis- tance at the time B and sum of semi-diameters, finding anew the value of t given by the preceding proportion, and adding it to, or subtracting it from, as the case may be, the time B. But prepara tory to the calculation of the exact times, it will suffice, in general, to take the first approximation. The end of the eclipse will fall between the last two of the several instants for which the apparent distance of the centres of the moon and sun have been computed. The approximate time of the end is found by the same method as that of the beginning.! * The second differences may easily be taken into the account in finding the quantities a and A' for the time b. Thus, let k = variation of a for the interval of an hour comprised between the instants above mentioned, k" = same for the suc- ceeding hour, and i = interval between A and the nearer of the two instants, (in f^ f^ 1^1 minutes.; Then, if we put /= 77, c = — ^-^ — , and v = var. of o in interval i, o 00 .■1/^(^ + 1) j 10 The upper sign is to be used when the time b is nearer the first than the second instant, the lower when it is nearer the second than the first c is to be used with its sign. The error by this method will not exceed the number c, (supposing the changes of k, k', from 10m. to 10m. to increase or decrease by equal degrees.) The general formula for interpolation is Q =g + - 'n which q is the first of a series of values, found at equal intervals, of the quantity whose value Q at the time ( Is sought, t is reck- oned from the time for which q is found. A is one of the equal intervals, d', d", d'", Sec, are the first, second, third, &c., differences. If we make h = 1, we hare Cl^q+td' + iS^d"+iSL=-^^d"-^^0. t In effecting the reductions of the quantilies a and X' to the first approximate time of end, f must stand for the variation of a during the hour preceding that comprised between the la^t two instants, and the last instant must be snbstiinted for the first. (See Note above.) TO CA.CULATE A SOLAR ECLIPSE. 879 The middle of the interval between the approximate times ol the beginning and end of the eclipse, will be a first approximation to the time of greatest obscuration. Note. When the object is merely to prepare for an ooservation results sufficiently near the truth may be obtained by a graphical construction. The elements of the construction are the difference of the apparent longitudes of the moon and sun, and the apparent latitude of the moon, found as above, for two or more instants du- ring the continuance of the eclipse. Draw a right line EF, (Fig. 119,) to represent the ecliptic, assume on it some point C for the Fig. 119. position of the sun at the instant of apparent conjunction, and lay off CA, CA', equal to the two riifferences of apparent longitude ; and to the right or left, according as the moon is to the west oi east of the sun at the instants for which the calculations have been made. Erect the perpendiculars kp, k'p', and mark off Aa, A'a' equal to the two apparent latitudes. Through a, a', draw a right line, and it will be the apparent relative orbit of the moon, or will differ but little from it. From C let fall the perpendicular Cm upon the relative orbit, m will be the apparent place of the moon at the instant of greatest obscuration. Take a distance in the di- viders equal to the sum of the apparent semi-diameters of the moon and sun, and placing one foot of it at C, mark off with the other the points/,/, for the beginning and end of the eclipse, and by ~ means of a square mark on EF the points b, e, which answer to the beginning and end. If the eclipse be total or annular, mark the points of immersion and emersion, g, g', with an opening in the dividers equal to the difference of the semi-diameters, and find the corresponding points b', e' on the line EF. If the calculations are made from hour to hour, the distance AA' is the apparent relative hourly motion of the sun and moon in lon- gitude. This distance laid off repeatedly to the right and left will determine the pomts 1, 2, &c., answering to Ih., 2h., &c. before 380 ASTRONOMICAL PROBLEMS. and after the times for which the calculations are made. If tlie spaces in which the points b, e, answering to the beginning and end of the eclipse, occur, be divided into quarters, and then sub- divided into three equal parts or five-minute spaces, the approxi- mate times of the beginning and end of the eclipse will become known. From the point m, as a centre, describe the lunar disc ; and from the point C, as a centre, describe the sun's disc, and we shall have the figure of the greatest eclipse. The quantity of the eclipse will result from the proportion SN : MN : : 12 : number of digits eclipsed. Draw from the centre C to the place of commencement/, tlie line C/; and through the same point C raise a perpendicular to the ecliptic. With the longitude of the sun at the time of the be- ginning, calculate its angle of position by Problem XIII, and lay it off in the figure, placing the circle of declination CP to the left if the tangent of the angle of position be positive, to the right if it be negative. Compute also for the time of beginning the angle of the vertical circle of the sun with the circle of declination, that is, the angle PSZ in Fig. 6 p. 13, for which we have in the triangle PSZ the side PS = co-dechnation, the side PZ = co-latitude, and the included angle ZPS. (The requisite formulse are given in the Ap- pendix.) Form this angle in the figure at the point C, placing CZ to the right or left of CP, according as the time is in the forenoon or afternoon ; CZ will be the vertical, and Z the vertex, or highest point of the sun. The arc Zt on the limb of the sun will be the angular distance from the vertex of the point on the limb at which the eclipse commences. For the True Times of Beginning, Greatest Obscuration, and End. The approximate times of beginning, greatest obscuration, and end of the eclipse, being calculated by the rules which have been given, find from the tables, or from the Nautical Almanac, (see Problem XXXI,) the moon's longitude, latitude, equatorial paral- lax, semi-diameter, and hourly motions in longitude and latitude, for the approximate time of greatest obscuration.* With the moon's longitude and latitude, and hourly motions in longitude and latitude, found for this time, calculate the longitude and latitude for the ap- proximate times of beginning and end. The parallax and semi- diameter may, without material error, be considered the same during the eclipse. With the moon's true longitude, latitude, and semi-diameter at the approximate times of beginning, greatest ob- scuration, and end, calculate its apparent longitude and latitude, * It will, in general, suffice to calculate the moon's longitude and latitude from tlie elements already found for the approximate time of full moon, if these have been accurately determined. The equatorial parallax and semi-diameter may be found by interpolation from the N autical Almanac. ' TO CALCULATE A SCLAR ECLIPSE. 381 and augmented semi-diameter, for these several times, (making usa of the relative parallax.) With the sun's longitude and hourly mo- tion previously found for the approximate time of new moon, find his longitude at the approximate times of beginning, greatest ob- scuration, and end. The sun's semi-diameter found for the ap- proximate time of new moon, will serve also for any time during the eclipse. With the data thus obtained, calculate by the formu- lae given on page 375 the apparent distance of the centres of the sun and moon at the approximate times of the three phases. Note. When very great accuracy is required, the moon's longi- tude, latitude, equatorial parallax, semi-diameter, and hourly mo- tions in longitude and latitude, must be calculated directly from the tables, or from the Nautical Almanac, for the approximate times of the beginning and end, as well as for that of the greatest obscuration. For the Beginning. Subtract the apparent longitude of the moon at the approximate time of beginning from the true longitude of the sun at the same time, and denote the difference by a. Do the same for the approx- imate time of greatest obscuration. Subtract the latter result from the former, paying attention to the signs, and call the remainder k. Next, take the difference between the apparent latitudes of the moon at the approximate times of beginning and greatest obscura- tion, if they are of the same name ; their sum, if they are of oppo- site names ; and denote the difference or sum, as the case may be, by n. This done, compute the correction to be apphed to the ap- proximate time of beginning by means of the following formulae : log. 6 =log. a +log. k + ar. co. log. n — 10 ; c=\' -b,S = d + 5-5"; log t — log. (S + A) + log. (S - A) -I- ar. co. log. n + ar. CO. log. c -1- log. L + 1.47712 - 20 : in which t — Correction of approx. time of beginn. (required) ; a = Diff. of appar. long, of Moon and Sun at approx. time ; L= Half duration of eclipse in minutes (known approximately) ; k = Appar. relative motion of Sun and Moon in long, in the in- terval L ; n = Moon's appar. m.otion in lat. in same interval ; \'= Moon's appar. lat. ; d = Augmented semi-diameter of the Moon ; 6 = Semi-diam. of Sun ; A = Appar. distance of centres of Sun and Moon. b and c are auxiliary quantities. First find the value of b by the first equation, and substitute it in the second. Then derive the values of c and S from the second ASTRONOMICAL PROBLEMS. ' and tiud equations, and substitute them in the fourth, and it wil] make known the value of t, which is to be apphed to the approxi- mate time of the beginning of the eclipse according to its sign. The quantities a, k, n, &c., are all to be expressed in seconds. The apparent latitude X' must be affected with the negative sign when it is south. The motion in latitude, n, must also have the negative sign in case the moon is apparently receding from the north pole, a and k are always positive.* The result may be verified, and corrected, by computing the ap- parent distance of the centres at the time found, and comparing it with the sum of the semi-diameters minus 5". Note. When great precision is desired, the quantities k and n must be found for some shorter interval than the half duration of ihe eclipse. Let some instant be fixed upon, some five or ten minutes before or after the approximate time of the beginning of ihe eclipse, according as the contact takes place before or after. For this time deduce the longitude and latitude of the moon, from the longitude and latitude at the approximate time of beginning, by means of their hourly variations ; and thence calculate the ap- parent longitude and latitude, and the augmented semi-diameter. Find the longitude of the sun for the time in question, from its longitude and hourly motion already known for the approximate time of beginning. Then proceed according to the rule given above, only using the quantities thus found for the time assumed, in place of the corresponding quantities answering to the approxi- mate time of greatest obscuration. L will always represent the interval for which k and n are rlpterroined. For the End. Subtract the longitude of the sun at the approximate time of the end from the apparent longitude of the moon at the same time. Do the same for the approximate time of greatest obscuration Then proceed according to the rule for the beginning, only substi- tuting everywhere the approximate time of the end for the approx imate time of the beginning, and taking in place of the formula c = X' — h, the following : c=X'-t-6. * It will be somewhat more accurate to use in place of k and 7/, as above de- jt )z' jt k k' it fined, the values of the following expressions : — 2J — ~- — or — — 31 — -- — j 36 6 36 -X 24 — ^^ — °^ ~fi ^i — OR — ' '^^^ ^^®^ ^^ each of these pairs of expressiona is to be used in f ase the true time of beginning is after the approximate time ; — the second in the other case. V and n' are the apparent relative motions in longi- tude and latitude during the last half of L. In case these expressions are used the following constant logarithm is to be employed instead of that above given, viz. 0.69897. In the calculation of the end of the eclipse, M and -n will answer to the last hall of L, and k' and n' to the first half. TO CALCULATE A SOLAR ECLIPSE. 383 For the Greatest Obscuration, Take the sum of the distances of the moon from the sun in ap> parent longitude at the approximate times of the beginning and end of the echpse, and call it k. Take the difference ol the apparent latitudes of the moon at the same times, if the two are of the same name ; but if they are of different names, take their sum. Denote the difference or sum by n. Let a' = the distance of ihe moon from the sun in apparent longitude at the true time of greatest ob- scuration ; X' = the apparent latitude of the moon at the approxi- mate time of greatest obscuration. k:n::'K':a'. Find the value of a' by this proportion, affecting X', n, k, always with the positive sign. Ascertain whether the greatest obscuration has place before or after the apparent conjunction, by observing whether the apparent latitude of the moon is increasing or decreasing about this time , the rule being, that when it is increasing, the greatest obscuration will occur before apparent conjunction ; when it is decreasing, after. If the approximate and true times of greatest obscuration are both before or both after apparent conjunction, from the value found for a' subtract the distance of the moon from the sun in ap- parent longitude at the approximate time ; but if one of the times is before and the other after apparent conjunction, take the sum of the same quantities. Denote the difference or sum by m. Also, let D = duration of eclipse, and t = correction to be applied to the approximate time of greatest obscuration. Then to find t, we have the proportion k : m : : D : t. If the apparent latitude of the moon is decreasing, t is to be applied according to the sign of m ; but if the apparent latitude is increasing, it is to be applied according to the opposite sign. A still more exact result may be had by repeating the foregoing calculations, making use now of the apparent latitude at the time just found. When the greatest accuracy is required, the values of k and n may be found more exactly after the same manner as foi the beginning or end. For the Quantity of the Eclipse. Find by interpolation the apparent latitude of the moon at the true time of greatest obscuration. With this, and the distance in longitude a' obtained by the proportion above given, compute by the formulae on page 375, the apparent distance of the centres oi the sun and moon at the time of greatest obscuration. Sublracl this distance from the sum of the apparent semi-diameters of the 384 ASTRONOMICAL PROBLEMS. two bodies, diminished by 5", and denote the reniaiader by R Then, Sun's semi-diam. (diminished by 3") : R : : 6 digits : number ot digits eclipsed. When the apparent distance of the centres of the sun and moon at the time of greatest obscuration is less than the difference be- tween the sun's semi-diameter and the augmented semi-diameter of the moon, the eclipse is either annular or total ; annular, when the sun's semi-diameter is the greater of the two ; total, when it is the less. For the Beginning and End of the Annular or Total Eclipse. The times of the beginning and end of the annular or total eclipse may be found as follows : the greatest obscuration will take place very nearly at the middle of the echpse in question, and will not differ, at most, more than five or eight minutes (acfcording as the eclipse is total or annular) from the beginning and end : to obtain the half duration of the eclipse, and thence the times of the beginning and end, we have the formulae log. tang ^ — ] og. X' -1- ar. CO . log. a, log. ^' = log. A -1- ar . CO. log. sin e : S=.S-d- 1", oiS=d — S + l"; loff c = log-(S+A)-Hog.(S-A) _ log. t = ar. CO. log. k' + log. c + log. D + 1.77815 — 10 ; Time of Begin. =M. — t, Time of End = M + « : in which M = Time of greatest obscuration ; X' = Moon's apparent latitude at that time ; a = Distance of moon from sun in appar. long. ; k — Variation of this distance during the whole eclipse, or rela live mot. in appar. long, during this interval ; k' = Moon's appar. mot. on relative orbit for same interval ; ^ = Inclination of relative orbit ; & = Semi-diameter of sun ; d — Augm. semi-diam. of moon ; A = Appar. distance of centres ; D = Duration of eclipse, (partial and annular or total ;) t = Half duration of annular or total eclipse. The Jirst value of S is used when the eclipse is annular, the second when it is total. The quantities may all be regarded as positive. The results may be verified and corrected by finding directly the apparent distance of the centres for the times obtained, and comparing it with the value of S. TO CALCULATE A SOLAR ECLIPSE. 385 For the Point of the Sun's Limb at which the Eclipse commences. Find the angle of position of the sun, and the angle between its vertical circle and circle of declination, at the beginning oi the eclipse, as explained at page 380. Let the former be denoted by p, and the latter by v. Give to each the negative sign, if laid on towards the right ; the positive sign if laid off towards the left. Let a =■ distance of the moon from the sun in apparent longitude at the beginning of thp eclipse ; X' = the moon's apparent latitude at the same time ; and ^ = angular distance of the point of contact from the ecliptic. Compute the angle 6 by the formula log. tang 6 = log. V + ar. co. log. a ; taking it always less than 90°, apd positive or negative accordmg to the sign of its tangent. X' is negative when south ; a is always positive. Let A = distance on the limb of the point of contact from the vertex. The above operations being performed, the value of A results from the equation k=p + v+90° — 6; p, V, and 6 being taken with their signs. If the result is affected with the positive sign, the point first touched will lie to the right of the vertex. If with the negative sign, it will lie to the left of the vertex. Note. The circumstances of an occultation of a fixed star by the moon may be calculated in nearly the same manner as those of a solar eclipse. The star in the occultation holds the place of the sun in the eclipse. The immersion and emersion of the star correspond to the beginning and end of the eclipse. The elements which ascertain the relative apparent place and motion of the moon and star, take the place of those which ascertain the relative appa- rent place and motion of the moon and sun. Thus the star's lon- gitude, corrected for aberration and nutation, (see Problem XXIII,) must be used instead of the sun's longitudes ; the apparent dis- tances of the moon from the star in latitude, instead of the moon's apparent latitudes ; and the moon's augmented semi-diameter, in- stead of the sum of the semi-diameters of the sun and moon. The difference of the longitudes, and the relative motion in longitude, must also now be reduced to a parallel to the ecliptic passing through the star, (see Appendix, page 431.) If X = apparent lati- tude of star, a = diff. of appar. longitudes of moon and star, and k = relative motion in longitude, we must substitute in the formu- las for the eclipse, for X', X' — X ; for a, a cos X ; and for k, k cos X. n will stand for the relative motion in latitude, or for the variation of X' — X. Example. Required to calculate an eclipse of the sun, for the 25 386 ASTRONOMICAL PROBLEMS. latitude and meridian of New York, that will occur on the 18th of Septemjer, 1838. Fm- the Approonimate Times of the Phases. Approximate time of New Moon. Sept. 18^- 8''- 49"'- Sun's longitude, . Do. hourly motion, . . Do. semi-diameter, . . Moon's longitude, . • Do. latitude. Do. equatorial parallax, . Do. semi-diameter, . . Do. hor. mot. in long. . Do. hor. mot. in lat. . Do. appar. long. (Prob. XVII), Do. appar. lat. (X'), Do. augm. semi-diameter, . Diff. of appar. long, (a), Appar. dist. of cen. (a), Sum of semi-diameters, T-h. 4911.. Sun's longitude, . Moon's appar. long. . Do. appar. lat. (X') Do. augm. semi-diameter, Diff. of appar. long, (a), Appar. dist. of cen. ('^), Sum of semi-diameters, gh. 4911 Sun's longitude, . Moon's appar. long. . Do. appar. lat. (X'), Do. augm. semi-diameter, Diff. of appar. long, (a), Appar. dist. of cen. (a), Sum of semi-diameters. 7"- 49'' 8 49 9 49 10 49 2281" 1025 377 1925 diff. or k. 1256" 1402 1.548 492" N 145 N 138 S 337 S 175° 27' 31".4 2 26 .7 15 57 .0 175 29 19 47 47 53 53 14 41 29 29 2 41 175 10 26 2 25 N. 14 47 17 5 17 15 30 44 175 °25' 4" 174 47 3 8 12 N. 14 49 38 1 38 53 30 46 175 °29 58" 175 36 15 2 18 S. 14 44 6 17 6 42 30 41 diff.. 347" 283 219 2333" 1035 402 1958 diff. Isuin seml-d. 1298" 1556 1846" 1844 1841 1839 TO CALCULATE A -SOLAR ECLIPSE. 387 For the Approximate Time of Beginning, A = 1298", d = 2333" - J 846" = 487" ; 1298" : 487" : : eO™- : t = 22™'.6 7h. 49m. 22 1st Approxi. S""- 11"- 7*-49"- . a = 2281" . X' Corrections for 22"'- 447 = 492' 133 'N. (See Note, p, gh. iiin. ^ a = 1834 . \' = 359 N. a = 1834" ar. co. log. 6.73660 . X' = 359 . log. 2.55509 • log. 3.26340 « = 11° 4' 30" . tan. 9.29169 , ar. CO . COS. log. 0.00817 Appar. dist. of cen. A = 1 869" , Sum of semi-diam. . 1846: 3.27157 487" : 23" : gh. um. + 1 : 22'"- :f = V^iS- 2d Approxi. 8''- 12'"' For the Approximate- Time of the End. h = 1556", d = 1958" - 1839" = 119", 1556" : 119": :60"'- : t = 4nu .6. 10"^ 49™. -5 1st Approxi. lO*- 44"* 101-.49™- . a = 1925" . Corrections for 5"'- 132 • • X'=357"«. 17 IC-- 44'"- . a = 1793 • X' = 340 S. a = 1793" . ar. co. log. 6.74642 V= 340 . . log. 2.53148 • log. 3.25358 6 — . . .tan. 9.27790 . ar. CO. cos, 0.00767 Appar. dist. of cen. A = 1825" . 3.26125 1839 133": ;14": : 5'":« = Cr.5. 388 ASTRONOMICAL PROBLEMS, lO"-- 44"- .5 2dApproxi. lO"^ 44'"-.5 For the Approximate Time of Greatest Obscuration Approx. time of begin. . S*- 12"* Approx. time of end, . 10 44 2 ) 18 56 IstApproxi. . 9 28 For the True Times of the Phases. Approx. time of Beginning. gh. i2'>'- Sun's longitude, 1 75" 26' 1 ".0 Do. semi-diam., 15 57 .0 Moon's app. Ion. 174 55 36 .7 Do. app. lat. 5 45 .3 N. Do.augm.semid. 14 48 .0 gh. 12-- 9 28 to 44 Approx. time of Greatest Obscur. gh. 2gn.. Approx. time of End. jOh-44"'- 175° 29' 6".8 175''32' 12".6 15 57 .0 15 57 .0 175 27 7 .7 176 2 17 .2 43 .5S. 5 32 .4 8 14 45 .1 14 41 .7 a k X' n A S 1824".3 119 .1 1804 .6 1705".2 1923 .7 345".3 N 43 .5S 332 .4S 388".8 288 .9 1856".7 1835 .0 1840".0 1833 .7 For the True Time of Beginning. a • 1824".3 , k , 1705 .2 , n • 388 .8 • b = — . 8001 .1 V ■b = • c = 345 .3 V- = 8346 .4 • s + A • 3696 .7 • 8- A , -16 .7 a n • • , • L • , 76m. . log. 3.26109 . log. 3.23178 ar. CO. log. 7.41028- log. 3 90315— ar. CO. log. 6.07850 . log. 3.56781 . log. 1.22272— ar. CO. log. 7.41028— . log. 1.88081 Const, log. 1.47712 Corr. of approx time, + 43'- .4 log. 1.63724 + TO CALCULATE A SOLAR ECLIPSE. 389 Corr. d approx. time, + 43°- .4 Approx. time, . 8*^ 12"^ .0 True time of begin. 8 12 43 .4, in Greenwich time DifT ofmerid. . 4 56 4 True time of begin. 3 16 39 .4, in New York time. For the True Time of End. a . . 1804".6 k . . 1923 .7 n . . 288 .9 b= - 12016 .3 X' . — 332 .4 • 1 ■ • « ■ 10"^ • • • • • • • • • -3'- 44™- . log. 3.25638 . log. 3.28414 ar. CO. log. 7.53925— . log. 4.07977— X' + 6=c=-12348 .7 S+A . . 3668 .7 S -A . . —1 .3 n . . . L . . . 76m. Corr. of approx. time, Approx. time. ar. CO. log. 5.90838— . log; 3.56451 . , log. 0.11394— ar. CO. log. 7.53925 — . log. 1.88081 . Const, log. 1.47712 . log. 0.48401 — .0 True time of end, . Diff. of merid. 10 4 43 56 57 4 .0, in Greenwich time. True time of end, . 5 47 53, in New York time. For the True Time of Greatest Obscuration. True tin Do. ne of of beginning, . . . S*- end, . . . .10 12"^ 43'- 43 57 .4 .0 2)18 56 40 A 2d Approx. 9 28 20 Ji gh. 49m. ^ . X' = 138" S. 9 28 • . X' = 43 .5 S. iff. 21 Diff. 94 .5 21m. J 20'- : : 94".5 : 1".5 20'- 43 .6 gh. 28'"- X'-45. 890 ASTRONOMICAL PROBLEMS. 1705".2 388".8 1923 .7 288 .9 k = 3628 .9 - : n = 677 ,7 : : V = 45".0 : a' = 8".4 Time of beginn. S"-- 12'"<'43" .4, at O''- 28"'- a'= 119".l Time of end, 10 43 57.0 a'= 8.4 D= 2 31 13 .6 »» = - 110 .7 3628".9 • 110".7 : : 2'-- 31"'- 13»-.6 : ^^^ 36»- .8 9''-28 .0 ■ True time (nearly) 9 32 36 .8 21"^ : 4"°- 37'- : : 94".5 : 20".8 43 .5 At 9^- 32™- 37'-, X' = 64 .3 3628".9 : fi7r'.7 : : 64".4 : 12".0 ; at &"■ 32"'- 37'-, a = 8".4 a' =12 .0 »»= 3 .6 8628".9 : 3".6 : : 2^- 31"- 13'-.6 : 9'-.0 gh- 32™- 36 .8 9 32 27 .8 True time of greatest obscur. . 9''- 32"'- 27'-.8, in Greenw. time Diff. of merid. . . . 4 56 4 True time of greatest obscur. . 4 36 23 .8, in N. Y. time. For the Quantity of the Eclipse. gh. 32.". 37». . X' = 64".3 21'"- : 9'- : : 94" .5 : .6 M nearest approach of centres, . X' = 63 .7 " " . . a = 12 .0 a . 12" .0 .) ar. CO. log. 8.92082, . . log. 1.07918 X' . 63 .7 . . . 1.80414 tan. 0.72496, . ar. co. cos. 0.73253 Shortest distance of centres, 64".8 . . log. 1.81171 Sum of semi-diaiheters, ' 1837 .0 '"; 1772 .2 15' 54" : 1772".2 : : 6 : 11,14 digits eclipsed. TO CALCrLATE A SOLAR ECLIPSE. 391 For the Situation of the Point at which the Obscuration com- mences. S"-- 12'"- . . a =1824", , . X'r=345".3N TG""- : 43'- : : 1705" : 16, !&"■ : 43'- : : 389" : 3 .7 a =1808, . log. 6.742; log. 2.53352 X' = 341 .6 ar. CO. log. 6.74280 At the beginn. a . 1808 . X' . 341.6 « = 10° 41' 57" . . tan. 9.27632 Obliq eclip.(Prob.X),23° 27' 47" . sin. 9.60005 . tan. 9.63753 Sun's longitude, 175 26 3 . sin. 8.90093 . cos. 9.99862 - sin>8.50098, tan. 9.63615- Sun's declination, 1° 49' 0" ; Angle of pes. 23° 23' 50". Meantime of begin. 3^- 16"'- 39'-, Lat. 40° 42' 40", Dec. 1°49' 0" Equa. oftime, . 5 58 90 90 Appar. time, 3 22 37,PZ=49 17 20,PS=88 11 60 4)202 37 Hour angle P = 50° 39' 15" Co. lat. PZ = 49 17 20 TO = 36° 23' 0" . Co. dec. PS =88 11 COS. 9.80210 tan. 0.06526 m'=51 48 m=36 23 P=50 39 15 S=42 38 10 Angle of position. Angle from eclip. (6), tan. 9.86736 ar. CO. sin. 0.10466 sin. 9.77320 tan. 0.08627 tan. 9.96413 — 23° 23' 50" — 10 41 50 . Angle of dec. circle from vertex (S), 42 38 10 90 Angular dist. of point first touched from vertex, 98 32, to the right • For the Beginning and End of the Annular Eclipse. Approx. time, 9''- 32"'- 27'-.8 =true time of greatest obscur. At this time, a = 12".2, X' = 63".7. a = 12".2 . ar. CO. log. 8.91364 . . log. 1.08636 X'=63 .7 . . log. 1 .80414 «r=79°9'30" . tan. 0.71778 . ar. co. cos. 0.72564 tan. 0.71778 A = 64".9 log. 1.81200 S92 ASTRONOMICAL PROBLEMS. S + A = 135".8 . log. 2.13290, «= 79° 9' 30" . ar. co. sin. 0.00783 S - A= 6 .2 . log. 0.79239, /c=3628".9 . log. 3.55977 2 ) 2.92529, k' 1.46264 D=152"'- • IP' 23 • • • .6 .8 ar. CO. log. 6.43240 1.46264 .log. 2.18184 Const log. 1.77815 i = 0'>- 1™' Time of greatest obscur. . 4 36 log. 1.85503 Formation of ring, . . 4 35 12 .2, New York time. Rupture of do. . .4 37 35 .4 « (( PROBLEM XXXI. To find the MoorCs Longitude, Latitude, Hourly Motions, Equa- torial Parallax, and Semi-diameter, for a given time, from the Nautical Almanac. Reduce the given time to mean time at Greenwich ; then, For the Longitude. Take from the Nautical Almanac the calculated longitudes an- swering to the noon and midnight, or midnight and noon, next pre- ceding and next following the given time. Commencing with the longitude answering to the first noon or midnight, subtract each longitude from the next following one : the three remainders will be xhe first differences. Also subtract each first difference from the following for the second differences, which will have the plus or minus sign, according as the first differences increase or de- crease. Find the quantity to be added to the second longitude by rea- son of the first differences, by the proportion, 12''- : excess of given time above time of second longitude : : second first differear-e : fourth term. With the given time from noon or midnight at the side, take from Table XCIII the quantities corresponding to the minutes, tens of seconds, and seconds, of the mean or half sum of the two second differences, at the top : the sum of these will be the correction f 07 second differences, which must have the contrary sign to the mean. The sum of the second longitude, the fourth term, and the cor rection for second differences, will be the longitude required. TO FIND moon's LONG., ETC., FROM NAUTICAL ALMANAC. 393 For the Latitude, Prefix to north latitudes the positive sign, but to south latitudes tlie negative sign, and proceed according to the rules for the lon- gitude, only that attention must now be paid to the signs of the first differences, which may either be plus or minus. The sign of the resulting latitude will ascertain whether it ia north or south. For the Hourly Motion in Longitude. Solve the proportion, 1 2"'' : given time from noon or midnight . : half sum of second differences : a fourth term ; which must have the same sign as the half sum of the second differences. Take the sum of the second first difference, half the mean of the second differences, with its sign changed, and this fourth term, and divide it by 12 : the quotient will be the required hourly mo- tion in longitude. For the Hourly Motion in Latitude. With the given time from noon or midnight, the second first difference of latitude, and the mean of the second differences, find the hourly motion in latitude in the same manner as directed fol finding the hourly motion in longitude. When the hourly motion is positive, the moon is tending north ; and when it is negative, she is tending south. For the Semi-diameter and Equatorial Parallax. The moon's semi-diameter and equatorial parallax may be taken from the Nautical Almanac, with sufficient accuracy, by simple proportion, the correction for second differences being too small to be taken into account, unless great precision is required. Corrections for Third and Fourth Differences. When the moon's longitude and latitude are required with great precision, corrections must also be applied for the third and fourth differences. To determine these, take from the Almanac the three longitudes or latitudes immediately preceding the given time, and the three immediately following it, and find the first, second, third, and fourth differences, subtracting always each number from the following one, and paying attention to the signs. With the given time from noon or midnight at the side, and the middle third difference at the top, take from Table XCIV the correction for third differences, which must have the same sign as the middle third difference when the given time from noon or midnight is less than 6 hours ; the contrary sign, when the given time is more than 6 hours. 394 ASTRONOMICAL PROBLEMS. With the given time, and half sum of fourth differences, take from Table XCV the correction lor fourth differences, giving il always the same sign as the half sum. The sum of the third longitude or latitude, the proportional pari of the middle first difference answering to the given time from noon or midnight, and the corrections for second, third, and fourth differences, having regard to the signs of all the quantities, wil be the longitude or latitude required. APPENDIX. TRIOONOMETRICAL FORMUL-fi* I. Relative to a Single Arc or Angle a. 1. sin* a + cos* a = 1 2. sin a = tan a cos a tan a 3. sin a - = 4. cos a =1 5. tan a = 6. cot a = tan a sin a 7. sin a = 2 sin ^ a cos ^ a 8. cos a = 1 — 2 sin* i a 9. cos a = 2 cos* i a — ] ■■« ^ I sin a 10. tania=-— ; 1 + cos a 1, .1 sin a 11. cot i a v^l+tan»a 1 ^^ 1 + tan* o sin a cos a 1 cos a 12. 1 — cos a 1 — cos a 1 + cos a 13. sin 2 a = 2 sin a cos a 14. cos 2 a = 2 cos* o — 1 = 1 — 8 sin* a II. Relative to Two Arcs a and b, op which a is supposed TO BE THE GREATER. 15. sin (a + 6) = sin a cos 6 + sin 6 cos a 16. sin (a — 6) = sin a cos 6 — sin b cos a 17. cos (o + 6) = cos a cos 6 — sin a sin b * The radius is supposed to be equal to unity in all of the formulae. 396 APPENDIX. IS. COS (a — b) ■■= cos a cos 6 + sin a sin 6 , , . tan a + tan b 19. tan(a + i) = - ; ; — r ^ 1 — tan a tan o , . tan a — tan b 20. tan (a — 6) = -— ; — r 1+tanatano 21. sin a + sin 6 = 2 sin 1^ (a + b) cos ^ {a — b) 22. sin a — sin 6 = 2 sin ^ (o — 6) cos ^ (a + 6) 23. cos a +C0S& = 2 cos |(a + b) cos i (a — t) 24. cos 6 — cosa = 2 sin ^ (a + 6) sin | (a — b) sin (a + 6) 25. tan a + tan b 26. tan o — tan b 29. 30. 31. 32. 38. 39 40. cos a cos b sin (a — b) cos a cos b , sin (a + b) 27. cot a + cot 6 = . . / sin a sin 6 7 sin (a — b) 28. cot 6 — cot a = ^ — -. — r- sin a sm o sin a + sin 6 _ tan ^ (a + 6) sin a — sin 6 tan i {a — h) cos 6 + cos a _ cot 5 (a + h) cos b — cos a tan ^ (a — 6) tan a + tan 6 cot b + cot a _ sin (a + fe) tan a — tan b cot 6 — cot a sin (a — 6) cot 6 — tan a _ cot a — tan b _ cos (a + fc) cot 6 + tan a cot a + tan b cos (a — 6) 33. sin^ a — sin^ b = sin (a + 6) sin (a — b) 34. cos^ a — sin^ b — cos (a + 6) cos (a — b) 35. 1 ± sin a = 2 sin^ (45° ± \ a) 36. L^-4^=tan^(45°±i«: 1 =F sm a ^ " ' 37 li?HLf = tan(45<'±ia) cos a V » / 1 — sin g _ sin° (45° — \ a) 1 — cos a sin' ^ a 1 + sin fc ^ sin' (45° + 1 fe) 1 + cosa cos' I a 1 + tan 6 " ,,,„.,, r^- — 5= tan (45° + 6) 1 — tan o It 1 — tan i , .,„ .. 41. :r— r= 1^(45" — ft) 1 + tan6 ^ ' TRIGONOMETRICAL FORMULiE. 397 42. ■ sin a C03 b =1 sin (a + 6) + i sin {a — b) 43. cos a sin ft = J sin {a + b) — ^ sin (a — 6) 44. sin a sin 6 — J cos (a—b) — I cos (a + b) 45. cos a cos fe = i cos (a + 6) + ^ cos (a — 6) ITT. Trigonometrical Series. r,3 sin a = a a" a- 2.3 2.3.4.5 /,4 — &c. 46.^ cos a = 1 — tan a = + 1 a° cot a = a 2 3 a 3' + 2.3.4 2a' 2. 3. 4. 5. 6 17a' &c. 3.5 (^ _ 3^5 3\ 5. 7 2a^ !^ 5. 7 ~ &c. &c. Let a = length of an arc of a circle of which the radius is 1, and (a") = number of seconds in this arc, then to replace an arc ex- pressed by its length, by the number of seconds contained in it, we have the formula 47. a = (a") sin 1" ; log. sin 1" ^.685574867. IV. Differences of Trigonometrical Lines. 48. A sin a; = + 2 sin 5- A a;, cos (a; + -i A a:) 49. A cos X = — 2 sin i A a;, sin (a: + i A a;) sin A X 50. A tan a; = + cos X. cos (a; + A a;) sin A X 51. Acota;= — - , ^ ^ sin X. sin {x + Ax) V. Resolution op Right-angled Spherical Triangles * Table of Solutions. Given. Required. Solution. Hypothen. f side op. giv. ang. 52 sin a; = sin A . sin a and •< side adj. giv. ang. 53 tan x = tan h . cos a an angle (^ the other angle 54 cot a; = cos A .tana the other side 55 cos x=- Hypothen. and a side cos s ang. adj. giv. side 56 cos x = tan s . cot A •J -~ sin s ang. op. giv. side 57 sm x = -. — 7 * Baily'B Astroncmical Tables and Formals. 398 APPENDIX. the hvpothen. 58 sin x ■ •"^ sm a , . _ , lilt, ilV LIVLll^^lla ^^ K>±x* .*/ . I V A side and •'^ sm « | | the angle J the other side 59 sin x = tan s . cot a ^ | PP the other ansle 60 sin x = = ° cos s ) £ A. side and C the hypothen. 61 cot x — cos a . cot » the angle < the other side 62 tanx = tan a . sin s adjacent I. the other angle 63 cos x = sin a . cos s C the hypothen. 64 cos x = rectang. cos. of the The two < giv. sides sides (_ an angle 65 cot x = sin adj. side x cot. op. side f the hypothen. 66 cos a; = rectang. cot. of the The two J given angles ^'^S^'^ la side 67' cosx =^-?^^:^- ^. sm. adj.ang. In these formulae, x denotes the quantity sought. a = the given angle s = the given side h = the hypothenuse. Napier's rules. The formulae for the resolution of right-angled spherical tnan- gles are all embraced in two rules discovered by Lord Napier, and called Napier's Rules for the Circular Parts. The circular parts, so called, are the two legs of the triangle, or sides which form the right angle, the complement of the hypothenuse, and the comple- ments of the acute angles. The right angle is omitted. In re- solving a right-angled spherical triangle, there are always three of the circular parts under consideration, namely, the two given parts and the required part. When the three parts in question are con- tiguous to each other, the middle one is called the middle part, a,ni the others the adjacent parts. When two of them are contiguous, and the third is separated from these by a part on each side, the part thus separated is called the middle part, and the other two the opposite parts. The rules for the use of the circular parts are (the radius being taken = 1 ), 1 . Sine of the middle part = the rectangle of the tangents of the adjacent parts. 2. Sine of the middle part = the rectangle of the cosines of the opposite parts. particular cases of right-angled spherical triangles. Equations 52 to 67, or Napier's rules, are Sufficient to resolve all the cases of right-angled spherical triangles ; but they lack pre- cision if the unknown quantity is very small and determined by RESOLUTION OF SPHERICAL TRIANGLES. 399 means of its cosine or cotangent ; or, if the unknown quantity is near 90°, and given by a sine or a tangent : in these cases the fol- lowing formulae may be used : _ ^ 3, cos(B + C) 68. tan'Ja=-3^^^-^ 69. tan'' IB =^^J sm (a + c) 70. tan' ic = tan i (a + &) tan i{a — b) 71 . tan (45° - ^b) = V tan (45° - a?), tan a; = sin a sin B 72. tan«i&=tan(^:=-5+45°) tan (^±^-45°). a is the hypothenuse, B, C, the acute angles, and b, c, the si lea opposite the acute angles. VI. Resolution of Oblique-Angled Spherical Triangles. General Furmulce. Let A, B, C, denote the three angles of a spherical triangle, and a, b, c, the sides which are opposite to them respectively. ^„ sin A sinB • sin C sin a sin b sin c or, the sines of the angles are proportional to the sines of the op- posite sides. 74. cos c = cos a cos 6 + sin a sin b cos C 75. cos c = cos (a — 6) — 2 sin a sin b sin^ |C 76. cos C = sin A sin B cos c — cos A cos B 77. sin a cos c = sin c cos a cos B + sin 6 cos C 78. sin a cot c = cos a cos B + sin B cot C 79. sin a cos B = sin c cos b — sin 6 cos c cos A >.• > Case I. Given the three sides, a, b, c. To find one of the angles. 80. sin'H^ ^'^^'^^"^^"'^ sm sin c "si. cosMA^-^'^^^'"^^-'^^ 82. k = sin b sin c a + b+c 2 Case II. Given the three angles, A, B, C To find one of the sides. . . — cos K cos (K — A) 83. sin*ia= ■ p ■ ^ ' * sm B sm,-0 r 400 APPENDIX. or. 84. cos^ la = 85. K = _ cos(K - B) cos (K - C ) sin B sin C A + B + C Case III. Given two sides a and b, and the included angle C. 1°. To find the two other angles A and B. cos 5 {a—b)' > Napier's Analogies 86. tani(A + B)=cotiC. 87. tani(A--B)=cotiC. 2°. To find the third side c. r tan \c = tan \ {a—b) 88. i or, tan ic= tan h{a+ b). or equa. 73. Case IV. Given two angles A and B, and the adjacent side c. 1°. To find the other two sides, a and b. cosHA-B)-] cos^ {a+b) sin i (« — b) sini (a +6) , sin i (A + B) ■sini(A-B) cosHA+B) cos HA -B) 89. tan ^ (a+6)=tanic 90. tan ^(a— 6)=tan Jc cos 5(A+ B) sin i (A- B) ;» Napier's Analogies. sin i (A+B) S". To find the third angle C. sin i (a + b) sin i {a— b) cos i (a+ b) cos i {a—b) cotiC=tani(A— B). 91. 90°). When if 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, h, and another arc or angle. Case 1 . Given two sides and the included angle ; or b, c, A. Equation 92 makes known m, 94 m', 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, b. Equation 93 makes known n, 95 n', which may be negative (what the calculation shows,) 97 B, 99 a ; finally, equation 73 (page 399) gives c, which is known in kind. • Francoeur's Practical Astronomy. 26 402 APPENDIX. Case 3. Given two sides and an opposite angle; crb, a, A. Equation 92 gives m, 96 m', 94 c, 98 and 73 B and C ; or else, 93 gives n, 99 n', 95 C, 97 and 73 B and c. This problem admits in general of two solutions. In effect, the arc 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, h. Equation 92 gives m, 98 m', 94 c, 96 a, and equation 73 makes known C ; or else 93 gives n, 97 n', 95 C, 99 and 73 a and c. There are also two solutions in this case ; for, m' or n' is given by a sin., and therefore two supplementary arcs satisfy the ques- tion. Thus 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 ob- tained. 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 i A sin 6 102. tan 5 a = tan b cos B 103. cos b = cot B cot i A 104. cos 5 A = cos 5 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 FORMULA. FormulcE for the Parallax in Right Ascension and Declination, and in Longitude and Latitude. (See Article 93, page 65.) ^'S- ^21 ^ Let s (Fig. 121) be the true place ^ of a star seen from the centre of the earth, s'the apparent place, seen from a point on the surface of which z is the zenith, the latitude being /. The displacement ss' = p is the parallax ■m altitude, which takes effect in the vertical circle zs' ; p is the VARALLAX IN RIGHT ASCENSION AND DECLINATION. 403 pole ; the hour angle zps =q\a changed into zps', and sps' = a is the variation of the hour angle, or the parallax in right ascen sion ; 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 pa; (90°— I), * Multiplying, term by term, we obtain sin s' sin (? + a) : sin (i sin Z : : sin a sin s' : sin p cos Z : , . sin p cos I . , , , whence, sm a = . ', . — = sm (o + a). smasinZ ^ Or, substituting for p its value given by equa. (8,) p. 62, and replacing H by P, sin P cos Z . . , . ... sm a = : — - — sm (o + a) . . , (A). sm 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 + «), (For. 15, p. 395), and putting sin P cos I ^-> — =m, sm a sma = m (sin q cos a + sin a cos q), or, dividing by sin a, 1 =m (sin gf cot a + cos q) ; whence, by transformation, wi sin o . , „ . , , . tan a = — =Tnsmq +m' sin q cos q (very nearly.) 1— mcosg' ^ a\j J ' Restoring the value of m, sin P cos Z . , /sinPcosZXa . tan a = : — ; — sin (7 + I ■. — -j — I sm q cos q. sm a ^ \ sin a / 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 Z . , /P cos Z\2 . . ,,, ._. a = — : — r sm o + I — ■■ — T- I sm q cos q sin 1" . , . (B). smd ^ \ smd / The parallax in declination (S) is the difference of the arcs ps (=i d) and ps' {=d + 5.) Let zs = z, and zs' = Z. The trian- gles zps and zps' give (For. 74 and 73), cos . ., tan 5 = [?in ( ^ V cos V / cos y . . . ^ PQg y sin {d — y) cos (d—y) (very nearly ;) or, replacing tan S and sin P by i5 and P, expressing these arcs in seconds, (For. 47, p. 397), and reducing by For. 13, p. 395, . (5=- lZ sin (d — y) + / P sin l \ \ cos y ' ZX^sin 1' sin 2 (d — y) (G.) cos y " ■" \ cos y / 2 If the place of a body be referred to the ecliptic, similar formu lae will give the parallax in latitude and longitude, 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 Non- agesimal Degree, or the Nonagesimal Fig. 122. Let K (Fig. 122) be the pole of the eclipticyZ), p the pole of the equator /"a ; /is the vernal equinox, the ori- gin of longitudes and of right ascensions ; kbs is the eastern horizon, b the hor oscope, or the point of the eclipticwhich is just rising; pz = 90° — / (the latitude of given place) ; Kp = u the obhquity of the ecliptic. The circle Kznv is at the same time perpendicular at n to the ecliptic /6, and at V to the horizon hb ; it is a circle of latitude and a vertical cir- cle, since it passes through the pole K and the zenith z: b is 90° from all the points of the circle Knv ; zn is the latitude of the ze- nith, /w its longitude ; the point n is the noaagesimal, since hn = 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 ; thus/??. =N the longitude of the nonagesimal, and nv =h the altitude of the nonagesimal, designate the situation of this point, and conse- quently ascertain the position of the ecliptic and its pole at the moment of observation.* ' FrancoBur's Uranography, p. 421 INGITUDE 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 arc /i =90°, since the plane Kpz, passing through the poles K and ■p, is at the same time perpendicular to the ecliptic and to the equator; the arc mi=fi—fm = 'dO°—s; then the angle zpK. = 180°— zpi - 180°— mi = 90°+ s* Now, in the spherical triangle pKz we know the sides Kp = u, zp = 90°— I = ki, and the included angle zpK=-90° + s; and may therefore find Kz = h the altitude of the nonagesimal, and the angle pKz = nc =fc~fn = 90°— N = complement of the longi- tude N of the nonagesimal. Let S = sum of the angles Kzp and zKp, then, (For. 86, page 400,) tan iS =5^5^). cot i (90° + s), cos^(H+w) ^ , „ cos i (H — u) oi, tan hS = p^ir- — f. tan i (90°— s): cosi(H+u) but, tan ^S =-tan (ISC-^S), and tan i (90''-s) =-tan i (s-QO") j substituting, and denoting (180° — ^S) by E, we have tanE=^g=4-tani(*-90°) • • • (H). cos i (H+ u) ^ ^ ' Again, letD =zKp—Kzp, then, (For. 87,) tan ^D = ^'"fg-"\ . cotH90°+5); sm i (H + u) ^ whence, by transforming as above, and denoting (180°— 5D) by F we have tanF = !i"4-Sx4- tani (*-90°) . . . (I). sm 4 (H+ cj) / \ / Now, iS + iJ) =pKz = 90°— N ; whence, N = 90°- (i S + iD) , or, N = 360° +90°- (4S + W) = 180°-iS +180°-iD +90°; consequently, N = E + F + 90° . . . (J), rejecting 360° when the sum exceeds that number. Next, for the altitude of the nonagesimal, we have, (For. 88,) , , cos ^S , /Tx . ^ tan in = p=-. tan 2 (H + u), cos iD '^""•'^v tan HH +")... (K). cos F' N and h being known, to obtain the formulcefor the parallax I longitude and latitude, we have only to replace in the formulas * FranccEur's Uranography, p. 421. 408 APPENDIX. for the parallax in right ascension and declinatioi , the altitude I of the pole of the equator by that 90°— A of the pole K of the eclip tic, and the distance im of the star s from the meridian by the dis- tance nc to the vertical through the nonagesimal. Let us change then in formulae (A), (B), (C), (D), (E), (F), and (G), 1 into 90°— h, and q into fc —fn = L— N, L being the longitude /c 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 n, and the parallax in latitude by *, we obtain in terms of the apparent longi- tude and latitude, . „ sin P sin A . . - at t ^\ /t \ smn = — . — -, — (smL— N + n) . . . (L), sina ^ , , , , sin (L— N + n) / sin P cos h \ „,. cot (d + *) = '^ ^^ { Cotd r-T— ) . . . (M), sm(L — JN) \ smd / sin P cos k ,^^^ tan X = : — z — . . . (N), sm d ^ ' ., , , sm(L— N + n)cos ((^ + a;) cot(d+*)= V ^, \' - .^ ' . . . (O), sm (L— N)smacosa? ^ ' sin * = sin P cos h sin (d+rr) — cos (/ \^i +co^/ " V V(l + e)Tl + cos u)) These two expressions (Z) and (ot), that is, nf = e sin M + u, lan V / (\ — e\ u 2 = vVrT-J''"i' 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 express- ing the true anomaly in terms of the mean. MG is drawn parallel to DC. (1.) Find the half difference of SOLUTION OF KEPLER'S PROBLEM. 421 the angles at the base EM of the triangle ECM, from this ex- pres.sion, tan i (GEM — CME) = tan i (GEM + CME) x ^^Zf, in which GEM + GME = ACM. the mean anomaly. (2.) Find GEM by adding | (GEM + GME) and ^ (GEM — CME) and use this angle as an approximate value to the eccen- tric anomaly DCA, from which, however, it reallv differs bv Z EMO. ■^ (3.) Use this approximate value of /DCA = /EOT in com- puting ET which equals the arc DM ; for, since (see p. 419), p t = ^.— p X DEA, and (the body being supposed to revolve area circle p in the circle ADM) = ^_- x ACM, area AED = area ACM. area circle ' or, area DEC + area AGD = area DGM + area ACD ; con- sequently the area DEC = the area DGM, and, expressing their values, ET x DC DM X DC , ,, „„ ^,^ X = 75 , and thus, ET = DM. Having then computed ET = DM, find the sine of the resulting arc DM, which sine = OT ; the difference of the arc and sine (ET — OT) gives EG. (4.) Use EG in computing the angle EMO, the real difference between the eccentric anomaly DCA and the /MEG; add the computed Z EMO to / MEG, in order to obtain / DCA. The result, however, is not the exact value of /DCA, since /EMO has been computed only approximately; that is, by a process which commenced by assuming /MEG for the value of the /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 / DCA = / ECT, ET, EO, and / EMO must be again computed. Formula for calculating the Pa/rallax in Altitude oj a Heavenly Body from its Tru« Zenith 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 oo-ordinatea. For this purpose there is required an eX' pression for the paraUax in altitude in terms of the true zenith distance. If we make Z=z+p in equation (8) p. 62, we shall have J sin^ Bin o = sm H sin (z +p), or sin H =-: — ; p ; ■^ ^ ■^•" sm {z+p) ' irheuce . si n p _ 8in (z +p) + sin p 422 APPENDIX. aod Dividing, or, whence, 1— sin H = 1 — amp sin (z+p) —Bmp sin (z+i)) sin (z +p) ' 1 + sin H _sin (z +p) + sin p . 1— sin H ~sin (i!+jp)— sin j) teng» (45°+ i H) J^^K3J±p1. (App. For. 36, 29); tang i 2 Fig. 129. tang(i z+p)=tang i z tang' (45° + J H). . . .(a). This equation maizes known i z+p, from which we may obtain p by subtract ing^z. Formulce for computing the Annual Variations in the Sight Ascension and De- clination of a Heavenly Body. (See Art. 119, p. 88.) Let VLA (Fig. 129) be the ecliptic, K its pole, PP'f" the circle described by the mean pole, P the mean pole, and TQA the mean equator at any given time, P' the mean polo and V'Q'A' the mean equator a year afterwards, and s a star. Draw Y'r perpendicular to the declination circle 7sa. We have an. var. in dec.=:sa'— 5a=P5— P'«=Pr; but since PPV may be considered as a right-angled plane triangle, Pr=PP' cos P'Pr=PP' sin QPa (a). Regarding KPP' as a right-angled isosceles triangle, we obtain sin KPP' or 1 : sm KP' :: sin PKP' : sin PP' ; whence, sm PP'=sin PKP' sin KP , or PP'=PKP' sm KP' (nearly), substituting iu equation (a), there results, P/-=PKP' sia KP' sin QPa. PKP' = 50 ".24 ; KP' = obliquity of the ediptic = w ; yPa=VQ— ya=90°— R (R designating the right ascension of the star s). Tbusi finally, an. var. in dec. = 50".24 sin «) cos R (c). Next, we have an. var. in r. asc.= but, V'm=TV' cosTV'«i=50".24 cos m; and since the right-angled triangles sPV and sia' are similar. .(») =T'o'— Ta=Y'a'— m6=T'wi + 6a'. •Wi whence, sin sr or sin sP' (nearly) : sin P'r :: sin sal : sin Jo' ; sin 6a'=sin PV sin P s or 6a' = P') sm sa sin P's (nearly). The triangle PPV gives P'r=PP' sm P'Pr=PP' cos QPo=PKP' sin KP' oos QPa (equa. 6); and sinP'«=eos sa'. Substituting, we obtain ha =PKP' sin KP' cos QPa- :PKP' sin KP' cos QPo tang so'. HELIOCENTRIC LONG. AND LAT. 423 Replaoing PKP', KP', and QPa by their values, as above, and taking the dedina- tion sa for sa' and denoting it by D, there results, Ja'=50".24 sin w sin R tang D. Now, substituting in equation (d) the values of V'm, and ba', we have an. var. in r. aso.=50".24 cos u + 50".24 sin oi sin R tang D (e) The results of formulae (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. Y'm=50".!i4 cos a,=50".24 cos 23°27'=48".0, Is the annual retrograde motion of the equinoctial points along the equator. Formvlcefor computing the Heliocentric Longitude and Latitude, a/nd Radius-vector of a Planet, from its Geocentric Longitude and Latitude. (Referred to in Art. 17T, p. 119.) The longitude of the node and the inclination of the orbit are supposed to be known. Let NP (Pig. 130) be part of the orbit of a planet, SNC the plane of the / V Fig. 130. sdiptic, IT the ascending node, S the sun, E the earth, and P the planet ; also, let Pn- be a perpendicular let fall from P upon the plane of the ecliptic, and BY, ST, the direction of the vernal equinox. Let X = PBjt the geocentric latitude of the planet ; I = PSi its heliocentric latitude ; G = TBi its geocentric longitude ; L = VSir its heliocentric longitude ; S = VBS 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 a = SP the radius-vector of the planet. The point t is called the reduced place of the planet, and Sa- its curtate distance. All the angles of the triangle SEir have also received particular appellations ; SirB the angle subtended at the reduced place of the planet by the radius of the earth's orbit, is called the Annual Parallax, SBa- the Elongation, and ES» the Commutaiion. Let A = StB, E = SEir and = ESt. Draw Sr' parallel to Eir: then A = :rSir' = VSt — TSt'=VS^— VB-r = L— G; E = TB^ — VES = G— S; C=TSB — VS^ = 180° ^- VSE' — VSir - 180° + VES — TS» = 180° ^-S — L = T — L (puttine;T=180° + S). (1.) Jiw the Wir) : tan i (EiS — SErr), or, r + V coa I : r — v coal :: tang J (A + E) : tang J (A — E) ; whence, « cos J r — V cos I *■ , tang i (A - E) = ,—-^3^ tang i(A + E) = ^^^^ tang i (A + E). V cos I Let tang 6 = : then, tangi(A-E) = [^;'-^;'tangHA+ E) ; or, tang i (A — E) = tang (45° — e) tang i { A + E) (a) But, A + E = 180°— C, andE=i(A + E) — i(A— B). Next, to find the geocentric latUitde. Sir tang ; = Pit = Eir tang X , Sir tang X whence, ° but. Sir : Eir and therefore tang X^-^-^^'.... (5), When a planet is in conjunction or opposition, the sines of the angles of eloag» tion and commutation 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, (Pig. 56, p. 120 ;) then, Pir Pir *a°g^ = E7 = ES + Sir' But the triangle SPrr gives Vt!= V sin I and Sr =: 1; cos I, and ES = r; V sin I hence, tang X = -^ ^ ^.^-^ . . . . (c). To find the distance of the planet from the earth, represent the distance by D ; then, from the triangles PirS and EPir, we have Pir = EP sin PEir =: D sin X, and ' Pit = SP sin PSir = vain I; V sin I sin X The distance of a planet being known, its horizontal parallax may be computed from the equatioa Bit ~ tang I • Sir : sin E : sin C, or ^ sin B — sin ' sin E tang X sin C ~ tang I sin E tang I whence, D = tttT • • ' • ^ ■' sinH = ^....(e.) (Art. 88). 4:26 AFFENDII. CALCULATION OP AN ECLIPSE OF THE SUN.. (1). Of the circumstances of the general eclipse. It is a simple inference from what haa been established in Art. 333, that an eclipse of the sun will begin and end upon the earth, at the times before and alter conjunction, when the distance of the centres of the moon and sun is equal to F—p + 6 + d; that the total eclipse will begin and end when this distance is equal to F—p—S + d; and the annular echpse when the distance is equal to P—p + 6 —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 (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 during the eclipse, be the same as that whicli ob- tains in the actual state of the ease. Let N'C'L' represent the orbit that would be described by the moon if it had such a motion, which is called the Selative Orbit Let CM be drawn perpendicular to it; and let Cf=Gf = P — p + l + d, and Cg = Cff'= P — p — 6 + d, or P — p + i — 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 ;/andy the places at tlie 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, tanff I I ^^ = * cos I. CM = X sini ^ M— m 3600s. X sin I cos I , , Interval from con. to mid. = ;7 ■ • • W- M — m Interval from middle to beginning or end 36003. cos I , . - M — m "^^^'+ ^''°^ ^) (^'—^ ''°^ I) . . . (6). Interval for total eclipse 3600s. cos I , = ~M^r^ir'^(*" + ^ <=°^ I) (*"— ^ cos I) . . . (c). Interval for annular eclipse Seoos. cosi , , - "M^T"^ V (i'" + a cos I) (A'"— A cos I) . . . (d). 6 (ft' — X cos T) Quantity = ^ . . . (e). k' = 'P—p + 6 + d, k" = P—p — S + d,Jc"' = 'P—p + l — d ...(/). The letters A, M, m, &c., represent quantities of the same name as in the forMu.a for a lunar eclipse; but they designate the values of these quantities at the tune of CALCULATION OF AN ECLIPSE OF THE SUN. 42 ■? conjunction, instead of opposition. These values are in practice obtained from tables of tlie 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 instanf at a given place, is indi- cated 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 calcula- tion 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 seeli 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, llien, 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 semi- diameter of the moon, for any given time and place, see Problem XVII. According to the celebrated astronomer Dusfejour, in order to malco the observa- tions 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 ap- parent 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 allow- ance for an Inflexion of the light which passes near tlie border of this luminary, sup- posed to be produced by its atmosphere. It must be observed, however, that the asl ronomers 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 longi- tude 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 the parallax. But tlie customary mode of proceeding is a little different from this : the true longitude and latitude of the sun are employed in- stead of the apparent, and the parallax of the sun is referred to the moon ; 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, semi- diameter, and hourly motions in longitude, and latitude. Subtract the sun's hori' zontal 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 lon« 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 liourly motions in longitude and lati- tude, at the time T, calculate the longitudes and tho moon's latitude for the time T'; and (or this time also calculate the moon's apparent 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 Bun 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 re- duced from the equator to the given place. (See Prob. XV.) 428 APPENDIX. ""^^^ it by rH. In the same manner as at the time T, we find whether apparent conjuno- tion occurs before or after the time T'. If it occurs between the times T and T', tlie sura of B and re', 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 mooni at the times T and T', will make known the apparent relative hourly motion in latitude. With the relative hourly motion in longitude arid the difference 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, witli the apparent latitude of the moon at the time T and its hourly motion in latitude, find the appar- ent latitude at the time of apparent conjunction thus determined. Then, knowing the relative hourly motion of the sun and moon in longitude and latitude, togethei with the time of apparent conjunction, and the apparent latitude at that time, and re' garding the apparent relative orbit of the moon as a right line (which it is nearly), it is plain that the time of beginning, greatest obscuration, and end, as well as tlie quan- tity of the eclipse, may he calculated after the same manner as in the general eclipse ; the discof thesunansweringto the section of the luminous frustum mentioned in Art. 331, and the apparent elements answering to the true. Let C (Fig. 132) represent the centre of the sun supposed stationary, CC the apparent latitude of the moon at apparent conjunction, N'C the apparent relative orbit of the moon, determined by its passing through the point C and making a determinate angle with the ecliptic N'F, or by its pass- ing through the situations of the moon at the times T and T'. Also, let RKFK' be the bolder of the sun's disc ; / /' the positions of the moon's centre at the begin- ning and end of the eclipse, determined by describing a circle around C as a centre, with a radius equal to the sura of the apparent serai-diameters of tlie sun and moon; and M (the foot of the perpendicular let fall from G upon N'C) its position at the time of greatest obscuration. If the eclipse shonld 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 G 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 clo.ser 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 may also be had by construction, after the same manner as in a lunar eclipse (330). In order to be able to observe the beginning or end of a solar eclipse, it is neces- sary to know the position of the point on the sun's limb where the first or last eon- tact 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. 132,1 on the lines Cf, Of. 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 be- tween 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 (Pig. 6, p. 13) be the zenith, P the elevated pole, and S the sun ; then in the trian- gle ZPS we shall know ZP the co-latitude, ZPS the hour angle of the sun, and we may deduce PS, the co-declination of the sun, from the longitude of the sun as de- rived from the tables (equ. 24). These three quantities being known, ZSP, the angle made by the vertical through the sun with its circle of declination, may be computed ; and being laid ofi' in the figure to the right or left of CP (Fig. 132), ac- cording as the time of beginning is before or after noon, the point Z or Z', .is 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 arc Zt, Z't, on the limb, will ascertain the situation of /, the first point of ooiitacn Willi respect to it. The situation of the last point of contact may be found by the same mode of proceeding. Let us now show how to find the exact times of the Veginning, greatest oiscwation, and end of the eclipse, the approximate times being Isnown. Let B designate the approximate time of beginning, taken to the nearest minute. Calculate for the time B by means of the tables, tjje sun's longitude, hourly motion, and semi- diameter; 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 .ipparent longitude of the moon from the true longitude of the sun ; the difl'erence will be the apparent distance of the moon 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, 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 Mm =i 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 i\'=a' + \' (g).* Having computed the value of A, we find, by com- paring 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. B~ Fix upon a time some 4 or 5 minutes before or after B, according as the beginning is before or after, and call it FiO. 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 betweeu this and the same distance a at the time B, and we shall have the ap- parent 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 ( = the inter- val between the approximate and true times of the beginning of the eclipse, and suppose S and M (Fig. 1 33) to be the situations of the sun and moon at the true time of beginning. In the time i, the apparent relative motions in longitude and latitude will be, respectively, equal to Id and nt, and accordingly we shall have Sm = a— A(, Mm =>^ + nt. The small right-angled triangle SMm may be considered as a plane triangle; the hypothenuse SM=i//=the sum of the apparent semi-diameters of the sun and moon, mmus 6".5 (^. 427). "We have then, to find t, the equation or, developing and transposing, (n^ + V) t^-2 (aA;-X«) « = ;p« -(..«+ X") = ^p^—A^; * In place of equation (g) the following equations may be employed in logarith mic computation : tang9 = --' A = -3^^' where 9 la an auxiliary arc. 430 APPENDIX. making A.= i//'— A ', and B = aJc — >.n, we have (n* + J') <' — 2B< = A, and t= „«+fce " ' " ^"^^ The negative sign must he prefixed to the radical, for, if we suppose A to be equal to zero, t must he equal to zero. Multiplying the numerator and denomi- nator by B + V B^+A(n'^+k^), and restoring the value of A we obtain 3600s. (A '—i/-^) (m seconds), t — 5- ,- . . . (,«)■ Although this equation has been investigated for the beginning of the eclipse, it is plain that it wiU answer equally well for the determination of the other phases, if we give the proper values and signs to 1//, a, A, 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 i 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 i^' 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 echpse, which is the approximate time of greatest obscuration; then compute the inclination of the relative orbit by the equation n tang 1 = -J- . . . 0> after which ip wiU result from the equation ip = A cos I ... (4). X is the moon's latitude at the time of apparent conjunction, which is easily cal- culated, by means of the values of k and ra, 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 ecUpse, we have, li = 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 moon — 1".5. If the value of ip, 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 station- ary position of the sun's centre, and M the place of the moon's centre at the instant of its nearest approach to 0. Also, let a =z CR the apparent distance of the moon from the sun in longitude at the time of the near- est approach of the centres, A' = EM the moon's apparent latitude at the same time, fc = Mi the apparent relative motion in Fia. 134. longitude in some short interval about this time, and n=z kn the moon's apparent motion in latitude during the same intervaL The right-angled triangles iink and CMR are similar, for their sides are respect- ively perpendicular to each other ; whence Wc : MR :: kn : CR; M n and OR = MR^'or,<. = X'-^ . . . (Z). * Developing the radical in equation Qi), and neglecting all the terms after the second, as being very small, we obtain for the beginning and end of the ecUpan the more con venient formula _ 18 00s. (a'— /") t= g — CALCULATION OF AK ECLIPSE OF THE SUK. 431 If the moon's apparent latitude be found for the approximate time of greatest obscuration, and substituted for X' in equation (I), this equation will give ver^ nearly the apparent distance (a) 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 longi- tude, the true time of greatest obscuration may be found nearly by simple propor- tion. A more accurate result may then be had by finding the moon's apparent latitude for the time obtained, substituting it for \' in equation (I) and then repeat- ing the calculations. _ A simpler, though less accurate method than that already given, of finding the timesof 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 2an 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 M.n, the distance passed over in some short interval (L). Let e = < MnJc the complement of the incUnation of the relative orbit, k = Uk, k' =: Mn, and t = half duration of total or annular eclipse. The triangles Unk, CRM, give Un = — ^'^ ,orfc' = _j_. . .(m): sin Unk sin and tang ROM = tang Unk = ™ , or, tang 6 = }L . . . (n). CR a Finding the value of B by tlie last equation, and substituting it in equation (m), we obtain the value of k' ; and then, to find t, we have k':h :: Kg : t, or t = liJLM k' whence, t -- L-v/-/-"— A2_Lv'(,/,-fA) (i^— A) K ft' The apparent distance of the centres of the two bodies at the time of greatest ob^ Bcuration being known, the quantity of the eclipse may be readily found. We have but to subtract the apparent distance from the sum of the apparent semi-diametera, and state the proportion, as the sun's apparent diameter : the remainder :: 12 digits : the digits eclipsed. (For a more particular description of the method of calculat- ing a solar eclipse, see Prob. XXX.) CALCULATION" OF AN OCOULTATION. 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 the ecliptic, we have, in place of the lati- tude of the moon, employed in solar eclipses, the differeuce 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 longi- tude, these quantities referred to an arc passing through the star and parallel to the ecliptic. Thus, if EC (Pig. 133) re- present the ecliptic, K its polo, s the situation of the star, M that of the moon, and sm! an arc passing through s and parallel to the arc EC, we have in place of mM, m'M = «iM — mm', and in place of Sm, am'. The hourly variation of Sm must also be reduced to the arc «m'. The reduction of the difference of longitude of the moon and star, to the parallel to the ecliptic, passing through the star, is effected by multiplying the difference by the co- sine of the latitude of the .star. For, let AB (Fig. 135) be an arc of the ecliptic, and A'B' the corresponding arc of a 482 APPENDIX. circle parallel to it, then, since similar arcs of circles are proportional to theii radii we liave BC : B'C :: AB : A'B'= :^1^121. BC But, B'C — Caz= B'C cos BOB' = BC cos BB': hence, A'B' = AB.BC cos BB' ^ ^g ^^^ gg-^ BO The reduction of the relative hourly motion in longitude to the parallel in question, is obviously effected in the same manner. JSrOTE I. CONSTRTJCTION 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 computa- tion 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 longi- tude, the equation of the centre, the amounts Of the various inequalities in longitude, and the nutation in longitude, at any a.ssumed 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 tropi- cal year, is 360°. From this we may find by proportion the mean motions in a com- mon year of 365 dnys, 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 XTIII. contains the mean longitude of the sun at the beginning of each of the years in- serted in the first column. The third column of this table contains the mean longi- tude 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 from 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 for hours from 1 to 34 ; 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 bo added to the epoch for the given year, laken out from Table XVIII, the result will be the mean longitude at the given time. (See Problem 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 liuown 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 9 + B sin 29 -)- sin 38 + &o. ; * The quantities in Table XTIH are called Epochs. The Epoch of a quantltv 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 S the mean anomaly. Now, by giving to the mean anomaly 6 in this equation a series of values increasing by small equal differences (of 1", for instance,) from zero to 360°, and computing the correspond- ing values of the equation of the centre, then registering in a column the diiVerent values assigned to 6, and in another column to the rij;ht of this the computed values of the equation of the centre, we sliall obtain a tabic which will give on inspection the equation of the centre corresponding to any purticular mean anomaly. In this manner was constructed Table XXV. In '.his table, however, tor the sake of com- pactness, the values of the equation, instead of being registered ni one column, are put in as many different columns as there may be different numbers of signs in the value of the mffiin 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 tl.'e value found for it talie 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-gumevt of that quantity. Accordingly the. mean anomaly is the argument of the equation of the centre in Table XXV". (3.) For the ivequatilies. — 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 longitude 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 entry, 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 longitude 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 cor- responding 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 moan lon- gitude of the sun may be found by the solar tables (XVIII to XXIi, and thence the mean heliocentric longitude of the earth by subtracting 180° ; and the mean longitude of the disturbing planet may be had from similar tables. The columns of Table XVIII, marked I, II, III, IT, V, VI, VII, contain the arguments 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 con- tain the variations of the arguments for months, days, and hours. Their varia- tions for minutes and seconds are too small to be taken into account. With these tables, and Table XVlII. the values of the arguments at any given time may bo found, and by means of the arguments the perturbations may be taken from Tables, XXTIII, XXIX, XXXII,' XXXI, XXX, and .XXXIII. (4.) Fm- (he 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 360°) 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, result^ 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 radius- vector. We have True rad.-vector = eUiptio 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 expres- sions (Art. 214). The tables of the perturbations will have the same .Tgumeuta 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 ciise of these bodies, for the computation of the ecliptic longitude and the latitude. 28 484 APPENDIX. Tlie difference Ijetweeu the orbit longitude and the ecliptic longitude is called th* 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 lon- gitude of the node (or, what amounts to the same, the orliit longitude plus the supplemeut of the lougitude of the node to 360°). If this formula bo reduced tc a table, by taking the reduction from the tal)le and adding it to the orbit longitude, we shaU have the ecUptic lougitude. Table LUX is a table of reduction tor the moon. For Hie 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 (ruclina.) A table constructed from this formula will have for its argument the orbit lon^ gitudo minus the longitude of the node, which is also the argument of the reduction, (See Table LV.) The tables of the perturbations in latitude are constructed upon the same priu. ciples as the tables of the perturbations in longitude and radius-vector. NOTE II. {Referred io onp. 175.) 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 circumstauoe 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 in 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 elactric 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 ceutral portions of this opening, the lower envelope will still be in a comparatively quiescent condition, and retain its natural depth of sh&de. Subsequently tlie same process of evolution is repeated at this envelope; an open- ing is made in it that lias 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 readied. In the onse of the larger spots and of long continuance, the same pro- cess 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 througji DEVELOPMENT OP SUN'S SPOTS. 435 the first and second, or through the firgt, 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 helow 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 penumbrtil 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 fcl- lowed by the encroachment of the luminous margiu upon the penumbra. But it is conceivable, also, that the closing up of a spot may 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 indica- tions of the beginning of such condensatiouK. 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 mag- netic 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 super- ficial 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 al&o develops- similar magnetic cur- rents, 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 tlie motion of revolution of a single planet would have but little direct effect to change its magnetic action on tlie sun as a whole ; except so Far as this may vary by reason of the varying distance of the planet. But in point of fact the efl'ective action of a planet will change with its distance in longi- tude from the point towards which the pun is movini^ in space. It will be at its maximum when the planet is in heliocentric conjunction with this point, and at a niinimu[D when it is iu opposition to it. In the former case its heliocentric longitude will be 253°, and in tlie latter 73°. 6. The joint magnetic action of two planets varies with their relative position ; it has its mjiximum 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 mininaum of spots, since the magnetic intensity at tlie surface of the photosphere is on the increase before every such epoch. The approximate coincidence of planetary conjunctions with special epochs of minimum of spots is a recognized fact in the -case of Jupiter and the earth, and Venus and the earth. On the other baud, the opposition of two ■planets tends to determine a maximum of spots. 8. Jupiter and Venus are the most influential planets. The period of the spots is de- termined mainly by tlie motion of revolution of Jupiter, but appears to be modified by the varying planetiiry 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 variatiims of distance from the sun, but it seems improbable that effects so marked should result from so slight a cause. The mean period of the spots is, according to Wolf, nearly one year, and according to Schwabe, nearly two years less than Jupiter's period of revolution. This difference cannot be explained oy any mere reciureuce of planetary configurations. In fad, 436 APPENDIX. epochs of maximum and minimum of spots have occurred under every variety of configurations; and also when Jupiter lias 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-coudition of the sun. Mean Heliocentric Mean Heliocentric Epochs of Maxima. Relative Numbers. Longitude. Epochs of Minima. Obs'd Min. -Mean Min. Longitude. Jupiter. Saturn. Jupiter. Saturn. 1150.0 68.2 4° 231° 1144.5 + 0.558 197° 164° ] 161.6 15.0 353 12 1155.1 + 0.639 111 301 1710.0 19.4 251 116 1766.5 + 0.S20 145 13 1179.5 99.2 119 232 1775.8 —1.499 67 187 1188.5 90.6 93 342 1784.8 —3.618 340 297 1804.0 10.0 203 112 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 1831.2 lU.O 131 218 1833.8 + 0.906 28 177 1848.6 100.4 111 358 1844.0 —0.013 338 301 1860.5 98.6 . 119 143 1856.2 + 1.068 348 91 It will be seen that since 1161 the m.ixima have occurred when Jupiter was in the second or third quadrant of longitude; and that since 1155, the minima have occurred when lie 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 ordi- nary effect of the planets is, at intervals, wliolly reversed. 9. The highest maxima of spots have occurred when Jupiter was in the second quadrant of longitude; that is. after he lias passed a certain distance beyond the point (long. 73°) where his magnetic action on the sua 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 eartli'a 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 causea of the disturbances experienced by the magnetic needle on the earth's surface. NOTE III. {Referred to on page 215.) A remarkable analogy iu the periods of rotation of the primary planets was diff covered a few years since (1848) by Daniel Kirkwood, of Pottsville, Pennsylvania This analogy is now generally known by the name of Kirkwuod's Law, and is aa follows : " Let P be the point of equal attraction between any planet and the one next in- terior, 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 maa- ner; KIRKWOOD'S ANALOGY. 437 n =r the number of sidereal rotationa performed by the former during one sidereal revolution around the sun ; n — the number performed by the latter ; then it will be found that D»,D'=; or« = 7i'g)l That is, the square of the number of rotations made by a planet dwing one revolution round the sun, is proportional 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 exami- nation by Sears C. Walker, with the following result : " We may therefore conclude," says he, "tliat whether Kirkwood's Analogy is or is not the expression of a physi- cal 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. 19-36.) There are but tliree planets, viz., Venus, tlie Earth, and Saturn, for which all the elements embraced iu this law are known. The diameters of the spheres of attraction of Mercury and Neptune are, from the nature of the case, incapable of determina- tion. The mass of the oue planet into which the planetods are supposed once to have been united is not known witli certainty, as there maybe planetoids yet undis- covered, and its period of rotation is hypothetical only. Tlie diameters of the spheres 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 discov- ery was heightened by the fact, that it was at once perceived thai it furnished a new and powerful argument in support of the nebular hjtpothesis (or cosmogony) devised by Laplace. iSee 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. {Eeferred to on page 27'7.) It remains to deduce, if possible, the known law of the distribution of the incli- nations of the cometary orbits. This law is, that the number of orbital inclinations of different values increases with the angle from 0° to 50° or 60°, and then de- creases ; as appears from the following tabular statement, from which the comets of short period have been deducted. 0° to 10° 10° to 20° 20° to 80° 80° to 40° 40° to 80° 50° to 60° 60° to 70° 70° to 80° 80° to 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 ve- locity must bS 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 ex- pelling 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 httle from 180°, whatever directiou may be assumed for 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, ii, or a little greater, or a little less. 438 APPENDIX. Under these oiroumstances, 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 answering to the assumed maximum initial velocity, V. Now iif we conceive the point at which a discharge occurs to lie in any latitude {I), the'velocity of rotation will be less in the proportion of cos Z to 1 ; and making use of the parallelogram of velocities as before, and retaining the same effective ve- locity 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 (I) the orbits described by the masses detached may have every variety of inchnation from 1° to 90° ; but the larger inclinations wUl result from an action of the expelling force exerted in a direction incUned 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 be- ing within its surface at the point of discharge. In view of what has now been stated, it may be seen that the actual lavi of disiri- hution of (he inclinations might result if the frequency of discharge were to decrease with the angle included between the line of direction of the operating force and the meridian. This theoretical result suggests, as the possible origin of the separation of frag- ments from the surface of the nebulous body, (ho flow of electric currents in all di- rections from points near the equator ; similar to the currents we have conceived tc 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 liabihty 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 would fall, for each latitude, I, on the meridian, between I' 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 accor- dance 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 neut 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 sur- face 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, aa . playing an important part in the production of the phenomena observed. Analogy would then lead us to infer that the exci'ing cause of the electric currents sup- posed to have furnishad 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 haa ORIGIN OF SIDEREAL SYSTEMS. 439 been operative in the other eases. The same process may have continued in opera- tion down to (he present epoch, originating, in the later ages, meteorites rather than true cometary bodies. In fact we have seen that a continual process of die- charge of nebulous magnetic matter from the suu is in operation in the present age. (Art. 293.) NOTE V. ORIGIN OP SIDEREAL SYSTEMS. We propose, in the present note, to develop very briefly a theory of the possi- ble evohition of all sidereal systems from primordial, rotating, nebulous masses. Fundamental hypoOiesis, and circmnsianxxs ' — — 9 — 15 — 17 — 15 — 8 2 1 9 15 17 14 8 4 1 10 16 17 14 7 6 2 10 16 17 14 7 8 2 11 16 17 13 6 10 3 11 16 17 13 6 12 4 12 17 17 12 5 14 4 12 17 16 12 6 16 6 13 17 16 12 4 18 5 13 17 16 11 3 20 6 13 17 16 11 3 22 7 14 17 16 10 8 24 7 14 17 15 10 3 26 8 15 17 15 9 1 28 8 15 17 15 9 I 30 — 9 — 15 — 17 — 15 — 8 — TABLE XXV. Equation of the Sun^s Centre. Argument. Sun's Mean Anomaly. 2] VI» VII» VIII, IX» x« XI, 11» W 11» 11» 11» 11, o o ' " O ' " Q r If o , „ o ' " ' " 29 59 13.9 29 2 35.2 28 20 23.0 28 3 53.7 28 18 17.1 29 29.3 . 1 29 57 15.6 29 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 7 41.5 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 29 47 25.2 28 52 33.8 28 14 45.4 28 4 16.6 28 24 45.1 29 11 23.8 7 29 45 27.7 28 50 57.5 28 13 55.6 28 4 27.7 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 3B.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 29 33 48.2 28 41 45.4 28 9 36.5 28 6 19.1 28 33 38.9 29 24 48.9 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 29 28 4.0 28 37 27.2 28 7 53.2 28 7 43.1 28 37 52.6 29 30 44.6 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 29 24 17.0 28 34 42.1 28 6 54.4 28 8 49.4 28 40 49.6 29 34 44.7 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 16 50.8 28 29 30.1 28 5 21.1 28 11 26.6 28 47 1.3 29 42 50.1 23 29 15 0.9 28 28 15.9 28 5 2.9 28 12 11.0 28 48 37.7 29 44 52.5 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 29 11 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 55 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 2 35.2 28 20 23.0 28 3 53.7 i 28 18 17.1 29 29.3 29 59 13.9 TABLE XXVL Secular Variation of Equation of Sun^s Centre. Argument. Sun's Mean Anomaly. VI» VII» VIIIs IX» Xs XI. o ff // /, /r /' • + + 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 3 28 6 13 17 16 11 2 24 7 14 17 16 10 « 26 7 14 17 16 10 1 .28 8 14 17 15 9 1 30 + 8 + 15 + 17 + 15 + 9 + 30 22 TABLE XXVII. Nutations. Argutnont . Supplement of the Nod e, or N. Solar Nh,tatton. N. Long. R. Asc. Obliq. N. Long. R. Asc. Obliq. Long. Obliq. + 0.0 +0.0 + 9.2 500 — 0.0 — 00 — 9.3 Jan. ff j ff 10 1.0 1.0 9.1 510 1.1 1.0 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 3.0 9.0 530 3.3 2.9 9.2 21 l.ll 0.2 40 4.2 4.0 8.9 540 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 1.3 +0.1 70 7.2 6.9 8.3 570 7.5 6.6 8.4 30 1.0 0.3 80 8.2 7.8 8.1 580 8.5 7.5 8.1 March. o 90 9.1 8.7 7.8 590 9.5 8.4 7.8 0.7 + 0.3 0.4 05 100 + 10 + 9.4 + 7.5 600 — 10.4 — 9.1 — 7.5 12 110 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 April. 1 11 21 130 12.4 11.7 6.3 630 12.8 11.4 6.3 0.5 08 1.1 0.5 0.2 0.2 140 13.1 12.4 5.9 610 13.5 12.0 5.9 ,150 + 13.8 + 13.0 + 5.5 650 — 14.2 -12.6 — 5.4 160 14.4 13.6 5.0 660 14.8 13.2 4.9 May. 1 11 21 31 170 15.0 14.1 4.5 670 15.3 13 8 44 1.2 1.2 1.1 0.8 + 0.1 — 0.1 0.3 0.4 180 15.5 14.5 4.0 680 15.8 14.2 3.9 190 15.9 14.S 3.5 690 16.2 14.7 3.3 200 + 16.3 + 15.1 + 2.9 700 — 16.6 — 15.0 — 2.8 210 166 15 4 2.4 710 16.9 15.3 22 June. 10 20 30 220 16.9 15.6 1.8 720 17.1 15.4 1.6 0.4 — 0.0 + 0.4 0.5 230 17.1 15.7 1 2 730 17.2 15.7 1.1 0.5 0.5 240 17.2 15.9 0.7 740 17.3 15.9 — 0.5 250 + 17.3 + 15.9 + 0.1 750 — 17.3 — 15.9 + 0.1 260 17.3 15.9 — 0.5 7B0 17.2 15.9 0.7 July. 10 20 30 0.7 1.0 12 0.4 0.3 — 0.1 270 17.2 15.7 1.1 770 17.1 15.7 1.2 280 17.1 15.6 1.6 780 16.9 15.4 1.8 290 16.9 15.4 2.2 790 16.6 15.3 2.4 300 + 16.6 + 15.1 _ 2.8 800 _16.3 — 15.0 + 2.9 4.Ufr. 310 16.2 14.8 3.3 810 15.9 14.7 Z^ 9 19 29 1.3 1.2 0.9 + 0.0 0.4 0.4 320 15 8 14.5 3.9 820 15.5 14.2 4.y 330 15.3 14.1 4.4 830 15.0 13.8 4.0 340 14.8 13.6 4.9 840 14.4 13.2 50 Sept. 350 + 14.2 4- 13.0 — 5.4 850 _13.8 — 12.6 + 5.5 8 06 0.5 360 13.5 12.4 5.9 860 13.1 12.0 59 18 28 + 2 — 0.2 0.5 0.5 370 12,8 11.7 6.3 970 12.4 11.4 6.3 380 12.0 11.1 6.7 880 11.6 lO.fi 6.7 Oct. 390 11.2 10.3 7.1 890 10.8 9.9 7.1 8 0.6 0.5 400 + 10 4 + 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 85 7.8 8.1 920 8.2 7.5 8.1 Nov. 430 7.5 69 8.4 930 7.2 6.6 8.3 7 1.2: + o.ol 440 6.5 60 8.7 940 6.2 5.7 8.5 17 1 2 0.2| 450 + 5.5 + 4.9 — 8.9 950 — 5.2 — 4.8 + 8.7 27 1.0 0.4 460 4.4 4.0 9.0 960 4.2 3.9 8.9 Dec. 470 3.3 30 92 970 3.2 29 9.0 7 06 0.5 480 2.2 2.1 9.3 980 2.1 2.0 9.1 17 — 0.2 0.5 490 1.1 1.0 9.3 990 1.0 1.0 9.1 27 + 0.3 0.5 500 + 0.0 + 00 — 9.3 1000 — 0.0 - 0.0 + 0.2 37 + 0.6 -0.5 TABLE XXVIII. TABLE XXIX 23 jMnar Equation, 1 si pi2i t. Argument I. Lunar Equation, 2d part. Arguments I. and VI. I. I Equa I Equ vi_ 50 100 150 200 250 300 f/ 350 400 450 COO /. ~ ,7 7.5' 500 7.5 1.3 1.2 1.2 1.1 1.0 1.0 1.0 1.1 1.2 1.2 1.3 10 8.0 510 7.0 1 50 1.5 1.5 1.5 1.3 1.1 1.0 0.9 1.0 1.1 1.1 1.1 20 8.4 520 6.6 100 1.7 1.8 1.7 1.4 1.2 1.1 1.0 0.9 0.9 0.9 0.9 30 8.9 530 6.1 150 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 5.6 200 1.9 2.0 2.0 1.7 1.5 1.4 1.0 0,8 0.8 0.8 0.7 50 9.8 550 5.2 250 2.0 2.0 2.0 1.8 1.6 1.5 1.1 0,9 0.7 0.7 0.6 60 10.3 560 4.7 300 1.9 1.9 1.9 1.9 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 1.6 1.4 1.0 1.0 0.9 0.8 80 11.1 580 3.9 400 1.6 1.7 1.8 1.9 ..7 1.6 1.4 1.2 1.1 1.0 1.0 90 11.5 590 3.5 450 1.5 1.5 1.6 1.7 1.7 1.7 1.6 1.4 1.2 1.2 1.1 100 11.9 600 3.1 500 1.3 1.4 1.4 1.5 1.7 1.7 1.7 1.5 1.4 1.4 1.3 no 12,3 610 2.7 550 1.1 1.2 1.2 1.4' 1.6 1.7 1.7 i 7 1.6 1.5 1.5 120 12.6 620 2.4 600 1.0 1.0 1 1 1.2 1.4 1.6 1.8 1.8 1.8 1.7 1.6 130 13.0 630 2.0 650 0.8 0.9 1.0 1.1 1.3 1.5 1.7 1.8 1.9 1.9 1.8 140 13.3 640 . 7 700 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 09 1.1 1.2 1.5 1.8 2.0 1.9 1.9 170 14.1 670 0.9 850 0.7 0.8 0.8 0.9 0.9 1.1 1.4 1.7 1.8 1.8 1.9 ISO 14.3 680 0.7 900 0.9 0.9 0.9 0.9 1.0 1.1 1.2 1.5 1.7 1.7 1.7 190 14 5 690 0.5 950 1.1 1.0 1.1 1.0 1.0 1.0 1.1 1.3 1.4 1.6 1.6 SOO 210 320 230 14.6 14.8 14.9 14.9 700 710 720 730 0.4 0.2 0.1 0.1 1.3 1.2 1.2 1.1 1.0 1.0 1.0 1.1 1.2 1.2 1.3 I. 1 VI 500 550 600 6£0 700 750 800 850 900 950 1000 240 15.0 740 0.0 „ ~"^~ // '~~~ "^"^ 250 )5.0 750 0.0 1.3 1.4 1.4 1.5 1.6 1.6 1.6 1.5 1.4 1.4 1.3 260 15.0 760 0.0 50 1.1 1.1 1.2 1.3 1.5 1.5 1.7 1.6 1.5 1.5 1.5 270 14.9 770 0.1 100 0.9 0.9 0.9 1.1 1.3 1.5 1.6 1.7 1.7 1.7 1.7 280 14.9 780 0.1 150 0.7 0.8 0.8 0.9 1.2 1.4" 1.6 1.9 1,8 1.8 1.9 290 14.8 790 0.2 200 0.7 0.7 0.6 0.8 l.I 1.2 1.6 1.8 1,8 1.8 1.9 300 14.6 800 0.4 250 0.6 0.6 0.7 0.7 1.0 1.1 1.5 1.7 1,9 1.9 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 1320 14.2 820 0.7 350 0.8 0.7 0.7 0.8 0.9 1.0 1.4 1.6 1.6 1.7 1.8 330 14.1 830 0.9 400 1.0 0.9 0.8 08 0.9 1.0 1.2 1.4 1.5 1.6 1.6 340 13.8 840 1.2 450 1.1 1.1 1.0 0.9 0.9 0.9 1.0 1.2 1.4 1.4 1.5 350 13.6 850 1.4 500 1.3 1.2 1.2 1.1 0.9 0,9 0.9 1.1 1.2 1.2 1.3 360 13.3 860 1.7 550 1.5 1.1 1.4 1.2 1.0 0.9 0.9 0.9 1.0 1.1 1.1 370 13.0 870 2.0 600 1.6 1.6 1.5 1.4 1.2 1.0 0.8 0.8 0.8 0.9 1.0 380 12.G 880 2.4 650 1.8 1.7 1.6 1.6 1.3 1.1 0.9 0.8 0.7 0.7 0.8 390 12.3 890 2.7 700 1.9 l.S 1.8 1.6 1.4' 1,2 0.9 0.7 0.7 0.7 0.7 4U0 11.9 900 3.1 750 2.0 1.9 1.9 1.7 1.5 1.3 1.0 0.7 0.7 0.6 0.6 410 11.5 910 3.5 800 1.9 1.8 1.8 1.8 1.6 1.4 1.1 0.8 0.6 0.7 0.7 420 11.1 920 3.9 850 ! 1.9 1.8 1.8 1.8 1.6 1.5 1.2 0.9 0.8 0.8 0.7 430 10.7 930 4.3 900 1.7 1.7 1.7 1.7 1,6 1,5 1.3 1.1 0.9 0.9 0.9 440 10.3 940 4.7 950 1.5 1.5 1.5 1.6 1.7 1.6 1.5 1.3 1.2 1.1 1.1 450 460 9.8 9.4 950 960 5.2 5.6 1.3 1.4 1.4 1.5 1.6 1.6 1.6 1.5 1.4 1.4 1.3 1 Constant 1".3 470 8.9 970 6.1 480 84 980 6.6 490 80 990 7.0 SOO 7 5 1 1000 7.5 1 1 24 TABLE XXX. Perturbations -produced by Venus. Arguments II and III. m. 11. 10 20 30 1 40 1 50 60 70 14.7 80 90 100 128 110 120' ^1 fi 20.8 19.8 19.0 17.9 16,8 15.9 14.0 13.2 12.5 1Z.3 20 23' 1 40 23.5 69 22 2 80 '-"^ ^ 22.7 21.6 21.0 20.1 19.3 18.4 17.4 !6.4 15.5 14.5 13.81 13.4 23.2 22.9 22.7 22.0 21.1 20.4 19.'j 18.7 17.9 16.9 ' 16.1 15.3 1 22.5 20.7 23.1 22.7 22.8 22.5 21.9 21.3 20.5 19.9 19.1 18.2 17.4 21.4 21.7 22.1 22.3 22.2 22.2 21.7 21.3 20.7 19.9 19.3 100 17.6 18.6 19.2 19.9 20.5 21.0 21.6 21.7 21.6 21.6 21.5 21.1 20.5 120 140 160 180 200 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 13.6 14.2 14.8 15.5 16.2 17.0 17.6 18.3 19.0 19.4 20.0 20.0 20.4 12.7 13.2 13.6 14.1 14.6 15.0 1.5.7 16.4 17.0 17.3 18.1 18.7 19.2 12.7 12.9 13.1 13.5 13.9 14.0 14.5 14.8 1.5.0 15.8 16.4 16.8 17.2 13.2 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 14.3 14.5 14.6 14.6 14.7 14.8 240 13.6 13.8 14.1 14.4 14.6 14.S 14.8 14.9 15.1 15.1 15.1 14.9 14,8 260 12.8 133 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 14.6 15.1 15.4 16.0 16.2 16.2 16.3 16.2 300 10.1 10.9 113 12.1 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 14.4 380 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 12.1 11.5 10.5 9.6 9.0 8.5 8.0 7.3 6.8 6.6 6.5 6.4 6.5 480 13.6 12.8 11.9 11.0 10.4 9.6 8.8 8.2 7.7 7.2 6.8 6.4 6.5 500 15.1 14.4 13.4 12.4 11.6 10.8 10.1 9.3 8.6 8.1 7.5 7.1 6.8 520 16.5 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 18.1 17.5 16.4 15.5 14.5 13.7 12.8 11.8 11.1 10.4 9.7 8.9 8.2 560 20.4 19.3 18.2 17.6 16.5 15.4 14.4 13.4 12.7 11.6 10.8 10.2 92 580 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 22.2 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 14.7 040 29.0 28.5 27 7 26.9 26.2 25.1:24.1 22.9 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 34.5 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 23.8 29.2 29.3 29.5 29.5 29.5 29.2 28.8 28.4 27.8 27.2 26.4 25.2 720 26.9 27.6 28.3 29.0 29.2 29.4 '29.4 29.3 29.1 28.9 28.4 27.9 27.3 740 24.7 25.7 26.6 27.3 27.9 28.5 29.1 29.0 29.2 29.3 29.1 28.8 28.4 760 23.2 23.5 24.3 25.3 26.2 27.0!27.B 28.3 28.6 28.7 28.9 29.1 29.0 780 19.6 21.0 22.0 23.2 24.2 25.1 125.9 26.7 27.3 27.8 28.4 28.5 28.7 800 17.2 18.5 19.3 20.9 21.8 22.9 23.9 25.0 25.8 26 4 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 26.3 26.6 840 13.2 14.0 15 16.0 17.0 18.2 18.8 20.3 21.7 22.7 23.6 24.5 25.3 800 11.5 12.2 13.0 13 9 14.9 15.9 17.1 18.0 18.9 20.3 31.4 22.6 23,5 880 11.0 11.2 11.5 12.2 13.0 13.7 14.8 15.7 16.8 18.1 19.1 20.2 21,1 000 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 U.C 11.5 11.1 11.2 11.3 11.7 12.1 12.7 13.4 14 4 15.2 16.4 940 14.0 13.3 12.6 12.3 11.6 11.5 11.3 11.4 11.6 12.0 12.8 13.3 14.2 960 16.7 15.0 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 12. -2 12.0 11.9 11.8 11.8 1000 21,6 20.8 19.8 j 19.0 17.9 40 16.8 15.9 14.7 14.0 13.2 12.8 12.5 110 12.2 L 1 1 ^^ 20 1 30 53 60 71 80 90 00 120 _i TABLE XXX. 25 Perturbations produced by Venus. Arguments II and III. III. II. 1 120 130 I 140 1 160 160 170 ISO 190 200 210 220 230 240 12.2 12.2 12.3 12.4 12 8 13 3 -' 1 " 1 " 13.9 ; 14.7 15.6 16.5 ft 17.7 18.8 20.1 20 13.4 12.9 12.6 12.3 12.2 12.4 12.9 13.3 14.0 14.6 15.5 16.4 17.3 40 15.3 144 14.0 13.5 13.0 12.9 12.6 12.6 13.1 13.5 140 14.4 15.4 60 17.4 16.7 16.0 15.2 14.5 140 13.6 13.3 13.2 13.2 13.4 13.5 14.1 80 19.3 18.7 17.7 17 1 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 14.6 14.3 120 20.8 20.7 20.4 20.0 ■ 19.7 19 2 18.5 18.0 17.3 16.9 10.5 16.2 15.6 140 20.4 ' 20.4 20.2 20.0,20.1 19.7 19.5 19.3 18.8 18.2 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 17.7 18.5 18.5 18.5 18.8 18.4 18.8 19.0 19.0 18.9 18.6 18.5 200 16.0 16.2 16.6 16.8 17.5 17.6 17.7 17.9 18.1 18.2 18.3 18.3 18.3 220 14.8 15.0 15.3 15.7 16.1 16.2 16.6 16.8 17.1 17.5 17.1 17.4 17.5 240 14.8 147 14.8 15.0 15.1 15.4 15.7 15.8 16.0 16.1 16.1 16° 16. t 260 15.6 15.7 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 15.1 149 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 1,.8 17.9 IS.l 18.3 18.2 18.2 17.9 17.5 17.3 16 8 360 14.4 15.2 16.0 16.7 17.4 18.1 18.4 18.6 18.8 18.8 18.8 18.7 18.4 380 12.4 13.4 14.3 153 16.1 16.9 17.5 18.1 18.6 19.1 19.3 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 13.0 141 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 12.9 13.8 149 16.0 16.7 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 13.5 600 6.8 6.7 6.5 6.3 6.5 6.6 7.0 7.4 '8.2 8.6 9.4 10.4 11.3 520 7.4 7.0 6.8 6.a 6.3 6.1 6.3 6.6 7.0 7.5 8.0 88 9.3 540 8.2 V.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 66 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 60 600 12.6 11.4 10.5 9.5 8.7 8.1 7.4 7.0 6.4 6.1 5.8 5.5 5.6 620 14.7 13.5 12.4 11.4 104 9.5 8.7 7.9 7.3 6.7 6.2 5.6 5.2 640 17.2 16.2 149 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 9.6 8.6 7.7 6.9 680 22.8 21.7 20.4 19.3 18.1 16.8 15.7 142 13.0 11.9 10.7 9.6 8.6 700 25.2 24.3 23.3 22.1 20.7 19.7 18.5 17.3 16 143 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 13.9 740 28.4 27.7 27.4 20.6 25.9 249 240 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 23.0 21.7 20.7 19.7 7S0 28.7 28.7 28.8 28.7 28.3 28.0 27.2 26.1 20.1 25.2 243 23.3 22.2 800 26.1 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 27.8 28.1 28.3 28.1 28.1 28.0 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 27.9 28.0 27.9 27 ? 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 27.7 900 18.5 20.1 21.3 22.1 23.1 24.7 25.0 25.7 26.3 26.9 27.3 27.5 27.6 920 16 4 17.7 18.4 20.0 21.0 22.2 23.0 23.9 24.9 25.7 26.2 26.9 27.3 940 !4.2 149 16.1 17.5 18.2 19.6 20.8 21.9 23.0 239 24.7 25.7 26.1 960 12.6 13.3 141 14.4 15.9 17.2 17.9 19.5 20.5 21.7 22.7 23.9 247 9S0 11.8 13.1 12.7 13.3 14.1 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 12.8 13.3 170 13.9 180 147 190 15.6 16.5 17.6 18.8 20.1 l^i 130 140 150 160 1 200 210 220 230 24) 26 TABLE XXX. Perturbations ■produced by Venus, Arguments II. and III. III. II. |240 250 260 270 1 280 290 300 310 320 330 340 |350 j 360 i „ 20.1 21.1 22.2 " 1 " 23.4 24.3 25 2 25.8 26.6 27.2 27.6 27.7 27.6 1 27.6 20 17.3 18.6 19.7 20.9 21.9 23.0 24.2 24.9 25.8 28.6 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 14.6 15.2 16.3 17.2 18.1 18.9 20.3 21.2 22.3 23.4 24.5 25.3 80 13.6 14.0 14.5 14.9 15.5 16.3 17.3 18.2 19.0 20.0 21.1 22.0 23.1 ioa 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 'l5.6 15.2 14.8 14.8 1.5.0 14.9 15.0 15.2 1.5.9 16.3 17.0 17.7 1S.5 140 ' 17.0 16.6 16.4 15.8 15.5 15.4 156 15.6 15.5 15.6 16.1 16.7 17.1 ;eo 18.1 17.7 17.5 17.3 16.9 16.6 16.3 15.9 16.1 16,3 16.3 16.2 16.5 1«'0 IS.o 18.5 18.3 18.1 17.9 17.6 17.5 17.3 17.0 16.9 16.7 16.8 16.9. 200 1 18.3 18.4 18.2 18.2 18.2 18.2 18.1 18.1 17.8 17.7 17.6 17.5; 17.71 220 17.5 17.6 17.8 17.8 1?0 18.0 18.2 18.1 18.1 18.3 18.4 18.3 il8.3 240 16 4 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 15.5 15.5 15.6 15.8 16.1 16.4 16.6 16.8 16.9 17.4 17.7 18.2 280 15.0 14.9 14.9 11.9 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 13.7 13.1 13.0 12.7 12.6 12.6 360 18.4 17.9 i 17.5 19.2(18.9 17.0 16.5 15.9 15.4 14.9 14.3 13.7 13.0 12.6 12.3 380 19.5 18.5 17.9 177 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 18.11 17.5 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 16.5 440 17.8 18.7 19 2 19.7 20.1 20,4 20.7 20.7 20.5 20.2 19.8 19.5 i8.e 460 15.9 16.8 17.6 18.6 19.2 19.9 20.3 20.6 21.0 20,9 20.9 20.8 20.3 480 13.5 14.6 15.5 16.6 17.7 18.5 19.3 19.9 20.5 20.8 21.1 21.2 21.2 500 11.3 12.4 13.4 14.4 15.5 15.5 17.7 18.6 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 12.1 13.1 142 15.3 16.3 17.4 18.3 19.2 560 6.9 7.2 7.8 8.4 9.2 10.1 11.0 119 13 1 14.1 15.2 16.2 17.2 580 6.0 6.3 6.6 7.0 7.6 1 8.4 9.1 99 10.9 11.9 12.9 14.1 15.0 600 5.6 5.6 5.8 6.1 6 5 6.8 7.4 8.1 8,8 9.9 10.7 11.8 12.8 620 5.2 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 5.9 5.6 5.2 4.9 5.01 5.0 5.2 5.5 5.8 6.4 7.0 7.6 8.5 660 6.9 6.3 5.7 5.4 5.0! 4.8 4.5 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 8.7 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 4'i 760 19.7 18 5 17.2 15.9 14.7 13.5 12.2 10 8 9.8 8.9 7.6 6.7 5.9 780 22.2 21.2 20.1 19.0 17.6 16.3 15.1 14.0 12.6 11.6 10.2 9.2 8.1 800 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 2.5.1 24.4 23.3 22.3 21.6 20.4 19.4 18.2 17.2 15.9 14 6 13.6 840 27.2 26.6 25.8 25.0 24.3 23.5 22.4 21.6 20.5 19.4 18.4 17.3 16.4 860 27.5 27.1 26.8 26.4 25.5 24.8 24.3 23.3 22,2 21.5 20.5 19.6 18.4 880 27.7 27.5 27.2 27.0 28.5 26 25 5 24.7 i 24.1 23.2 22.0 21.4 20.4 900 27.6 27.8 27 9 27.6 27.1 28.7 26.5 1 25.7 1 25.3 24.6 23.9 23.0 22.01 920 27.3 27.5 27,5 27.6 27.7 27.5 27,2, 26.7 26.3 25.7 25.1 24.3 23.6 940 26.1 26.7 27.2 27.4 27.7 27.7 27.6 127.5 27.1 26,6 26.2 25.6 25 5 960 24.7 25.4 26.2 26.6 27.2 27.5 27.7 127,7 27.6 27.4 27,1 27.0 20 2 980 22 6 23.7 24.6 25.3 25.9 26.8 27 2 ' 27.5 27 7 27.8 27.6 27.5 27.1 1000 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' 240 250 260 270 280 290 300 i 310 1 320 1 330 340 , 350 360 I TABLE XXX 27 Perturbations produced by Venvs. Arguments II. and III. III. II. 360 370 380 390 400 410 420 430 440 450 460 1 470 480 27.6 27.7 27.3 26.7 26.2 25.5 24.7 33.8 23.1 '22.3 21.3 20.3 i 19.3 20 27.7 27.8 27.8 1 27 6 27.4 26.8 26.3 35.6 24.8 24.0 23.1 32.0 20.9 40 26.9 27.3 27.6 127.9 27.9 27.7 27.5 37.1 26.3 35.6 24.9 24.0 23.3 60 25.3 26.0 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 27.5 27.9 28.2 28.0 27.6 27.5 27.2 100 20.8 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 22.4 23.2 241 25.1 25.8 26.4 26.9 27.3 i 27.5 140 17.1 17.9 18.6 19.3 20.3 31.3 23.0 23.9 23.7 24.7 25.5 26.0 26.7 160 16.5 17.1 17.4 18.1 18.8 19.3 20.1 21.0 21.9 22.6 23.5 24.3 25.1 180 16.9 17.0 17.1 17.4 18.0 18.4 18.9 19.4 20.1 20.7 21.2 32.3 23.0 200 17.7 17.5 17.7 17.7 17.6 18.1 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 18.3 18.6 18.7 18.9 19.3 19.5 20.0 20.4 240 18.6 18.8 IS. 9 18.9 18.9 19.0 19.2 19.1 19.2 19.5 19.6 19.7 19.9 260 18.2 18.5 18.7 18.8 19.0 19.3 19.5 19.6 19.9 19.9 20.0 20.1 20.2 280 16.8 17.4 17.9 18.3 18.7 19.1 19.3 19.8 20.0 20.3 20.4 20.6 20.8 300 15.5 15.8 16.2 16.6 17.6 18.1 18.5 19.2 19.4 19.9 20.6 30.8 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 12.6 12.9 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 14.3 149 15.7 16.5 17.3 380 13.1 12.5 11.9 11.6 11.5 11.4 11.6 11.7 13.3 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 lO.S 10.5 10.6 10.7 440 18.6 17.9 17.1 16.1 15.6 144 13.5 12.8 11.9 11.1 10.6 10.3 10.3 460 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 21.3 21.1 20.8 20.3 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 17.8 17.0 15.7 14.4 520 20.6 21.2 21.7 21.7 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 21.1 21.8 22.0 31.8 21.7 21.5 21.2 20.9 30.3 19.6 560 17.2 18.4 19.0 20.0 20.8 21.1 33.7 21.9 22.2 22.1 21.9 31.7 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 23.3 22.1 600 12.8 13.9 15.1 15.9 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 22.0 640 8.5 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 15.8 17.1 18.1 19.0 680 5.1 5.7 6.4 7.1 7.9 8.7 9.7 11.0 12.1 13.1 14.1 15.7 16.8 700 4.2 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 3.8 3.9 4.4 4.9 5.7 6.4 7.4 8.9 760 5.9 5.1 4.4 4.0 3.6 3.4 3.4 3.5 3.9 4.3 4.7 5.2 5.9 780 8.1 7.1 6.1 5.3 4.6 41 3.7 3.3 3.3 3.1 3.4 3.6 4.1 800 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 16.4 15.1 13.7 12.9 11.7 10.6 9.5 8.6 7.5 6.6 5.7 49 44 800 IS. 4 17.5 16.6 15.4 14.3 13.1 12.1 11.1 10.0 9.1 7.9 7.0 6.3 880 20.4 19.6 18.7 17.5 16.6 15.6 14.5 13.6 12.5 11.5 10.4 9.5 8.6 900 22.0 21.1 20.2 19.4 18.7 17.7 16.5 15.7 14.7 13.8 12.5 11.9 10 9 920 236 22.7 21.7 21.1 20.1 19.4 18.4 17.5 16.7 15 6 148 13.9 13.1 940 25.5 24.1 23.4 22.4 21.4 20.6 19.9 19.0 18.2 173 16.6 15.7 148 960 26.2 25.H 24.7 24.1 23.8 32.3 21.3 20.6, 19,0 18.9 17.9 17.1 16.3 980 27.1 26.7 26.3 25.5 24.9 23.8 23.4 22.2 21.0 20.4 19.4 18.6 17.7 1000 27.6 360 27.7 27.3 380 26.7 26.3 25.5 24.7 23.8 23.1 33.3 21.3 460 20.2 470 19.3 370 390 400 i 410 i 420 430 1 440 450' 480 28 TABLE XXX. Perturbations prodnced by Venus. Arguments II and III. m. ! II- 480 490 500 510 I 520 530 1 640 550 560 |670 mo 590 600 10 s 1 1 19.3 18.3 17.4 16.6 15.7 15.0 ' 14.2 13.6 13.1 12.3 11.7 11.3 so 20.9 20 2 19.1 18.2 17.1 162 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 15.1 14.4 13.7 13.0 12.3 60 1 VS.O 24.0 23 2 22.0 20.7 19.9 18.9 17.7 16.8 15.8 14.9 14.0 ; 13.3 80 , ;?7.2 26 4 25.6 24.1 23.2 22.1 20.8 20.0 18.7 17.9 16.6 15.6 14.8 iOO 27.4 27.2 26.8 26.3 25.4 24.5 23.5 22.2 20.9 20.0 18.6 17.6 16.6 120 27.5 27.5 27.6 27.1 20.8 26.3 ' 25.4 24.6 23.7 22.4 21.0 20.1 18 8 j'tv; 26.T 27.0 27.2 27.4 27.3 27.4 26.9 26.2 25.4 24.6 23.9 22.6 ',211 u;o 25.1 25.6 26.1 I2B.7 26.9 27.3 27.1 27.0 26.9 26.4 25.5 24.7 23.9 ISO 23.0 23.8 24.5 ! 25.0 25.7 26.3 26.7 26.8 27.0 26.8 26.6 26.2 25.6 200 21.5 22.2 22.8 [23.5 24.1 24.7 1 25.5 25.8 26.3 26.6 26.6 26.6 26.4 220 20.4 21.0 21.5 22.0 22.6 23.2 ' 23.8 24.5 25.0 25.4 25.8 26.0 26.2 240 19.9 20.4 20.8 21.2 21.6 21.8,22.2 22.6 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 22 3 22.7 23.1 23,3 23.6 280 20.8 20.8 21.0 21.1 21.3 21.4 121.5 21.8 22.0 22.2 22.7 23,0 23.3 300 20.9 21.0 21.5 21.7 21.7 22.0 22.0 22.1 22 1 22.2 22 4 22 6 22.8 320 20.8 21.2 21.5 31.6 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 18.4 19.5 20.0 20.6 21.5 22.2 22.7 23.0 23.7 23.7 24,0 24.2 380 15.0 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 1.5.4 16.2 17.3 18.1 19.2 20.3 21.4 22 4 23.0 23.7 420 10.7 11.2 12.0 12.5 13.5 14.5 15.6 16.7 17.7 18.7 20 1 21.0 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 10.1 10.7 11.3 12.2 13.0 140 1.5.1 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 13.4 500 14.4 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 11.9 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 11.3 10.5 10.1 9.5 9.3 500 21.1 20.4 19.8 19.0 18.2 17.2 16.0 14.8 13.7 12.7 11.7 10.9 10.2 580 22.1 21.8 21.5 20.9 20.3 19.3 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 17.7 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 20.7 21.7 22.0 22.3 22.6 22.5 22 6 1 22.4 22.0 21.6 21.1 20 3 19.6 660 19.0 20,0 20.8 21.3 22.1 22.3 22 22.8 22.7 22.6 22.2 21.8 21.3 680 16.8 18.0 19.0 19.9 20.8 21.5 22 1 22.6 22.7 23.0 23.0 22.8 22 4 700 14.1 15.2 16.8 17.9 18.8 20 22 1 21.5 22.2 22.6 23.9 23 23.S 720 11.5 12.7 13.9 1.5.0 16.4 17.9 18.6 19.7 20.8 21.6 22 3 22 7 23.0 740 8.9 9.8 10.9 12.2 13.6 14 8 16.2 17.5 18.7 19.5 20.6 21.6 22.3 760 5.9 6.8 8.0 9.3 10.3 11.8 13 2 14.5 15.9 17.4 18.2 19.5 20.5 780 4.1 4.9 5.6 6.4 7.5 8.6 9.9 U.l 12.6 14.0 15.6 16.8 18.1 800 3.1 3.3 4.4 4.8 5.5 6.1 6.9 7.9 9.4 10.7 12.1 13.4 14.9 820 3.1 3.1 3.2 3.1 3.6 3.9 4.8 5.7 65 75 8.7 10.0 11.5 840 4.4 3.7 3.5 3.2 3.2 3.1 3.4 3.7 4.1 .5,0 6.2 7.0 8.3' 860 6.3 5.5 4.6 4.1 3.6 3.4 3.3 3.2 3.4 3.4 4.0 4.5 5.6 880 8.6 7.6 6.7 5.9 5.2 4.5 4.1 3.8 3.5 3 4 3.4 3.6 3.9 900 1 10.9 10.0 9.1 8.3 7.2 6.5 5.8 5.1 4.4 42 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 14.8 14.1 13.1 12.4 11.5 10.8 9.8 9.1 8.3 7.6 6.8 6.5 5.9 960 16.3 15.4 14.6 14.0 13.2 12.6 11.7 11.0 10.1 9.6 8.8 8.1 7.5 980 17.7 16.8 16.2 1,5.2 14.5 13.9 13.1 12.5 1 11.8 11.3 10.5 9.7 9.3 1000 19.3 480 lrf.3 17.4 16.6 15.7 15.0 14.2 13.6 13.1 12.3 11.7 11.3 .0.8 1 490 500 510 1 520 1 530 1 540 1 550 560 570 580 590 000 TABLE XXX. 29 Periurhations produced by Venus. Arguments II. and III. UL n. 600 1 610 I 620 1 630 640 650 660 7,4 670 600 690 700 710 720 5,4 10.8 10.2 9.5 9,1 8.4 7.9 " 1 " 7.0 6.6 6.3 5.9 5.5 20 11.5 11.3 10.7 10,4 9.8 9.4 8,9 8.5 7.9 7.7 7.3 6.7 6.6 40 12.3 12.0 11.5 11,0 10.7 10.3 10,0 9.6 9.3 8.9 8.5 8.1 7.8 60 13.3 12.7 12.1 11,6 11.2 10.9 10.5 10.2 10.0 9.8 9.5 9.2 8.9 80 14.8 13.6 12.9 12,4 11.8 11.3 10.9 10.7 10.3 9.9 9.8 9.8 9.6 100 16.6 15.4 14.4 13,4 12.6 12.1 11.5 11.0 10.6 10.2 10.0 9.9 9.6 120 18.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 21.1 20.1 18.9 17,7 16.5 15,2 14.2 13.0 12.3 11.6 11.1 10.3 S.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 25.6 24.8 23.6 22,9 21.6 20,6 19.1 18.0 16.7 15.5 14.3 12.9 12.0 200 26.4 26.0 25.6 24,9 24.0 22,9 21.7 20.8 19.3 18.1 16.9 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 17.1 240 24.6 25.1 25.1 25,3 25.2 25,1 24.7 24.3 24.0 23.0 21.9 21.3 20.2 260 23.6 23.9 24.2 24,5 24.7 24,8 24.9 24.6 24.3 23.8 23.4 22.9 21.6 280 23.3 23.6 23.9 24,2 24.7 24,8 25.0 24.9 24.9 24.8 24.4 24.0 23.5 300 22.8 23.0 23.3 23.4 23.8 24.0 24.1 24.5 24.5 24.6 24.5 24.4 24.0 320 22.9 23.0 23.1 23.2 23.4 23,3 23.6 23.8 24.0 23.9 24.2 24.2 24.2 340 23.5 23.5 23.5 23.4 23.5 23,6 23 6 23.5 23.5 23.6 23.9 23.8 23.8 360 24.2 24.2 24.3 24.2 24.2 24.0 23.7 23.9 24.0 23.7 23.7 23.6 23.6 380 24.5 24.6 24.8 25.1 24.8 24.9 25.0 24.9 24.6 24.5 24.6 24.3 24.0 400 23.7 24.3 24.7 25.0 25.4 25.7 25.7 25.5 25.5 25.4 25.2 24.8 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 20.8 21.7 22.7 23.7 24.6 25.4 26.0 26.5 26.7 26.9 27.0 26.9 460 16,5 17.8 19,0 20.1 21.4 22.3 23.5 24.8 25.4 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 11.1 12.0 13.0 13.8 14.9 16.3 17.9 19.1 20.5 21.6 22.9 24.2 25.1 520 9.6 9.8 10.5 11.5 12.4 13.4 14.4 15.5 17.1 18.4 19.9 21.2 22.3 l>iO 9.3 9.0 9.2 9.6 10.3 11.0 11.9 12.8 13.9 15.1 16.5 17.9 19.4 560 10.2 9.7 9.3 9.1 9.1 9.4 10.0 10.6 11.5 12.4 13.3 14.5 16.0 580 12.2 11.3 10.4 9.9 9.4 9.0 9,2 9.3 9.7 10.4 11.0 12.0 12.7 600 14.4 13.3 12.5 11.6 10.8 10.1 9,6 9.4 9.1 9.3 9.9 10.0 10.8 620 16.9 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 16.3 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 10.2 680 22 4 22.0 21.5 20.8 20.2 19.0 18.1 17,0 15.8 14.7 13.7 12.8 12.0 roo 23.2 23.2 22.6 22.2 21,7 21.0 20.5 19.3 18.3 17.3 16.0 15.0 14.1 720 23.0 23 3 23.2 23.4 23.1 22.4 21.9 21.3 20.8 19.5 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 22.5 22.8 23.3 23,7 236 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 14.9 16.4 17.7 19.1 20.1 21,2 21.1 22,9 23.4 23.8 1 24.1 24.2 23.9 820 11.5 12.9 14.3 15.8 17.8 18,7 20.0 20,9 22.0 22.7 23.5 23.9 24.0 840 8.2 9.5 10.8 12.2 13.8 15,2 16,6 18,1 19.5 20.0 21.7 22.6 23.3 8R0 5.0 6.8 7.7 8.8 10.2 11,5 13.2 14,7 16.0 17.4 19.0 20.2 21.3 880 3.9 4.4 5.2 6.1 7.2 8,2 9.7 10,9 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 3.8 3,9 39 4.0 4,3 4.7 5,4 6.4 7,3 8.6 9,8 11.2 940 5.9 5.1 4,0 4,4 4.2 4,3 4.3 4,3 4.9 5.3 6.3 7,0 8.0 960 7.5 6.9 6,3 58 5.3 4,7 47 4,6 4.6' 4.6 4.9 5,4 6.0 980 9.3 8.7 7,9 7.4 6.8 6,4 6.0 5,6 5.2 5.0 4.9 !5,1 5.1 1000 10.8 10.2 610 9,5 9.1 8.4 7,9 7.4 660 7.0 6.6 1 6.3 5.9 5,5 710 5.4 720 600 620 630 640 650 670 680 690 700 80 TABLE XXX. Perturbations produced by Vbtius. Arguments II. and III. m. IL 720 730 740 750 6.0 760 6.3 770 6.8 780 790 1 800 810 820 1 830 840 5.4 5.5 5.8 7.6 8.4 9.3 10.4 11.7 12.9 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 11.21 40 7.8 7.4 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 8.1 7.8 7.6 7.4 7.4 7.3 7.4 7.4 7.7 8.3 80 9.6 9.5 9.1 9.1 9.0 88 8.4 8.2 8.1 8.1 8.0 8.1 8.2 100 9.6 9.5 9.6 9.5 9.5 9.3 9.3 9.2 9.2 9.0 8.7 8.7 8.7 120 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 99 9.5 9.6 9.4 9.3; 9.3 9.0 9.3 9.5 9.8 9.7 9.8 10.0 l60 10.5 9.9 9.5 9.1 8.9 9.0 8.9 9.0 9.0 9.0 9.5 9.6 9.9 ISO 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 1 17.1 15.7 14.6 13.2 12.0 10.9 10.2 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 19.8 18.8 17.3 16.1 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 2i.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 20.9 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 23.7 23 5 23.3 23.1 23.1 22.7 22.4 22.2 21.6 20.8 20.0 400 ^ 24.6 24.4 24.4 24.0 23.8 23.4 23.2 23.0 22.8 22.4 22.1 21.6 21.3 420 '25.9 25.6 25.2 24.8 24.7 24.3 23.9 23.6 23.3 22.9 22.7 22.3 21.7 440 i 26.9 26.6 26.4 2i3.2 25.9 25.5 25.2 24.9 24.5 23 8 23.4 23.0 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 24.2 23.7 480 26.8 27.4 27 6 28.0 28.1 28 2 27.7 27.4 27.3 26.6 26.2 25.7 25.1 500 25.1 26.1 26.8 27.5 28.1 28 2 28.6 28.5 28.4 28.3 27.6 27.2 26.7 520 22.3 23.9 24.8 25.9 26.8 27.5 28.1 28.5 28.7 29.0 28.8 28.6 28.4 540 19.4 20.7 22'. 1 23.4 24.6 25.6 26.5 27.4 28.0 28.7 28.9 29.1 29.2 560 16.0 17.3 18.6 19.9 21.4 22.9 24.1 25.5 26.4 27.3 28.2 28.6 29.2 580 12.7 14.1 15.5 16.8 18.0 19.3 20 9 22.2 23 5 24.9 26.1 27.0 27.8 600 10.8 11.6 12 7 13.6 14.9 16.2 17.5 18.7 20.2 21.8 23.0 24.4 25.5 620 9.7 10.0 10.5 10.7 12.2 13.2 14.4 15.6 17.0 18.3 19.6 21.2 22.6 640 9.5 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 11.2 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 10.1 9.7 9.7 9.9 9.9 10.4 10.9 11.5 720 16.4 15.3 14.4 13.3 12.2 11.6 10.9 10.2 10.1 9.9 10.0 10.1 10.4 740 18.8 17.7 16.7 15.6 14.4 13.5 12.4 11.5 11.1 10.7 10.1 10.0 10.3 760 21.3 20.1 ! 19.2 18.1 16.6 15.6 14.7 13.fi 12.8 11.9 11.3 10.7 10.3 780 23.0 22.3 '21.5 20.5 19.4 18.4 172 15.8 14.9 14.0 130 12.2 11.3 800 23.9 23.9 23.4 22.6 21.9 20.7 19.8 18.3 17.5 16.2 15.1 14.2 13.4 820 24.0 24.5 ' 24.2 23.9 23.3 22.6 22.3 21.3 20.3 19.4 18.3 17.3 16.2 840 23.3 24.0 24 ■! 24.5 24.4 24.3 23.8 23.4 22.7 21.7 20.8 19.6 18.3 860 21.3 22.3 23.3 23.9 24.2 24.7 24,5 2i.ri 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 24.0 23.5 900 14 8 16.1 17.6 19.0 20.6 21.5 22.5 23.2 24.1 24.5 24.2 24.8 24.5 920 11.2 12.6 14.0 15.5 17.0 18.4 19.9 21.0 22.0 22.9 23.5 24.5 24.5 940 8.0 9.3 lo.r 12.0 13.3 14 8 16.4 17.6 19.1 20.4 21.4 22.4 23.2 960 6.0 6.9 7.8 8.6 10.2' 11.5 12.7 14.1 15.6 16.9 18.5 19.5 20.7 980 5.1 5.5 6.0 6.7 7.7 8.5 S.7 10.9 12.2 13.6 14.8 16.1 17.6 1000 5.4 720 5.5 5.8 5.8 750 6.3 6.8 7.6 8.4 790 9.3 800 10.5 810 11.7 12.9 14.3 — — 1 730 740 760 770 780 820 830 840 ' TABLE XXX. 31 Perturbations produced by Venus. Arguments II. and III. ra. H. 840 S'li.i 860 1 870 880 ; 890 j 900 910 j 920 930 940 950 960 1 / ,/ „ , - \ .. // ,/ „ f* ,, „ „ 14.3 15.5 16.9 18.2 49.2 20.2 21.4 22.5 23.0 23.5 24.0 24.2 24.2 20 11.2 12.4 13.6 14.9 16.2 17.3 18.6 19.6 20.5 21.5 22.4 23.1 23.6 40 9.3 10.2 10.9 11.8 13.3 14.2 15.5 16.6 17.8 18.8 19,7 20.7 21.6 60 8.3 8.7 9.5 10.1 10.8 11.6 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 12,4 129 14.0 120 9.6 9.5 9.3 9.6 9.6 9.7 9.9 9.8 10.4 1.0.9 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 1O.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 10.0 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 0.7 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 13.8 128 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 19.0 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 17.3 16,7 15.7 180 25.1 24.4 23.9 23.3 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 23.6 23.0 22,3 21.4 20.7 20.3 19,1 18.1 520 28.4 27.8 273 26.8 26.3 25,6 24.7 23,9 23.3 22,6 21.8 20,8 20.1 540 29.2 i 29.2 28.9 28.5 27.8 27.4 26.8 26,1 25.3 24,4 23.7 23.0 22.0 860 29.2 1 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 29.8 293 28.0 28,7 27.9 27,3 26.6 600 ■1- ' 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 26.2 27.1 27.9 28.8 29.3 29.6 29,8 30.1 29.8 29.6 640 19.2 20.6 21.6 23.3 24.6 25.2 26.6 27.8 28.3 28.9 29.4 29.7 29.9 660 16.21 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 19.5 20.7 22,0 23,1 2-1,2 25.1 720 10.4 11.0 11.4 12.3 13.3 14.3 15.6 16.4 17.7 19,3 19,9 21,8 22.6 7-;o 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 10,7 11.1 11.5 12,3 13,2 14.0 15.0 800 13.4 12.5 11.7 11.0 10.6 10.3 10,3 10.4 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 11,9 11.4 11.0 10.9 880 23.5 22.4 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 ,10.9 19,7 18.6 17.2 16,4 15.3 14.1 13.3 920 24.5 24.8 24.7 24 3 24.1 '23.2 22.3 21.3 20.0 19.3 18.0 16.7 15.7 940 23.2 24.0 24.5 24.6 24.5 24.5 24,2 23.5 22.7 21.8 20.6 19.5 18.4 960 20.7 21.9 22.8 23.6 24.0 24.5 24,5 24.2 24.3 23.7 22.9 22.1 21.0 980 17.6 18.7 20.1 21.2 22.2 23.1 23.6 24.0 24.3 24.3 24.3 23.7 23.0 1000 14.3 15.5 16.9 18.2 19.2 20.2 21,4 22.5 23.0 23.5 24.0 24.2 050 24.2 I 840 850 860 870 880 1 890 900 910 920 930 940 960 52 TABLE XXX. XXXI. Perturbations by Venus. Arguments II and III. III. Perturbations by Mars. Arguments II and IV. IV, II. 960 1 970 1 980 990 , 32.5 lOOOi ] If 9.5 10 20 j 30, 40 1 50 60 70l 24.2 23.7 33.1 21.6 10.2 10.8 11.2 .11.5 11.7 11.8 " ! 11. .5 20 23.6 23.7 34.0 23.4 33.1 8.3 9.1 9.8 10.5 10.9 11.2 11.5 11.6 40 21.6 22.4 33.9 33.5 23.5 7.1 7.9 8.8 9.4 10.0 10.6 10.8 11.2 60 19.1 20.1 20.7 31.5 23.3 5.8 6.7 7.6 8.4 9.1 9.8 103 10.5 80 16.2 17.3 18.4 19.7 20.0 4.3 5.3 6.4 7.3 8.0 8.9 9.3 9.9 100 14.0 14.8 15.6 16.5 17.6 3.3 4.2 5.0 5.9 6.S 7.6 8.4 9.1 130 12.3 12.9 13.7 14.3 15.3 2.4 3.1 3.9 4.8 5.6 6.4 7.3 8.0 140 11.5 12.0 13.6 13.8 13.6 2.1 2.4 2.9 3.8 4.6 5.5 6.3 7.0 160 11.6 11.8 12.1 12.3 12.7 3.0 2.2 2.4 3.7 3.5 4.4 5.1 5.9 180 12.2 12.2 13.3 12.5 13.7 1.9 2.0 2.3 3.6 2.9 3.4 3.9 4.9 200 12.4 12.7 12.8 13.1 13.2 3.3 2.2 2.2 3.4 2.7 3.0 3.4 3.8 220 12.3 12.7 13.0 13.3 13.5 3.0 2.6 2.5 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 11.5 12,2 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 10.1 6.2 5.8 5.6 5.0 4.8 4.3 3.9 3.8 320 6.7 6.8 7.3 7.8 8.3 6.9 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 6.3 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 3S0 8.9 8.2 7.5 6.9 6.8 7.5 76 7.3 7.3 7.3 7.1 6.7 6.5 400 10.9 10.0 9.0 3,3 7.5 7.3 7.3 7.5 7.4 7.4 7.4 7.1 7.0 420 12.6 11.6 10.7 9,9 9.1 6.9 7.0 7.3 7.4 7.4 7.4 7.3 7.5 410 14.3 133 12 5 11.6 10.6 6.5 6.8 6.8 7.1 7.3 7.3 7.3 7.4 460 15.7 14.8 13 9 13.0 12.1 6.3 0.2 6.5 6.7 6.8 7.1 7.1 7.3 430 16.9 16.3 15.5 14.5 13.6 5.8 5.9 6.0 6.2 6.4 6.5 7.0 6.9 500 18.1 17.6 16.6 15.8 15.1 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 16.5 5.1 5.1 5.1 53 5.4 5.6 5.S 6.0 540 32 21.0 20.2 19.2 18.1 4.7 4.8 4.8 4.8 5.0 5.1 5.4 5.5 560 24.3 23.5 23.6 21.5 30.6 4.4 4.5 4.6 4.6 4.7 4.8 4.8 5.0 5S0 26.6 35.7 34.9 23.8 23.0 4.3 4.3 4.4 4.3 4.5 4.4 4.4 4.5 600 28.5 27.8 27.0 26.3 35.4 4.0 4.2 4.3 4.2 4.2 4.3 1.2 4.3 620 39,6 29.2 28.8 38.3 27.4 4.2 4.0 4.1 4.0 4.0 4.0 40 3.9 640 29.9 30.0 29.9 29.5 29.5 4.3 4.2 4.1 4.0 4.1 4.0 3.9 3.9 660 29.2 33.5 29.7 39.8 39.9 4.6 4.4 4.3 4.1 4.1 41 4,0 3,8 680 37.5 28.6 38.9 39.3 29.7 4.8 4.6 4.5 4.3 4.2 4.1 4.0 3,9 700 25.1 26.4 37.3 27.8 28.7 5.3 5.0 4.8 4.5 4.6 4.0 4,1 4.1 720 22.6 23.9 25.0 26.1 26.8 5.8 5.5 5.1 5.0 4.7 4.5 4.1 4.1 740 19.5 21.3 22.5 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 22.1 7.4 6.7 6.4 6.0 5.6 5.3 5.1 5.0 780 15.0 15.8 17.1 18.5 19.3 8.2 7.6 6.9 6.5 6.4 5.8 5.6 53 800 12.2 14.1 14.8 15.9 17.0 9.2 8.5 8.0, 7.3 6.8 6.5 6.1 5.8 820 11.4 13.0 12.5 13.4 15.4 10.1 9.6 8.8 8.2 7.6 7.1 6.7 6.5 840 11,2 11.3 11.7 12.2 13.2 10.9 10.4 9.8 9.1 8.4 7.9 7.5 6.9 860 10.9 10,8 10.9 11.3 11.5 11.7 11.0 10.4 10.0 -9.4 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 9.7 8.9 8.4 900 13.3 12.3 12.9 11.3 11.2 12.4 12.2 11.8 11.6 10.8 10.3 9.7 9.3 920 15.7 14.6 13.7 12.8 12.1 1 12.3 12.3 12.2 11.9' 11.6 11.0 10.5 9.9 940 18.4 17.3 1 16 3 14.5 14.0 12.1 12.1 12.2 12.2 11.8 11.4 11.0 10.6 960 21.0 20.0 18 9 17.9 16.7 i 11.4 11.9 11.9 12.0 12.0 11.7 11.4 11.0 980 23.0 22.4 21.4 20.3 19.5 10.6 11.1 11.6 11.8 11.9 11.9 11.7 11 4 iOOO 24.2 23,7,23.1 22.5 31.6 9.5 10.2 10.8 11.2 11.6 11.7 11.8 11.5 960 970 1 980 990 1000 10 20 1 30 1 40 50 60 70 TABLE XXXI. 33 Perturbations produced hy Mars Arguments 11 and IV. TV. TI, 70 11.2 90 11.0 100 110 120 130 9.5 140 9.0 150 160 170 180 190 7.6 300 -, 4 11.5 10.6 10.1 9.9 8.6 8.2 8.1 7.8 20 11.6 11.4 11.0 10.9 10.6 10.2 9.7 9.1 9.1 8.8 8.4 8.1 7.9 7 S 4U 11.2 11.3 11.2 11.0 10.8 10.5 10.3 9.8 9.4 9.3 9.1 8.7 8.4 8.2 60 10.5 10.9 11.1 10.9 11.0 10.9 10.4 10.0 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 10.6 105 10.4 10.3 10.1 9.9 9.6 9.3 9.0 8.8 120 8.0 8.S 9.3 9.5 9.9 10.2 10.2 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 99 9.9 9.7 9.7 9.4 9.3 8.9 160 6.9 6.5 7.2 80 8.5 8.9 9.2 9.6 9.5 9.6 9.5 9 5 9.3 9.1 180 4.9 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 3.8 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 35 39 4.4 5.1 5.8 6.4 7.1 7.6 7.9 8.4 8.6 88 8.8 8.7 240 3.2 3.6 4.0 4.4 5.0 5.5 6.2 6.8 7.4 7.6 8.1 8.4 8.4 8.5 260 3.1 32 3.8 4.1 4.5 49 5.4 5.9 6.6 7.1 7.5 7.7 8.0 8.2 280 34 3 4 3.5 3.8 4.2 4.5 4.9 5.5 5.6 62 C.8 7.1 7.5 7.8 300 38 37 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 4.3 4.2 4.1 4.0 4.1 4.2 4.4 4.7 5.0 5.1 5.8 6.0 6.4 6.6 340 5.1 4.9 4.6 4.4 4.4 4.3 4.5 4.5 5.0 5.2 5.5 5.8 60 63 36(1 5.8 5.6 5.3 5.0 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 5.9 5.7 5.5 5.4 5.1 5.1 51 5.1 5.4 5.5 5.7 .5.8 400 7.0 6.7 6.7 6.3 6.1 5.9 57 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 6.3 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 6.3 460 7.3 7.4 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 69 7.1 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 6.0 6.8 6.9 7.2 73 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 7.1 540 5.5 5.7 6.0 6.3 6.6 69 7.1 7 3 7.4 7.7 7.9 8.0 8 2 8.3 560 5.0 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 5.7 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 6.4 6.1 6.6 6.9 7.4 640 3.9 3.8 3.8 3.8 3.9 3.9 4.1 4.3 4.5 5 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 6.0 53 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 33 3.2 3.3 3.2 3.5 3,6 3,8 4.2 720 4.1 4.1 4.0 3.8 3.6 3.5 33 3.2 3.3 3.2 3.0 3,2 .3,4 3.6 740 4.3 4.3 4.2 4.0 3.8 37 3.5 3.2 3.0 3 2.9 2,8 2.9 3.1 700 5.0 4.7 4.4 4.3 4.1 3.8 3.7 34 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 29 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 2,7 820 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 0.6 6.2 6.2 5.5 5.4 5.2 6.0 4.8 4.4 4.1 36 880 84 7.9 7.6 7.1 6.9 6.4 6.4 5.8 5.7 5.4 5.2 5 4.6 43 900 9.J 8.7 8.3 7.7 7.4 7.1 0,7 6.6 6.1 6.0 5.0 5.4 5.2 4.9 9.0 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 54 940 10.6 10.1 9.5 8.9 8.7 8.2 7.8 7.6 7.2 7.1 6.5 6.0 6.3 59 960 ! 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 66 980 11.4 11.0 106 10.2 9.8 9.2 8.9 8.4 S.l 8.0 7.6 7.3 7.2 6.!* 1000 11.5 112 11.0 10.6 10.0 110 9.9 9.5 9.0 8.6 8.2 160 8.1 170 7.4 7.6 190 74 70 80 90 100 120 1 130 140 150 180 1 20U 84 TABLE XXXI. Perturbations produced by Mars. Arguments II. and IV. IV. II. 200 210 820 230 240 250 260 270 280 290 300 310 330 74 7.2 7.0 6.6 64 6.2 .5.7 5.3 4.9 4.7 4.1 3.8 3.4 20 78 7.2 7.3 7.2 7.0 6.6 6.3 6.0 .5.7 5.3 5.0 4.4 39 40 8.2 8.1 7.6 7.5 7.3 7.2 0.8 6.6 6.2 5.9 5.6 .5.2 4.7 60 8.4 8.0 7.9 7.8 7.6 7.5 7.3 7.1 6.8 6.4 6.1 5.8 5.4 80 8.6 8.5 8.2 8,0 7.6 7.7 7.6 7.4 7,1 7.0 6.7 6 3 6.0 100 8.8 8.5 8.6 8.4 8.2 7.6 7,7 7.8 7.6 7.3 7.2 6.9 66 1 120 8.9 8.7 8.4 8.4 8.3 8.3 8.0 7.9 7.7 7.6 7,5 7.3 7.0 140 8.9 8.7 8.4 8.3 82 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 83 8.3 82 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 80 7.8 200 8.9 8.8 8.0 8.4 8.4 8.3 8.1 8.0 7.9 7.8 7.8 7.9 7.9 220 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 78 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 73 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 68 360 6.1 6.2 6.4 6.5 69 6.9 7.0 7.0 0.9 6.8 6.7 6.6 6.5 380 5.8 6.1 6.3 6.4 6.6 6.7 6.0 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 6.5 6.5 6.4 440 6.3 6.4 64 6.6 6.5 6.6 6.5 6.5 6.5 6.5 6.3 63 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 69 69 69 7.0 7.0 6.8 6.7 6.6 6.5 6.3 500 7.4 7.5 7.4 7.4 7.3 7.2 7.3 7.2 7 1 6.9 6.8 6,8 6.6 520 7.9 7.8 7.8 7.8 7.8 7.6 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 86 8,4 86 8.7 8.5 8.5 84 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 89 9.1 9.1 92 9.2 9.1 9.0 8.8 8.7 620 7.4 7.6 8.0 8.5 8.7 9.0 92 9.5 9.5 9.5 9.4 9.3 9.2 640 6.7 7.2 7.5 7.9 8 3 87 9.0 9.3 9.5 98 9.8 9.7 9.7 660 60 6.3 7.0 7.3 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 8.1 8.5 9.0 9.3 9.8 10.0 10.2 700 4.2 4.5 5.2 58 6.4 6.8 7.4 8.0 8.5 8.9 9.2 9,8 10.1 720 3.6 39 4.3 4.7 .5.3 5,9 6.6 7.0 7.8 8.3 8,8 9 1 9.7 740 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 2.8 3.0 33 3.6 4,0 4,4 5,1 5.8 6.5 7.2 "8 8.4 780 2.5 2.6 2.5 2.7 3.1 3.3 3,7 4.1 4.8 5.4 6 1 6.J 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 66 820 29 2.6 2.4 2.3 2.2 2.3 2.6 2,8 3.1 3.4 4.1 4.7 5.4 840 3.1 2.8 2,6 2.4 2.3 2.2 2.3 2.4 2.6 2.8 3.2 3.8 4.3 860 3.6 3.3 30 2.7 2.4 2.3 2.1 2.2 2.3 2.5 2.7 3.0 34 880 4.3 38 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 2.4 920 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 47 4.3 3.8 3.4 3.0 2.7 2.4 2.1 2.0 960 6.5 6.2 59 5.5 5.1 4.9 4.5 4.0 34 3.1 2.8 2.4 23 980 6.9 6,8 6.4 6.1 5.8 5.4 5.1 4.8 4.3 3.9 3.5 3.0 27 1000 7.4 7.3 210 7.0 6.6 6.4 6,2 5.7 5.3 49 4.7 4.1 3.8 34 1 200 220 230 240 250 260 270 280 290 300 310 320 TABLE XXXI. 85 Perturbations produced hy Mars, Arguments II. and IV. IV. (I. 320 330 340 350 360 370 380 390 400 410 420 430 41« 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 23 2.5 2.7 3.0 3.3 3.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 60 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 80 6.0 5.7 5.4 4.8 4.4 4.0 3.6 3.4 3.1 29 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 3.4 140 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 50 200 7.9 7.8 7.7 7.6 7.5 7.3 7.1 6.9 6.6 6.4 6.1 5.6 5.5 220 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 7.0 7.6 7.6 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 7.1 7.1 7.1 7.0 7.2 7.3 7.3 7.3 7.2 7.2 7.3 7.2 7.1 320 6.8 6.8 6.9 6.9 6.8 7.0 7.1 7.1 7.1 7.1 7.1 7.0 7.2 340 6.8 6.7 6.6 6.6 6.6 6.8 6.9 6.9 7.0 7.0 6.9 6.9 6.9 360 6.5 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 6.7 400 6.4 6.2 6.2 6.0 6.1 6.0 6.0 6.0 6.0 6.1 6.2 6.3 6.4 420 6.4 6.2 6.1 6.0 5.9 5.8 5.9 5.9 5.9 5.9 5.9 6.0 6.0 440 6.2 6.1 6.0 5.8 5.8 5.7 5.6 5.6 5.6 5.7 5.7 5.8 5.9 460 6.2 6.0 5.9 5.8 5.7 5.5 5.5 5.4 5.5 5.4 5.5 5,3 5.4 4S0 63 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 6.6 6.4 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 7.2 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 8-4 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 6.0 5.6 5.3 5.1 620 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 8.2 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 10.1 10.3 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 10.0 10.3 10.6 10.7 10.8 10.9 10.6 ;o.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 10.8 10.7 800 6.6 7.3 7.9 8.5 9.2 9.8 10.1 10.6 10.8 11.1 11.3 11.1 11.0 820 5.4 6.0 7.0 7.6 8.2 8.9 96 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 10.9 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.r 920 2.2 2.3 2.3 2.8 3.3 3.7 4,3 5.0 5.8 6.5 7.2 8.0 8.7 040 2.0 2.1 2.3 2.3 2.7 2.9 3,4 4.1 4.7 5.5 6.1 7.0 77 960 2.3 2.2 2.2 2.3 2.3 2.5 2,8 3.2 3,9 4.5 5.1 5.7 65 980 2,7 2.4 2.2 2.3 2,3 2.4 2,5 2.8 3.0 3.6 4.1 4.7 5 5 1000 S.4 2.8 2.6 340 2.4 2.2 2.3 2.3 2.5 390 27 400 2.9 410 3.4 420 4.0 45 320 330 3£0 360 370 380 430 410 36 TABLE XXXI. Perturbations produced by Mars. Arguments II and IV. IV. f • — n. 440 450 460 470 480 490 500 510 520 530 10.0 540 550 500 4.5 5.2 5 9 6.6 7.3 8.0 8.5 9.0 9.5 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 40 3.2 3.7 42 4.8 5.4 5.9 6.6 7.3 7.9 8.4 8.9 9.4 9.8 60 3.0 3.2 3.6 4.0 4.5 5.1 5.7 63 6.9 7.5 8.0 8.6 9.1 80 2.9 3.1 33 3.5 39 4.4 4.9 5.4 5.9 6.5 7.1 7.7 8.2 100 3.0 3.1 32 3.5 3.6 3.8 4.2 4.8 5.3 5.9 6.4 6.9 7.4 120 34 3.3 3.3 3.4 3.5 3.6 3.9 4.2 4.7 5.1 5.6 6.0 6.6 140 3.9 3.8 36 36 3.6 3.7 4.0 4.0 4.2 4.6 5.0 54 5.9 160 4.4 4.2 3.9 4,1 3.8 3.7 4.0 4.1 4.2 4.5 4.6 4.9 5.3 180 5.0 4.8 4.4 42 4.2 4.2 4.0 4.1 4.3 4.4 4.4 4.7 5.0 200 5.5 5.2 5.1 4.8 4.6 4.5 4.5 4.4 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-i 5,9 58 5.6 54 5.3 5.2 5.1 5.1 5.1 5.2 5.2 260 67 6.0 6.4 6.1 6.0 5.9 5.8 5 r 5.6 5.5 5.4 5.4 5.4 280 6.9 6.8 67 6.5 6.3 6.2 6.1 6.0 5.9 5.9 5.9 5.8 5.8 300 7.1 7.0 6.8 6.8 6.6 6.5 6.4 6.3 6.2 6.2 6.2 6.2 6.2 320 7.2 7.1 69 6.8 6.8 6.7 6.6 6.5 6.5 6.5 6.5 6.6 6.6 340 6.9 6.9 7.0 6.9 6.9 6.8 6.7 6.8 6.7 6.6 e.'i 6.8 6.9 360 6.9 6.8 68 6.8 6.8 6.7 6.7 6.6 6.6 6.8 6.8 6.8 6.9 380 6.7 6.5 6.5 6.6 6.7 6.6 6.6 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 68 420 6.0 62 6.3 6.3 6.2 6.2 6.3 6.3 6.3 6.3 6.5 6.6 67 440 5.9 5.9 6.0 6.0 6.0 6.0 6.0 6.1 6.0 6.1 6.2 6.2 6.4 460 5.4 5.5 5.7 5.8 5.8 58 ■5.8 5.8 5.8 5.8 5.9 6.0 C.l 480 53 53 5.5 5.5 5.5 5.6 5.5 5.6 5.4 5.6 5.7 5.5 5.8 500 5.0 5.0 5.1 5.2 5.3 5.3 5.3 5.2 5.2 5.2 5.3 5.4 5.4 520 4.9 4.9 49 4.8 5.0 5.1 5.1 5.1 5.1 5.1 5.0 6.0 5.1 540 4.8 48 4.7 4.8 4.8 4.9 4.9 5.0 4.9 4.8 4.8 4.9 4.8 560 4.7 4,6 4.6 4.7 4.7 4.6 4.7 4.7 4.7 4.7 4.6 4.6 4.6 580 4.9 46 4.5 4.5 4.6 4.5 4.4 4.4 4.5 4.5 4.5 4.4 4.4 600 5.1 4.9 4.6 4.5 4.4 4.4 4.4 4.3 43 4.3 4.3 4.3 4.3 620 5.3 5.1 4.9 4.7 4.6 4.4 4.3 4.1 4.2 4.2 4.2 4.2 4.1 640 5.8 5.4 5.2 5.0 4.7 4.6 4.4 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? 4.5 4.3 4.2 4.2 4.1 4.0 680 7.1 6.6 6.2 5.7 5.4 5.1 4.9 4.7 4.5 4.4 4.3 40 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 86 8.0 7.6 7,1 6.6 6.2 5.7 5.5 5.2 4.9 4.6 4.6 4.3 740 9.4 9.0 3.4 8.0 7.4 6.9 63 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 7.2 6.5 6.2 5.8 5.5 5.2 4.9 780 10.7 10.5 9.9 9.6 9.0 8.5 V.8 7.4 7.0 6.4 6.1 5.7 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 10.2 9.9 9.4 8.8 8.2 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 11.2 11.0 10.9 10.7 10.2 10.0 9.4 920 87 9.3 9.9 10.3 10.8 11.0 11.1 11.2 11.2 11.0 10.7 10.4 10.1 940 77 8.2 8.8 9.5 10.1 10.4 10.9 11.0 11.2 11.2 11.0 10 7 10.5 960 «.5 7.3 8.1 86 9.3 9.8 10.2 10.6 10.8 11.1 11.2 10.9 10.8 980 5.5 62 7.0 7.7 8.3 8.9 9.5 10.0 10.4 10.6 10.8 110 10.9 1000 4.5 5.2 5.9 6.6 7.3 8.0 8.5 500 9.0 510 9.5 530 10 10.4 10.7 10.9 440 450 460 470 ! 480 490 530 540 550 560 TABLE XXXI. 37 Perturbations produced by Mars. Arguments II and TV. IV. n. 560 10 9 570 10,8 580 590 600 610 10.0 620 630 640 650 660 670 680 7.7 10.6 10.4 10.3 9.7 9.2 8.9 8.5 8.1 7.9 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 10.4 10.4 i 10.5 10.3 10.2 9.9 9.6 9.4 9.1 8.9 8.5 60 9.1 9.4 98 10.2 10.2 10.3 10.2 10.1 9.9 9.6 93 9,0 8.8 80 8.2 8.7 9.0 9.3 9.6 9.8 10 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 6.6 6.9 7.6 8 1 83 8.6 9.0 9.2 9.4 9.5 0.5 9.4 9.3 140 5.9 6.3 0.8 7.2 7.7 8.0 8.3 8.7 8.9 9.1 9.2 9.3 9.3 160 5.3 5.8 6.0 6.5 6.9 7.4 7.7 8.0 8.4 8.5 8.8 8.9 9.0 180 5,0 5.2 5.6 6.0 6.3 6.7 7.1 7.2 7.7 8.1 8.1 8.4 8.6 200 4.8 5.0 5.3 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 .^.2 5.2 5.3 5.3 5.4 5.5 5.7 5.9 6.1 6.4 6.0 6.8 7.1 260 5.4 5.5 5.5 5.5 5.5 ,5.5 5.5 5.7 5.8 6.0 6.3 6.4 6.5 280 5,8 5.8 5.8 5.9 5.8 5.8 5.8 5.9 5.9 5.9 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 5.9 5.9 6.0 6.1 320 6,6 6.5 6.6 6.6 65 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 0.9 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 7.7 7.7 7.6 7,5 7.4 7.2 400 6.8 7.0 7.1 7.3 7,6 7.9 8.0 8.0 8.1 8.1 8,1 79 7.8 420 6.7 6.9 7,0 72 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 8.4 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 7.9 ■ 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 6.4 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.S 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 3.3 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 4.1 3.8 3,6 3.3 3.1 2,9 2.8 2.6 780 5.5 5.1 4,9 4,5 4.4 4.3 4.1 3,9 3.8 34 32 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 4.6 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 6,0 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 5.7 940 10.5 10.2 9.8 9.4 8.8 8.5 8.0 7.6 7,3 6.9 6.6 6,2 C 1 960 10,8 10.5 10.2 10.0 9.5 9,1 8.6 8.2 7,8 7.5 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 7.4 72 1000 10.9 10.8 10.6 10.4 590 10,3 600 10,0 610 9.7 620 9.2 8,9 8.5 8.1 6fi0 7.9 670 77 560 570 580 630 ' 640 650 680 31 88 TABLE XXXI. Perturbations produced by Mars. Arguments II. and IV'. IV. 1[. 680 1 fiOO 700 710 720 730 1 740 750 760 770 780 790 800 , 7.4 69 6.7 1 6.4 5.5 5.2 4.8 4.4 7.7 6.8 6.1 5.8 ft 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 40 8.5 8.3 7.8 7.5 72 7.1 7.0 6.9 66 6.4 6.1 5.8 5 3 fiO 8.8 8.6 8.3 8.1 7.8 7.6 7.5 7.4 7.1 6.9 6.7 6.3 6.0 80 9.1 8.9 8.7 8.4 8.1 8.0 7.8 7.6 7.4 7.3 7.1 6.9 6.5 100 9.2 8.9 8.8 8.7 8.6 8.3 8.0 7.7 7.6 7.6 7.6 7.3 7.0 130 9.3 9.3 9.0 8.7 8.6 8.4 8.2 8.1 7.9 7.8 7.7 7.6 7.5 140 93 9.2 9.0 9.0 8,7 8.5 8.4 8.3 8.0 7.8 7.7 7.7 7.7 IBO 9.0 9.0 8.9 8.8 8.7 8.6 8.5 8.4 8.3 80 7.9 7.8 78 180 8.6 8.6 8.7 8.7 8.7 8.6 8.5 8.3 8.3 8.0 8.2 7.8 7.9 200 8.0 8.2 8.3 8.3 8.5 8.4 8.4 8.4 8.3 8.1 8.1 8.1 7.9 220 7.5 7.7 7.9 8.1 8.2 8.2 8.1 8.2 8.2 8.0 8.1 8.0 8.0 240 7.1 7.2 7.4 7.5 7.6 7.7 7.8 7.8 7.9 8.0 8.0 7.8 7.8 260 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 67 6.9 7.1 7.2 7.3 7.3 7.3 7.3 7.4 300 6.1 6.0 6.2 6.4 6.4 6.5 6.6 6.7 6.9 6.9 6.9 7.1 7.1 320 6.0 6.0 60 6.0 6.2 6.1 6.2 6.3 6.5 6.5 6.6 6.6 6.8 340 6.3 6.2 CO 6.0 6.0 6.0 0.1 6.1 6.2 6.2 6.3 6.3 6.4 360 67 6.6 6.4 61 6.0 5.9 6.0 5.9 59 .5.9 6.0 6.1 6.2 380 7.2 7.1 6,8 6.6 6.4 6.2 6.1 .5.9 5.8 5.7 5.6 5.8 5.9 400 7.8 7.7 7.4 7.1 6.8 6,6 6.4 6.1 6.0 5.8 5.6 5.5 5.6 420 84 8.2 80 7.8 7.5 7.3 6.8 6.5 6.2 6.0 5.7 5.5 5.4 440 8.9 8.8 8.7 8.4 8.2 7.8 7.5 7.1 6.6 6.2 6.0 5.7 5.6 160 9.2 9.2 9.2 90 8.8 8.5 8.2 7.9 7.5 69 6.5 6.3 6.0 480 9.3 9.5 9.6 9.6 9.4 9.2 9.1 8.6 8.3 7,8 7.2 6.9 6.5 500 9.4 9.6 9.8 10,0 9.9, 9.8 9.6 9.4 9.1 8.7 8.2 7.6 7.2 520 9.0 9.5 9.8 10.1 10.2 10.3 10.3 10.0 9.8 9.5 9.1 8.5 8.0 540 8.5 9.1 9.5 10.0, 10.3 10.5 10.6 10.6 10.4 10.1 9.8 9.5 9.0 560 7.8 8.5 9.0 9.5 99 10.4 10.8 10.8 10.9 10.8 10.6 10.2 9.9 580 6.9 7.6 8.3 9.0 9.7 10.0 10.4 10.7 11.1 11.2 11.0 11.0 10.6 600 6.2 6.8 74 8.0 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 7.1 7.8 8.6 9.4 10.3 10.6 11.0 11.5 11.7 11.7 640 4.5 5.0 .5.5 6.2 68 7.6 8,4 9.3 10.0 10,7 11.1 11.6 11.8 660 3.7 4.1 4.7 5.2 5.9 6.5 7.3 8.3 9.1 9.8 10.5 ,11.2 11.5 680 3.1 3.4 3.8 4.3 4.8 5.5 6.2 7.0 7.8 8.7 9.6 1C.2 11.0 700 2.9 2.8 30 3.4 3.9 4.5 5.2 6.0 6.7 7.5 8.£ 9.4 10.1 720 2.7 2.6 2 5 2.7 1 3.1 3.5 4.0 4.8 5.6 6.4 7.3 82 9.1 740 3.5 2.4 2.4 2.4 1 2 5 2.7 3.1 3.6 4.5 5.3 6.1 6.9 7.8 760 2.5 2.3' 2 2 2.1 2.1 2.3 2.4 3.8 3.2 4.1 4.7 5.7 6.6 780 2.7 2..-): 3.3 2,1 2.0 1.9 2.1 2.2 2.5 2.9 3.6 4.4 S.2 800 2.8 2.7 1 2.4 2.2 2.0 1.8 1.8 1.8 2.0 2.3 2.5 3.3 4.0 h20 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 3,5! 3 2.6 23 2.1 1.9 1.6 1.5 1.5 1.6 1.7 3.2 860 4.4 4.0: 3 5 3.2 1 3.8 2.3 1.9 1.7 1.4 1.3 1.3 1.4 1.6 880 4.7 4.4 4.1 3.7 3.3 3.0 2,5 2.1 1.7 1.4 1.3 1.3 1.2 900 52 5.0 4.6 4.3 4.0 3.6 3.2 2.7 2.2 1.0 1.3 1.2 1.1 920 5.7 5.3 5.1 5.0 4.6 4.2 3.8 3.4 2.9 23 1.9 1.3 1.1 940 ! 6.1 5.9 56 5.4 5.2 4.8 4.5 3.9 3.5 3,1 2.6 2.1 1.5 •360; 6.6 6.4 6,2 5.9 5.6 1 5.4 5.1 4.7 4.3 3.7 3.3 2.8 2.3 ] 980: y.2 6.9 66 6.4 6,2 .5.9 56 5.3 5.0 4.6 4.0 3.5 3.0 lUOO 7.7 680 7.4 1 090 6.9 '700 es 6.7 6.4 730 6.1 5.8 5.5 (760 5.3 4.8 4.4 790 [3.7 1 800 1 110 1 720 740 750 770 780 TABLE XXXI. 39 Pertvrhations produced hy Mars Arguments II. and IV. IV. II. 800 810 820 830 840 1 850 i 860 870 880 890 900 910 920 22 37 3.2 2.6 2,1 1,7 1.3 0.9 0.7 0,7 1.0 1,2 1,6 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 6.0 5.7 5 2 4,7 4.1 3.6 3,1 2,6 2,0 1.5 1.2 0,9 1,0 80 6.5 6.3 6.0 5,5, 5.0 4.6 4.0 3,4 2.7 2.2 1,8 1,5 1.3 100 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 6.8 6.5 6.2 5,7 5.1 4.7 4.1 3,5' 2.9 24 140 7 7 7.7 7,5 7.3 7.0 6.7 6,4 6,0 5.6 5,1 4.5 38 33 160 7.8 7.9 7.7 7.6 7.4 7.2 7,0 6,8 6.3 5,8 5,4 4,8 42 180 79 7.8 7,9 7,9 7.7 7.6 7.5 7,1 7.0 6,6 6,1 5.7 5.8 200 7.9 7.9 7,8 7,9 7.8 7.7 7,6 7,5 7.5 7,1 6,8 6.3 6.1 220 8,0 7.9 7,8 7,8 7.8 7.8 7.8 7,8 7.6 7,5 7.4 7,1 6.7 240 7.8 7,7 7,7 7,7 7.7 7.7 7,8 7,8 7.7 7,6 7.6 7.5 7,2 260 7.8 7.7 7.7 7.6 7 7 7.7 7,7 7,7 7.7 7,7 7,8 7.8 7,6 280 7.4 74 7.5 7.5 7.5 7.5 7,5 7,5 7.5 7,6 7,6 7.8 7,7 300 7.1 7.2 7.3 7.3 7.3 7.3 7,3 7,4 7,5 7,4 7,5 7.5 7.7 320 68 6.9 6.8 7.0 7.1 7,1 7,1 7,1 7,3 7,3 7,3 7.4 7.4 340 64 6.5 6.6 6.6 6.7 6,7 6,8 6,9 7.0 7,1 7,2 7.2 7.8 360 6.2 6.2 6.2 6.3 6.4 6,4 6,5 6,6 6.7 6.7 6,9 6.9 7.1 380 5 9 5.8 5.8 5.9 6.0 6,1 6,2 6,3 6.4 6,4 6,4 6.6 6.8 400 56 5.6 5.6 5.7 5.7 5,7 6,8 5,9 5.9 6,0 6.1 6.2 64 420 5 4 5.4 5,5 5,5 5.5 5.5 5,5 5,5 5.6 5.6 5.6 5.7 5.8 440 56 5.3 53 5,3 5.3 5.2 5,2 5.2 5.2 5,1 5.0 5.3 5.5 460 60 5.6 5.4 5.3 5.2 52 5,1 5.0 5.1 5,2 5.2 5.2 53 480 65 6.0 5.7 5.4 5.2 5.2 5,1 4.9 4.9 4,9 4,9 5.0 50 500 7.2 6.8 6.3 5.9 5.6 5.3 5,0 4.8 4.9 4,8 4,8 4.8 49 520 8.0 7.4 7.0 6.5 6.1 5.5 5.4 5.1 49 4,7 4,7 4.7 4.8 540 9.0 8.4 7.8 7.3 6.7 6.3 5.8 5.4 5,2 4,9 4,7 4.7 4.7 560 9.9 9.5 8.8 8.2 7,7 7.1 6.5 6.0 5,7 5.3 5.0 4.8 4.6 580 10.6 10.2 98 93 8,8 8.1 7.2 6.8 6.4 6.0 5.6 5,1 4.9 600 11.2 11.0 10.7 10.3 9,6 9.1 8.5 7.7 7.1 6.4 6.1 5,6 5,3 620 11.7 11.5 11.4 11.0 10,6 9.9 9.5 8.9 8.1 7.4 6.8 6.3 5,9 640 11.8 11,9 11,8 11.7 113 11,0 10.4 9.8 9.3 8.5 7.8 7.1 6,6 660 11.5 11.8 12.0 12,1 11,9 11,6 11.2 10.8 10.2 9.6 8.9 8.2 7,5 680 11.0 11.6 12.1 12,2 12,1 12,2 12.1 11.5 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 9,7 780 9.1 10,0 10.6 11,4 11,9 12,4 12,6 12,5 12,4 120 11.6 11.2 10,8 740 7.8 8.8 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 12,6 12.8 12,7 12.5 12.1 780 .5.2 6,3 7.1 8.1 9,2 10.1 10.7 11,6 12,0 12.4 12.8 12.9 12,8 800 4.0 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 9.1 10,0 108 11,4 12.0 860 1.6 1.6 2,2 2,9 3,6 4,6 5,6 6.6 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 92 10.1 900 1.1 1.1 1.2 1,3 1,7 2,2 29 3.8 4,8 5,7 6,8 7.9 8.8 920 1.1 1,0 1,0 1 1 1,1 1.4 1,9 2.6 3.4 4,4 5,3 6,3 7.4 940 1.5 1,1 0,8 9,9 1,0 1.1 1,3 1.6 2.3 3,1 3,9 5.0 6.9 960 23 1.7 1,3 0,9 0,7 0,8 0,9 1.2 1.4 2,0 2,8 35 4.6 9S0 30 2,5 1,9 1,4 1.2 1,0 0.8 0,9 1 1 1,4 1,7 2,4 3.3 1000 37 800 3.2 810 2,6 2,1 830 1.7 1,3 0.9 0,7 at 1,0 1,2 1,6 2.2 920 . 820 840 850 860 870 880 890 900 910 40 TABLE XXXI. TABLE XXXIl. Perturbations by Mars. Perfs . by Jupiter Arguments II. and IV. Arg's. II. and V. I\ II. 920 22 930 ft 3.0 940 950 4.8 960 970 6.9 980 7.8 990 1000 J 1 10 1 20 1 30 \ 3.8 5.8 8.4 " i 9.5 15.3 f 1 " 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 1.0 1.1 1.3 1.8 2.3 3.0 3.7 4.8 5.8 14.4 14.4 14.4 14.4 80 1.3 1.1 1.2 1.4 1.6 2.2 2.7 3.6 4.5 13.4 13.9 14.0 14.2 100 1.7 1.3 1.3 i.2 1.3 1.6 2.0 2.6 3.3 1 13.2 13.4 13.6 13.7 120 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 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 44 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 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 7.5 V.3 7.1 6.7 6.3 5.9 5.5 3.9 4.1 4.7 5.2 300 7.7 77 7.7 7.7 7.4 7.2 7.0 6.6 6.1 1 3.4 3.5 3.9 4.3 320 7.4 7.4 7.6 7.7 7.6 76 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 7.6 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 2.9 380 68 6.9 7.0 7.0 7.0 7.1 7.3 7.5 7.5 4.5 4.0 3.4 3.1 too 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 63 6.6 6.5 6.7 6.7 6.9 6.1 5.2 46 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 5.8 4.9 460 5.3 5.3 5.4 5.7 5.7 5.7 5.9 6.1 6.2 9.0 7.9 7.0 6.3 480 5.0 5.0 5.0 5.1 5.3 5.4 5.5 5.6 5.8 10.5 9.5 8.5 7.6 500 4.!) 4.9 5.0 5.0 5.0 5.1 5.2 5.3 5.3 12.3 11.3 10.0 9.1 '.■%0 4.8 4.8 48 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 12.3 560 4.6 4.5 4.5 4.4 4.5 4.5 4.5 4.5 4.4 17.1 Ifi.l 15.1 14.0 580 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 bOO 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 18.4 640 6.C 6.1 5.6 5.4 5.0 4.7 4.6 4.5 4.3 21.6 20.9 20.2 19.5 060 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 i;j.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 10.2 9.7 8.9 8.2 7.6 6.9 6.5 21.6 21.6 21.5 21.5 760 12.1 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 12.9 125 12.1 11.7 11.2 10.5 9.8 92 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 11.0 11.5 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 9.9 10.7 11.2 11.7 12.1 15.8 15.7 15.7 15.6 i 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 7.3 8.2 8.9 9.9 10.6 15.3 15.2 15.2 15,1 i 1000 2.2 920 3.0 930 3.8 940 4.8 950 5.8 960 6.9 970 7.8 980 8.7 990 9.5 15.3 15.1 15.0 15.0 1000 1 10 80 30 TABLE XXXII. 41 Perturbations froduced by Jupiter. Arguments II. and V. V. u. 30 40 50 60 70 80 , 90 100 1 110 120 130 140 IfOl 15.0 14.8 14,7 14.7 14.6 „ 1 -, 14.5 '14.5 14.4 14,5 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 14.1 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 13.8 138 13.7 60 14.4 14.3 14.3 14.2 14.1 13.9 13.8 13.6 l.T.'i 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 13.9 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 13.0 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 12.9 12.9 13.0 12.9 12.8 12.7 12.5 180 10,7 11.1 11.6 11.9 12.2 12.3 12.5 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 12,3 220 8.3 8.8 9.5 99 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 9.4 9.8 10.3 10.6 11.0 11,3 11.5 11,7 11.8 260 6.1 6.5 7.: 7.6 8.3 8.8 9.3 9.7 10.1 10.5 10.6 il.O 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 3.8 4.1 4.5 5.0 5.4 6,1 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 2.8 2.8 27 2.8 2.9 3.0 3.2 3,5 4,1 4.6 5,0 5.6 400 3 5 3.1 2.9 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 3.3 3.1 2.8 2.7 2.8 2,9 3.1 3,2 3.5 3,8 43 440 4.9 44 3.9 34 3.1 2.7 2.8 2.7 28 3.1 3.1 3.2 3,5 460 6.3 5.4 i 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 6.7^ 5 9 5.2 4.6 4.1 3.6 3.1 3.0 2.8 2.8 2.6 2.7 500 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 1 7.7 6,9 6.1 5.5 4.8 4.2 3.8 3,5 3,2 31 540 12.3 11,1 10.2 9.1 8.4 7.4 6.6 5.9 5.3 4.7 4,1 3,8 3.5 560 14.0 13,0 11,9 10.8 9.9 8.7 7.9 7.1 6,4 5.8 5,2 4.5 4.1 580 15.7 14.5 13.6 125 11.4 10.4 9.3 8.3 7,7 6,9 6,2 5.5 5.0 600 17.0 16,0 16.0 14.0 13.1 12.0 11.0 10.1 9.2 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 9 8 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 VA 10,8 9.4 8,7 660 20.4 19.7 18.9 1 18.1 17.4 16.3 15.6 14.6 13.7 12.8 11.9 no 10,1 680 21.2 20,5 19.9 19.1 18.5 17.6 16.8 16.0 15.1 14.2 13.5 12.5 11.6 700 21.5 21.0 20.8 20 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 20,5 20.0 19,4 18.9 18.4 17.7 17.2 16.8 15.7 760 21.2 21,0 21.0 20.8 20,7 20.3 20,0 19 4 19.0 18.6 17.9 17.4 16.7 7S0 20.7 20.7 20.7 20.6 20,6 20.3 20.2 19.8 19.4 19.1 18,7 18.1 17.6 8u0 20.1 20.2 120.3 20.3 20,4 20.3 -0,1 19,9 19,7 19.3 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 18,9 18 7 840 18.6 18 8 18.9 19.0 19,2 19.3 nu 19.4 19.4 19.4 19,4 19.0 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 17,6 17.9 18.0 18.2 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 17,4 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 15.5 15.6 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 15.3 15.2 15.2 15,2 15.3 15,4 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 14,7 14,7 14,6 14.5 14.5 14.4 14.5 110 14.5 120 14.6 1,30 14,7 140 14.8 30 40 50 60 70 80 90 100 150 42 TABLE XXXII. Perturbations produced by Jupiter. Arguments II. and V. V. 11. IflO 160 170 180 190 200 1 210 220 230 240 250 260 270 14.8 15.0 15.3 15.5 15.8 15.9 16.2 16.3 16.7 17.0 17.1 17.3 17.5 20 14.2 14.3 14.6 118 14.9 15.3 15.5 15.7 15.9 16.3 16.6 16.8 17.1 40 13.7 13.7 13.9 14.1 14.3 14.5 14.8 15.0 15.3 155 15,8 16.2 16.4 60 13.3 13.2 13.4 13.5 13.6 13.8 14.1 14.3 14.6 14.8 15,1 15.5 15.8 80 13.1 130 13.0 13.0 13.1 13.1 i 13.3 13.5 13.8 14.1 14 4 14.5 15.1 100 12.7 13.7 12.7 12.6 12.7 12.6 12.8 12.9 13.1 13.4 13.7 14.0 14.2 120 12.6 12.5 ii.5 12.4 12.3 12.2 12.3 12.3 12.6 12.8 13.0 13.3 13.6 140 12.6 12.4 12.4 12.3 12.1 12.0 12 12.0 12.1 12.1 12.3 13.5 13.8 160 12.5 12.3 12.2 12.1 12.1 11.9 11.8 11.8 U.S 11.8 11.9 12.0 12.2 180 12.5 12.3 12.2 12.1 11.9 11.8 11.7 i 11.5 11.5 11.5 11.6 11.7 11.8 200 12.3 12.2 12.2 12.0 11.9 11.7 11.7 11.5 11.4 11.3 11.3 11.3 11.5 220 12.0 12.0 12.1 12.0 11.8 11,6 11.6 11.5 il.4 11.3 11.2 11.1 11.1 240 11.8 11.8 11.9 11.9 11.8 11.6 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 11.3 11.3 11.3 11.2 11.1 11.0 109 280 10.6 10.8 11.1 11.2 11.2 11.2 11.3 11.3 11.2 11.2 11.1 11.0 IP 9 300 9.9 10.1 10.5 10.8 10.9 11.0 11 1 11.0 11.0 11.0 11.0 U.l 10.9 320 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 9.3 9.6 9.9 10.2 10.3 10.5 10.6 10.6 10.7 10.7 360 7.1 7.5 8.0 8.4 8.9 9.2 9.5 9.8 10.1 10.3 10.4 10.5 10.5 380 5.6 6.2 6.8 7.6 7.8 8.3 8.9 9.3 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 8.8 9.1 94 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 5.4 5.7 6.2 6.7 7.1 7.6 7.9 8.4 8.7 460 3.2 3.3 3.8 4.1 4.5 4.9 5.4 5.7 6.3 6.7 7 2 7.7 8.0 480 2.7 29 3.2 3 6 3.9 4.3 47 5.0 5.4 5 9 0,3 6.8 73 500 2.7 2.7 2.9 3 1 3.4 3.6 4.0 4.4 4,8 5 2 5.7 5.9 6.4 520 3 1 2.8 2.9 3.0 3.1 3.2 35 3.8 4.2 4.7 4 1) 5.4 5.7 540 3.5 3 3 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 3.6 33 32 3.2 3.2 33 3.5 37 4.0 4.3 4.5 580 5.0 46 4.2 4.0 36 3.5 3.3 3.2 3.4 3 5 3.7 4.(1 4.2 600 6.0 5.4 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 6.6 6,0 5.6 5.1 4.6 4.2 4.0 3.9 3.8 39 3.9 4.0 040 8.7 7.8 T.3| 6.6 6.1 5.5 5.3 4.7 4.4 4.2 4.0 4.0 4.1 660 10.1 9.3 8,6 7.7 7.2 6.5 6.3 5.9 53 4.9 4.6 4.5 4.4 680 11.6 10.8 100 9.3 8.5 7.5 7.3 6.7 6.3 5.8 5.5 5.2 4.9 700 13.0 12.1 11.5 10.7 9.9 9.0 8.5 7.8 7.4 6.9 6.3 6.0 5.8 720 14.3 13.5 12.8 12.1 11.3 10.6 9.8 9.1 87 8.0 7.C 7.0 6.6 740 15.7 14.9 14.2 13.4 12.7 12.0 11.2 10.5 97 9.3 8.9 8.3 7.7 760 16.7 15.9 15.5 14.7 13.9 133 12.6 11.8 11 3| 10.5 10.0 9.5 9.0 780 17.6 17.0 16.4 157 15.1 146 13.8 13.2 12.6 11.9 11.2 109 102 800 18.2 17.8 17.3 IU.8 16.2 16.0 15.0 14.3 13.7 13 1 12.6 12.0 11.5 820 18.7 18 3 18.0 17.6 17.0 16.6 16.0 15,3 14.9 14.3 13.7 13.1 12.6 840 18.9 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 1S.7 18,6 18.4 183 17.9 17.4 17.1 16.7 16.3 15.9 15,4 150 SSO 18.7 13.5 18 6 18.5 18 3 18.2 18.0 17.7 17.4 17.1 16.6 .6 3 15.9 900 18.2 18.2 18.3 18.3 183 18.1 18.1 18.0 17.8 17.6 17 3 17.0 16.7 920 17.7 17.9 18.0 18.0 18.1 18.1 18.0 18.0 18.0 17.8 17.7 17.6 17.3 940 17.1 17.1 17.4 17.6 17.6 17.7 17.8 17.8 17.9 18.0 17.8 17.8 17.7 960 16.3 16.5 16.8 16.9 17.1 17.2 17.4 17.5 17.6 17.8 17.9 18.0 17.9 980 15.5 15.7 16.1 16.3 16.5 16.7 16,8 17.0 17,2 1 17 3 17.6 17.7 17.9, 1000 14.8 15.0 15.3 15 5 15,8 15.9 200 16 2 16.3 230 16.7 j 17.0 17.1 17.3 17.5 ' 150 160 170 180 190 210 230 1 240 350 260 270] TABLE XXXII. 43 Perturbations produced by Jupiter. Arguments II. and V V Jl. 279 ; 280 290 300 310 320 330 340 360 360 370 380 390 17.5 17.5 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 18.0 18.0 40 16.4 16.8 16.9 17.2 17.6 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 18.5 18.5 18.7 80 15.1 15.4 15.7. 16.1 16.4 16.7 17.0 17.5 17.8 18,0 18.3 18.5 18.8 100 14.2 14.6 15.1 15.0 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 14.5 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 15.5 220 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 10.6 10.7 10.6 10.8 11.0 11.2 11.3 11,5 11.8 12.2 300 10.9 10.8 10.7 10.6 10.6 10.5 10.6 10.7 10.8 10.9 11,1 11.4 11.8 320 10.9 10.7 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 10.7 10.6 lO.-i 10.5 10,4 10.5 10.5 10.6 10.5 10,6 10.7 10.8 360 10.5 10.5 10.5 J0.5 10.5 10,4 10.4 10.4 10.4 10.3 10,5 10.6 10,8 380 102 10.3 10 3 10.3 10.4 10.3 10.4 10.4 10.4 10.3 10.3 10.4 10,6 400 9.9 10.0 100 10.2 10.3 10.2 10.2 10.3 10.4 10.3 10.3 10.3 10,5 420 9.4 96 9.8 9.9 10.1 10.2 10.1 10.2 10.2 10,2 10.3 10 3 10,4 440 8.7 9.0 9.2 9.4 9.7 9.8 10.0 10.1 10.2 10,1 10.1 10.2 10,4 460 8.0 8.4 8.6 8.8 9.1 9.3 9.6 9.9 10.1 10,0 10 10.2 10,3 480 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 6.1 6.6 6.9 7.3 7.6 7.9 8.3 8.6 8,9 9.1 9.4 9,7 i>40 5.1 5.4 5.8 6.2 6.7 7.0 7.4 7.7 8.0 8.3 8.6 8.9 9,2 560 4.5 4 9 5.1 5.5 6.0 6.3 6.7 7.2 7.5 7.7 8.0 8.3 8,7 5S0 4.2 4.4 4,8 5.0 5.3 5.7 6.1 6.6 6.9 7.1 7.4 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 7,6 620 4.0 4.0 4.1 4.3 4.7 4.8 5.1 5.5 5.8 6 1 6.4 6.7 7.0 640 4.1 4.1 4.2 4.2 4.4 4.6 4.8 5.1 5.4 5.6 5.9 6.3 6.6 660 4.4 4.3 4.3 4.3 4.5 4.5 4.7 4.9 5.1 5.3 .5.5 5.8 6.2 680 4.9 4.9 4.7 4.6 4.7 4.5 4.6 4.8 5.0 5.1 5.3 5.5 5.8 700 5.8 5.4 5.2 5.1 5.0 4.9 4.9 4.9 5.1 5,2 5,3 5.4 5.6 720 6.6 6.2 5.9 5.7 5.6 5.5 5.4 5.3 5.3 5,3 53 5.4 5.5 740 7.7 7.2 6.8 6.5 6.4 6.1 6,0 5.9 5.8 5,7 5.6 5.5 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 59 6.0 780 10.2 9.7 9.1 8,4 8.2 7,7 7.6 7.4 7.2 6.9 6.6 6.5 65 eoo 11.5 11.0 10.4 9.8 9.4 8.7 8.5 8.3 8.0 7.7 7.6 73 7.1 020 12.6 12,1 11.7 11.2 10.6 10.1 9.7 9.2 9.1 8.6 8.3 8.1 7.9 640 13 8 13.2 12.8 12.3 11.9 11.3 10.9 10.5 10.2 9.6 94 9,1 8.9 660 15.0 14.4 13.8 13.5 13.1 12.6 12.1 11.7 11.2 10.7 10.4 10.1 10.0 880 15.9 1.^)4 15.0 14.4 14.2 13.7 13.4 12 9 12.5 12.0 11.5 113 11.1 900 16.7 16.4 15.9 1,5.5 15.2 14.8 14.4 14 1 13.7 13.2 12.8 12 4 12.2 920 17.3 17.1 16.8 16.5 16.2 15.7 15.5 15.2 14.8 143 14.0 136 13.3 940 17,7 17.5 17.3 17.1 16.9 166 16.3 16.1 16.0 15.5 15,0 14.7 14.5 960 !7." 17.8 17.6 17.5 17.4 17.2 17.0 16.9 16,8 164 16.2 15 8 15.6 980 17.9 17.8 17.8 17.8 17.8 17.8 17.6 17.5 17.3 17.2 17.0 16.8 1.66 1000 17.5 17.7 17.7 17.8 17.9 17.9 320 18.0 330 18.0 340 17.9 350 17.7 300 17.6 370 17.5 360 17.5 " 27C 280 290 300 310 390 ii TABLE XXXII. Perturbations produced by Jupiter. Arguments II. and V. V. II. 390 400 410 420 430 440 450 460 470 480 490 500 filO 17.5 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 18.9 18.9 18.7 18.8 18.6 18.7 18.4 18.1 17.9 17.7 80 18.8 18.9 19.2 193 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 19.7 120 140 160 180 200 18.3 18.6 18.9 19.2 19.5 19.8 20.0 20,1 20.3 20.3 20 4 20.4 20.4 17.7 18.2 18.6 18.9 19.2 19.6 20.0 20.3 20.5 20.6 20. r 20.8 21.0 17.1 17.6 17.9 18.5 19.0 19.3 19.8 20.2 20.5 20.6 20.9 21.1 21.2 16.3 in 8 17.3 17.9 18.3 18.8 19.3 19.8 20,3 20.6 20.9 21.1 21.4 15.5 16.0, 16.5 17.1 17.7 18.2 18.6 19.1 19,8 20.2 20.7 21.0 21.4 220 240 260 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 13.8 14.2 14.7 15.2 15.9 16.5 17.1 17.7 18,4 18.9 19.5 20.1 20.7 13.0 13.4 13.9 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 1 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 1.5.3 16.0 16.6 17.4 340 10.8 11 2i 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 10.8 11.0 11.2 11.6 11.9 12.3 12.6 13.3 13.6 14.2 14.8 1,5.5 380 10.6 10.6 10.7 10.9 11.2 11.4 11.9 12.2 12.6 12.9 13.5 13.9 14.51 400 10.5 10.5 10.6 10.6 10.9 11.1 11.4 11.8 12.2 12.5 12.9 13.3 13.8 420 10.4 10.4 10.5 10.6 10.7 10.9 11.2 11.3 11.7 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 11.2 11.3 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 10.2 10.4 10.5 10.7 10.8 10.9 11.0 11 2 11.3 117 520 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 S60 8.7 89 9.1 93 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 7.3 7.6 7.9 8.2 8.5 8.8 9.0 9.4 9.6 10.0 10.1 10.4 640 6.6 6.8 7.1 7.4 7.7 7.9 8.2 8.6 8.9 9.1 9.4 9.7 10.1 660 62 64 66 6.9 7.3 7.6 7.9 8.1 8.3 8.6 8.9 9.2 9.5 680 58 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 6.0 6.2 6.4 6.6 6.9 7.1 7.4 7.6 7.9 8.2 8.5 720 5.5 5.6 5.7 5.9 62 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 6.1 6.2 6.4 6.7 6.9 7.1 7.2 7.5 760 6.0 60 60 6.0 6 61 6.2 6.3 6.4 6.5 6.7 6.8 7.1 780 6.5 63 6.2 6.2 6.3 6.3 6.3 6.3 6.4 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 87 820 7.9 7.6 7.5 7.3 7.2 7.0 7.0 6.8 6.8 6.7 6.6 6.6 6.7 840 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 90 87 8.4 8.2 8.1 7.9 7.7 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 8.5 920 133 13.0 12.6 12.3 12.1 11.5 11.3 11.0 10.6 10.2 10.1 9.7 9.4 940 i 14-5 14.1 13.8 13.5 132 12.8 12.5 11.9 11.8 11.3 11.0 10.7 10.4 960 ' 15.6 15.3 14.9 14.6', 14.4 14.0 13.7 13.3 13.0 12.5 12.1 11.8 11.5 980 16.fi 16.3 16.0 15.7 1 15.6 15 2 14.9 14.6 14.2 13.8 13.6 12.9 12.7 1000 17.5 171 17 16.7 420 16.5 16.3 440 16.1 15.8 15.6 15.1 1 14.6 14.3 13.9 1390 400 410 430 150 460 470 480 1 490 500 510 TABLE XXXII. Perturbations produced by Jupiter. Arguments II. and V. V. 11. 610 520 530 540 S50 560 570 580 690 600 610 620 630 13.9 1 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 15.3, 14.9 14.4 13.9 13.5 13.1 12,5 12.1 11,5 11,0 10,4 10.0 9.4 40 16.5 1 16.3 15,7 15.4 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 18.1 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 19,7 19,5 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 21.5 21.6 21.5 21.3 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 1 22,5 1 22,5 22,5 22.4 22,3 22,1 21,8 21,5 220 21.1 21.5 21.8 22.2 22.5 22,8,23,1 23,1 22,9 22,8 22,9 22,6 22.5 240 20.7 21.1 21.5 21.9 22.3 22,7 23.0 23,3 23,4 235 23.4 23,3 23.2! 260 20 20.6 21.0 21.6 22.0 22.4 22.8 23,2 23.5 23.8 23.8 23,8 23.9 280 19.1 19.9 20.4 20.9 21,5 22.0 22.4 23,0 23.3 23.7 24.0 24,1 24.1 300 18.2 19.0 19.6 20.3 20.7 21,3 21,8 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.fi 18.5 19,2 19.9 20,4 21,1 21.6 22.2 22.8 23.3 23.7 360 15.5 16.2 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 14.2 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 13.5 13,9 14,4 15,0 15.6 16.1 16.9 17.5 18,2 18,7 480 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 5fi0 11.2 11.4 11.5 11.9 12,1 12,4 12.7 13.1 13.4 13.8 14.1 14,5 14.9 .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 13.4 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 11,5 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 7.4 7,6 7,9 8.1 8,4 8,7 840 6.8 G.8 6.8 6.8 6.8 6,9 6,9 7,1 72 7,4 7.6 7,9 8.1 860 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 7.7 7.5 7.4 7.3 7,1 7.0 6,8 6,8 67 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 6,7 960 11.5 11.2 10.7 104 9.8 9.5 9,1 8.8 8.5 8.1 7.7 7.4 7,1 980 12.7 12.3 11.8 11.5 11,1 10.6 10 9.7 9.2 8.9 8.5 8.1 7,7 1000 13.9 13.4 520 13.1 12.7 12,1 550 U.8 11.3 570 10.8 580 10.2 9.9 9.4 8,9 8,4 510 530 540 560 5,90 600 610 620 630 46 TABLE XXXIL Perturbations produced by Jupiter. Arguments II. and V. V. II. 630 8.4 640 650 660 670 7.3 6.9 680 690 700 710 720 730 6.2 6.2 740 750 ! 8.0 7.7 6.7 6.5 6.5 6.3 6.4 6.5 20 9.4 9.0 8.4 8.0 i 7.5 7.1 6.9 6.7 6.4 6.3 6.0 6.1 6.1 40 10.5 10.1 9.4 8.9 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 8.8 160 19.1 18.6 17.9 17.3 16.6 15.7 15.0 14.2 13.3 12.6 11.7 10.9 10.0 ISO 20.3 19.9 19.4 18.8 18.0 17.3 16.7 15.8 15.0 14.1 13.2 12.4 11.5 200 21.5 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 20.3 19.7 19.0 18.2 17.5 16.6 15.5 14.71 240 23.2 230 22.9 22.5 22,0 21.6 21.1 20.5 19.8 19.1 18.2 17.3 16.4' 260 239 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 24.4 24.2 23.9 23.6 23.1 22.5 21.9 21.2 320 24.2 24.5 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 23.7 360 23.2 23.7 24.2 24.5 24.7 25.0 25,1 25.3 25.4 25.3 25.1 24.9 24.5 380 22 5 23.1 23.6 24.1 24.4 24.7 2.5.1 25.2 25,4 25.5 25.4 25,3 25.2 400 21.5 22.3 22.8 23 4 23.9 24.3 24,7 25.1 25,2 25.4 25.6 25.6 25.5 420 20.6 21.3 22.0 22.6 23.1 23.6 24,1 24.5 25.0 26.2 25.4 25.6 25.7 440 19.6 20.3 21.0 21.8 22.3 22.9 23,4 23.9 24.3 24.8 25.0 25.2 25.6 460 18.7 19.4 20.1 20.7 21.3 21.9 22,6 23.3 23.6,24.1 24.6 24.8 25.1 480 17.9 18,5 19.1 19.7 20.3 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 19,9 20.7 21.4 21.9 22.5 22.9 23.4 23.9 520 16.2 16.8 17.3 17.9 18.4 19.0 19.7 20.4 21.0 21.6 21.1 22.6 23.0 540 15.4 16.1 16.6 17.2 17.5 18.1 18.7 19.3 19.9 20.5 21.2 22.7 22.2 560 14.9 15.4 16.0 16.5 16.9 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 14.2 14.6 14.9 15.3 15.8 16.3 16.6 17.0 17.4 17.9 18.3 18.9 19.4 620 138 14.2 14.6 14.9 1.5.1 15.7 16,2 16.6 16.9 17,3 17.6 18,0 18 5 640 13.5 14.0 14.2 14.6 14.8 15.1 15,6 16.1 16.5 16,8 17.1 17,5 17 9 660 13.2 13.5 13.9 14.3 14.6 14.9 15,2 1.5.6 15.9 16.4 16,6 17.0 17 3 680 12.8 13.2 13.5 13.9 14.2 14.5 14.9 15.2 15.6 16.0 16,2 16.5 10.8 700 12.4 12.9 13.3 13.5 13.8 14.2 14.5 14.9 15.1 15.6 15,9 16.2 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 16.1 740 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 13.2 13.7 14.1 14.5 14.7 15.0 15 4 780 10.2 10.6 11.2 11.6 11.9 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 14.5 820 8.7 9.3 9.7 10.0 10.5 10.9 11.4 11.9 12.3 12.8 13,2 13.6 14.0 840 8.1 8.4 8.8 9.3 9.6 10.1 10.6 11,1 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 U.S 900 6.8 7.0 7.1 7.3 7.4 7.8 8.2 8,5 8.9 9.4 9,8 10.3 10.8, 920 6.7 6.8 6.8 6.9 7.0 7.0 7.4 7.8 8.1 86 8,9 9.4 99; 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 89 960 7.1 7.0 6.8 6.7 6.5 6.5 6.6 6.7 6.8 7 1 73 7.7 8.0 980 7.7 7.4 7.1 6.9 6.6 6.5 6.4 6.4 6.3 65 6,f 6.9 7,3 1000 8.4 8.0 7.7 7.3 6.9 6.7 6.5 6.5 6.3 710 6.2 720 6,2 73) 6.4 740 6,5 633 R40 650 660 670 680 690 700 750. TABLE XXXII. 47 Perturbations produced by Jupiter. Arguments II. and Y. V. II. 1 750 1 760 770 780 790 800 810 1 820 1 830 840 85U 860 870 6.5 6.8 7.2 7.5 8.0 8.4 8.8 9.5 10.1 10.6 11.0 11.6 12.4 20 6.1 6.2 6.5 6.7 7.0 7.4 79 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 6.4 5.4 5.6 5.9 6.3 6.8 120 7.6 7.4 6.5 6.0 5.7 6.5 5.1 5.2 5.1 5.1 5.2 6.5 5.8 140 8.8 8.1 7.4 6.8 6.2 6.8 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 6.5 6.1 5.9 4.7 4.7 4.7 180 11.5 10.6 9.7 9.0 8.2 7.6 6.9 6.3 5.8 5.2 4.8 4.7 4.6 200 13.1 12.2 11.2 10.4 9.5 8.8 7.9 7.1 6.5 6.9 5.3 5.0 4.7 220 14.7 13.8 12.9 12.0 11.1 10.2 9.3 8.4 7.6 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 6.6 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 16.0 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 9.1 320 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 23.0 23.0 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 24.6 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 25.6 25.6 25.5 25.4 25.2 24.9 24.5 24.1 23.5 SCO 23.9 24.2 24.7 25.0 25.3 25.4 25.6 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 23.2 22.6 23.2 23.6 24.0 24.4 24.6 24.9 25.1 25.0 25.1 25.1 25 560 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 16.3 16.5 16.9 17 2 17.6 17.8 18.0 18.3 18.6 18.7 191 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 18.2 780 14.9 15.3 15.6 16.9 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 1.5.8 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 1 860 12.6 13.1 13.5 13.9 14.3 14.8 15.2 15.5 15.6 18.8 16.0 16.3 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 16.1 900 10.8 11.3 11.9 12.4 13.0 13 4 13.7 14.2 14.7 15.0 15.2 15.5 16.7 920 9.9 10.3 10.8 11.4 12.0 12.5 12.9 13.4 14.0 14.3 14.7 16.0 16.3 940 8.9 9.4 9.9 10.4 11.0 11.6 12.1 12.5 13.0 13.6 13.9 14.4 14.71 960 8.0 8.3 8.8 9.-1 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 8.9 9.5 9.9 10.5 11.1 11.6 12.1 12.8 13 3 1000 6.5 750 6.8 760 7.2 7.5 780 8.0 8.4 8.8 9.6 10.0 105 840 11.0 11.6 12.4 770 790 1 800 I 810 1 820 1 830 8.50 £60 870 TABLE XXXII. Perturbations produced by Jupiter. Arguments 11. and V. V. 11. 870 880 890 13.2 900 13.6 910 920 1 930 940 950 960 970 980 990 15.2 1000 12.4 12.9 13.9 14.2 14.4 14.8 15.0 15.1 15.1 15.2 15,3 20 1 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 14.9 \[) 10.0 10.5 11.1 11.7 12.3 12.6 13.0 13.4 13.7 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 12.1 12.5 12.8 13.2 13,5 13.8 100 6.8 7.2 7.6 8.1 8.6 9.4 9.9 105 10.9 11.4 12.0 12.4 12,8 13.2 120 5.8 6.1 66 7.1 7.6 1 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 0.3 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 3.4 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 4.1 4.0 3.8 3.9 4.0 4.3 4.3 4.7 5.2 5.7 260 6.9 6.1 5.4 4.9 4.4! 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 5.2 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 49 4.3 3.8 3.5 380 16.5 15.4 14.2 1.3.0 11.8 10.8 9.7 8.7 7,8 6.9 6.1 5.4 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 95 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 10.0 9,0 480 23 5 22.7 22 21.1 20.2 19.3 18.2 17,3 16 2 15.0 13.8 12-8 11.6 10,5 500 24.3 23.8 23 22.3 21.6 20.7 197 18,8 178 16 7 15.4 14.5 13.4 12,3 520 24.8 24.3 23 7 23.2 22.7 31.9 21.1 20,2 19,2 18.3 17.2 16.1 150 14.0 540 25.0 24,8 24.3 23 9 23.4 22 8 22 1 21,3 20.0 19.7 18.7 17,6 16.6 15.6 560 24.9 24 8 24.7 24-4 24 23.6 22 9 22,4 21.6 20.8 20.0 19.1 18.2 17.1 580 24.7 24.7 24.6 24.5 24.3 23.9 23.5 23,1 22,5 21,9 21.1 20.3 19.5 18.6 600 24.3 24.3 24.3 24.3 24.3 24.1 23.8 235 23 22,5 22.0 21.4 20.6 19.8 620 23.6 23.7 23.9 24.0 24.1 24.1 23.9 23,7 23 l 23.1 22.6 22.1 21.4 20.8 640 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 22.3 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 20.7 20.9 21.2 21.5 21.7 21.9 22,2 22.3 22.5 22,5 22,5 22.4 22.2 720 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 22.0 740 18.8 19.0 19.2 19.6 19.9 20.2 20.5 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 1.94 19.6 19.9 20.1 20.3 20,5 20.S 21.0 21.2 780 17.7 17.8 18.0 18.1 18.4 18.7 18.8 19.1 19.3 19,5 19,7 20.0 20.2 20.4 800 17.3 17.4 17.4 17.7 17.9 18.0 18.1 18.4 18.6 18.9 18,9 19.1 19.4 19.6 820 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 17.9 181 860 16.4 16.5 16.5 16.6 166 16.7 16 8 16.9 16.9 17.0 17.0 17.1 17.2 174 880 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 1.5.6 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 15.7 15.8 960 14.1 14.3 145 14.8 152 15 5 15.5 1.5.7 15.7 15,7 15.6 15.6 15.5 15.5 980 13 3 12.7 139 14.2 145 14.8 15.1 15.3 1.5,4 15.5 15.4 15.4 15.4 15.3 1000 12.4 870 12.9 880 13.2 890 13.6 900 13.9 910 14.2 920 14.4 930 14 8 940 15.0 15.1 15.1 970 15.2 980 15.2 15 3 1000 950 960 990 TABLE XXXIII. Perturbations pioduced by Saturn. Arguments II and VII. VII. 49 II 100 300 300 400 500 600 700 800 900 1000 // // n // // // // /, „ ,/ ff 1.2 1.5 1.4 1.0 0.7 06 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 o.a 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 1.6 900 1.5 1.4 1.1 0.9 0.7 0.6 0.3 0.2 0.6 1.2 1.6 1000 1.2 1.5 1.4 1.0 0.7 0.6 0.5 0.5 0.4 0.8 1.8 Constant, l."0 TABLE XXXIV. Variable Part of Suri's Aberration. Argument, Sun's Mean Anomaly. 0« I» II» III» IV« V» o 0.0 0.0 0.1 0.3 0.5 0.6 o 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 34 9 0.0 0.0 0.2 0.3 0.5 0.6 91 12 0.0 0.1 0.2 0.4 05 0.6 18 16 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 a 30 0.0 0,1 0.3 0.5 0.6 0.6 XI. X' IX« VUU VII» VI» Constiint, 0."3 50 TABLE XXXV, Mbort's Epochs. Teibb. 1 i S 4 B 9885 6 0635 T 5979 8 9921 t 7633 10 219 11 226 12 458 18 468 1830 00174 4541 4461 4638 1831 00103 1749 4127 9381 2357 6432 7040 2378 6487 826 587 177 940 1832 B 00032 8957 3793 4125 4829 2229 8100 4835 5351 432 948 897 413 1833 00^35 6816 4499 9156 7636 -399 9219 7683 4239 108 340 687 920 1834 00164 J 024 4164 3900 0107 4196'0279 0140 3103 715 701 406 393 1835 00093 1232 3830 8644 2579 9993 1340 2598 1967 321 (161 125 866 1836 B 00022 8-141 3496 3388 6051 579l'24no 6055 0831 928 4-.'2 845 339 1837 00224 6299 4202 8419 7858 196'!3518 7903 9719 C05 814 636 846 1838 00153 3508 3868 3163 0329 7757 457H 0360 8583 211 175 354 319 1839 00082 0716 3534 7907 2801 3555 5639 2818 7447 818 536 074 792 1840 B 00011 7935 3199 2651 5273 9352 6700 6275 6310 424 896 793 265 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 (1200 3237 7169 3023 711P 9939 3038 2926 314 010 022 718 1844 B 00000 7408 2903 1913 5495 2914 lOOO 5495 1790 920 371 741 191 1845 00203 6266 3609 6944 8302 9083 2118 8343 067 .s 597 763 531 698 1846 00132 2475 3275 1688 0773 4S'8( 3179 0800 9542 203 123 250 171 1847 OOObl 9683 2941 6432 3245 0678 4239 3257 fi406 810 484 970 644 1848 B 99990 6892 2606 1176 5717 6475 5300 6716 7270 41t 845 689 117 1849 01)192 4750 3312 6207 ^524 2644 6418 8563 6158 093 237 479 624 1850 00121 1958 2978 0951 0996 8442 7479 1030 5022 700 697 199 097 1851 00050 9167 2644 5695 3467 4239 8539 3477 3885 306 958 918 570 1852 B 99979 6375 331(1 0439 5939 0036 960(1 5935 3749 9l:s 319 637 043 1853 00181 4233 3016 5469 8740 6206 0718 8782 1637 589 711 427 650 1854 iiOUO 1442 2681 0213 1217 3003 1778 1240 0501 196 (i72 147 023 1855 00039 b650 2347 4967 3689 7801 2839 3697 9365 602 432 86K 496 1 856 B 99968 5859 2013 9701 6160 3598 3899 6156 8229 409 793 586 969 1857 uoni 3717 2719 4732 8968 9767 5018 90(12 7117 (186 185 375 476 1858 00100 0925 2386 9476 1439 5565 6078 1460 6981 692 546 1195 949 1859 00029 8134 2051 4220 3911 1362 7139 3917 4845 299 9ii7 814 422 1860 B 99958 5342 1716 8964 6383 7159 8199 6374 3709 906 267 534 896 1861 00160 32011 2423 3995 9190 3329 9317 9222 2597 581 659 323 402 1863 00089 04119 2^l8^ 8739 1661 9126 J37.S 1679 1461 188 020 043 876 1863 00018 7617 1754 3483 4133 4923 1438 4137 1324 79.i 381 76- 348 1864 B 99947 4826 1420 8227 6005 0721 2499 6594 9188 401 74-.^ 482 821 1865 00149 26S4 2126 3357 941-.: 6890 36)7 9442 8(i76 (J7b 134 272 328 18B6 00078 9898 1792 8001 1883 3687 467S 1899 6940 683 494 991 801 18117 00007 7101 1457 2745 4355 8486 5738 4357 5 b 04 291 855 711 274 1868 B 99936 43(19 1123 7489 (i827 4283 6799 6814 4668 898 216 431 747 1869 00138 2168 1829 2520 9634 1432 7917 9662 355(i 574 H08 220 254 1870 00O67 9376 1495 7264 2105 6249 8978 2119 2420 181 968 94(1 727 • 1871 99996 6585 1161 2008 4577 2046 0038 4576 1283 7 87 329 659 200 1872 B 99923 3793 0827 6752 7049 7H43 1099 7034 0147 394 69(1 378 673 1873 00127 1651 1633 1782 9856 4013 2217 9881 9035 (i70 082 168 189 1874 OOOoK 9860 1198 6526 2327 981" 3277 2339 7899 677 443 888 653 1875 99985 6068 0864 1270 4799 5608 4338 4796 6763 283 803 007 126 1876 B 99914 3277 0530 6014 7280 1405 5S98 7254 6627 890 164 327 599 1877 00117 1135 1236 1045 0078 7574 6517 0101 4515 567 556 116 106 1878 O0046 8343 0902 5789 2549 3372 7577 2559 3379 173 917 836 579 1879 99975 5652 0568 (1533 6021 9169 8638 5016 2243 780 278 565 052 1 88iJ B 99904 276(. 0233 5277 7493 4966 9698 7473 1107 386 638 275 525 1881 10106 0618 0940 0308 0300 1136 0816 0321 9995 062 080 064 032 1882 00035 7827 0605 6052 2771 6933 1877 2798 8859 669 391 784 505 1883 99964 5035 0271 Z 148 152 477 70O 549 OUO 1854 9 21 19 8.8 5 14 30 634 636 456 459 557 639 735 276 1855 10 10 38 51.1 9 6 2 047 048 763 766 637 579 9il 563 186SB 10 29 58 33.3 25 34 461 461 071 071 717 518 107 830 1857 U 19 2126.2 4 27 15 909 912 407 411 832 463 334 140 1858 8 41 8.4 8 17 47 323 325 716 718 912 402 520 417 1859 28 (I0U.7 8 19 736 737 023 024 992 342 706 694 1860 B 117 20 32,9 3 28 51 150 150 330 330 072 281 8a2 971 1861 2 6 43 25.8 8 32 698 601 666 670 187 226 119 381 1862 2 26 3 8.0 1121 4 012 014 974 977 267 166 303 598 1863 3 15 22 50.1 3 1136 426 426 282 283 347 105 491 834 1864 B 4 4 42 32.3 7 2 8 839 839 59U 589 427 044 677 ill 1865 4 24 5 25.2 11 3 49 287 291 926 929 542 989 904 422 1866 5 13 25 7.3 2 24 21 701 703 233 236 622 928 090 698 1867 6 2 44 49.5 6 14 53 115 116 641 542 702 868 276 975 1868 B 6 22 4 31.7 10 5 26 529 528 849 848 782 807 462 252 1869 7 11 27 246 2 7 7 977 980 186 188 897 762 689 662 1870 8 47 6.7 6 27 39 390 392 493 495 977 691 875 839 1871 8 20 6 49.0 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 25 666 668 445 447 252 616 474 703 1874 10 18 9 6.3 9 57 080 081 753 754 332 454 660 979 1875 U 7 28 48.6 21 29 493 493 060 060 412 394 846 256 1876 B 11 26 48 30.8 412 1 907 906 368 866 492 333 032 533 1877 16 11 23.7 ^13 42 355 357 704 706 607 278 259 843 1878 1 6 31 5.9 4 14 769 770 012 013 687 217 446 120 1879 1 24 50 48.2 3 24 46 182 183 320 319 767 157 631 397 1880 B 2 14 10 30.4 7 15 18 696 595 6i7 623 847 096 817 674 1881 3 3 33 23.3 11 16 59 044 046 963 965 962 041 044 984 1882 3 22 53 5 5 3 781 458 459 271 272 042 980 230 261 1883 4 12 12 47.6 6 28 3 872 871 679 578 122 920 416 537 1884 B 6 1 32 29.8 10 18 36 2s5 284 887 884 202 859 602 814 1885 6 20 55 23.0 2 20 16 733 736 223 224 317 804 829 125 32 54 TABLE XXXVI. Moon's Motions for Months. Months. January February March April May June July Aug. Sept. Oct. Nov. Dec. ; Com. Bis. ( Com. \ Bis. ( Com. t Bis. I Com. 1 B.S. ( Com. iBis. 5 Com. \ Bis. ( Com. \ Bis. ( Com. \ Bis. j Com. \ Bis. ( Com. \ Bis. 00000 08487 16153 16427 24640 24914 32853 33127 41340 41614 49554 49828 58041 58315 66528 66802 74741 75015 83228 8350.] 91442 91716 2 0000 0146 8343 8993 0000 2246 1371 2411 0000 8896 6931 7218 5827 6114 4436 8490 3616 9140j4657 798614822 8636,5862 4723 8133 7067 3332 8783 8107 3619 7629 8279 7776 8426 7922 8572 7419 8069 7565 8215 7062 7712 8273 9313 0518 1558 2764 3804 3969 5009 6215 7255 7420 8460 1942 2228 0838 1125 9734 0021 8343 8630 7239 7526 5848 6135 6 0000 0402 9797 0132 0199 0534 0265 0600 0666 1002 0732 7341 1068 7713 1134 8874 1470 9246 1536 0408 1871 0780 0000 1533 1951 2323 3484 3856 4646 5018 6179 6551 1602 1938 2004 2339 2070 2405 1569 1941 3102 3475 4264 4636 0000 1789 3404 3462 5193 5251 6924 6982 8713 8771 0444 0502 2233 2290 4021 4079 5752 5810 7541 7599 9272 9330 0000 2099 3027 3418 5126 5517 6835 7226 8934 9325 3691 0643 4396 1034 4420 2742 5148 31.33 5173 4842 5901 5232.5925 0000 0753 1433 1457 2186 2210 2914 2938 3667 10 11 12 -13 000 175 139 209 314 384 419 489 593 663 698 768 873 943 048 118 6550 6630 6941:6654 8649 7382 9040 ! 7407 1 397 0358 811l'432 0749!8135 502 152 222 327 000.000 9651184 836 157 868 228 801 832 735 766 700 731 634 665 599 630 563 595 497 528 462 493 427 342 412 456 526 640 710 754 824 938 009 123 193 237 .307 421 492 396 .535 606 000 059 016 050 076 110 101 135 160 194 185 219 245 279 304 338 329 363 388 423 414 448 TABLE XXXVL Moon' s Motions for Months. Months. Evection. Anomaly. Variation. Longitude. , O ' " J O ' '/ . o ' " s O ' /' January 0.0 0.0 February 11 20 48 42 1 15 53.1 17 54 48 1 18 28 5.8 March I Com. Us. 10 7 40 26 1 20 50 4.2 11 29 15 15 1 27 24 26.6 10 18 59 26 2 3 .53 58.2 11 26 42 2 10 35 1.6 April ( Com. ': Bis. 9 28 29 8 3 5 50 57.3 17 10 3 3 15 52 32.5 10 9 48 8 3 18 54 51.2 29 21 29 3 29 3 7.5 May Com. ■ Bis. 9 7 58 51 4 7 47 56.4 22 53 24 4 21 10 3.3 9 19 17 50 4 20 51 50.3 1 5 4 50 5 4 20 38 3 June Com. ': Bis. 8 28 47 33 5 22 48 49.4 1 10 48 11 6 9 38 9.1 9 10 6 33 6 5 52 43.4 1 22 59 38 6 22 48 44.1 July ( Com. i Bis. 8 8 17 16 6 24 45 48.5 1 16 31 32 7 14 55 39.9 8 19 36 15 7 7 49 42.5 1 28 42 59 7 28 6 15.0 Aug. ( Com. \ Bis. 7 29 5 59 8 9 46 41.6 2 4 26 20 9 3 23 45.8 8 10 24 58 8 22 50 35.5 2 16 37 47 9 16 34 20.8 Sept. ( Co/r.. ';Bis. 7 19 54 41 9 24 47 34.6 2 22 21 7 10 21 51 51.6 8 1 13 40 10 7 51 28.6 3 4 32 34 11 5 2 26.7 Oct. ( Com. I Bis 6 29 24 24 10 26 44 33.7 2 28 4 28 11 27 9 22.4 7 10 43 23 11 9 48 27.7 3 10 15 55 10 19 57.5 Not. Com. ' B's. 6 20 13 6 11 45 26.8 3 15 59 16 1 15 37 28.3 7 1 32 5 24 49 20.7 3 28 10 43 1 28 48 3.3 iDec. ( Com. 1 Bis. 5 29 42 49 1 13 42 25.9 3 21 42 37 2 20 54 59.1 6 11 1 48 1 26 46 19.8 4 3 54 4 3 4 5 34.1 TABLE XXXVI. MoorHs Motions for Months. m Months. January February March {^r April May June July Aug. Sept. Oct. Nov. Dec. 5 Com. ^Bis. ( Com. 39 3 10 25 52.7 20 9 25 50 10 53 15.0 ) 9 32 10 58 49.2 21 9 54 7 11 25 54.7 ) 40 1 11 31 45.6 23 10 22 24 U 58 34.5 I 10 29 12 4 42.1 23 10 50 42 12 31 14.2 I 40 58 12 37 38.6 *4 11 « 1 59 13 3 54.0 _L J 11 27 13 10 35.0 _ TABLE. XXXVIII, 61 Moon's Motions for Hours. H. 14 4 15 1 16 2 17 2 18 1 1 20 21 22 23 34 25 26 37 28 39 1 3 8 3 3 4 3 1 1 3 12 4 5 5 3 1 1 4 16 5 6 7 4 3 1 1 1 5 21 6 8 9 5 3 1 1 6 25 8 9 11 6 3 1 3 7 29 9 11 12 8 3 1 3 8 33 10 13 14 9 3 1 2 9 37 11 14 16 10 4 2 1 3 10 41 13 15 18 11 4 2 3 3 3 11 45 14 17 19 13 5 3 2 3 2 12 49 15 18 21 13 1 1 5 2 2 3 3 2 13 54 16 80 23 14 5 1 2 2 4 3 3 14 58 18 21 25 15 6 2 2 2 4 3 3 15 62 19 23 26 10 6 2 3 3 4 3 2 16 66 20 25 28 17 1 1 7 2 3 2 5 3 2 17 70 21 26 30 18 1 7 2 3 3 5 3 2 18 74 23 28 33 19 2 8 2 3 3 5 3 2 19 78 24 29 33 31 2 8 2 3 3 6 3 3 20 83 25 31 35 32 3 8 2 3 3 6 3 3 21 87 26 32 37 23 3 9 2 4 3 6 4 3 22 91 28 34 39 24. 2 9 2 4 3 6 4 3 23 95 29 35 40 25 2 10 3 4 4 7 4 3 24 99 31 37 42 26 2 10 3 4 4 7 4 3 H H. Sup. of Nod II V V( VII 1 nu IX X XI XII 1 7.9 o 28 1 2 1 1 1 2 1 2 15.9 56 3 3 2 3 3 3 3 3 23.8 1 24 .4 5 4 4 4 1 5 4 4 31.8 1 52 6 7 5 6 6 1 7 6 5 39.7 3 19 7 8 6 7 7 1 9 7 6 47.7 2 47 9 10 7 9 9 1 10 9 7 55.6 3 15 10 12 8 10 10 3 12 10 8 1 3.6 3 43 11 13 9 U 13 3 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 27.4 5 7 16 18 13 15 16 2 19 15 12 1 35.3 5 35 17 30 14 17 18 3 21 17 13 1 43.3 6 2 18 31 15 IS 19 3 33 18 14 1 51.2 6 30 30 23 16 19 31 3 34 19 15 1 59.2 6 58 31 25 IS 21 22 3 36 21 16 2 7.1 7 26 33 26 19 22 24 4 38 32 17 2 15.0 7 54 34 28 20 24 35 4 39 24 18 3 23.0 8 22 36 29 21 25 37 4 31 25 19 2 30.9 8 50 27 31 22 27 28 4 33 27 20 2 38.9 9 18 38 32 24 28 30 4 35 28 31 3 46.8 9 45 30 34 25 29 31 S 37 29 22 2 54.8 10 13 31 36 26 31 33 5 38 31 33 3 2.7 10 41 33 38 27 32 34 5 40 33 24 3 10.6 •1 9 34 39 28 34 1 36 5 43 34 TABLE XXXIX. Moon's Motions for Minutes. 1 8 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 _»_ ^_— — ' — — 3ll 6 14 22 6 7 8 1 1 8 1 1 2 32 6 14 23 6 7 8 1 9 2 2 2 33 6 15 24 7 8 »1 1 9 2 3 3 34 6 15 25 7 8 9 ! 1 9 3 3 3 35 7 10 25 7 8 9 1 10 3 3 3 36 7 16 26 7 8 9 1 10 3 3 3 37| 7 17 27 7 9 10 1 10 3 3 3 38 7 17 27 8 9 10 2 10 2 2 3 1 39 7 18 28 8 9 10 2 11 3 2 3 40 8 18 29 8 9 10 2 11 3 2 3 41 8 19 30 8 10 11 2 11 2 2 3 42 8 19 30 8 10 11 2 11 3 2 3 43 8 19 31 9 10 11 3 13 3 3 3 441 8 20 32 9 10 11 3 12 2 2 3 45 9 20 32 9 10 13 2 12 2 2 3 46 9 21 33 9 11 12 2 12 2 3 3 471 9 21 34 9 11 12 2 13 3 3 3 48: 9 22 35 10 11 12 2 13 3 3 3 49 9 22 35 10 11 13 2 13 3 2 3 50 9 23 36 10 11 13 3 13 3 3 3 51 10 23 37 10 12 13 2 14 3 2 4 52 10 24 38 10 12 13 2 14 3 3 4 53 10 24 38 11 12 14 3 14 3 3 4 54 10 24 39 11 12 i4 3 14 3 3 4 2 55 10 25 40 11 13 14 2 15 3 3 4 8 56 25 40 11 13 14 2 15 3 3 4 8 67 26 41 11 13 15 3 15 3 3 4 3 58 26 42 12 13 15 2 16 3 3 4 8 8 59 27 43 12 14 15 3 16 3 3 4 8 2 1 60 87 43 12 14 15 3 16 1 3 1 3 4 1 8 8 1 TABLE XXXIX. 63 MoorCs Motions for Minutes. Sup. 1 Min. Evec. Anom. Varia. Long. Nod. II V VI VII vin IX XI XII 1 28 / If 32.7 30 32.9 0.1 2 57 1 5.3 1 1 1 5.9 0.3 1 3 1 25 1 38.0 1 31 1 38.8 0.4 1 4 1 53 2 10.6 2 2 2 11.8 0.5 3 5 2 21 2 43.3 2 32 2 44.7 0.7 2 6 2 50 3 16.0 3 3 3 17.6 0.8 3 7 3 18 3 48.6 3 33 3 50.6 0.9 3 8 3 46 4 21.3 4 4 4 23.5 1.1 4 9 4 15 4 54.0 4 34 4 56.5 1.2 4 10 4 43 5 26.6 5 5 5 29.4 1.3 5 11 5 11 5 59.3 5 35 6 2.4 1.5 5 12 5 40 6 31.9 6 6 6 35.3 1.6 6 13 6 8 7 4.6 6 36 7 8.2 1.7 6 14 6 36 7 37.3 7 7 7 41.2 1.9 7 15 7 4 8 9.9 7 37 8 14.1 2.0 7 16 7 33 8 42.6 8 8 8 47.1 2.1 7 17 8 1 9 15.3 8 38 9 20.0 2.3 8 18 8 29 9 47.9 9 9 9 52.9 2.4 8 19 8 58 10 20.6 9 39 10 25.9 2.5 9 20 9 26 10 53.2 10 ,10 10 58.8 2.6 9 1 21 9 54 11 25.9 10 40 11 31.8 3.8 10 22 10 22 11 58.6 11 11 12 4.7 3.9 10 23 10 51 12 31.2 11 41 12 37.6 3.0 11 24 11 19 13 3.9 12 12 13 10.6 3.2 11 25 11 47 13 36.6 12 42 13 43.5 3.3 12 26 \2 16 14 9.2 13 13 14 16.5 3.4 13 27 13 44 14 41.9 13 43 14 49.4 3.6 13 28 13 12 15 14.6 14 13 15 22.3 3.7 13 89 13 40 15 47.2 14 44 15 55.3 3.8 13 .8^. 14 9 16 19.9 15 14 16 38.2 4.0 14 64 TABLE XXXIX. MoorCs Motions for Minutes. 3 4 5 6 7 8 9 10 11 12 13 14 16 16 17 18 19 20 21 22 23 24 25 |26 !27 128 IS9 [so I 2 3 1 1 1 1 2 2 3 2 4 3 4 3 5 2 4 6 2 4 6 2 5 7 2 5 8 2 5 9 2 6 9 3 6 10 3 7 11 3 7 12 3 8 12 3 8 13 4 9 14 4 9 14 4 10 15 4 10 16 4 10 17 5 11 17 5 11 18 5 12 19 5 12 19 5 13 20 6 13 21 6 14 22 6 7 10 11 12 13 14 15 1 1 1 1 1 1 1 16 u 18. TABLE XXXIX. 65 Maori's Motions for Minutes. Sup. -\ Min. Evcc. Anom. Varia. Long. Nod. II V VI VII vm IX XI XII 31 14 37 16 52.5 15 45 17 1.2 PI 4.1 14 32 15 5 17 25.2 16 15 17 34.1 4.2 15 33 15 34 17 57.9 16 46 18 7.1 4.4 15 34 16 2 18 30.6 17 16 18 40.0 4.5 16 35 16 30 19 3.2 17 47 19 12.9 4.7 16 36 16 58 19 35.8 18 17 19 45.9 4.8 17 37 17 27 20 8.5 18 48 20 18.8 4.9 17 38 17 55 20 41.2 19 18 20 51.8 6.0 18 39 18 23 21 13.8 19 49 21 24.7 6.2 18 40 18 52 21 46.5 30 19 21 67.6 6.3 19 41 19 20 22 19.2 20 50 22 30.6 5.4 19 42 19 48 22 51.8 21 20 23 3.5 5.6 20 43 20 16 23 24.5 21 51 23 36.5 5.7 20 44 20 45 23 57.1 23 21 24 9.4 5.8 21 45 21 13 24 29.8 22 53 24 42.3 6.0 21 .. 46 21 41 25 2.5 33 22 35 15.3 6.1 21 1 47 22 10 25 35.1 33 53 35 48.3 6.2 22 1 48 22 38 26 7.8 24 33 36 31.2 6.4 22 49 23 6 26 40.5 34 54 36 54.1 6.5 23 50 23 34 27 13.1 25 24 37 37.0 6.6 23 51 24 3 37 45.8 25 55 38 0.0 6.8 34 52 24 31 28 18.6 26 25 28 33.9 6.9 34 53 24 59 28 51.1 26 56 39 5.9 7.0 35 54 25 28 29 23.8 27 26 39 38.8 7.1 25 2 55 25 66 29 56.4 27 56 30 11.8 7.3 26 1 2 66 26 24 30 29 1 28 27 30 44.7 7.4 26 1 S 57 26 63 31 1.8 28 57 31 17.6 7.5 27 2 2 58 27 21 31 34.4 29 28 31 60 6 7.7 27 2 3 59 37 49 32 7.1 29 58 32 2^.5 7.8 28 2 2 60 28 17 32 39.8 SO 29 B2 56.5 7.9 28 1 1 8 1 1 t 2 66 TABLE XL. Moon's Mntwns for Seconds. Sec. Evec. Anoin. Var. Long. Sec. Evec. Anom. Var. Long. 1 0.5 1 0.5 31 15 16.9 16 17.0 S 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 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 6.4 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 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 31 24.5 33 24.7 16 8 8.7 8 8.8 46 22 25.0 2-J 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 U.O 50 24 27.2 35 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 14.1 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 l.'i.O 5'; 28 32.1 30 32.4 80 14 16.3 15 16.5 60 28 32.7 30 38.9 1 TAB^E XLI. 67 First Equation of Moon's Lora^ j^wde.— ^Argument 1 . Arg. 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500 1550 1600 1650 1700 1750 1800 1850 1900 1950 2000 2050 2100 2150 2200 2250 2300 2350 2400 2450 2500 Diff. for 10 40.0 18.8 57.7 36.6 15.6 54.7 33.9 13.2 52.6 32,3 12.1 52.1 32.4 13.0 53.8 34.9 16.4 58.2 40.3 22.8 5.7 49.0 32.8 17.0 1.6 46.7 4 32.3 4 18.4 4 5.0 3 52.2 3 39.9 3 28.1 3 16.9 3 6.3 2 56.3 2 46.8 2 38.0 2 29.7 2 22.1 2 15.1 2 8.8 2 3.1 1 58.0 1 53.6 1 49.8 1 46.7 1 44.2 1 42.3 1 41 1 1 40.6 1 40.7 4.24 4.22 4.22 4.20 4.18 4.16- 4.14 4.12 4.06 4.04 4.00 3.94 3.88 3.84 3.78 3.70 364 3.58 3.50 3.42 3.34 3.24 3.16 3.08 2.98 2.88 2.78 2.68 2.56 2.46 2.36 2.24 2.12 2.00 1.90 1.76 1.66 1.52 1.40 1.26 1.14 1.02 0.88 0.76 0.62 0.50 0.38 0.24 O.IO Arg. 0.02 2500 2550 2600 2650 2700 2750 2800 2850 2900 2950 3000 3050 3100 3150 3200 3250 3300 3350 3400 3450 3500 3550 3600 3650 3700 3750 3800 3850 3900 3950 4000 4050 4100 4150 4200 4250 4300 4350 4400 4450 4500 4560 4600 4650 4700 4750 4800 4850 4900 4950 5000 DifF. for 10 Arg. 1 40.7 1 41.5 1 42.9 1 45.0 1 47.7 1 51.0 1 55.0 1 59.6 2 4.8 2 10.7 2 17.1 2 24.2 2 31.9 2 40.1 2 48.9 2 58.3 3 8.2 3 18.7 3 29.7 3 41.3 3 53.4 4 5.9 4 19.0 4 32.5 4 46.5 5 0.9 5 15.8 5 31.0 5 46.7 6 2.8 6 19.2 6 36.0 6 53.1 7 10.6 7 28.4 7 46.4 8 4.7 8 23.3 8 42.2 9 1.2 9 20.4 9 39.9 9 59.5 10 19.2 10 39.1 10 59.1 11 19.1 U 39.3 11 59.5 12 19.7 12 40.0 0.16 0.28 0.42 0.54 0.60 0.80 0.92 1.04 1.18 1.28 1.42 1.54 1.64 1.76 1.88 1.98 2.10 2.20 2.32 2.42 2.50 2.62 2.70 2.80 2.88 2.98 3.04 3.14 3.22 3.28 3.36 3.42 3.50 3.56 3.60 3.66 3.72 3.78 3.80 3.84 3.90 3.92 3.94 3.98 4.00 4.00 4.04 4>04 4.04 4.06 Diff. for 10 Arg. 5000 5050 5100 5150 5200 5250 5300 5350 5400 5450 5500 5550 5600 5650 5700 5750 5800 5850 5900 5950 6000 6050 6100 6150 6200 6250 6300 6350 6400 6450 6500 6550 6600 6650 6700 6750 6800 6850 6900 6950 7000 7050 23 7100 23 7150 23 7200 23 7250,23 12 13 13 13 14 14 14 15 15 15 15 16 16 16 17 17 17 18 18 18 19 19 19 19 20 20 20 20 21 21 21 21 21 22 22 22 22 22 22 22 23 7300 23 7350 23 7400 23 7450 23 7500 23 40.0 0.3 20.5 40.7 0.9 20.9 40.9 0.8 20.5 40.1 59.6 18.8 37.8 56.7 15.3 33.6 51.6 9.4 26.9 44.0 0.8 17.2 33.3 49.0 4.2 19.1 33.5 47.5 1.0 14.1 26.6 38.7 50.3 1.3 11.8 21.7 31.1 39.9 48.1 55.8 2.9 9.3 15.2 20.4 25.0 29.0 32.3 35.0 37.1 38.5 39.3 4.06 4.04 4.04 4.04 4.00 4.00 3.98 3.94 3.92 3.90 3.84 3.80 3.78 3 72 3.66 3.60 3.56 3.50 3.42 3.36 3.28 3.22 3.14 3.04 2.98 2.88 2.80 2.70 2.62 2.50 2.42 2.32 2.20 2.10 1.98 1.88 1.7S 1.64 1.54 1.42 1.28 1.18 1.04 0.92 0.80 0.66 0.54 0.42 0.28 0.16 Di VIIIs IXs X" XI" Difl. Diff.,^^ Difi. 0== Diff. 0° Diff. 0= DiffJ 30 00 28 37.0 27 14.1 25 51.2 24 28.3 523 6.6 21 43.0 30 20.5 IS 5S.2 17 36.1 10 14.2 il 14 52 5 12I1331 2 13 12 10 1 10 49.3 9 28.8 16 17 18 19 30 I 21 22 8 8.7 6 49.0 5 29.7 4 10.8 2 52.4 134.4 17.0 23 59 0.1 24 57 43.7 25 56 27.8 86 55 12.6 27 53 58.0 88 52 44.0 89,51 30.7 80150 18.0 83.0 82.9 82.9 82.9 82.7 82.6 82.5 82.3 83.1 81.9 81.7 81.3 81 1 80.8 80.5 80.1 79.7 79.3 78.9 78.4 78.0 77.4 76.9 76.4 75.9 75.2 74.6 74.0 73.3 72.7 50 18.0 49 6.0 47 54.8 46 44.3 45 34.5 44 25.6 43 17.4 42 10.1 41 3.6 39 57.9 38 53.2 37 49.3 36 46.4 35 44.4 34 43.4 33 43.4 32 44.3 31 46.3 30 49.3 39 53.3 38 58.4 38 4.6 37 11.9 36 30.3 25 39.8 24 40.5 3 53.4 23 5.4 22 19.6 31 35.1 30 51.7 72.0 71.3 70.5 69.8 68.9 68 3 67.3 66.5 65.7 64.7 63.9 62.9 63.0 61.0 60.0 59.1 58.0 57.0 56.0 54.9 53.8 52.7 51.6 50.5 49.3 48.1 47.0 45.8 44.5 43.4 0" 20 51.7 30 9.6 19 38.8 18 49.3 18 10.8 17 33.8 16 5S.1 16 33.6 15 50.5 15 18.8 14 48.4 14 19.3 13 51.5 13 35.2 13 0.3 12 36.7 13 14.5 11 53.7 11 34.4 11 16.5 10 59.8 10 44.9 10 31.3 10 19.0 10 8.3 9 59.0 9 51.2 9 44.8 9 39.9 9 36.5 9 34.5 42.1 40.8 39.6 38.4 37.0 35.7 34.5 33.1 31.7 30.4 29.1 27.8 26.3 25.0 23.5 22.3 30.8 19.3 17.9 16.7 14.9 13.7 12 2 10.7 9.3 7.8 6.4 4.9 3.4 3.0 0" 9 34.5 9 34.0 9 35.0 9 37.4 9 41.3 9 46.7 9 53.6 10 1.9 10 11.7 10 22.9 10 35.6 10 49.9 11 .5.5 11 33.6 1141.3 12 1.2 12 33.6 12 45.5 13 ! 13 35.5 14 3.7 14 31.3 15 1.3 15 33 .f, 16 5.3 16 39.4 17 14.9 17 51.7 18 29.9 19 9.4 19 50.3 0.5 1.0 2.4 3.9 5.4 6.9 8.3 9.8 11.3 13.7 14.3 15.6 17.1 1S.6 20,0 21.4 33.9 34.3 35.7 37.2 28.6 29.9 31.4 32.7 34.1 35.5 36.8 38.2 39.5 40.9 19 50.3 30 32.4 31 15.8 22 0.6 22 46.6 33 33.8 24 23.3 35 13.1 36 3.0 36 55.2 37 48.5 38 43.0 39 38.6 30 35.4 31 33 4 32 32.4 33 33.4 34 33.6 35 35.8 36 39.0 37 43.3 38 48.5 39 54.7 41 1.8 43 9.9 43 18.9 44 38.7 45 39.4 46 51.0 48 3.3 49 16.5 42.1 43.4 44.8 46.0 47.2 48.5 49.8 50.9 52.2 53.3 54.5 55.6 56.8 58.0 59.0 60.0 61.2 62.2 63.3 64.3 65.2 66.2 67.1 68.1 69.0 69.8 70.7 71.6 72.3 73.2 49 16.5 50 30.4 5145.1 53 0.6 54 16.7 55 33.5 56 50.9 58 9.0 59 2TVr 47.0 2 6.8 3 27.1 4 48.0 6 9.3 731.1 8 53.3 10 15.9 11 38.9 13 2.3 14 26.0 15 49.9 17 14.2 18 38.6 20 3.3 31 28.2 22 53.3 24 18.4 35 43.7 37 9.1 38 34.5 30 0.0 73.9 74.7 175.5 76.1 76.8 77.4 78.1 78.7 79.3 79.8 80.3 80.9 81.3 81.8 82.2 83.6 83.0 83.4 83.7 83.9 84.3 84.4 84.7 184 9 185.1 8.5.1 85.3 85.4 85.4 85.5 lo 74 TABLE LI. Equation of Moon's Centre. Argument. Anomaly corrected. 0« Diff forlO 0.0 332.6 7 .5.2 10 37.8 14 10.3 17 42.7 21 15.0 2447.3 2819.4 3151.2 35 23.0 10° Diff forlO 30 1 30 2 30 3 30 4 30 5 30 38 54.5 6 42 25.8 30 45.56.9 7 49 27.7 301,52 58.2 8 0!56 28.5 30 59 .58.4 9 70.9 70 9 70 9 70.8 70 8 70.8 70.8 70.7 70.6 70.6 70.5 70.4 70.4 70.3 70.2 70.1 30 10 30 U 3 28.0 6 57.2 10 26.0 13 54.5 17 22.5 30120 50.1 12 0|2417.3 30|2744.0 13 31 10.2 30 34 35.8 14 0138 1.0 3014125.6 15 4449.6 8° 70.0 69.9 69.7 69.6 69.5 69.3 69.2 69.1 68.9 68.7 68.5 68.4 |68.2 68.0 20 57.9 23 55.6 26 .52.2 29 47.7 3242.0 35 35.2 38 27.1 41 18.0 44 7.6 46 56.0 49 43.2 52 29.1 55 13.8 57 57.2 39.3 320.1 5 59.7 8 37.9 11 14.8 13 .50.3 16 24.5 18 57.3 2128.8 23 .58.8 26 27.5 28 .54.7 59.2 58.9 58.5 58.1 57.7 57.3 .57.0 56.5 56.1 55.7 55.3 II» 12° Diff forlO III" 13° 31 20.5 3344.9 36 7.9 38 29.4 4049.3 11° 54.9 54.5 54.0 .53.6 53.2 .52.7 52.3 51.8 51.4 50.9 50.5 50.0 49.6 49.1 48.6 48.1 ,47.7 47.2 46.6 3843.6 40 14.0 41 42.7 43 9.6 4434.9 4558.4 47 20.2 48 40.3 49 58.7 51 15.3 52 30.2 5343.3 .5454.7 56 4.4 57 12.3 5818.5 59 22.9 25.6 126 5 2 25.7 3 23.0 418.7 512.5 6 4.6 6 54.9 743.5 30.1 29.6 290 28.4 27.8 27.3 26.7 26.1 25.5 25.0 TVs Diff I forlO 12° 24.4 23.8 23.2 22.6 22.1 21.5 20.9 20.3 19.7 19.1 18.6 17.9 J 7.4 16.8 16.2 15.6 8 30.3 9 15.4 9 58.8 1040.1 1119.9 13° 1735.2 1720 9 17 4.8 16 47.1 16 27.6 16 6.5 15 43.7 15 19.2 14 53.1 1425.2 13 55.8 13 24.7 1251.9 12 17.4 1141.4 11 3.7 10 24.3 9 43.4 9 0.8 8 16.6 7 30.8 6 43.4 5 54.4 5 3.9 411.7 318.0 15.0 14.4 13.8 13.3 2 22.7 125.8 I 27.4 |59 27.4 158 25.9 12° 4.8! 5.4 5.9 6.5 7.0 7.6 8.2 8.7 9.3 9.8 10.4 10.9 11.5 12.0 12.6 13.1 13.6 14.2 14.7 15.3 15.8 16.3 16.8 17.4 17.9 18.4 19.0 19.5 20.0 20.5 16 20.8 1435.3 12 48.5 11 0.4 911.1 7 20.5 5 28.7 3 35.6 141.3 Diff forlO 9° V» Diff for 10 59 45.8 57 49.1 5551.1 53 52.0 5151.7 49 50.3 47 47.6 35.2 35.6 36.0 36.4 36.9 37.3 37.7 38.1 38.5 38.9 39.3 39.7 40.1 40.5 40.9 58 28.9 55 43.8 52 .58.0 .5011.6 47 24.5 44 36.8 4148.5' 55.0 55.3 55.5 55.7 55.9 |56.1 56.3 38 59.5i.g5 36 10.0Lb7 33 19.8'^''-^ 30 29.1 45 43.8 43 38.9 41 32.8 39 25.6 37 17.3 35 7.9 32 57.4 30 45.8 28 33.1 26 19.4 24 4.6 21 48.8 1931.9 11714.1 11455.2 11° 41.3 41,7 42.0 42.4 42.S 43.1 43.5 43.9 44.2 44.6 44.9 45.3 45.6 45.9 46.3 27 37.8 24 45.9 21 53.5 19 0.6 16 7.1 13 13.1 10 18.6 7 23.6 4 28.1 132.2 58 35.8 55 38.9 52 41.7 49 43.9 46 45.8 43 47.3 40 48.4 37 49.1J 34 49.51 l31 49 4 8° 56.9 57.1 57.3 I 57.5 .57.0 57.8 58.0 58.2 58.3 58.5 58.6 58.8 .59.0 .59.1 59.3 59.4 59.5 59.6 i59.3 59 9 60.0 TABLE LI. Equation of Moon's Centre. Argument. Anomaly corrected. 75 vis VII« VIIIs IX» x» XI* DifF forlO 30 1 30 2 30 3 30 4 30 5 30 6 30 7 30 8 30 9 30 10 0.0 56 54.6 53 49.2 5043.9 47 38.6 4433.4 4128.1 38 23.0 35 18.0 32 13.0 29 8.1 26 3.4 22 58.8 19 54.3 16 50.0 13 45.8 1041.9 738.0 434.4 131.0 58 27.8 30 5524.9 II 52 22.2 30 13 30 13 30 14 30 15 4919.7 4617.5 43 15.6 40 14.0 3712.6 3411.6 31 10.9 28 10.6 61.8 61.8 61.8 61.8 61.7 61.8 61 7 61.7 61.7 61.6 61.6 61.5 61.5 61.4 61.4 61.3 61.3 61.2 61.1 61.1 61.0 60.9 60.8 60.7 60.6 60.5 60.5 60.3 60.2 60.1 Diff forlO 1° Diff for 10 0° Diff forlO 1° 131.1 58 46.7 56 3.0 53 20.0 50 37.7 47 56.2 45 15.4 42 35.3 39 56.0 37 17.4 34 39.6 32 2.7 29 26.5 26 51.1 2416 6 21 42.9 19 10.0 16 33.0 14 6.9 1136.6 9 7.3 6 38.9 411.3 144.7 59 18.9 56 54.2 5430.4 52 7.5 49 45.6 4724.7 45 4.8 54.8 54 6 54 3 54.1 53 8 53.6 53.4 53.1 .52.9 52.6 .52.3 52.1 51.8 51.5 51.2 51.0 50.7 50.4 50.1 49.8 49.5 49.2 48.9 48.6 48.2 47.9 47.6 47.3 47.0 46.6 43 39.2 4155.0 40 12.0 38 30.5 36 50.3 3511.3 33 33.7 31 57.5 30 22.6 28 49.0 2716.8 2546.1 2416.7 22 48.7 2122.1 19 56.9 18 33.1 1710.8 15 49.8 1430.4 13 12.5 1155.9 1040.9 9 27.3 815.2 7 4.6 5 55.4 447.8 341.7 237.1 134.1 1° 34.7 34.3 33.8 J3.4 33.0 32.5 32.1 31.6 31.2 30.7 30.2 29.8 29 3 28.9 28.4 27.9 27.4 27.0 266 26.0 25.5 25.0 24.5 24.0 23.5 23.1 22.5 22.0 21.5 21.0 42 24.8 4212.1 42 1.2 41 52.0 41 44.4 41 38.7 41 34.6 41 32.2 4131.6 4132 7 41 35.6 4140.1 4146.4 4154.5 42 4.3 42 15.9 42 29.2 42 44.2 43 1 43 19.6 43 39.9 44 2.0 4425.9 4451.5 4518.8 45 48.0 46 18.9 46 51.5 47 26.0 48 2.2 48 40.1 4.2 3.6 3.1 2.5 1.9 1.4 0.8 0.2 0.4 1.0 1.5 2.1 2.7 3.3 3.9 4.4 .5.0 5.6 6.2 6.8 7.4 8.0 8.5 9.1 9.7 10.3 10.9 11.5 12.1 12.6 Diff" for 10 2116.4 22 48.5 24 22.2 25 57.7 27 34.8 29 13.7 30 54.2 32 36.3 34 20.2 36 5.6 37 52.8 39 41.5 4132.0 43 24.0 45 17.7 4712.9 49 51 8.3 .53 8.4 .5510.1 57 13.3 5918.2 124:5 3 32.4 541.9 752.9 10 5,5 1219.5 1435.1 1652.1 1910.7 30.7 31.2 31.8 32.4 33.0 33.5 34.0 34.6 35.1 35.7 36.2 36.8 37.3 37.9 38.4 39.0 39.5 40.0 40.6 41.1 41.6 42.1 42.6 43.2 43.7 44.2 44,7 45.2 45.7 46.2 3° tU.JO 57 OJ 0i5.8 321.8 6 28.8 9 38.8 12 45.7 15 55.5 19 6.2 22 17.8 25 30.3 28 43,7 31 57,8 3512.9 38 28.7 4145.2 45 2.6 48 20.7 5139,6 .5459.1 .58 19.3 140.3 5 1.9 8 24.1 1146.9 15104 &9.6 60.0 60.3 60.7 61.0 61.3 61.7 62,0 62,3 62.7 63,0 63.3 63,6 63.9 64.2 64.5 64.7 65.0 66,3 65.6 65.8 66.0 66.3 66.5 66.7 67.2 67.4 67.6 67.8 76 TABLE Ll. Equation of Moon's Centre Argument. Anomaly corrected. lis Ills IVs V» |8° Diff forlO U° Diff! forlO, 13° Diff forlO 12° Diff ,,o for 10/1 Diff forlO 8° Diff for 10 5458. 58 19 140. 5 0. 8 20. 1139. 1457. 18 14.1 37 42.: 40 53.1 44 4.1 4714.; 50 23.; 66.5 66.3 66.0 65.8 65.5 6.5.3 65.0 64.7 64.5 64.2 60.7 60.3 60.0 59.6 4049.3 43 7.9 45 24.9 4740.5 49 54.5 52 7.1 5418.1 56 27.6 58 35.5 041.8 246.7 449.9 651.6 851.7 10 50.2 12 47.1 1442 3 16 36.0 18 28.0 2018.5 22 7.2 46.2 45.7 45.2 44.7 44.2 43.7 43.2 42.6 42.1 41.6 41.1 40.6 40.0 39.5 39.0 38.4 37.9 37.3 36.8 36.2 35.7 35.1 34.6 34.1 23 54.4 25 39.8 27 23.7 29 5.8 30 46.3i"'' 33.0 32 25 2' 34 2 3^f* 35 37.8^ ■« 37 11.5 ^i* 38 43.6 |l8° 30.7 11 19.9 1157.8 12 34.0 13 8.5 1341.1 1412.0 1441.2 15 8.5 15 34.1 15 5S.0 16 20.1 16 40.4 16.58.9 1715.8 1730.8 1744.1 17 55.7 18 5.5 18 13.6 18 19.9 18 24.4 18 27.3 18 28.4 18 27.8 18 2.5.4 18 21.3 18 1.5.6 18 8.0 1758.8 1747.9 17 35.2 1 13° 12.6 12.1 11.5 10.9 10.3 9.7 9.1 8.5 8.0 7.4 6.8 6.2 5.6 5.0 44 3.9 33 2.7 2.1 1.5 1.0 0.4 0.2 0.8 1.4 1.9 2.5 3.1 3.6 4.2 58 25.9 57 22.9 5618.3 55 12.2 54 4.6 52 55.4 5144.8 .50 32.7 49 19.1 48 4.1 46 47.5 45 29.6 4410.2 42 49.2 4126.9 40 3.1 38 37.9 3711.3 35 43 3 ;3413.9 32 43.2 31 11.0 29 37.4 28 2.5 26 26.3 2448.7 23 9.7 21 29.5 19 48.0 18 5.0 16 20.8 12° 21.0 21.5 22.0 22.5 23.1 23.5 24') 24.5 25.0 25.5 26.0 26.5 27.0 27.4 27.9 28.4 28.9 29.3 29.8 30.2 30.7 31.2 31.6 32.1 32.5 33.0 33.4 33.8 34.3 34.7 1455.2 12 35.3 10 14.4 752.5 5 29.6 3 5.8 041.1 58 15.3 5548.7 53 21.1 50 52.7 48 23.4 45 53.1 43 22.0 40 50.0 38 17.1 35 43.4 33 8.9 30 33.5 2757.3 25 20.4 22 42.6 20 4.0 17 24.7 1444.7 12 3.8 46.6 47.0 47.3 47.7 47.9 48.3 48.6 48.9 49.2 49.5 49.8 50.1 50.4 .50.7 51.0 51.2 51.5 51.8 52.1 52.3 52.6 .52.9 53.1 3.3 53.6 53.8 9 22.3 6 40.0 •^*- 1 3 57.0 54-3 1 13.3 '54 6 58 28.9 ^*-^ 9° 31 49.4 28 49.1 2548.4 22 47.4 19 46.0 16 44.4 13 42.5 10 40.3 737.8 435.1 132.2 .58 29.0 55 25.6 52 22.0 49 18.1 46 14.2 43 10.0 40 5.7 37 1.2 33 56.6 30 51.9 2747.0 2442.0 21 37.0 18 31.8 15 26.6 1221.4 9 16.1 610.8 3 54 0.0 7° 60.1 60.2 60.3 60.5 60.5 60.6 60.7 60.8 60.9 61.0 61.1 61.1 61.2 61.3 61.3 61.4 61.4 61.5, 61.5 61.6; 61.6 61.7 61.7 61.7 61.7 61.7 61.8 61.8 61.8 61.8 TABLE LI 7.7 Equation of Moon^s Centre. Argument. Anomaly corrected. I ' VI» 5° forlO 15 28 10. 30 25 10. 16 2210. 30 17 30 18 30 19 30 19 11.6 16 12. 1314. 1016 718, 421 124 20 0,58 27.8 30 55 31 21 52 36 30 49 41. 22 46 46. 30 43 52. 23 o'40 59. 30 38 6. 24 035 14. 30|32 22 25 29 30. 30 26 30 27 (I 30 1 30 39 30 30 Diff„o VII» 2° 26 40. 23 50 21 0. 18 11 15 23. 12 35. 9 48. 7 2. 416. 131. 60.0 59.9 69.8 596 59.5 59.4 59.3 59.1 59.0 58.8 58.6 53.5 58.3 58.2 !58.0 57.8 57.6 57.5 57.3 57.1 1 56.9 56.7 56.5 56.3 56.1 55.9 55.7 55.5 55.3 55.0 Diff! o forlO,! 45 4.8 42 45.9 40 28.1 38 11.2 35 55.4 33 40.6 31 26.9 29 14 27 2.6 24 53.1 22 42 7 20 34.4 18 27.2 1621.1 1416.2 12 12.4 9.7 10 8 6 4 210.9 14.2 8.0 8.9 VIIIs Diff forlO 58 18.7 50 24.4 &4 31.3 52 39.5 5048.9 48 59.6 4711.5 45 24.7 43 39.2 1° 46.3 45.9 45.6 45.3 44.9 44.6 4t.2 43.9 43.5 43.1 42.8 42.4 42.0 41.6 41.3 40.9 40.5 40.1 39.7 39.3 138.9 38.5 38.1 37.7 37.3 36.9 36.4 36.0 35.6 35.2 134.1 32.6 59 32.6 58 34.2 57 37.3 50 42.0 55 48.3 54 56.1 54 5.6 53 16.(> 52 29.2 51 43.4 50 59.2 50 16.fi 49 35.7 48 .56.3 48 18.6 4743.6 47 8.1 46 35.3 46 4.2 45 34.8 45 6.9 4440.8 44 16.3 43 53.5 43 32.4 43 12.9 42 55.2 42 39.1 42 24.8 0° 20.5 20.0 19.5 19.0 18.4 17.9 17.4 16.8 163 1.5.8 15.3 14.7 14.2 13.6 13.1 12.6 12.0 11 5 10 9 10.4 9.8 9.3 8.7 8.2 7.6 7.0 6.5 5.9 5.4 4.8 IX» x» Diff I forlO 48 40.1 49 19.9 50 1.4 50 44.6 5129.7 53 16 5 53 5.1 ,53 .55.4 5447.5 .55 41.3 56 37.0 57 34.3 58 33.5 59 34^4 37.] 1 41.5 2 47.7 3 55.6 5 5.3 6 16.7 7 29.8 8 44.7 10 1.3: 11 19.7| 12 39.8 14 1.6 1525.1 16 50.4 18 17.3 19 46.0 21 16.4 1° 13.3 13.8 14.4 15.0 15,6 16.2 16.8 17.4 17.9 18.6 19.1 19.7 20.3 20.9 21.5 22.1 22.6 23.2 23.8 24.4 25.0 25.5 126.1 !26.7 27.3 27.8 28.4 29.0 29.6 30 1 Difi for 10 19 10.7 21 30.6 23.52.1 26 15.1 28 39.5 31 5.3 33 32.5 36 1.2 38 31.2 41 2.7 43 35.5 46 9.7 48 45.2 5122.1 54 0.3 56 39.9 59 20.7 2 2.8 446.2 730.9 1016.8 13 4.0 15 52.4. 18 42.0. 2132.9! 2424.8 '27 18.0 130 12.3 46.6 47.2 47.7 48.1 48.6 49.1 49.6 .50.0 •50.5 50.9 51.4 51.8 52.3 52.7 53:2 53.6 54.0 54.5 54.9 55.3 55.7 .56.1 56 5 57.0 57.3 157.7 58.1 33 136 7.8 4.4 58.5 39 2.1 .58.9 59.2 XI» Diffi forlO 15 10.4 8 34.4 21 59.0 25 24.2 28 49.8 32 16.0 35 42.7 39 9.9 42 37.5 46 5.5 49 34.0 53 2.8 56 31.9 1.6 331.5 7 1 10 32.5 14 3.1 1734.2 21 5.5 2437.0 28 8.8 3140 7 35 12.8 38 45.1 4217.3 4549.7 49 22.2 52 54.8 50 27^ 0.6 68.0 68.3 68.4 68.5 68.7 68.9 69.1 69.2 69.3 69.5 69.6 69.7 69.9 70.0 70.1 70.3 70.3 70.4 70.4 70.5 70.6 70.6 70.7 70.7 70.8 70.8 70.8 70.9 70.9 70.9 78 TABLE LIL Variation. Argument. Variation, corrected. 0« 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 3D 21 23 23 34 25 26' 37 28 29 30 38 39 13 40 26 41 39 42 52 44 4. 45 16 46 27. 47 38. 48 48. 49 57. 51 5. 52 12, 53 18. 54 23 55 27 56 29 57 30 58 30 59 28 24 1 IS. 2 11. 3 2 3 51 4 38. 5 23 6 6 6 46 7 25. S 1 1°"" DifE 73.3 73.3 73.0 72.7 72.3 71.9 71.3 70.7 69.9 69.1 2 67.2 66.1 64.9 63.7 62.3 60.9 59.4 57.9 56.2 54.5 52.7 50.9 48.9 47,0 44.9 42.9 40,7 38,5 36.3 8 1.5 8 35.5 9 7.2 9 36.5 10 3,4 10 27 9 10 49.9 11 9.4 11 26.4 11 40.9 11 52.9 12 2.2 12 9.0 12 13.2 12 14.8 12 13.9 12 10.3 12 4.2 11 55.5 11 44.2 11 30.5 11 14.1 10 55.3 10 34,0 10 10.2 9 44.0 9 15.4 8 44.5 8 11.2 7 35.7 6 57.9 Diff. 34.0 31.7 29.3 26.9 24.5 22.0 19.5 17.0 14.5 12.0 9.3 6.8 4.2 1.6 0.9 3.6 6.1 8.7 11.3 13.7 16.4 18.8 21.3 23.8 26.2 28.0 30.9 53.3 35.5 37.8 II> 1° 6 57.9 6 18.0, 5 35.9 4 51.7 4 5,5 3 17.3, 2 27.2! 1 35.3! 41,6 Ills Diff. 59 46.1 58 49.0 57 50.2 56 50,0 55 48,3 ■54 45.2 53 40.9 52 35.3 51 28.5 50 20.7 49 11.9 48 2.2 46 51.7 45 40.5 44 28.6 43 IB.l 42 3.2 39.9 42.1 44.2 46.2 48.2 50.1 51.9 53.7 55.5 57.1 58.8 60.2 61.7 63.1 64.3 65.6 66.8 67.8 68.8 69.7 70.5 71.2 71.9 72.5 72.9 I40 49.9U,, 39 36.2 38 22.4I 37 8.4' 35 54.4 0° 0" Diff. [73.3 3.7 1 74.0 74.0 35 54.4 34 40.4 33 26.6 32 13.0 30 59.6 29 46.7 28 34.3 27 22.4 26 11.2 25 0.7 23 51.1 22 42.3 21 34.5 20 27.9 19 22.3 18 18.0 17 15.0 16 13.4 15 13.2 14 14.6 13 17.5 12 22.2 11 28.5 10 36.7 9 46.8 8 58.S 8 12.7 7 28,7 6 46.8 6 7.1 5 29,5 00 iv» 0" 74.0 73.8 73.6 73,4 72,9 72.4! 71.9 71.2 70.5 68.8 67.8 66,6 65.6 64,3 63.0 61.6 60.2 58.6 57.1 55.3 53.7 51.8 49.9 48.0 46.1 44.0 41.9 39 7 .37.6 5 29.5 4 54,2 4 21.3 3 50.6 3 32,3 2 56.5 3 33.1 3 12,1 153.7 1 37,8 1 24.5 1 13.7 1 5.5 1 0.0 57,0 ;0 56.7 59.0 jl 3.9 !l 11.5 121.6 134.4 1 49.8 2 7.8 2 28.3 2 51.4 3 16.9 v» Diff. 0° 35.3 32.9 30,7 28.3 25.8 23.4 21.0 18.4 15,9 13.3 10.8 8.3 5.5 3.0 0.3 3.3 4.9 7.6 10.1 12.8 15,4 18.0 205 23,1 2.').5 28,1 30.6 32,9 3 45.0 4 15.6 4 48,5„. . 5 23.9',:^° * 6 1.6"^'-' 6 1.6 641.6 7 23.9 8 8.4 8 55 9 43.7 10 34.5 1 1 27.3 12 22.0 13 18.6 14 16.9 15 17.0 16 18.7 17 22.0 18 26.9 19 33.1 20 40.7 21 49.6 22 59.6 24 10.8 25 22.9 26 35.9 27 49.8 29 4.5 30 19.7 31 35.6 33 51.9 34 8.6 35 25.6 36 42.7 38 0.0 0° Diff. 40.0 42.3 44.5 46.6 48.7 50 8 52.8 54.7 56.6 58.3 60.1 61.7 63.3 64.9 66.2 67.6 68.9. 70.0: 71.2 72.1 73.0 73.9 74.7 75,2 75,9 76,3 76.7 77.0 77.1 77,3 TABLE LIL Variation. Argument. Variation corrected. VI» VII* VIII» IX» X' XI« 0° Diff. 1° Diff. 1° Diff.00 Diff. 0° Diff. Diff. i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 38 0. 39 17. 40 34. 41 .51. 43 8. 4424. 4540, 46 55 48 10 49 24 50 37 51 49, 53 5410 5519 56 26 57 33 .58 38 5941 043 143 3 41. £38. 432. 525. 616. 7 5 7.51 836 918 9 58 77.3 77.1 77.0 76.7 76.3 75.9 75.2 74.7 3.9 73.0 72.1 71.2 70.0 68.9 67.6 66.2 64.9 63.3 61.7 60.1 58.3 56.6 54.7 f.2.8 50.8 48.7 46.6 44.5 42.3 40.0 9 58.4 1036.1 1111.5 1144.4 12 15.0 1243.1 13 8.6 13 3J.7 13 52.2 14 10.2 1425.6 1438.4 1448.5 1456.1 15 1.0 15 3.3 3.0 0.0 1454.5 1446.3 1435.5 1422.2 14 6.3 13 47.9 1326.9 13 3.5 12 37.7 12 9.4 1138.7 11 0.8 10 30.5 1°~ 37.7 35.4 32.9 30.6 28.1 25.5 23.1 20.5 18.0 15.4 12.8 10.1 7.6 4.9 2.3 0.3 3.0 5.5 8.2 10.8 13.3 15.9 18.4 21.0 23.4 25.8 28.3 30.7 32.9 35.3 10.30.5 9 52.9! 913.2 831.3 747.3 7 1.2 613.2 523.3 431.5 3 37.8 242.5 145.4 46.8 5946.6 58 45.0 5742.0 56 37.7 55 32.1 54 25.5 53 17.7 52 8.9 50 59.3 149 48.8 48 37.6 47 25.7 46 13.3 45 0.4 4347.0 ]42 33.4 i41 19.6 40 5.6 QO 37.6 39.7 41.9 44.0 46.1 48.0 49.9 51.8 53.7 56.3 57.1 58.6 60.2 61.6 63.0 64.3 65.6 66.6 67.8 68.8 69.6 70.5 71.2 71.9 72.4 72.9 73.4 73.6 73.8 74.0 40 5.6 38 51.6 3737.6 36 23.8 3510.1 33 56.8 3243.9 3131.4 30 19.5 29 8.3 2757.8 26 48.1 25 39.3 2431.5 23 24.7 22 19.1 21 14.8 2011.7 19 10.0 18 9.8 1711.0 16 13.9 15 18.4 1424.7 13 32.8 1242.7 1154.5 11 8.3 1024.1 942.0 9 2.1 Oo 74.0 74.0 73.8 73.7 73.3 72.9 72.5 71.9 71.2 70.5 69.7 68.8 67.8 66.8 65.6 64.3 63.1 61.7 60.2 58.8 57.1 55.5 53.7 51.9 50.1 48.2 46.2 44.2 42.1 39.9 9 2.1 824.3 748.8 715.5 644.6 616.0 5 49.8 5 26.0 5 4.7 445.9 4 29.5 415.8 4 4.5 3 55.8 3 49.7 3 46.1 3 45.2 3 46.8 3 51.0 3 57.8 4 7.1 419 1 433.6 4 50.6 5 10.1 5 32.1 5 56.6 6 23.5 6 52.8 724.5 758.5 0° 37.8 35.5 33.3 30.9 28.6 26.2 23.8 21.3 18.8 16.4 13.7 11.3 8.7 6.1 3.6 0.9 1.6 4.2 6.8 9.3 12.0 14.5 17.0 19.5 22.0 24.5 26.9 29.3 31.7 34.0 7 .58.5 8 34.S 913.3 9 54.0 1036.9 1121.8 12 8.8 12 57.7 1348.6 1441.3 1535.8 16 32.0 1729.9 18 29.3 19 30.2 20 32.5 21 36.2 22 41.1 23 47.2 2454.4 26 2.0 2711.7 28 21.6 29 32.3 30 43.6 31 55.5 33 7.8 34 20.5 35 33.5 36 46.7 0.0 36.3 38.5 40.7 42 9 44.9 47.0 48.9 50.9 52.7 54.5 56.2 57.9 59.4 i 60.9 62.3 63.7 64.9 66.1 67.2 68.2 69.1 69.9 70.7 71.3 71.9 72.3 72.7 73.0" 73.2 73.3 80 'lABLE LIII. Reduction. Argument. Supplomeiit of Node+Mooii's Orbit Longitude. OsVIs Diff. IsVIIsDiiE IlsVnis Diff. lIIsIXs 7 0.0 6 45.6 6 31.2 6 16.9 6 2.6 5 ASA ; 5 34.3 r5 20.3 1 5 6.4 52.6 39.0 25.6 12.3 59.3 3 46.5 3 33.9 3 21.6 3 9.5 2 57.7 2 46.2 2 35.0 24.2 13.7 3.5 53.7 44.2 35.2 26.0 18 3 10.4 14.4 14.4 14.3 14.3 14.2 14.1 14.0 13.9 13.8 13.6 13.4 13.3 13.0 12.8 12.6 12.3| 12.1 \ 11.8 ; 11.5; 11.2; 3.0 560 49.5 43.4 37.8 32.7 28.2 23.9 20.0 16.8 4) 14.1 30|l 3 0| 10.8 10.5 10.2 9.8 9.5 9.0 8.7 82 7.9 7.4 11.8 10.1 8.8 8.1 7.8 8.1 8.8 10.1 11.8 14.1 16.8 20.0 23.9 28.2 32.7 37.8 43.4 49.6 56.0 3.0 7.0 6.5 6.1 5.0 5.1 4.5 4.3 3.9 3.2 2 7 2.3 1.7 1.3 0.7 0.3 I0.3 0.7 I 1.7 2.3 2.7 3.2 3.9 4.3 4.5 5.1 5 6 6.1 6.5 7.0 1 3.0 1 10.4 1 18.3 1 26.5 1 35.2 1 44.2 1 53.7 2 3.5 2 13.7 2 24.2 2 35 2 46.2 2 57.7 3 9.5 3 21.6 3 33.9 46.5 59.3 12.3 25.6 39.0 52 6 6.4 20.3 34.3 48.4 6 2.6 6 16.9 6 31.2 6 45.6 7 0.0 7.4 7.9 8.2 8.7 9.0 9.5 9.3 10.2 10.5 lO.S 11.2 11.5 11.8 12.1 12.3 12.6 12.8 13.0 '13.3 13.4 jl3.6l 13.8 13.9 14 14.1 14.2 14.3 14.3 14.4 !l4.4' Diff IVsXs 7 0.0 7 14.4 7 28.8 7 43.1 , 7 57.4 8 11.6 8 25.7 8 39.7 8 53.6 9 7.4 9 21.0 9 34.4 9 47.7 10 0.7 10 13.5 10 26.1 10 38.4 10 50.5 11 2.3 11 13.8 11 25.0 11 35.8 11 46.3 11 56.5 12 6.3 12 15.8 12 24.8 12 33.5 12 41.7 12 49.6 12 57.0 14.4 14.4 14.3 14.3 14 2 14.11 14.0 13.9 13.8 13.6 13.4 13.3 13.0 12.8 12.6 12.3 12.1 11.8 11.5 11.2 10.8 10.5 10.2 9.8 9.5 9.0 8.7 8.2 7.9 7.4 Diff. VsXIs 57.0 4.0 10.5 16.6 22.2! 27.3 31.8 36.1 4o.o; 43.2 45.9 48.2 49.9 51.2 51.9 .52.2 51.9 51.2 49.9 48 45.9 43.2 40.0 36.1 31.8 27.3 22.2 16.6 10.5 4.0 57.0 y.o 6.5 6.1 5.6 5.1 4.5 4.3 3.9 3.2 2.7 2.3 1.7 1.3 0.7 0.3 0.3 0.7 1.3 1.7 2.3 2.7 3.2 3.9 4.3 4.5 5.1 .5.6 6.1 6.5 7.0 12 57.0 12 49.6 2 41.7 12 33.5 12 24.8 12 15.8 6.3 56.5 46.3 35.8 25.0 Diff. 11 13.8 11 2.3 10 50.5 10 38.4 10 26.1 10 13.5 10 0.7 9 47.7 9 34.4 9 21.0 9 7.4 8 53.6 8 39.7 8 25.7 8 11.6 57.4 43.1 28.8 14 41 0| 7.4 7.9 %X 8.7 9.0 9.0 9.8 10.2 10.5 10.8 11.2 11.5 11.8 12.1 12.3 12.6 12.8 13.0 13.3 13.4 13.6 13.8 13.9 14.0 14.1 14.2 14.3 14.3 14 4 14 4 TABLE LIV. Lunar Nutation m LonHiude. Argument. Supplement of the Node. 0. h II» IITs IV« Vi + + + + + + 0.0 8.5 14.8 17.3 15.2 8.8 30 3 0.6 9.0 15.1 17.2 14.9 8.1 28 4 1.2 9.4 15.4 17.2 14.5 7.7 26 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 17.1 13.5 6.1 20 12 3.5 11.4 16.3 17.0 13.0 5.4 18 14 4.1 11.8 16.5 16.9 12.6 4.8 16 16 4.6 12.2 16.7 16.7 12.2 4.3 14 18 5.2 12.6 16.8 16.5 11.8 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 XlB X. IX. VIII. VII. Vis TABLE LV. 81 MomCs Distance from the North Pole of the Ecliptic Argument. Supplement of Node+Moon'a Orbit Longitude. 84° III* IV» 85° Ditr. for 10 87° V» VI» Diff. o for 10 °^ Diff. for 10 92° vn» I Diff. Ifor 10 vin» 94° 30 ! 30 ■i SO 3 (» 30 4 30 5 30 6 30 7 30 8 30 9 30 10 30 11 30 12 30 13 30 14 30 15 3916.0 39 16,7 39 18.8 39 22.4 39 27.3 39 33.7 39 41.5 39 50.6 40 1.2 40 13.2 40 26.7 40 41.5 40 57.7 41 15.4 4134.4 41 54.8 4216.7 42 39.9 43 4.6 43 30.6 43 58.1 44 26.9 44.57.1 4528.8 46 1.8 46 36.1 4711.9 4749.0 48 27.5 49 7.4 49 48.7 84° II* 20 42.7 22 4.2 23 27.0 24 51.0 26 16.2 27 42.6 29 10.1 30 38.9 32 8.8 33 39.9 3512.2 36 45.6 38 20.1 39 55.8 41 32.7 43 10.6 44 49.7 46 29.9 4811.2 49 53.5 51 37.0 .53 21.6 55 7.1 56 53.8 58 41.0 30.3 2 20.1 411.0 6 2.9 755.7 9 49.6 27.2 27.6 28.0 28.4 28.8 29.2 29.r. 30.0 30.4 30.8 31.1 31.5 31.9 32.3 32.6 33.0 33.4 33.8 34.1 34.5 34.9 35.2 35.7 35.9 36.2 36.6 37.0 37.3 37.6 38.0 86° I» 1346.6 16 6.9 18 27.8 20 49.5 23 11.8 25 34.8 27 .58.5 30 22.8 32 47.7 3513.2 37 39.3 40 6.1 42 33.4 45 1.2 47 29.6 49 58.6 5228.1 54 58.2 5728.7 59 59.8 2 31.3 5 3.3 7 35.8 10 8.8 1242.1 1516.0 17 50.2 20 24.9 22 59.9 25 35.3 28 11.1 0» 46.8 47.0 47.2 47.4 47.7 47.9 48.1 48.3 48.5 48.7 48.9 49.1 49.3 49.5 49.7 49.8 50.0 50.2 50 4 50.5 50.7 50.8 51.0 51.1 51.3 51.4 51.6 51.7 51.8 51.9 48 0.0 .5041.4 53 22.9 56 4.3 58 45.7 127.0 4 8.3 649.5 9 30.6 1211.6 1452.5 17 33 3 20 14.0 22 54.4 25 34.8 28 14.9 30 54.9 33 34.7 36 14.3 38 53.7 41 32.8 4411.7 46 50.4 49 28.7 52 6.8 ,5444.6 15722.1 59 59.3 2 36.2 512.7 748.9 91° XI» 53.8 53.8 53.8 53.8 53.8 53.8 53.7 53.7 .53.7 53.6 53.6 .53.6 53.5 53.5 .53.4 53.3 53.3 53,2 53.1 53.0 53.0 52.9 8 5-.i.7 52.6 52.5 52.4 .52.3 52.2 52.1 2213.4l4„g 24 33- 1146 4 26 53.2! 29 10.2 31 27.5 3344.2 36 0.2 38 15.3 40 29.7 42 43.3 44 56.2 47 8.1 49 19.4 51 29.7 53 39.3 55 48.0 57 55.8 2.8 2 8.9 414.1 618.4 821.8 10 24.3 12 25.9 14 26.6 16 26.3 18 25.0 20 22.8 22 19.7 24 15.5 2610.4 93° X« 46.0 45.8 45.6 45.3 45.0 44.8 44.5 44.3 44.0 43.8 43.4 43.2 42.9 42 6 42.3 42.0 41.7 41.5 41.1 40.8 40.5 40.2 39.9 39.6 139.3 38.0 386 38.3 1517.3 16 37.7 17 56.8 1914.6 20 31.3 21 46.7 23 0.8 2413.7 25 25.3 26 35.7 2744.8 28 52.6 29 59.0 31 4.3 32 8.2 33 10.9 3412.2 35 12.2 36 10.9 37 8.3 38 4.4 38 59.1 39 52.5 40 44.6 41 35.3 4224,7 4312.7 43 59.4 44 44.7 45 28.7 4611.3 94° IX» 30 30 29 30 28 30 27 30 26 30 25 80 24 30 23 30 22 30 21 30 20 30 19 30 18 30 17 30 16 30 15 82 TABLE LV. Moon's Distance from the North Pole of the Ecliptic- Argument. Supplement of Node+Moon's Orbit Longitude. III« 84° 86° 15 49 ?0 50 16 51 30 52 17 52 30 53 I 18 54 30 55 19 56 30 57 20 57 3058 21 59 30 22 1 30 2 48.7 31 3 15.3 0.6 47.3 35.3 IV» Diff. for 10 15 4 23 7.5; 25 0.9 27 55.6 29 23 30 24 30 25 30 26 30 27 51.7 49.1 47!8'35 47.8 38 49.1 40 51.8 42 55.7 144 1.0 46 7.4 ',48 15.2 51 30,14 28 15 30 16 29 18 30 19 30 20 24.3 34.7 46.3 59.2 13.3 28.7 45.4 3.2 22.3 42.7 85' RO I ir» i« 49.6 44.5 40.3 37 2 3o.O 33.7 33.4 341 35.7 38.2 41.6 45.9 51.1 57.2 4.2 12.0 20.7 30.3 40.6 51.9 3.8 16.7 30.3 44.7 59.8 15.8 32.5 49.8 7.S 26 9 46.6 87° 88° 38.3 38.6 39.0 H9.3 39.6 39.9 40.2 40.5 40.8 41.1 41.4 41.7 42.0 42.3 42.6 42.9 43.2 43 4 43.6 44.0 44.3 44.5 44 8 450 45 3 45.6 45.8 46.0 46.4 46.6 28 11.1 30 47.3 33 23 8 6 0.7 38 37.9 41 15.4 43 53.2 46 31.3 49 9.6 51 48.3 54 27.2 57 6.3 59 45.7 V» 2 25.3 5 5.1 7 45.1 10 25.2 13 5.6 15 46.0 18 26.7 21 7.5 23 48.4 26 29.4 29 10.5 31 51.7 34 33.0 37 143 39 55.7 42 37.1 45 18.6 48 0.0 89° 1 Os DifF. for 10 91° 52.1 52.2 52.3 52.4 52.5 52.6 .52.7 52.8 52.9 53.0 53.0 53.1 53.2 53.3 53.3 53.4 53.5 53.5 53.6 53.6 53.6 53.7 .53.7 53.7 53.7 53.8 53.8 53.8 53.8 53.8 VI» Diff: for ID 7 48.9 10 247 13 0.1 15 35.1 18 9.8 20 44.0 23 17.9 25 51.2 28 24.2 30 56.7 33 28.7 36 0.2 38 31.3 41 1.8 43 31.9 46 1.4 48 30.4 .50 58.8 53 26.6 56 53.9 58 20.7 46 8 3 12 3 5 37.2 8 1.5 10 25.2 |l2 48.2 ;15 10.5 17 32.2 19 53 1 ,22 13.4 9a° XI« 93° 51.9 51.8 51.7 51.6 51.4 51.3 Sl.l 51.0 50.8 50.7 50.5 50.4 50.2 50.0 49.8 49.7 49.5 49 3 49.1 48.9 48.7 48.5 48.3 48.2 47.9 47.7 47.4 47.2 47.0 46.7 VII» 26 10.4 28 4.3 29 57.1 31 49.0 33 39.9 35 29.7 37 18.4 39 6.2 40 52.9 42 38.4 44 23.0 46 6.5 47 48.8 49 30.1 .51 10.3 52 49.4 54 27.3 .56 4.2 .57 39.9 .59 14.4 ~0T7.8 2 20.1 3 51 2 5 21.1 6 49.9 8 17.4 9 43.8 11 9.0 12 33 13 55.S 15 17.3 94° Xe Diff: for 10 38.0 37.6 37.3 37.0 36.6 36.2 35.9 35.6 35.2 349 34.5 34.1 33.8 33.4 33.0 32.6 32.3 31.9 315 31.1 30.8 30.4 30.0 29.6 29.2 28.8 28.4 28 27.6 27.2 VIII» 94° 46 11.3 46 52.6 47 32.5 48 11.0 48 48.1 49 23.9 49 58.2 50 31.2 51 2.9 51 33.1 52 1.9 52 29.4 52 55.4 53 20.1 53 43.3 54 5.'2 54 25.6 54 446 55 2.3 .55 18.5 55 33.3 55 46.8 55 58.8 56 9.4 56 18.5 56 26.3 .56 32.7 56 37.6 .56 41.2 .56 43.3 56 440 94° IX» 15 30 14 30 13 30 12 30 U 30 10 30 9 30 8 30 7 30 6 30 5 30 4 30 3 30 2 30 1 30 TABLE LVI. f\.0 10.7 6,9'23,2 8,2 3.8 1,9 150 100 1.6 600 360 3.7 50.0 11.6 7,7^22,8 9,2 4,4 2.1 140 110 1.5 610 370 4.3 48.9 12.6 8,7 22.4 10,3 5,1 2,3 130 120 1,5 620 380 4.9 47.7 13.6 9,7|21,9 11,5 5.8 2,5 120 130 1.5 630 390 5.6 46 6 14.8 10,721,4 12,8 6.6 2,8 no 140 1,5 640 400 6.4 45.2 16.0 11,8 20,9 14.1 7,4 3,0 100 150 1.6 650 410 7.1 43.9 17.2 13,0 20,4 15,5 8,3 3.3 90 160 1.7 660 420 7.9 42.5 18.5 14.2' 19.9 17.0 9.1 3,5 80 170 1.9 670 430 8.8 41.6 19.8 15.5! 19.3 18,5 10,1 3,8 70 180 2,1 680 440 9.6 39.5 21.2 16.8 18.7 20,1 11,0 4.1 60 190 2,3 690 450 10.5 38.0 22.6 18,1 18.1 21,7 12,0 4,4 50 200 2,5 700 460 11.3 36.4 24.1 19,4 17,5 23.3 12.9 4.7 40 210 2,8 710 470 12.2 34.9 25.5 20,8 16.9 24.9 13,9 5.0 30 220 3.1 720 480 13.2 33.2 27.0 22.2 16.3 26.6 15,0 5,4 20 230 3.4 730 490 14.1 31.6 28.5 23,6 15,6 28.3 16,0 5,7 10 240 3,7 740 500 15 30.0 30 25 15,0 30.0 17,0 6,0 250 4,0 750 .510 15 9 28 4 31.5 26,4 14.4 31.7 18.0 6,3 990 260 4.3 760 520 16.8 26.8 33.0 27.8 13,7 33.4 19,0 6.6 980 270 4,6 770 530 17.8 25.) 34.5 29,2 13.1 35.1 20,1 7,0 970 280 49 780 540 18.7 23.6 35.9 30.6 12.5 36,7 21,1 7,3 960 290 5,2 790 550 19.5 22.0 37.4 31.9 11.9 38,3 22,0 7.6 950 300 5.5 800 560 20.4 20.5 38.8 33.2 11.3 39,9 23,0 7,9 940 310 5,7 810 570 21.2 19.0 40.2 34.5 10.7 41,5 23,9 8,2 930 320 5,'J 820 580 22.1 17.5 41.5'35.8 10.1 43,0 24,9 8,5 920 330 0.1 830 590 22.9 16.1 42.837.0 9.6 44,5 25,7 8.7 910 340 6.3 840 600 23.6 14.8 44.038.2 9.1 45,9 26,6 9,0 900 350 6,4 850 610 24.4 13.5 45.2 39.3 8,6 47,2 27,4 9.2 890 360 6,5 860 S20 25.1 12.3 46.440.3 8,1 48,5 28,2 9.5 880 370 6.5 870 630 25.7 11.1 47.4 41.3 7,6 49,7 28,9 9.7 870 380 6,5 880 640 26.3 10.0 48.4142.3 7,2 50,8 29.6 9.9 860 390 6.5 890 650 26.9 9.0 49.3 43.1 6.8 51,8 30,2 10.1 850 400 6,4 900 660 27.4 8.1 50.243.9 6.5 52.8 30,8 10.3 840 410 6,3 910 670 27.9 7.3 50.9 44.6 6.2 53,6 31.3 10,5 830 420 6.1 920 680 28.3 6.6 51.6 45.3 5.9 54,4 31,8 10,6 820 430 5.9 930 690 28.7 5.9 52.2 45.8 5.6 55.1 32,2 10.7 810 440 5.7 940 700 29.0 5.4 52.7 46.3 5,3 55,7 32,5 10.8 800 450 5.5 950 960 970 980 990 1000 710 29.2 4.9 53.1 46.7 5.2 56,2 32,8| 10.9 790 460 5.2 720 29.4 4.6 53.5 47.0 5.1 56,5 33,0 11,0 780 470 4.9 730 29.6 4.3 53.7 47.2 5.0 56.8 33.2, 11.0 770 480 4.6 740129. 7 4.2 53.8 47.3 4.9 56.9 33.31 11.1 760 490 4.3 750 29.7 4.1 53.9 47,4 4,9i57,0 33,3'll.l|750| 500 4.0 Con.stnnt 10" TABLE LX. TABLE LXL 85 Small Equations of Moon! s Parallax. Mooris Equatorial Parallax. Args., 1, 2, 4, 5, 6, 8, 9, 13, 13, of Long. Argument. Arg. of Evection. 18 0.2 10.3 240.4 270.5 30 0.5 33|0.fi 36;0 7 39,0.7 42 0.8 45 8 48 0.8 50 0.8 2 4 '/ „ 1.6 0.6 1.6 0.6 1.5 0.6 1.5 0.6 1.4 0.5 1,3 0.5 1.1 0.4 1.0 0.4 0.9 0.3 0.7 0.3 0.6 0.2 0.4 0.2 0.3 0.1 0.2 0.1 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 5 6 8 9 12 13 , ,/ ' ~ If „ 1.0 1.9 0.0 3.6 1.4 2.0 1.6 1.9 0.0 3.5 1.4 2.0 1.5 1.8 0.0 3.1 1.4 1.9 1.5 1.8 0.1 2.6 1.3 1.8 1.4 1.7 0.2 1.9 1.2 1.7 1.3 1.6 0.2 1.3 1.1 1.6 1.1 1.4 0.3 0.7 1.0 1.4 1.0 1.3 0.5 0.2 0.9 1.2 0.9 1.2 0.6 0.0 0.7 1.0 0.7,1.0 0.7 0.1 0.6 0.9 0.6 0.9 0.8 0.4 0.5 0.7 0.4|0.7 0.9 0.8 0.4 0.5 0.3 0.6 1.0 1.5 0.3 0.4 0.2 0.5 1.1 2.1 ;o.2 0.2 0.1 0.4 1.1 3.8,0.1 0.1 0.0 0.3 1.2 3.2^0.0 0.0 0.0 0.3 1.2 3.5|0.0 0.0 0.0 0.3 1.2 3.6 0.0 0.0 100 97 94 91 88 85 82 79 76 73 70 67 64 61 58 55 52 50 Constsnt 7" The first two figures only of the Argunrents are taken. 0» 1 20.8 1 20.8 1 20.8 1 20.7 120.7 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 IS 19 20 21 22 23 24 25 26 1 16.9 27 1 16.6 88 1 16.2 29 1 15.9 3011 15.fi 1 15.6 1 15.2 1 14.9 1 14.5 1 14.2 1 20.6 1 13.8 1 20.6 1 13.4 1 20.5 1 13.0 1 20.4 1 12.6 1 20.3 1 12.2 130.2 1 11.7 1 20.l'l 11.3 1 19.9 1 10.8 1 19.8 1 10.4 1 19.6 1 9.9 1 19.5,1 1 19.3 1 1 19.1 1 1 18.9 1 1 18.7 1 1 18.4 1 1 18.2 1 1 18.0 1 1 17.7 1 1 17.4 1 1 17.1 1 1 1 1 1 1 X]» 9.4 9,0 8,5 8.0 7.5 7.0 6.5 5.9 5.4 4.8 4.3 3.8 3.2 2.6 2.1 1.5 II» III> 1.5 0.9 0.3 59.7 59.2 58.6 57.9 57.3 56.7 56.1 55.5 54.9 54,2 53,6 53,0 53.3 51.7 51.1 50.4 49.8 49.1 48.5 47.8 47.2 46.5 45.9 45,2 44,6 43,9 43,3 42,6 42,6 41,9 41,3 40,6 40,0 39,4 38,7 1 37,4 5.8 B.l 35.5 34,9 34.2 33.6 33.0 32.4 <:i.7 31.1 30.;i 29,t IV» V« 24,1 23,6 23,0 22,5 21,9 21.4 20,9 20,4 19,9 19.4 18.9 18.4 17.9 17.5 17.0 16.6 16.1 15.7 15.2 14.8 14.4 39.3 14.0 28.7 28.1 27.5 36.9 26.3 25.8 25.2 24.7 24.1 13.6 13.2 12.9 12.5 12.1 11.8 11.5 11.1 10.8 10.8 10.5 10.2 9,9 9,6 9:4 9,1 8.8 8,6 8.4 8.2 8.0 7.8 7.6 7.4 7.2 7.1 14 6.9 13 30 29 28 27 26 25 24 231 221 21 20' 19| isi 171 16! 15 6.8 6.7 6.5 6.4 6.3 6.3 6.2 6.1 6.1 6.1 6.0 6.0 6.0 X» IX» VIII«VII»VI. 34 TABLE LXIl. Moon's Equatorial Parallax. Argument. Anomaly. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 (U diff I» diff 58 57.7 58 57.7 58 57.6 58 57.4 58,57.1 58 56.8 0.4 58.56.4lo4 58 56.0Qg 58 55.4 58 54.8 58 54.21 58 53.41, 58 ,52. el 58 51.8 58 50.8 58 49.8 58 48.7 58 47.6 58 46.4 .58 45.1 58 43.8 .58 42.4 58 40.9 58 39.4 58 37.8 58 36.2 58 34.4 88 32.7 58 30.9 58 29.0 C8 27,0 0.6 0.6 10.8 0.8 0.8 1.0 1.0 1.1 1.1 1.2 1.3 1.3 1.4 1.5 1.5 l.fi 1.6 1.8 1.7 18 1.9 20 58 27.0 58 25.0 .58 23.0 58 20.9 58 18.7 58 16.5 58 14.3 .58 12.0 58 9.6 II» diff Ills diff IV» diff V» 58 2.3 .57 59.8 5757.2 57 54.6 5751.9 57 49.2 57 46.4 5743,7 57 40.8 57 38.0 ,5735 1 57 32.2 57 29.3 ,57 26.3 57 23.3 57 20.2 ,57 17.2 5714.1 5711.0 57 7.9 X» 2.0 2.0 1 2.1 1 2.2 2.2 2.2 23 2.4 2.4 •2.4 |2.5 2.5 2.6 2.6 2.7 2.7 2.8 2.7 2.9 2.8 2.9 2.9 2.9 3.0 3.0 3.0 3.0 3.1 3.1 31 57 7 57 4.8 57 1.6 56 58.4 I56 55.2 56 52.0 56 48.8 56 45.5 56 42.3 56 39.0 56 35.7 56 32.4 56 29.1 56 25.8 56 22.5 .56 19.2 .56 15.9 5612.6 .56 9.3 56 6.0 56 2.7 ,55 59.3 55 56.0 55 52.7 55 49.4 55 46.1 5542 8 55 39.6 55 36.4 55 33.1 55 29.8 IX» 3.1 3.2 3.2 3.2 3.2 3.2 3.3 3 2 3.3 3.3 3.3 3.3 3.3 3.3 3.3 3.3 3.3 3.3 3.3 3.3 3.4 3.3 3.3 3.3 3.3 3.3 3.2 3.2 3.3 3.3 5529.8 55 26.6 55 23.4 55 20.2 55 17.0 55 13.8 55 10.6 55 7.5 55 4.4 55 1.3 54 58.2 5455.1 5452.1 5449.1 5446.1 5443.1 5440.2 54 37.3 54 34.4 5431.5 54 28.7 ,54 25.9 54 23.1 54 20.3 54 17.6 5414.9 5412.2 54 54 7.0 54 4.4 54 1.9 VIIIs 3.2 3.2 3.2 3.2 3.2 3.2 3.1 3.1 3.1 3.1 3.1 3.0 3.0 3.0 3.0 2.9 2.9 2.9 2.9 2.8 2.8 2.8 2.8 2.7 7 2.7 2.6 2.6 2.6 2.5 diff I 54 1.9 53 59.4 .53 .56.9 53 54.5 53 52.1 53 49.7 5347.4 53 45.1 ,53 42.9 ,53 40.6 .53 38.5 53 36.3 ,53 34.2 53 32.1 53 30.1 53 28.1 53 26.2 53 24.3 ,53 22.4 53 20.6 .53 18.8 5317.0 53 15.3 53 13.7 .53 12.0 53 10.4 5 2,5 2.4 2.4 2.4 2.3 2.3 2.2 2.3 2.1 2.2 2.1 2.1 2.0 2.0 1.9 1.9 1.9 1.8 1.8 1.8 1.7 1.6 1.7 1.6 1.5 1.5 1.5 1.4 1 3 VII» 3 2,4 1-^ 3 .52 59.3 J* 52 58.1 Jf 52 57.0,'-' 1.2 1.0 11.0 ll.O 0.9 0.9 52 55.8 ,52 54.8 52 53.8 52 52.8 52 51.9 52 51.0 ,52 50.1 52 49.3 ,52 48,6 52 47.9 52 47.2 .5246.6 52 46.0 ,52 45.5 52 45.0 52 44.6 ,52 44,2 52 43 8 52 43,5 52 43.3 52 43.1 5242,9 52 42,8 5242,7 52 42,7 I VI» 0,9 0,8 0,7 0.7 0.7 0.6 0.6 0,5 0.5 0.4 0.4 0.4 0,3 0,2 0.2 0,2 0.1 0.1 0.0 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 TABLE LXIII. 87 Maoris Equatorial Parallax. Argument. Argument of the Variation. 0> \> II* III* IV« V» o f. „ /, // „ // o 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 S 55.2 38.1 12.4 4.3 21.9 47.7 26 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 8 546 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 6.6 26.3 50.9 20 11 53.7 32.7 8.7 6.0 27.2 61.5 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 63.6 16 16 51.5 28.2 6.3 8.2 31.8 64.0 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 45 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 66.4 6 23 46.1 20.1 3.9 13.8 39.9 66.6 6 26 45.4 19.3 3.8 14.5 40.8 66.8 4 27 44.6 18.5 3.7 15.3 41.6 66.9 3 28 43.9 17.6 3.7 16.1 42.4 56.9 3 . 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 XI» X. IX» VIII» VII* A^» 88 TABLE LXIV. TABLE LXV. Reduction of the Parallax, and also of the Latitude. Argument. Latitude. Moon's Semi-diameter. A/gument. Equatorial Parallax. Lai Red. of par Red. of Lat. Eq.Par Semidia. Eq.Par Semidia Eq.Par Semidia. seel Pro. 1 Par. 53 14 26.6 56 15 15.6 59 16 4.6 1 0.3 « / /f 0.0 0.0 53 10 14 29.3 58 10 15 18.3 59 10 16 7.4 2 0.5 3 0.0 1 11.8 53 20 14 32.0 56 20 15 21.0 59 20 16 lO.l 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 15 26.5 59 40 16 15.6 5 1.4 12 15 0.5 0.7 4 39.3 5 43.4 53 50 14 40.2 56 50 15 29.2 59 50 16 18.3 6 1.6 54 14 42.9 57 15 31.9 60 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 22 21 1.4 7 39.7 54 20 14 48.3 57 20 15 37.4 60 20 16 26.4 9 24 24 1.8 8 30.7 54 30 14 51.1 57 30 15 40.1 fiO 30 16 29.2 1012.7 1 27 31) 2.3 2.7 9 16.1 9 55.4 54 40 14 53.8 57 40 15 42.8 60 40 16 31.9 54 50 14 56.5 57 50 15 45.6 60 50 16 34.6 33 3.3 10 28.3 55 14 59.2 58 15 48.3 61 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 53.7 61 20 16 42.8 42 45 4.9 5.5 11 24.7 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 15 15.6 59 16 4.6 62 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 LXVL 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 2 23.7 84 11.0 Horizon. Semi-diameter.| | Horizon. Semi-diameter. 87 Alt. Alt. 11.1 1 i'Z.'i 90 11.1 0.0 14'30" 15- 16 17 14' 30" 15' 16 17 Subsidiary Table. o 2 4 0.6 1.0 0.6 1.1 0.7 1.3 0.8 1.5 o 42 45 9.2 9.7 9.8 10.4 11.2 11.8 12.6 13.3 Lat. + 3' — 3' 6 1.5 1.6 1.9 2.1 48 10.2 10.9 11.4 12.4 13.0 14.0 14.7 '/ " 8 2.0 2.1 2.4 2.7 51 10^6 + 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 12 0.0 0.0 12 2.9 3.1 3.6 4.0 57 1 11.5 12.3 14.0 15.8 15 0.0 0.0 14 3.4 3.6 4.1 4.7 60 11 8 12.7 14.4 16.3 18 0.1 0.1 16 3.8 4.1 4.7 5.3 63 12.2 13.0 14.9 16.8 24 0.1 0.1 18 4.3 4.6 5.2 5.9 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 36 0.2 0.2 24 5.6 6.0 6.8 7.7 72 13.0 13.9 15.9 17.9 42 0.2 0.2 27 6.2 6.7 7.6 8.6 75 132 14.1 16.1 18.2 48 0.3 0.3 30 6.9 7.3 8.4 9.5 78 13.4 14.3 16.3 18.4 54 03 0.3 33 7.5 8.0 9.1 10.3 81 13.5 14.4 16.5 18.6 36 8.1 8.6 9.8 11.1 84 13.6 14.5 16.6 18.7 60 72 78 0.4 0.5 0.6 0.4 0.5 0.6 39 8.6 9.2 10.5 11.9 90 13.7 14.6 16.7 18.8 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. % 3 4 5 6 1 7 S 0.00 9 0.16 5.0 0.0 2.9 1.9 0.0 0.00 0.00 S 5.0 0.0 2.8 1.9 0.0 0.00 0.00 0.00 0.15 4 4.9 0.0 2.8 1.9 0.0 0.01 0.00 0.02 0.15 6 4.8 0.1 2.8 1.9 0.1 0.03 0.01 0.05 0.14 « 4.7 0.2 2.7 1.8 0.1 0.06 0.01 0.09 0.12 10 4.5 0.3 S.B 1.7 0.2 0.09 0.02 0.14 0.10 12 4.3 0.4 2.5 1.7 0.2 0.13 0.02 0.19 0.U9 14 4.1 0.6 23 1.6 0.3 0.18 0.03 0.26 0.07 16 3.8 0.7 2.2 1.5 0.4 0.23 0.04 0.33 0.05 18 3.6 0.9 2.0 1.4 0.5 0.28 0.05 0.41 0.03 SO 3.3 1.1 1.9 1.3 0.6 0.34 0.06 0.50 0.02 22 3.0 1.3 1.7 1.1 0.7 0.40 0.07 0.58 0.01 24 2.7 1.5 1.5 1.0 0.8 0.46 0.08 0.67 0.00 26 2.3 1.7 1.3 0.9 0.9 0.52 0.10 077 TOO 28 2.0 1.9 1.2 0.8 1.0 0.58 0.11 0.86 0.00 30 1.7 2.1 1.0 0.7 1.1 0.63 0.12 0.94 0.01 32 1.4 2.2 0.8 0.5 1.2 0.69 0.13 1.03 0.01 34 1.2 2.4 0.7 0.4 1.3 0.74 0.14 1.11 0.03 36 0.9 2.6 0.5 03 1.3 0.78 0.15 1.18 0.05 88 0.7 2.7 0.4 0.3 1.4 0.82 0.16 1.25 0.06 40 0.5 2.8 0.3 0.2 1.5 0.86 0.16 1.30 0.U8 42 0.3 2.9 0.2 0.1 1.5 0.89 0.17 1.35 0.10 44 0.2 3.0 0.1 O.I 1.6 0.91 0.17 1.39 0.11 46 0.1 3.1 0.0 0.0 1.6 0.93 0.18 1.42 0.12 48 0.0 3.1 0.0 0.0 1.6 0.94 0.18 1.44 0.13 50 0.0 3.1 0.0 0.0 1.6 0.94 0.18 1.44 0.13 Arg, 100 98 96 94 92 90 88 86 84 82 80 78 76 74 72 70 68 66 64 62 60 58 56 54 52 50 Arg. 10 11 12 13 14 15 16 17 18 Aig, S-rg. 10 11 0.00 0.26 2 0.00 0.25 4 0.02 0.24 6 0.04 0.22 8 0.08 0.20 10 0.12 0.17 12 0.16 0.14 14 0.20 0.11 16 0.24 0.08 18 0.28 0.05 30 0.31 0.03 22 0.34 0.01 24 0.35 0.00 26 0.36 0.00 28 035 0.01 30 0.34 0.02 32 0.32 0.04 34 0.29 0.06 36 0.26 0.09 38 0.22 0.11 40 0.18 0.14 42 0.15 0.16 44 0.12 0.19 46 0.10 0.21 48 0.09 0.22 SO 0.08 0.22 0.00 0.00 0.01 0.03 0.04 0.07 0.09 0.12 0.16 0.19 0.23 0.27 0.31 035 0.39 0.43 0.47 0.50 0.54 0.57 0.59 0.62 0.63 0.65 0.66 0.66 0.00 0.00 0.00 0.01 0.02 0.03 0.04 0.06 0.07 0.09 0.11 0.13 0.15 0.17 0.19 0.21 0.23 0.25 0.26 0.28 0.29 0.30 0.31 0.32 0.32 0.32 0.00 0.00 0.01 0.C2 0.04 0.06 0.09 0.12 0.15 0.19 0.22 0.26 0.30 0.34 0.38 0.42 0.45 0.49 0.52 0.55 0.58 0.60 0.62 0.63 0.64 0.64 0.00 0.00 0.00 0.01 0.01 0.02 0.02 0.03 0.04 0.05 O.OG 0.07 0.08 0.08 0.09 0.10 O.U 0.12 0.13 0.14 0.14 0.15 0.15 0.16 0.16 0.16 16 17 18 0.21 0.26 0.00 26 0.00 0.20 0.C6 0.00 0.20 0.25 0.00 0.20 0.25 0.01 0.20 0.24 0.01 0.20 0.22 0.02 0.19 0.21 0,02 0.19 0.20 0.03 0.18 0.19 0.04 0.18 0.17 0.05 0.17 0.15 0.06 0.17 0.14 0.07 0.16 0.12 0.07 0.16 O.U 0.08 0.15 0.09 0.09 0.15 0.07 0.10 0.14 0.06 O.U 0.14 0.05 0.12 0.13 0.04 0.12 0.13 0.02 0.13 0.18 0.01 0.13 0.12 0.01 0.14 0.12 0.00 0.14 0.12 0.00 0.14 ! 12) 0.00 0.14 O.U 1 90 TABLE LXVIIL Moon's Horary Motion in Longitude. Argument. Argument of the Eveclion. 0» I» II» III» IV» V. 1 o 80.3 74.7 59.6 39.4 19.8 .■).£ o 30 1 80.3 74.3 58.9 38.7 19.3 5.6 29 8 80.3 73.9 58.3 38.0 18.7 6.3 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 80 77.8 65.4 46.4 26.0 9.7 1.4 10 81 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 70.7 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 86 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 i XI» x» IX» VIIIs VII« VI. 1 TABLE LXIX. Moon's Horary Motion m Longitude. ^Tg^lmo^*.s. Sum of Equations, 2, 3, &c., and Eveclion -jcrrocted 0" 10" 20" s I II III IV V VI o 00 0.0 0.1 0.2 0.3 0.4 0.5 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.5 0.4 0.3 0.2 0.1 0.0 0.0 » o XII XI X IX VIII VII VI 0" 10" 80" TABLE LXX. 91 Moon's Horary Motion in Longitude. Arguments. Sum of preceding equations, and Anomaly corrected 1 0" 10" 20" 30" 40" 50" 60" 70" 80" 90" 100" 4.1 5.3 6.5 7.6 8.8 10,0 11.2 12.4 13.5 14.7 15.9 XII 6 4.1 5.3 6.5 7.7 8,8 10.0 11.2 12.3 13.5 14.7 15.9 25 10 ^.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 U.l 12.3 13.4 14,5 15.7 15 20 4.5 5.6 6.7 7,8 8.9 10,0 U.l 12,2 13.3 14.4 15.5 10 25 4.8 5.8 6.9 7.9 9.0 10,0 U.O 12,1 13.1 14,2 15.2 5 I 5.1 6.0 7.0 8.0 9.0 10.0 U.O 12.0 13,0 14.0 14.9 XI 6 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 ib.o 10,6 11.2 11.8 12,4 13.0 5 II 7.5 8.0 8.5 9.0 9.5 10.0 10,5 U.O 11.5 12,0 12.5 X 5 7.9 8.4 8.8 9.2 9.6 to.o 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 U.l 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* 10.0 10.0 10,0 10.1 10.1 10.1 5 III 10.4 10.3 10.2 10.1 10,1 10.0 9.9 9.9 9,8 9.7 9.6 IX 5 10.8 10.7 10,5 10,3 10,2 10.0 9.8 9.7 9.5 9.3 9.2 85 10 11.3 U.O 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 U.O 10.5 10.0 9.5 9.0 8.5 8.0 7.5 5 IV 12.9 12.3 11.7 11.2 10.6 10.0 9.4 8,8 8.3 7.7 7.1 VIII 5 13.3 12.6 11.9 11.3 10,6 10.0 9.4 8.7 8.1 74 6.7 25 lU 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 14.6 13.7 12.7 11.8 10.9 10.0 9.1 8.2 7,3 6.3 5.4 VII 6 14.7 13.8 12.8 11.9 10.9 10.0 9.1 8.1 7.2 6,2 5.3 25 10 14.9 13.9 12.9 12,0 U.O 10.0 9.0 8,0 7.1 6.1 5.1 20 15 15.0 14.0 13.0 12.0 U.O 10,0 9.0 8,0 7.0 6,0 5,0 15 SO 15.1 14.1 13.0 12.0 U.O 10.0 9.0 8.0 7.0 5,9 4,9 10 25 15.1 14.1 13.1 12.0 U.O 10,0 9.0 80 6.9 5,9 4.9 5 VI 15.1 14.1 10" 13.1 20' 12.1 U.O 10,0 50" 9,0 8.0 70" 6.9 80' 5.9 90' 4.9 100" VI i. 0' 30" 40" 60" 92 TABLE LXXI. Moon^s Horary Motion in Longitude. Argument. Anomaly corrected. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 2.5 26 27 28 29 30 O diff. 441.5 441.5 441.3 441.1 440.8 440.4 439.9 439.4 438.7 438.0 437.2 436.3 435.3 434.2 433.1 431.8 430.5 429.1 427.fi 426.1 424.5 422.7 421.0 419.1 417.2 415.2 413.1 410.9 408.7 406.4 404.1 XI» 0.0 0.1 0.2 0.3 0.4 0.5 0.5 0.7 0.7 0.8 0.9 1.0 1.1 1.1 1.3 1.3 1.4 1.5 1.5 1.6 1.7 1.7 1.9 1.9 2.0 2.1 2.2 2.2 2.3 2.3 I» diff. I lis 'diff 404.1 401.6 399.2 396.6 394.0 391.3 388.6 385.8 383.0 380.1 377.1 374.1 371.1 368.0 364.8 361.6 358.4 355.1 351.8 348.4 345.0 341.6 338.1 334.6 331.1 327.5 324.0 320.3 316.7 313.0 309.3 ; Xs 2.5 2.4 2.6, 2.6 2.7 2.7 3.8 2.8 2.9 3.0 3.0 3.0 3.1 3.2 3.2 3.2 3.3 3.3 3.4 3.4 3.4 3.5 3.5 3.5 3.6 3.5 3.7 3.6 3.7 3.7 309.3 305.6 301.9 298.1 294.4 290.6 286.8 283.0 279.2 275.4 271.5 267.7 263.8 260.0 256.2 252.3 248.5 244,6 240.8 236.9 233.1 229.3 225.4 221.6 217.8 214.0 210.3 206.5 202.8 199.0 195.3 IXs Ills diff 3.7 3.7 3.8 3.7 3.8! 3.8 3.8 3.8 3.8 3.9 3.8 3.9 3.8 3.8 3.9 3.8 3.9 3.8 3.9 3.8 3.8 3.9 3.8 I 3.8 I 3.8 3.7 3.8 3.7 3.8 3.7 195.3 191.6 187.9 184.3 180.6 177.0 173.4 169.8 166.3 162.8 159.3 155.8 153.4 148.9 145.5 142.2 138.S 135.6 132.3 129.1 125.9 122.7 119.6 116.5 113.4 110.4 107.4 104.5 101.6 98.7 95.8 vni» JV« diff 3.7 3.7 3.6 3.7 3.6 3.6 3.6 3.5 3.5 3.5 3.5 3.4 3.5 3.4 3.3 3.3 3.3 3.3 3.2 3.2 3.2 3.1 3.1 3.1 3.0 3.0 2.9 2.9 2.9 2.9 95.8 93.0 90.2 87.6 84.9 82.3 79.7 77.1 74.6 72.1 69.7 67.3 65.0 62.7 60.4 58.2 56.1 53.9 51.9 49.8 47.9 45.9 44.0 43.3 40.4 38.7 37.0 35.3 33.7 32.1 30.6 VII» 2.8 2.8 2.6 2.7 3.6 2.6 3.6 3.5 3.5 2.4 3.4 3.3 3.3 2,3 3.3 3.1 2.2 2.0 3.1 1.9 3.0 1.9 1.8 1.8 1.7 1.7 1.7 1.6 1.6 1.5 V* diff. 30.6 39.3 37,8 26,4 25.1 23.8 23.6 31.4 20.3 19.2 18.3 17.2 16.3 15.4 14.6 13.8 13.1 12.4 11.8 11.3 10.7 10.2 9.8 9.4 9.1 8.8 8.6 8.4 8.3 8.2 8.2 VI« 1,4 1,4 1,4 1.3 1.3 1.3 1.3 l.l 1.1 1.0 1.0 0,9 0.9 0.8 0.8 0.7 0.7 0.6 0,6 0,5 0.5 0,4 0.4 0.3 0.3 0.2 0.3 0.1 0.1 0,0 30 29 28 27 36 25 34 23 23 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 6 4 3 S I TABLE LXXII. MoorCs Horary Motion in Longitude. Arguments. Sum of preceding Equations, and Arg. of Variation 93 ** f ,, „ ,, „ ir /* „ ,^ *f ft /» 50 100 150 200 250 300 360 400 450 13.7 500 14.7 550 15.7 600 16.7 O 4.5 5.5 6.5 7.6 8.6 9.6 10.6 11.6 12.6 a ° XII 5 4.6 5.6 6.6 7.6 8.6 9.6 10.6 11.6 12.6 13.6 14.6 16.6 16.6 25 10 4.8 6.8 6.8 7.7 8.7 9.6 10.6 11.5 12.5 13.4 14.4 1.5.3 16.3 20 15 5.3 6.1 7.0 7.9 8.8 9.7 10.5 11.4 12.3 13.1 14.0 14.9 15 8 15 20 5.8 6.6 7.4 8.2 8.9 9.7 10.6 11.2 12.0 12.8 13.5 14.3 15.1 10 85 6.6 7.2 7.8 8.5 9.1 9.7 10.4 11.0 11.7 12.3 12.9 13.6 14.2 S I 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 5 8.3 8.6 8.9 9.2 9.5 9.9 10.2 10.5 10.8 11.2 11.6 11.8 12.1 25 10 92 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 15 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.E 9-2 9.0 8.8 8.6 10 25 12.1 11.7 113 10.9 10.6 10.2 9.8 9.4 9.0 8.6 8.3 7.9 7.5 5 II 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 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 11.1 103 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 20 15.3 14.3 133 12.3 11.4 10.4 9.4 8.4 7.5 6.5 5.5 4 5 3.6 10 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 15.6 14.5 13.6 12.5 11.4 10.4 9.4 8.4 7.3 6.3 5.3 4.2 3.2 IX 5 15.4 14.4 13.4 12.4 11.4 10.4 9.4 8.4 7.4 6.4 54 4.4 33 25 10 15.2 14.2 13.3 12.3 11.3 10.4 9.4 8.5 7.5 6.5 5.6 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 4.2 15 20 14.2 13.4 12.6 11.9 11.1 10.3 9.5 8.8 8.0 7.2 6.4 5.6 4.9 10 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 6.7 5 IV 12.7 12.2 11.7 11.2 10.7 10.2 9.7 9.2 8.7 8.2 7.7 7.2 6.7 vnio 5 11.9 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 99 10.0 100 10.0 10.0 10.0 100 10 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 10.5 10.7 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 12.5 5 V 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 6 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 20 15 5.0 5.9 6.8 7.8 8.7 9.6 10.6 11.5 12.4 13.3 14.3 15.2 16.1 15 20 4.6 5.6 6.6 7.6 8.6 9.6 10.6 11.6 12.6 13 6 14.6 15.7 16.7 10 25 4.3 5.4 6.4 7.5 8.5 9.6 10.6 11.7 12.7 13.8 14.9 15.9 17.0 S VI 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 50 100 150 200 250 " 1 " 300 350 400 450 500 550 600 04 TABLE LXXIIL Moon's Horary Motion in Longitude. 1 Argument. Argument of the Variation. ... 0« I« 11" III» IV» v» o „ ., „ /^ „ „ o 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 6 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 16 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 5.3 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 44 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 49.6 77.7 8 23 65.3 28.6 3.3 14.2 51.1 78.1 7 24 64.4 27.3 3.1 15.1 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 XI» X» IXo VIIIo VII» W> TABLE LXXIV. 95 MoorCs Horary Motion in Longitude, Arguments. Arg. ot Reduction and Sum of preceding Equations tf ,. " 1 " ..\,.\ .. II 1 f» „ H 1 // „ tf 50 100 150 200 250 300 1 350 400 1 450 500 650 600 650 « II ./ „ ~7r /. // ; ,/ ./ „ „ „ ./ s 3.3 3.1 2.9 2.7 2.5 2.3 2.1 1.9 1.7 1.5 1.3 1.1 0.9 0.7 XII n 3.3 3.1 2.9 2.7 2.5 2.3 2.1 1.9 1.7 1.5 1.3 1.1 0.9 0.7 26 10 3.2 3.0 2.8 2.6 2.4 2.3 2.1 1.9 1.7 1.5 1.3 1.1 1.0 0.8 20 15 3.1 2.9 2.8 2.6 2.4 2.2 2.1 1.9 1.7 1.5 1.4 1.2 1.0 0.9 16 20 3.0 28 2.7 2.5 2.4 2.2 2.1 1.9 1.8 1.6 1.5 1.3 1.1 1.0 10 25 2.8 2.7 2.6 2.4 2.3 2.2 2.1 1.9 1.8 1.7 1.5 1.4 1.3 1.2 6 I 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 1.3 XI 5 2.4 2.4 2.3 2.2 2.2 2.] 2.0 2.0 1.9 1.8 1.8 1.7 1.6 1.6 26 10 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 20 15 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 16 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 6 n 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 5 1.2 1.3 1.4 1.6 1.7 1.8 1.9 2.1 22 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 17 1.9 2.1 2.3 •ib 2.7 2.9 i?.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 6 III 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 6 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 26 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 3.2 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 20 1.0 1.2 1.3 i.S 1.6 l.S 1.9 2.1 2.2 2.4 2.5 2.7 2.9 3.0 10 25 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 5 IV 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 vin 5 1.6 1.6 1.7 l.S 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 2.2 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 2.0 2.0 ,2.0 15 SO 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 6 V 2.6 2.5 3.4 2.3 2.2 2.1 2.0 1.9 1.8 1.7 1.6 1.5 1.4 1.3 vn 5 2.8 2.7 2.6 2.4 2.3 2.2 2.1 1.9 1.8 1.7 1.5 1.4 1.3 1.2 25 10 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 20 15 3.1 2.9 2.8 2.6 2.4 2.2 2.1 1.9 1.7 1.5 1.4 1.2 1.0 0.9 16 20 3.2 3.0 2.8 2.6 2.4 2.3 2.1 1.9 1.7 1.5 1.3 1.1 1.0 0.8 10 35 3.3 3.1 2.9 2.7 2.5 2.3 2.1 1.9 1.7 1.5 1.3 1.1 0.9 0.7 6 VI 3.3 3.1 2.9 2.7 2.5 2.3 2.1 1.9 1.7 1.5 1.3 1.1 0.9 0.7 \\ 60 100 " 150 200 250 300 350 400 450 500 650 600 650 96 TABLE LXXV. A/i)OJi'« Horary Motion in Long. Arg. Arg. of Reduction. TABLE LXXVL Moon's Horary Motion in Long (Equation of the second order.) Arguments. Arg's.of Table LXX, Os Vis Is Vis IlsVIIls /- -. tf o 2.1 6.0 14.0 30 1 2.1 6.3 14.2 29 2 2.1 6.5 14.4 28 3 2.1 6.8 147 27 4 2.2 7.0 149 26 5 2.2 7.3 15.1 25 6 2.2 7.5 1.5.3 24 7 2.3 7.8 15.5 23 8 2.4 8.1 15.7 22 9 25 8.4 15.9 21 10 2.5 8.6 16.1 20 11 2.6 8.9 16.2 19 12 2.7 9.2 16.4 18 13 2.9 9.4 16.6 17 14 3.0 9.7 16.7 16 15 3.1 10.0 16.9 15 16 3.3 10.3 17.0 14 17 3.4 10.6 17.1 13 18 3.6 10.8 17.3 12 19 3.8 n.i 17.4 11 20 3.9 11.4 17.5 10 21 4.1 11.6 17.5 9 82 4.3 11.9 17.6 8 23 4.5 12.2 17.7 7 54 4.7 12.5 17.8 6 25 49 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 XIsV. XsIVs IXsUIs *t „ „ Arg 50 100 o „ /. " 1 0.05 0.05 0.05 I 0.08 0.05 0.02 ir 0.10 0.05 0.00 m 0.10 0.05 0.00 IV 0.09 0.05 0.01 V 0.07 0.05 0.03 VI 0.05 0.05 0.05 VII 0.03 0.05 0.07 viir 0.01 0.06 0.09 IX 0.00 0.05 0.10 X 0.00 0.05 0.10 XI 0.02 0.05 0.08 XII 0.05 0.05 O.OS tf ft // 50 100 Constant to be added 27'24".0. TABLE LXXVII. MoorCs Horary Motion in Longitude. (Equiitions of the second order.) Arguments. Arguments of Tables LXXIl and LXXIV. Variation. Reduction. 100 200 300 400 500 600 600 0.03 « 0. * o VI. 0.14 0.14 0.14 0.14 0.14 0.14 14 03 1. VII. 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. 0.22 0.19 0.16 O.W 0.10 0.07 0.03 0.01 0.05 ni. IX. 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.03 0.03 IV. X. 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 0.18 0.23 0.28 0.05 0.00 V. XI. 0.06 0.09 0.12 10.15 0.18 0.22 0.26 0.05 0.01 VI. XII. 0.14 0.14 0.14 |0.14 0.14 C.14 0.14 0.03 0.03 TABLE LXXVIII. 97 Moon's Horary Motion in Longitude (Equations of the second order.) Arguments. Args. of Evection, Anomaly, Variation, Reduction. r ■ Evec. Anom. Var. Red. Evec. Anom. Var. Red. 1 o o 0.16 1.05 0.34 0.08 0.16 1.05 0.34 0.08 s o XII 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 1 20 0.10 0.59 0.13 0.13 0.23 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 I 0.08 0.40 0.05 0.14 0.25 1.70 0.63 0.01 XI 5 0.07 0.31 0.02 0.15 0.36 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 IS 0.04 0.17 0.01 0.15 0.38 1.93 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 3.02 0.65 0.01 5 n 0.02 0.04 0.06 0.14 0.31 2.05 0.63 0.01 X 5 0.01 0.02 0.09 0.13 0.32 2.08 0.59 0.03 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 0.00 0.04 0.35 0.08 0.33 2.06 0.33 0.08 IX 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 33 2.00 0.23 0.10 20 15 0.01 0.14 0.51 0.04 0.33 1.96 0.17 Oil 15 20 001 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 0.03 0.28 0.63 0.01 0.30 1.82 0.05 0.14 VIII 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 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 0.09 0.64 0.64 0.01 0.24 1.45 0.04 0.14 VII 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 IJ SO 0.14 0.91 0.46 0.05 0.19 1.18 0.33 0.10 10 35 0.15 0.98 0.40 0.06 0.18 1.12 0.28 0,09 5 VI 0.16 1.05 0.34 0.08 0.16 1.05 0.34 0.08 VI 98 TABLE LXXIX. MoorCs Horary Motion in Latitude. Argument. Arg. I of Latitude. O I» II. III» IV' V. o 378.0 rf 3543 289.2 200.0 110.8 45.7 o 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 8 376.3 340.5 266.9 175.1 90.2 34.9 28 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 16 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 IS 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 S 26 360.1 299.8 212.5 121.7 52.3 22.4 4 27 358.7 297.2 209.3 119.0 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 XI» X« IX» VIIIs VII» Vis 1 TABLE LXXX. Moon's Horary Motion in Latitude. Arguments. Args. V, VI, VII, VIII, IX, X, XI, and XII, of Latitude. Arg. V VI VII 0.34 vni 0.00 IX X 0.04 XI XII 0.08 Arg. 1000 0.00 0.50 0.50 0.12 50 0.01 0.49 0.33 0.00 0.49 0.04 0.120.07 9.50 100 0.04 0.45 0.30 0.02 0.45 0.04 0.11 0.05 900 150 0.09 0.40 0.27 0.04 0.40 0.03 0.10 03 850 200 0.16 0.33 0.22 0.06 0.33 0.03 0.080.01 800 250 0.23 0.25 0.17 0.09 0.25 0.02 0.06 0.00 750 300 0.30 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 6.50 400 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 0.00,0.07 5.50 600 0.46 0.00 0.00 0.18; 00 0.00 0.00'0.08 500 TABLE LXXXI. Moon's Horary Motion in Latitude. 99 Arguments. Preceding equation, and Sum of equations of Horary Motion in Longitude, except the last two. Pr. eq. 20 0" 50" 100" 150" 200" 250" 300" 350" 400" «0" 500" 650" 600" 650" 1 1".6 59.0 1".4 54.5 l."l 0".9 0".6 0".4 0".l 0''.2 0".4 0".7 0".9 1".2 1".4 1".7 Diff. 50.0 45.4 40.9 36.4 31.8 27.3 22.8 18.2 13.7 9.1 4.6 If 0.1 ff 4.5 30 57.4 53.1 48.9 44.6 40.3 36.0 31.7 27.4 23.2 18.9 14.6 10.3 6.0 1.7 4.3 40 55.8 51.8 47.7 43.7 39.7 35.6 31.6 27.6 23.6 19.5 15.5 11.5 7.4 3.4 4.0 50 54.2 50.4 46.6 42.9 39.1 35.3 31.5 27.7 24.0 20.2 16.4 12.6 8.8 6.1 3.8 HO 52.6 49.1 45.5 42.0 38.5 34.9 31.4 27.9 24.4 20.8 17.3 13.8 10.2 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 16.1 13.1 10.0 3.0 90 47.7 45.0 42.2 39.4 36.7 33.9 31.1 28.3 25.6 22.8 20.0 17.3 14.5 11.7 2.8 100 46.1 43.6 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 17.3 15.0 2.3 120 42.9 40.9 38.9 36.9 34.8 32.8 30.8 28.8 26.8 24.8 22.7 20.7 18.7 16.7 2.0 130 41.3 39.5 37.8 36.0 34.2 32.5 30.7 28.9 27.2 25.4 23.7 21.9 20.1 18.4 1.8 140 39.7 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 35.5 34.3 33.0 31.8 30.5 29.2 28.0 26.7 25.5 24.2 23.0 21.7 1.3 160 36.5 35.4 34.4 33.4 32.4 31.4 30.4 29.4 28.4 27.4 26.4 25.4 24.4 23.3 1.0 170 34.8 34.1 33.3 32.6 31.8 31.1 30.3 29.5 28.8 28.0 27.3 26.5 25.8 25.0 0.8 180 33.2 32.7 32.2 31.7 31.2 30.7 30.2 29.7 29.2 28.7 28.2 27.7 27.2 26.7 0.5 190 31.6 31.4 31.1 30.9 30.6 30.4 30.1 29.8 29.6 29.3 29.1 28 a 28.6 28.3 0.3 200 30.0 30.0 30.0 30.0 30.0 30.0 30.0 30.0 30.0 80.0 30.0 30 30.0 30.0 0.0 210 28.4 28.6 28.9 29.1 29.4 29.6 29.9 30.2 30.4 30.7 30.9 31.2 3L.4 31.7 0.3 220 26.8!27.3 27.8 28.3 28.8 29.3 29.8 30.3 30.8 31.3 31.8 32.3 32.8 33.3 0.5 230 25.2 25.9 26.7 27.4 28.2 28.9 29.7 30.5 31.2 32.0 32.7 33.5 34.2 35.0 0.8 240 23.524.6 25.6 26.6 27.6 28.6 29.6 30.6 31.6 32.6 33.6 34.6 35.6 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 1.3 260 20.3 21.8 23.3 24.9 26.4 27.9 29.4 30.9 32.4 33.9 35.4 37.0 38.6 40.0 1.5 270 18.7 20.5 22.2 24.0 25.8 27.5 29.3 31.1 32.8 34.6 36.3 38.1 39.9 41.6 1.8 280 17.1 19. 1| 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 16.5 17.8 20.0 22.3 84.6 26.8 29.1 31.4 33.6 35.9 38.2 40.4 42.7 45.0 2.3 300 13.9 16.4 18.9 21.4 24.0 26.5 29.0 31.5 34.0 36.6 39.1 41.6 44.1 46.6 2.5 310 12.3 15.0 17.8 20.6 23.3 26.1 28.9 31.7 34.4 37.2 40.0 42.7 45.5 48.3 2.8 320 10.7 13.7 16.7 19.7 22.7 25.8 28.8 31.8 34.8 37.9 40.9 43.9 46.9 60.0 3.0 330 9.0 12.3 15.6 18.9 22.1 25.4 28.7 32.0 35.2 38.5 41.8 45.1 48.3 61.6 33 340 7.4 10.9 14.5 18.0 21.5 25.1 28.6 32.1135.6 39.2 42.7 46.2 49.8 63.3 35 350 5.8 9.6 13.4 17.1 20.9 24.7 28.5 32.3 36.0 39.8 43.6 47.4 51.2 54.9 38 360 4.2 8.2 12.3 16.3 20.3 24.4 23.4 32.4 36.4 40.5 44.5 48.5 52.6 56.6 4.0 370 2.6 6.9 11.1 15.4 19.7 24.0 28.3 32.6 36.8 41.1 45.4 49.7 54.0 58.3 4.3 380 1.0 5.5 10.0 14.6 19.1 23.6 28.2 32.7 1 37.2 41.8 46.3 50.9 55.4 59.9 4.5 0" 50" lO'y 150" 200" 250" 300" 350" 400" 450" 500" 550" 600" 650" TABLE LXXXI L Moon's Horary Motion in Latitude. Argument. Arg IL of Latitude. o 0» Is II« Ills IV« V* 9.3 8.7 7.1 5.0 2.9 1.3 o 30 3 9.3 8.6 6.9 4.8 2.7 1.2 27 6 9.2 8.5 6.7 4.6 2.5 1.1 24 9 9.2 8.3 6.5 4.3 2.3 1.0 21 12 B.2 8.2 6.3 4.1 2.1 0.9 18 IS 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 12 81 9.0 7.7 5.7 3.5 1.7 0.8 9 84 8.9 7.5 6.4 3.3 1.5 0.8 6 87 e8 7.3 5.2 3.1 1.4 0.7 3 30^ S7 7.1 5.0 2.9 1.3 vfi« 0.7 XI» X;, IX., vnis Wis 100 TABLE LXXXIII. Moon's Horary Motion in Latitude. Arguments. Preceding equation, and Sum of equations of Horary Motion in Longi- tude, except the last two. Pnv f9 " " " // /* " « equ. lUO 200 300 400 bOO 600 700 2.1 1.8 1.5 1.2 0.9 0.6 0.3 0.0 T 1.9 1.6 1.4 0.9 0.7 0.4 0.2 2 1.7 1.5 1.3 1.0 0.8 0.6 0.3 a 1.5 1.4 1.2 1.0 0.9 0.8 0.6 4 1.3 1.2 1.2 1.1 1.0 0.9 0.9 6 1.1 1.1 1.1 1.1 1.1 1.1 1.1 6 0.9 1.0 1.0 1.1 1.2 1.3 1.3 7 0.7 0.8 1.0 1.2 1.3 1.4 1.6 8 0.5 0.7 0.9 1.2 1.4 1.6 1.9 9 0.3 0.6 0.8 1.3 1.5 1.8 2.0 10 0.1 0.4 0.7 1.0 1.3 1.6 1.9 2.2 100 200 300 400 500 600 /- 700 Constant to be subtracted 337" .3. TABLE LXXXV. Moon's Horary Motion in Latitude. (Equations of second • order.) Arguments. Preceding equation, and Sum of equations of Horary Motion in Longi- tude, except the last two. Prec. n " " " " " " tr equ. 0.65 100 0.57 300 0.48 300 0.39 400 0.31 500 0.21 600 0.13 700 0.00 0.00 0.10 0.62 0.55 0.47 |0.39 0.31 0.23 0.15 04 0.20 0.69 0.53 0.46 0.39 0.32 0.25 O.IR 0.09 0.30 0.66 0.51 0.45 0.39 0.33 0.27 0.31 0.13 0.40 0.63 0.48 0.44 0.39 0.34 0.29 0.34 017 0.50 0.50 0.46 0.43 0.38 0.35 0.30 0.27 0.21 0.60 0.47 0.44 0.42 0.38 0.36 0.32 0.29 0.25 0.70 0.44 0.42 0.40 0.38 0.36 0.34 0.33 0.30 0.80 0.41 0.40 0.39 0.38 0.37 0.36 0.35 034 0.90 0.38 0.38 0.38 0.38 0.38 0.38 0.3S 38 1.00 0.35 0.36 0.37 0.38 0.39 0.40 0.41 43 1.10 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 1.20 0.29 0.32 0.34 0.38 0.40 0.44 0.47 051 1.30 0.2B 0.30 0.33 0.38 0.41 0.46 0.49 55 1.40 0.23 0.28 0.32 0.37 0.42 0.47 0!i2 0.59 1.50 0.30 0.25 0.31 0.37 0.43 0.49 55 63 1.60 0.17 0.23 0.30 0.37 0.44 0.51 O.W 67 1.70 0.14 0.21 0.29 0.37 0.45 0.53 O.fil 0.72 1.80 0.11 0.19 0.28 0.37 0.45 0.55 0.64 0.76 no 200 300 400 500 600 700 TABLE LXXXIV. Moon's Hot. Motionin LcH (Equa. of second order.) Argument. Arg. I of Lat. I I • o " " « o O 0.90 0.90 XII 5 0.83 0.97 85 10 0.75 1.05 30 15 0.68 1.12 15 20 0.61 1.19 10 25 0.54 1.26 5 I 0.47 1.33 XI 5 0.41 1.39 25 10 0.35 1.45 20 15 0.29 1.51 15 20 0.24 1.56 10 25 0.20 1.60 5 n 0.16 1.64 X 5 0.12 1.68 25 10 0.09 1.71 20 15 0.07 1.73 15 20 0.05 1.75 10 £5 0.04 1.76 5 III 0.04 1.76 IX 6 0.04 1.76 25 10 0.05 1.75 20 15 0.07 1.73 15 20 0.09 1.71 10 25 0.12 1.68 5 IV 0.16 1.64 VIII 5 0.20 1.60 25 10 0.24 1.56 20 15 0.29 1.51 15 20 0.35 1.45 10 25 '0.41 1.39 5 V 0.47 1.33 VII 5 0.54 1.26 25 10 0.61 1.19 30 15 0.68 1.12 15 20 0.75 1.05 10 25 0.83 0.97 5 VI |0.90 0.90 VI TABLE LXXXVI. New Moons and Arguments, in January. 101 Mean New Tears. Moon In January. I. II. JII. IV. N. d. h. m. 1836 R 17 10 32 0169 124 1 17 08 669 18.7 5 19 -0 ,0171 98B2 00 97 692 183=! 24 16 53 81 917S 99 85 799 1839 14 1 :2 .0383 7780 82 74 822 1840 B 3 10 30 0085 6386 66 63 844 1S41 21 8 3 0,i95 5709 68 61 951 1842 10 IC, 51 02,17 4)14 46 40 974 18 3 29 14 24 0S07 3637 44 28 081 1844 B 18 23 13 U,u9 2 i, •Ai 17 1..4 ls4a 7 8 1 02 1 1 0848 11 Oij 126 1846 26 5 34 0721 0171 09 94 234 1847 15 14 2i 0423 87(7 92 84 ■.;56 1848 15 4 23 11 0126 73-2 76 73 278 1849 22 2 1 -13 0635 li70.-, 73 Oi 386 185U 12 5 32 0337 6311 6J 50 4i.8 1851 1 14 21 38 3916 40 39 431 1852 r? -0 11 53 0.549 3239 38 27 638 1853 8 20 42 1 251 1845 21 16 660 1854 27 18 14 0761 1168 19 04 668 1856 17 "^ 3 0463 9773 02 93 690 1856 B 6 11 51 0164 8379 85 82 713 1-57 24 9 24 067.1 TlO-1 84 70 820 1858 13 18 13 0376 6307 67 69 843 1859 3 3 1 0078 4913 .iO 48 865 1860 B 22 34 0588 4286 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 1864B 8 32 201 9376 96 91 146 1865 25 22 6 714 8699 94 79 254 1866 15 6 34 416 730 1 77 68 276 1S67 4 15 42 117 6910 60 57 299 18K8B 23 13 1.. 628 5234 68 46 406 1869 11 22 3 329 3838 41 36 428 1870 30 19 86 810 8161 40 23 636 1871 20 4 25 641 1767 23 12 558 1872 B 9 13 13 243 372 6 1 581 1873 27 10 46 753 9696 4 89 688 1874 16 19 34 455 8301 87 78 710 1875 6 4 23 157 6906 70 67 733 1876 B 25 1 66 667 6229 69 66 840 1877 13 10 44 369 4836 52 44 862 1878 2 19 33 71 3441 36 33 885 1879 21 17 5 581 2763 33 21 993 1880 B 11 1 54 283 1869 16 10 15 1881 28 23 27 793 692 14 99 123 1882 18 8 15 4ii6 9297 98 88 145 1883 7 17 4 197 79.3 81 77 167 1884 B 26 14 86 707 7226 79 65 275 1885 14 23 25 409 5832 62 54 297 35 102 TABLE LXXXMI Mean Lunations and Changes of the ArguraonU. Mum liUnations. I. II. III. IV. N. d. h m 14 18 22 404 5359 58 50 43 1 29 12 44 808 717 15 99 85 3 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 S3 89 22 13 383 21 33 510 9323 98 88 108 TABLE LXXXVm. Xumlxir of Days from the commencemert of the vc'-ar to the first of each month- Months. Com. Bis. January February March . Days. 31 59 Days. 31 60 April . May . 90 120 91 121 June 151 152 Ju.V . 181 182 August . 212 213 September October . 243 273 244 274 November 304 305 December 334 335 TABLE LXXXli:. Equations for New and Full Moon. 103 ■ Arg. M " || Arg. I II /krg [11^ IV Arg h m\ h m h m h m m m i 20, 10 10 5000 4 20 10 10 25 3 31 25 leu 4 36 9 36 5100 4 5 10 50 26 3 31 24 200 ' 4 52 9 2 5200 3 49 11 30 27 3 30 23 300 5 8 8 28 5300 3 34 12 9 28 3 30 22 400 ^ 5 24 7 55 5400 3 19 12 48 29 3 30 21 SOO'5 40 7 -2? 5500 3 4 13 26 30 3 30 20 600 5 55 6 49 5600 2 49 14 3 31 3 30 19 700 6 10 6 17 5700 2 35 14 39 32 4 30 18 800 6 24 5 46 5800 2 21 15 13 33 4 29 17 900 6 38 5 15 5900 2 8 15 46 34 4 29 16 1000 ' 6 51 4 46 6000 1 55 16 18 35 4 29 15 1100 7 4 4 17 6100 1 42 16 48 36 5 28 14 1200 7 15 3 50 6200 1 31 17 16 37 5 28 13 1300 7 27 3 24 6300 1 19 17 42 38 5 27 12 1400 7 37 2 59 6400 1 9 18 6 39 5 27 11 1500 7 47 2 35 6500 59 18 28 40 6 26 10 1600 7 55 2 14 6600 50 18 48 41 6 26 9 1700 8 3 1 53 6700 42 19 6 42 7 25 8 1800 8 10 1 35 6800 34 19 21 43 7 25 r 1900 8 16 1 18 6900 28 19 33 44 7 24 6 2000 8 21 1 3 7000 ' 22 19 44 45 8 23 6 2100 8 25 51 7100 17 19 52 46 8 23 4 2200 8 29 40 7200 14 19 57 47 9 22 3 2300 8 31 32 7300 11 20 48 9 21 2 2400 8 32 25 7400 9 20 1 49 10 21 1 2500 8 32 21 7500 8 19 59 50 10 20 2600 8 31 19 7000 8 19 55 51 10 19 99 2700 8 29 20 7700 9 19 48 52 11 19 98 2800 8 26 23 7800 11 19 40 53 11 18 97 2900 8 23 28 7900 15 19 29 54 12 17 96 3000 8 18 36 8000 19 19 17 55 12 17 95 3100 8 12 47 8100 24 19 2 56 13 16 94 3200 8 6 59 8200 30 18 45 57 13 15 93 3300 7 58 1 14 8300 37 18 27 58 13 15 92 3400 7 50 1 32 8400 45 18 6 59 14 14 91 3500 7 41 1 52 8500 53 17 45 60 14 14 90 3600 7 31 2 14 8600 1 3 17 21 61 15 13 89 3700 7 21 2 38 8700 1 13 16 56 62 15 13 88 3800 7 9 3 4 8800 1 25 16 30 63 15 12 87 3900 6 58 3 32 8900 1 36 16 3 64 15 12 86 4000 6 45 4 2 9000 1 49 15 34 65 16 11 85 4100 6 32 4 34 9100 2 2 15 5 66 16 11 84 4aeo 6 19 5 7 9200 2 16 14 34 67 16 11 83 4300 6 5 5 41 9300 2 30 14 3 68 16 10 82 4400 5 51 6 17 9400 2 45 13 31 69 17 10 81 4500 5 36 6 54 9500 3 12 58 70 17 10 80 4600 5 21 7 32 9600 3 16 12 25 71 17 10 79 4700 5 6 8 11 9700 3 32 U 52 72 17 10 78 4800 4 51 8 50 9800 3 48 11 18 73 17 10 77 4900 4 35 9 30 9900 4 4 10 44 74 17 9 76 5000 4 20 10 10 10000 4 20 10 10 75 17 9 75 101 TABLE XC. Mean Right Ascensions and Declinations of 50 principal Fixed Stars, for the beginning of 1 840. Stars' Name. Aig Right Ascen.j AnnualVar. Declination. Ann. Var. 1 Algemb 2 iSAndromedae 3 Polaris 4 Achemar 5 oArietis 2.3 2 23 1 3 h in a 6 0.31 1 46.7 1 2 10.38 1 31 44.88 1 58 9.94 + 3*0775 3.309 16.1962 2.2351 3.3457 o / " 14 17 38.82 N 34 46 17.2 N 88 27 21.96 N 58 3 5.13 S 22 42 11.81 N + 20 051 19.35 19.339 — 18.473 + 17.4.'>5 6 aCeti 7 a Persei 8 Aldebaran 9 Capella 10 Rigel 23 2.3 1 1 1 2 53 55.34 3 12 55.97 4 26 44.77 5 4 52.67 5 6 51.09 + 3.1257 4.2280 3.4264 4.4066 2.8783 3 27 30.09 N 49 17 8.74 N 16 10 56.82 N 45 49 42.81 N 8 23 29.29 S + 14.561 13.371 7.949 4.793 — 4.620 11 /STauri 12 y Orionis 13 a ColumbaR 14 o Orionis 15 Canopus 2 2 2 1 1 5 16 10.96 5 16 33.1 6 33 51.52 5 46 30.71 6 20 24.18 + 3.7820 3.210 2 1688 3.2430 1.3278 28 27 58.20 N 6 11 .55.3 N 34 9 47.41 S 7 22 17.14N 52 36 38.42 S + 3.825 + 3.82 — 2.291 + 1.191 1.778 16 Sirius 17 Castor 18 Procyon 19 Pollux 20 aHydrae 1 3 1.2 2 2 6 38 5.76 7 24 23.06 7 30 55.53 7 35 31.07 9 19 43.57 + 2.6458 3.8572 3.1448 3.6840 2.9500 16 30 4.79 S 32 13 58.89 N 5 37 48.92 N 28 24 25.57 N 7 .58 4.83 S + 4.449 — 7.206 8.720 8.107 + 15.341 21 Regulus 22 a Ursae Majoris 23 /SLeonis 24 ^Virginis 25 y Ursae Majoris 1 1.2 23 3.4 2 9 59 50.93 10 53 47.98 11 40 53.69 n 42 21.4 11 45 22.93 + 3.2220 3.8077 3.0660 3.124 3.1914 12 44 49.70 N 62 36 48.93 N 15 28 1.16N 2 40 2 6 N .54 35 4 67 N — 17.3.56 19.221 19.985 19.98 20.014 26 a 8 Crucis 27 Spica 28 fi Centauri 29 a Draconis 30 Arcturus 2 1 2 3.4 1 12 17 43.7 13 16 4636 13 57 18.0 14 2.8 14 8 21.96 + 3.258 3.1502 3.491 1.625 2.7335 62 12 47. 9S 10 19 24.39 S 35 34 41.9 S 65 8 32.1 N 20 1 7.67 N + 19.99 18.945 17.499 — 17.37 18.956 31 o 2 Centauri 32 a 2 Librae 33 /J Ursae Minoris 34 y a Ursae Minoris 35 a Goronae Borealis 1 3 3 3.4 2 14 28 47.84 14 42 2.44 14 51 14.66 15 21 1.3 15 27 54.87 + 4.0086 3.3088 — 0.2787 — 0.179 + 2.5277 60 10 6.24 S 15 22 18.25 S 74 48 34.18 N 72 24 14.1 N 27 15 27.71 N + 15.1.52 15.256 — 14.713 12.81 12.361 36 a Serpentis 37 ^Scorpii 38 Antares J9 a Herculis 40 a Ophiuchi 2.3 2 1 3.4 2 15 36 23.43 15 56 8.68 16 19 36.49 17 7 21.30 17 27 30.56 + 2.9386 3.4729 3.0625 2.7317 2.7724 6 56 2 80N 19 21 38 82S 26 4 13.13S 14 34 41.43N 12 40 58.65 N — 11.770 + 10.330 8.519 — 4.576 2.844 41 i) Ursae Minoris 42 Ycga 43 Altair 44 a a Capricomi 45 a Cygni 3 1 1 3 1 18 23 56.48 18 SI 31.19 19 42 58.61 20 9 10.34 20 35 58.80 — 19.';072 + 2 0116 2.9255 3.3323 2.0416 86 35 28.89 N 3S 38 16.85 N 8 27 0.21 N 13 2 5.57 S 44 42 41.38 N + 2.161 2.742 8.701 — 10 705 + 12.614 46 a Aquarii 47 Fomalkaut 4S ^Pegasi 49 Markah 60 a Andromcdae 3 1 2 2 1 21 57 33.93 22 48 47.67 22 56 1.1 22 56 47.75 24 7.72 + 3.0835 3 3114 2.878 2.9771 3.0704 1 5 38.00 S 30 28 4.91 S 27 13 1.7 N 14 20 46.92 N 28 12 27.06 N — 17.256 19.092 + 19 2.55 19 295 20 056 TABLE XCI. 105 Constants for the Aberration and Nutatim in Right Ascension and Declination of the Stars in the preceding Catalogue ■ Aberration. | Nutaticn.